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1 | \input texinfo @c -*-texinfo-*- |
2 | @comment %**start of header (This is for running Texinfo on a region.) | |
3 | @c smallbook | |
4 | @setfilename ../info/calc | |
5 | @c [title] | |
0d48e8aa | 6 | @settitle GNU Emacs Calc 2.02g Manual |
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7 | @setchapternewpage odd |
8 | @comment %**end of header (This is for running Texinfo on a region.) | |
9 | ||
10 | @tex | |
11 | % Some special kludges to make TeX formatting prettier. | |
12 | % Because makeinfo.c exists, we can't just define new commands. | |
13 | % So instead, we take over little-used existing commands. | |
14 | % | |
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15 | % Suggested by Karl Berry <karl@@freefriends.org> |
16 | \gdef\!{\mskip-\thinmuskip} | |
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17 | % Redefine @cite{text} to act like $text$ in regular TeX. |
18 | % Info will typeset this same as @samp{text}. | |
19 | \gdef\goodtex{\tex \let\rm\goodrm \let\t\ttfont \turnoffactive} | |
20 | \gdef\goodrm{\fam0\tenrm} | |
21 | \gdef\cite{\goodtex$\citexxx} | |
22 | \gdef\citexxx#1{#1$\Etex} | |
23 | \global\let\oldxrefX=\xrefX | |
24 | \gdef\xrefX[#1]{\begingroup\let\cite=\dfn\oldxrefX[#1]\endgroup} | |
5d67986c | 25 | |
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26 | % Redefine @c{tex-stuff} \n @whatever{info-stuff}. |
27 | \gdef\c{\futurelet\next\mycxxx} | |
28 | \gdef\mycxxx{% | |
29 | \ifx\next\bgroup \goodtex\let\next\mycxxy | |
30 | \else\ifx\next\mindex \let\next\relax | |
31 | \else\ifx\next\kindex \let\next\relax | |
32 | \else\ifx\next\starindex \let\next\relax \else \let\next\comment | |
33 | \fi\fi\fi\fi \next | |
34 | } | |
35 | \gdef\mycxxy#1#2{#1\Etex\mycxxz} | |
36 | \gdef\mycxxz#1{} | |
37 | @end tex | |
38 | ||
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39 | @c Fix some other things specifically for this manual. |
40 | @iftex | |
41 | @finalout | |
42 | @mathcode`@:=`@: @c Make Calc fractions come out right in math mode | |
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43 | @tex |
44 | \gdef\coloneq{\mathrel{\mathord:\mathord=}} | |
5d67986c | 45 | |
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46 | \gdef\beforedisplay{\vskip-10pt} |
47 | \gdef\afterdisplay{\vskip-5pt} | |
48 | \gdef\beforedisplayh{\vskip-25pt} | |
49 | \gdef\afterdisplayh{\vskip-10pt} | |
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50 | @end tex |
51 | @newdimen@kyvpos @kyvpos=0pt | |
52 | @newdimen@kyhpos @kyhpos=0pt | |
53 | @newcount@calcclubpenalty @calcclubpenalty=1000 | |
5d67986c | 54 | @ignore |
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55 | @newcount@calcpageno |
56 | @newtoks@calcoldeverypar @calcoldeverypar=@everypar | |
57 | @everypar={@calceverypar@the@calcoldeverypar} | |
58 | @ifx@turnoffactive@undefinedzzz@def@turnoffactive{}@fi | |
59 | @ifx@ninett@undefinedzzz@font@ninett=cmtt9@fi | |
60 | @catcode`@\=0 \catcode`\@=11 | |
61 | \r@ggedbottomtrue | |
62 | \catcode`\@=0 @catcode`@\=@active | |
5d67986c | 63 | @end ignore |
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64 | @end iftex |
65 | ||
18f952d5 | 66 | @copying |
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67 | This file documents Calc, the GNU Emacs calculator. |
68 | ||
0d48e8aa | 69 | Copyright (C) 1990, 1991, 2001, 2002 Free Software Foundation, Inc. |
d7b8e6c6 | 70 | |
18f952d5 | 71 | @quotation |
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72 | Permission is granted to copy, distribute and/or modify this document |
73 | under the terms of the GNU Free Documentation License, Version 1.1 or | |
74 | any later version published by the Free Software Foundation; with the | |
75 | Invariant Sections being just ``GNU GENERAL PUBLIC LICENSE'', with the | |
76 | Front-Cover texts being ``A GNU Manual,'' and with the Back-Cover | |
77 | Texts as in (a) below. | |
d7b8e6c6 | 78 | |
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79 | (a) The FSF's Back-Cover Text is: ``You have freedom to copy and modify |
80 | this GNU Manual, like GNU software. Copies published by the Free | |
81 | Software Foundation raise funds for GNU development.'' | |
18f952d5 KB |
82 | @end quotation |
83 | @end copying | |
84 | ||
85 | @dircategory Emacs | |
86 | @direntry | |
87 | * Calc: (calc). Advanced desk calculator and mathematical tool. | |
88 | @end direntry | |
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89 | |
90 | @titlepage | |
91 | @sp 6 | |
92 | @center @titlefont{Calc Manual} | |
93 | @sp 4 | |
0d48e8aa | 94 | @center GNU Emacs Calc Version 2.02g |
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95 | @c [volume] |
96 | @sp 1 | |
0d48e8aa | 97 | @center January 2002 |
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98 | @sp 5 |
99 | @center Dave Gillespie | |
100 | @center daveg@@synaptics.com | |
101 | @page | |
102 | ||
103 | @vskip 0pt plus 1filll | |
0d48e8aa | 104 | Copyright @copyright{} 1990, 1991, 2001, 2002 Free Software Foundation, Inc. |
18f952d5 | 105 | @insertcopying |
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106 | @end titlepage |
107 | ||
108 | @c [begin] | |
109 | @ifinfo | |
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110 | @node Top, , (dir), (dir) |
111 | @chapter The GNU Emacs Calculator | |
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112 | |
113 | @noindent | |
0d48e8aa | 114 | @dfn{Calc} is an advanced desk calculator and mathematical tool |
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115 | that runs as part of the GNU Emacs environment. |
116 | ||
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117 | This manual is divided into three major parts: ``Getting Started,'' |
118 | the ``Calc Tutorial,'' and the ``Calc Reference.'' The Tutorial | |
119 | introduces all the major aspects of Calculator use in an easy, | |
120 | hands-on way. The remainder of the manual is a complete reference to | |
121 | the features of the Calculator. | |
d7b8e6c6 EZ |
122 | |
123 | For help in the Emacs Info system (which you are using to read this | |
124 | file), type @kbd{?}. (You can also type @kbd{h} to run through a | |
125 | longer Info tutorial.) | |
126 | ||
127 | @end ifinfo | |
128 | @menu | |
129 | * Copying:: How you can copy and share Calc. | |
130 | ||
131 | * Getting Started:: General description and overview. | |
b275eac7 | 132 | * Interactive Tutorial:: |
d7b8e6c6 EZ |
133 | * Tutorial:: A step-by-step introduction for beginners. |
134 | ||
135 | * Introduction:: Introduction to the Calc reference manual. | |
136 | * Data Types:: Types of objects manipulated by Calc. | |
137 | * Stack and Trail:: Manipulating the stack and trail buffers. | |
138 | * Mode Settings:: Adjusting display format and other modes. | |
139 | * Arithmetic:: Basic arithmetic functions. | |
140 | * Scientific Functions:: Transcendentals and other scientific functions. | |
141 | * Matrix Functions:: Operations on vectors and matrices. | |
142 | * Algebra:: Manipulating expressions algebraically. | |
143 | * Units:: Operations on numbers with units. | |
144 | * Store and Recall:: Storing and recalling variables. | |
145 | * Graphics:: Commands for making graphs of data. | |
146 | * Kill and Yank:: Moving data into and out of Calc. | |
147 | * Embedded Mode:: Working with formulas embedded in a file. | |
148 | * Programming:: Calc as a programmable calculator. | |
149 | ||
150 | * Installation:: Installing Calc as a part of GNU Emacs. | |
151 | * Reporting Bugs:: How to report bugs and make suggestions. | |
152 | ||
153 | * Summary:: Summary of Calc commands and functions. | |
154 | ||
155 | * Key Index:: The standard Calc key sequences. | |
156 | * Command Index:: The interactive Calc commands. | |
157 | * Function Index:: Functions (in algebraic formulas). | |
158 | * Concept Index:: General concepts. | |
159 | * Variable Index:: Variables used by Calc (both user and internal). | |
160 | * Lisp Function Index:: Internal Lisp math functions. | |
161 | @end menu | |
162 | ||
163 | @node Copying, Getting Started, Top, Top | |
164 | @unnumbered GNU GENERAL PUBLIC LICENSE | |
165 | @center Version 1, February 1989 | |
166 | ||
167 | @display | |
168 | Copyright @copyright{} 1989 Free Software Foundation, Inc. | |
169 | 675 Mass Ave, Cambridge, MA 02139, USA | |
170 | ||
171 | Everyone is permitted to copy and distribute verbatim copies | |
172 | of this license document, but changing it is not allowed. | |
173 | @end display | |
174 | ||
175 | @unnumberedsec Preamble | |
176 | ||
177 | The license agreements of most software companies try to keep users | |
178 | at the mercy of those companies. By contrast, our General Public | |
179 | License is intended to guarantee your freedom to share and change free | |
180 | software---to make sure the software is free for all its users. The | |
181 | General Public License applies to the Free Software Foundation's | |
182 | software and to any other program whose authors commit to using it. | |
183 | You can use it for your programs, too. | |
184 | ||
185 | When we speak of free software, we are referring to freedom, not | |
186 | price. Specifically, the General Public License is designed to make | |
187 | sure that you have the freedom to give away or sell copies of free | |
188 | software, that you receive source code or can get it if you want it, | |
189 | that you can change the software or use pieces of it in new free | |
190 | programs; and that you know you can do these things. | |
191 | ||
192 | To protect your rights, we need to make restrictions that forbid | |
193 | anyone to deny you these rights or to ask you to surrender the rights. | |
194 | These restrictions translate to certain responsibilities for you if you | |
195 | distribute copies of the software, or if you modify it. | |
196 | ||
197 | For example, if you distribute copies of a such a program, whether | |
198 | gratis or for a fee, you must give the recipients all the rights that | |
199 | you have. You must make sure that they, too, receive or can get the | |
200 | source code. And you must tell them their rights. | |
201 | ||
202 | We protect your rights with two steps: (1) copyright the software, and | |
203 | (2) offer you this license which gives you legal permission to copy, | |
204 | distribute and/or modify the software. | |
205 | ||
206 | Also, for each author's protection and ours, we want to make certain | |
207 | that everyone understands that there is no warranty for this free | |
208 | software. If the software is modified by someone else and passed on, we | |
209 | want its recipients to know that what they have is not the original, so | |
210 | that any problems introduced by others will not reflect on the original | |
211 | authors' reputations. | |
212 | ||
213 | The precise terms and conditions for copying, distribution and | |
214 | modification follow. | |
215 | ||
216 | @iftex | |
217 | @unnumberedsec TERMS AND CONDITIONS | |
218 | @end iftex | |
219 | @ifinfo | |
220 | @center TERMS AND CONDITIONS | |
221 | @end ifinfo | |
222 | ||
223 | @enumerate | |
224 | @item | |
225 | This License Agreement applies to any program or other work which | |
226 | contains a notice placed by the copyright holder saying it may be | |
227 | distributed under the terms of this General Public License. The | |
228 | ``Program'', below, refers to any such program or work, and a ``work based | |
229 | on the Program'' means either the Program or any work containing the | |
230 | Program or a portion of it, either verbatim or with modifications. Each | |
231 | licensee is addressed as ``you''. | |
232 | ||
233 | @item | |
234 | You may copy and distribute verbatim copies of the Program's source | |
235 | code as you receive it, in any medium, provided that you conspicuously and | |
236 | appropriately publish on each copy an appropriate copyright notice and | |
237 | disclaimer of warranty; keep intact all the notices that refer to this | |
238 | General Public License and to the absence of any warranty; and give any | |
239 | other recipients of the Program a copy of this General Public License | |
240 | along with the Program. You may charge a fee for the physical act of | |
241 | transferring a copy. | |
242 | ||
243 | @item | |
244 | You may modify your copy or copies of the Program or any portion of | |
245 | it, and copy and distribute such modifications under the terms of Paragraph | |
246 | 1 above, provided that you also do the following: | |
247 | ||
248 | @itemize @bullet | |
249 | @item | |
250 | cause the modified files to carry prominent notices stating that | |
251 | you changed the files and the date of any change; and | |
252 | ||
253 | @item | |
254 | cause the whole of any work that you distribute or publish, that | |
255 | in whole or in part contains the Program or any part thereof, either | |
256 | with or without modifications, to be licensed at no charge to all | |
257 | third parties under the terms of this General Public License (except | |
258 | that you may choose to grant warranty protection to some or all | |
259 | third parties, at your option). | |
260 | ||
261 | @item | |
262 | If the modified program normally reads commands interactively when | |
263 | run, you must cause it, when started running for such interactive use | |
264 | in the simplest and most usual way, to print or display an | |
265 | announcement including an appropriate copyright notice and a notice | |
266 | that there is no warranty (or else, saying that you provide a | |
267 | warranty) and that users may redistribute the program under these | |
268 | conditions, and telling the user how to view a copy of this General | |
269 | Public License. | |
270 | ||
271 | @item | |
272 | You may charge a fee for the physical act of transferring a | |
273 | copy, and you may at your option offer warranty protection in | |
274 | exchange for a fee. | |
275 | @end itemize | |
276 | ||
277 | Mere aggregation of another independent work with the Program (or its | |
278 | derivative) on a volume of a storage or distribution medium does not bring | |
279 | the other work under the scope of these terms. | |
280 | ||
281 | @item | |
282 | You may copy and distribute the Program (or a portion or derivative of | |
283 | it, under Paragraph 2) in object code or executable form under the terms of | |
284 | Paragraphs 1 and 2 above provided that you also do one of the following: | |
285 | ||
286 | @itemize @bullet | |
287 | @item | |
288 | accompany it with the complete corresponding machine-readable | |
289 | source code, which must be distributed under the terms of | |
290 | Paragraphs 1 and 2 above; or, | |
291 | ||
292 | @item | |
293 | accompany it with a written offer, valid for at least three | |
294 | years, to give any third party free (except for a nominal charge | |
295 | for the cost of distribution) a complete machine-readable copy of the | |
296 | corresponding source code, to be distributed under the terms of | |
297 | Paragraphs 1 and 2 above; or, | |
298 | ||
299 | @item | |
300 | accompany it with the information you received as to where the | |
301 | corresponding source code may be obtained. (This alternative is | |
302 | allowed only for noncommercial distribution and only if you | |
303 | received the program in object code or executable form alone.) | |
304 | @end itemize | |
305 | ||
306 | Source code for a work means the preferred form of the work for making | |
307 | modifications to it. For an executable file, complete source code means | |
308 | all the source code for all modules it contains; but, as a special | |
309 | exception, it need not include source code for modules which are standard | |
310 | libraries that accompany the operating system on which the executable | |
311 | file runs, or for standard header files or definitions files that | |
312 | accompany that operating system. | |
313 | ||
314 | @item | |
315 | You may not copy, modify, sublicense, distribute or transfer the | |
316 | Program except as expressly provided under this General Public License. | |
317 | Any attempt otherwise to copy, modify, sublicense, distribute or transfer | |
318 | the Program is void, and will automatically terminate your rights to use | |
319 | the Program under this License. However, parties who have received | |
320 | copies, or rights to use copies, from you under this General Public | |
321 | License will not have their licenses terminated so long as such parties | |
322 | remain in full compliance. | |
323 | ||
324 | @item | |
325 | By copying, distributing or modifying the Program (or any work based | |
326 | on the Program) you indicate your acceptance of this license to do so, | |
327 | and all its terms and conditions. | |
328 | ||
329 | @item | |
330 | Each time you redistribute the Program (or any work based on the | |
331 | Program), the recipient automatically receives a license from the original | |
332 | licensor to copy, distribute or modify the Program subject to these | |
333 | terms and conditions. You may not impose any further restrictions on the | |
334 | recipients' exercise of the rights granted herein. | |
335 | ||
336 | @item | |
337 | The Free Software Foundation may publish revised and/or new versions | |
338 | of the General Public License from time to time. Such new versions will | |
339 | be similar in spirit to the present version, but may differ in detail to | |
340 | address new problems or concerns. | |
341 | ||
342 | Each version is given a distinguishing version number. If the Program | |
343 | specifies a version number of the license which applies to it and ``any | |
344 | later version'', you have the option of following the terms and conditions | |
345 | either of that version or of any later version published by the Free | |
346 | Software Foundation. If the Program does not specify a version number of | |
347 | the license, you may choose any version ever published by the Free Software | |
348 | Foundation. | |
349 | ||
350 | @item | |
351 | If you wish to incorporate parts of the Program into other free | |
352 | programs whose distribution conditions are different, write to the author | |
353 | to ask for permission. For software which is copyrighted by the Free | |
354 | Software Foundation, write to the Free Software Foundation; we sometimes | |
355 | make exceptions for this. Our decision will be guided by the two goals | |
356 | of preserving the free status of all derivatives of our free software and | |
357 | of promoting the sharing and reuse of software generally. | |
358 | ||
359 | @iftex | |
360 | @heading NO WARRANTY | |
361 | @end iftex | |
362 | @ifinfo | |
363 | @center NO WARRANTY | |
364 | @end ifinfo | |
365 | ||
366 | @item | |
367 | BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY | |
368 | FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN | |
369 | OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES | |
370 | PROVIDE THE PROGRAM ``AS IS'' WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED | |
371 | OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF | |
372 | MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS | |
373 | TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE | |
374 | PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, | |
375 | REPAIR OR CORRECTION. | |
376 | ||
377 | @item | |
378 | IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL | |
379 | ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR | |
380 | REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, | |
381 | INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES | |
382 | ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT | |
383 | LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES | |
384 | SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE | |
385 | WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN | |
386 | ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. | |
387 | @end enumerate | |
388 | ||
b275eac7 | 389 | @node Getting Started, Tutorial, Copying, Top |
d7b8e6c6 | 390 | @chapter Getting Started |
d7b8e6c6 EZ |
391 | @noindent |
392 | This chapter provides a general overview of Calc, the GNU Emacs | |
393 | Calculator: What it is, how to start it and how to exit from it, | |
394 | and what are the various ways that it can be used. | |
395 | ||
396 | @menu | |
397 | * What is Calc:: | |
398 | * About This Manual:: | |
399 | * Notations Used in This Manual:: | |
400 | * Using Calc:: | |
401 | * Demonstration of Calc:: | |
402 | * History and Acknowledgements:: | |
403 | @end menu | |
404 | ||
405 | @node What is Calc, About This Manual, Getting Started, Getting Started | |
406 | @section What is Calc? | |
407 | ||
408 | @noindent | |
409 | @dfn{Calc} is an advanced calculator and mathematical tool that runs as | |
410 | part of the GNU Emacs environment. Very roughly based on the HP-28/48 | |
411 | series of calculators, its many features include: | |
412 | ||
413 | @itemize @bullet | |
414 | @item | |
415 | Choice of algebraic or RPN (stack-based) entry of calculations. | |
416 | ||
417 | @item | |
418 | Arbitrary precision integers and floating-point numbers. | |
419 | ||
420 | @item | |
421 | Arithmetic on rational numbers, complex numbers (rectangular and polar), | |
422 | error forms with standard deviations, open and closed intervals, vectors | |
423 | and matrices, dates and times, infinities, sets, quantities with units, | |
424 | and algebraic formulas. | |
425 | ||
426 | @item | |
427 | Mathematical operations such as logarithms and trigonometric functions. | |
428 | ||
429 | @item | |
430 | Programmer's features (bitwise operations, non-decimal numbers). | |
431 | ||
432 | @item | |
433 | Financial functions such as future value and internal rate of return. | |
434 | ||
435 | @item | |
436 | Number theoretical features such as prime factorization and arithmetic | |
5d67986c | 437 | modulo @var{m} for any @var{m}. |
d7b8e6c6 EZ |
438 | |
439 | @item | |
440 | Algebraic manipulation features, including symbolic calculus. | |
441 | ||
442 | @item | |
443 | Moving data to and from regular editing buffers. | |
444 | ||
445 | @item | |
446 | ``Embedded mode'' for manipulating Calc formulas and data directly | |
447 | inside any editing buffer. | |
448 | ||
449 | @item | |
450 | Graphics using GNUPLOT, a versatile (and free) plotting program. | |
451 | ||
452 | @item | |
453 | Easy programming using keyboard macros, algebraic formulas, | |
454 | algebraic rewrite rules, or extended Emacs Lisp. | |
455 | @end itemize | |
456 | ||
457 | Calc tries to include a little something for everyone; as a result it is | |
458 | large and might be intimidating to the first-time user. If you plan to | |
459 | use Calc only as a traditional desk calculator, all you really need to | |
460 | read is the ``Getting Started'' chapter of this manual and possibly the | |
461 | first few sections of the tutorial. As you become more comfortable with | |
462 | the program you can learn its additional features. In terms of efficiency, | |
463 | scope and depth, Calc cannot replace a powerful tool like Mathematica. | |
d7b8e6c6 EZ |
464 | But Calc has the advantages of convenience, portability, and availability |
465 | of the source code. And, of course, it's free! | |
466 | ||
467 | @node About This Manual, Notations Used in This Manual, What is Calc, Getting Started | |
468 | @section About This Manual | |
469 | ||
470 | @noindent | |
471 | This document serves as a complete description of the GNU Emacs | |
472 | Calculator. It works both as an introduction for novices, and as | |
473 | a reference for experienced users. While it helps to have some | |
474 | experience with GNU Emacs in order to get the most out of Calc, | |
475 | this manual ought to be readable even if you don't know or use Emacs | |
476 | regularly. | |
477 | ||
478 | @ifinfo | |
479 | The manual is divided into three major parts:@: the ``Getting | |
480 | Started'' chapter you are reading now, the Calc tutorial (chapter 2), | |
481 | and the Calc reference manual (the remaining chapters and appendices). | |
482 | @end ifinfo | |
483 | @iftex | |
484 | The manual is divided into three major parts:@: the ``Getting | |
485 | Started'' chapter you are reading now, the Calc tutorial (chapter 2), | |
486 | and the Calc reference manual (the remaining chapters and appendices). | |
487 | @c [when-split] | |
488 | @c This manual has been printed in two volumes, the @dfn{Tutorial} and the | |
489 | @c @dfn{Reference}. Both volumes include a copy of the ``Getting Started'' | |
490 | @c chapter. | |
491 | @end iftex | |
492 | ||
493 | If you are in a hurry to use Calc, there is a brief ``demonstration'' | |
494 | below which illustrates the major features of Calc in just a couple of | |
495 | pages. If you don't have time to go through the full tutorial, this | |
496 | will show you everything you need to know to begin. | |
497 | @xref{Demonstration of Calc}. | |
498 | ||
499 | The tutorial chapter walks you through the various parts of Calc | |
500 | with lots of hands-on examples and explanations. If you are new | |
501 | to Calc and you have some time, try going through at least the | |
502 | beginning of the tutorial. The tutorial includes about 70 exercises | |
503 | with answers. These exercises give you some guided practice with | |
504 | Calc, as well as pointing out some interesting and unusual ways | |
505 | to use its features. | |
506 | ||
507 | The reference section discusses Calc in complete depth. You can read | |
508 | the reference from start to finish if you want to learn every aspect | |
509 | of Calc. Or, you can look in the table of contents or the Concept | |
510 | Index to find the parts of the manual that discuss the things you | |
511 | need to know. | |
512 | ||
513 | @cindex Marginal notes | |
514 | Every Calc keyboard command is listed in the Calc Summary, and also | |
515 | in the Key Index. Algebraic functions, @kbd{M-x} commands, and | |
516 | variables also have their own indices. @c{Each} | |
517 | @asis{In the printed manual, each} | |
518 | paragraph that is referenced in the Key or Function Index is marked | |
519 | in the margin with its index entry. | |
520 | ||
521 | @c [fix-ref Help Commands] | |
522 | You can access this manual on-line at any time within Calc by | |
523 | pressing the @kbd{h i} key sequence. Outside of the Calc window, | |
524 | you can press @kbd{M-# i} to read the manual on-line. Also, you | |
525 | can jump directly to the Tutorial by pressing @kbd{h t} or @kbd{M-# t}, | |
526 | or to the Summary by pressing @kbd{h s} or @kbd{M-# s}. Within Calc, | |
527 | you can also go to the part of the manual describing any Calc key, | |
528 | function, or variable using @w{@kbd{h k}}, @kbd{h f}, or @kbd{h v}, | |
529 | respectively. @xref{Help Commands}. | |
530 | ||
531 | Printed copies of this manual are also available from the Free Software | |
532 | Foundation. | |
533 | ||
534 | @node Notations Used in This Manual, Demonstration of Calc, About This Manual, Getting Started | |
535 | @section Notations Used in This Manual | |
536 | ||
537 | @noindent | |
538 | This section describes the various notations that are used | |
539 | throughout the Calc manual. | |
540 | ||
541 | In keystroke sequences, uppercase letters mean you must hold down | |
542 | the shift key while typing the letter. Keys pressed with Control | |
543 | held down are shown as @kbd{C-x}. Keys pressed with Meta held down | |
544 | are shown as @kbd{M-x}. Other notations are @key{RET} for the | |
545 | Return key, @key{SPC} for the space bar, @key{TAB} for the Tab key, | |
546 | @key{DEL} for the Delete key, and @key{LFD} for the Line-Feed key. | |
547 | ||
548 | (If you don't have the @key{LFD} or @key{TAB} keys on your keyboard, | |
549 | the @kbd{C-j} and @kbd{C-i} keys are equivalent to them, respectively. | |
550 | If you don't have a Meta key, look for Alt or Extend Char. You can | |
551 | also press @key{ESC} or @key{C-[} first to get the same effect, so | |
5d67986c | 552 | that @kbd{M-x}, @kbd{@key{ESC} x}, and @kbd{C-[ x} are all equivalent.) |
d7b8e6c6 EZ |
553 | |
554 | Sometimes the @key{RET} key is not shown when it is ``obvious'' | |
5d67986c | 555 | that you must press @key{RET} to proceed. For example, the @key{RET} |
d7b8e6c6 EZ |
556 | is usually omitted in key sequences like @kbd{M-x calc-keypad @key{RET}}. |
557 | ||
558 | Commands are generally shown like this: @kbd{p} (@code{calc-precision}) | |
559 | or @kbd{M-# k} (@code{calc-keypad}). This means that the command is | |
560 | normally used by pressing the @kbd{p} key or @kbd{M-# k} key sequence, | |
561 | but it also has the full-name equivalent shown, e.g., @kbd{M-x calc-precision}. | |
562 | ||
563 | Commands that correspond to functions in algebraic notation | |
564 | are written: @kbd{C} (@code{calc-cos}) [@code{cos}]. This means | |
565 | the @kbd{C} key is equivalent to @kbd{M-x calc-cos}, and that | |
566 | the corresponding function in an algebraic-style formula would | |
567 | be @samp{cos(@var{x})}. | |
568 | ||
569 | A few commands don't have key equivalents: @code{calc-sincos} | |
570 | [@code{sincos}].@refill | |
571 | ||
572 | @node Demonstration of Calc, Using Calc, Notations Used in This Manual, Getting Started | |
573 | @section A Demonstration of Calc | |
574 | ||
575 | @noindent | |
576 | @cindex Demonstration of Calc | |
577 | This section will show some typical small problems being solved with | |
578 | Calc. The focus is more on demonstration than explanation, but | |
579 | everything you see here will be covered more thoroughly in the | |
580 | Tutorial. | |
581 | ||
582 | To begin, start Emacs if necessary (usually the command @code{emacs} | |
5d67986c | 583 | does this), and type @kbd{M-# c} (or @kbd{@key{ESC} # c}) to start the |
d7b8e6c6 EZ |
584 | Calculator. (@xref{Starting Calc}, if this doesn't work for you.) |
585 | ||
586 | Be sure to type all the sample input exactly, especially noting the | |
587 | difference between lower-case and upper-case letters. Remember, | |
5d67986c | 588 | @key{RET}, @key{TAB}, @key{DEL}, and @key{SPC} are the Return, Tab, |
d7b8e6c6 EZ |
589 | Delete, and Space keys. |
590 | ||
591 | @strong{RPN calculation.} In RPN, you type the input number(s) first, | |
592 | then the command to operate on the numbers. | |
593 | ||
594 | @noindent | |
5d67986c | 595 | Type @kbd{2 @key{RET} 3 + Q} to compute @c{$\sqrt{2+3} = 2.2360679775$} |
d7b8e6c6 EZ |
596 | @asis{the square root of 2+3, which is 2.2360679775}. |
597 | ||
598 | @noindent | |
599 | Type @kbd{P 2 ^} to compute @c{$\pi^2 = 9.86960440109$} | |
600 | @asis{the value of `pi' squared, 9.86960440109}. | |
601 | ||
602 | @noindent | |
5d67986c | 603 | Type @key{TAB} to exchange the order of these two results. |
d7b8e6c6 EZ |
604 | |
605 | @noindent | |
606 | Type @kbd{- I H S} to subtract these results and compute the Inverse | |
607 | Hyperbolic sine of the difference, 2.72996136574. | |
608 | ||
609 | @noindent | |
5d67986c | 610 | Type @key{DEL} to erase this result. |
d7b8e6c6 EZ |
611 | |
612 | @strong{Algebraic calculation.} You can also enter calculations using | |
613 | conventional ``algebraic'' notation. To enter an algebraic formula, | |
614 | use the apostrophe key. | |
615 | ||
616 | @noindent | |
5d67986c | 617 | Type @kbd{' sqrt(2+3) @key{RET}} to compute @c{$\sqrt{2+3}$} |
d7b8e6c6 EZ |
618 | @asis{the square root of 2+3}. |
619 | ||
620 | @noindent | |
5d67986c | 621 | Type @kbd{' pi^2 @key{RET}} to enter @c{$\pi^2$} |
d7b8e6c6 EZ |
622 | @asis{`pi' squared}. To evaluate this symbolic |
623 | formula as a number, type @kbd{=}. | |
624 | ||
625 | @noindent | |
5d67986c | 626 | Type @kbd{' arcsinh($ - $$) @key{RET}} to subtract the second-most-recent |
d7b8e6c6 EZ |
627 | result from the most-recent and compute the Inverse Hyperbolic sine. |
628 | ||
629 | @strong{Keypad mode.} If you are using the X window system, press | |
630 | @w{@kbd{M-# k}} to get Keypad mode. (If you don't use X, skip to | |
631 | the next section.) | |
632 | ||
633 | @noindent | |
634 | Click on the @key{2}, @key{ENTER}, @key{3}, @key{+}, and @key{SQRT} | |
635 | ``buttons'' using your left mouse button. | |
636 | ||
637 | @noindent | |
638 | Click on @key{PI}, @key{2}, and @t{y^x}. | |
639 | ||
640 | @noindent | |
641 | Click on @key{INV}, then @key{ENTER} to swap the two results. | |
642 | ||
643 | @noindent | |
644 | Click on @key{-}, @key{INV}, @key{HYP}, and @key{SIN}. | |
645 | ||
646 | @noindent | |
647 | Click on @key{<-} to erase the result, then click @key{OFF} to turn | |
648 | the Keypad Calculator off. | |
649 | ||
650 | @strong{Grabbing data.} Type @kbd{M-# x} if necessary to exit Calc. | |
651 | Now select the following numbers as an Emacs region: ``Mark'' the | |
5d67986c | 652 | front of the list by typing @kbd{C-@key{SPC}} or @kbd{C-@@} there, |
d7b8e6c6 EZ |
653 | then move to the other end of the list. (Either get this list from |
654 | the on-line copy of this manual, accessed by @w{@kbd{M-# i}}, or just | |
655 | type these numbers into a scratch file.) Now type @kbd{M-# g} to | |
656 | ``grab'' these numbers into Calc. | |
657 | ||
d7b8e6c6 | 658 | @example |
5d67986c | 659 | @group |
d7b8e6c6 EZ |
660 | 1.23 1.97 |
661 | 1.6 2 | |
662 | 1.19 1.08 | |
d7b8e6c6 | 663 | @end group |
5d67986c | 664 | @end example |
d7b8e6c6 EZ |
665 | |
666 | @noindent | |
667 | The result @samp{[1.23, 1.97, 1.6, 2, 1.19, 1.08]} is a Calc ``vector.'' | |
668 | Type @w{@kbd{V R +}} to compute the sum of these numbers. | |
669 | ||
670 | @noindent | |
671 | Type @kbd{U} to Undo this command, then type @kbd{V R *} to compute | |
672 | the product of the numbers. | |
673 | ||
674 | @noindent | |
675 | You can also grab data as a rectangular matrix. Place the cursor on | |
676 | the upper-leftmost @samp{1} and set the mark, then move to just after | |
677 | the lower-right @samp{8} and press @kbd{M-# r}. | |
678 | ||
679 | @noindent | |
680 | Type @kbd{v t} to transpose this @c{$3\times2$} | |
681 | @asis{3x2} matrix into a @c{$2\times3$} | |
682 | @asis{2x3} matrix. Type | |
683 | @w{@kbd{v u}} to unpack the rows into two separate vectors. Now type | |
5d67986c | 684 | @w{@kbd{V R + @key{TAB} V R +}} to compute the sums of the two original columns. |
d7b8e6c6 EZ |
685 | (There is also a special grab-and-sum-columns command, @kbd{M-# :}.) |
686 | ||
687 | @strong{Units conversion.} Units are entered algebraically. | |
5d67986c RS |
688 | Type @w{@kbd{' 43 mi/hr @key{RET}}} to enter the quantity 43 miles-per-hour. |
689 | Type @w{@kbd{u c km/hr @key{RET}}}. Type @w{@kbd{u c m/s @key{RET}}}. | |
d7b8e6c6 EZ |
690 | |
691 | @strong{Date arithmetic.} Type @kbd{t N} to get the current date and | |
692 | time. Type @kbd{90 +} to find the date 90 days from now. Type | |
5d67986c | 693 | @kbd{' <25 dec 87> @key{RET}} to enter a date, then @kbd{- 7 /} to see how |
d7b8e6c6 EZ |
694 | many weeks have passed since then. |
695 | ||
696 | @strong{Algebra.} Algebraic entries can also include formulas | |
5d67986c | 697 | or equations involving variables. Type @kbd{@w{' [x + y} = a, x y = 1] @key{RET}} |
d7b8e6c6 EZ |
698 | to enter a pair of equations involving three variables. |
699 | (Note the leading apostrophe in this example; also, note that the space | |
5d67986c | 700 | between @samp{x y} is required.) Type @w{@kbd{a S x,y @key{RET}}} to solve |
d7b8e6c6 EZ |
701 | these equations for the variables @cite{x} and @cite{y}.@refill |
702 | ||
703 | @noindent | |
704 | Type @kbd{d B} to view the solutions in more readable notation. | |
705 | Type @w{@kbd{d C}} to view them in C language notation, and @kbd{d T} | |
706 | to view them in the notation for the @TeX{} typesetting system. | |
707 | Type @kbd{d N} to return to normal notation. | |
708 | ||
709 | @noindent | |
5d67986c | 710 | Type @kbd{7.5}, then @kbd{s l a @key{RET}} to let @cite{a = 7.5} in these formulas. |
d7b8e6c6 EZ |
711 | (That's a letter @kbd{l}, not a numeral @kbd{1}.) |
712 | ||
713 | @iftex | |
714 | @strong{Help functions.} You can read about any command in the on-line | |
715 | manual. Type @kbd{M-# c} to return to Calc after each of these | |
716 | commands: @kbd{h k t N} to read about the @kbd{t N} command, | |
5d67986c | 717 | @kbd{h f sqrt @key{RET}} to read about the @code{sqrt} function, and |
d7b8e6c6 EZ |
718 | @kbd{h s} to read the Calc summary. |
719 | @end iftex | |
720 | @ifinfo | |
721 | @strong{Help functions.} You can read about any command in the on-line | |
722 | manual. Remember to type the letter @kbd{l}, then @kbd{M-# c}, to | |
723 | return here after each of these commands: @w{@kbd{h k t N}} to read | |
5d67986c | 724 | about the @w{@kbd{t N}} command, @kbd{h f sqrt @key{RET}} to read about the |
d7b8e6c6 EZ |
725 | @code{sqrt} function, and @kbd{h s} to read the Calc summary. |
726 | @end ifinfo | |
727 | ||
5d67986c | 728 | Press @key{DEL} repeatedly to remove any leftover results from the stack. |
d7b8e6c6 EZ |
729 | To exit from Calc, press @kbd{q} or @kbd{M-# c} again. |
730 | ||
731 | @node Using Calc, History and Acknowledgements, Demonstration of Calc, Getting Started | |
732 | @section Using Calc | |
733 | ||
734 | @noindent | |
735 | Calc has several user interfaces that are specialized for | |
736 | different kinds of tasks. As well as Calc's standard interface, | |
737 | there are Quick Mode, Keypad Mode, and Embedded Mode. | |
738 | ||
739 | @c [fix-ref Installation] | |
740 | Calc must be @dfn{installed} before it can be used. @xref{Installation}, | |
741 | for instructions on setting up and installing Calc. We will assume | |
742 | you or someone on your system has already installed Calc as described | |
743 | there. | |
744 | ||
745 | @menu | |
746 | * Starting Calc:: | |
747 | * The Standard Interface:: | |
748 | * Quick Mode Overview:: | |
749 | * Keypad Mode Overview:: | |
750 | * Standalone Operation:: | |
751 | * Embedded Mode Overview:: | |
752 | * Other M-# Commands:: | |
753 | @end menu | |
754 | ||
755 | @node Starting Calc, The Standard Interface, Using Calc, Using Calc | |
756 | @subsection Starting Calc | |
757 | ||
758 | @noindent | |
759 | On most systems, you can type @kbd{M-#} to start the Calculator. | |
760 | The notation @kbd{M-#} is short for Meta-@kbd{#}. On most | |
761 | keyboards this means holding down the Meta (or Alt) and | |
762 | Shift keys while typing @kbd{3}. | |
763 | ||
764 | @cindex META key | |
765 | Once again, if you don't have a Meta key on your keyboard you can type | |
766 | @key{ESC} first, then @kbd{#}, to accomplish the same thing. If you | |
767 | don't even have an @key{ESC} key, you can fake it by holding down | |
768 | Control or @key{CTRL} while typing a left square bracket | |
769 | (that's @kbd{C-[} in Emacs notation).@refill | |
770 | ||
771 | @kbd{M-#} is a @dfn{prefix key}; when you press it, Emacs waits for | |
772 | you to press a second key to complete the command. In this case, | |
773 | you will follow @kbd{M-#} with a letter (upper- or lower-case, it | |
774 | doesn't matter for @kbd{M-#}) that says which Calc interface you | |
775 | want to use. | |
776 | ||
777 | To get Calc's standard interface, type @kbd{M-# c}. To get | |
778 | Keypad Mode, type @kbd{M-# k}. Type @kbd{M-# ?} to get a brief | |
779 | list of the available options, and type a second @kbd{?} to get | |
780 | a complete list. | |
781 | ||
782 | To ease typing, @kbd{M-# M-#} (or @kbd{M-# #} if that's easier) | |
783 | also works to start Calc. It starts the same interface (either | |
784 | @kbd{M-# c} or @w{@kbd{M-# k}}) that you last used, selecting the | |
785 | @kbd{M-# c} interface by default. (If your installation has | |
786 | a special function key set up to act like @kbd{M-#}, hitting that | |
787 | function key twice is just like hitting @kbd{M-# M-#}.) | |
788 | ||
789 | If @kbd{M-#} doesn't work for you, you can always type explicit | |
790 | commands like @kbd{M-x calc} (for the standard user interface) or | |
791 | @w{@kbd{M-x calc-keypad}} (for Keypad Mode). First type @kbd{M-x} | |
792 | (that's Meta with the letter @kbd{x}), then, at the prompt, | |
793 | type the full command (like @kbd{calc-keypad}) and press Return. | |
794 | ||
795 | If you type @kbd{M-x calc} and Emacs still doesn't recognize the | |
796 | command (it will say @samp{[No match]} when you try to press | |
797 | @key{RET}), then Calc has not been properly installed. | |
798 | ||
799 | The same commands (like @kbd{M-# c} or @kbd{M-# M-#}) that start | |
800 | the Calculator also turn it off if it is already on. | |
801 | ||
802 | @node The Standard Interface, Quick Mode Overview, Starting Calc, Using Calc | |
803 | @subsection The Standard Calc Interface | |
804 | ||
805 | @noindent | |
806 | @cindex Standard user interface | |
807 | Calc's standard interface acts like a traditional RPN calculator, | |
808 | operated by the normal Emacs keyboard. When you type @kbd{M-# c} | |
809 | to start the Calculator, the Emacs screen splits into two windows | |
810 | with the file you were editing on top and Calc on the bottom. | |
811 | ||
d7b8e6c6 | 812 | @smallexample |
5d67986c | 813 | @group |
d7b8e6c6 EZ |
814 | |
815 | ... | |
816 | --**-Emacs: myfile (Fundamental)----All---------------------- | |
817 | --- Emacs Calculator Mode --- |Emacs Calc Mode v2.00... | |
818 | 2: 17.3 | 17.3 | |
819 | 1: -5 | 3 | |
820 | . | 2 | |
821 | | 4 | |
822 | | * 8 | |
823 | | ->-5 | |
824 | | | |
825 | --%%-Calc: 12 Deg (Calculator)----All----- --%%-Emacs: *Calc Trail* | |
d7b8e6c6 | 826 | @end group |
5d67986c | 827 | @end smallexample |
d7b8e6c6 EZ |
828 | |
829 | In this figure, the mode-line for @file{myfile} has moved up and the | |
830 | ``Calculator'' window has appeared below it. As you can see, Calc | |
831 | actually makes two windows side-by-side. The lefthand one is | |
832 | called the @dfn{stack window} and the righthand one is called the | |
833 | @dfn{trail window.} The stack holds the numbers involved in the | |
834 | calculation you are currently performing. The trail holds a complete | |
835 | record of all calculations you have done. In a desk calculator with | |
836 | a printer, the trail corresponds to the paper tape that records what | |
837 | you do. | |
838 | ||
839 | In this case, the trail shows that four numbers (17.3, 3, 2, and 4) | |
840 | were first entered into the Calculator, then the 2 and 4 were | |
841 | multiplied to get 8, then the 3 and 8 were subtracted to get @i{-5}. | |
842 | (The @samp{>} symbol shows that this was the most recent calculation.) | |
843 | The net result is the two numbers 17.3 and @i{-5} sitting on the stack. | |
844 | ||
845 | Most Calculator commands deal explicitly with the stack only, but | |
846 | there is a set of commands that allow you to search back through | |
847 | the trail and retrieve any previous result. | |
848 | ||
849 | Calc commands use the digits, letters, and punctuation keys. | |
850 | Shifted (i.e., upper-case) letters are different from lowercase | |
851 | letters. Some letters are @dfn{prefix} keys that begin two-letter | |
852 | commands. For example, @kbd{e} means ``enter exponent'' and shifted | |
853 | @kbd{E} means @cite{e^x}. With the @kbd{d} (``display modes'') prefix | |
854 | the letter ``e'' takes on very different meanings: @kbd{d e} means | |
855 | ``engineering notation'' and @kbd{d E} means ``@dfn{eqn} language mode.'' | |
856 | ||
857 | There is nothing stopping you from switching out of the Calc | |
858 | window and back into your editing window, say by using the Emacs | |
859 | @w{@kbd{C-x o}} (@code{other-window}) command. When the cursor is | |
860 | inside a regular window, Emacs acts just like normal. When the | |
861 | cursor is in the Calc stack or trail windows, keys are interpreted | |
862 | as Calc commands. | |
863 | ||
864 | When you quit by pressing @kbd{M-# c} a second time, the Calculator | |
865 | windows go away but the actual Stack and Trail are not gone, just | |
866 | hidden. When you press @kbd{M-# c} once again you will get the | |
867 | same stack and trail contents you had when you last used the | |
868 | Calculator. | |
869 | ||
870 | The Calculator does not remember its state between Emacs sessions. | |
871 | Thus if you quit Emacs and start it again, @kbd{M-# c} will give you | |
872 | a fresh stack and trail. There is a command (@kbd{m m}) that lets | |
873 | you save your favorite mode settings between sessions, though. | |
874 | One of the things it saves is which user interface (standard or | |
875 | Keypad) you last used; otherwise, a freshly started Emacs will | |
876 | always treat @kbd{M-# M-#} the same as @kbd{M-# c}. | |
877 | ||
878 | The @kbd{q} key is another equivalent way to turn the Calculator off. | |
879 | ||
880 | If you type @kbd{M-# b} first and then @kbd{M-# c}, you get a | |
881 | full-screen version of Calc (@code{full-calc}) in which the stack and | |
882 | trail windows are still side-by-side but are now as tall as the whole | |
883 | Emacs screen. When you press @kbd{q} or @kbd{M-# c} again to quit, | |
884 | the file you were editing before reappears. The @kbd{M-# b} key | |
885 | switches back and forth between ``big'' full-screen mode and the | |
886 | normal partial-screen mode. | |
887 | ||
888 | Finally, @kbd{M-# o} (@code{calc-other-window}) is like @kbd{M-# c} | |
889 | except that the Calc window is not selected. The buffer you were | |
890 | editing before remains selected instead. @kbd{M-# o} is a handy | |
891 | way to switch out of Calc momentarily to edit your file; type | |
892 | @kbd{M-# c} to switch back into Calc when you are done. | |
893 | ||
894 | @node Quick Mode Overview, Keypad Mode Overview, The Standard Interface, Using Calc | |
895 | @subsection Quick Mode (Overview) | |
896 | ||
897 | @noindent | |
898 | @dfn{Quick Mode} is a quick way to use Calc when you don't need the | |
899 | full complexity of the stack and trail. To use it, type @kbd{M-# q} | |
900 | (@code{quick-calc}) in any regular editing buffer. | |
901 | ||
902 | Quick Mode is very simple: It prompts you to type any formula in | |
903 | standard algebraic notation (like @samp{4 - 2/3}) and then displays | |
904 | the result at the bottom of the Emacs screen (@i{3.33333333333} | |
905 | in this case). You are then back in the same editing buffer you | |
906 | were in before, ready to continue editing or to type @kbd{M-# q} | |
907 | again to do another quick calculation. The result of the calculation | |
908 | will also be in the Emacs ``kill ring'' so that a @kbd{C-y} command | |
909 | at this point will yank the result into your editing buffer. | |
910 | ||
911 | Calc mode settings affect Quick Mode, too, though you will have to | |
912 | go into regular Calc (with @kbd{M-# c}) to change the mode settings. | |
913 | ||
914 | @c [fix-ref Quick Calculator mode] | |
915 | @xref{Quick Calculator}, for further information. | |
916 | ||
917 | @node Keypad Mode Overview, Standalone Operation, Quick Mode Overview, Using Calc | |
918 | @subsection Keypad Mode (Overview) | |
919 | ||
920 | @noindent | |
921 | @dfn{Keypad Mode} is a mouse-based interface to the Calculator. | |
7d8c2d57 EZ |
922 | It is designed for use with terminals that support a mouse. If you |
923 | don't have a mouse, you will have to operate keypad mode with your | |
924 | arrow keys (which is probably more trouble than it's worth). Keypad | |
d7b8e6c6 EZ |
925 | mode is currently not supported under Emacs 19. |
926 | ||
927 | Type @kbd{M-# k} to turn Keypad Mode on or off. Once again you | |
928 | get two new windows, this time on the righthand side of the screen | |
929 | instead of at the bottom. The upper window is the familiar Calc | |
930 | Stack; the lower window is a picture of a typical calculator keypad. | |
931 | ||
932 | @tex | |
933 | \dimen0=\pagetotal% | |
934 | \advance \dimen0 by 24\baselineskip% | |
935 | \ifdim \dimen0>\pagegoal \vfill\eject \fi% | |
936 | \medskip | |
937 | @end tex | |
938 | @smallexample | |
939 | |--- Emacs Calculator Mode --- | |
940 | |2: 17.3 | |
941 | |1: -5 | |
942 | | . | |
943 | |--%%-Calc: 12 Deg (Calcul | |
944 | |----+-----Calc 2.00-----+----1 | |
945 | |FLR |CEIL|RND |TRNC|CLN2|FLT | | |
946 | |----+----+----+----+----+----| | |
947 | | LN |EXP | |ABS |IDIV|MOD | | |
948 | |----+----+----+----+----+----| | |
949 | |SIN |COS |TAN |SQRT|y^x |1/x | | |
950 | |----+----+----+----+----+----| | |
951 | | ENTER |+/- |EEX |UNDO| <- | | |
952 | |-----+---+-+--+--+-+---++----| | |
953 | | INV | 7 | 8 | 9 | / | | |
954 | |-----+-----+-----+-----+-----| | |
955 | | HYP | 4 | 5 | 6 | * | | |
956 | |-----+-----+-----+-----+-----| | |
957 | |EXEC | 1 | 2 | 3 | - | | |
958 | |-----+-----+-----+-----+-----| | |
959 | | OFF | 0 | . | PI | + | | |
960 | |-----+-----+-----+-----+-----+ | |
961 | @end smallexample | |
d7b8e6c6 EZ |
962 | |
963 | Keypad Mode is much easier for beginners to learn, because there | |
964 | is no need to memorize lots of obscure key sequences. But not all | |
965 | commands in regular Calc are available on the Keypad. You can | |
966 | always switch the cursor into the Calc stack window to use | |
967 | standard Calc commands if you need. Serious Calc users, though, | |
968 | often find they prefer the standard interface over Keypad Mode. | |
969 | ||
970 | To operate the Calculator, just click on the ``buttons'' of the | |
971 | keypad using your left mouse button. To enter the two numbers | |
972 | shown here you would click @w{@kbd{1 7 .@: 3 ENTER 5 +/- ENTER}}; to | |
973 | add them together you would then click @kbd{+} (to get 12.3 on | |
974 | the stack). | |
975 | ||
976 | If you click the right mouse button, the top three rows of the | |
977 | keypad change to show other sets of commands, such as advanced | |
978 | math functions, vector operations, and operations on binary | |
979 | numbers. | |
980 | ||
d7b8e6c6 EZ |
981 | Because Keypad Mode doesn't use the regular keyboard, Calc leaves |
982 | the cursor in your original editing buffer. You can type in | |
983 | this buffer in the usual way while also clicking on the Calculator | |
984 | keypad. One advantage of Keypad Mode is that you don't need an | |
985 | explicit command to switch between editing and calculating. | |
986 | ||
987 | If you press @kbd{M-# b} first, you get a full-screen Keypad Mode | |
988 | (@code{full-calc-keypad}) with three windows: The keypad in the lower | |
989 | left, the stack in the lower right, and the trail on top. | |
990 | ||
991 | @c [fix-ref Keypad Mode] | |
992 | @xref{Keypad Mode}, for further information. | |
993 | ||
994 | @node Standalone Operation, Embedded Mode Overview, Keypad Mode Overview, Using Calc | |
995 | @subsection Standalone Operation | |
996 | ||
997 | @noindent | |
998 | @cindex Standalone Operation | |
999 | If you are not in Emacs at the moment but you wish to use Calc, | |
1000 | you must start Emacs first. If all you want is to run Calc, you | |
1001 | can give the commands: | |
1002 | ||
1003 | @example | |
1004 | emacs -f full-calc | |
1005 | @end example | |
1006 | ||
1007 | @noindent | |
1008 | or | |
1009 | ||
1010 | @example | |
1011 | emacs -f full-calc-keypad | |
1012 | @end example | |
1013 | ||
1014 | @noindent | |
1015 | which run a full-screen Calculator (as if by @kbd{M-# b M-# c}) or | |
1016 | a full-screen X-based Calculator (as if by @kbd{M-# b M-# k}). | |
1017 | In standalone operation, quitting the Calculator (by pressing | |
1018 | @kbd{q} or clicking on the keypad @key{EXIT} button) quits Emacs | |
1019 | itself. | |
1020 | ||
1021 | @node Embedded Mode Overview, Other M-# Commands, Standalone Operation, Using Calc | |
1022 | @subsection Embedded Mode (Overview) | |
1023 | ||
1024 | @noindent | |
1025 | @dfn{Embedded Mode} is a way to use Calc directly from inside an | |
1026 | editing buffer. Suppose you have a formula written as part of a | |
1027 | document like this: | |
1028 | ||
d7b8e6c6 | 1029 | @smallexample |
5d67986c | 1030 | @group |
d7b8e6c6 EZ |
1031 | The derivative of |
1032 | ||
1033 | ln(ln(x)) | |
1034 | ||
1035 | is | |
d7b8e6c6 | 1036 | @end group |
5d67986c | 1037 | @end smallexample |
d7b8e6c6 EZ |
1038 | |
1039 | @noindent | |
1040 | and you wish to have Calc compute and format the derivative for | |
1041 | you and store this derivative in the buffer automatically. To | |
1042 | do this with Embedded Mode, first copy the formula down to where | |
1043 | you want the result to be: | |
1044 | ||
d7b8e6c6 | 1045 | @smallexample |
5d67986c | 1046 | @group |
d7b8e6c6 EZ |
1047 | The derivative of |
1048 | ||
1049 | ln(ln(x)) | |
1050 | ||
1051 | is | |
1052 | ||
1053 | ln(ln(x)) | |
d7b8e6c6 | 1054 | @end group |
5d67986c | 1055 | @end smallexample |
d7b8e6c6 EZ |
1056 | |
1057 | Now, move the cursor onto this new formula and press @kbd{M-# e}. | |
1058 | Calc will read the formula (using the surrounding blank lines to | |
1059 | tell how much text to read), then push this formula (invisibly) | |
1060 | onto the Calc stack. The cursor will stay on the formula in the | |
1061 | editing buffer, but the buffer's mode line will change to look | |
1062 | like the Calc mode line (with mode indicators like @samp{12 Deg} | |
1063 | and so on). Even though you are still in your editing buffer, | |
1064 | the keyboard now acts like the Calc keyboard, and any new result | |
1065 | you get is copied from the stack back into the buffer. To take | |
1066 | the derivative, you would type @kbd{a d x @key{RET}}. | |
1067 | ||
d7b8e6c6 | 1068 | @smallexample |
5d67986c | 1069 | @group |
d7b8e6c6 EZ |
1070 | The derivative of |
1071 | ||
1072 | ln(ln(x)) | |
1073 | ||
1074 | is | |
1075 | ||
1076 | 1 / ln(x) x | |
d7b8e6c6 | 1077 | @end group |
5d67986c | 1078 | @end smallexample |
d7b8e6c6 EZ |
1079 | |
1080 | To make this look nicer, you might want to press @kbd{d =} to center | |
1081 | the formula, and even @kbd{d B} to use ``big'' display mode. | |
1082 | ||
d7b8e6c6 | 1083 | @smallexample |
5d67986c | 1084 | @group |
d7b8e6c6 EZ |
1085 | The derivative of |
1086 | ||
1087 | ln(ln(x)) | |
1088 | ||
1089 | is | |
1090 | % [calc-mode: justify: center] | |
1091 | % [calc-mode: language: big] | |
1092 | ||
1093 | 1 | |
1094 | ------- | |
1095 | ln(x) x | |
d7b8e6c6 | 1096 | @end group |
5d67986c | 1097 | @end smallexample |
d7b8e6c6 EZ |
1098 | |
1099 | Calc has added annotations to the file to help it remember the modes | |
1100 | that were used for this formula. They are formatted like comments | |
1101 | in the @TeX{} typesetting language, just in case you are using @TeX{}. | |
1102 | (In this example @TeX{} is not being used, so you might want to move | |
1103 | these comments up to the top of the file or otherwise put them out | |
1104 | of the way.) | |
1105 | ||
1106 | As an extra flourish, we can add an equation number using a | |
5d67986c | 1107 | righthand label: Type @kbd{d @} (1) @key{RET}}. |
d7b8e6c6 | 1108 | |
d7b8e6c6 | 1109 | @smallexample |
5d67986c | 1110 | @group |
d7b8e6c6 EZ |
1111 | % [calc-mode: justify: center] |
1112 | % [calc-mode: language: big] | |
1113 | % [calc-mode: right-label: " (1)"] | |
1114 | ||
1115 | 1 | |
1116 | ------- (1) | |
1117 | ln(x) x | |
d7b8e6c6 | 1118 | @end group |
5d67986c | 1119 | @end smallexample |
d7b8e6c6 EZ |
1120 | |
1121 | To leave Embedded Mode, type @kbd{M-# e} again. The mode line | |
1122 | and keyboard will revert to the way they were before. (If you have | |
1123 | actually been trying this as you read along, you'll want to press | |
1124 | @kbd{M-# 0} [with the digit zero] now to reset the modes you changed.) | |
1125 | ||
1126 | The related command @kbd{M-# w} operates on a single word, which | |
1127 | generally means a single number, inside text. It uses any | |
1128 | non-numeric characters rather than blank lines to delimit the | |
1129 | formula it reads. Here's an example of its use: | |
1130 | ||
1131 | @smallexample | |
1132 | A slope of one-third corresponds to an angle of 1 degrees. | |
1133 | @end smallexample | |
1134 | ||
1135 | Place the cursor on the @samp{1}, then type @kbd{M-# w} to enable | |
1136 | Embedded Mode on that number. Now type @kbd{3 /} (to get one-third), | |
1137 | and @kbd{I T} (the Inverse Tangent converts a slope into an angle), | |
1138 | then @w{@kbd{M-# w}} again to exit Embedded mode. | |
1139 | ||
1140 | @smallexample | |
1141 | A slope of one-third corresponds to an angle of 18.4349488229 degrees. | |
1142 | @end smallexample | |
1143 | ||
1144 | @c [fix-ref Embedded Mode] | |
1145 | @xref{Embedded Mode}, for full details. | |
1146 | ||
1147 | @node Other M-# Commands, , Embedded Mode Overview, Using Calc | |
1148 | @subsection Other @kbd{M-#} Commands | |
1149 | ||
1150 | @noindent | |
1151 | Two more Calc-related commands are @kbd{M-# g} and @kbd{M-# r}, | |
1152 | which ``grab'' data from a selected region of a buffer into the | |
1153 | Calculator. The region is defined in the usual Emacs way, by | |
1154 | a ``mark'' placed at one end of the region, and the Emacs | |
1155 | cursor or ``point'' placed at the other. | |
1156 | ||
1157 | The @kbd{M-# g} command reads the region in the usual left-to-right, | |
1158 | top-to-bottom order. The result is packaged into a Calc vector | |
1159 | of numbers and placed on the stack. Calc (in its standard | |
1160 | user interface) is then started. Type @kbd{v u} if you want | |
1161 | to unpack this vector into separate numbers on the stack. Also, | |
1162 | @kbd{C-u M-# g} interprets the region as a single number or | |
1163 | formula. | |
1164 | ||
1165 | The @kbd{M-# r} command reads a rectangle, with the point and | |
1166 | mark defining opposite corners of the rectangle. The result | |
1167 | is a matrix of numbers on the Calculator stack. | |
1168 | ||
1169 | Complementary to these is @kbd{M-# y}, which ``yanks'' the | |
1170 | value at the top of the Calc stack back into an editing buffer. | |
1171 | If you type @w{@kbd{M-# y}} while in such a buffer, the value is | |
1172 | yanked at the current position. If you type @kbd{M-# y} while | |
1173 | in the Calc buffer, Calc makes an educated guess as to which | |
1174 | editing buffer you want to use. The Calc window does not have | |
1175 | to be visible in order to use this command, as long as there | |
1176 | is something on the Calc stack. | |
1177 | ||
1178 | Here, for reference, is the complete list of @kbd{M-#} commands. | |
1179 | The shift, control, and meta keys are ignored for the keystroke | |
1180 | following @kbd{M-#}. | |
1181 | ||
1182 | @noindent | |
1183 | Commands for turning Calc on and off: | |
1184 | ||
1185 | @table @kbd | |
1186 | @item # | |
1187 | Turn Calc on or off, employing the same user interface as last time. | |
1188 | ||
1189 | @item C | |
1190 | Turn Calc on or off using its standard bottom-of-the-screen | |
1191 | interface. If Calc is already turned on but the cursor is not | |
1192 | in the Calc window, move the cursor into the window. | |
1193 | ||
1194 | @item O | |
1195 | Same as @kbd{C}, but don't select the new Calc window. If | |
1196 | Calc is already turned on and the cursor is in the Calc window, | |
1197 | move it out of that window. | |
1198 | ||
1199 | @item B | |
1200 | Control whether @kbd{M-# c} and @kbd{M-# k} use the full screen. | |
1201 | ||
1202 | @item Q | |
1203 | Use Quick Mode for a single short calculation. | |
1204 | ||
1205 | @item K | |
1206 | Turn Calc Keypad mode on or off. | |
1207 | ||
1208 | @item E | |
1209 | Turn Calc Embedded mode on or off at the current formula. | |
1210 | ||
1211 | @item J | |
1212 | Turn Calc Embedded mode on or off, select the interesting part. | |
1213 | ||
1214 | @item W | |
1215 | Turn Calc Embedded mode on or off at the current word (number). | |
1216 | ||
1217 | @item Z | |
1218 | Turn Calc on in a user-defined way, as defined by a @kbd{Z I} command. | |
1219 | ||
1220 | @item X | |
1221 | Quit Calc; turn off standard, Keypad, or Embedded mode if on. | |
1222 | (This is like @kbd{q} or @key{OFF} inside of Calc.) | |
1223 | @end table | |
1224 | @iftex | |
1225 | @sp 2 | |
1226 | @end iftex | |
1227 | ||
d7b8e6c6 EZ |
1228 | @noindent |
1229 | Commands for moving data into and out of the Calculator: | |
1230 | ||
1231 | @table @kbd | |
1232 | @item G | |
1233 | Grab the region into the Calculator as a vector. | |
1234 | ||
1235 | @item R | |
1236 | Grab the rectangular region into the Calculator as a matrix. | |
1237 | ||
1238 | @item : | |
1239 | Grab the rectangular region and compute the sums of its columns. | |
1240 | ||
1241 | @item _ | |
1242 | Grab the rectangular region and compute the sums of its rows. | |
1243 | ||
1244 | @item Y | |
1245 | Yank a value from the Calculator into the current editing buffer. | |
1246 | @end table | |
1247 | @iftex | |
1248 | @sp 2 | |
1249 | @end iftex | |
d7b8e6c6 | 1250 | |
d7b8e6c6 EZ |
1251 | @noindent |
1252 | Commands for use with Embedded Mode: | |
1253 | ||
1254 | @table @kbd | |
1255 | @item A | |
1256 | ``Activate'' the current buffer. Locate all formulas that | |
1257 | contain @samp{:=} or @samp{=>} symbols and record their locations | |
1258 | so that they can be updated automatically as variables are changed. | |
1259 | ||
1260 | @item D | |
1261 | Duplicate the current formula immediately below and select | |
1262 | the duplicate. | |
1263 | ||
1264 | @item F | |
1265 | Insert a new formula at the current point. | |
1266 | ||
1267 | @item N | |
1268 | Move the cursor to the next active formula in the buffer. | |
1269 | ||
1270 | @item P | |
1271 | Move the cursor to the previous active formula in the buffer. | |
1272 | ||
1273 | @item U | |
1274 | Update (i.e., as if by the @kbd{=} key) the formula at the current point. | |
1275 | ||
1276 | @item ` | |
1277 | Edit (as if by @code{calc-edit}) the formula at the current point. | |
1278 | @end table | |
1279 | @iftex | |
1280 | @sp 2 | |
1281 | @end iftex | |
d7b8e6c6 | 1282 | |
d7b8e6c6 EZ |
1283 | @noindent |
1284 | Miscellaneous commands: | |
1285 | ||
1286 | @table @kbd | |
1287 | @item I | |
1288 | Run the Emacs Info system to read the Calc manual. | |
1289 | (This is the same as @kbd{h i} inside of Calc.) | |
1290 | ||
1291 | @item T | |
1292 | Run the Emacs Info system to read the Calc Tutorial. | |
1293 | ||
1294 | @item S | |
1295 | Run the Emacs Info system to read the Calc Summary. | |
1296 | ||
1297 | @item L | |
1298 | Load Calc entirely into memory. (Normally the various parts | |
1299 | are loaded only as they are needed.) | |
1300 | ||
1301 | @item M | |
5d67986c | 1302 | Read a region of written keystroke names (like @kbd{C-n a b c @key{RET}}) |
d7b8e6c6 EZ |
1303 | and record them as the current keyboard macro. |
1304 | ||
1305 | @item 0 | |
1306 | (This is the ``zero'' digit key.) Reset the Calculator to | |
1307 | its default state: Empty stack, and default mode settings. | |
1308 | With any prefix argument, reset everything but the stack. | |
1309 | @end table | |
d7b8e6c6 EZ |
1310 | |
1311 | @node History and Acknowledgements, , Using Calc, Getting Started | |
1312 | @section History and Acknowledgements | |
1313 | ||
1314 | @noindent | |
1315 | Calc was originally started as a two-week project to occupy a lull | |
1316 | in the author's schedule. Basically, a friend asked if I remembered | |
1317 | the value of @c{$2^{32}$} | |
1318 | @cite{2^32}. I didn't offhand, but I said, ``that's | |
1319 | easy, just call up an @code{xcalc}.'' @code{Xcalc} duly reported | |
1320 | that the answer to our question was @samp{4.294967e+09}---with no way to | |
1321 | see the full ten digits even though we knew they were there in the | |
1322 | program's memory! I was so annoyed, I vowed to write a calculator | |
1323 | of my own, once and for all. | |
1324 | ||
1325 | I chose Emacs Lisp, a) because I had always been curious about it | |
1326 | and b) because, being only a text editor extension language after | |
1327 | all, Emacs Lisp would surely reach its limits long before the project | |
1328 | got too far out of hand. | |
1329 | ||
1330 | To make a long story short, Emacs Lisp turned out to be a distressingly | |
1331 | solid implementation of Lisp, and the humble task of calculating | |
1332 | turned out to be more open-ended than one might have expected. | |
1333 | ||
1334 | Emacs Lisp doesn't have built-in floating point math, so it had to be | |
1335 | simulated in software. In fact, Emacs integers will only comfortably | |
1336 | fit six decimal digits or so---not enough for a decent calculator. So | |
1337 | I had to write my own high-precision integer code as well, and once I had | |
1338 | this I figured that arbitrary-size integers were just as easy as large | |
1339 | integers. Arbitrary floating-point precision was the logical next step. | |
1340 | Also, since the large integer arithmetic was there anyway it seemed only | |
1341 | fair to give the user direct access to it, which in turn made it practical | |
1342 | to support fractions as well as floats. All these features inspired me | |
1343 | to look around for other data types that might be worth having. | |
1344 | ||
1345 | Around this time, my friend Rick Koshi showed me his nifty new HP-28 | |
1346 | calculator. It allowed the user to manipulate formulas as well as | |
1347 | numerical quantities, and it could also operate on matrices. I decided | |
1348 | that these would be good for Calc to have, too. And once things had | |
1349 | gone this far, I figured I might as well take a look at serious algebra | |
1350 | systems like Mathematica, Macsyma, and Maple for further ideas. Since | |
1351 | these systems did far more than I could ever hope to implement, I decided | |
1352 | to focus on rewrite rules and other programming features so that users | |
1353 | could implement what they needed for themselves. | |
1354 | ||
1355 | Rick complained that matrices were hard to read, so I put in code to | |
1356 | format them in a 2D style. Once these routines were in place, Big mode | |
1357 | was obligatory. Gee, what other language modes would be useful? | |
1358 | ||
1359 | Scott Hemphill and Allen Knutson, two friends with a strong mathematical | |
1360 | bent, contributed ideas and algorithms for a number of Calc features | |
1361 | including modulo forms, primality testing, and float-to-fraction conversion. | |
1362 | ||
1363 | Units were added at the eager insistence of Mass Sivilotti. Later, | |
1364 | Ulrich Mueller at CERN and Przemek Klosowski at NIST provided invaluable | |
1365 | expert assistance with the units table. As far as I can remember, the | |
1366 | idea of using algebraic formulas and variables to represent units dates | |
1367 | back to an ancient article in Byte magazine about muMath, an early | |
1368 | algebra system for microcomputers. | |
1369 | ||
1370 | Many people have contributed to Calc by reporting bugs and suggesting | |
1371 | features, large and small. A few deserve special mention: Tim Peters, | |
1372 | who helped develop the ideas that led to the selection commands, rewrite | |
1373 | rules, and many other algebra features; @c{Fran\c cois} | |
1374 | @asis{Francois} Pinard, who contributed | |
1375 | an early prototype of the Calc Summary appendix as well as providing | |
1376 | valuable suggestions in many other areas of Calc; Carl Witty, whose eagle | |
1377 | eyes discovered many typographical and factual errors in the Calc manual; | |
1378 | Tim Kay, who drove the development of Embedded mode; Ove Ewerlid, who | |
1379 | made many suggestions relating to the algebra commands and contributed | |
1380 | some code for polynomial operations; Randal Schwartz, who suggested the | |
1381 | @code{calc-eval} function; Robert J. Chassell, who suggested the Calc | |
1382 | Tutorial and exercises; and Juha Sarlin, who first worked out how to split | |
1383 | Calc into quickly-loading parts. Bob Weiner helped immensely with the | |
1384 | Lucid Emacs port. | |
1385 | ||
1386 | @cindex Bibliography | |
1387 | @cindex Knuth, Art of Computer Programming | |
1388 | @cindex Numerical Recipes | |
1389 | @c Should these be expanded into more complete references? | |
1390 | Among the books used in the development of Calc were Knuth's @emph{Art | |
1391 | of Computer Programming} (especially volume II, @emph{Seminumerical | |
1392 | Algorithms}); @emph{Numerical Recipes} by Press, Flannery, Teukolsky, | |
1393 | and Vetterling; Bevington's @emph{Data Reduction and Error Analysis for | |
1394 | the Physical Sciences}; @emph{Concrete Mathematics} by Graham, Knuth, | |
1395 | and Patashnik; Steele's @emph{Common Lisp, the Language}; the @emph{CRC | |
1396 | Standard Math Tables} (William H. Beyer, ed.); and Abramowitz and | |
1397 | Stegun's venerable @emph{Handbook of Mathematical Functions}. I | |
1398 | consulted the user's manuals for the HP-28 and HP-48 calculators, as | |
1399 | well as for the programs Mathematica, SMP, Macsyma, Maple, MathCAD, | |
1400 | Gnuplot, and others. Also, of course, Calc could not have been written | |
1401 | without the excellent @emph{GNU Emacs Lisp Reference Manual}, by Bil | |
1402 | Lewis and Dan LaLiberte. | |
1403 | ||
1404 | Final thanks go to Richard Stallman, without whose fine implementations | |
1405 | of the Emacs editor, language, and environment, Calc would have been | |
1406 | finished in two weeks. | |
1407 | ||
1408 | @c [tutorial] | |
1409 | ||
1410 | @ifinfo | |
1411 | @c This node is accessed by the `M-# t' command. | |
1412 | @node Interactive Tutorial, , , Top | |
1413 | @chapter Tutorial | |
1414 | ||
1415 | @noindent | |
1416 | Some brief instructions on using the Emacs Info system for this tutorial: | |
1417 | ||
1418 | Press the space bar and Delete keys to go forward and backward in a | |
1419 | section by screenfuls (or use the regular Emacs scrolling commands | |
1420 | for this). | |
1421 | ||
1422 | Press @kbd{n} or @kbd{p} to go to the Next or Previous section. | |
1423 | If the section has a @dfn{menu}, press a digit key like @kbd{1} | |
1424 | or @kbd{2} to go to a sub-section from the menu. Press @kbd{u} to | |
1425 | go back up from a sub-section to the menu it is part of. | |
1426 | ||
1427 | Exercises in the tutorial all have cross-references to the | |
1428 | appropriate page of the ``answers'' section. Press @kbd{f}, then | |
1429 | the exercise number, to see the answer to an exercise. After | |
1430 | you have followed a cross-reference, you can press the letter | |
1431 | @kbd{l} to return to where you were before. | |
1432 | ||
1433 | You can press @kbd{?} at any time for a brief summary of Info commands. | |
1434 | ||
1435 | Press @kbd{1} now to enter the first section of the Tutorial. | |
1436 | ||
1437 | @menu | |
1438 | * Tutorial:: | |
1439 | @end menu | |
1440 | @end ifinfo | |
1441 | ||
1442 | @node Tutorial, Introduction, Getting Started, Top | |
1443 | @chapter Tutorial | |
1444 | ||
1445 | @noindent | |
1446 | This chapter explains how to use Calc and its many features, in | |
1447 | a step-by-step, tutorial way. You are encouraged to run Calc and | |
1448 | work along with the examples as you read (@pxref{Starting Calc}). | |
1449 | If you are already familiar with advanced calculators, you may wish | |
1450 | @c [not-split] | |
1451 | to skip on to the rest of this manual. | |
1452 | @c [when-split] | |
1453 | @c to skip on to volume II of this manual, the @dfn{Calc Reference}. | |
1454 | ||
1455 | @c [fix-ref Embedded Mode] | |
1456 | This tutorial describes the standard user interface of Calc only. | |
1457 | The ``Quick Mode'' and ``Keypad Mode'' interfaces are fairly | |
1458 | self-explanatory. @xref{Embedded Mode}, for a description of | |
1459 | the ``Embedded Mode'' interface. | |
1460 | ||
1461 | @ifinfo | |
1462 | The easiest way to read this tutorial on-line is to have two windows on | |
1463 | your Emacs screen, one with Calc and one with the Info system. (If you | |
1464 | have a printed copy of the manual you can use that instead.) Press | |
1465 | @kbd{M-# c} to turn Calc on or to switch into the Calc window, and | |
1466 | press @kbd{M-# i} to start the Info system or to switch into its window. | |
1467 | Or, you may prefer to use the tutorial in printed form. | |
1468 | @end ifinfo | |
1469 | @iftex | |
1470 | The easiest way to read this tutorial on-line is to have two windows on | |
1471 | your Emacs screen, one with Calc and one with the Info system. (If you | |
1472 | have a printed copy of the manual you can use that instead.) Press | |
1473 | @kbd{M-# c} to turn Calc on or to switch into the Calc window, and | |
1474 | press @kbd{M-# i} to start the Info system or to switch into its window. | |
1475 | @end iftex | |
1476 | ||
1477 | This tutorial is designed to be done in sequence. But the rest of this | |
1478 | manual does not assume you have gone through the tutorial. The tutorial | |
1479 | does not cover everything in the Calculator, but it touches on most | |
1480 | general areas. | |
1481 | ||
1482 | @ifinfo | |
1483 | You may wish to print out a copy of the Calc Summary and keep notes on | |
1484 | it as you learn Calc. @xref{Installation}, to see how to make a printed | |
1485 | summary. @xref{Summary}. | |
1486 | @end ifinfo | |
1487 | @iftex | |
1488 | The Calc Summary at the end of the reference manual includes some blank | |
1489 | space for your own use. You may wish to keep notes there as you learn | |
1490 | Calc. | |
1491 | @end iftex | |
1492 | ||
1493 | @menu | |
1494 | * Basic Tutorial:: | |
1495 | * Arithmetic Tutorial:: | |
1496 | * Vector/Matrix Tutorial:: | |
1497 | * Types Tutorial:: | |
1498 | * Algebra Tutorial:: | |
1499 | * Programming Tutorial:: | |
1500 | ||
1501 | * Answers to Exercises:: | |
1502 | @end menu | |
1503 | ||
1504 | @node Basic Tutorial, Arithmetic Tutorial, Tutorial, Tutorial | |
1505 | @section Basic Tutorial | |
1506 | ||
1507 | @noindent | |
1508 | In this section, we learn how RPN and algebraic-style calculations | |
1509 | work, how to undo and redo an operation done by mistake, and how | |
1510 | to control various modes of the Calculator. | |
1511 | ||
1512 | @menu | |
1513 | * RPN Tutorial:: Basic operations with the stack. | |
1514 | * Algebraic Tutorial:: Algebraic entry; variables. | |
1515 | * Undo Tutorial:: If you make a mistake: Undo and the trail. | |
1516 | * Modes Tutorial:: Common mode-setting commands. | |
1517 | @end menu | |
1518 | ||
1519 | @node RPN Tutorial, Algebraic Tutorial, Basic Tutorial, Basic Tutorial | |
1520 | @subsection RPN Calculations and the Stack | |
1521 | ||
1522 | @cindex RPN notation | |
1523 | @ifinfo | |
1524 | @noindent | |
1525 | Calc normally uses RPN notation. You may be familiar with the RPN | |
1526 | system from Hewlett-Packard calculators, FORTH, or PostScript. | |
1527 | (Reverse Polish Notation, RPN, is named after the Polish mathematician | |
1528 | Jan Lukasiewicz.) | |
1529 | @end ifinfo | |
1530 | @tex | |
1531 | \noindent | |
1532 | Calc normally uses RPN notation. You may be familiar with the RPN | |
1533 | system from Hewlett-Packard calculators, FORTH, or PostScript. | |
1534 | (Reverse Polish Notation, RPN, is named after the Polish mathematician | |
1535 | Jan \L ukasiewicz.) | |
1536 | @end tex | |
1537 | ||
1538 | The central component of an RPN calculator is the @dfn{stack}. A | |
1539 | calculator stack is like a stack of dishes. New dishes (numbers) are | |
1540 | added at the top of the stack, and numbers are normally only removed | |
1541 | from the top of the stack. | |
1542 | ||
1543 | @cindex Operators | |
1544 | @cindex Operands | |
1545 | In an operation like @cite{2+3}, the 2 and 3 are called the @dfn{operands} | |
1546 | and the @cite{+} is the @dfn{operator}. In an RPN calculator you always | |
1547 | enter the operands first, then the operator. Each time you type a | |
1548 | number, Calc adds or @dfn{pushes} it onto the top of the Stack. | |
1549 | When you press an operator key like @kbd{+}, Calc @dfn{pops} the appropriate | |
1550 | number of operands from the stack and pushes back the result. | |
1551 | ||
1552 | Thus we could add the numbers 2 and 3 in an RPN calculator by typing: | |
1553 | @kbd{2 @key{RET} 3 @key{RET} +}. (The @key{RET} key, Return, corresponds to | |
1554 | the @key{ENTER} key on traditional RPN calculators.) Try this now if | |
1555 | you wish; type @kbd{M-# c} to switch into the Calc window (you can type | |
1556 | @kbd{M-# c} again or @kbd{M-# o} to switch back to the Tutorial window). | |
1557 | The first four keystrokes ``push'' the numbers 2 and 3 onto the stack. | |
1558 | The @kbd{+} key ``pops'' the top two numbers from the stack, adds them, | |
1559 | and pushes the result (5) back onto the stack. Here's how the stack | |
1560 | will look at various points throughout the calculation:@refill | |
1561 | ||
d7b8e6c6 | 1562 | @smallexample |
5d67986c | 1563 | @group |
d7b8e6c6 EZ |
1564 | . 1: 2 2: 2 1: 5 . |
1565 | . 1: 3 . | |
1566 | . | |
1567 | ||
5d67986c | 1568 | M-# c 2 @key{RET} 3 @key{RET} + @key{DEL} |
d7b8e6c6 | 1569 | @end group |
5d67986c | 1570 | @end smallexample |
d7b8e6c6 EZ |
1571 | |
1572 | The @samp{.} symbol is a marker that represents the top of the stack. | |
1573 | Note that the ``top'' of the stack is really shown at the bottom of | |
1574 | the Stack window. This may seem backwards, but it turns out to be | |
1575 | less distracting in regular use. | |
1576 | ||
1577 | @cindex Stack levels | |
1578 | @cindex Levels of stack | |
1579 | The numbers @samp{1:} and @samp{2:} on the left are @dfn{stack level | |
1580 | numbers}. Old RPN calculators always had four stack levels called | |
1581 | @cite{x}, @cite{y}, @cite{z}, and @cite{t}. Calc's stack can grow | |
1582 | as large as you like, so it uses numbers instead of letters. Some | |
1583 | stack-manipulation commands accept a numeric argument that says | |
1584 | which stack level to work on. Normal commands like @kbd{+} always | |
1585 | work on the top few levels of the stack.@refill | |
1586 | ||
1587 | @c [fix-ref Truncating the Stack] | |
1588 | The Stack buffer is just an Emacs buffer, and you can move around in | |
1589 | it using the regular Emacs motion commands. But no matter where the | |
1590 | cursor is, even if you have scrolled the @samp{.} marker out of | |
1591 | view, most Calc commands always move the cursor back down to level 1 | |
1592 | before doing anything. It is possible to move the @samp{.} marker | |
1593 | upwards through the stack, temporarily ``hiding'' some numbers from | |
1594 | commands like @kbd{+}. This is called @dfn{stack truncation} and | |
1595 | we will not cover it in this tutorial; @pxref{Truncating the Stack}, | |
1596 | if you are interested. | |
1597 | ||
1598 | You don't really need the second @key{RET} in @kbd{2 @key{RET} 3 | |
1599 | @key{RET} +}. That's because if you type any operator name or | |
1600 | other non-numeric key when you are entering a number, the Calculator | |
1601 | automatically enters that number and then does the requested command. | |
1602 | Thus @kbd{2 @key{RET} 3 +} will work just as well.@refill | |
1603 | ||
1604 | Examples in this tutorial will often omit @key{RET} even when the | |
1605 | stack displays shown would only happen if you did press @key{RET}: | |
1606 | ||
d7b8e6c6 | 1607 | @smallexample |
5d67986c | 1608 | @group |
d7b8e6c6 EZ |
1609 | 1: 2 2: 2 1: 5 |
1610 | . 1: 3 . | |
1611 | . | |
1612 | ||
5d67986c | 1613 | 2 @key{RET} 3 + |
d7b8e6c6 | 1614 | @end group |
5d67986c | 1615 | @end smallexample |
d7b8e6c6 EZ |
1616 | |
1617 | @noindent | |
1618 | Here, after pressing @kbd{3} the stack would really show @samp{1: 2} | |
1619 | with @samp{Calc:@: 3} in the minibuffer. In these situations, you can | |
1620 | press the optional @key{RET} to see the stack as the figure shows. | |
1621 | ||
1622 | (@bullet{}) @strong{Exercise 1.} (This tutorial will include exercises | |
1623 | at various points. Try them if you wish. Answers to all the exercises | |
1624 | are located at the end of the Tutorial chapter. Each exercise will | |
1625 | include a cross-reference to its particular answer. If you are | |
1626 | reading with the Emacs Info system, press @kbd{f} and the | |
1627 | exercise number to go to the answer, then the letter @kbd{l} to | |
1628 | return to where you were.) | |
1629 | ||
1630 | @noindent | |
1631 | Here's the first exercise: What will the keystrokes @kbd{1 @key{RET} 2 | |
1632 | @key{RET} 3 @key{RET} 4 + * -} compute? (@samp{*} is the symbol for | |
1633 | multiplication.) Figure it out by hand, then try it with Calc to see | |
1634 | if you're right. @xref{RPN Answer 1, 1}. (@bullet{}) | |
1635 | ||
1636 | (@bullet{}) @strong{Exercise 2.} Compute @c{$(2\times4) + (7\times9.4) + {5\over4}$} | |
1637 | @cite{2*4 + 7*9.5 + 5/4} using the | |
1638 | stack. @xref{RPN Answer 2, 2}. (@bullet{}) | |
1639 | ||
1640 | The @key{DEL} key is called Backspace on some keyboards. It is | |
1641 | whatever key you would use to correct a simple typing error when | |
1642 | regularly using Emacs. The @key{DEL} key pops and throws away the | |
1643 | top value on the stack. (You can still get that value back from | |
1644 | the Trail if you should need it later on.) There are many places | |
1645 | in this tutorial where we assume you have used @key{DEL} to erase the | |
1646 | results of the previous example at the beginning of a new example. | |
1647 | In the few places where it is really important to use @key{DEL} to | |
1648 | clear away old results, the text will remind you to do so. | |
1649 | ||
1650 | (It won't hurt to let things accumulate on the stack, except that | |
1651 | whenever you give a display-mode-changing command Calc will have to | |
1652 | spend a long time reformatting such a large stack.) | |
1653 | ||
1654 | Since the @kbd{-} key is also an operator (it subtracts the top two | |
1655 | stack elements), how does one enter a negative number? Calc uses | |
1656 | the @kbd{_} (underscore) key to act like the minus sign in a number. | |
1657 | So, typing @kbd{-5 @key{RET}} won't work because the @kbd{-} key | |
1658 | will try to do a subtraction, but @kbd{_5 @key{RET}} works just fine. | |
1659 | ||
1660 | You can also press @kbd{n}, which means ``change sign.'' It changes | |
1661 | the number at the top of the stack (or the number being entered) | |
1662 | from positive to negative or vice-versa: @kbd{5 n @key{RET}}. | |
1663 | ||
1664 | @cindex Duplicating a stack entry | |
1665 | If you press @key{RET} when you're not entering a number, the effect | |
1666 | is to duplicate the top number on the stack. Consider this calculation: | |
1667 | ||
d7b8e6c6 | 1668 | @smallexample |
5d67986c | 1669 | @group |
d7b8e6c6 EZ |
1670 | 1: 3 2: 3 1: 9 2: 9 1: 81 |
1671 | . 1: 3 . 1: 9 . | |
1672 | . . | |
1673 | ||
5d67986c | 1674 | 3 @key{RET} @key{RET} * @key{RET} * |
d7b8e6c6 | 1675 | @end group |
5d67986c | 1676 | @end smallexample |
d7b8e6c6 EZ |
1677 | |
1678 | @noindent | |
1679 | (Of course, an easier way to do this would be @kbd{3 @key{RET} 4 ^}, | |
1680 | to raise 3 to the fourth power.) | |
1681 | ||
1682 | The space-bar key (denoted @key{SPC} here) performs the same function | |
1683 | as @key{RET}; you could replace all three occurrences of @key{RET} in | |
1684 | the above example with @key{SPC} and the effect would be the same. | |
1685 | ||
1686 | @cindex Exchanging stack entries | |
1687 | Another stack manipulation key is @key{TAB}. This exchanges the top | |
1688 | two stack entries. Suppose you have computed @kbd{2 @key{RET} 3 +} | |
1689 | to get 5, and then you realize what you really wanted to compute | |
1690 | was @cite{20 / (2+3)}. | |
1691 | ||
d7b8e6c6 | 1692 | @smallexample |
5d67986c | 1693 | @group |
d7b8e6c6 EZ |
1694 | 1: 5 2: 5 2: 20 1: 4 |
1695 | . 1: 20 1: 5 . | |
1696 | . . | |
1697 | ||
5d67986c | 1698 | 2 @key{RET} 3 + 20 @key{TAB} / |
d7b8e6c6 | 1699 | @end group |
5d67986c | 1700 | @end smallexample |
d7b8e6c6 EZ |
1701 | |
1702 | @noindent | |
1703 | Planning ahead, the calculation would have gone like this: | |
1704 | ||
d7b8e6c6 | 1705 | @smallexample |
5d67986c | 1706 | @group |
d7b8e6c6 EZ |
1707 | 1: 20 2: 20 3: 20 2: 20 1: 4 |
1708 | . 1: 2 2: 2 1: 5 . | |
1709 | . 1: 3 . | |
1710 | . | |
1711 | ||
5d67986c | 1712 | 20 @key{RET} 2 @key{RET} 3 + / |
d7b8e6c6 | 1713 | @end group |
5d67986c | 1714 | @end smallexample |
d7b8e6c6 EZ |
1715 | |
1716 | A related stack command is @kbd{M-@key{TAB}} (hold @key{META} and type | |
1717 | @key{TAB}). It rotates the top three elements of the stack upward, | |
1718 | bringing the object in level 3 to the top. | |
1719 | ||
d7b8e6c6 | 1720 | @smallexample |
5d67986c | 1721 | @group |
d7b8e6c6 EZ |
1722 | 1: 10 2: 10 3: 10 3: 20 3: 30 |
1723 | . 1: 20 2: 20 2: 30 2: 10 | |
1724 | . 1: 30 1: 10 1: 20 | |
1725 | . . . | |
1726 | ||
5d67986c | 1727 | 10 @key{RET} 20 @key{RET} 30 @key{RET} M-@key{TAB} M-@key{TAB} |
d7b8e6c6 | 1728 | @end group |
5d67986c | 1729 | @end smallexample |
d7b8e6c6 EZ |
1730 | |
1731 | (@bullet{}) @strong{Exercise 3.} Suppose the numbers 10, 20, and 30 are | |
1732 | on the stack. Figure out how to add one to the number in level 2 | |
1733 | without affecting the rest of the stack. Also figure out how to add | |
1734 | one to the number in level 3. @xref{RPN Answer 3, 3}. (@bullet{}) | |
1735 | ||
1736 | Operations like @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/}, and @kbd{^} pop two | |
1737 | arguments from the stack and push a result. Operations like @kbd{n} and | |
1738 | @kbd{Q} (square root) pop a single number and push the result. You can | |
1739 | think of them as simply operating on the top element of the stack. | |
1740 | ||
d7b8e6c6 | 1741 | @smallexample |
5d67986c | 1742 | @group |
d7b8e6c6 EZ |
1743 | 1: 3 1: 9 2: 9 1: 25 1: 5 |
1744 | . . 1: 16 . . | |
1745 | . | |
1746 | ||
5d67986c | 1747 | 3 @key{RET} @key{RET} * 4 @key{RET} @key{RET} * + Q |
d7b8e6c6 | 1748 | @end group |
5d67986c | 1749 | @end smallexample |
d7b8e6c6 EZ |
1750 | |
1751 | @noindent | |
1752 | (Note that capital @kbd{Q} means to hold down the Shift key while | |
1753 | typing @kbd{q}. Remember, plain unshifted @kbd{q} is the Quit command.) | |
1754 | ||
1755 | @cindex Pythagorean Theorem | |
1756 | Here we've used the Pythagorean Theorem to determine the hypotenuse of a | |
1757 | right triangle. Calc actually has a built-in command for that called | |
1758 | @kbd{f h}, but let's suppose we can't remember the necessary keystrokes. | |
1759 | We can still enter it by its full name using @kbd{M-x} notation: | |
1760 | ||
d7b8e6c6 | 1761 | @smallexample |
5d67986c | 1762 | @group |
d7b8e6c6 EZ |
1763 | 1: 3 2: 3 1: 5 |
1764 | . 1: 4 . | |
1765 | . | |
1766 | ||
5d67986c | 1767 | 3 @key{RET} 4 @key{RET} M-x calc-hypot |
d7b8e6c6 | 1768 | @end group |
5d67986c | 1769 | @end smallexample |
d7b8e6c6 EZ |
1770 | |
1771 | All Calculator commands begin with the word @samp{calc-}. Since it | |
1772 | gets tiring to type this, Calc provides an @kbd{x} key which is just | |
1773 | like the regular Emacs @kbd{M-x} key except that it types the @samp{calc-} | |
1774 | prefix for you: | |
1775 | ||
d7b8e6c6 | 1776 | @smallexample |
5d67986c | 1777 | @group |
d7b8e6c6 EZ |
1778 | 1: 3 2: 3 1: 5 |
1779 | . 1: 4 . | |
1780 | . | |
1781 | ||
5d67986c | 1782 | 3 @key{RET} 4 @key{RET} x hypot |
d7b8e6c6 | 1783 | @end group |
5d67986c | 1784 | @end smallexample |
d7b8e6c6 EZ |
1785 | |
1786 | What happens if you take the square root of a negative number? | |
1787 | ||
d7b8e6c6 | 1788 | @smallexample |
5d67986c | 1789 | @group |
d7b8e6c6 EZ |
1790 | 1: 4 1: -4 1: (0, 2) |
1791 | . . . | |
1792 | ||
5d67986c | 1793 | 4 @key{RET} n Q |
d7b8e6c6 | 1794 | @end group |
5d67986c | 1795 | @end smallexample |
d7b8e6c6 EZ |
1796 | |
1797 | @noindent | |
1798 | The notation @cite{(a, b)} represents a complex number. | |
1799 | Complex numbers are more traditionally written @c{$a + b i$} | |
1800 | @cite{a + b i}; | |
1801 | Calc can display in this format, too, but for now we'll stick to the | |
1802 | @cite{(a, b)} notation. | |
1803 | ||
1804 | If you don't know how complex numbers work, you can safely ignore this | |
1805 | feature. Complex numbers only arise from operations that would be | |
1806 | errors in a calculator that didn't have complex numbers. (For example, | |
1807 | taking the square root or logarithm of a negative number produces a | |
1808 | complex result.) | |
1809 | ||
1810 | Complex numbers are entered in the notation shown. The @kbd{(} and | |
1811 | @kbd{,} and @kbd{)} keys manipulate ``incomplete complex numbers.'' | |
1812 | ||
d7b8e6c6 | 1813 | @smallexample |
5d67986c | 1814 | @group |
d7b8e6c6 EZ |
1815 | 1: ( ... 2: ( ... 1: (2, ... 1: (2, ... 1: (2, 3) |
1816 | . 1: 2 . 3 . | |
1817 | . . | |
1818 | ||
1819 | ( 2 , 3 ) | |
d7b8e6c6 | 1820 | @end group |
5d67986c | 1821 | @end smallexample |
d7b8e6c6 EZ |
1822 | |
1823 | You can perform calculations while entering parts of incomplete objects. | |
1824 | However, an incomplete object cannot actually participate in a calculation: | |
1825 | ||
d7b8e6c6 | 1826 | @smallexample |
5d67986c | 1827 | @group |
d7b8e6c6 EZ |
1828 | 1: ( ... 2: ( ... 3: ( ... 1: ( ... 1: ( ... |
1829 | . 1: 2 2: 2 5 5 | |
1830 | . 1: 3 . . | |
1831 | . | |
1832 | (error) | |
5d67986c | 1833 | ( 2 @key{RET} 3 + + |
d7b8e6c6 | 1834 | @end group |
5d67986c | 1835 | @end smallexample |
d7b8e6c6 EZ |
1836 | |
1837 | @noindent | |
1838 | Adding 5 to an incomplete object makes no sense, so the last command | |
1839 | produces an error message and leaves the stack the same. | |
1840 | ||
1841 | Incomplete objects can't participate in arithmetic, but they can be | |
1842 | moved around by the regular stack commands. | |
1843 | ||
d7b8e6c6 | 1844 | @smallexample |
5d67986c | 1845 | @group |
d7b8e6c6 EZ |
1846 | 2: 2 3: 2 3: 3 1: ( ... 1: (2, 3) |
1847 | 1: 3 2: 3 2: ( ... 2 . | |
1848 | . 1: ( ... 1: 2 3 | |
1849 | . . . | |
1850 | ||
5d67986c | 1851 | 2 @key{RET} 3 @key{RET} ( M-@key{TAB} M-@key{TAB} ) |
d7b8e6c6 | 1852 | @end group |
5d67986c | 1853 | @end smallexample |
d7b8e6c6 EZ |
1854 | |
1855 | @noindent | |
1856 | Note that the @kbd{,} (comma) key did not have to be used here. | |
1857 | When you press @kbd{)} all the stack entries between the incomplete | |
1858 | entry and the top are collected, so there's never really a reason | |
1859 | to use the comma. It's up to you. | |
1860 | ||
1861 | (@bullet{}) @strong{Exercise 4.} To enter the complex number @cite{(2, 3)}, | |
1862 | your friend Joe typed @kbd{( 2 , @key{SPC} 3 )}. What happened? | |
1863 | (Joe thought of a clever way to correct his mistake in only two | |
1864 | keystrokes, but it didn't quite work. Try it to find out why.) | |
1865 | @xref{RPN Answer 4, 4}. (@bullet{}) | |
1866 | ||
1867 | Vectors are entered the same way as complex numbers, but with square | |
1868 | brackets in place of parentheses. We'll meet vectors again later in | |
1869 | the tutorial. | |
1870 | ||
1871 | Any Emacs command can be given a @dfn{numeric prefix argument} by | |
1872 | typing a series of @key{META}-digits beforehand. If @key{META} is | |
1873 | awkward for you, you can instead type @kbd{C-u} followed by the | |
1874 | necessary digits. Numeric prefix arguments can be negative, as in | |
1875 | @kbd{M-- M-3 M-5} or @w{@kbd{C-u - 3 5}}. Calc commands use numeric | |
1876 | prefix arguments in a variety of ways. For example, a numeric prefix | |
1877 | on the @kbd{+} operator adds any number of stack entries at once: | |
1878 | ||
d7b8e6c6 | 1879 | @smallexample |
5d67986c | 1880 | @group |
d7b8e6c6 EZ |
1881 | 1: 10 2: 10 3: 10 3: 10 1: 60 |
1882 | . 1: 20 2: 20 2: 20 . | |
1883 | . 1: 30 1: 30 | |
1884 | . . | |
1885 | ||
5d67986c | 1886 | 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u 3 + |
d7b8e6c6 | 1887 | @end group |
5d67986c | 1888 | @end smallexample |
d7b8e6c6 EZ |
1889 | |
1890 | For stack manipulation commands like @key{RET}, a positive numeric | |
1891 | prefix argument operates on the top @var{n} stack entries at once. A | |
1892 | negative argument operates on the entry in level @var{n} only. An | |
1893 | argument of zero operates on the entire stack. In this example, we copy | |
1894 | the second-to-top element of the stack: | |
1895 | ||
d7b8e6c6 | 1896 | @smallexample |
5d67986c | 1897 | @group |
d7b8e6c6 EZ |
1898 | 1: 10 2: 10 3: 10 3: 10 4: 10 |
1899 | . 1: 20 2: 20 2: 20 3: 20 | |
1900 | . 1: 30 1: 30 2: 30 | |
1901 | . . 1: 20 | |
1902 | . | |
1903 | ||
5d67986c | 1904 | 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u -2 @key{RET} |
d7b8e6c6 | 1905 | @end group |
5d67986c | 1906 | @end smallexample |
d7b8e6c6 EZ |
1907 | |
1908 | @cindex Clearing the stack | |
1909 | @cindex Emptying the stack | |
5d67986c | 1910 | Another common idiom is @kbd{M-0 @key{DEL}}, which clears the stack. |
d7b8e6c6 EZ |
1911 | (The @kbd{M-0} numeric prefix tells @key{DEL} to operate on the |
1912 | entire stack.) | |
1913 | ||
1914 | @node Algebraic Tutorial, Undo Tutorial, RPN Tutorial, Basic Tutorial | |
1915 | @subsection Algebraic-Style Calculations | |
1916 | ||
1917 | @noindent | |
1918 | If you are not used to RPN notation, you may prefer to operate the | |
1919 | Calculator in ``algebraic mode,'' which is closer to the way | |
1920 | non-RPN calculators work. In algebraic mode, you enter formulas | |
1921 | in traditional @cite{2+3} notation. | |
1922 | ||
1923 | You don't really need any special ``mode'' to enter algebraic formulas. | |
1924 | You can enter a formula at any time by pressing the apostrophe (@kbd{'}) | |
1925 | key. Answer the prompt with the desired formula, then press @key{RET}. | |
1926 | The formula is evaluated and the result is pushed onto the RPN stack. | |
1927 | If you don't want to think in RPN at all, you can enter your whole | |
1928 | computation as a formula, read the result from the stack, then press | |
1929 | @key{DEL} to delete it from the stack. | |
1930 | ||
1931 | Try pressing the apostrophe key, then @kbd{2+3+4}, then @key{RET}. | |
1932 | The result should be the number 9. | |
1933 | ||
1934 | Algebraic formulas use the operators @samp{+}, @samp{-}, @samp{*}, | |
1935 | @samp{/}, and @samp{^}. You can use parentheses to make the order | |
1936 | of evaluation clear. In the absence of parentheses, @samp{^} is | |
1937 | evaluated first, then @samp{*}, then @samp{/}, then finally | |
1938 | @samp{+} and @samp{-}. For example, the expression | |
1939 | ||
1940 | @example | |
1941 | 2 + 3*4*5 / 6*7^8 - 9 | |
1942 | @end example | |
1943 | ||
1944 | @noindent | |
1945 | is equivalent to | |
1946 | ||
1947 | @example | |
1948 | 2 + ((3*4*5) / (6*(7^8)) - 9 | |
1949 | @end example | |
1950 | ||
1951 | @noindent | |
1952 | or, in large mathematical notation, | |
1953 | ||
1954 | @ifinfo | |
d7b8e6c6 | 1955 | @example |
5d67986c | 1956 | @group |
d7b8e6c6 EZ |
1957 | 3 * 4 * 5 |
1958 | 2 + --------- - 9 | |
1959 | 8 | |
1960 | 6 * 7 | |
d7b8e6c6 | 1961 | @end group |
5d67986c | 1962 | @end example |
d7b8e6c6 EZ |
1963 | @end ifinfo |
1964 | @tex | |
1965 | \turnoffactive | |
1966 | \beforedisplay | |
1967 | $$ 2 + { 3 \times 4 \times 5 \over 6 \times 7^8 } - 9 $$ | |
1968 | \afterdisplay | |
1969 | @end tex | |
1970 | ||
1971 | @noindent | |
1972 | The result of this expression will be the number @i{-6.99999826533}. | |
1973 | ||
1974 | Calc's order of evaluation is the same as for most computer languages, | |
1975 | except that @samp{*} binds more strongly than @samp{/}, as the above | |
1976 | example shows. As in normal mathematical notation, the @samp{*} symbol | |
1977 | can often be omitted: @samp{2 a} is the same as @samp{2*a}. | |
1978 | ||
1979 | Operators at the same level are evaluated from left to right, except | |
1980 | that @samp{^} is evaluated from right to left. Thus, @samp{2-3-4} is | |
1981 | equivalent to @samp{(2-3)-4} or @i{-5}, whereas @samp{2^3^4} is equivalent | |
1982 | to @samp{2^(3^4)} (a very large integer; try it!). | |
1983 | ||
1984 | If you tire of typing the apostrophe all the time, there is an | |
1985 | ``algebraic mode'' you can select in which Calc automatically senses | |
1986 | when you are about to type an algebraic expression. To enter this | |
1987 | mode, press the two letters @w{@kbd{m a}}. (An @samp{Alg} indicator | |
1988 | should appear in the Calc window's mode line.) | |
1989 | ||
1990 | Press @kbd{m a}, then @kbd{2+3+4} with no apostrophe, then @key{RET}. | |
1991 | ||
1992 | In algebraic mode, when you press any key that would normally begin | |
1993 | entering a number (such as a digit, a decimal point, or the @kbd{_} | |
1994 | key), or if you press @kbd{(} or @kbd{[}, Calc automatically begins | |
1995 | an algebraic entry. | |
1996 | ||
1997 | Functions which do not have operator symbols like @samp{+} and @samp{*} | |
1998 | must be entered in formulas using function-call notation. For example, | |
1999 | the function name corresponding to the square-root key @kbd{Q} is | |
2000 | @code{sqrt}. To compute a square root in a formula, you would use | |
2001 | the notation @samp{sqrt(@var{x})}. | |
2002 | ||
2003 | Press the apostrophe, then type @kbd{sqrt(5*2) - 3}. The result should | |
2004 | be @cite{0.16227766017}. | |
2005 | ||
2006 | Note that if the formula begins with a function name, you need to use | |
2007 | the apostrophe even if you are in algebraic mode. If you type @kbd{arcsin} | |
2008 | out of the blue, the @kbd{a r} will be taken as an Algebraic Rewrite | |
2009 | command, and the @kbd{csin} will be taken as the name of the rewrite | |
2010 | rule to use! | |
2011 | ||
2012 | Some people prefer to enter complex numbers and vectors in algebraic | |
2013 | form because they find RPN entry with incomplete objects to be too | |
2014 | distracting, even though they otherwise use Calc as an RPN calculator. | |
2015 | ||
2016 | Still in algebraic mode, type: | |
2017 | ||
d7b8e6c6 | 2018 | @smallexample |
5d67986c | 2019 | @group |
d7b8e6c6 EZ |
2020 | 1: (2, 3) 2: (2, 3) 1: (8, -1) 2: (8, -1) 1: (9, -1) |
2021 | . 1: (1, -2) . 1: 1 . | |
2022 | . . | |
2023 | ||
5d67986c | 2024 | (2,3) @key{RET} (1,-2) @key{RET} * 1 @key{RET} + |
d7b8e6c6 | 2025 | @end group |
5d67986c | 2026 | @end smallexample |
d7b8e6c6 EZ |
2027 | |
2028 | Algebraic mode allows us to enter complex numbers without pressing | |
2029 | an apostrophe first, but it also means we need to press @key{RET} | |
2030 | after every entry, even for a simple number like @cite{1}. | |
2031 | ||
2032 | (You can type @kbd{C-u m a} to enable a special ``incomplete algebraic | |
2033 | mode'' in which the @kbd{(} and @kbd{[} keys use algebraic entry even | |
2034 | though regular numeric keys still use RPN numeric entry. There is also | |
2035 | a ``total algebraic mode,'' started by typing @kbd{m t}, in which all | |
2036 | normal keys begin algebraic entry. You must then use the @key{META} key | |
2037 | to type Calc commands: @kbd{M-m t} to get back out of total algebraic | |
2038 | mode, @kbd{M-q} to quit, etc. Total algebraic mode is not supported | |
2039 | under Emacs 19.) | |
2040 | ||
2041 | If you're still in algebraic mode, press @kbd{m a} again to turn it off. | |
2042 | ||
2043 | Actual non-RPN calculators use a mixture of algebraic and RPN styles. | |
2044 | In general, operators of two numbers (like @kbd{+} and @kbd{*}) | |
2045 | use algebraic form, but operators of one number (like @kbd{n} and @kbd{Q}) | |
2046 | use RPN form. Also, a non-RPN calculator allows you to see the | |
2047 | intermediate results of a calculation as you go along. You can | |
2048 | accomplish this in Calc by performing your calculation as a series | |
2049 | of algebraic entries, using the @kbd{$} sign to tie them together. | |
2050 | In an algebraic formula, @kbd{$} represents the number on the top | |
2051 | of the stack. Here, we perform the calculation @c{$\sqrt{2\times4+1}$} | |
2052 | @cite{sqrt(2*4+1)}, | |
2053 | which on a traditional calculator would be done by pressing | |
2054 | @kbd{2 * 4 + 1 =} and then the square-root key. | |
2055 | ||
d7b8e6c6 | 2056 | @smallexample |
5d67986c | 2057 | @group |
d7b8e6c6 EZ |
2058 | 1: 8 1: 9 1: 3 |
2059 | . . . | |
2060 | ||
5d67986c | 2061 | ' 2*4 @key{RET} $+1 @key{RET} Q |
d7b8e6c6 | 2062 | @end group |
5d67986c | 2063 | @end smallexample |
d7b8e6c6 EZ |
2064 | |
2065 | @noindent | |
2066 | Notice that we didn't need to press an apostrophe for the @kbd{$+1}, | |
2067 | because the dollar sign always begins an algebraic entry. | |
2068 | ||
2069 | (@bullet{}) @strong{Exercise 1.} How could you get the same effect as | |
2070 | pressing @kbd{Q} but using an algebraic entry instead? How about | |
2071 | if the @kbd{Q} key on your keyboard were broken? | |
2072 | @xref{Algebraic Answer 1, 1}. (@bullet{}) | |
2073 | ||
2074 | The notations @kbd{$$}, @kbd{$$$}, and so on stand for higher stack | |
5d67986c | 2075 | entries. For example, @kbd{' $$+$ @key{RET}} is just like typing @kbd{+}. |
d7b8e6c6 EZ |
2076 | |
2077 | Algebraic formulas can include @dfn{variables}. To store in a | |
2078 | variable, press @kbd{s s}, then type the variable name, then press | |
2079 | @key{RET}. (There are actually two flavors of store command: | |
2080 | @kbd{s s} stores a number in a variable but also leaves the number | |
2081 | on the stack, while @w{@kbd{s t}} removes a number from the stack and | |
2082 | stores it in the variable.) A variable name should consist of one | |
2083 | or more letters or digits, beginning with a letter. | |
2084 | ||
d7b8e6c6 | 2085 | @smallexample |
5d67986c | 2086 | @group |
d7b8e6c6 EZ |
2087 | 1: 17 . 1: a + a^2 1: 306 |
2088 | . . . | |
2089 | ||
5d67986c | 2090 | 17 s t a @key{RET} ' a+a^2 @key{RET} = |
d7b8e6c6 | 2091 | @end group |
5d67986c | 2092 | @end smallexample |
d7b8e6c6 EZ |
2093 | |
2094 | @noindent | |
2095 | The @kbd{=} key @dfn{evaluates} a formula by replacing all its | |
2096 | variables by the values that were stored in them. | |
2097 | ||
2098 | For RPN calculations, you can recall a variable's value on the | |
2099 | stack either by entering its name as a formula and pressing @kbd{=}, | |
2100 | or by using the @kbd{s r} command. | |
2101 | ||
d7b8e6c6 | 2102 | @smallexample |
5d67986c | 2103 | @group |
d7b8e6c6 EZ |
2104 | 1: 17 2: 17 3: 17 2: 17 1: 306 |
2105 | . 1: 17 2: 17 1: 289 . | |
2106 | . 1: 2 . | |
2107 | . | |
2108 | ||
5d67986c | 2109 | s r a @key{RET} ' a @key{RET} = 2 ^ + |
d7b8e6c6 | 2110 | @end group |
5d67986c | 2111 | @end smallexample |
d7b8e6c6 EZ |
2112 | |
2113 | If you press a single digit for a variable name (as in @kbd{s t 3}, you | |
2114 | get one of ten @dfn{quick variables} @code{q0} through @code{q9}. | |
2115 | They are ``quick'' simply because you don't have to type the letter | |
2116 | @code{q} or the @key{RET} after their names. In fact, you can type | |
2117 | simply @kbd{s 3} as a shorthand for @kbd{s s 3}, and likewise for | |
2118 | @kbd{t 3} and @w{@kbd{r 3}}. | |
2119 | ||
2120 | Any variables in an algebraic formula for which you have not stored | |
2121 | values are left alone, even when you evaluate the formula. | |
2122 | ||
d7b8e6c6 | 2123 | @smallexample |
5d67986c | 2124 | @group |
d7b8e6c6 EZ |
2125 | 1: 2 a + 2 b 1: 34 + 2 b |
2126 | . . | |
2127 | ||
5d67986c | 2128 | ' 2a+2b @key{RET} = |
d7b8e6c6 | 2129 | @end group |
5d67986c | 2130 | @end smallexample |
d7b8e6c6 EZ |
2131 | |
2132 | Calls to function names which are undefined in Calc are also left | |
2133 | alone, as are calls for which the value is undefined. | |
2134 | ||
d7b8e6c6 | 2135 | @smallexample |
5d67986c | 2136 | @group |
d7b8e6c6 EZ |
2137 | 1: 2 + log10(0) + log10(x) + log10(5, 6) + foo(3) |
2138 | . | |
2139 | ||
5d67986c | 2140 | ' log10(100) + log10(0) + log10(x) + log10(5,6) + foo(3) @key{RET} |
d7b8e6c6 | 2141 | @end group |
5d67986c | 2142 | @end smallexample |
d7b8e6c6 EZ |
2143 | |
2144 | @noindent | |
2145 | In this example, the first call to @code{log10} works, but the other | |
2146 | calls are not evaluated. In the second call, the logarithm is | |
2147 | undefined for that value of the argument; in the third, the argument | |
2148 | is symbolic, and in the fourth, there are too many arguments. In the | |
2149 | fifth case, there is no function called @code{foo}. You will see a | |
2150 | ``Wrong number of arguments'' message referring to @samp{log10(5,6)}. | |
2151 | Press the @kbd{w} (``why'') key to see any other messages that may | |
2152 | have arisen from the last calculation. In this case you will get | |
2153 | ``logarithm of zero,'' then ``number expected: @code{x}''. Calc | |
2154 | automatically displays the first message only if the message is | |
2155 | sufficiently important; for example, Calc considers ``wrong number | |
2156 | of arguments'' and ``logarithm of zero'' to be important enough to | |
2157 | report automatically, while a message like ``number expected: @code{x}'' | |
2158 | will only show up if you explicitly press the @kbd{w} key. | |
2159 | ||
2160 | (@bullet{}) @strong{Exercise 2.} Joe entered the formula @samp{2 x y}, | |
2161 | stored 5 in @code{x}, pressed @kbd{=}, and got the expected result, | |
2162 | @samp{10 y}. He then tried the same for the formula @samp{2 x (1+y)}, | |
2163 | expecting @samp{10 (1+y)}, but it didn't work. Why not? | |
2164 | @xref{Algebraic Answer 2, 2}. (@bullet{}) | |
2165 | ||
2166 | (@bullet{}) @strong{Exercise 3.} What result would you expect | |
2167 | @kbd{1 @key{RET} 0 /} to give? What if you then type @kbd{0 *}? | |
2168 | @xref{Algebraic Answer 3, 3}. (@bullet{}) | |
2169 | ||
2170 | One interesting way to work with variables is to use the | |
2171 | @dfn{evaluates-to} (@samp{=>}) operator. It works like this: | |
2172 | Enter a formula algebraically in the usual way, but follow | |
2173 | the formula with an @samp{=>} symbol. (There is also an @kbd{s =} | |
2174 | command which builds an @samp{=>} formula using the stack.) On | |
2175 | the stack, you will see two copies of the formula with an @samp{=>} | |
2176 | between them. The lefthand formula is exactly like you typed it; | |
2177 | the righthand formula has been evaluated as if by typing @kbd{=}. | |
2178 | ||
d7b8e6c6 | 2179 | @smallexample |
5d67986c | 2180 | @group |
d7b8e6c6 EZ |
2181 | 2: 2 + 3 => 5 2: 2 + 3 => 5 |
2182 | 1: 2 a + 2 b => 34 + 2 b 1: 2 a + 2 b => 20 + 2 b | |
2183 | . . | |
2184 | ||
5d67986c | 2185 | ' 2+3 => @key{RET} ' 2a+2b @key{RET} s = 10 s t a @key{RET} |
d7b8e6c6 | 2186 | @end group |
5d67986c | 2187 | @end smallexample |
d7b8e6c6 EZ |
2188 | |
2189 | @noindent | |
2190 | Notice that the instant we stored a new value in @code{a}, all | |
2191 | @samp{=>} operators already on the stack that referred to @cite{a} | |
2192 | were updated to use the new value. With @samp{=>}, you can push a | |
2193 | set of formulas on the stack, then change the variables experimentally | |
2194 | to see the effects on the formulas' values. | |
2195 | ||
2196 | You can also ``unstore'' a variable when you are through with it: | |
2197 | ||
d7b8e6c6 | 2198 | @smallexample |
5d67986c | 2199 | @group |
d7b8e6c6 EZ |
2200 | 2: 2 + 5 => 5 |
2201 | 1: 2 a + 2 b => 2 a + 2 b | |
2202 | . | |
2203 | ||
5d67986c | 2204 | s u a @key{RET} |
d7b8e6c6 | 2205 | @end group |
5d67986c | 2206 | @end smallexample |
d7b8e6c6 EZ |
2207 | |
2208 | We will encounter formulas involving variables and functions again | |
2209 | when we discuss the algebra and calculus features of the Calculator. | |
2210 | ||
2211 | @node Undo Tutorial, Modes Tutorial, Algebraic Tutorial, Basic Tutorial | |
2212 | @subsection Undo and Redo | |
2213 | ||
2214 | @noindent | |
2215 | If you make a mistake, you can usually correct it by pressing shift-@kbd{U}, | |
5d67986c | 2216 | the ``undo'' command. First, clear the stack (@kbd{M-0 @key{DEL}}) and exit |
d7b8e6c6 EZ |
2217 | and restart Calc (@kbd{M-# M-# M-# M-#}) to make sure things start off |
2218 | with a clean slate. Now: | |
2219 | ||
d7b8e6c6 | 2220 | @smallexample |
5d67986c | 2221 | @group |
d7b8e6c6 EZ |
2222 | 1: 2 2: 2 1: 8 2: 2 1: 6 |
2223 | . 1: 3 . 1: 3 . | |
2224 | . . | |
2225 | ||
5d67986c | 2226 | 2 @key{RET} 3 ^ U * |
d7b8e6c6 | 2227 | @end group |
5d67986c | 2228 | @end smallexample |
d7b8e6c6 EZ |
2229 | |
2230 | You can undo any number of times. Calc keeps a complete record of | |
2231 | all you have done since you last opened the Calc window. After the | |
2232 | above example, you could type: | |
2233 | ||
d7b8e6c6 | 2234 | @smallexample |
5d67986c | 2235 | @group |
d7b8e6c6 EZ |
2236 | 1: 6 2: 2 1: 2 . . |
2237 | . 1: 3 . | |
2238 | . | |
2239 | (error) | |
2240 | U U U U | |
d7b8e6c6 | 2241 | @end group |
5d67986c | 2242 | @end smallexample |
d7b8e6c6 EZ |
2243 | |
2244 | You can also type @kbd{D} to ``redo'' a command that you have undone | |
2245 | mistakenly. | |
2246 | ||
d7b8e6c6 | 2247 | @smallexample |
5d67986c | 2248 | @group |
d7b8e6c6 EZ |
2249 | . 1: 2 2: 2 1: 6 1: 6 |
2250 | . 1: 3 . . | |
2251 | . | |
2252 | (error) | |
2253 | D D D D | |
d7b8e6c6 | 2254 | @end group |
5d67986c | 2255 | @end smallexample |
d7b8e6c6 EZ |
2256 | |
2257 | @noindent | |
2258 | It was not possible to redo past the @cite{6}, since that was placed there | |
2259 | by something other than an undo command. | |
2260 | ||
2261 | @cindex Time travel | |
2262 | You can think of undo and redo as a sort of ``time machine.'' Press | |
2263 | @kbd{U} to go backward in time, @kbd{D} to go forward. If you go | |
2264 | backward and do something (like @kbd{*}) then, as any science fiction | |
2265 | reader knows, you have changed your future and you cannot go forward | |
2266 | again. Thus, the inability to redo past the @cite{6} even though there | |
2267 | was an earlier undo command. | |
2268 | ||
2269 | You can always recall an earlier result using the Trail. We've ignored | |
2270 | the trail so far, but it has been faithfully recording everything we | |
2271 | did since we loaded the Calculator. If the Trail is not displayed, | |
2272 | press @kbd{t d} now to turn it on. | |
2273 | ||
2274 | Let's try grabbing an earlier result. The @cite{8} we computed was | |
2275 | undone by a @kbd{U} command, and was lost even to Redo when we pressed | |
2276 | @kbd{*}, but it's still there in the trail. There should be a little | |
2277 | @samp{>} arrow (the @dfn{trail pointer}) resting on the last trail | |
2278 | entry. If there isn't, press @kbd{t ]} to reset the trail pointer. | |
2279 | Now, press @w{@kbd{t p}} to move the arrow onto the line containing | |
2280 | @cite{8}, and press @w{@kbd{t y}} to ``yank'' that number back onto the | |
2281 | stack. | |
2282 | ||
2283 | If you press @kbd{t ]} again, you will see that even our Yank command | |
2284 | went into the trail. | |
2285 | ||
2286 | Let's go further back in time. Earlier in the tutorial we computed | |
2287 | a huge integer using the formula @samp{2^3^4}. We don't remember | |
2288 | what it was, but the first digits were ``241''. Press @kbd{t r} | |
2289 | (which stands for trail-search-reverse), then type @kbd{241}. | |
2290 | The trail cursor will jump back to the next previous occurrence of | |
2291 | the string ``241'' in the trail. This is just a regular Emacs | |
2292 | incremental search; you can now press @kbd{C-s} or @kbd{C-r} to | |
2293 | continue the search forwards or backwards as you like. | |
2294 | ||
2295 | To finish the search, press @key{RET}. This halts the incremental | |
2296 | search and leaves the trail pointer at the thing we found. Now we | |
2297 | can type @kbd{t y} to yank that number onto the stack. If we hadn't | |
2298 | remembered the ``241'', we could simply have searched for @kbd{2^3^4}, | |
2299 | then pressed @kbd{@key{RET} t n} to halt and then move to the next item. | |
2300 | ||
2301 | You may have noticed that all the trail-related commands begin with | |
2302 | the letter @kbd{t}. (The store-and-recall commands, on the other hand, | |
2303 | all began with @kbd{s}.) Calc has so many commands that there aren't | |
2304 | enough keys for all of them, so various commands are grouped into | |
2305 | two-letter sequences where the first letter is called the @dfn{prefix} | |
2306 | key. If you type a prefix key by accident, you can press @kbd{C-g} | |
2307 | to cancel it. (In fact, you can press @kbd{C-g} to cancel almost | |
2308 | anything in Emacs.) To get help on a prefix key, press that key | |
2309 | followed by @kbd{?}. Some prefixes have several lines of help, | |
2310 | so you need to press @kbd{?} repeatedly to see them all. This may | |
2311 | not work under Lucid Emacs, but you can also type @kbd{h h} to | |
2312 | see all the help at once. | |
2313 | ||
2314 | Try pressing @kbd{t ?} now. You will see a line of the form, | |
2315 | ||
2316 | @smallexample | |
2317 | trail/time: Display; Fwd, Back; Next, Prev, Here, [, ]; Yank: [MORE] t- | |
2318 | @end smallexample | |
2319 | ||
2320 | @noindent | |
2321 | The word ``trail'' indicates that the @kbd{t} prefix key contains | |
2322 | trail-related commands. Each entry on the line shows one command, | |
2323 | with a single capital letter showing which letter you press to get | |
2324 | that command. We have used @kbd{t n}, @kbd{t p}, @kbd{t ]}, and | |
2325 | @kbd{t y} so far. The @samp{[MORE]} means you can press @kbd{?} | |
28665d46 | 2326 | again to see more @kbd{t}-prefix commands. Notice that the commands |
d7b8e6c6 EZ |
2327 | are roughly divided (by semicolons) into related groups. |
2328 | ||
2329 | When you are in the help display for a prefix key, the prefix is | |
2330 | still active. If you press another key, like @kbd{y} for example, | |
2331 | it will be interpreted as a @kbd{t y} command. If all you wanted | |
2332 | was to look at the help messages, press @kbd{C-g} afterwards to cancel | |
2333 | the prefix. | |
2334 | ||
2335 | One more way to correct an error is by editing the stack entries. | |
2336 | The actual Stack buffer is marked read-only and must not be edited | |
2337 | directly, but you can press @kbd{`} (the backquote or accent grave) | |
2338 | to edit a stack entry. | |
2339 | ||
2340 | Try entering @samp{3.141439} now. If this is supposed to represent | |
2341 | @c{$\pi$} | |
2342 | @cite{pi}, it's got several errors. Press @kbd{`} to edit this number. | |
2343 | Now use the normal Emacs cursor motion and editing keys to change | |
2344 | the second 4 to a 5, and to transpose the 3 and the 9. When you | |
2345 | press @key{RET}, the number on the stack will be replaced by your | |
2346 | new number. This works for formulas, vectors, and all other types | |
2347 | of values you can put on the stack. The @kbd{`} key also works | |
2348 | during entry of a number or algebraic formula. | |
2349 | ||
2350 | @node Modes Tutorial, , Undo Tutorial, Basic Tutorial | |
2351 | @subsection Mode-Setting Commands | |
2352 | ||
2353 | @noindent | |
2354 | Calc has many types of @dfn{modes} that affect the way it interprets | |
2355 | your commands or the way it displays data. We have already seen one | |
2356 | mode, namely algebraic mode. There are many others, too; we'll | |
2357 | try some of the most common ones here. | |
2358 | ||
2359 | Perhaps the most fundamental mode in Calc is the current @dfn{precision}. | |
2360 | Notice the @samp{12} on the Calc window's mode line: | |
2361 | ||
2362 | @smallexample | |
2363 | --%%-Calc: 12 Deg (Calculator)----All------ | |
2364 | @end smallexample | |
2365 | ||
2366 | @noindent | |
2367 | Most of the symbols there are Emacs things you don't need to worry | |
2368 | about, but the @samp{12} and the @samp{Deg} are mode indicators. | |
2369 | The @samp{12} means that calculations should always be carried to | |
2370 | 12 significant figures. That is why, when we type @kbd{1 @key{RET} 7 /}, | |
2371 | we get @cite{0.142857142857} with exactly 12 digits, not counting | |
2372 | leading and trailing zeros. | |
2373 | ||
2374 | You can set the precision to anything you like by pressing @kbd{p}, | |
2375 | then entering a suitable number. Try pressing @kbd{p 30 @key{RET}}, | |
2376 | then doing @kbd{1 @key{RET} 7 /} again: | |
2377 | ||
d7b8e6c6 | 2378 | @smallexample |
5d67986c | 2379 | @group |
d7b8e6c6 EZ |
2380 | 1: 0.142857142857 |
2381 | 2: 0.142857142857142857142857142857 | |
2382 | . | |
d7b8e6c6 | 2383 | @end group |
5d67986c | 2384 | @end smallexample |
d7b8e6c6 EZ |
2385 | |
2386 | Although the precision can be set arbitrarily high, Calc always | |
2387 | has to have @emph{some} value for the current precision. After | |
2388 | all, the true value @cite{1/7} is an infinitely repeating decimal; | |
2389 | Calc has to stop somewhere. | |
2390 | ||
2391 | Of course, calculations are slower the more digits you request. | |
2392 | Press @w{@kbd{p 12}} now to set the precision back down to the default. | |
2393 | ||
2394 | Calculations always use the current precision. For example, even | |
2395 | though we have a 30-digit value for @cite{1/7} on the stack, if | |
2396 | we use it in a calculation in 12-digit mode it will be rounded | |
2397 | down to 12 digits before it is used. Try it; press @key{RET} to | |
2398 | duplicate the number, then @w{@kbd{1 +}}. Notice that the @key{RET} | |
2399 | key didn't round the number, because it doesn't do any calculation. | |
2400 | But the instant we pressed @kbd{+}, the number was rounded down. | |
2401 | ||
d7b8e6c6 | 2402 | @smallexample |
5d67986c | 2403 | @group |
d7b8e6c6 EZ |
2404 | 1: 0.142857142857 |
2405 | 2: 0.142857142857142857142857142857 | |
2406 | 3: 1.14285714286 | |
2407 | . | |
d7b8e6c6 | 2408 | @end group |
5d67986c | 2409 | @end smallexample |
d7b8e6c6 EZ |
2410 | |
2411 | @noindent | |
2412 | In fact, since we added a digit on the left, we had to lose one | |
2413 | digit on the right from even the 12-digit value of @cite{1/7}. | |
2414 | ||
2415 | How did we get more than 12 digits when we computed @samp{2^3^4}? The | |
2416 | answer is that Calc makes a distinction between @dfn{integers} and | |
2417 | @dfn{floating-point} numbers, or @dfn{floats}. An integer is a number | |
2418 | that does not contain a decimal point. There is no such thing as an | |
2419 | ``infinitely repeating fraction integer,'' so Calc doesn't have to limit | |
2420 | itself. If you asked for @samp{2^10000} (don't try this!), you would | |
2421 | have to wait a long time but you would eventually get an exact answer. | |
2422 | If you ask for @samp{2.^10000}, you will quickly get an answer which is | |
2423 | correct only to 12 places. The decimal point tells Calc that it should | |
2424 | use floating-point arithmetic to get the answer, not exact integer | |
2425 | arithmetic. | |
2426 | ||
2427 | You can use the @kbd{F} (@code{calc-floor}) command to convert a | |
2428 | floating-point value to an integer, and @kbd{c f} (@code{calc-float}) | |
2429 | to convert an integer to floating-point form. | |
2430 | ||
2431 | Let's try entering that last calculation: | |
2432 | ||
d7b8e6c6 | 2433 | @smallexample |
5d67986c | 2434 | @group |
d7b8e6c6 EZ |
2435 | 1: 2. 2: 2. 1: 1.99506311689e3010 |
2436 | . 1: 10000 . | |
2437 | . | |
2438 | ||
5d67986c | 2439 | 2.0 @key{RET} 10000 @key{RET} ^ |
d7b8e6c6 | 2440 | @end group |
5d67986c | 2441 | @end smallexample |
d7b8e6c6 EZ |
2442 | |
2443 | @noindent | |
2444 | @cindex Scientific notation, entry of | |
2445 | Notice the letter @samp{e} in there. It represents ``times ten to the | |
2446 | power of,'' and is used by Calc automatically whenever writing the | |
2447 | number out fully would introduce more extra zeros than you probably | |
2448 | want to see. You can enter numbers in this notation, too. | |
2449 | ||
d7b8e6c6 | 2450 | @smallexample |
5d67986c | 2451 | @group |
d7b8e6c6 EZ |
2452 | 1: 2. 2: 2. 1: 1.99506311678e3010 |
2453 | . 1: 10000. . | |
2454 | . | |
2455 | ||
5d67986c | 2456 | 2.0 @key{RET} 1e4 @key{RET} ^ |
d7b8e6c6 | 2457 | @end group |
5d67986c | 2458 | @end smallexample |
d7b8e6c6 EZ |
2459 | |
2460 | @cindex Round-off errors | |
2461 | @noindent | |
2462 | Hey, the answer is different! Look closely at the middle columns | |
2463 | of the two examples. In the first, the stack contained the | |
2464 | exact integer @cite{10000}, but in the second it contained | |
2465 | a floating-point value with a decimal point. When you raise a | |
2466 | number to an integer power, Calc uses repeated squaring and | |
2467 | multiplication to get the answer. When you use a floating-point | |
2468 | power, Calc uses logarithms and exponentials. As you can see, | |
2469 | a slight error crept in during one of these methods. Which | |
2470 | one should we trust? Let's raise the precision a bit and find | |
2471 | out: | |
2472 | ||
d7b8e6c6 | 2473 | @smallexample |
5d67986c | 2474 | @group |
d7b8e6c6 EZ |
2475 | . 1: 2. 2: 2. 1: 1.995063116880828e3010 |
2476 | . 1: 10000. . | |
2477 | . | |
2478 | ||
5d67986c | 2479 | p 16 @key{RET} 2. @key{RET} 1e4 ^ p 12 @key{RET} |
d7b8e6c6 | 2480 | @end group |
5d67986c | 2481 | @end smallexample |
d7b8e6c6 EZ |
2482 | |
2483 | @noindent | |
2484 | @cindex Guard digits | |
2485 | Presumably, it doesn't matter whether we do this higher-precision | |
2486 | calculation using an integer or floating-point power, since we | |
2487 | have added enough ``guard digits'' to trust the first 12 digits | |
2488 | no matter what. And the verdict is@dots{} Integer powers were more | |
2489 | accurate; in fact, the result was only off by one unit in the | |
2490 | last place. | |
2491 | ||
2492 | @cindex Guard digits | |
2493 | Calc does many of its internal calculations to a slightly higher | |
2494 | precision, but it doesn't always bump the precision up enough. | |
2495 | In each case, Calc added about two digits of precision during | |
2496 | its calculation and then rounded back down to 12 digits | |
269b7745 | 2497 | afterward. In one case, it was enough; in the other, it |
d7b8e6c6 EZ |
2498 | wasn't. If you really need @var{x} digits of precision, it |
2499 | never hurts to do the calculation with a few extra guard digits. | |
2500 | ||
2501 | What if we want guard digits but don't want to look at them? | |
2502 | We can set the @dfn{float format}. Calc supports four major | |
2503 | formats for floating-point numbers, called @dfn{normal}, | |
2504 | @dfn{fixed-point}, @dfn{scientific notation}, and @dfn{engineering | |
2505 | notation}. You get them by pressing @w{@kbd{d n}}, @kbd{d f}, | |
2506 | @kbd{d s}, and @kbd{d e}, respectively. In each case, you can | |
2507 | supply a numeric prefix argument which says how many digits | |
2508 | should be displayed. As an example, let's put a few numbers | |
2509 | onto the stack and try some different display modes. First, | |
5d67986c | 2510 | use @kbd{M-0 @key{DEL}} to clear the stack, then enter the four |
d7b8e6c6 EZ |
2511 | numbers shown here: |
2512 | ||
d7b8e6c6 | 2513 | @smallexample |
5d67986c | 2514 | @group |
d7b8e6c6 EZ |
2515 | 4: 12345 4: 12345 4: 12345 4: 12345 4: 12345 |
2516 | 3: 12345. 3: 12300. 3: 1.2345e4 3: 1.23e4 3: 12345.000 | |
2517 | 2: 123.45 2: 123. 2: 1.2345e2 2: 1.23e2 2: 123.450 | |
2518 | 1: 12.345 1: 12.3 1: 1.2345e1 1: 1.23e1 1: 12.345 | |
2519 | . . . . . | |
2520 | ||
2521 | d n M-3 d n d s M-3 d s M-3 d f | |
d7b8e6c6 | 2522 | @end group |
5d67986c | 2523 | @end smallexample |
d7b8e6c6 EZ |
2524 | |
2525 | @noindent | |
2526 | Notice that when we typed @kbd{M-3 d n}, the numbers were rounded down | |
2527 | to three significant digits, but then when we typed @kbd{d s} all | |
2528 | five significant figures reappeared. The float format does not | |
2529 | affect how numbers are stored, it only affects how they are | |
2530 | displayed. Only the current precision governs the actual rounding | |
2531 | of numbers in the Calculator's memory. | |
2532 | ||
2533 | Engineering notation, not shown here, is like scientific notation | |
2534 | except the exponent (the power-of-ten part) is always adjusted to be | |
2535 | a multiple of three (as in ``kilo,'' ``micro,'' etc.). As a result | |
2536 | there will be one, two, or three digits before the decimal point. | |
2537 | ||
2538 | Whenever you change a display-related mode, Calc redraws everything | |
2539 | in the stack. This may be slow if there are many things on the stack, | |
2540 | so Calc allows you to type shift-@kbd{H} before any mode command to | |
2541 | prevent it from updating the stack. Anything Calc displays after the | |
2542 | mode-changing command will appear in the new format. | |
2543 | ||
d7b8e6c6 | 2544 | @smallexample |
5d67986c | 2545 | @group |
d7b8e6c6 EZ |
2546 | 4: 12345 4: 12345 4: 12345 4: 12345 4: 12345 |
2547 | 3: 12345.000 3: 12345.000 3: 12345.000 3: 1.2345e4 3: 12345. | |
2548 | 2: 123.450 2: 123.450 2: 1.2345e1 2: 1.2345e1 2: 123.45 | |
2549 | 1: 12.345 1: 1.2345e1 1: 1.2345e2 1: 1.2345e2 1: 12.345 | |
2550 | . . . . . | |
2551 | ||
5d67986c | 2552 | H d s @key{DEL} U @key{TAB} d @key{SPC} d n |
d7b8e6c6 | 2553 | @end group |
5d67986c | 2554 | @end smallexample |
d7b8e6c6 EZ |
2555 | |
2556 | @noindent | |
2557 | Here the @kbd{H d s} command changes to scientific notation but without | |
2558 | updating the screen. Deleting the top stack entry and undoing it back | |
2559 | causes it to show up in the new format; swapping the top two stack | |
5d67986c | 2560 | entries reformats both entries. The @kbd{d @key{SPC}} command refreshes the |
d7b8e6c6 EZ |
2561 | whole stack. The @kbd{d n} command changes back to the normal float |
2562 | format; since it doesn't have an @kbd{H} prefix, it also updates all | |
2563 | the stack entries to be in @kbd{d n} format. | |
2564 | ||
2565 | Notice that the integer @cite{12345} was not affected by any | |
2566 | of the float formats. Integers are integers, and are always | |
2567 | displayed exactly. | |
2568 | ||
2569 | @cindex Large numbers, readability | |
2570 | Large integers have their own problems. Let's look back at | |
2571 | the result of @kbd{2^3^4}. | |
2572 | ||
2573 | @example | |
2574 | 2417851639229258349412352 | |
2575 | @end example | |
2576 | ||
2577 | @noindent | |
2578 | Quick---how many digits does this have? Try typing @kbd{d g}: | |
2579 | ||
2580 | @example | |
2581 | 2,417,851,639,229,258,349,412,352 | |
2582 | @end example | |
2583 | ||
2584 | @noindent | |
2585 | Now how many digits does this have? It's much easier to tell! | |
2586 | We can actually group digits into clumps of any size. Some | |
2587 | people prefer @kbd{M-5 d g}: | |
2588 | ||
2589 | @example | |
2590 | 24178,51639,22925,83494,12352 | |
2591 | @end example | |
2592 | ||
2593 | Let's see what happens to floating-point numbers when they are grouped. | |
2594 | First, type @kbd{p 25 @key{RET}} to make sure we have enough precision | |
2595 | to get ourselves into trouble. Now, type @kbd{1e13 /}: | |
2596 | ||
2597 | @example | |
2598 | 24,17851,63922.9258349412352 | |
2599 | @end example | |
2600 | ||
2601 | @noindent | |
2602 | The integer part is grouped but the fractional part isn't. Now try | |
2603 | @kbd{M-- M-5 d g} (that's meta-minus-sign, meta-five): | |
2604 | ||
2605 | @example | |
2606 | 24,17851,63922.92583,49412,352 | |
2607 | @end example | |
2608 | ||
2609 | If you find it hard to tell the decimal point from the commas, try | |
2610 | changing the grouping character to a space with @kbd{d , @key{SPC}}: | |
2611 | ||
2612 | @example | |
2613 | 24 17851 63922.92583 49412 352 | |
2614 | @end example | |
2615 | ||
2616 | Type @kbd{d , ,} to restore the normal grouping character, then | |
2617 | @kbd{d g} again to turn grouping off. Also, press @kbd{p 12} to | |
2618 | restore the default precision. | |
2619 | ||
2620 | Press @kbd{U} enough times to get the original big integer back. | |
2621 | (Notice that @kbd{U} does not undo each mode-setting command; if | |
2622 | you want to undo a mode-setting command, you have to do it yourself.) | |
2623 | Now, type @kbd{d r 16 @key{RET}}: | |
2624 | ||
2625 | @example | |
2626 | 16#200000000000000000000 | |
2627 | @end example | |
2628 | ||
2629 | @noindent | |
2630 | The number is now displayed in @dfn{hexadecimal}, or ``base-16'' form. | |
2631 | Suddenly it looks pretty simple; this should be no surprise, since we | |
2632 | got this number by computing a power of two, and 16 is a power of 2. | |
2633 | In fact, we can use @w{@kbd{d r 2 @key{RET}}} to see it in actual binary | |
2634 | form: | |
2635 | ||
2636 | @example | |
2637 | 2#1000000000000000000000000000000000000000000000000000000 @dots{} | |
2638 | @end example | |
2639 | ||
2640 | @noindent | |
2641 | We don't have enough space here to show all the zeros! They won't | |
2642 | fit on a typical screen, either, so you will have to use horizontal | |
2643 | scrolling to see them all. Press @kbd{<} and @kbd{>} to scroll the | |
2644 | stack window left and right by half its width. Another way to view | |
2645 | something large is to press @kbd{`} (back-quote) to edit the top of | |
2646 | stack in a separate window. (Press @kbd{M-# M-#} when you are done.) | |
2647 | ||
2648 | You can enter non-decimal numbers using the @kbd{#} symbol, too. | |
2649 | Let's see what the hexadecimal number @samp{5FE} looks like in | |
2650 | binary. Type @kbd{16#5FE} (the letters can be typed in upper or | |
2651 | lower case; they will always appear in upper case). It will also | |
2652 | help to turn grouping on with @kbd{d g}: | |
2653 | ||
2654 | @example | |
2655 | 2#101,1111,1110 | |
2656 | @end example | |
2657 | ||
2658 | Notice that @kbd{d g} groups by fours by default if the display radix | |
2659 | is binary or hexadecimal, but by threes if it is decimal, octal, or any | |
2660 | other radix. | |
2661 | ||
2662 | Now let's see that number in decimal; type @kbd{d r 10}: | |
2663 | ||
2664 | @example | |
2665 | 1,534 | |
2666 | @end example | |
2667 | ||
2668 | Numbers are not @emph{stored} with any particular radix attached. They're | |
2669 | just numbers; they can be entered in any radix, and are always displayed | |
2670 | in whatever radix you've chosen with @kbd{d r}. The current radix applies | |
2671 | to integers, fractions, and floats. | |
2672 | ||
2673 | @cindex Roundoff errors, in non-decimal numbers | |
2674 | (@bullet{}) @strong{Exercise 1.} Your friend Joe tried to enter one-third | |
2675 | as @samp{3#0.1} in @kbd{d r 3} mode with a precision of 12. He got | |
2676 | @samp{3#0.0222222...} (with 25 2's) in the display. When he multiplied | |
2677 | that by three, he got @samp{3#0.222222...} instead of the expected | |
2678 | @samp{3#1}. Next, Joe entered @samp{3#0.2} and, to his great relief, | |
2679 | saw @samp{3#0.2} on the screen. But when he typed @kbd{2 /}, he got | |
2680 | @samp{3#0.10000001} (some zeros omitted). What's going on here? | |
2681 | @xref{Modes Answer 1, 1}. (@bullet{}) | |
2682 | ||
2683 | @cindex Scientific notation, in non-decimal numbers | |
2684 | (@bullet{}) @strong{Exercise 2.} Scientific notation works in non-decimal | |
2685 | modes in the natural way (the exponent is a power of the radix instead of | |
2686 | a power of ten, although the exponent itself is always written in decimal). | |
2687 | Thus @samp{8#1.23e3 = 8#1230.0}. Suppose we have the hexadecimal number | |
2688 | @samp{f.e8f} times 16 to the 15th power: We write @samp{16#f.e8fe15}. | |
2689 | What is wrong with this picture? What could we write instead that would | |
2690 | work better? @xref{Modes Answer 2, 2}. (@bullet{}) | |
2691 | ||
2692 | The @kbd{m} prefix key has another set of modes, relating to the way | |
2693 | Calc interprets your inputs and does computations. Whereas @kbd{d}-prefix | |
2694 | modes generally affect the way things look, @kbd{m}-prefix modes affect | |
2695 | the way they are actually computed. | |
2696 | ||
2697 | The most popular @kbd{m}-prefix mode is the @dfn{angular mode}. Notice | |
2698 | the @samp{Deg} indicator in the mode line. This means that if you use | |
2699 | a command that interprets a number as an angle, it will assume the | |
2700 | angle is measured in degrees. For example, | |
2701 | ||
d7b8e6c6 | 2702 | @smallexample |
5d67986c | 2703 | @group |
d7b8e6c6 EZ |
2704 | 1: 45 1: 0.707106781187 1: 0.500000000001 1: 0.5 |
2705 | . . . . | |
2706 | ||
2707 | 45 S 2 ^ c 1 | |
d7b8e6c6 | 2708 | @end group |
5d67986c | 2709 | @end smallexample |
d7b8e6c6 EZ |
2710 | |
2711 | @noindent | |
2712 | The shift-@kbd{S} command computes the sine of an angle. The sine | |
2713 | of 45 degrees is @c{$\sqrt{2}/2$} | |
2714 | @cite{sqrt(2)/2}; squaring this yields @cite{2/4 = 0.5}. | |
2715 | However, there has been a slight roundoff error because the | |
2716 | representation of @c{$\sqrt{2}/2$} | |
2717 | @cite{sqrt(2)/2} wasn't exact. The @kbd{c 1} | |
2718 | command is a handy way to clean up numbers in this case; it | |
2719 | temporarily reduces the precision by one digit while it | |
2720 | re-rounds the number on the top of the stack. | |
2721 | ||
2722 | @cindex Roundoff errors, examples | |
2723 | (@bullet{}) @strong{Exercise 3.} Your friend Joe computed the sine | |
2724 | of 45 degrees as shown above, then, hoping to avoid an inexact | |
2725 | result, he increased the precision to 16 digits before squaring. | |
2726 | What happened? @xref{Modes Answer 3, 3}. (@bullet{}) | |
2727 | ||
2728 | To do this calculation in radians, we would type @kbd{m r} first. | |
2729 | (The indicator changes to @samp{Rad}.) 45 degrees corresponds to | |
2730 | @c{$\pi\over4$} | |
2731 | @cite{pi/4} radians. To get @c{$\pi$} | |
2732 | @cite{pi}, press the @kbd{P} key. (Once | |
2733 | again, this is a shifted capital @kbd{P}. Remember, unshifted | |
2734 | @kbd{p} sets the precision.) | |
2735 | ||
d7b8e6c6 | 2736 | @smallexample |
5d67986c | 2737 | @group |
d7b8e6c6 EZ |
2738 | 1: 3.14159265359 1: 0.785398163398 1: 0.707106781187 |
2739 | . . . | |
2740 | ||
2741 | P 4 / m r S | |
d7b8e6c6 | 2742 | @end group |
5d67986c | 2743 | @end smallexample |
d7b8e6c6 EZ |
2744 | |
2745 | Likewise, inverse trigonometric functions generate results in | |
2746 | either radians or degrees, depending on the current angular mode. | |
2747 | ||
d7b8e6c6 | 2748 | @smallexample |
5d67986c | 2749 | @group |
d7b8e6c6 EZ |
2750 | 1: 0.707106781187 1: 0.785398163398 1: 45. |
2751 | . . . | |
2752 | ||
2753 | .5 Q m r I S m d U I S | |
d7b8e6c6 | 2754 | @end group |
5d67986c | 2755 | @end smallexample |
d7b8e6c6 EZ |
2756 | |
2757 | @noindent | |
2758 | Here we compute the Inverse Sine of @c{$\sqrt{0.5}$} | |
2759 | @cite{sqrt(0.5)}, first in | |
2760 | radians, then in degrees. | |
2761 | ||
2762 | Use @kbd{c d} and @kbd{c r} to convert a number from radians to degrees | |
2763 | and vice-versa. | |
2764 | ||
d7b8e6c6 | 2765 | @smallexample |
5d67986c | 2766 | @group |
d7b8e6c6 EZ |
2767 | 1: 45 1: 0.785398163397 1: 45. |
2768 | . . . | |
2769 | ||
2770 | 45 c r c d | |
d7b8e6c6 | 2771 | @end group |
5d67986c | 2772 | @end smallexample |
d7b8e6c6 EZ |
2773 | |
2774 | Another interesting mode is @dfn{fraction mode}. Normally, | |
2775 | dividing two integers produces a floating-point result if the | |
2776 | quotient can't be expressed as an exact integer. Fraction mode | |
2777 | causes integer division to produce a fraction, i.e., a rational | |
2778 | number, instead. | |
2779 | ||
d7b8e6c6 | 2780 | @smallexample |
5d67986c | 2781 | @group |
d7b8e6c6 EZ |
2782 | 2: 12 1: 1.33333333333 1: 4:3 |
2783 | 1: 9 . . | |
2784 | . | |
2785 | ||
5d67986c | 2786 | 12 @key{RET} 9 / m f U / m f |
d7b8e6c6 | 2787 | @end group |
5d67986c | 2788 | @end smallexample |
d7b8e6c6 EZ |
2789 | |
2790 | @noindent | |
2791 | In the first case, we get an approximate floating-point result. | |
2792 | In the second case, we get an exact fractional result (four-thirds). | |
2793 | ||
2794 | You can enter a fraction at any time using @kbd{:} notation. | |
2795 | (Calc uses @kbd{:} instead of @kbd{/} as the fraction separator | |
2796 | because @kbd{/} is already used to divide the top two stack | |
2797 | elements.) Calculations involving fractions will always | |
2798 | produce exact fractional results; fraction mode only says | |
2799 | what to do when dividing two integers. | |
2800 | ||
2801 | @cindex Fractions vs. floats | |
2802 | @cindex Floats vs. fractions | |
2803 | (@bullet{}) @strong{Exercise 4.} If fractional arithmetic is exact, | |
2804 | why would you ever use floating-point numbers instead? | |
2805 | @xref{Modes Answer 4, 4}. (@bullet{}) | |
2806 | ||
2807 | Typing @kbd{m f} doesn't change any existing values in the stack. | |
2808 | In the above example, we had to Undo the division and do it over | |
2809 | again when we changed to fraction mode. But if you use the | |
2810 | evaluates-to operator you can get commands like @kbd{m f} to | |
2811 | recompute for you. | |
2812 | ||
d7b8e6c6 | 2813 | @smallexample |
5d67986c | 2814 | @group |
d7b8e6c6 EZ |
2815 | 1: 12 / 9 => 1.33333333333 1: 12 / 9 => 1.333 1: 12 / 9 => 4:3 |
2816 | . . . | |
2817 | ||
5d67986c | 2818 | ' 12/9 => @key{RET} p 4 @key{RET} m f |
d7b8e6c6 | 2819 | @end group |
5d67986c | 2820 | @end smallexample |
d7b8e6c6 EZ |
2821 | |
2822 | @noindent | |
2823 | In this example, the righthand side of the @samp{=>} operator | |
2824 | on the stack is recomputed when we change the precision, then | |
2825 | again when we change to fraction mode. All @samp{=>} expressions | |
2826 | on the stack are recomputed every time you change any mode that | |
2827 | might affect their values. | |
2828 | ||
2829 | @node Arithmetic Tutorial, Vector/Matrix Tutorial, Basic Tutorial, Tutorial | |
2830 | @section Arithmetic Tutorial | |
2831 | ||
2832 | @noindent | |
2833 | In this section, we explore the arithmetic and scientific functions | |
2834 | available in the Calculator. | |
2835 | ||
2836 | The standard arithmetic commands are @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/}, | |
2837 | and @kbd{^}. Each normally takes two numbers from the top of the stack | |
2838 | and pushes back a result. The @kbd{n} and @kbd{&} keys perform | |
2839 | change-sign and reciprocal operations, respectively. | |
2840 | ||
d7b8e6c6 | 2841 | @smallexample |
5d67986c | 2842 | @group |
d7b8e6c6 EZ |
2843 | 1: 5 1: 0.2 1: 5. 1: -5. 1: 5. |
2844 | . . . . . | |
2845 | ||
2846 | 5 & & n n | |
d7b8e6c6 | 2847 | @end group |
5d67986c | 2848 | @end smallexample |
d7b8e6c6 EZ |
2849 | |
2850 | @cindex Binary operators | |
2851 | You can apply a ``binary operator'' like @kbd{+} across any number of | |
2852 | stack entries by giving it a numeric prefix. You can also apply it | |
2853 | pairwise to several stack elements along with the top one if you use | |
2854 | a negative prefix. | |
2855 | ||
d7b8e6c6 | 2856 | @smallexample |
5d67986c | 2857 | @group |
d7b8e6c6 EZ |
2858 | 3: 2 1: 9 3: 2 4: 2 3: 12 |
2859 | 2: 3 . 2: 3 3: 3 2: 13 | |
2860 | 1: 4 1: 4 2: 4 1: 14 | |
2861 | . . 1: 10 . | |
2862 | . | |
2863 | ||
5d67986c | 2864 | 2 @key{RET} 3 @key{RET} 4 M-3 + U 10 M-- M-3 + |
d7b8e6c6 | 2865 | @end group |
5d67986c | 2866 | @end smallexample |
d7b8e6c6 EZ |
2867 | |
2868 | @cindex Unary operators | |
2869 | You can apply a ``unary operator'' like @kbd{&} to the top @var{n} | |
2870 | stack entries with a numeric prefix, too. | |
2871 | ||
d7b8e6c6 | 2872 | @smallexample |
5d67986c | 2873 | @group |
d7b8e6c6 EZ |
2874 | 3: 2 3: 0.5 3: 0.5 |
2875 | 2: 3 2: 0.333333333333 2: 3. | |
2876 | 1: 4 1: 0.25 1: 4. | |
2877 | . . . | |
2878 | ||
5d67986c | 2879 | 2 @key{RET} 3 @key{RET} 4 M-3 & M-2 & |
d7b8e6c6 | 2880 | @end group |
5d67986c | 2881 | @end smallexample |
d7b8e6c6 EZ |
2882 | |
2883 | Notice that the results here are left in floating-point form. | |
2884 | We can convert them back to integers by pressing @kbd{F}, the | |
2885 | ``floor'' function. This function rounds down to the next lower | |
2886 | integer. There is also @kbd{R}, which rounds to the nearest | |
2887 | integer. | |
2888 | ||
d7b8e6c6 | 2889 | @smallexample |
5d67986c | 2890 | @group |
d7b8e6c6 EZ |
2891 | 7: 2. 7: 2 7: 2 |
2892 | 6: 2.4 6: 2 6: 2 | |
2893 | 5: 2.5 5: 2 5: 3 | |
2894 | 4: 2.6 4: 2 4: 3 | |
2895 | 3: -2. 3: -2 3: -2 | |
2896 | 2: -2.4 2: -3 2: -2 | |
2897 | 1: -2.6 1: -3 1: -3 | |
2898 | . . . | |
2899 | ||
2900 | M-7 F U M-7 R | |
d7b8e6c6 | 2901 | @end group |
5d67986c | 2902 | @end smallexample |
d7b8e6c6 EZ |
2903 | |
2904 | Since dividing-and-flooring (i.e., ``integer quotient'') is such a | |
2905 | common operation, Calc provides a special command for that purpose, the | |
2906 | backslash @kbd{\}. Another common arithmetic operator is @kbd{%}, which | |
2907 | computes the remainder that would arise from a @kbd{\} operation, i.e., | |
2908 | the ``modulo'' of two numbers. For example, | |
2909 | ||
d7b8e6c6 | 2910 | @smallexample |
5d67986c | 2911 | @group |
d7b8e6c6 EZ |
2912 | 2: 1234 1: 12 2: 1234 1: 34 |
2913 | 1: 100 . 1: 100 . | |
2914 | . . | |
2915 | ||
5d67986c | 2916 | 1234 @key{RET} 100 \ U % |
d7b8e6c6 | 2917 | @end group |
5d67986c | 2918 | @end smallexample |
d7b8e6c6 EZ |
2919 | |
2920 | These commands actually work for any real numbers, not just integers. | |
2921 | ||
d7b8e6c6 | 2922 | @smallexample |
5d67986c | 2923 | @group |
d7b8e6c6 EZ |
2924 | 2: 3.1415 1: 3 2: 3.1415 1: 0.1415 |
2925 | 1: 1 . 1: 1 . | |
2926 | . . | |
2927 | ||
5d67986c | 2928 | 3.1415 @key{RET} 1 \ U % |
d7b8e6c6 | 2929 | @end group |
5d67986c | 2930 | @end smallexample |
d7b8e6c6 EZ |
2931 | |
2932 | (@bullet{}) @strong{Exercise 1.} The @kbd{\} command would appear to be a | |
2933 | frill, since you could always do the same thing with @kbd{/ F}. Think | |
2934 | of a situation where this is not true---@kbd{/ F} would be inadequate. | |
2935 | Now think of a way you could get around the problem if Calc didn't | |
2936 | provide a @kbd{\} command. @xref{Arithmetic Answer 1, 1}. (@bullet{}) | |
2937 | ||
2938 | We've already seen the @kbd{Q} (square root) and @kbd{S} (sine) | |
2939 | commands. Other commands along those lines are @kbd{C} (cosine), | |
2940 | @kbd{T} (tangent), @kbd{E} (@cite{e^x}) and @kbd{L} (natural | |
2941 | logarithm). These can be modified by the @kbd{I} (inverse) and | |
2942 | @kbd{H} (hyperbolic) prefix keys. | |
2943 | ||
2944 | Let's compute the sine and cosine of an angle, and verify the | |
2945 | identity @c{$\sin^2x + \cos^2x = 1$} | |
2946 | @cite{sin(x)^2 + cos(x)^2 = 1}. We'll | |
2947 | arbitrarily pick @i{-64} degrees as a good value for @cite{x}. With | |
2948 | the angular mode set to degrees (type @w{@kbd{m d}}), do: | |
2949 | ||
d7b8e6c6 | 2950 | @smallexample |
5d67986c | 2951 | @group |
d7b8e6c6 EZ |
2952 | 2: -64 2: -64 2: -0.89879 2: -0.89879 1: 1. |
2953 | 1: -64 1: -0.89879 1: -64 1: 0.43837 . | |
2954 | . . . . | |
2955 | ||
5d67986c | 2956 | 64 n @key{RET} @key{RET} S @key{TAB} C f h |
d7b8e6c6 | 2957 | @end group |
5d67986c | 2958 | @end smallexample |
d7b8e6c6 EZ |
2959 | |
2960 | @noindent | |
2961 | (For brevity, we're showing only five digits of the results here. | |
2962 | You can of course do these calculations to any precision you like.) | |
2963 | ||
2964 | Remember, @kbd{f h} is the @code{calc-hypot}, or square-root of sum | |
2965 | of squares, command. | |
2966 | ||
2967 | Another identity is @c{$\displaystyle\tan x = {\sin x \over \cos x}$} | |
2968 | @cite{tan(x) = sin(x) / cos(x)}. | |
d7b8e6c6 | 2969 | @smallexample |
5d67986c | 2970 | @group |
d7b8e6c6 EZ |
2971 | |
2972 | 2: -0.89879 1: -2.0503 1: -64. | |
2973 | 1: 0.43837 . . | |
2974 | . | |
2975 | ||
2976 | U / I T | |
d7b8e6c6 | 2977 | @end group |
5d67986c | 2978 | @end smallexample |
d7b8e6c6 EZ |
2979 | |
2980 | A physical interpretation of this calculation is that if you move | |
2981 | @cite{0.89879} units downward and @cite{0.43837} units to the right, | |
2982 | your direction of motion is @i{-64} degrees from horizontal. Suppose | |
2983 | we move in the opposite direction, up and to the left: | |
2984 | ||
d7b8e6c6 | 2985 | @smallexample |
5d67986c | 2986 | @group |
d7b8e6c6 EZ |
2987 | 2: -0.89879 2: 0.89879 1: -2.0503 1: -64. |
2988 | 1: 0.43837 1: -0.43837 . . | |
2989 | . . | |
2990 | ||
2991 | U U M-2 n / I T | |
d7b8e6c6 | 2992 | @end group |
5d67986c | 2993 | @end smallexample |
d7b8e6c6 EZ |
2994 | |
2995 | @noindent | |
2996 | How can the angle be the same? The answer is that the @kbd{/} operation | |
2997 | loses information about the signs of its inputs. Because the quotient | |
2998 | is negative, we know exactly one of the inputs was negative, but we | |
2999 | can't tell which one. There is an @kbd{f T} [@code{arctan2}] function which | |
3000 | computes the inverse tangent of the quotient of a pair of numbers. | |
3001 | Since you feed it the two original numbers, it has enough information | |
3002 | to give you a full 360-degree answer. | |
3003 | ||
d7b8e6c6 | 3004 | @smallexample |
5d67986c | 3005 | @group |
d7b8e6c6 EZ |
3006 | 2: 0.89879 1: 116. 3: 116. 2: 116. 1: 180. |
3007 | 1: -0.43837 . 2: -0.89879 1: -64. . | |
3008 | . 1: 0.43837 . | |
3009 | . | |
3010 | ||
5d67986c | 3011 | U U f T M-@key{RET} M-2 n f T - |
d7b8e6c6 | 3012 | @end group |
5d67986c | 3013 | @end smallexample |
d7b8e6c6 EZ |
3014 | |
3015 | @noindent | |
3016 | The resulting angles differ by 180 degrees; in other words, they | |
3017 | point in opposite directions, just as we would expect. | |
3018 | ||
3019 | The @key{META}-@key{RET} we used in the third step is the | |
3020 | ``last-arguments'' command. It is sort of like Undo, except that it | |
3021 | restores the arguments of the last command to the stack without removing | |
3022 | the command's result. It is useful in situations like this one, | |
3023 | where we need to do several operations on the same inputs. We could | |
3024 | have accomplished the same thing by using @kbd{M-2 @key{RET}} to duplicate | |
3025 | the top two stack elements right after the @kbd{U U}, then a pair of | |
3026 | @kbd{M-@key{TAB}} commands to cycle the 116 up around the duplicates. | |
3027 | ||
3028 | A similar identity is supposed to hold for hyperbolic sines and cosines, | |
3029 | except that it is the @emph{difference} | |
3030 | @c{$\cosh^2x - \sinh^2x$} | |
3031 | @cite{cosh(x)^2 - sinh(x)^2} that always equals one. | |
3032 | Let's try to verify this identity.@refill | |
3033 | ||
d7b8e6c6 | 3034 | @smallexample |
5d67986c | 3035 | @group |
d7b8e6c6 EZ |
3036 | 2: -64 2: -64 2: -64 2: 9.7192e54 2: 9.7192e54 |
3037 | 1: -64 1: -3.1175e27 1: 9.7192e54 1: -64 1: 9.7192e54 | |
3038 | . . . . . | |
3039 | ||
5d67986c | 3040 | 64 n @key{RET} @key{RET} H C 2 ^ @key{TAB} H S 2 ^ |
d7b8e6c6 | 3041 | @end group |
5d67986c | 3042 | @end smallexample |
d7b8e6c6 EZ |
3043 | |
3044 | @noindent | |
3045 | @cindex Roundoff errors, examples | |
3046 | Something's obviously wrong, because when we subtract these numbers | |
3047 | the answer will clearly be zero! But if you think about it, if these | |
3048 | numbers @emph{did} differ by one, it would be in the 55th decimal | |
3049 | place. The difference we seek has been lost entirely to roundoff | |
3050 | error. | |
3051 | ||
3052 | We could verify this hypothesis by doing the actual calculation with, | |
3053 | say, 60 decimal places of precision. This will be slow, but not | |
3054 | enormously so. Try it if you wish; sure enough, the answer is | |
3055 | 0.99999, reasonably close to 1. | |
3056 | ||
3057 | Of course, a more reasonable way to verify the identity is to use | |
3058 | a more reasonable value for @cite{x}! | |
3059 | ||
3060 | @cindex Common logarithm | |
3061 | Some Calculator commands use the Hyperbolic prefix for other purposes. | |
3062 | The logarithm and exponential functions, for example, work to the base | |
3063 | @cite{e} normally but use base-10 instead if you use the Hyperbolic | |
3064 | prefix. | |
3065 | ||
d7b8e6c6 | 3066 | @smallexample |
5d67986c | 3067 | @group |
d7b8e6c6 EZ |
3068 | 1: 1000 1: 6.9077 1: 1000 1: 3 |
3069 | . . . . | |
3070 | ||
3071 | 1000 L U H L | |
d7b8e6c6 | 3072 | @end group |
5d67986c | 3073 | @end smallexample |
d7b8e6c6 EZ |
3074 | |
3075 | @noindent | |
3076 | First, we mistakenly compute a natural logarithm. Then we undo | |
3077 | and compute a common logarithm instead. | |
3078 | ||
3079 | The @kbd{B} key computes a general base-@var{b} logarithm for any | |
3080 | value of @var{b}. | |
3081 | ||
d7b8e6c6 | 3082 | @smallexample |
5d67986c | 3083 | @group |
d7b8e6c6 EZ |
3084 | 2: 1000 1: 3 1: 1000. 2: 1000. 1: 6.9077 |
3085 | 1: 10 . . 1: 2.71828 . | |
3086 | . . | |
3087 | ||
5d67986c | 3088 | 1000 @key{RET} 10 B H E H P B |
d7b8e6c6 | 3089 | @end group |
5d67986c | 3090 | @end smallexample |
d7b8e6c6 EZ |
3091 | |
3092 | @noindent | |
3093 | Here we first use @kbd{B} to compute the base-10 logarithm, then use | |
3094 | the ``hyperbolic'' exponential as a cheap hack to recover the number | |
3095 | 1000, then use @kbd{B} again to compute the natural logarithm. Note | |
3096 | that @kbd{P} with the hyperbolic prefix pushes the constant @cite{e} | |
3097 | onto the stack. | |
3098 | ||
3099 | You may have noticed that both times we took the base-10 logarithm | |
3100 | of 1000, we got an exact integer result. Calc always tries to give | |
3101 | an exact rational result for calculations involving rational numbers | |
3102 | where possible. But when we used @kbd{H E}, the result was a | |
3103 | floating-point number for no apparent reason. In fact, if we had | |
3104 | computed @kbd{10 @key{RET} 3 ^} we @emph{would} have gotten an | |
3105 | exact integer 1000. But the @kbd{H E} command is rigged to generate | |
3106 | a floating-point result all of the time so that @kbd{1000 H E} will | |
3107 | not waste time computing a thousand-digit integer when all you | |
3108 | probably wanted was @samp{1e1000}. | |
3109 | ||
3110 | (@bullet{}) @strong{Exercise 2.} Find a pair of integer inputs to | |
3111 | the @kbd{B} command for which Calc could find an exact rational | |
3112 | result but doesn't. @xref{Arithmetic Answer 2, 2}. (@bullet{}) | |
3113 | ||
3114 | The Calculator also has a set of functions relating to combinatorics | |
3115 | and statistics. You may be familiar with the @dfn{factorial} function, | |
3116 | which computes the product of all the integers up to a given number. | |
3117 | ||
d7b8e6c6 | 3118 | @smallexample |
5d67986c | 3119 | @group |
d7b8e6c6 EZ |
3120 | 1: 100 1: 93326215443... 1: 100. 1: 9.3326e157 |
3121 | . . . . | |
3122 | ||
3123 | 100 ! U c f ! | |
d7b8e6c6 | 3124 | @end group |
5d67986c | 3125 | @end smallexample |
d7b8e6c6 EZ |
3126 | |
3127 | @noindent | |
3128 | Recall, the @kbd{c f} command converts the integer or fraction at the | |
3129 | top of the stack to floating-point format. If you take the factorial | |
3130 | of a floating-point number, you get a floating-point result | |
3131 | accurate to the current precision. But if you give @kbd{!} an | |
3132 | exact integer, you get an exact integer result (158 digits long | |
3133 | in this case). | |
3134 | ||
3135 | If you take the factorial of a non-integer, Calc uses a generalized | |
3136 | factorial function defined in terms of Euler's Gamma function | |
3137 | @c{$\Gamma(n)$} | |
3138 | @cite{gamma(n)} | |
3139 | (which is itself available as the @kbd{f g} command). | |
3140 | ||
d7b8e6c6 | 3141 | @smallexample |
5d67986c | 3142 | @group |
d7b8e6c6 EZ |
3143 | 3: 4. 3: 24. 1: 5.5 1: 52.342777847 |
3144 | 2: 4.5 2: 52.3427777847 . . | |
3145 | 1: 5. 1: 120. | |
3146 | . . | |
3147 | ||
5d67986c | 3148 | M-3 ! M-0 @key{DEL} 5.5 f g |
d7b8e6c6 | 3149 | @end group |
5d67986c | 3150 | @end smallexample |
d7b8e6c6 EZ |
3151 | |
3152 | @noindent | |
3153 | Here we verify the identity @c{$n! = \Gamma(n+1)$} | |
3154 | @cite{@var{n}!@: = gamma(@var{n}+1)}. | |
3155 | ||
3156 | The binomial coefficient @var{n}-choose-@var{m}@c{ or $\displaystyle {n \choose m}$} | |
3157 | @asis{} is defined by | |
3158 | @c{$\displaystyle {n! \over m! \, (n-m)!}$} | |
3159 | @cite{n!@: / m!@: (n-m)!} for all reals @cite{n} and | |
3160 | @cite{m}. The intermediate results in this formula can become quite | |
3161 | large even if the final result is small; the @kbd{k c} command computes | |
3162 | a binomial coefficient in a way that avoids large intermediate | |
3163 | values. | |
3164 | ||
3165 | The @kbd{k} prefix key defines several common functions out of | |
3166 | combinatorics and number theory. Here we compute the binomial | |
3167 | coefficient 30-choose-20, then determine its prime factorization. | |
3168 | ||
d7b8e6c6 | 3169 | @smallexample |
5d67986c | 3170 | @group |
d7b8e6c6 EZ |
3171 | 2: 30 1: 30045015 1: [3, 3, 5, 7, 11, 13, 23, 29] |
3172 | 1: 20 . . | |
3173 | . | |
3174 | ||
5d67986c | 3175 | 30 @key{RET} 20 k c k f |
d7b8e6c6 | 3176 | @end group |
5d67986c | 3177 | @end smallexample |
d7b8e6c6 EZ |
3178 | |
3179 | @noindent | |
3180 | You can verify these prime factors by using @kbd{v u} to ``unpack'' | |
3181 | this vector into 8 separate stack entries, then @kbd{M-8 *} to | |
3182 | multiply them back together. The result is the original number, | |
3183 | 30045015. | |
3184 | ||
3185 | @cindex Hash tables | |
3186 | Suppose a program you are writing needs a hash table with at least | |
3187 | 10000 entries. It's best to use a prime number as the actual size | |
3188 | of a hash table. Calc can compute the next prime number after 10000: | |
3189 | ||
d7b8e6c6 | 3190 | @smallexample |
5d67986c | 3191 | @group |
d7b8e6c6 EZ |
3192 | 1: 10000 1: 10007 1: 9973 |
3193 | . . . | |
3194 | ||
3195 | 10000 k n I k n | |
d7b8e6c6 | 3196 | @end group |
5d67986c | 3197 | @end smallexample |
d7b8e6c6 EZ |
3198 | |
3199 | @noindent | |
3200 | Just for kicks we've also computed the next prime @emph{less} than | |
3201 | 10000. | |
3202 | ||
3203 | @c [fix-ref Financial Functions] | |
3204 | @xref{Financial Functions}, for a description of the Calculator | |
3205 | commands that deal with business and financial calculations (functions | |
3206 | like @code{pv}, @code{rate}, and @code{sln}). | |
3207 | ||
3208 | @c [fix-ref Binary Number Functions] | |
3209 | @xref{Binary Functions}, to read about the commands for operating | |
3210 | on binary numbers (like @code{and}, @code{xor}, and @code{lsh}). | |
3211 | ||
3212 | @node Vector/Matrix Tutorial, Types Tutorial, Arithmetic Tutorial, Tutorial | |
3213 | @section Vector/Matrix Tutorial | |
3214 | ||
3215 | @noindent | |
3216 | A @dfn{vector} is a list of numbers or other Calc data objects. | |
3217 | Calc provides a large set of commands that operate on vectors. Some | |
3218 | are familiar operations from vector analysis. Others simply treat | |
3219 | a vector as a list of objects. | |
3220 | ||
3221 | @menu | |
3222 | * Vector Analysis Tutorial:: | |
3223 | * Matrix Tutorial:: | |
3224 | * List Tutorial:: | |
3225 | @end menu | |
3226 | ||
3227 | @node Vector Analysis Tutorial, Matrix Tutorial, Vector/Matrix Tutorial, Vector/Matrix Tutorial | |
3228 | @subsection Vector Analysis | |
3229 | ||
3230 | @noindent | |
3231 | If you add two vectors, the result is a vector of the sums of the | |
3232 | elements, taken pairwise. | |
3233 | ||
d7b8e6c6 | 3234 | @smallexample |
5d67986c | 3235 | @group |
d7b8e6c6 EZ |
3236 | 1: [1, 2, 3] 2: [1, 2, 3] 1: [8, 8, 3] |
3237 | . 1: [7, 6, 0] . | |
3238 | . | |
3239 | ||
3240 | [1,2,3] s 1 [7 6 0] s 2 + | |
d7b8e6c6 | 3241 | @end group |
5d67986c | 3242 | @end smallexample |
d7b8e6c6 EZ |
3243 | |
3244 | @noindent | |
3245 | Note that we can separate the vector elements with either commas or | |
3246 | spaces. This is true whether we are using incomplete vectors or | |
3247 | algebraic entry. The @kbd{s 1} and @kbd{s 2} commands save these | |
3248 | vectors so we can easily reuse them later. | |
3249 | ||
3250 | If you multiply two vectors, the result is the sum of the products | |
3251 | of the elements taken pairwise. This is called the @dfn{dot product} | |
3252 | of the vectors. | |
3253 | ||
d7b8e6c6 | 3254 | @smallexample |
5d67986c | 3255 | @group |
d7b8e6c6 EZ |
3256 | 2: [1, 2, 3] 1: 19 |
3257 | 1: [7, 6, 0] . | |
3258 | . | |
3259 | ||
3260 | r 1 r 2 * | |
d7b8e6c6 | 3261 | @end group |
5d67986c | 3262 | @end smallexample |
d7b8e6c6 EZ |
3263 | |
3264 | @cindex Dot product | |
3265 | The dot product of two vectors is equal to the product of their | |
3266 | lengths times the cosine of the angle between them. (Here the vector | |
3267 | is interpreted as a line from the origin @cite{(0,0,0)} to the | |
3268 | specified point in three-dimensional space.) The @kbd{A} | |
3269 | (absolute value) command can be used to compute the length of a | |
3270 | vector. | |
3271 | ||
d7b8e6c6 | 3272 | @smallexample |
5d67986c | 3273 | @group |
d7b8e6c6 EZ |
3274 | 3: 19 3: 19 1: 0.550782 1: 56.579 |
3275 | 2: [1, 2, 3] 2: 3.741657 . . | |
3276 | 1: [7, 6, 0] 1: 9.219544 | |
3277 | . . | |
3278 | ||
5d67986c | 3279 | M-@key{RET} M-2 A * / I C |
d7b8e6c6 | 3280 | @end group |
5d67986c | 3281 | @end smallexample |
d7b8e6c6 EZ |
3282 | |
3283 | @noindent | |
3284 | First we recall the arguments to the dot product command, then | |
3285 | we compute the absolute values of the top two stack entries to | |
3286 | obtain the lengths of the vectors, then we divide the dot product | |
3287 | by the product of the lengths to get the cosine of the angle. | |
3288 | The inverse cosine finds that the angle between the vectors | |
3289 | is about 56 degrees. | |
3290 | ||
3291 | @cindex Cross product | |
3292 | @cindex Perpendicular vectors | |
3293 | The @dfn{cross product} of two vectors is a vector whose length | |
3294 | is the product of the lengths of the inputs times the sine of the | |
3295 | angle between them, and whose direction is perpendicular to both | |
3296 | input vectors. Unlike the dot product, the cross product is | |
3297 | defined only for three-dimensional vectors. Let's double-check | |
3298 | our computation of the angle using the cross product. | |
3299 | ||
d7b8e6c6 | 3300 | @smallexample |
5d67986c | 3301 | @group |
d7b8e6c6 EZ |
3302 | 2: [1, 2, 3] 3: [-18, 21, -8] 1: [-0.52, 0.61, -0.23] 1: 56.579 |
3303 | 1: [7, 6, 0] 2: [1, 2, 3] . . | |
3304 | . 1: [7, 6, 0] | |
3305 | . | |
3306 | ||
5d67986c | 3307 | r 1 r 2 V C s 3 M-@key{RET} M-2 A * / A I S |
d7b8e6c6 | 3308 | @end group |
5d67986c | 3309 | @end smallexample |
d7b8e6c6 EZ |
3310 | |
3311 | @noindent | |
3312 | First we recall the original vectors and compute their cross product, | |
3313 | which we also store for later reference. Now we divide the vector | |
3314 | by the product of the lengths of the original vectors. The length of | |
3315 | this vector should be the sine of the angle; sure enough, it is! | |
3316 | ||
3317 | @c [fix-ref General Mode Commands] | |
3318 | Vector-related commands generally begin with the @kbd{v} prefix key. | |
3319 | Some are uppercase letters and some are lowercase. To make it easier | |
3320 | to type these commands, the shift-@kbd{V} prefix key acts the same as | |
3321 | the @kbd{v} key. (@xref{General Mode Commands}, for a way to make all | |
3322 | prefix keys have this property.) | |
3323 | ||
3324 | If we take the dot product of two perpendicular vectors we expect | |
3325 | to get zero, since the cosine of 90 degrees is zero. Let's check | |
3326 | that the cross product is indeed perpendicular to both inputs: | |
3327 | ||
d7b8e6c6 | 3328 | @smallexample |
5d67986c | 3329 | @group |
d7b8e6c6 EZ |
3330 | 2: [1, 2, 3] 1: 0 2: [7, 6, 0] 1: 0 |
3331 | 1: [-18, 21, -8] . 1: [-18, 21, -8] . | |
3332 | . . | |
3333 | ||
5d67986c | 3334 | r 1 r 3 * @key{DEL} r 2 r 3 * |
d7b8e6c6 | 3335 | @end group |
5d67986c | 3336 | @end smallexample |
d7b8e6c6 EZ |
3337 | |
3338 | @cindex Normalizing a vector | |
3339 | @cindex Unit vectors | |
3340 | (@bullet{}) @strong{Exercise 1.} Given a vector on the top of the | |
3341 | stack, what keystrokes would you use to @dfn{normalize} the | |
3342 | vector, i.e., to reduce its length to one without changing its | |
3343 | direction? @xref{Vector Answer 1, 1}. (@bullet{}) | |
3344 | ||
3345 | (@bullet{}) @strong{Exercise 2.} Suppose a certain particle can be | |
3346 | at any of several positions along a ruler. You have a list of | |
3347 | those positions in the form of a vector, and another list of the | |
3348 | probabilities for the particle to be at the corresponding positions. | |
3349 | Find the average position of the particle. | |
3350 | @xref{Vector Answer 2, 2}. (@bullet{}) | |
3351 | ||
3352 | @node Matrix Tutorial, List Tutorial, Vector Analysis Tutorial, Vector/Matrix Tutorial | |
3353 | @subsection Matrices | |
3354 | ||
3355 | @noindent | |
3356 | A @dfn{matrix} is just a vector of vectors, all the same length. | |
3357 | This means you can enter a matrix using nested brackets. You can | |
3358 | also use the semicolon character to enter a matrix. We'll show | |
3359 | both methods here: | |
3360 | ||
d7b8e6c6 | 3361 | @smallexample |
5d67986c | 3362 | @group |
d7b8e6c6 EZ |
3363 | 1: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ] |
3364 | [ 4, 5, 6 ] ] [ 4, 5, 6 ] ] | |
3365 | . . | |
3366 | ||
5d67986c | 3367 | [[1 2 3] [4 5 6]] ' [1 2 3; 4 5 6] @key{RET} |
d7b8e6c6 | 3368 | @end group |
5d67986c | 3369 | @end smallexample |
d7b8e6c6 EZ |
3370 | |
3371 | @noindent | |
3372 | We'll be using this matrix again, so type @kbd{s 4} to save it now. | |
3373 | ||
3374 | Note that semicolons work with incomplete vectors, but they work | |
3375 | better in algebraic entry. That's why we use the apostrophe in | |
3376 | the second example. | |
3377 | ||
3378 | When two matrices are multiplied, the lefthand matrix must have | |
3379 | the same number of columns as the righthand matrix has rows. | |
3380 | Row @cite{i}, column @cite{j} of the result is effectively the | |
3381 | dot product of row @cite{i} of the left matrix by column @cite{j} | |
3382 | of the right matrix. | |
3383 | ||
3384 | If we try to duplicate this matrix and multiply it by itself, | |
3385 | the dimensions are wrong and the multiplication cannot take place: | |
3386 | ||
d7b8e6c6 | 3387 | @smallexample |
5d67986c | 3388 | @group |
d7b8e6c6 EZ |
3389 | 1: [ [ 1, 2, 3 ] * [ [ 1, 2, 3 ] |
3390 | [ 4, 5, 6 ] ] [ 4, 5, 6 ] ] | |
3391 | . | |
3392 | ||
5d67986c | 3393 | @key{RET} * |
d7b8e6c6 | 3394 | @end group |
5d67986c | 3395 | @end smallexample |
d7b8e6c6 EZ |
3396 | |
3397 | @noindent | |
3398 | Though rather hard to read, this is a formula which shows the product | |
3399 | of two matrices. The @samp{*} function, having invalid arguments, has | |
3400 | been left in symbolic form. | |
3401 | ||
3402 | We can multiply the matrices if we @dfn{transpose} one of them first. | |
3403 | ||
d7b8e6c6 | 3404 | @smallexample |
5d67986c | 3405 | @group |
d7b8e6c6 EZ |
3406 | 2: [ [ 1, 2, 3 ] 1: [ [ 14, 32 ] 1: [ [ 17, 22, 27 ] |
3407 | [ 4, 5, 6 ] ] [ 32, 77 ] ] [ 22, 29, 36 ] | |
3408 | 1: [ [ 1, 4 ] . [ 27, 36, 45 ] ] | |
3409 | [ 2, 5 ] . | |
3410 | [ 3, 6 ] ] | |
3411 | . | |
3412 | ||
5d67986c | 3413 | U v t * U @key{TAB} * |
d7b8e6c6 | 3414 | @end group |
5d67986c | 3415 | @end smallexample |
d7b8e6c6 EZ |
3416 | |
3417 | Matrix multiplication is not commutative; indeed, switching the | |
3418 | order of the operands can even change the dimensions of the result | |
3419 | matrix, as happened here! | |
3420 | ||
3421 | If you multiply a plain vector by a matrix, it is treated as a | |
3422 | single row or column depending on which side of the matrix it is | |
3423 | on. The result is a plain vector which should also be interpreted | |
3424 | as a row or column as appropriate. | |
3425 | ||
d7b8e6c6 | 3426 | @smallexample |
5d67986c | 3427 | @group |
d7b8e6c6 EZ |
3428 | 2: [ [ 1, 2, 3 ] 1: [14, 32] |
3429 | [ 4, 5, 6 ] ] . | |
3430 | 1: [1, 2, 3] | |
3431 | . | |
3432 | ||
3433 | r 4 r 1 * | |
d7b8e6c6 | 3434 | @end group |
5d67986c | 3435 | @end smallexample |
d7b8e6c6 EZ |
3436 | |
3437 | Multiplying in the other order wouldn't work because the number of | |
3438 | rows in the matrix is different from the number of elements in the | |
3439 | vector. | |
3440 | ||
3441 | (@bullet{}) @strong{Exercise 1.} Use @samp{*} to sum along the rows | |
3442 | of the above @c{$2\times3$} | |
3443 | @asis{2x3} matrix to get @cite{[6, 15]}. Now use @samp{*} to | |
3444 | sum along the columns to get @cite{[5, 7, 9]}. | |
3445 | @xref{Matrix Answer 1, 1}. (@bullet{}) | |
3446 | ||
3447 | @cindex Identity matrix | |
3448 | An @dfn{identity matrix} is a square matrix with ones along the | |
3449 | diagonal and zeros elsewhere. It has the property that multiplication | |
3450 | by an identity matrix, on the left or on the right, always produces | |
3451 | the original matrix. | |
3452 | ||
d7b8e6c6 | 3453 | @smallexample |
5d67986c | 3454 | @group |
d7b8e6c6 EZ |
3455 | 1: [ [ 1, 2, 3 ] 2: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ] |
3456 | [ 4, 5, 6 ] ] [ 4, 5, 6 ] ] [ 4, 5, 6 ] ] | |
3457 | . 1: [ [ 1, 0, 0 ] . | |
3458 | [ 0, 1, 0 ] | |
3459 | [ 0, 0, 1 ] ] | |
3460 | . | |
3461 | ||
5d67986c | 3462 | r 4 v i 3 @key{RET} * |
d7b8e6c6 | 3463 | @end group |
5d67986c | 3464 | @end smallexample |
d7b8e6c6 EZ |
3465 | |
3466 | If a matrix is square, it is often possible to find its @dfn{inverse}, | |
3467 | that is, a matrix which, when multiplied by the original matrix, yields | |
3468 | an identity matrix. The @kbd{&} (reciprocal) key also computes the | |
3469 | inverse of a matrix. | |
3470 | ||
d7b8e6c6 | 3471 | @smallexample |
5d67986c | 3472 | @group |
d7b8e6c6 EZ |
3473 | 1: [ [ 1, 2, 3 ] 1: [ [ -2.4, 1.2, -0.2 ] |
3474 | [ 4, 5, 6 ] [ 2.8, -1.4, 0.4 ] | |
3475 | [ 7, 6, 0 ] ] [ -0.73333, 0.53333, -0.2 ] ] | |
3476 | . . | |
3477 | ||
3478 | r 4 r 2 | s 5 & | |
d7b8e6c6 | 3479 | @end group |
5d67986c | 3480 | @end smallexample |
d7b8e6c6 EZ |
3481 | |
3482 | @noindent | |
3483 | The vertical bar @kbd{|} @dfn{concatenates} numbers, vectors, and | |
3484 | matrices together. Here we have used it to add a new row onto | |
3485 | our matrix to make it square. | |
3486 | ||
3487 | We can multiply these two matrices in either order to get an identity. | |
3488 | ||
d7b8e6c6 | 3489 | @smallexample |
5d67986c | 3490 | @group |
d7b8e6c6 EZ |
3491 | 1: [ [ 1., 0., 0. ] 1: [ [ 1., 0., 0. ] |
3492 | [ 0., 1., 0. ] [ 0., 1., 0. ] | |
3493 | [ 0., 0., 1. ] ] [ 0., 0., 1. ] ] | |
3494 | . . | |
3495 | ||
5d67986c | 3496 | M-@key{RET} * U @key{TAB} * |
d7b8e6c6 | 3497 | @end group |
5d67986c | 3498 | @end smallexample |
d7b8e6c6 EZ |
3499 | |
3500 | @cindex Systems of linear equations | |
3501 | @cindex Linear equations, systems of | |
3502 | Matrix inverses are related to systems of linear equations in algebra. | |
3503 | Suppose we had the following set of equations: | |
3504 | ||
3505 | @ifinfo | |
3506 | @group | |
3507 | @example | |
3508 | a + 2b + 3c = 6 | |
3509 | 4a + 5b + 6c = 2 | |
3510 | 7a + 6b = 3 | |
3511 | @end example | |
3512 | @end group | |
3513 | @end ifinfo | |
3514 | @tex | |
3515 | \turnoffactive | |
3516 | \beforedisplayh | |
3517 | $$ \openup1\jot \tabskip=0pt plus1fil | |
3518 | \halign to\displaywidth{\tabskip=0pt | |
3519 | $\hfil#$&$\hfil{}#{}$& | |
3520 | $\hfil#$&$\hfil{}#{}$& | |
3521 | $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr | |
3522 | a&+&2b&+&3c&=6 \cr | |
3523 | 4a&+&5b&+&6c&=2 \cr | |
3524 | 7a&+&6b& & &=3 \cr} | |
3525 | $$ | |
3526 | \afterdisplayh | |
3527 | @end tex | |
3528 | ||
3529 | @noindent | |
3530 | This can be cast into the matrix equation, | |
3531 | ||
3532 | @ifinfo | |
3533 | @group | |
3534 | @example | |
3535 | [ [ 1, 2, 3 ] [ [ a ] [ [ 6 ] | |
3536 | [ 4, 5, 6 ] * [ b ] = [ 2 ] | |
3537 | [ 7, 6, 0 ] ] [ c ] ] [ 3 ] ] | |
3538 | @end example | |
3539 | @end group | |
3540 | @end ifinfo | |
3541 | @tex | |
3542 | \turnoffactive | |
3543 | \beforedisplay | |
3544 | $$ \pmatrix{ 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 6 & 0 } | |
3545 | \times | |
3546 | \pmatrix{ a \cr b \cr c } = \pmatrix{ 6 \cr 2 \cr 3 } | |
3547 | $$ | |
3548 | \afterdisplay | |
3549 | @end tex | |
3550 | ||
3551 | We can solve this system of equations by multiplying both sides by the | |
3552 | inverse of the matrix. Calc can do this all in one step: | |
3553 | ||
d7b8e6c6 | 3554 | @smallexample |
5d67986c | 3555 | @group |
d7b8e6c6 EZ |
3556 | 2: [6, 2, 3] 1: [-12.6, 15.2, -3.93333] |
3557 | 1: [ [ 1, 2, 3 ] . | |
3558 | [ 4, 5, 6 ] | |
3559 | [ 7, 6, 0 ] ] | |
3560 | . | |
3561 | ||
3562 | [6,2,3] r 5 / | |
d7b8e6c6 | 3563 | @end group |
5d67986c | 3564 | @end smallexample |
d7b8e6c6 EZ |
3565 | |
3566 | @noindent | |
3567 | The result is the @cite{[a, b, c]} vector that solves the equations. | |
3568 | (Dividing by a square matrix is equivalent to multiplying by its | |
3569 | inverse.) | |
3570 | ||
3571 | Let's verify this solution: | |
3572 | ||
d7b8e6c6 | 3573 | @smallexample |
5d67986c | 3574 | @group |
d7b8e6c6 EZ |
3575 | 2: [ [ 1, 2, 3 ] 1: [6., 2., 3.] |
3576 | [ 4, 5, 6 ] . | |
3577 | [ 7, 6, 0 ] ] | |
3578 | 1: [-12.6, 15.2, -3.93333] | |
3579 | . | |
3580 | ||
5d67986c | 3581 | r 5 @key{TAB} * |
d7b8e6c6 | 3582 | @end group |
5d67986c | 3583 | @end smallexample |
d7b8e6c6 EZ |
3584 | |
3585 | @noindent | |
3586 | Note that we had to be careful about the order in which we multiplied | |
3587 | the matrix and vector. If we multiplied in the other order, Calc would | |
3588 | assume the vector was a row vector in order to make the dimensions | |
3589 | come out right, and the answer would be incorrect. If you | |
3590 | don't feel safe letting Calc take either interpretation of your | |
3591 | vectors, use explicit @c{$N\times1$} | |
3592 | @asis{Nx1} or @c{$1\times N$} | |
3593 | @asis{1xN} matrices instead. | |
3594 | In this case, you would enter the original column vector as | |
3595 | @samp{[[6], [2], [3]]} or @samp{[6; 2; 3]}. | |
3596 | ||
3597 | (@bullet{}) @strong{Exercise 2.} Algebraic entry allows you to make | |
3598 | vectors and matrices that include variables. Solve the following | |
3599 | system of equations to get expressions for @cite{x} and @cite{y} | |
3600 | in terms of @cite{a} and @cite{b}. | |
3601 | ||
3602 | @ifinfo | |
3603 | @group | |
3604 | @example | |
3605 | x + a y = 6 | |
3606 | x + b y = 10 | |
3607 | @end example | |
3608 | @end group | |
3609 | @end ifinfo | |
3610 | @tex | |
3611 | \turnoffactive | |
3612 | \beforedisplay | |
3613 | $$ \eqalign{ x &+ a y = 6 \cr | |
3614 | x &+ b y = 10} | |
3615 | $$ | |
3616 | \afterdisplay | |
3617 | @end tex | |
3618 | ||
3619 | @noindent | |
3620 | @xref{Matrix Answer 2, 2}. (@bullet{}) | |
3621 | ||
3622 | @cindex Least-squares for over-determined systems | |
3623 | @cindex Over-determined systems of equations | |
3624 | (@bullet{}) @strong{Exercise 3.} A system of equations is ``over-determined'' | |
3625 | if it has more equations than variables. It is often the case that | |
3626 | there are no values for the variables that will satisfy all the | |
3627 | equations at once, but it is still useful to find a set of values | |
3628 | which ``nearly'' satisfy all the equations. In terms of matrix equations, | |
3629 | you can't solve @cite{A X = B} directly because the matrix @cite{A} | |
3630 | is not square for an over-determined system. Matrix inversion works | |
3631 | only for square matrices. One common trick is to multiply both sides | |
3632 | on the left by the transpose of @cite{A}: | |
3633 | @ifinfo | |
3634 | @samp{trn(A)*A*X = trn(A)*B}. | |
3635 | @end ifinfo | |
3636 | @tex | |
3637 | \turnoffactive | |
3638 | $A^T A \, X = A^T B$, where $A^T$ is the transpose \samp{trn(A)}. | |
3639 | @end tex | |
3640 | Now @c{$A^T A$} | |
3641 | @cite{trn(A)*A} is a square matrix so a solution is possible. It | |
3642 | turns out that the @cite{X} vector you compute in this way will be a | |
3643 | ``least-squares'' solution, which can be regarded as the ``closest'' | |
3644 | solution to the set of equations. Use Calc to solve the following | |
3645 | over-determined system:@refill | |
3646 | ||
3647 | @ifinfo | |
3648 | @group | |
3649 | @example | |
3650 | a + 2b + 3c = 6 | |
3651 | 4a + 5b + 6c = 2 | |
3652 | 7a + 6b = 3 | |
3653 | 2a + 4b + 6c = 11 | |
3654 | @end example | |
3655 | @end group | |
3656 | @end ifinfo | |
3657 | @tex | |
3658 | \turnoffactive | |
3659 | \beforedisplayh | |
3660 | $$ \openup1\jot \tabskip=0pt plus1fil | |
3661 | \halign to\displaywidth{\tabskip=0pt | |
3662 | $\hfil#$&$\hfil{}#{}$& | |
3663 | $\hfil#$&$\hfil{}#{}$& | |
3664 | $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr | |
3665 | a&+&2b&+&3c&=6 \cr | |
3666 | 4a&+&5b&+&6c&=2 \cr | |
3667 | 7a&+&6b& & &=3 \cr | |
3668 | 2a&+&4b&+&6c&=11 \cr} | |
3669 | $$ | |
3670 | \afterdisplayh | |
3671 | @end tex | |
3672 | ||
3673 | @noindent | |
3674 | @xref{Matrix Answer 3, 3}. (@bullet{}) | |
3675 | ||
3676 | @node List Tutorial, , Matrix Tutorial, Vector/Matrix Tutorial | |
3677 | @subsection Vectors as Lists | |
3678 | ||
3679 | @noindent | |
3680 | @cindex Lists | |
3681 | Although Calc has a number of features for manipulating vectors and | |
3682 | matrices as mathematical objects, you can also treat vectors as | |
3683 | simple lists of values. For example, we saw that the @kbd{k f} | |
3684 | command returns a vector which is a list of the prime factors of a | |
3685 | number. | |
3686 | ||
3687 | You can pack and unpack stack entries into vectors: | |
3688 | ||
d7b8e6c6 | 3689 | @smallexample |
5d67986c | 3690 | @group |
d7b8e6c6 EZ |
3691 | 3: 10 1: [10, 20, 30] 3: 10 |
3692 | 2: 20 . 2: 20 | |
3693 | 1: 30 1: 30 | |
3694 | . . | |
3695 | ||
3696 | M-3 v p v u | |
d7b8e6c6 | 3697 | @end group |
5d67986c | 3698 | @end smallexample |
d7b8e6c6 EZ |
3699 | |
3700 | You can also build vectors out of consecutive integers, or out | |
3701 | of many copies of a given value: | |
3702 | ||
d7b8e6c6 | 3703 | @smallexample |
5d67986c | 3704 | @group |
d7b8e6c6 EZ |
3705 | 1: [1, 2, 3, 4] 2: [1, 2, 3, 4] 2: [1, 2, 3, 4] |
3706 | . 1: 17 1: [17, 17, 17, 17] | |
3707 | . . | |
3708 | ||
5d67986c | 3709 | v x 4 @key{RET} 17 v b 4 @key{RET} |
d7b8e6c6 | 3710 | @end group |
5d67986c | 3711 | @end smallexample |
d7b8e6c6 EZ |
3712 | |
3713 | You can apply an operator to every element of a vector using the | |
3714 | @dfn{map} command. | |
3715 | ||
d7b8e6c6 | 3716 | @smallexample |
5d67986c | 3717 | @group |
d7b8e6c6 EZ |
3718 | 1: [17, 34, 51, 68] 1: [289, 1156, 2601, 4624] 1: [17, 34, 51, 68] |
3719 | . . . | |
3720 | ||
3721 | V M * 2 V M ^ V M Q | |
d7b8e6c6 | 3722 | @end group |
5d67986c | 3723 | @end smallexample |
d7b8e6c6 EZ |
3724 | |
3725 | @noindent | |
3726 | In the first step, we multiply the vector of integers by the vector | |
3727 | of 17's elementwise. In the second step, we raise each element to | |
3728 | the power two. (The general rule is that both operands must be | |
3729 | vectors of the same length, or else one must be a vector and the | |
3730 | other a plain number.) In the final step, we take the square root | |
3731 | of each element. | |
3732 | ||
3733 | (@bullet{}) @strong{Exercise 1.} Compute a vector of powers of two | |
3734 | from @c{$2^{-4}$} | |
3735 | @cite{2^-4} to @cite{2^4}. @xref{List Answer 1, 1}. (@bullet{}) | |
3736 | ||
3737 | You can also @dfn{reduce} a binary operator across a vector. | |
3738 | For example, reducing @samp{*} computes the product of all the | |
3739 | elements in the vector: | |
3740 | ||
d7b8e6c6 | 3741 | @smallexample |
5d67986c | 3742 | @group |
d7b8e6c6 EZ |
3743 | 1: 123123 1: [3, 7, 11, 13, 41] 1: 123123 |
3744 | . . . | |
3745 | ||
3746 | 123123 k f V R * | |
d7b8e6c6 | 3747 | @end group |
5d67986c | 3748 | @end smallexample |
d7b8e6c6 EZ |
3749 | |
3750 | @noindent | |
3751 | In this example, we decompose 123123 into its prime factors, then | |
3752 | multiply those factors together again to yield the original number. | |
3753 | ||
3754 | We could compute a dot product ``by hand'' using mapping and | |
3755 | reduction: | |
3756 | ||
d7b8e6c6 | 3757 | @smallexample |
5d67986c | 3758 | @group |
d7b8e6c6 EZ |
3759 | 2: [1, 2, 3] 1: [7, 12, 0] 1: 19 |
3760 | 1: [7, 6, 0] . . | |
3761 | . | |
3762 | ||
3763 | r 1 r 2 V M * V R + | |
d7b8e6c6 | 3764 | @end group |
5d67986c | 3765 | @end smallexample |
d7b8e6c6 EZ |
3766 | |
3767 | @noindent | |
3768 | Recalling two vectors from the previous section, we compute the | |
3769 | sum of pairwise products of the elements to get the same answer | |
3770 | for the dot product as before. | |
3771 | ||
3772 | A slight variant of vector reduction is the @dfn{accumulate} operation, | |
3773 | @kbd{V U}. This produces a vector of the intermediate results from | |
3774 | a corresponding reduction. Here we compute a table of factorials: | |
3775 | ||
d7b8e6c6 | 3776 | @smallexample |
5d67986c | 3777 | @group |
d7b8e6c6 EZ |
3778 | 1: [1, 2, 3, 4, 5, 6] 1: [1, 2, 6, 24, 120, 720] |
3779 | . . | |
3780 | ||
5d67986c | 3781 | v x 6 @key{RET} V U * |
d7b8e6c6 | 3782 | @end group |
5d67986c | 3783 | @end smallexample |
d7b8e6c6 EZ |
3784 | |
3785 | Calc allows vectors to grow as large as you like, although it gets | |
3786 | rather slow if vectors have more than about a hundred elements. | |
3787 | Actually, most of the time is spent formatting these large vectors | |
3788 | for display, not calculating on them. Try the following experiment | |
3789 | (if your computer is very fast you may need to substitute a larger | |
3790 | vector size). | |
3791 | ||
d7b8e6c6 | 3792 | @smallexample |
5d67986c | 3793 | @group |
d7b8e6c6 EZ |
3794 | 1: [1, 2, 3, 4, ... 1: [2, 3, 4, 5, ... |
3795 | . . | |
3796 | ||
5d67986c | 3797 | v x 500 @key{RET} 1 V M + |
d7b8e6c6 | 3798 | @end group |
5d67986c | 3799 | @end smallexample |
d7b8e6c6 EZ |
3800 | |
3801 | Now press @kbd{v .} (the letter @kbd{v}, then a period) and try the | |
3802 | experiment again. In @kbd{v .} mode, long vectors are displayed | |
3803 | ``abbreviated'' like this: | |
3804 | ||
d7b8e6c6 | 3805 | @smallexample |
5d67986c | 3806 | @group |
d7b8e6c6 EZ |
3807 | 1: [1, 2, 3, ..., 500] 1: [2, 3, 4, ..., 501] |
3808 | . . | |
3809 | ||
5d67986c | 3810 | v x 500 @key{RET} 1 V M + |
d7b8e6c6 | 3811 | @end group |
5d67986c | 3812 | @end smallexample |
d7b8e6c6 EZ |
3813 | |
3814 | @noindent | |
3815 | (where now the @samp{...} is actually part of the Calc display). | |
3816 | You will find both operations are now much faster. But notice that | |
3817 | even in @w{@kbd{v .}} mode, the full vectors are still shown in the Trail. | |
3818 | Type @w{@kbd{t .}} to cause the trail to abbreviate as well, and try the | |
3819 | experiment one more time. Operations on long vectors are now quite | |
3820 | fast! (But of course if you use @kbd{t .} you will lose the ability | |
3821 | to get old vectors back using the @kbd{t y} command.) | |
3822 | ||
3823 | An easy way to view a full vector when @kbd{v .} mode is active is | |
3824 | to press @kbd{`} (back-quote) to edit the vector; editing always works | |
3825 | with the full, unabbreviated value. | |
3826 | ||
3827 | @cindex Least-squares for fitting a straight line | |
3828 | @cindex Fitting data to a line | |
3829 | @cindex Line, fitting data to | |
3830 | @cindex Data, extracting from buffers | |
3831 | @cindex Columns of data, extracting | |
3832 | As a larger example, let's try to fit a straight line to some data, | |
3833 | using the method of least squares. (Calc has a built-in command for | |
3834 | least-squares curve fitting, but we'll do it by hand here just to | |
3835 | practice working with vectors.) Suppose we have the following list | |
3836 | of values in a file we have loaded into Emacs: | |
3837 | ||
3838 | @smallexample | |
3839 | x y | |
3840 | --- --- | |
3841 | 1.34 0.234 | |
3842 | 1.41 0.298 | |
3843 | 1.49 0.402 | |
3844 | 1.56 0.412 | |
3845 | 1.64 0.466 | |
3846 | 1.73 0.473 | |
3847 | 1.82 0.601 | |
3848 | 1.91 0.519 | |
3849 | 2.01 0.603 | |
3850 | 2.11 0.637 | |
3851 | 2.22 0.645 | |
3852 | 2.33 0.705 | |
3853 | 2.45 0.917 | |
3854 | 2.58 1.009 | |
3855 | 2.71 0.971 | |
3856 | 2.85 1.062 | |
3857 | 3.00 1.148 | |
3858 | 3.15 1.157 | |
3859 | 3.32 1.354 | |
3860 | @end smallexample | |
3861 | ||
3862 | @noindent | |
3863 | If you are reading this tutorial in printed form, you will find it | |
3864 | easiest to press @kbd{M-# i} to enter the on-line Info version of | |
3865 | the manual and find this table there. (Press @kbd{g}, then type | |
3866 | @kbd{List Tutorial}, to jump straight to this section.) | |
3867 | ||
3868 | Position the cursor at the upper-left corner of this table, just | |
3869 | to the left of the @cite{1.34}. Press @kbd{C-@@} to set the mark. | |
5d67986c | 3870 | (On your system this may be @kbd{C-2}, @kbd{C-@key{SPC}}, or @kbd{NUL}.) |
d7b8e6c6 EZ |
3871 | Now position the cursor to the lower-right, just after the @cite{1.354}. |
3872 | You have now defined this region as an Emacs ``rectangle.'' Still | |
3873 | in the Info buffer, type @kbd{M-# r}. This command | |
3874 | (@code{calc-grab-rectangle}) will pop you back into the Calculator, with | |
3875 | the contents of the rectangle you specified in the form of a matrix.@refill | |
3876 | ||
d7b8e6c6 | 3877 | @smallexample |
5d67986c | 3878 | @group |
d7b8e6c6 EZ |
3879 | 1: [ [ 1.34, 0.234 ] |
3880 | [ 1.41, 0.298 ] | |
3881 | @dots{} | |
d7b8e6c6 | 3882 | @end group |
5d67986c | 3883 | @end smallexample |
d7b8e6c6 EZ |
3884 | |
3885 | @noindent | |
3886 | (You may wish to use @kbd{v .} mode to abbreviate the display of this | |
3887 | large matrix.) | |
3888 | ||
3889 | We want to treat this as a pair of lists. The first step is to | |
3890 | transpose this matrix into a pair of rows. Remember, a matrix is | |
3891 | just a vector of vectors. So we can unpack the matrix into a pair | |
3892 | of row vectors on the stack. | |
3893 | ||
d7b8e6c6 | 3894 | @smallexample |
5d67986c | 3895 | @group |
d7b8e6c6 EZ |
3896 | 1: [ [ 1.34, 1.41, 1.49, ... ] 2: [1.34, 1.41, 1.49, ... ] |
3897 | [ 0.234, 0.298, 0.402, ... ] ] 1: [0.234, 0.298, 0.402, ... ] | |
3898 | . . | |
3899 | ||
3900 | v t v u | |
d7b8e6c6 | 3901 | @end group |
5d67986c | 3902 | @end smallexample |
d7b8e6c6 EZ |
3903 | |
3904 | @noindent | |
3905 | Let's store these in quick variables 1 and 2, respectively. | |
3906 | ||
d7b8e6c6 | 3907 | @smallexample |
5d67986c | 3908 | @group |
d7b8e6c6 EZ |
3909 | 1: [1.34, 1.41, 1.49, ... ] . |
3910 | . | |
3911 | ||
3912 | t 2 t 1 | |
d7b8e6c6 | 3913 | @end group |
5d67986c | 3914 | @end smallexample |
d7b8e6c6 EZ |
3915 | |
3916 | @noindent | |
3917 | (Recall that @kbd{t 2} is a variant of @kbd{s 2} that removes the | |
3918 | stored value from the stack.) | |
3919 | ||
3920 | In a least squares fit, the slope @cite{m} is given by the formula | |
3921 | ||
3922 | @ifinfo | |
3923 | @example | |
3924 | m = (N sum(x y) - sum(x) sum(y)) / (N sum(x^2) - sum(x)^2) | |
3925 | @end example | |
3926 | @end ifinfo | |
3927 | @tex | |
3928 | \turnoffactive | |
3929 | \beforedisplay | |
3930 | $$ m = {N \sum x y - \sum x \sum y \over | |
3931 | N \sum x^2 - \left( \sum x \right)^2} $$ | |
3932 | \afterdisplay | |
3933 | @end tex | |
3934 | ||
3935 | @noindent | |
3936 | where @c{$\sum x$} | |
3937 | @cite{sum(x)} represents the sum of all the values of @cite{x}. | |
3938 | While there is an actual @code{sum} function in Calc, it's easier to | |
3939 | sum a vector using a simple reduction. First, let's compute the four | |
3940 | different sums that this formula uses. | |
3941 | ||
d7b8e6c6 | 3942 | @smallexample |
5d67986c | 3943 | @group |
d7b8e6c6 EZ |
3944 | 1: 41.63 1: 98.0003 |
3945 | . . | |
3946 | ||
3947 | r 1 V R + t 3 r 1 2 V M ^ V R + t 4 | |
3948 | ||
d7b8e6c6 | 3949 | @end group |
5d67986c | 3950 | @end smallexample |
d7b8e6c6 | 3951 | @noindent |
d7b8e6c6 | 3952 | @smallexample |
5d67986c | 3953 | @group |
d7b8e6c6 EZ |
3954 | 1: 13.613 1: 33.36554 |
3955 | . . | |
3956 | ||
3957 | r 2 V R + t 5 r 1 r 2 V M * V R + t 6 | |
d7b8e6c6 | 3958 | @end group |
5d67986c | 3959 | @end smallexample |
d7b8e6c6 EZ |
3960 | |
3961 | @ifinfo | |
3962 | @noindent | |
3963 | These are @samp{sum(x)}, @samp{sum(x^2)}, @samp{sum(y)}, and @samp{sum(x y)}, | |
3964 | respectively. (We could have used @kbd{*} to compute @samp{sum(x^2)} and | |
3965 | @samp{sum(x y)}.) | |
3966 | @end ifinfo | |
3967 | @tex | |
3968 | \turnoffactive | |
3969 | These are $\sum x$, $\sum x^2$, $\sum y$, and $\sum x y$, | |
3970 | respectively. (We could have used \kbd{*} to compute $\sum x^2$ and | |
3971 | $\sum x y$.) | |
3972 | @end tex | |
3973 | ||
3974 | Finally, we also need @cite{N}, the number of data points. This is just | |
3975 | the length of either of our lists. | |
3976 | ||
d7b8e6c6 | 3977 | @smallexample |
5d67986c | 3978 | @group |
d7b8e6c6 EZ |
3979 | 1: 19 |
3980 | . | |
3981 | ||
3982 | r 1 v l t 7 | |
d7b8e6c6 | 3983 | @end group |
5d67986c | 3984 | @end smallexample |
d7b8e6c6 EZ |
3985 | |
3986 | @noindent | |
3987 | (That's @kbd{v} followed by a lower-case @kbd{l}.) | |
3988 | ||
3989 | Now we grind through the formula: | |
3990 | ||
d7b8e6c6 | 3991 | @smallexample |
5d67986c | 3992 | @group |
d7b8e6c6 EZ |
3993 | 1: 633.94526 2: 633.94526 1: 67.23607 |
3994 | . 1: 566.70919 . | |
3995 | . | |
3996 | ||
3997 | r 7 r 6 * r 3 r 5 * - | |
3998 | ||
d7b8e6c6 | 3999 | @end group |
5d67986c | 4000 | @end smallexample |
d7b8e6c6 | 4001 | @noindent |
d7b8e6c6 | 4002 | @smallexample |
5d67986c | 4003 | @group |
d7b8e6c6 EZ |
4004 | 2: 67.23607 3: 67.23607 2: 67.23607 1: 0.52141679 |
4005 | 1: 1862.0057 2: 1862.0057 1: 128.9488 . | |
4006 | . 1: 1733.0569 . | |
4007 | . | |
4008 | ||
4009 | r 7 r 4 * r 3 2 ^ - / t 8 | |
d7b8e6c6 | 4010 | @end group |
5d67986c | 4011 | @end smallexample |
d7b8e6c6 EZ |
4012 | |
4013 | That gives us the slope @cite{m}. The y-intercept @cite{b} can now | |
4014 | be found with the simple formula, | |
4015 | ||
4016 | @ifinfo | |
4017 | @example | |
4018 | b = (sum(y) - m sum(x)) / N | |
4019 | @end example | |
4020 | @end ifinfo | |
4021 | @tex | |
4022 | \turnoffactive | |
4023 | \beforedisplay | |
4024 | $$ b = {\sum y - m \sum x \over N} $$ | |
4025 | \afterdisplay | |
4026 | \vskip10pt | |
4027 | @end tex | |
4028 | ||
d7b8e6c6 | 4029 | @smallexample |
5d67986c | 4030 | @group |
d7b8e6c6 EZ |
4031 | 1: 13.613 2: 13.613 1: -8.09358 1: -0.425978 |
4032 | . 1: 21.70658 . . | |
4033 | . | |
4034 | ||
4035 | r 5 r 8 r 3 * - r 7 / t 9 | |
d7b8e6c6 | 4036 | @end group |
5d67986c | 4037 | @end smallexample |
d7b8e6c6 EZ |
4038 | |
4039 | Let's ``plot'' this straight line approximation, @c{$y \approx m x + b$} | |
4040 | @cite{m x + b}, and compare it with the original data.@refill | |
4041 | ||
d7b8e6c6 | 4042 | @smallexample |
5d67986c | 4043 | @group |
d7b8e6c6 EZ |
4044 | 1: [0.699, 0.735, ... ] 1: [0.273, 0.309, ... ] |
4045 | . . | |
4046 | ||
4047 | r 1 r 8 * r 9 + s 0 | |
d7b8e6c6 | 4048 | @end group |
5d67986c | 4049 | @end smallexample |
d7b8e6c6 EZ |
4050 | |
4051 | @noindent | |
4052 | Notice that multiplying a vector by a constant, and adding a constant | |
4053 | to a vector, can be done without mapping commands since these are | |
4054 | common operations from vector algebra. As far as Calc is concerned, | |
4055 | we've just been doing geometry in 19-dimensional space! | |
4056 | ||
4057 | We can subtract this vector from our original @cite{y} vector to get | |
4058 | a feel for the error of our fit. Let's find the maximum error: | |
4059 | ||
d7b8e6c6 | 4060 | @smallexample |
5d67986c | 4061 | @group |
d7b8e6c6 EZ |
4062 | 1: [0.0387, 0.0112, ... ] 1: [0.0387, 0.0112, ... ] 1: 0.0897 |
4063 | . . . | |
4064 | ||
4065 | r 2 - V M A V R X | |
d7b8e6c6 | 4066 | @end group |
5d67986c | 4067 | @end smallexample |
d7b8e6c6 EZ |
4068 | |
4069 | @noindent | |
4070 | First we compute a vector of differences, then we take the absolute | |
4071 | values of these differences, then we reduce the @code{max} function | |
4072 | across the vector. (The @code{max} function is on the two-key sequence | |
4073 | @kbd{f x}; because it is so common to use @code{max} in a vector | |
4074 | operation, the letters @kbd{X} and @kbd{N} are also accepted for | |
4075 | @code{max} and @code{min} in this context. In general, you answer | |
4076 | the @kbd{V M} or @kbd{V R} prompt with the actual key sequence that | |
4077 | invokes the function you want. You could have typed @kbd{V R f x} or | |
4078 | even @kbd{V R x max @key{RET}} if you had preferred.) | |
4079 | ||
4080 | If your system has the GNUPLOT program, you can see graphs of your | |
4081 | data and your straight line to see how well they match. (If you have | |
4082 | GNUPLOT 3.0, the following instructions will work regardless of the | |
4083 | kind of display you have. Some GNUPLOT 2.0, non-X-windows systems | |
4084 | may require additional steps to view the graphs.) | |
4085 | ||
5d67986c | 4086 | Let's start by plotting the original data. Recall the ``@var{x}'' and ``@var{y}'' |
d7b8e6c6 EZ |
4087 | vectors onto the stack and press @kbd{g f}. This ``fast'' graphing |
4088 | command does everything you need to do for simple, straightforward | |
4089 | plotting of data. | |
4090 | ||
d7b8e6c6 | 4091 | @smallexample |
5d67986c | 4092 | @group |
d7b8e6c6 EZ |
4093 | 2: [1.34, 1.41, 1.49, ... ] |
4094 | 1: [0.234, 0.298, 0.402, ... ] | |
4095 | . | |
4096 | ||
4097 | r 1 r 2 g f | |
d7b8e6c6 | 4098 | @end group |
5d67986c | 4099 | @end smallexample |
d7b8e6c6 EZ |
4100 | |
4101 | If all goes well, you will shortly get a new window containing a graph | |
4102 | of the data. (If not, contact your GNUPLOT or Calc installer to find | |
4103 | out what went wrong.) In the X window system, this will be a separate | |
4104 | graphics window. For other kinds of displays, the default is to | |
4105 | display the graph in Emacs itself using rough character graphics. | |
4106 | Press @kbd{q} when you are done viewing the character graphics. | |
4107 | ||
4108 | Next, let's add the line we got from our least-squares fit: | |
4109 | ||
d7b8e6c6 | 4110 | @smallexample |
5d67986c | 4111 | @group |
d7b8e6c6 EZ |
4112 | 2: [1.34, 1.41, 1.49, ... ] |
4113 | 1: [0.273, 0.309, 0.351, ... ] | |
4114 | . | |
4115 | ||
5d67986c | 4116 | @key{DEL} r 0 g a g p |
d7b8e6c6 | 4117 | @end group |
5d67986c | 4118 | @end smallexample |
d7b8e6c6 EZ |
4119 | |
4120 | It's not very useful to get symbols to mark the data points on this | |
4121 | second curve; you can type @kbd{g S g p} to remove them. Type @kbd{g q} | |
4122 | when you are done to remove the X graphics window and terminate GNUPLOT. | |
4123 | ||
4124 | (@bullet{}) @strong{Exercise 2.} An earlier exercise showed how to do | |
4125 | least squares fitting to a general system of equations. Our 19 data | |
4126 | points are really 19 equations of the form @cite{y_i = m x_i + b} for | |
4127 | different pairs of @cite{(x_i,y_i)}. Use the matrix-transpose method | |
4128 | to solve for @cite{m} and @cite{b}, duplicating the above result. | |
4129 | @xref{List Answer 2, 2}. (@bullet{}) | |
4130 | ||
4131 | @cindex Geometric mean | |
4132 | (@bullet{}) @strong{Exercise 3.} If the input data do not form a | |
4133 | rectangle, you can use @w{@kbd{M-# g}} (@code{calc-grab-region}) | |
4134 | to grab the data the way Emacs normally works with regions---it reads | |
4135 | left-to-right, top-to-bottom, treating line breaks the same as spaces. | |
4136 | Use this command to find the geometric mean of the following numbers. | |
4137 | (The geometric mean is the @var{n}th root of the product of @var{n} numbers.) | |
4138 | ||
4139 | @example | |
4140 | 2.3 6 22 15.1 7 | |
4141 | 15 14 7.5 | |
4142 | 2.5 | |
4143 | @end example | |
4144 | ||
4145 | @noindent | |
4146 | The @kbd{M-# g} command accepts numbers separated by spaces or commas, | |
4147 | with or without surrounding vector brackets. | |
4148 | @xref{List Answer 3, 3}. (@bullet{}) | |
4149 | ||
4150 | @ifinfo | |
4151 | As another example, a theorem about binomial coefficients tells | |
4152 | us that the alternating sum of binomial coefficients | |
4153 | @var{n}-choose-0 minus @var{n}-choose-1 plus @var{n}-choose-2, and so | |
4154 | on up to @var{n}-choose-@var{n}, | |
4155 | always comes out to zero. Let's verify this | |
4156 | for @cite{n=6}.@refill | |
4157 | @end ifinfo | |
4158 | @tex | |
4159 | As another example, a theorem about binomial coefficients tells | |
4160 | us that the alternating sum of binomial coefficients | |
4161 | ${n \choose 0} - {n \choose 1} + {n \choose 2} - \cdots \pm {n \choose n}$ | |
4162 | always comes out to zero. Let's verify this | |
4163 | for \cite{n=6}. | |
4164 | @end tex | |
4165 | ||
d7b8e6c6 | 4166 | @smallexample |
5d67986c | 4167 | @group |
d7b8e6c6 EZ |
4168 | 1: [1, 2, 3, 4, 5, 6, 7] 1: [0, 1, 2, 3, 4, 5, 6] |
4169 | . . | |
4170 | ||
5d67986c | 4171 | v x 7 @key{RET} 1 - |
d7b8e6c6 | 4172 | |
d7b8e6c6 | 4173 | @end group |
5d67986c | 4174 | @end smallexample |
d7b8e6c6 | 4175 | @noindent |
d7b8e6c6 | 4176 | @smallexample |
5d67986c | 4177 | @group |
d7b8e6c6 EZ |
4178 | 1: [1, -6, 15, -20, 15, -6, 1] 1: 0 |
4179 | . . | |
4180 | ||
5d67986c | 4181 | V M ' (-1)^$ choose(6,$) @key{RET} V R + |
d7b8e6c6 | 4182 | @end group |
5d67986c | 4183 | @end smallexample |
d7b8e6c6 EZ |
4184 | |
4185 | The @kbd{V M '} command prompts you to enter any algebraic expression | |
4186 | to define the function to map over the vector. The symbol @samp{$} | |
4187 | inside this expression represents the argument to the function. | |
4188 | The Calculator applies this formula to each element of the vector, | |
4189 | substituting each element's value for the @samp{$} sign(s) in turn. | |
4190 | ||
4191 | To define a two-argument function, use @samp{$$} for the first | |
5d67986c | 4192 | argument and @samp{$} for the second: @kbd{V M ' $$-$ @key{RET}} is |
d7b8e6c6 EZ |
4193 | equivalent to @kbd{V M -}. This is analogous to regular algebraic |
4194 | entry, where @samp{$$} would refer to the next-to-top stack entry | |
5d67986c | 4195 | and @samp{$} would refer to the top stack entry, and @kbd{' $$-$ @key{RET}} |
d7b8e6c6 EZ |
4196 | would act exactly like @kbd{-}. |
4197 | ||
4198 | Notice that the @kbd{V M '} command has recorded two things in the | |
4199 | trail: The result, as usual, and also a funny-looking thing marked | |
4200 | @samp{oper} that represents the operator function you typed in. | |
4201 | The function is enclosed in @samp{< >} brackets, and the argument is | |
4202 | denoted by a @samp{#} sign. If there were several arguments, they | |
4203 | would be shown as @samp{#1}, @samp{#2}, and so on. (For example, | |
4204 | @kbd{V M ' $$-$} will put the function @samp{<#1 - #2>} on the | |
4205 | trail.) This object is a ``nameless function''; you can use nameless | |
4206 | @w{@samp{< >}} notation to answer the @kbd{V M '} prompt if you like. | |
4207 | Nameless function notation has the interesting, occasionally useful | |
4208 | property that a nameless function is not actually evaluated until | |
4209 | it is used. For example, @kbd{V M ' $+random(2.0)} evaluates | |
4210 | @samp{random(2.0)} once and adds that random number to all elements | |
4211 | of the vector, but @kbd{V M ' <#+random(2.0)>} evaluates the | |
4212 | @samp{random(2.0)} separately for each vector element. | |
4213 | ||
4214 | Another group of operators that are often useful with @kbd{V M} are | |
4215 | the relational operators: @kbd{a =}, for example, compares two numbers | |
4216 | and gives the result 1 if they are equal, or 0 if not. Similarly, | |
4217 | @w{@kbd{a <}} checks for one number being less than another. | |
4218 | ||
4219 | Other useful vector operations include @kbd{v v}, to reverse a | |
4220 | vector end-for-end; @kbd{V S}, to sort the elements of a vector | |
4221 | into increasing order; and @kbd{v r} and @w{@kbd{v c}}, to extract | |
4222 | one row or column of a matrix, or (in both cases) to extract one | |
4223 | element of a plain vector. With a negative argument, @kbd{v r} | |
4224 | and @kbd{v c} instead delete one row, column, or vector element. | |
4225 | ||
4226 | @cindex Divisor functions | |
4227 | (@bullet{}) @strong{Exercise 4.} The @cite{k}th @dfn{divisor function} | |
4228 | @tex | |
4229 | $\sigma_k(n)$ | |
4230 | @end tex | |
4231 | is the sum of the @cite{k}th powers of all the divisors of an | |
4232 | integer @cite{n}. Figure out a method for computing the divisor | |
4233 | function for reasonably small values of @cite{n}. As a test, | |
4234 | the 0th and 1st divisor functions of 30 are 8 and 72, respectively. | |
4235 | @xref{List Answer 4, 4}. (@bullet{}) | |
4236 | ||
4237 | @cindex Square-free numbers | |
4238 | @cindex Duplicate values in a list | |
4239 | (@bullet{}) @strong{Exercise 5.} The @kbd{k f} command produces a | |
4240 | list of prime factors for a number. Sometimes it is important to | |
4241 | know that a number is @dfn{square-free}, i.e., that no prime occurs | |
4242 | more than once in its list of prime factors. Find a sequence of | |
4243 | keystrokes to tell if a number is square-free; your method should | |
4244 | leave 1 on the stack if it is, or 0 if it isn't. | |
4245 | @xref{List Answer 5, 5}. (@bullet{}) | |
4246 | ||
4247 | @cindex Triangular lists | |
4248 | (@bullet{}) @strong{Exercise 6.} Build a list of lists that looks | |
4249 | like the following diagram. (You may wish to use the @kbd{v /} | |
4250 | command to enable multi-line display of vectors.) | |
4251 | ||
d7b8e6c6 | 4252 | @smallexample |
5d67986c | 4253 | @group |
d7b8e6c6 EZ |
4254 | 1: [ [1], |
4255 | [1, 2], | |
4256 | [1, 2, 3], | |
4257 | [1, 2, 3, 4], | |
4258 | [1, 2, 3, 4, 5], | |
4259 | [1, 2, 3, 4, 5, 6] ] | |
d7b8e6c6 | 4260 | @end group |
5d67986c | 4261 | @end smallexample |
d7b8e6c6 EZ |
4262 | |
4263 | @noindent | |
4264 | @xref{List Answer 6, 6}. (@bullet{}) | |
4265 | ||
4266 | (@bullet{}) @strong{Exercise 7.} Build the following list of lists. | |
4267 | ||
d7b8e6c6 | 4268 | @smallexample |
5d67986c | 4269 | @group |
d7b8e6c6 EZ |
4270 | 1: [ [0], |
4271 | [1, 2], | |
4272 | [3, 4, 5], | |
4273 | [6, 7, 8, 9], | |
4274 | [10, 11, 12, 13, 14], | |
4275 | [15, 16, 17, 18, 19, 20] ] | |
d7b8e6c6 | 4276 | @end group |
5d67986c | 4277 | @end smallexample |
d7b8e6c6 EZ |
4278 | |
4279 | @noindent | |
4280 | @xref{List Answer 7, 7}. (@bullet{}) | |
4281 | ||
4282 | @cindex Maximizing a function over a list of values | |
4283 | @c [fix-ref Numerical Solutions] | |
4284 | (@bullet{}) @strong{Exercise 8.} Compute a list of values of Bessel's | |
4285 | @c{$J_1(x)$} | |
4286 | @cite{J1} function @samp{besJ(1,x)} for @cite{x} from 0 to 5 | |
4287 | in steps of 0.25. | |
4288 | Find the value of @cite{x} (from among the above set of values) for | |
4289 | which @samp{besJ(1,x)} is a maximum. Use an ``automatic'' method, | |
4290 | i.e., just reading along the list by hand to find the largest value | |
4291 | is not allowed! (There is an @kbd{a X} command which does this kind | |
4292 | of thing automatically; @pxref{Numerical Solutions}.) | |
4293 | @xref{List Answer 8, 8}. (@bullet{})@refill | |
4294 | ||
4295 | @cindex Digits, vectors of | |
4296 | (@bullet{}) @strong{Exercise 9.} You are given an integer in the range | |
4297 | @c{$0 \le N < 10^m$} | |
4298 | @cite{0 <= N < 10^m} for @cite{m=12} (i.e., an integer of less than | |
4299 | twelve digits). Convert this integer into a vector of @cite{m} | |
4300 | digits, each in the range from 0 to 9. In vector-of-digits notation, | |
4301 | add one to this integer to produce a vector of @cite{m+1} digits | |
4302 | (since there could be a carry out of the most significant digit). | |
4303 | Convert this vector back into a regular integer. A good integer | |
4304 | to try is 25129925999. @xref{List Answer 9, 9}. (@bullet{}) | |
4305 | ||
4306 | (@bullet{}) @strong{Exercise 10.} Your friend Joe tried to use | |
4307 | @kbd{V R a =} to test if all numbers in a list were equal. What | |
4308 | happened? How would you do this test? @xref{List Answer 10, 10}. (@bullet{}) | |
4309 | ||
4310 | (@bullet{}) @strong{Exercise 11.} The area of a circle of radius one | |
4311 | is @c{$\pi$} | |
4312 | @cite{pi}. The area of the @c{$2\times2$} | |
4313 | @asis{2x2} square that encloses that | |
5d67986c | 4314 | circle is 4. So if we throw @var{n} darts at random points in the square, |
d7b8e6c6 EZ |
4315 | about @c{$\pi/4$} |
4316 | @cite{pi/4} of them will land inside the circle. This gives us | |
4317 | an entertaining way to estimate the value of @c{$\pi$} | |
4318 | @cite{pi}. The @w{@kbd{k r}} | |
4319 | command picks a random number between zero and the value on the stack. | |
4320 | We could get a random floating-point number between @i{-1} and 1 by typing | |
4321 | @w{@kbd{2.0 k r 1 -}}. Build a vector of 100 random @cite{(x,y)} points in | |
4322 | this square, then use vector mapping and reduction to count how many | |
4323 | points lie inside the unit circle. Hint: Use the @kbd{v b} command. | |
4324 | @xref{List Answer 11, 11}. (@bullet{}) | |
4325 | ||
4326 | @cindex Matchstick problem | |
4327 | (@bullet{}) @strong{Exercise 12.} The @dfn{matchstick problem} provides | |
4328 | another way to calculate @c{$\pi$} | |
4329 | @cite{pi}. Say you have an infinite field | |
4330 | of vertical lines with a spacing of one inch. Toss a one-inch matchstick | |
4331 | onto the field. The probability that the matchstick will land crossing | |
4332 | a line turns out to be @c{$2/\pi$} | |
4333 | @cite{2/pi}. Toss 100 matchsticks to estimate | |
4334 | @c{$\pi$} | |
4335 | @cite{pi}. (If you want still more fun, the probability that the GCD | |
4336 | (@w{@kbd{k g}}) of two large integers is one turns out to be @c{$6/\pi^2$} | |
4337 | @cite{6/pi^2}. | |
4338 | That provides yet another way to estimate @c{$\pi$} | |
4339 | @cite{pi}.) | |
4340 | @xref{List Answer 12, 12}. (@bullet{}) | |
4341 | ||
4342 | (@bullet{}) @strong{Exercise 13.} An algebraic entry of a string in | |
4343 | double-quote marks, @samp{"hello"}, creates a vector of the numerical | |
4344 | (ASCII) codes of the characters (here, @cite{[104, 101, 108, 108, 111]}). | |
4345 | Sometimes it is convenient to compute a @dfn{hash code} of a string, | |
4346 | which is just an integer that represents the value of that string. | |
4347 | Two equal strings have the same hash code; two different strings | |
4348 | @dfn{probably} have different hash codes. (For example, Calc has | |
4349 | over 400 function names, but Emacs can quickly find the definition for | |
4350 | any given name because it has sorted the functions into ``buckets'' by | |
4351 | their hash codes. Sometimes a few names will hash into the same bucket, | |
4352 | but it is easier to search among a few names than among all the names.) | |
4353 | One popular hash function is computed as follows: First set @cite{h = 0}. | |
4354 | Then, for each character from the string in turn, set @cite{h = 3h + c_i} | |
4355 | where @cite{c_i} is the character's ASCII code. If we have 511 buckets, | |
4356 | we then take the hash code modulo 511 to get the bucket number. Develop a | |
4357 | simple command or commands for converting string vectors into hash codes. | |
4358 | The hash code for @samp{"Testing, 1, 2, 3"} is 1960915098, which modulo | |
4359 | 511 is 121. @xref{List Answer 13, 13}. (@bullet{}) | |
4360 | ||
4361 | (@bullet{}) @strong{Exercise 14.} The @kbd{H V R} and @kbd{H V U} | |
4362 | commands do nested function evaluations. @kbd{H V U} takes a starting | |
4363 | value and a number of steps @var{n} from the stack; it then applies the | |
4364 | function you give to the starting value 0, 1, 2, up to @var{n} times | |
4365 | and returns a vector of the results. Use this command to create a | |
4366 | ``random walk'' of 50 steps. Start with the two-dimensional point | |
4367 | @cite{(0,0)}; then take one step a random distance between @i{-1} and 1 | |
4368 | in both @cite{x} and @cite{y}; then take another step, and so on. Use the | |
4369 | @kbd{g f} command to display this random walk. Now modify your random | |
4370 | walk to walk a unit distance, but in a random direction, at each step. | |
4371 | (Hint: The @code{sincos} function returns a vector of the cosine and | |
4372 | sine of an angle.) @xref{List Answer 14, 14}. (@bullet{}) | |
4373 | ||
4374 | @node Types Tutorial, Algebra Tutorial, Vector/Matrix Tutorial, Tutorial | |
4375 | @section Types Tutorial | |
4376 | ||
4377 | @noindent | |
4378 | Calc understands a variety of data types as well as simple numbers. | |
4379 | In this section, we'll experiment with each of these types in turn. | |
4380 | ||
4381 | The numbers we've been using so far have mainly been either @dfn{integers} | |
4382 | or @dfn{floats}. We saw that floats are usually a good approximation to | |
4383 | the mathematical concept of real numbers, but they are only approximations | |
4384 | and are susceptible to roundoff error. Calc also supports @dfn{fractions}, | |
4385 | which can exactly represent any rational number. | |
4386 | ||
d7b8e6c6 | 4387 | @smallexample |
5d67986c | 4388 | @group |
d7b8e6c6 EZ |
4389 | 1: 3628800 2: 3628800 1: 518400:7 1: 518414:7 1: 7:518414 |
4390 | . 1: 49 . . . | |
4391 | . | |
4392 | ||
5d67986c | 4393 | 10 ! 49 @key{RET} : 2 + & |
d7b8e6c6 | 4394 | @end group |
5d67986c | 4395 | @end smallexample |
d7b8e6c6 EZ |
4396 | |
4397 | @noindent | |
4398 | The @kbd{:} command divides two integers to get a fraction; @kbd{/} | |
4399 | would normally divide integers to get a floating-point result. | |
4400 | Notice we had to type @key{RET} between the @kbd{49} and the @kbd{:} | |
4401 | since the @kbd{:} would otherwise be interpreted as part of a | |
4402 | fraction beginning with 49. | |
4403 | ||
4404 | You can convert between floating-point and fractional format using | |
4405 | @kbd{c f} and @kbd{c F}: | |
4406 | ||
d7b8e6c6 | 4407 | @smallexample |
5d67986c | 4408 | @group |
d7b8e6c6 EZ |
4409 | 1: 1.35027217629e-5 1: 7:518414 |
4410 | . . | |
4411 | ||
4412 | c f c F | |
d7b8e6c6 | 4413 | @end group |
5d67986c | 4414 | @end smallexample |
d7b8e6c6 EZ |
4415 | |
4416 | The @kbd{c F} command replaces a floating-point number with the | |
4417 | ``simplest'' fraction whose floating-point representation is the | |
4418 | same, to within the current precision. | |
4419 | ||
d7b8e6c6 | 4420 | @smallexample |
5d67986c | 4421 | @group |
d7b8e6c6 EZ |
4422 | 1: 3.14159265359 1: 1146408:364913 1: 3.1416 1: 355:113 |
4423 | . . . . | |
4424 | ||
5d67986c | 4425 | P c F @key{DEL} p 5 @key{RET} P c F |
d7b8e6c6 | 4426 | @end group |
5d67986c | 4427 | @end smallexample |
d7b8e6c6 EZ |
4428 | |
4429 | (@bullet{}) @strong{Exercise 1.} A calculation has produced the | |
4430 | result 1.26508260337. You suspect it is the square root of the | |
4431 | product of @c{$\pi$} | |
4432 | @cite{pi} and some rational number. Is it? (Be sure | |
4433 | to allow for roundoff error!) @xref{Types Answer 1, 1}. (@bullet{}) | |
4434 | ||
4435 | @dfn{Complex numbers} can be stored in both rectangular and polar form. | |
4436 | ||
d7b8e6c6 | 4437 | @smallexample |
5d67986c | 4438 | @group |
d7b8e6c6 EZ |
4439 | 1: -9 1: (0, 3) 1: (3; 90.) 1: (6; 90.) 1: (2.4495; 45.) |
4440 | . . . . . | |
4441 | ||
4442 | 9 n Q c p 2 * Q | |
d7b8e6c6 | 4443 | @end group |
5d67986c | 4444 | @end smallexample |
d7b8e6c6 EZ |
4445 | |
4446 | @noindent | |
4447 | The square root of @i{-9} is by default rendered in rectangular form | |
4448 | (@w{@cite{0 + 3i}}), but we can convert it to polar form (3 with a | |
4449 | phase angle of 90 degrees). All the usual arithmetic and scientific | |
4450 | operations are defined on both types of complex numbers. | |
4451 | ||
4452 | Another generalized kind of number is @dfn{infinity}. Infinity | |
4453 | isn't really a number, but it can sometimes be treated like one. | |
4454 | Calc uses the symbol @code{inf} to represent positive infinity, | |
4455 | i.e., a value greater than any real number. Naturally, you can | |
4456 | also write @samp{-inf} for minus infinity, a value less than any | |
4457 | real number. The word @code{inf} can only be input using | |
4458 | algebraic entry. | |
4459 | ||
d7b8e6c6 | 4460 | @smallexample |
5d67986c | 4461 | @group |
d7b8e6c6 EZ |
4462 | 2: inf 2: -inf 2: -inf 2: -inf 1: nan |
4463 | 1: -17 1: -inf 1: -inf 1: inf . | |
4464 | . . . . | |
4465 | ||
5d67986c | 4466 | ' inf @key{RET} 17 n * @key{RET} 72 + A + |
d7b8e6c6 | 4467 | @end group |
5d67986c | 4468 | @end smallexample |
d7b8e6c6 EZ |
4469 | |
4470 | @noindent | |
4471 | Since infinity is infinitely large, multiplying it by any finite | |
4472 | number (like @i{-17}) has no effect, except that since @i{-17} | |
4473 | is negative, it changes a plus infinity to a minus infinity. | |
4474 | (``A huge positive number, multiplied by @i{-17}, yields a huge | |
4475 | negative number.'') Adding any finite number to infinity also | |
4476 | leaves it unchanged. Taking an absolute value gives us plus | |
4477 | infinity again. Finally, we add this plus infinity to the minus | |
4478 | infinity we had earlier. If you work it out, you might expect | |
4479 | the answer to be @i{-72} for this. But the 72 has been completely | |
4480 | lost next to the infinities; by the time we compute @w{@samp{inf - inf}} | |
28665d46 | 4481 | the finite difference between them, if any, is undetectable. |
d7b8e6c6 EZ |
4482 | So we say the result is @dfn{indeterminate}, which Calc writes |
4483 | with the symbol @code{nan} (for Not A Number). | |
4484 | ||
4485 | Dividing by zero is normally treated as an error, but you can get | |
4486 | Calc to write an answer in terms of infinity by pressing @kbd{m i} | |
4487 | to turn on ``infinite mode.'' | |
4488 | ||
d7b8e6c6 | 4489 | @smallexample |
5d67986c | 4490 | @group |
d7b8e6c6 EZ |
4491 | 3: nan 2: nan 2: nan 2: nan 1: nan |
4492 | 2: 1 1: 1 / 0 1: uinf 1: uinf . | |
4493 | 1: 0 . . . | |
4494 | . | |
4495 | ||
5d67986c | 4496 | 1 @key{RET} 0 / m i U / 17 n * + |
d7b8e6c6 | 4497 | @end group |
5d67986c | 4498 | @end smallexample |
d7b8e6c6 EZ |
4499 | |
4500 | @noindent | |
4501 | Dividing by zero normally is left unevaluated, but after @kbd{m i} | |
4502 | it instead gives an infinite result. The answer is actually | |
4503 | @code{uinf}, ``undirected infinity.'' If you look at a graph of | |
4504 | @cite{1 / x} around @w{@cite{x = 0}}, you'll see that it goes toward | |
4505 | plus infinity as you approach zero from above, but toward minus | |
4506 | infinity as you approach from below. Since we said only @cite{1 / 0}, | |
4507 | Calc knows that the answer is infinite but not in which direction. | |
4508 | That's what @code{uinf} means. Notice that multiplying @code{uinf} | |
4509 | by a negative number still leaves plain @code{uinf}; there's no | |
4510 | point in saying @samp{-uinf} because the sign of @code{uinf} is | |
4511 | unknown anyway. Finally, we add @code{uinf} to our @code{nan}, | |
4512 | yielding @code{nan} again. It's easy to see that, because | |
4513 | @code{nan} means ``totally unknown'' while @code{uinf} means | |
4514 | ``unknown sign but known to be infinite,'' the more mysterious | |
4515 | @code{nan} wins out when it is combined with @code{uinf}, or, for | |
4516 | that matter, with anything else. | |
4517 | ||
4518 | (@bullet{}) @strong{Exercise 2.} Predict what Calc will answer | |
4519 | for each of these formulas: @samp{inf / inf}, @samp{exp(inf)}, | |
4520 | @samp{exp(-inf)}, @samp{sqrt(-inf)}, @samp{sqrt(uinf)}, | |
4521 | @samp{abs(uinf)}, @samp{ln(0)}. | |
4522 | @xref{Types Answer 2, 2}. (@bullet{}) | |
4523 | ||
4524 | (@bullet{}) @strong{Exercise 3.} We saw that @samp{inf - inf = nan}, | |
4525 | which stands for an unknown value. Can @code{nan} stand for | |
4526 | a complex number? Can it stand for infinity? | |
4527 | @xref{Types Answer 3, 3}. (@bullet{}) | |
4528 | ||
4529 | @dfn{HMS forms} represent a value in terms of hours, minutes, and | |
4530 | seconds. | |
4531 | ||
d7b8e6c6 | 4532 | @smallexample |
5d67986c | 4533 | @group |
d7b8e6c6 EZ |
4534 | 1: 2@@ 30' 0" 1: 3@@ 30' 0" 2: 3@@ 30' 0" 1: 2. |
4535 | . . 1: 1@@ 45' 0." . | |
4536 | . | |
4537 | ||
5d67986c | 4538 | 2@@ 30' @key{RET} 1 + @key{RET} 2 / / |
d7b8e6c6 | 4539 | @end group |
5d67986c | 4540 | @end smallexample |
d7b8e6c6 EZ |
4541 | |
4542 | HMS forms can also be used to hold angles in degrees, minutes, and | |
4543 | seconds. | |
4544 | ||
d7b8e6c6 | 4545 | @smallexample |
5d67986c | 4546 | @group |
d7b8e6c6 EZ |
4547 | 1: 0.5 1: 26.56505 1: 26@@ 33' 54.18" 1: 0.44721 |
4548 | . . . . | |
4549 | ||
4550 | 0.5 I T c h S | |
d7b8e6c6 | 4551 | @end group |
5d67986c | 4552 | @end smallexample |
d7b8e6c6 EZ |
4553 | |
4554 | @noindent | |
4555 | First we convert the inverse tangent of 0.5 to degrees-minutes-seconds | |
4556 | form, then we take the sine of that angle. Note that the trigonometric | |
4557 | functions will accept HMS forms directly as input. | |
4558 | ||
4559 | @cindex Beatles | |
4560 | (@bullet{}) @strong{Exercise 4.} The Beatles' @emph{Abbey Road} is | |
4561 | 47 minutes and 26 seconds long, and contains 17 songs. What is the | |
4562 | average length of a song on @emph{Abbey Road}? If the Extended Disco | |
4563 | Version of @emph{Abbey Road} added 20 seconds to the length of each | |
4564 | song, how long would the album be? @xref{Types Answer 4, 4}. (@bullet{}) | |
4565 | ||
4566 | A @dfn{date form} represents a date, or a date and time. Dates must | |
4567 | be entered using algebraic entry. Date forms are surrounded by | |
4568 | @samp{< >} symbols; most standard formats for dates are recognized. | |
4569 | ||
d7b8e6c6 | 4570 | @smallexample |
5d67986c | 4571 | @group |
d7b8e6c6 EZ |
4572 | 2: <Sun Jan 13, 1991> 1: 2.25 |
4573 | 1: <6:00pm Thu Jan 10, 1991> . | |
4574 | . | |
4575 | ||
5d67986c | 4576 | ' <13 Jan 1991>, <1/10/91, 6pm> @key{RET} - |
d7b8e6c6 | 4577 | @end group |
5d67986c | 4578 | @end smallexample |
d7b8e6c6 EZ |
4579 | |
4580 | @noindent | |
4581 | In this example, we enter two dates, then subtract to find the | |
4582 | number of days between them. It is also possible to add an | |
4583 | HMS form or a number (of days) to a date form to get another | |
4584 | date form. | |
4585 | ||
d7b8e6c6 | 4586 | @smallexample |
5d67986c | 4587 | @group |
d7b8e6c6 EZ |
4588 | 1: <4:45:59pm Mon Jan 14, 1991> 1: <2:50:59am Thu Jan 17, 1991> |
4589 | . . | |
4590 | ||
4591 | t N 2 + 10@@ 5' + | |
d7b8e6c6 | 4592 | @end group |
5d67986c | 4593 | @end smallexample |
d7b8e6c6 EZ |
4594 | |
4595 | @c [fix-ref Date Arithmetic] | |
4596 | @noindent | |
4597 | The @kbd{t N} (``now'') command pushes the current date and time on the | |
4598 | stack; then we add two days, ten hours and five minutes to the date and | |
4599 | time. Other date-and-time related commands include @kbd{t J}, which | |
4600 | does Julian day conversions, @kbd{t W}, which finds the beginning of | |
4601 | the week in which a date form lies, and @kbd{t I}, which increments a | |
4602 | date by one or several months. @xref{Date Arithmetic}, for more. | |
4603 | ||
4604 | (@bullet{}) @strong{Exercise 5.} How many days until the next | |
4605 | Friday the 13th? @xref{Types Answer 5, 5}. (@bullet{}) | |
4606 | ||
4607 | (@bullet{}) @strong{Exercise 6.} How many leap years will there be | |
4608 | between now and the year 10001 A.D.? @xref{Types Answer 6, 6}. (@bullet{}) | |
4609 | ||
4610 | @cindex Slope and angle of a line | |
4611 | @cindex Angle and slope of a line | |
4612 | An @dfn{error form} represents a mean value with an attached standard | |
4613 | deviation, or error estimate. Suppose our measurements indicate that | |
4614 | a certain telephone pole is about 30 meters away, with an estimated | |
4615 | error of 1 meter, and 8 meters tall, with an estimated error of 0.2 | |
4616 | meters. What is the slope of a line from here to the top of the | |
4617 | pole, and what is the equivalent angle in degrees? | |
4618 | ||
d7b8e6c6 | 4619 | @smallexample |
5d67986c | 4620 | @group |
d7b8e6c6 EZ |
4621 | 1: 8 +/- 0.2 2: 8 +/- 0.2 1: 0.266 +/- 0.011 1: 14.93 +/- 0.594 |
4622 | . 1: 30 +/- 1 . . | |
4623 | . | |
4624 | ||
5d67986c | 4625 | 8 p .2 @key{RET} 30 p 1 / I T |
d7b8e6c6 | 4626 | @end group |
5d67986c | 4627 | @end smallexample |
d7b8e6c6 EZ |
4628 | |
4629 | @noindent | |
4630 | This means that the angle is about 15 degrees, and, assuming our | |
4631 | original error estimates were valid standard deviations, there is about | |
4632 | a 60% chance that the result is correct within 0.59 degrees. | |
4633 | ||
4634 | @cindex Torus, volume of | |
4635 | (@bullet{}) @strong{Exercise 7.} The volume of a torus (a donut shape) is | |
4636 | @c{$2 \pi^2 R r^2$} | |
4637 | @w{@cite{2 pi^2 R r^2}} where @cite{R} is the radius of the circle that | |
4638 | defines the center of the tube and @cite{r} is the radius of the tube | |
4639 | itself. Suppose @cite{R} is 20 cm and @cite{r} is 4 cm, each known to | |
4640 | within 5 percent. What is the volume and the relative uncertainty of | |
4641 | the volume? @xref{Types Answer 7, 7}. (@bullet{}) | |
4642 | ||
4643 | An @dfn{interval form} represents a range of values. While an | |
4644 | error form is best for making statistical estimates, intervals give | |
4645 | you exact bounds on an answer. Suppose we additionally know that | |
4646 | our telephone pole is definitely between 28 and 31 meters away, | |
4647 | and that it is between 7.7 and 8.1 meters tall. | |
4648 | ||
d7b8e6c6 | 4649 | @smallexample |
5d67986c | 4650 | @group |
d7b8e6c6 EZ |
4651 | 1: [7.7 .. 8.1] 2: [7.7 .. 8.1] 1: [0.24 .. 0.28] 1: [13.9 .. 16.1] |
4652 | . 1: [28 .. 31] . . | |
4653 | . | |
4654 | ||
4655 | [ 7.7 .. 8.1 ] [ 28 .. 31 ] / I T | |
d7b8e6c6 | 4656 | @end group |
5d67986c | 4657 | @end smallexample |
d7b8e6c6 EZ |
4658 | |
4659 | @noindent | |
4660 | If our bounds were correct, then the angle to the top of the pole | |
4661 | is sure to lie in the range shown. | |
4662 | ||
4663 | The square brackets around these intervals indicate that the endpoints | |
4664 | themselves are allowable values. In other words, the distance to the | |
4665 | telephone pole is between 28 and 31, @emph{inclusive}. You can also | |
4666 | make an interval that is exclusive of its endpoints by writing | |
4667 | parentheses instead of square brackets. You can even make an interval | |
4668 | which is inclusive (``closed'') on one end and exclusive (``open'') on | |
4669 | the other. | |
4670 | ||
d7b8e6c6 | 4671 | @smallexample |
5d67986c | 4672 | @group |
d7b8e6c6 EZ |
4673 | 1: [1 .. 10) 1: (0.1 .. 1] 2: (0.1 .. 1] 1: (0.2 .. 3) |
4674 | . . 1: [2 .. 3) . | |
4675 | . | |
4676 | ||
4677 | [ 1 .. 10 ) & [ 2 .. 3 ) * | |
d7b8e6c6 | 4678 | @end group |
5d67986c | 4679 | @end smallexample |
d7b8e6c6 EZ |
4680 | |
4681 | @noindent | |
4682 | The Calculator automatically keeps track of which end values should | |
4683 | be open and which should be closed. You can also make infinite or | |
4684 | semi-infinite intervals by using @samp{-inf} or @samp{inf} for one | |
4685 | or both endpoints. | |
4686 | ||
4687 | (@bullet{}) @strong{Exercise 8.} What answer would you expect from | |
4688 | @samp{@w{1 /} @w{(0 .. 10)}}? What about @samp{@w{1 /} @w{(-10 .. 0)}}? What | |
4689 | about @samp{@w{1 /} @w{[0 .. 10]}} (where the interval actually includes | |
4690 | zero)? What about @samp{@w{1 /} @w{(-10 .. 10)}}? | |
4691 | @xref{Types Answer 8, 8}. (@bullet{}) | |
4692 | ||
4693 | (@bullet{}) @strong{Exercise 9.} Two easy ways of squaring a number | |
5d67986c | 4694 | are @kbd{@key{RET} *} and @w{@kbd{2 ^}}. Normally these produce the same |
d7b8e6c6 EZ |
4695 | answer. Would you expect this still to hold true for interval forms? |
4696 | If not, which of these will result in a larger interval? | |
4697 | @xref{Types Answer 9, 9}. (@bullet{}) | |
4698 | ||
5d67986c | 4699 | A @dfn{modulo form} is used for performing arithmetic modulo @var{m}. |
d7b8e6c6 EZ |
4700 | For example, arithmetic involving time is generally done modulo 12 |
4701 | or 24 hours. | |
4702 | ||
d7b8e6c6 | 4703 | @smallexample |
5d67986c | 4704 | @group |
d7b8e6c6 EZ |
4705 | 1: 17 mod 24 1: 3 mod 24 1: 21 mod 24 1: 9 mod 24 |
4706 | . . . . | |
4707 | ||
5d67986c | 4708 | 17 M 24 @key{RET} 10 + n 5 / |
d7b8e6c6 | 4709 | @end group |
5d67986c | 4710 | @end smallexample |
d7b8e6c6 EZ |
4711 | |
4712 | @noindent | |
4713 | In this last step, Calc has found a new number which, when multiplied | |
5d67986c RS |
4714 | by 5 modulo 24, produces the original number, 21. If @var{m} is prime |
4715 | it is always possible to find such a number. For non-prime @var{m} | |
d7b8e6c6 EZ |
4716 | like 24, it is only sometimes possible. |
4717 | ||
d7b8e6c6 | 4718 | @smallexample |
5d67986c | 4719 | @group |
d7b8e6c6 EZ |
4720 | 1: 10 mod 24 1: 16 mod 24 1: 1000000... 1: 16 |
4721 | . . . . | |
4722 | ||
5d67986c | 4723 | 10 M 24 @key{RET} 100 ^ 10 @key{RET} 100 ^ 24 % |
d7b8e6c6 | 4724 | @end group |
5d67986c | 4725 | @end smallexample |
d7b8e6c6 EZ |
4726 | |
4727 | @noindent | |
4728 | These two calculations get the same answer, but the first one is | |
4729 | much more efficient because it avoids the huge intermediate value | |
4730 | that arises in the second one. | |
4731 | ||
4732 | @cindex Fermat, primality test of | |
4733 | (@bullet{}) @strong{Exercise 10.} A theorem of Pierre de Fermat | |
4734 | says that @c{\w{$x^{n-1} \bmod n = 1$}} | |
4735 | @cite{x^(n-1) mod n = 1} if @cite{n} is a prime number | |
4736 | and @cite{x} is an integer less than @cite{n}. If @cite{n} is | |
4737 | @emph{not} a prime number, this will @emph{not} be true for most | |
4738 | values of @cite{x}. Thus we can test informally if a number is | |
4739 | prime by trying this formula for several values of @cite{x}. | |
4740 | Use this test to tell whether the following numbers are prime: | |
4741 | 811749613, 15485863. @xref{Types Answer 10, 10}. (@bullet{}) | |
4742 | ||
4743 | It is possible to use HMS forms as parts of error forms, intervals, | |
4744 | modulo forms, or as the phase part of a polar complex number. | |
4745 | For example, the @code{calc-time} command pushes the current time | |
4746 | of day on the stack as an HMS/modulo form. | |
4747 | ||
d7b8e6c6 | 4748 | @smallexample |
5d67986c | 4749 | @group |
d7b8e6c6 EZ |
4750 | 1: 17@@ 34' 45" mod 24@@ 0' 0" 1: 6@@ 22' 15" mod 24@@ 0' 0" |
4751 | . . | |
4752 | ||
5d67986c | 4753 | x time @key{RET} n |
d7b8e6c6 | 4754 | @end group |
5d67986c | 4755 | @end smallexample |
d7b8e6c6 EZ |
4756 | |
4757 | @noindent | |
4758 | This calculation tells me it is six hours and 22 minutes until midnight. | |
4759 | ||
4760 | (@bullet{}) @strong{Exercise 11.} A rule of thumb is that one year | |
4761 | is about @c{$\pi \times 10^7$} | |
4762 | @w{@cite{pi * 10^7}} seconds. What time will it be that | |
4763 | many seconds from right now? @xref{Types Answer 11, 11}. (@bullet{}) | |
4764 | ||
4765 | (@bullet{}) @strong{Exercise 12.} You are preparing to order packaging | |
4766 | for the CD release of the Extended Disco Version of @emph{Abbey Road}. | |
4767 | You are told that the songs will actually be anywhere from 20 to 60 | |
4768 | seconds longer than the originals. One CD can hold about 75 minutes | |
4769 | of music. Should you order single or double packages? | |
4770 | @xref{Types Answer 12, 12}. (@bullet{}) | |
4771 | ||
4772 | Another kind of data the Calculator can manipulate is numbers with | |
4773 | @dfn{units}. This isn't strictly a new data type; it's simply an | |
4774 | application of algebraic expressions, where we use variables with | |
4775 | suggestive names like @samp{cm} and @samp{in} to represent units | |
4776 | like centimeters and inches. | |
4777 | ||
d7b8e6c6 | 4778 | @smallexample |
5d67986c | 4779 | @group |
d7b8e6c6 EZ |
4780 | 1: 2 in 1: 5.08 cm 1: 0.027778 fath 1: 0.0508 m |
4781 | . . . . | |
4782 | ||
5d67986c | 4783 | ' 2in @key{RET} u c cm @key{RET} u c fath @key{RET} u b |
d7b8e6c6 | 4784 | @end group |
5d67986c | 4785 | @end smallexample |
d7b8e6c6 EZ |
4786 | |
4787 | @noindent | |
4788 | We enter the quantity ``2 inches'' (actually an algebraic expression | |
4789 | which means two times the variable @samp{in}), then we convert it | |
4790 | first to centimeters, then to fathoms, then finally to ``base'' units, | |
4791 | which in this case means meters. | |
4792 | ||
d7b8e6c6 | 4793 | @smallexample |
5d67986c | 4794 | @group |
d7b8e6c6 EZ |
4795 | 1: 9 acre 1: 3 sqrt(acre) 1: 190.84 m 1: 190.84 m + 30 cm |
4796 | . . . . | |
4797 | ||
5d67986c | 4798 | ' 9 acre @key{RET} Q u s ' $+30 cm @key{RET} |
d7b8e6c6 | 4799 | |
d7b8e6c6 | 4800 | @end group |
5d67986c | 4801 | @end smallexample |
d7b8e6c6 | 4802 | @noindent |
d7b8e6c6 | 4803 | @smallexample |
5d67986c | 4804 | @group |
d7b8e6c6 EZ |
4805 | 1: 191.14 m 1: 36536.3046 m^2 1: 365363046 cm^2 |
4806 | . . . | |
4807 | ||
4808 | u s 2 ^ u c cgs | |
d7b8e6c6 | 4809 | @end group |
5d67986c | 4810 | @end smallexample |
d7b8e6c6 EZ |
4811 | |
4812 | @noindent | |
4813 | Since units expressions are really just formulas, taking the square | |
4814 | root of @samp{acre} is undefined. After all, @code{acre} might be an | |
4815 | algebraic variable that you will someday assign a value. We use the | |
4816 | ``units-simplify'' command to simplify the expression with variables | |
4817 | being interpreted as unit names. | |
4818 | ||
4819 | In the final step, we have converted not to a particular unit, but to a | |
4820 | units system. The ``cgs'' system uses centimeters instead of meters | |
4821 | as its standard unit of length. | |
4822 | ||
4823 | There is a wide variety of units defined in the Calculator. | |
4824 | ||
d7b8e6c6 | 4825 | @smallexample |
5d67986c | 4826 | @group |
d7b8e6c6 EZ |
4827 | 1: 55 mph 1: 88.5139 kph 1: 88.5139 km / hr 1: 8.201407e-8 c |
4828 | . . . . | |
4829 | ||
5d67986c | 4830 | ' 55 mph @key{RET} u c kph @key{RET} u c km/hr @key{RET} u c c @key{RET} |
d7b8e6c6 | 4831 | @end group |
5d67986c | 4832 | @end smallexample |
d7b8e6c6 EZ |
4833 | |
4834 | @noindent | |
4835 | We express a speed first in miles per hour, then in kilometers per | |
4836 | hour, then again using a slightly more explicit notation, then | |
4837 | finally in terms of fractions of the speed of light. | |
4838 | ||
4839 | Temperature conversions are a bit more tricky. There are two ways to | |
4840 | interpret ``20 degrees Fahrenheit''---it could mean an actual | |
4841 | temperature, or it could mean a change in temperature. For normal | |
4842 | units there is no difference, but temperature units have an offset | |
4843 | as well as a scale factor and so there must be two explicit commands | |
4844 | for them. | |
4845 | ||
d7b8e6c6 | 4846 | @smallexample |
5d67986c | 4847 | @group |
d7b8e6c6 EZ |
4848 | 1: 20 degF 1: 11.1111 degC 1: -20:3 degC 1: -6.666 degC |
4849 | . . . . | |
4850 | ||
5d67986c | 4851 | ' 20 degF @key{RET} u c degC @key{RET} U u t degC @key{RET} c f |
d7b8e6c6 | 4852 | @end group |
5d67986c | 4853 | @end smallexample |
d7b8e6c6 EZ |
4854 | |
4855 | @noindent | |
4856 | First we convert a change of 20 degrees Fahrenheit into an equivalent | |
4857 | change in degrees Celsius (or Centigrade). Then, we convert the | |
4858 | absolute temperature 20 degrees Fahrenheit into Celsius. Since | |
4859 | this comes out as an exact fraction, we then convert to floating-point | |
4860 | for easier comparison with the other result. | |
4861 | ||
4862 | For simple unit conversions, you can put a plain number on the stack. | |
4863 | Then @kbd{u c} and @kbd{u t} will prompt for both old and new units. | |
4864 | When you use this method, you're responsible for remembering which | |
4865 | numbers are in which units: | |
4866 | ||
d7b8e6c6 | 4867 | @smallexample |
5d67986c | 4868 | @group |
d7b8e6c6 EZ |
4869 | 1: 55 1: 88.5139 1: 8.201407e-8 |
4870 | . . . | |
4871 | ||
5d67986c | 4872 | 55 u c mph @key{RET} kph @key{RET} u c km/hr @key{RET} c @key{RET} |
d7b8e6c6 | 4873 | @end group |
5d67986c | 4874 | @end smallexample |
d7b8e6c6 EZ |
4875 | |
4876 | To see a complete list of built-in units, type @kbd{u v}. Press | |
4877 | @w{@kbd{M-# c}} again to re-enter the Calculator when you're done looking | |
4878 | at the units table. | |
4879 | ||
4880 | (@bullet{}) @strong{Exercise 13.} How many seconds are there really | |
4881 | in a year? @xref{Types Answer 13, 13}. (@bullet{}) | |
4882 | ||
4883 | @cindex Speed of light | |
4884 | (@bullet{}) @strong{Exercise 14.} Supercomputer designs are limited by | |
4885 | the speed of light (and of electricity, which is nearly as fast). | |
4886 | Suppose a computer has a 4.1 ns (nanosecond) clock cycle, and its | |
4887 | cabinet is one meter across. Is speed of light going to be a | |
4888 | significant factor in its design? @xref{Types Answer 14, 14}. (@bullet{}) | |
4889 | ||
4890 | (@bullet{}) @strong{Exercise 15.} Sam the Slug normally travels about | |
4891 | five yards in an hour. He has obtained a supply of Power Pills; each | |
4892 | Power Pill he eats doubles his speed. How many Power Pills can he | |
4893 | swallow and still travel legally on most US highways? | |
4894 | @xref{Types Answer 15, 15}. (@bullet{}) | |
4895 | ||
4896 | @node Algebra Tutorial, Programming Tutorial, Types Tutorial, Tutorial | |
4897 | @section Algebra and Calculus Tutorial | |
4898 | ||
4899 | @noindent | |
4900 | This section shows how to use Calc's algebra facilities to solve | |
4901 | equations, do simple calculus problems, and manipulate algebraic | |
4902 | formulas. | |
4903 | ||
4904 | @menu | |
4905 | * Basic Algebra Tutorial:: | |
4906 | * Rewrites Tutorial:: | |
4907 | @end menu | |
4908 | ||
4909 | @node Basic Algebra Tutorial, Rewrites Tutorial, Algebra Tutorial, Algebra Tutorial | |
4910 | @subsection Basic Algebra | |
4911 | ||
4912 | @noindent | |
4913 | If you enter a formula in algebraic mode that refers to variables, | |
4914 | the formula itself is pushed onto the stack. You can manipulate | |
4915 | formulas as regular data objects. | |
4916 | ||
d7b8e6c6 | 4917 | @smallexample |
5d67986c | 4918 | @group |
d7b8e6c6 EZ |
4919 | 1: 2 x^2 - 6 1: 6 - 2 x^2 1: (6 - 2 x^2) (3 x^2 + y) |
4920 | . . . | |
4921 | ||
5d67986c | 4922 | ' 2x^2-6 @key{RET} n ' 3x^2+y @key{RET} * |
d7b8e6c6 | 4923 | @end group |
5d67986c | 4924 | @end smallexample |
d7b8e6c6 | 4925 | |
5d67986c RS |
4926 | (@bullet{}) @strong{Exercise 1.} Do @kbd{' x @key{RET} Q 2 ^} and |
4927 | @kbd{' x @key{RET} 2 ^ Q} both wind up with the same result (@samp{x})? | |
d7b8e6c6 EZ |
4928 | Why or why not? @xref{Algebra Answer 1, 1}. (@bullet{}) |
4929 | ||
4930 | There are also commands for doing common algebraic operations on | |
4931 | formulas. Continuing with the formula from the last example, | |
4932 | ||
d7b8e6c6 | 4933 | @smallexample |
5d67986c | 4934 | @group |
d7b8e6c6 EZ |
4935 | 1: 18 x^2 + 6 y - 6 x^4 - 2 x^2 y 1: (18 - 2 y) x^2 - 6 x^4 + 6 y |
4936 | . . | |
4937 | ||
5d67986c | 4938 | a x a c x @key{RET} |
d7b8e6c6 | 4939 | @end group |
5d67986c | 4940 | @end smallexample |
d7b8e6c6 EZ |
4941 | |
4942 | @noindent | |
4943 | First we ``expand'' using the distributive law, then we ``collect'' | |
4944 | terms involving like powers of @cite{x}. | |
4945 | ||
4946 | Let's find the value of this expression when @cite{x} is 2 and @cite{y} | |
4947 | is one-half. | |
4948 | ||
d7b8e6c6 | 4949 | @smallexample |
5d67986c | 4950 | @group |
d7b8e6c6 EZ |
4951 | 1: 17 x^2 - 6 x^4 + 3 1: -25 |
4952 | . . | |
4953 | ||
5d67986c | 4954 | 1:2 s l y @key{RET} 2 s l x @key{RET} |
d7b8e6c6 | 4955 | @end group |
5d67986c | 4956 | @end smallexample |
d7b8e6c6 EZ |
4957 | |
4958 | @noindent | |
4959 | The @kbd{s l} command means ``let''; it takes a number from the top of | |
4960 | the stack and temporarily assigns it as the value of the variable | |
4961 | you specify. It then evaluates (as if by the @kbd{=} key) the | |
4962 | next expression on the stack. After this command, the variable goes | |
4963 | back to its original value, if any. | |
4964 | ||
4965 | (An earlier exercise in this tutorial involved storing a value in the | |
4966 | variable @code{x}; if this value is still there, you will have to | |
5d67986c | 4967 | unstore it with @kbd{s u x @key{RET}} before the above example will work |
d7b8e6c6 EZ |
4968 | properly.) |
4969 | ||
4970 | @cindex Maximum of a function using Calculus | |
4971 | Let's find the maximum value of our original expression when @cite{y} | |
4972 | is one-half and @cite{x} ranges over all possible values. We can | |
4973 | do this by taking the derivative with respect to @cite{x} and examining | |
4974 | values of @cite{x} for which the derivative is zero. If the second | |
4975 | derivative of the function at that value of @cite{x} is negative, | |
4976 | the function has a local maximum there. | |
4977 | ||
d7b8e6c6 | 4978 | @smallexample |
5d67986c | 4979 | @group |
d7b8e6c6 EZ |
4980 | 1: 17 x^2 - 6 x^4 + 3 1: 34 x - 24 x^3 |
4981 | . . | |
4982 | ||
5d67986c | 4983 | U @key{DEL} s 1 a d x @key{RET} s 2 |
d7b8e6c6 | 4984 | @end group |
5d67986c | 4985 | @end smallexample |
d7b8e6c6 EZ |
4986 | |
4987 | @noindent | |
4988 | Well, the derivative is clearly zero when @cite{x} is zero. To find | |
4989 | the other root(s), let's divide through by @cite{x} and then solve: | |
4990 | ||
d7b8e6c6 | 4991 | @smallexample |
5d67986c | 4992 | @group |
d7b8e6c6 EZ |
4993 | 1: (34 x - 24 x^3) / x 1: 34 x / x - 24 x^3 / x 1: 34 - 24 x^2 |
4994 | . . . | |
4995 | ||
5d67986c | 4996 | ' x @key{RET} / a x a s |
d7b8e6c6 | 4997 | |
d7b8e6c6 | 4998 | @end group |
5d67986c | 4999 | @end smallexample |
d7b8e6c6 | 5000 | @noindent |
d7b8e6c6 | 5001 | @smallexample |
5d67986c | 5002 | @group |
d7b8e6c6 EZ |
5003 | 1: 34 - 24 x^2 = 0 1: x = 1.19023 |
5004 | . . | |
5005 | ||
5d67986c | 5006 | 0 a = s 3 a S x @key{RET} |
d7b8e6c6 | 5007 | @end group |
5d67986c | 5008 | @end smallexample |
d7b8e6c6 EZ |
5009 | |
5010 | @noindent | |
5011 | Notice the use of @kbd{a s} to ``simplify'' the formula. When the | |
5012 | default algebraic simplifications don't do enough, you can use | |
5013 | @kbd{a s} to tell Calc to spend more time on the job. | |
5014 | ||
5015 | Now we compute the second derivative and plug in our values of @cite{x}: | |
5016 | ||
d7b8e6c6 | 5017 | @smallexample |
5d67986c | 5018 | @group |
d7b8e6c6 EZ |
5019 | 1: 1.19023 2: 1.19023 2: 1.19023 |
5020 | . 1: 34 x - 24 x^3 1: 34 - 72 x^2 | |
5021 | . . | |
5022 | ||
5d67986c | 5023 | a . r 2 a d x @key{RET} s 4 |
d7b8e6c6 | 5024 | @end group |
5d67986c | 5025 | @end smallexample |
d7b8e6c6 EZ |
5026 | |
5027 | @noindent | |
5028 | (The @kbd{a .} command extracts just the righthand side of an equation. | |
5029 | Another method would have been to use @kbd{v u} to unpack the equation | |
5d67986c | 5030 | @w{@samp{x = 1.19}} to @samp{x} and @samp{1.19}, then use @kbd{M-- M-2 @key{DEL}} |
d7b8e6c6 EZ |
5031 | to delete the @samp{x}.) |
5032 | ||
d7b8e6c6 | 5033 | @smallexample |
5d67986c | 5034 | @group |
d7b8e6c6 EZ |
5035 | 2: 34 - 72 x^2 1: -68. 2: 34 - 72 x^2 1: 34 |
5036 | 1: 1.19023 . 1: 0 . | |
5037 | . . | |
5038 | ||
5d67986c | 5039 | @key{TAB} s l x @key{RET} U @key{DEL} 0 s l x @key{RET} |
d7b8e6c6 | 5040 | @end group |
5d67986c | 5041 | @end smallexample |
d7b8e6c6 EZ |
5042 | |
5043 | @noindent | |
5044 | The first of these second derivatives is negative, so we know the function | |
5045 | has a maximum value at @cite{x = 1.19023}. (The function also has a | |
5046 | local @emph{minimum} at @cite{x = 0}.) | |
5047 | ||
5048 | When we solved for @cite{x}, we got only one value even though | |
5049 | @cite{34 - 24 x^2 = 0} is a quadratic equation that ought to have | |
5050 | two solutions. The reason is that @w{@kbd{a S}} normally returns a | |
5051 | single ``principal'' solution. If it needs to come up with an | |
5052 | arbitrary sign (as occurs in the quadratic formula) it picks @cite{+}. | |
5053 | If it needs an arbitrary integer, it picks zero. We can get a full | |
5054 | solution by pressing @kbd{H} (the Hyperbolic flag) before @kbd{a S}. | |
5055 | ||
d7b8e6c6 | 5056 | @smallexample |
5d67986c | 5057 | @group |
d7b8e6c6 EZ |
5058 | 1: 34 - 24 x^2 = 0 1: x = 1.19023 s1 1: x = -1.19023 |
5059 | . . . | |
5060 | ||
5d67986c | 5061 | r 3 H a S x @key{RET} s 5 1 n s l s1 @key{RET} |
d7b8e6c6 | 5062 | @end group |
5d67986c | 5063 | @end smallexample |
d7b8e6c6 EZ |
5064 | |
5065 | @noindent | |
5066 | Calc has invented the variable @samp{s1} to represent an unknown sign; | |
5067 | it is supposed to be either @i{+1} or @i{-1}. Here we have used | |
5068 | the ``let'' command to evaluate the expression when the sign is negative. | |
5069 | If we plugged this into our second derivative we would get the same, | |
5070 | negative, answer, so @cite{x = -1.19023} is also a maximum. | |
5071 | ||
5072 | To find the actual maximum value, we must plug our two values of @cite{x} | |
5073 | into the original formula. | |
5074 | ||
d7b8e6c6 | 5075 | @smallexample |
5d67986c | 5076 | @group |
d7b8e6c6 EZ |
5077 | 2: 17 x^2 - 6 x^4 + 3 1: 24.08333 s1^2 - 12.04166 s1^4 + 3 |
5078 | 1: x = 1.19023 s1 . | |
5079 | . | |
5080 | ||
5d67986c | 5081 | r 1 r 5 s l @key{RET} |
d7b8e6c6 | 5082 | @end group |
5d67986c | 5083 | @end smallexample |
d7b8e6c6 EZ |
5084 | |
5085 | @noindent | |
5086 | (Here we see another way to use @kbd{s l}; if its input is an equation | |
5087 | with a variable on the lefthand side, then @kbd{s l} treats the equation | |
5088 | like an assignment to that variable if you don't give a variable name.) | |
5089 | ||
5090 | It's clear that this will have the same value for either sign of | |
5091 | @code{s1}, but let's work it out anyway, just for the exercise: | |
5092 | ||
d7b8e6c6 | 5093 | @smallexample |
5d67986c | 5094 | @group |
d7b8e6c6 EZ |
5095 | 2: [-1, 1] 1: [15.04166, 15.04166] |
5096 | 1: 24.08333 s1^2 ... . | |
5097 | . | |
5098 | ||
5d67986c | 5099 | [ 1 n , 1 ] @key{TAB} V M $ @key{RET} |
d7b8e6c6 | 5100 | @end group |
5d67986c | 5101 | @end smallexample |
d7b8e6c6 EZ |
5102 | |
5103 | @noindent | |
5104 | Here we have used a vector mapping operation to evaluate the function | |
5105 | at several values of @samp{s1} at once. @kbd{V M $} is like @kbd{V M '} | |
5106 | except that it takes the formula from the top of the stack. The | |
5107 | formula is interpreted as a function to apply across the vector at the | |
5108 | next-to-top stack level. Since a formula on the stack can't contain | |
5109 | @samp{$} signs, Calc assumes the variables in the formula stand for | |
5110 | different arguments. It prompts you for an @dfn{argument list}, giving | |
5111 | the list of all variables in the formula in alphabetical order as the | |
5112 | default list. In this case the default is @samp{(s1)}, which is just | |
5113 | what we want so we simply press @key{RET} at the prompt. | |
5114 | ||
5115 | If there had been several different values, we could have used | |
5116 | @w{@kbd{V R X}} to find the global maximum. | |
5117 | ||
5118 | Calc has a built-in @kbd{a P} command that solves an equation using | |
5119 | @w{@kbd{H a S}} and returns a vector of all the solutions. It simply | |
5120 | automates the job we just did by hand. Applied to our original | |
5121 | cubic polynomial, it would produce the vector of solutions | |
5122 | @cite{[1.19023, -1.19023, 0]}. (There is also an @kbd{a X} command | |
5123 | which finds a local maximum of a function. It uses a numerical search | |
5124 | method rather than examining the derivatives, and thus requires you | |
5125 | to provide some kind of initial guess to show it where to look.) | |
5126 | ||
5127 | (@bullet{}) @strong{Exercise 2.} Given a vector of the roots of a | |
5128 | polynomial (such as the output of an @kbd{a P} command), what | |
5129 | sequence of commands would you use to reconstruct the original | |
5130 | polynomial? (The answer will be unique to within a constant | |
5131 | multiple; choose the solution where the leading coefficient is one.) | |
5132 | @xref{Algebra Answer 2, 2}. (@bullet{}) | |
5133 | ||
5134 | The @kbd{m s} command enables ``symbolic mode,'' in which formulas | |
5135 | like @samp{sqrt(5)} that can't be evaluated exactly are left in | |
5136 | symbolic form rather than giving a floating-point approximate answer. | |
5137 | Fraction mode (@kbd{m f}) is also useful when doing algebra. | |
5138 | ||
d7b8e6c6 | 5139 | @smallexample |
5d67986c | 5140 | @group |
d7b8e6c6 EZ |
5141 | 2: 34 x - 24 x^3 2: 34 x - 24 x^3 |
5142 | 1: 34 x - 24 x^3 1: [sqrt(51) / 6, sqrt(51) / -6, 0] | |
5143 | . . | |
5144 | ||
5d67986c | 5145 | r 2 @key{RET} m s m f a P x @key{RET} |
d7b8e6c6 | 5146 | @end group |
5d67986c | 5147 | @end smallexample |
d7b8e6c6 EZ |
5148 | |
5149 | One more mode that makes reading formulas easier is ``Big mode.'' | |
5150 | ||
d7b8e6c6 | 5151 | @smallexample |
5d67986c | 5152 | @group |
d7b8e6c6 EZ |
5153 | 3 |
5154 | 2: 34 x - 24 x | |
5155 | ||
5156 | ____ ____ | |
5157 | V 51 V 51 | |
5158 | 1: [-----, -----, 0] | |
5159 | 6 -6 | |
5160 | ||
5161 | . | |
5162 | ||
5163 | d B | |
d7b8e6c6 | 5164 | @end group |
5d67986c | 5165 | @end smallexample |
d7b8e6c6 EZ |
5166 | |
5167 | Here things like powers, square roots, and quotients and fractions | |
5168 | are displayed in a two-dimensional pictorial form. Calc has other | |
5169 | language modes as well, such as C mode, FORTRAN mode, and @TeX{} mode. | |
5170 | ||
d7b8e6c6 | 5171 | @smallexample |
5d67986c | 5172 | @group |
d7b8e6c6 EZ |
5173 | 2: 34*x - 24*pow(x, 3) 2: 34*x - 24*x**3 |
5174 | 1: @{sqrt(51) / 6, sqrt(51) / -6, 0@} 1: /sqrt(51) / 6, sqrt(51) / -6, 0/ | |
5175 | . . | |
5176 | ||
5177 | d C d F | |
5178 | ||
d7b8e6c6 | 5179 | @end group |
5d67986c | 5180 | @end smallexample |
d7b8e6c6 | 5181 | @noindent |
d7b8e6c6 | 5182 | @smallexample |
5d67986c | 5183 | @group |
d7b8e6c6 EZ |
5184 | 3: 34 x - 24 x^3 |
5185 | 2: [@{\sqrt@{51@} \over 6@}, @{\sqrt@{51@} \over -6@}, 0] | |
5186 | 1: @{2 \over 3@} \sqrt@{5@} | |
5187 | . | |
5188 | ||
5d67986c | 5189 | d T ' 2 \sqrt@{5@} \over 3 @key{RET} |
d7b8e6c6 | 5190 | @end group |
5d67986c | 5191 | @end smallexample |
d7b8e6c6 EZ |
5192 | |
5193 | @noindent | |
5194 | As you can see, language modes affect both entry and display of | |
5195 | formulas. They affect such things as the names used for built-in | |
5196 | functions, the set of arithmetic operators and their precedences, | |
5197 | and notations for vectors and matrices. | |
5198 | ||
5199 | Notice that @samp{sqrt(51)} may cause problems with older | |
5200 | implementations of C and FORTRAN, which would require something more | |
5201 | like @samp{sqrt(51.0)}. It is always wise to check over the formulas | |
5202 | produced by the various language modes to make sure they are fully | |
5203 | correct. | |
5204 | ||
5205 | Type @kbd{m s}, @kbd{m f}, and @kbd{d N} to reset these modes. (You | |
5206 | may prefer to remain in Big mode, but all the examples in the tutorial | |
5207 | are shown in normal mode.) | |
5208 | ||
5209 | @cindex Area under a curve | |
5210 | What is the area under the portion of this curve from @cite{x = 1} to @cite{2}? | |
5211 | This is simply the integral of the function: | |
5212 | ||
d7b8e6c6 | 5213 | @smallexample |
5d67986c | 5214 | @group |
d7b8e6c6 EZ |
5215 | 1: 17 x^2 - 6 x^4 + 3 1: 5.6666 x^3 - 1.2 x^5 + 3 x |
5216 | . . | |
5217 | ||
5218 | r 1 a i x | |
d7b8e6c6 | 5219 | @end group |
5d67986c | 5220 | @end smallexample |
d7b8e6c6 EZ |
5221 | |
5222 | @noindent | |
5223 | We want to evaluate this at our two values for @cite{x} and subtract. | |
5224 | One way to do it is again with vector mapping and reduction: | |
5225 | ||
d7b8e6c6 | 5226 | @smallexample |
5d67986c | 5227 | @group |
d7b8e6c6 EZ |
5228 | 2: [2, 1] 1: [12.93333, 7.46666] 1: 5.46666 |
5229 | 1: 5.6666 x^3 ... . . | |
5230 | ||
5d67986c | 5231 | [ 2 , 1 ] @key{TAB} V M $ @key{RET} V R - |
d7b8e6c6 | 5232 | @end group |
5d67986c | 5233 | @end smallexample |
d7b8e6c6 EZ |
5234 | |
5235 | (@bullet{}) @strong{Exercise 3.} Find the integral from 1 to @cite{y} | |
5236 | of @c{$x \sin \pi x$} | |
5237 | @w{@cite{x sin(pi x)}} (where the sine is calculated in radians). | |
5238 | Find the values of the integral for integers @cite{y} from 1 to 5. | |
5239 | @xref{Algebra Answer 3, 3}. (@bullet{}) | |
5240 | ||
5241 | Calc's integrator can do many simple integrals symbolically, but many | |
5242 | others are beyond its capabilities. Suppose we wish to find the area | |
5243 | under the curve @c{$\sin x \ln x$} | |
5244 | @cite{sin(x) ln(x)} over the same range of @cite{x}. If | |
5d67986c | 5245 | you entered this formula and typed @kbd{a i x @key{RET}} (don't bother to try |
d7b8e6c6 EZ |
5246 | this), Calc would work for a long time but would be unable to find a |
5247 | solution. In fact, there is no closed-form solution to this integral. | |
5248 | Now what do we do? | |
5249 | ||
5250 | @cindex Integration, numerical | |
5251 | @cindex Numerical integration | |
5252 | One approach would be to do the integral numerically. It is not hard | |
5253 | to do this by hand using vector mapping and reduction. It is rather | |
5254 | slow, though, since the sine and logarithm functions take a long time. | |
5255 | We can save some time by reducing the working precision. | |
5256 | ||
d7b8e6c6 | 5257 | @smallexample |
5d67986c | 5258 | @group |
d7b8e6c6 EZ |
5259 | 3: 10 1: [1, 1.1, 1.2, ... , 1.8, 1.9] |
5260 | 2: 1 . | |
5261 | 1: 0.1 | |
5262 | . | |
5263 | ||
5d67986c | 5264 | 10 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x |
d7b8e6c6 | 5265 | @end group |
5d67986c | 5266 | @end smallexample |
d7b8e6c6 EZ |
5267 | |
5268 | @noindent | |
5269 | (Note that we have used the extended version of @kbd{v x}; we could | |
5d67986c | 5270 | also have used plain @kbd{v x} as follows: @kbd{v x 10 @key{RET} 9 + .1 *}.) |
d7b8e6c6 | 5271 | |
d7b8e6c6 | 5272 | @smallexample |
5d67986c | 5273 | @group |
d7b8e6c6 EZ |
5274 | 2: [1, 1.1, ... ] 1: [0., 0.084941, 0.16993, ... ] |
5275 | 1: sin(x) ln(x) . | |
5276 | . | |
5277 | ||
5d67986c | 5278 | ' sin(x) ln(x) @key{RET} s 1 m r p 5 @key{RET} V M $ @key{RET} |
d7b8e6c6 | 5279 | |
d7b8e6c6 | 5280 | @end group |
5d67986c | 5281 | @end smallexample |
d7b8e6c6 | 5282 | @noindent |
d7b8e6c6 | 5283 | @smallexample |
5d67986c | 5284 | @group |
d7b8e6c6 EZ |
5285 | 1: 3.4195 0.34195 |
5286 | . . | |
5287 | ||
5288 | V R + 0.1 * | |
d7b8e6c6 | 5289 | @end group |
5d67986c | 5290 | @end smallexample |
d7b8e6c6 EZ |
5291 | |
5292 | @noindent | |
5293 | (If you got wildly different results, did you remember to switch | |
5294 | to radians mode?) | |
5295 | ||
5296 | Here we have divided the curve into ten segments of equal width; | |
5297 | approximating these segments as rectangular boxes (i.e., assuming | |
5298 | the curve is nearly flat at that resolution), we compute the areas | |
5299 | of the boxes (height times width), then sum the areas. (It is | |
5300 | faster to sum first, then multiply by the width, since the width | |
5301 | is the same for every box.) | |
5302 | ||
5303 | The true value of this integral turns out to be about 0.374, so | |
5304 | we're not doing too well. Let's try another approach. | |
5305 | ||
d7b8e6c6 | 5306 | @smallexample |
5d67986c | 5307 | @group |
d7b8e6c6 EZ |
5308 | 1: sin(x) ln(x) 1: 0.84147 x - 0.84147 + 0.11957 (x - 1)^2 - ... |
5309 | . . | |
5310 | ||
5d67986c | 5311 | r 1 a t x=1 @key{RET} 4 @key{RET} |
d7b8e6c6 | 5312 | @end group |
5d67986c | 5313 | @end smallexample |
d7b8e6c6 EZ |
5314 | |
5315 | @noindent | |
5316 | Here we have computed the Taylor series expansion of the function | |
5317 | about the point @cite{x=1}. We can now integrate this polynomial | |
5318 | approximation, since polynomials are easy to integrate. | |
5319 | ||
d7b8e6c6 | 5320 | @smallexample |
5d67986c | 5321 | @group |
d7b8e6c6 EZ |
5322 | 1: 0.42074 x^2 + ... 1: [-0.0446, -0.42073] 1: 0.3761 |
5323 | . . . | |
5324 | ||
5d67986c | 5325 | a i x @key{RET} [ 2 , 1 ] @key{TAB} V M $ @key{RET} V R - |
d7b8e6c6 | 5326 | @end group |
5d67986c | 5327 | @end smallexample |
d7b8e6c6 EZ |
5328 | |
5329 | @noindent | |
5330 | Better! By increasing the precision and/or asking for more terms | |
5331 | in the Taylor series, we can get a result as accurate as we like. | |
5332 | (Taylor series converge better away from singularities in the | |
5333 | function such as the one at @code{ln(0)}, so it would also help to | |
5334 | expand the series about the points @cite{x=2} or @cite{x=1.5} instead | |
5335 | of @cite{x=1}.) | |
5336 | ||
5337 | @cindex Simpson's rule | |
5338 | @cindex Integration by Simpson's rule | |
5339 | (@bullet{}) @strong{Exercise 4.} Our first method approximated the | |
5340 | curve by stairsteps of width 0.1; the total area was then the sum | |
5341 | of the areas of the rectangles under these stairsteps. Our second | |
5342 | method approximated the function by a polynomial, which turned out | |
5343 | to be a better approximation than stairsteps. A third method is | |
5344 | @dfn{Simpson's rule}, which is like the stairstep method except | |
5345 | that the steps are not required to be flat. Simpson's rule boils | |
5346 | down to the formula, | |
5347 | ||
5348 | @ifinfo | |
5349 | @example | |
5350 | (h/3) * (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + ... | |
5351 | + 2 f(a+(n-2)*h) + 4 f(a+(n-1)*h) + f(a+n*h)) | |
5352 | @end example | |
5353 | @end ifinfo | |
5354 | @tex | |
5355 | \turnoffactive | |
5356 | \beforedisplay | |
5357 | $$ \displaylines{ | |
5358 | \qquad {h \over 3} (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + \cdots | |
5359 | \hfill \cr \hfill {} + 2 f(a+(n-2)h) + 4 f(a+(n-1)h) + f(a+n h)) \qquad | |
5360 | } $$ | |
5361 | \afterdisplay | |
5362 | @end tex | |
5363 | ||
5364 | @noindent | |
5365 | where @cite{n} (which must be even) is the number of slices and @cite{h} | |
5366 | is the width of each slice. These are 10 and 0.1 in our example. | |
5367 | For reference, here is the corresponding formula for the stairstep | |
5368 | method: | |
5369 | ||
5370 | @ifinfo | |
5371 | @example | |
5372 | h * (f(a) + f(a+h) + f(a+2h) + f(a+3h) + ... | |
5373 | + f(a+(n-2)*h) + f(a+(n-1)*h)) | |
5374 | @end example | |
5375 | @end ifinfo | |
5376 | @tex | |
5377 | \turnoffactive | |
5378 | \beforedisplay | |
5379 | $$ h (f(a) + f(a+h) + f(a+2h) + f(a+3h) + \cdots | |
5380 | + f(a+(n-2)h) + f(a+(n-1)h)) $$ | |
5381 | \afterdisplay | |
5382 | @end tex | |
5383 | ||
5384 | Compute the integral from 1 to 2 of @c{$\sin x \ln x$} | |
5385 | @cite{sin(x) ln(x)} using | |
5386 | Simpson's rule with 10 slices. @xref{Algebra Answer 4, 4}. (@bullet{}) | |
5387 | ||
5388 | Calc has a built-in @kbd{a I} command for doing numerical integration. | |
5389 | It uses @dfn{Romberg's method}, which is a more sophisticated cousin | |
5390 | of Simpson's rule. In particular, it knows how to keep refining the | |
5391 | result until the current precision is satisfied. | |
5392 | ||
5393 | @c [fix-ref Selecting Sub-Formulas] | |
5394 | Aside from the commands we've seen so far, Calc also provides a | |
5395 | large set of commands for operating on parts of formulas. You | |
5396 | indicate the desired sub-formula by placing the cursor on any part | |
5397 | of the formula before giving a @dfn{selection} command. Selections won't | |
5398 | be covered in the tutorial; @pxref{Selecting Subformulas}, for | |
5399 | details and examples. | |
5400 | ||
5401 | @c hard exercise: simplify (2^(n r) - 2^(r*(n - 1))) / (2^r - 1) 2^(n - 1) | |
5402 | @c to 2^((n-1)*(r-1)). | |
5403 | ||
5404 | @node Rewrites Tutorial, , Basic Algebra Tutorial, Algebra Tutorial | |
5405 | @subsection Rewrite Rules | |
5406 | ||
5407 | @noindent | |
5408 | No matter how many built-in commands Calc provided for doing algebra, | |
5409 | there would always be something you wanted to do that Calc didn't have | |
5410 | in its repertoire. So Calc also provides a @dfn{rewrite rule} system | |
5411 | that you can use to define your own algebraic manipulations. | |
5412 | ||
5413 | Suppose we want to simplify this trigonometric formula: | |
5414 | ||
d7b8e6c6 | 5415 | @smallexample |
5d67986c | 5416 | @group |
d7b8e6c6 EZ |
5417 | 1: 1 / cos(x) - sin(x) tan(x) |
5418 | . | |
5419 | ||
5d67986c | 5420 | ' 1/cos(x) - sin(x) tan(x) @key{RET} s 1 |
d7b8e6c6 | 5421 | @end group |
5d67986c | 5422 | @end smallexample |
d7b8e6c6 EZ |
5423 | |
5424 | @noindent | |
5425 | If we were simplifying this by hand, we'd probably replace the | |
5426 | @samp{tan} with a @samp{sin/cos} first, then combine over a common | |
5427 | denominator. There is no Calc command to do the former; the @kbd{a n} | |
5428 | algebra command will do the latter but we'll do both with rewrite | |
5429 | rules just for practice. | |
5430 | ||
5431 | Rewrite rules are written with the @samp{:=} symbol. | |
5432 | ||
d7b8e6c6 | 5433 | @smallexample |
5d67986c | 5434 | @group |
d7b8e6c6 EZ |
5435 | 1: 1 / cos(x) - sin(x)^2 / cos(x) |
5436 | . | |
5437 | ||
5d67986c | 5438 | a r tan(a) := sin(a)/cos(a) @key{RET} |
d7b8e6c6 | 5439 | @end group |
5d67986c | 5440 | @end smallexample |
d7b8e6c6 EZ |
5441 | |
5442 | @noindent | |
5443 | (The ``assignment operator'' @samp{:=} has several uses in Calc. All | |
5444 | by itself the formula @samp{tan(a) := sin(a)/cos(a)} doesn't do anything, | |
5445 | but when it is given to the @kbd{a r} command, that command interprets | |
5446 | it as a rewrite rule.) | |
5447 | ||
5448 | The lefthand side, @samp{tan(a)}, is called the @dfn{pattern} of the | |
5449 | rewrite rule. Calc searches the formula on the stack for parts that | |
5450 | match the pattern. Variables in a rewrite pattern are called | |
5451 | @dfn{meta-variables}, and when matching the pattern each meta-variable | |
5452 | can match any sub-formula. Here, the meta-variable @samp{a} matched | |
5453 | the actual variable @samp{x}. | |
5454 | ||
5455 | When the pattern part of a rewrite rule matches a part of the formula, | |
5456 | that part is replaced by the righthand side with all the meta-variables | |
5457 | substituted with the things they matched. So the result is | |
5458 | @samp{sin(x) / cos(x)}. Calc's normal algebraic simplifications then | |
5459 | mix this in with the rest of the original formula. | |
5460 | ||
5461 | To merge over a common denominator, we can use another simple rule: | |
5462 | ||
d7b8e6c6 | 5463 | @smallexample |
5d67986c | 5464 | @group |
d7b8e6c6 EZ |
5465 | 1: (1 - sin(x)^2) / cos(x) |
5466 | . | |
5467 | ||
5d67986c | 5468 | a r a/x + b/x := (a+b)/x @key{RET} |
d7b8e6c6 | 5469 | @end group |
5d67986c | 5470 | @end smallexample |
d7b8e6c6 EZ |
5471 | |
5472 | This rule points out several interesting features of rewrite patterns. | |
5473 | First, if a meta-variable appears several times in a pattern, it must | |
5474 | match the same thing everywhere. This rule detects common denominators | |
5475 | because the same meta-variable @samp{x} is used in both of the | |
5476 | denominators. | |
5477 | ||
5478 | Second, meta-variable names are independent from variables in the | |
5479 | target formula. Notice that the meta-variable @samp{x} here matches | |
5480 | the subformula @samp{cos(x)}; Calc never confuses the two meanings of | |
5481 | @samp{x}. | |
5482 | ||
5483 | And third, rewrite patterns know a little bit about the algebraic | |
5484 | properties of formulas. The pattern called for a sum of two quotients; | |
5485 | Calc was able to match a difference of two quotients by matching | |
5486 | @samp{a = 1}, @samp{b = -sin(x)^2}, and @samp{x = cos(x)}. | |
5487 | ||
5488 | @c [fix-ref Algebraic Properties of Rewrite Rules] | |
5489 | We could just as easily have written @samp{a/x - b/x := (a-b)/x} for | |
5490 | the rule. It would have worked just the same in all cases. (If we | |
5491 | really wanted the rule to apply only to @samp{+} or only to @samp{-}, | |
5492 | we could have used the @code{plain} symbol. @xref{Algebraic Properties | |
5493 | of Rewrite Rules}, for some examples of this.) | |
5494 | ||
5495 | One more rewrite will complete the job. We want to use the identity | |
5496 | @samp{sin(x)^2 + cos(x)^2 = 1}, but of course we must first rearrange | |
5497 | the identity in a way that matches our formula. The obvious rule | |
5498 | would be @samp{@w{1 - sin(x)^2} := cos(x)^2}, but a little thought shows | |
5499 | that the rule @samp{sin(x)^2 := 1 - cos(x)^2} will also work. The | |
5500 | latter rule has a more general pattern so it will work in many other | |
5501 | situations, too. | |
5502 | ||
d7b8e6c6 | 5503 | @smallexample |
5d67986c | 5504 | @group |
d7b8e6c6 EZ |
5505 | 1: (1 + cos(x)^2 - 1) / cos(x) 1: cos(x) |
5506 | . . | |
5507 | ||
5d67986c | 5508 | a r sin(x)^2 := 1 - cos(x)^2 @key{RET} a s |
d7b8e6c6 | 5509 | @end group |
5d67986c | 5510 | @end smallexample |
d7b8e6c6 EZ |
5511 | |
5512 | You may ask, what's the point of using the most general rule if you | |
5513 | have to type it in every time anyway? The answer is that Calc allows | |
5514 | you to store a rewrite rule in a variable, then give the variable | |
5515 | name in the @kbd{a r} command. In fact, this is the preferred way to | |
5516 | use rewrites. For one, if you need a rule once you'll most likely | |
5517 | need it again later. Also, if the rule doesn't work quite right you | |
5518 | can simply Undo, edit the variable, and run the rule again without | |
5519 | having to retype it. | |
5520 | ||
d7b8e6c6 | 5521 | @smallexample |
5d67986c RS |
5522 | @group |
5523 | ' tan(x) := sin(x)/cos(x) @key{RET} s t tsc @key{RET} | |
5524 | ' a/x + b/x := (a+b)/x @key{RET} s t merge @key{RET} | |
5525 | ' sin(x)^2 := 1 - cos(x)^2 @key{RET} s t sinsqr @key{RET} | |
d7b8e6c6 EZ |
5526 | |
5527 | 1: 1 / cos(x) - sin(x) tan(x) 1: cos(x) | |
5528 | . . | |
5529 | ||
5d67986c | 5530 | r 1 a r tsc @key{RET} a r merge @key{RET} a r sinsqr @key{RET} a s |
d7b8e6c6 | 5531 | @end group |
5d67986c | 5532 | @end smallexample |
d7b8e6c6 EZ |
5533 | |
5534 | To edit a variable, type @kbd{s e} and the variable name, use regular | |
5535 | Emacs editing commands as necessary, then type @kbd{M-# M-#} or | |
5536 | @kbd{C-c C-c} to store the edited value back into the variable. | |
5537 | You can also use @w{@kbd{s e}} to create a new variable if you wish. | |
5538 | ||
5539 | Notice that the first time you use each rule, Calc puts up a ``compiling'' | |
5540 | message briefly. The pattern matcher converts rules into a special | |
5541 | optimized pattern-matching language rather than using them directly. | |
5542 | This allows @kbd{a r} to apply even rather complicated rules very | |
5543 | efficiently. If the rule is stored in a variable, Calc compiles it | |
5544 | only once and stores the compiled form along with the variable. That's | |
5545 | another good reason to store your rules in variables rather than | |
5546 | entering them on the fly. | |
5547 | ||
5548 | (@bullet{}) @strong{Exercise 1.} Type @kbd{m s} to get symbolic | |
5549 | mode, then enter the formula @samp{@w{(2 + sqrt(2))} / @w{(1 + sqrt(2))}}. | |
5550 | Using a rewrite rule, simplify this formula by multiplying both | |
5551 | sides by the conjugate @w{@samp{1 - sqrt(2)}}. The result will have | |
5552 | to be expanded by the distributive law; do this with another | |
5553 | rewrite. @xref{Rewrites Answer 1, 1}. (@bullet{}) | |
5554 | ||
5555 | The @kbd{a r} command can also accept a vector of rewrite rules, or | |
5556 | a variable containing a vector of rules. | |
5557 | ||
d7b8e6c6 | 5558 | @smallexample |
5d67986c | 5559 | @group |
d7b8e6c6 EZ |
5560 | 1: [tsc, merge, sinsqr] 1: [tan(x) := sin(x) / cos(x), ... ] |
5561 | . . | |
5562 | ||
5d67986c | 5563 | ' [tsc,merge,sinsqr] @key{RET} = |
d7b8e6c6 | 5564 | |
d7b8e6c6 | 5565 | @end group |
5d67986c | 5566 | @end smallexample |
d7b8e6c6 | 5567 | @noindent |
d7b8e6c6 | 5568 | @smallexample |
5d67986c | 5569 | @group |
d7b8e6c6 EZ |
5570 | 1: 1 / cos(x) - sin(x) tan(x) 1: cos(x) |
5571 | . . | |
5572 | ||
5d67986c | 5573 | s t trig @key{RET} r 1 a r trig @key{RET} a s |
d7b8e6c6 | 5574 | @end group |
5d67986c | 5575 | @end smallexample |
d7b8e6c6 EZ |
5576 | |
5577 | @c [fix-ref Nested Formulas with Rewrite Rules] | |
5578 | Calc tries all the rules you give against all parts of the formula, | |
5579 | repeating until no further change is possible. (The exact order in | |
5580 | which things are tried is rather complex, but for simple rules like | |
5581 | the ones we've used here the order doesn't really matter. | |
5582 | @xref{Nested Formulas with Rewrite Rules}.) | |
5583 | ||
5584 | Calc actually repeats only up to 100 times, just in case your rule set | |
5585 | has gotten into an infinite loop. You can give a numeric prefix argument | |
5586 | to @kbd{a r} to specify any limit. In particular, @kbd{M-1 a r} does | |
5587 | only one rewrite at a time. | |
5588 | ||
d7b8e6c6 | 5589 | @smallexample |
5d67986c | 5590 | @group |
d7b8e6c6 EZ |
5591 | 1: 1 / cos(x) - sin(x)^2 / cos(x) 1: (1 - sin(x)^2) / cos(x) |
5592 | . . | |
5593 | ||
5d67986c | 5594 | r 1 M-1 a r trig @key{RET} M-1 a r trig @key{RET} |
d7b8e6c6 | 5595 | @end group |
5d67986c | 5596 | @end smallexample |
d7b8e6c6 EZ |
5597 | |
5598 | You can type @kbd{M-0 a r} if you want no limit at all on the number | |
5599 | of rewrites that occur. | |
5600 | ||
5601 | Rewrite rules can also be @dfn{conditional}. Simply follow the rule | |
5602 | with a @samp{::} symbol and the desired condition. For example, | |
5603 | ||
d7b8e6c6 | 5604 | @smallexample |
5d67986c | 5605 | @group |
d7b8e6c6 EZ |
5606 | 1: exp(2 pi i) + exp(3 pi i) + exp(4 pi i) |
5607 | . | |
5608 | ||
5d67986c | 5609 | ' exp(2 pi i) + exp(3 pi i) + exp(4 pi i) @key{RET} |
d7b8e6c6 | 5610 | |
d7b8e6c6 | 5611 | @end group |
5d67986c | 5612 | @end smallexample |
d7b8e6c6 | 5613 | @noindent |
d7b8e6c6 | 5614 | @smallexample |
5d67986c | 5615 | @group |
d7b8e6c6 EZ |
5616 | 1: 1 + exp(3 pi i) + 1 |
5617 | . | |
5618 | ||
5d67986c | 5619 | a r exp(k pi i) := 1 :: k % 2 = 0 @key{RET} |
d7b8e6c6 | 5620 | @end group |
5d67986c | 5621 | @end smallexample |
d7b8e6c6 EZ |
5622 | |
5623 | @noindent | |
5624 | (Recall, @samp{k % 2} is the remainder from dividing @samp{k} by 2, | |
5625 | which will be zero only when @samp{k} is an even integer.) | |
5626 | ||
5627 | An interesting point is that the variables @samp{pi} and @samp{i} | |
5628 | were matched literally rather than acting as meta-variables. | |
5629 | This is because they are special-constant variables. The special | |
5630 | constants @samp{e}, @samp{phi}, and so on also match literally. | |
5631 | A common error with rewrite | |
5632 | rules is to write, say, @samp{f(a,b,c,d,e) := g(a+b+c+d+e)}, expecting | |
5633 | to match any @samp{f} with five arguments but in fact matching | |
5634 | only when the fifth argument is literally @samp{e}!@refill | |
5635 | ||
5636 | @cindex Fibonacci numbers | |
5d67986c RS |
5637 | @ignore |
5638 | @starindex | |
5639 | @end ignore | |
d7b8e6c6 EZ |
5640 | @tindex fib |
5641 | Rewrite rules provide an interesting way to define your own functions. | |
5642 | Suppose we want to define @samp{fib(n)} to produce the @var{n}th | |
5643 | Fibonacci number. The first two Fibonacci numbers are each 1; | |
5644 | later numbers are formed by summing the two preceding numbers in | |
5645 | the sequence. This is easy to express in a set of three rules: | |
5646 | ||
d7b8e6c6 | 5647 | @smallexample |
5d67986c RS |
5648 | @group |
5649 | ' [fib(1) := 1, fib(2) := 1, fib(n) := fib(n-1) + fib(n-2)] @key{RET} s t fib | |
d7b8e6c6 EZ |
5650 | |
5651 | 1: fib(7) 1: 13 | |
5652 | . . | |
5653 | ||
5d67986c | 5654 | ' fib(7) @key{RET} a r fib @key{RET} |
d7b8e6c6 | 5655 | @end group |
5d67986c | 5656 | @end smallexample |
d7b8e6c6 EZ |
5657 | |
5658 | One thing that is guaranteed about the order that rewrites are tried | |
5659 | is that, for any given subformula, earlier rules in the rule set will | |
5660 | be tried for that subformula before later ones. So even though the | |
5661 | first and third rules both match @samp{fib(1)}, we know the first will | |
5662 | be used preferentially. | |
5663 | ||
5664 | This rule set has one dangerous bug: Suppose we apply it to the | |
5665 | formula @samp{fib(x)}? (Don't actually try this.) The third rule | |
5666 | will match @samp{fib(x)} and replace it with @w{@samp{fib(x-1) + fib(x-2)}}. | |
5667 | Each of these will then be replaced to get @samp{fib(x-2) + 2 fib(x-3) + | |
5668 | fib(x-4)}, and so on, expanding forever. What we really want is to apply | |
5669 | the third rule only when @samp{n} is an integer greater than two. Type | |
5d67986c | 5670 | @w{@kbd{s e fib @key{RET}}}, then edit the third rule to: |
d7b8e6c6 EZ |
5671 | |
5672 | @smallexample | |
5673 | fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2 | |
5674 | @end smallexample | |
5675 | ||
5676 | @noindent | |
5677 | Now: | |
5678 | ||
d7b8e6c6 | 5679 | @smallexample |
5d67986c | 5680 | @group |
d7b8e6c6 EZ |
5681 | 1: fib(6) + fib(x) + fib(0) 1: 8 + fib(x) + fib(0) |
5682 | . . | |
5683 | ||
5d67986c | 5684 | ' fib(6)+fib(x)+fib(0) @key{RET} a r fib @key{RET} |
d7b8e6c6 | 5685 | @end group |
5d67986c | 5686 | @end smallexample |
d7b8e6c6 EZ |
5687 | |
5688 | @noindent | |
5689 | We've created a new function, @code{fib}, and a new command, | |
5d67986c | 5690 | @w{@kbd{a r fib @key{RET}}}, which means ``evaluate all @code{fib} calls in |
d7b8e6c6 EZ |
5691 | this formula.'' To make things easier still, we can tell Calc to |
5692 | apply these rules automatically by storing them in the special | |
5693 | variable @code{EvalRules}. | |
5694 | ||
d7b8e6c6 | 5695 | @smallexample |
5d67986c | 5696 | @group |
d7b8e6c6 EZ |
5697 | 1: [fib(1) := ...] . 1: [8, 13] |
5698 | . . | |
5699 | ||
5d67986c | 5700 | s r fib @key{RET} s t EvalRules @key{RET} ' [fib(6), fib(7)] @key{RET} |
d7b8e6c6 | 5701 | @end group |
5d67986c | 5702 | @end smallexample |
d7b8e6c6 EZ |
5703 | |
5704 | It turns out that this rule set has the problem that it does far | |
5705 | more work than it needs to when @samp{n} is large. Consider the | |
5706 | first few steps of the computation of @samp{fib(6)}: | |
5707 | ||
d7b8e6c6 | 5708 | @smallexample |
5d67986c | 5709 | @group |
d7b8e6c6 EZ |
5710 | fib(6) = |
5711 | fib(5) + fib(4) = | |
5712 | fib(4) + fib(3) + fib(3) + fib(2) = | |
5713 | fib(3) + fib(2) + fib(2) + fib(1) + fib(2) + fib(1) + 1 = ... | |
d7b8e6c6 | 5714 | @end group |
5d67986c | 5715 | @end smallexample |
d7b8e6c6 EZ |
5716 | |
5717 | @noindent | |
5718 | Note that @samp{fib(3)} appears three times here. Unless Calc's | |
5719 | algebraic simplifier notices the multiple @samp{fib(3)}s and combines | |
5720 | them (and, as it happens, it doesn't), this rule set does lots of | |
5721 | needless recomputation. To cure the problem, type @code{s e EvalRules} | |
5722 | to edit the rules (or just @kbd{s E}, a shorthand command for editing | |
5723 | @code{EvalRules}) and add another condition: | |
5724 | ||
5725 | @smallexample | |
5726 | fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2 :: remember | |
5727 | @end smallexample | |
5728 | ||
5729 | @noindent | |
5730 | If a @samp{:: remember} condition appears anywhere in a rule, then if | |
5731 | that rule succeeds Calc will add another rule that describes that match | |
5732 | to the front of the rule set. (Remembering works in any rule set, but | |
5733 | for technical reasons it is most effective in @code{EvalRules}.) For | |
5734 | example, if the rule rewrites @samp{fib(7)} to something that evaluates | |
5735 | to 13, then the rule @samp{fib(7) := 13} will be added to the rule set. | |
5736 | ||
5d67986c | 5737 | Type @kbd{' fib(8) @key{RET}} to compute the eighth Fibonacci number, then |
d7b8e6c6 EZ |
5738 | type @kbd{s E} again to see what has happened to the rule set. |
5739 | ||
5740 | With the @code{remember} feature, our rule set can now compute | |
5741 | @samp{fib(@var{n})} in just @var{n} steps. In the process it builds | |
5742 | up a table of all Fibonacci numbers up to @var{n}. After we have | |
5743 | computed the result for a particular @var{n}, we can get it back | |
5744 | (and the results for all smaller @var{n}) later in just one step. | |
5745 | ||
5746 | All Calc operations will run somewhat slower whenever @code{EvalRules} | |
5d67986c | 5747 | contains any rules. You should type @kbd{s u EvalRules @key{RET}} now to |
d7b8e6c6 EZ |
5748 | un-store the variable. |
5749 | ||
5750 | (@bullet{}) @strong{Exercise 2.} Sometimes it is possible to reformulate | |
5751 | a problem to reduce the amount of recursion necessary to solve it. | |
5752 | Create a rule that, in about @var{n} simple steps and without recourse | |
5753 | to the @code{remember} option, replaces @samp{fib(@var{n}, 1, 1)} with | |
5754 | @samp{fib(1, @var{x}, @var{y})} where @var{x} and @var{y} are the | |
5755 | @var{n}th and @var{n+1}st Fibonacci numbers, respectively. This rule is | |
5756 | rather clunky to use, so add a couple more rules to make the ``user | |
5757 | interface'' the same as for our first version: enter @samp{fib(@var{n})}, | |
5758 | get back a plain number. @xref{Rewrites Answer 2, 2}. (@bullet{}) | |
5759 | ||
5760 | There are many more things that rewrites can do. For example, there | |
5761 | are @samp{&&&} and @samp{|||} pattern operators that create ``and'' | |
5762 | and ``or'' combinations of rules. As one really simple example, we | |
5763 | could combine our first two Fibonacci rules thusly: | |
5764 | ||
5765 | @example | |
5766 | [fib(1 ||| 2) := 1, fib(n) := ... ] | |
5767 | @end example | |
5768 | ||
5769 | @noindent | |
5770 | That means ``@code{fib} of something matching either 1 or 2 rewrites | |
5771 | to 1.'' | |
5772 | ||
5773 | You can also make meta-variables optional by enclosing them in @code{opt}. | |
5774 | For example, the pattern @samp{a + b x} matches @samp{2 + 3 x} but not | |
5775 | @samp{2 + x} or @samp{3 x} or @samp{x}. The pattern @samp{opt(a) + opt(b) x} | |
5776 | matches all of these forms, filling in a default of zero for @samp{a} | |
5777 | and one for @samp{b}. | |
5778 | ||
5779 | (@bullet{}) @strong{Exercise 3.} Your friend Joe had @samp{2 + 3 x} | |
5780 | on the stack and tried to use the rule | |
5781 | @samp{opt(a) + opt(b) x := f(a, b, x)}. What happened? | |
5782 | @xref{Rewrites Answer 3, 3}. (@bullet{}) | |
5783 | ||
5784 | (@bullet{}) @strong{Exercise 4.} Starting with a positive integer @cite{a}, | |
5785 | divide @cite{a} by two if it is even, otherwise compute @cite{3 a + 1}. | |
5786 | Now repeat this step over and over. A famous unproved conjecture | |
5787 | is that for any starting @cite{a}, the sequence always eventually | |
5788 | reaches 1. Given the formula @samp{seq(@var{a}, 0)}, write a set of | |
5789 | rules that convert this into @samp{seq(1, @var{n})} where @var{n} | |
5790 | is the number of steps it took the sequence to reach the value 1. | |
5791 | Now enhance the rules to accept @samp{seq(@var{a})} as a starting | |
5792 | configuration, and to stop with just the number @var{n} by itself. | |
5793 | Now make the result be a vector of values in the sequence, from @var{a} | |
5794 | to 1. (The formula @samp{@var{x}|@var{y}} appends the vectors @var{x} | |
5795 | and @var{y}.) For example, rewriting @samp{seq(6)} should yield the | |
5796 | vector @cite{[6, 3, 10, 5, 16, 8, 4, 2, 1]}. | |
5797 | @xref{Rewrites Answer 4, 4}. (@bullet{}) | |
5798 | ||
5799 | (@bullet{}) @strong{Exercise 5.} Define, using rewrite rules, a function | |
5800 | @samp{nterms(@var{x})} that returns the number of terms in the sum | |
5801 | @var{x}, or 1 if @var{x} is not a sum. (A @dfn{sum} for our purposes | |
5802 | is one or more non-sum terms separated by @samp{+} or @samp{-} signs, | |
5803 | so that @cite{2 - 3 (x + y) + x y} is a sum of three terms.) | |
5804 | @xref{Rewrites Answer 5, 5}. (@bullet{}) | |
5805 | ||
5806 | (@bullet{}) @strong{Exercise 6.} Calc considers the form @cite{0^0} | |
5807 | to be ``indeterminate,'' and leaves it unevaluated (assuming infinite | |
5808 | mode is not enabled). Some people prefer to define @cite{0^0 = 1}, | |
5809 | so that the identity @cite{x^0 = 1} can safely be used for all @cite{x}. | |
5810 | Find a way to make Calc follow this convention. What happens if you | |
5811 | now type @kbd{m i} to turn on infinite mode? | |
5812 | @xref{Rewrites Answer 6, 6}. (@bullet{}) | |
5813 | ||
5814 | (@bullet{}) @strong{Exercise 7.} A Taylor series for a function is an | |
5815 | infinite series that exactly equals the value of that function at | |
5816 | values of @cite{x} near zero. | |
5817 | ||
5818 | @ifinfo | |
5819 | @example | |
5820 | cos(x) = 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ... | |
5821 | @end example | |
5822 | @end ifinfo | |
5823 | @tex | |
5824 | \turnoffactive \let\rm\goodrm | |
5825 | \beforedisplay | |
5826 | $$ \cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - {x^6 \over 6!} + \cdots $$ | |
5827 | \afterdisplay | |
5828 | @end tex | |
5829 | ||
5830 | The @kbd{a t} command produces a @dfn{truncated Taylor series} which | |
5831 | is obtained by dropping all the terms higher than, say, @cite{x^2}. | |
5832 | Calc represents the truncated Taylor series as a polynomial in @cite{x}. | |
5833 | Mathematicians often write a truncated series using a ``big-O'' notation | |
5834 | that records what was the lowest term that was truncated. | |
5835 | ||
5836 | @ifinfo | |
5837 | @example | |
5838 | cos(x) = 1 - x^2 / 2! + O(x^3) | |
5839 | @end example | |
5840 | @end ifinfo | |
5841 | @tex | |
5842 | \turnoffactive \let\rm\goodrm | |
5843 | \beforedisplay | |
5844 | $$ \cos x = 1 - {x^2 \over 2!} + O(x^3) $$ | |
5845 | \afterdisplay | |
5846 | @end tex | |
5847 | ||
5848 | @noindent | |
5849 | The meaning of @cite{O(x^3)} is ``a quantity which is negligibly small | |
5850 | if @cite{x^3} is considered negligibly small as @cite{x} goes to zero.'' | |
5851 | ||
5852 | The exercise is to create rewrite rules that simplify sums and products of | |
5853 | power series represented as @samp{@var{polynomial} + O(@var{var}^@var{n})}. | |
5854 | For example, given @samp{1 - x^2 / 2 + O(x^3)} and @samp{x - x^3 / 6 + O(x^4)} | |
5855 | on the stack, we want to be able to type @kbd{*} and get the result | |
5856 | @samp{x - 2:3 x^3 + O(x^4)}. Don't worry if the terms of the sum are | |
5857 | rearranged or if @kbd{a s} needs to be typed after rewriting. (This one | |
5858 | is rather tricky; the solution at the end of this chapter uses 6 rewrite | |
5859 | rules. Hint: The @samp{constant(x)} condition tests whether @samp{x} is | |
5860 | a number.) @xref{Rewrites Answer 7, 7}. (@bullet{}) | |
5861 | ||
5862 | @c [fix-ref Rewrite Rules] | |
5863 | @xref{Rewrite Rules}, for the whole story on rewrite rules. | |
5864 | ||
5865 | @node Programming Tutorial, Answers to Exercises, Algebra Tutorial, Tutorial | |
5866 | @section Programming Tutorial | |
5867 | ||
5868 | @noindent | |
5869 | The Calculator is written entirely in Emacs Lisp, a highly extensible | |
5870 | language. If you know Lisp, you can program the Calculator to do | |
5871 | anything you like. Rewrite rules also work as a powerful programming | |
5872 | system. But Lisp and rewrite rules take a while to master, and often | |
5873 | all you want to do is define a new function or repeat a command a few | |
5874 | times. Calc has features that allow you to do these things easily. | |
5875 | ||
5876 | (Note that the programming commands relating to user-defined keys | |
5877 | are not yet supported under Lucid Emacs 19.) | |
5878 | ||
5879 | One very limited form of programming is defining your own functions. | |
5880 | Calc's @kbd{Z F} command allows you to define a function name and | |
5881 | key sequence to correspond to any formula. Programming commands use | |
5882 | the shift-@kbd{Z} prefix; the user commands they create use the lower | |
5883 | case @kbd{z} prefix. | |
5884 | ||
d7b8e6c6 | 5885 | @smallexample |
5d67986c | 5886 | @group |
d7b8e6c6 EZ |
5887 | 1: 1 + x + x^2 / 2 + x^3 / 6 1: 1 + x + x^2 / 2 + x^3 / 6 |
5888 | . . | |
5889 | ||
5d67986c | 5890 | ' 1 + x + x^2/2! + x^3/3! @key{RET} Z F e myexp @key{RET} @key{RET} @key{RET} y |
d7b8e6c6 | 5891 | @end group |
5d67986c | 5892 | @end smallexample |
d7b8e6c6 EZ |
5893 | |
5894 | This polynomial is a Taylor series approximation to @samp{exp(x)}. | |
5895 | The @kbd{Z F} command asks a number of questions. The above answers | |
5896 | say that the key sequence for our function should be @kbd{z e}; the | |
5897 | @kbd{M-x} equivalent should be @code{calc-myexp}; the name of the | |
5898 | function in algebraic formulas should also be @code{myexp}; the | |
5899 | default argument list @samp{(x)} is acceptable; and finally @kbd{y} | |
5900 | answers the question ``leave it in symbolic form for non-constant | |
5901 | arguments?'' | |
5902 | ||
d7b8e6c6 | 5903 | @smallexample |
5d67986c | 5904 | @group |
d7b8e6c6 EZ |
5905 | 1: 1.3495 2: 1.3495 3: 1.3495 |
5906 | . 1: 1.34986 2: 1.34986 | |
5907 | . 1: myexp(a + 1) | |
5908 | . | |
5909 | ||
5d67986c | 5910 | .3 z e .3 E ' a+1 @key{RET} z e |
d7b8e6c6 | 5911 | @end group |
5d67986c | 5912 | @end smallexample |
d7b8e6c6 EZ |
5913 | |
5914 | @noindent | |
5915 | First we call our new @code{exp} approximation with 0.3 as an | |
5916 | argument, and compare it with the true @code{exp} function. Then | |
5917 | we note that, as requested, if we try to give @kbd{z e} an | |
5918 | argument that isn't a plain number, it leaves the @code{myexp} | |
5919 | function call in symbolic form. If we had answered @kbd{n} to the | |
5920 | final question, @samp{myexp(a + 1)} would have evaluated by plugging | |
5921 | in @samp{a + 1} for @samp{x} in the defining formula. | |
5922 | ||
5923 | @cindex Sine integral Si(x) | |
5d67986c RS |
5924 | @ignore |
5925 | @starindex | |
5926 | @end ignore | |
d7b8e6c6 EZ |
5927 | @tindex Si |
5928 | (@bullet{}) @strong{Exercise 1.} The ``sine integral'' function | |
5929 | @c{${\rm Si}(x)$} | |
5930 | @cite{Si(x)} is defined as the integral of @samp{sin(t)/t} for | |
5931 | @cite{t = 0} to @cite{x} in radians. (It was invented because this | |
5932 | integral has no solution in terms of basic functions; if you give it | |
5933 | to Calc's @kbd{a i} command, it will ponder it for a long time and then | |
5934 | give up.) We can use the numerical integration command, however, | |
5935 | which in algebraic notation is written like @samp{ninteg(f(t), t, 0, x)} | |
5936 | with any integrand @samp{f(t)}. Define a @kbd{z s} command and | |
5937 | @code{Si} function that implement this. You will need to edit the | |
5938 | default argument list a bit. As a test, @samp{Si(1)} should return | |
5939 | 0.946083. (Hint: @code{ninteg} will run a lot faster if you reduce | |
5940 | the precision to, say, six digits beforehand.) | |
5941 | @xref{Programming Answer 1, 1}. (@bullet{}) | |
5942 | ||
5943 | The simplest way to do real ``programming'' of Emacs is to define a | |
5944 | @dfn{keyboard macro}. A keyboard macro is simply a sequence of | |
5945 | keystrokes which Emacs has stored away and can play back on demand. | |
5946 | For example, if you find yourself typing @kbd{H a S x @key{RET}} often, | |
5947 | you may wish to program a keyboard macro to type this for you. | |
5948 | ||
d7b8e6c6 | 5949 | @smallexample |
5d67986c | 5950 | @group |
d7b8e6c6 EZ |
5951 | 1: y = sqrt(x) 1: x = y^2 |
5952 | . . | |
5953 | ||
5d67986c | 5954 | ' y=sqrt(x) @key{RET} C-x ( H a S x @key{RET} C-x ) |
d7b8e6c6 EZ |
5955 | |
5956 | 1: y = cos(x) 1: x = s1 arccos(y) + 2 pi n1 | |
5957 | . . | |
5958 | ||
5d67986c | 5959 | ' y=cos(x) @key{RET} X |
d7b8e6c6 | 5960 | @end group |
5d67986c | 5961 | @end smallexample |
d7b8e6c6 EZ |
5962 | |
5963 | @noindent | |
5964 | When you type @kbd{C-x (}, Emacs begins recording. But it is also | |
5965 | still ready to execute your keystrokes, so you're really ``training'' | |
5966 | Emacs by walking it through the procedure once. When you type | |
5967 | @w{@kbd{C-x )}}, the macro is recorded. You can now type @kbd{X} to | |
5968 | re-execute the same keystrokes. | |
5969 | ||
5970 | You can give a name to your macro by typing @kbd{Z K}. | |
5971 | ||
d7b8e6c6 | 5972 | @smallexample |
5d67986c | 5973 | @group |
d7b8e6c6 EZ |
5974 | 1: . 1: y = x^4 1: x = s2 sqrt(s1 sqrt(y)) |
5975 | . . | |
5976 | ||
5d67986c | 5977 | Z K x @key{RET} ' y=x^4 @key{RET} z x |
d7b8e6c6 | 5978 | @end group |
5d67986c | 5979 | @end smallexample |
d7b8e6c6 EZ |
5980 | |
5981 | @noindent | |
5982 | Notice that we use shift-@kbd{Z} to define the command, and lower-case | |
5983 | @kbd{z} to call it up. | |
5984 | ||
5985 | Keyboard macros can call other macros. | |
5986 | ||
d7b8e6c6 | 5987 | @smallexample |
5d67986c | 5988 | @group |
d7b8e6c6 EZ |
5989 | 1: abs(x) 1: x = s1 y 1: 2 / x 1: x = 2 / y |
5990 | . . . . | |
5991 | ||
5d67986c | 5992 | ' abs(x) @key{RET} C-x ( ' y @key{RET} a = z x C-x ) ' 2/x @key{RET} X |
d7b8e6c6 | 5993 | @end group |
5d67986c | 5994 | @end smallexample |
d7b8e6c6 EZ |
5995 | |
5996 | (@bullet{}) @strong{Exercise 2.} Define a keyboard macro to negate | |
5997 | the item in level 3 of the stack, without disturbing the rest of | |
5998 | the stack. @xref{Programming Answer 2, 2}. (@bullet{}) | |
5999 | ||
6000 | (@bullet{}) @strong{Exercise 3.} Define keyboard macros to compute | |
6001 | the following functions: | |
6002 | ||
6003 | @enumerate | |
6004 | @item | |
6005 | Compute @c{$\displaystyle{\sin x \over x}$} | |
6006 | @cite{sin(x) / x}, where @cite{x} is the number on the | |
6007 | top of the stack. | |
6008 | ||
6009 | @item | |
6010 | Compute the base-@cite{b} logarithm, just like the @kbd{B} key except | |
6011 | the arguments are taken in the opposite order. | |
6012 | ||
6013 | @item | |
6014 | Produce a vector of integers from 1 to the integer on the top of | |
6015 | the stack. | |
6016 | @end enumerate | |
6017 | @noindent | |
6018 | @xref{Programming Answer 3, 3}. (@bullet{}) | |
6019 | ||
6020 | (@bullet{}) @strong{Exercise 4.} Define a keyboard macro to compute | |
6021 | the average (mean) value of a list of numbers. | |
6022 | @xref{Programming Answer 4, 4}. (@bullet{}) | |
6023 | ||
6024 | In many programs, some of the steps must execute several times. | |
6025 | Calc has @dfn{looping} commands that allow this. Loops are useful | |
6026 | inside keyboard macros, but actually work at any time. | |
6027 | ||
d7b8e6c6 | 6028 | @smallexample |
5d67986c | 6029 | @group |
d7b8e6c6 EZ |
6030 | 1: x^6 2: x^6 1: 360 x^2 |
6031 | . 1: 4 . | |
6032 | . | |
6033 | ||
5d67986c | 6034 | ' x^6 @key{RET} 4 Z < a d x @key{RET} Z > |
d7b8e6c6 | 6035 | @end group |
5d67986c | 6036 | @end smallexample |
d7b8e6c6 EZ |
6037 | |
6038 | @noindent | |
6039 | Here we have computed the fourth derivative of @cite{x^6} by | |
6040 | enclosing a derivative command in a ``repeat loop'' structure. | |
6041 | This structure pops a repeat count from the stack, then | |
6042 | executes the body of the loop that many times. | |
6043 | ||
6044 | If you make a mistake while entering the body of the loop, | |
6045 | type @w{@kbd{Z C-g}} to cancel the loop command. | |
6046 | ||
6047 | @cindex Fibonacci numbers | |
6048 | Here's another example: | |
6049 | ||
d7b8e6c6 | 6050 | @smallexample |
5d67986c | 6051 | @group |
d7b8e6c6 EZ |
6052 | 3: 1 2: 10946 |
6053 | 2: 1 1: 17711 | |
6054 | 1: 20 . | |
6055 | . | |
6056 | ||
5d67986c | 6057 | 1 @key{RET} @key{RET} 20 Z < @key{TAB} C-j + Z > |
d7b8e6c6 | 6058 | @end group |
5d67986c | 6059 | @end smallexample |
d7b8e6c6 EZ |
6060 | |
6061 | @noindent | |
6062 | The numbers in levels 2 and 1 should be the 21st and 22nd Fibonacci | |
6063 | numbers, respectively. (To see what's going on, try a few repetitions | |
6064 | of the loop body by hand; @kbd{C-j}, also on the Line-Feed or @key{LFD} | |
6065 | key if you have one, makes a copy of the number in level 2.) | |
6066 | ||
6067 | @cindex Golden ratio | |
6068 | @cindex Phi, golden ratio | |
6069 | A fascinating property of the Fibonacci numbers is that the @cite{n}th | |
6070 | Fibonacci number can be found directly by computing @c{$\phi^n / \sqrt{5}$} | |
6071 | @cite{phi^n / sqrt(5)} | |
6072 | and then rounding to the nearest integer, where @c{$\phi$ (``phi'')} | |
6073 | @cite{phi}, the | |
6074 | ``golden ratio,'' is @c{$(1 + \sqrt{5}) / 2$} | |
6075 | @cite{(1 + sqrt(5)) / 2}. (For convenience, this constant is available | |
6076 | from the @code{phi} variable, or the @kbd{I H P} command.) | |
6077 | ||
d7b8e6c6 | 6078 | @smallexample |
5d67986c | 6079 | @group |
d7b8e6c6 EZ |
6080 | 1: 1.61803 1: 24476.0000409 1: 10945.9999817 1: 10946 |
6081 | . . . . | |
6082 | ||
6083 | I H P 21 ^ 5 Q / R | |
d7b8e6c6 | 6084 | @end group |
5d67986c | 6085 | @end smallexample |
d7b8e6c6 EZ |
6086 | |
6087 | @cindex Continued fractions | |
6088 | (@bullet{}) @strong{Exercise 5.} The @dfn{continued fraction} | |
6089 | representation of @c{$\phi$} | |
6090 | @cite{phi} is @c{$1 + 1/(1 + 1/(1 + 1/( \ldots )))$} | |
6091 | @cite{1 + 1/(1 + 1/(1 + 1/( ...@: )))}. | |
6092 | We can compute an approximate value by carrying this however far | |
6093 | and then replacing the innermost @c{$1/( \ldots )$} | |
6094 | @cite{1/( ...@: )} by 1. Approximate | |
6095 | @c{$\phi$} | |
6096 | @cite{phi} using a twenty-term continued fraction. | |
6097 | @xref{Programming Answer 5, 5}. (@bullet{}) | |
6098 | ||
6099 | (@bullet{}) @strong{Exercise 6.} Linear recurrences like the one for | |
6100 | Fibonacci numbers can be expressed in terms of matrices. Given a | |
6101 | vector @w{@cite{[a, b]}} determine a matrix which, when multiplied by this | |
6102 | vector, produces the vector @cite{[b, c]}, where @cite{a}, @cite{b} and | |
6103 | @cite{c} are three successive Fibonacci numbers. Now write a program | |
6104 | that, given an integer @cite{n}, computes the @cite{n}th Fibonacci number | |
6105 | using matrix arithmetic. @xref{Programming Answer 6, 6}. (@bullet{}) | |
6106 | ||
6107 | @cindex Harmonic numbers | |
6108 | A more sophisticated kind of loop is the @dfn{for} loop. Suppose | |
6109 | we wish to compute the 20th ``harmonic'' number, which is equal to | |
6110 | the sum of the reciprocals of the integers from 1 to 20. | |
6111 | ||
d7b8e6c6 | 6112 | @smallexample |
5d67986c | 6113 | @group |
d7b8e6c6 EZ |
6114 | 3: 0 1: 3.597739 |
6115 | 2: 1 . | |
6116 | 1: 20 | |
6117 | . | |
6118 | ||
5d67986c | 6119 | 0 @key{RET} 1 @key{RET} 20 Z ( & + 1 Z ) |
d7b8e6c6 | 6120 | @end group |
5d67986c | 6121 | @end smallexample |
d7b8e6c6 EZ |
6122 | |
6123 | @noindent | |
6124 | The ``for'' loop pops two numbers, the lower and upper limits, then | |
6125 | repeats the body of the loop as an internal counter increases from | |
6126 | the lower limit to the upper one. Just before executing the loop | |
6127 | body, it pushes the current loop counter. When the loop body | |
6128 | finishes, it pops the ``step,'' i.e., the amount by which to | |
6129 | increment the loop counter. As you can see, our loop always | |
6130 | uses a step of one. | |
6131 | ||
6132 | This harmonic number function uses the stack to hold the running | |
6133 | total as well as for the various loop housekeeping functions. If | |
6134 | you find this disorienting, you can sum in a variable instead: | |
6135 | ||
d7b8e6c6 | 6136 | @smallexample |
5d67986c | 6137 | @group |
d7b8e6c6 EZ |
6138 | 1: 0 2: 1 . 1: 3.597739 |
6139 | . 1: 20 . | |
6140 | . | |
6141 | ||
5d67986c | 6142 | 0 t 7 1 @key{RET} 20 Z ( & s + 7 1 Z ) r 7 |
d7b8e6c6 | 6143 | @end group |
5d67986c | 6144 | @end smallexample |
d7b8e6c6 EZ |
6145 | |
6146 | @noindent | |
6147 | The @kbd{s +} command adds the top-of-stack into the value in a | |
6148 | variable (and removes that value from the stack). | |
6149 | ||
6150 | It's worth noting that many jobs that call for a ``for'' loop can | |
6151 | also be done more easily by Calc's high-level operations. Two | |
6152 | other ways to compute harmonic numbers are to use vector mapping | |
6153 | and reduction (@kbd{v x 20}, then @w{@kbd{V M &}}, then @kbd{V R +}), | |
6154 | or to use the summation command @kbd{a +}. Both of these are | |
6155 | probably easier than using loops. However, there are some | |
6156 | situations where loops really are the way to go: | |
6157 | ||
6158 | (@bullet{}) @strong{Exercise 7.} Use a ``for'' loop to find the first | |
6159 | harmonic number which is greater than 4.0. | |
6160 | @xref{Programming Answer 7, 7}. (@bullet{}) | |
6161 | ||
6162 | Of course, if we're going to be using variables in our programs, | |
6163 | we have to worry about the programs clobbering values that the | |
6164 | caller was keeping in those same variables. This is easy to | |
6165 | fix, though: | |
6166 | ||
d7b8e6c6 | 6167 | @smallexample |
5d67986c | 6168 | @group |
d7b8e6c6 EZ |
6169 | . 1: 0.6667 1: 0.6667 3: 0.6667 |
6170 | . . 2: 3.597739 | |
6171 | 1: 0.6667 | |
6172 | . | |
6173 | ||
5d67986c | 6174 | Z ` p 4 @key{RET} 2 @key{RET} 3 / s 7 s s a @key{RET} Z ' r 7 s r a @key{RET} |
d7b8e6c6 | 6175 | @end group |
5d67986c | 6176 | @end smallexample |
d7b8e6c6 EZ |
6177 | |
6178 | @noindent | |
6179 | When we type @kbd{Z `} (that's a back-quote character), Calc saves | |
6180 | its mode settings and the contents of the ten ``quick variables'' | |
6181 | for later reference. When we type @kbd{Z '} (that's an apostrophe | |
6182 | now), Calc restores those saved values. Thus the @kbd{p 4} and | |
6183 | @kbd{s 7} commands have no effect outside this sequence. Wrapping | |
6184 | this around the body of a keyboard macro ensures that it doesn't | |
6185 | interfere with what the user of the macro was doing. Notice that | |
6186 | the contents of the stack, and the values of named variables, | |
6187 | survive past the @kbd{Z '} command. | |
6188 | ||
6189 | @cindex Bernoulli numbers, approximate | |
6190 | The @dfn{Bernoulli numbers} are a sequence with the interesting | |
6191 | property that all of the odd Bernoulli numbers are zero, and the | |
6192 | even ones, while difficult to compute, can be roughly approximated | |
6193 | by the formula @c{$\displaystyle{2 n! \over (2 \pi)^n}$} | |
6194 | @cite{2 n!@: / (2 pi)^n}. Let's write a keyboard | |
6195 | macro to compute (approximate) Bernoulli numbers. (Calc has a | |
6196 | command, @kbd{k b}, to compute exact Bernoulli numbers, but | |
6197 | this command is very slow for large @cite{n} since the higher | |
6198 | Bernoulli numbers are very large fractions.) | |
6199 | ||
d7b8e6c6 | 6200 | @smallexample |
5d67986c | 6201 | @group |
d7b8e6c6 EZ |
6202 | 1: 10 1: 0.0756823 |
6203 | . . | |
6204 | ||
5d67986c | 6205 | 10 C-x ( @key{RET} 2 % Z [ @key{DEL} 0 Z : ' 2 $! / (2 pi)^$ @key{RET} = Z ] C-x ) |
d7b8e6c6 | 6206 | @end group |
5d67986c | 6207 | @end smallexample |
d7b8e6c6 EZ |
6208 | |
6209 | @noindent | |
6210 | You can read @kbd{Z [} as ``then,'' @kbd{Z :} as ``else,'' and | |
6211 | @kbd{Z ]} as ``end-if.'' There is no need for an explicit ``if'' | |
6212 | command. For the purposes of @w{@kbd{Z [}}, the condition is ``true'' | |
6213 | if the value it pops from the stack is a nonzero number, or ``false'' | |
6214 | if it pops zero or something that is not a number (like a formula). | |
6215 | Here we take our integer argument modulo 2; this will be nonzero | |
6216 | if we're asking for an odd Bernoulli number. | |
6217 | ||
6218 | The actual tenth Bernoulli number is @cite{5/66}. | |
6219 | ||
d7b8e6c6 | 6220 | @smallexample |
5d67986c | 6221 | @group |
d7b8e6c6 EZ |
6222 | 3: 0.0756823 1: 0 1: 0.25305 1: 0 1: 1.16659 |
6223 | 2: 5:66 . . . . | |
6224 | 1: 0.0757575 | |
6225 | . | |
6226 | ||
5d67986c | 6227 | 10 k b @key{RET} c f M-0 @key{DEL} 11 X @key{DEL} 12 X @key{DEL} 13 X @key{DEL} 14 X |
d7b8e6c6 | 6228 | @end group |
5d67986c | 6229 | @end smallexample |
d7b8e6c6 EZ |
6230 | |
6231 | Just to exercise loops a bit more, let's compute a table of even | |
6232 | Bernoulli numbers. | |
6233 | ||
d7b8e6c6 | 6234 | @smallexample |
5d67986c | 6235 | @group |
d7b8e6c6 EZ |
6236 | 3: [] 1: [0.10132, 0.03079, 0.02340, 0.033197, ...] |
6237 | 2: 2 . | |
6238 | 1: 30 | |
6239 | . | |
6240 | ||
5d67986c | 6241 | [ ] 2 @key{RET} 30 Z ( X | 2 Z ) |
d7b8e6c6 | 6242 | @end group |
5d67986c | 6243 | @end smallexample |
d7b8e6c6 EZ |
6244 | |
6245 | @noindent | |
6246 | The vertical-bar @kbd{|} is the vector-concatenation command. When | |
6247 | we execute it, the list we are building will be in stack level 2 | |
6248 | (initially this is an empty list), and the next Bernoulli number | |
6249 | will be in level 1. The effect is to append the Bernoulli number | |
6250 | onto the end of the list. (To create a table of exact fractional | |
6251 | Bernoulli numbers, just replace @kbd{X} with @kbd{k b} in the above | |
6252 | sequence of keystrokes.) | |
6253 | ||
6254 | With loops and conditionals, you can program essentially anything | |
6255 | in Calc. One other command that makes looping easier is @kbd{Z /}, | |
6256 | which takes a condition from the stack and breaks out of the enclosing | |
6257 | loop if the condition is true (non-zero). You can use this to make | |
6258 | ``while'' and ``until'' style loops. | |
6259 | ||
6260 | If you make a mistake when entering a keyboard macro, you can edit | |
6261 | it using @kbd{Z E}. First, you must attach it to a key with @kbd{Z K}. | |
6262 | One technique is to enter a throwaway dummy definition for the macro, | |
6263 | then enter the real one in the edit command. | |
6264 | ||
d7b8e6c6 | 6265 | @smallexample |
5d67986c | 6266 | @group |
d7b8e6c6 | 6267 | 1: 3 1: 3 Keyboard Macro Editor. |
5d67986c | 6268 | . . Original keys: 1 @key{RET} 2 + |
d7b8e6c6 EZ |
6269 | |
6270 | type "1\r" | |
6271 | type "2" | |
6272 | calc-plus | |
6273 | ||
5d67986c | 6274 | C-x ( 1 @key{RET} 2 + C-x ) Z K h @key{RET} Z E h |
d7b8e6c6 | 6275 | @end group |
5d67986c | 6276 | @end smallexample |
d7b8e6c6 EZ |
6277 | |
6278 | @noindent | |
6279 | This shows the screen display assuming you have the @file{macedit} | |
6280 | keyboard macro editing package installed, which is usually the case | |
6281 | since a copy of @file{macedit} comes bundled with Calc. | |
6282 | ||
6283 | A keyboard macro is stored as a pure keystroke sequence. The | |
6284 | @file{macedit} package (invoked by @kbd{Z E}) scans along the | |
6285 | macro and tries to decode it back into human-readable steps. | |
6286 | If a key or keys are simply shorthand for some command with a | |
6287 | @kbd{M-x} name, that name is shown. Anything that doesn't correspond | |
6288 | to a @kbd{M-x} command is written as a @samp{type} command. | |
6289 | ||
6290 | Let's edit in a new definition, for computing harmonic numbers. | |
6291 | First, erase the three lines of the old definition. Then, type | |
6292 | in the new definition (or use Emacs @kbd{M-w} and @kbd{C-y} commands | |
6293 | to copy it from this page of the Info file; you can skip typing | |
6294 | the comments that begin with @samp{#}). | |
6295 | ||
6296 | @smallexample | |
6297 | calc-kbd-push # Save local values (Z `) | |
6298 | type "0" # Push a zero | |
6299 | calc-store-into # Store it in variable 1 | |
6300 | type "1" | |
6301 | type "1" # Initial value for loop | |
5d67986c | 6302 | calc-roll-down # This is the @key{TAB} key; swap initial & final |
d7b8e6c6 EZ |
6303 | calc-kbd-for # Begin "for" loop... |
6304 | calc-inv # Take reciprocal | |
6305 | calc-store-plus # Add to accumulator | |
6306 | type "1" | |
6307 | type "1" # Loop step is 1 | |
6308 | calc-kbd-end-for # End "for" loop | |
6309 | calc-recall # Now recall final accumulated value | |
6310 | type "1" | |
6311 | calc-kbd-pop # Restore values (Z ') | |
6312 | @end smallexample | |
6313 | ||
6314 | @noindent | |
6315 | Press @kbd{M-# M-#} to finish editing and return to the Calculator. | |
6316 | ||
d7b8e6c6 | 6317 | @smallexample |
5d67986c | 6318 | @group |
d7b8e6c6 EZ |
6319 | 1: 20 1: 3.597739 |
6320 | . . | |
6321 | ||
6322 | 20 z h | |
d7b8e6c6 | 6323 | @end group |
5d67986c | 6324 | @end smallexample |
d7b8e6c6 EZ |
6325 | |
6326 | If you don't know how to write a particular command in @file{macedit} | |
6327 | format, you can always write it as keystrokes in a @code{type} command. | |
6328 | There is also a @code{keys} command which interprets the rest of the | |
6329 | line as standard Emacs keystroke names. In fact, @file{macedit} defines | |
6330 | a handy @code{read-kbd-macro} command which reads the current region | |
6331 | of the current buffer as a sequence of keystroke names, and defines that | |
6332 | sequence on the @kbd{X} (and @kbd{C-x e}) key. Because this is so | |
6333 | useful, Calc puts this command on the @kbd{M-# m} key. Try reading in | |
5d67986c | 6334 | this macro in the following form: Press @kbd{C-@@} (or @kbd{C-@key{SPC}}) at |
d7b8e6c6 EZ |
6335 | one end of the text below, then type @kbd{M-# m} at the other. |
6336 | ||
d7b8e6c6 | 6337 | @example |
5d67986c | 6338 | @group |
d7b8e6c6 | 6339 | Z ` 0 t 1 |
5d67986c | 6340 | 1 @key{TAB} |
d7b8e6c6 EZ |
6341 | Z ( & s + 1 1 Z ) |
6342 | r 1 | |
6343 | Z ' | |
d7b8e6c6 | 6344 | @end group |
5d67986c | 6345 | @end example |
d7b8e6c6 EZ |
6346 | |
6347 | (@bullet{}) @strong{Exercise 8.} A general algorithm for solving | |
6348 | equations numerically is @dfn{Newton's Method}. Given the equation | |
6349 | @cite{f(x) = 0} for any function @cite{f}, and an initial guess | |
6350 | @cite{x_0} which is reasonably close to the desired solution, apply | |
6351 | this formula over and over: | |
6352 | ||
6353 | @ifinfo | |
6354 | @example | |
6355 | new_x = x - f(x)/f'(x) | |
6356 | @end example | |
6357 | @end ifinfo | |
6358 | @tex | |
6359 | \beforedisplay | |
6360 | $$ x_{\goodrm new} = x - {f(x) \over f'(x)} $$ | |
6361 | \afterdisplay | |
6362 | @end tex | |
6363 | ||
6364 | @noindent | |
6365 | where @cite{f'(x)} is the derivative of @cite{f}. The @cite{x} | |
6366 | values will quickly converge to a solution, i.e., eventually | |
6367 | @c{$x_{\rm new}$} | |
6368 | @cite{new_x} and @cite{x} will be equal to within the limits | |
6369 | of the current precision. Write a program which takes a formula | |
6370 | involving the variable @cite{x}, and an initial guess @cite{x_0}, | |
6371 | on the stack, and produces a value of @cite{x} for which the formula | |
6372 | is zero. Use it to find a solution of @c{$\sin(\cos x) = 0.5$} | |
6373 | @cite{sin(cos(x)) = 0.5} | |
6374 | near @cite{x = 4.5}. (Use angles measured in radians.) Note that | |
6375 | the built-in @w{@kbd{a R}} (@code{calc-find-root}) command uses Newton's | |
6376 | method when it is able. @xref{Programming Answer 8, 8}. (@bullet{}) | |
6377 | ||
6378 | @cindex Digamma function | |
6379 | @cindex Gamma constant, Euler's | |
6380 | @cindex Euler's gamma constant | |
6381 | (@bullet{}) @strong{Exercise 9.} The @dfn{digamma} function @c{$\psi(z)$ (``psi'')} | |
6382 | @cite{psi(z)} | |
6383 | is defined as the derivative of @c{$\ln \Gamma(z)$} | |
6384 | @cite{ln(gamma(z))}. For large | |
6385 | values of @cite{z}, it can be approximated by the infinite sum | |
6386 | ||
6387 | @ifinfo | |
6388 | @example | |
6389 | psi(z) ~= ln(z) - 1/2z - sum(bern(2 n) / 2 n z^(2 n), n, 1, inf) | |
6390 | @end example | |
6391 | @end ifinfo | |
6392 | @tex | |
6393 | \let\rm\goodrm | |
6394 | \beforedisplay | |
6395 | $$ \psi(z) \approx \ln z - {1\over2z} - | |
6396 | \sum_{n=1}^\infty {\code{bern}(2 n) \over 2 n z^{2n}} | |
6397 | $$ | |
6398 | \afterdisplay | |
6399 | @end tex | |
6400 | ||
6401 | @noindent | |
6402 | where @c{$\sum$} | |
6403 | @cite{sum} represents the sum over @cite{n} from 1 to infinity | |
6404 | (or to some limit high enough to give the desired accuracy), and | |
6405 | the @code{bern} function produces (exact) Bernoulli numbers. | |
6406 | While this sum is not guaranteed to converge, in practice it is safe. | |
6407 | An interesting mathematical constant is Euler's gamma, which is equal | |
6408 | to about 0.5772. One way to compute it is by the formula, | |
6409 | @c{$\gamma = -\psi(1)$} | |
6410 | @cite{gamma = -psi(1)}. Unfortunately, 1 isn't a large enough argument | |
6411 | for the above formula to work (5 is a much safer value for @cite{z}). | |
6412 | Fortunately, we can compute @c{$\psi(1)$} | |
6413 | @cite{psi(1)} from @c{$\psi(5)$} | |
6414 | @cite{psi(5)} using | |
6415 | the recurrence @c{$\psi(z+1) = \psi(z) + {1 \over z}$} | |
6416 | @cite{psi(z+1) = psi(z) + 1/z}. Your task: Develop | |
6417 | a program to compute @c{$\psi(z)$} | |
6418 | @cite{psi(z)}; it should ``pump up'' @cite{z} | |
6419 | if necessary to be greater than 5, then use the above summation | |
6420 | formula. Use looping commands to compute the sum. Use your function | |
6421 | to compute @c{$\gamma$} | |
6422 | @cite{gamma} to twelve decimal places. (Calc has a built-in command | |
6423 | for Euler's constant, @kbd{I P}, which you can use to check your answer.) | |
6424 | @xref{Programming Answer 9, 9}. (@bullet{}) | |
6425 | ||
6426 | @cindex Polynomial, list of coefficients | |
6427 | (@bullet{}) @strong{Exercise 10.} Given a polynomial in @cite{x} and | |
6428 | a number @cite{m} on the stack, where the polynomial is of degree | |
6429 | @cite{m} or less (i.e., does not have any terms higher than @cite{x^m}), | |
6430 | write a program to convert the polynomial into a list-of-coefficients | |
6431 | notation. For example, @cite{5 x^4 + (x + 1)^2} with @cite{m = 6} | |
6432 | should produce the list @cite{[1, 2, 1, 0, 5, 0, 0]}. Also develop | |
6433 | a way to convert from this form back to the standard algebraic form. | |
6434 | @xref{Programming Answer 10, 10}. (@bullet{}) | |
6435 | ||
6436 | @cindex Recursion | |
6437 | (@bullet{}) @strong{Exercise 11.} The @dfn{Stirling numbers of the | |
6438 | first kind} are defined by the recurrences, | |
6439 | ||
6440 | @ifinfo | |
6441 | @example | |
6442 | s(n,n) = 1 for n >= 0, | |
6443 | s(n,0) = 0 for n > 0, | |
6444 | s(n+1,m) = s(n,m-1) - n s(n,m) for n >= m >= 1. | |
6445 | @end example | |
6446 | @end ifinfo | |
6447 | @tex | |
6448 | \turnoffactive | |
6449 | \beforedisplay | |
6450 | $$ \eqalign{ s(n,n) &= 1 \qquad \hbox{for } n \ge 0, \cr | |
6451 | s(n,0) &= 0 \qquad \hbox{for } n > 0, \cr | |
6452 | s(n+1,m) &= s(n,m-1) - n \, s(n,m) \qquad | |
6453 | \hbox{for } n \ge m \ge 1.} | |
6454 | $$ | |
6455 | \afterdisplay | |
6456 | \vskip5pt | |
6457 | (These numbers are also sometimes written $\displaystyle{n \brack m}$.) | |
6458 | @end tex | |
6459 | ||
6460 | This can be implemented using a @dfn{recursive} program in Calc; the | |
6461 | program must invoke itself in order to calculate the two righthand | |
6462 | terms in the general formula. Since it always invokes itself with | |
6463 | ``simpler'' arguments, it's easy to see that it must eventually finish | |
6464 | the computation. Recursion is a little difficult with Emacs keyboard | |
6465 | macros since the macro is executed before its definition is complete. | |
6466 | So here's the recommended strategy: Create a ``dummy macro'' and assign | |
6467 | it to a key with, e.g., @kbd{Z K s}. Now enter the true definition, | |
6468 | using the @kbd{z s} command to call itself recursively, then assign it | |
6469 | to the same key with @kbd{Z K s}. Now the @kbd{z s} command will run | |
6470 | the complete recursive program. (Another way is to use @w{@kbd{Z E}} | |
6471 | or @kbd{M-# m} (@code{read-kbd-macro}) to read the whole macro at once, | |
6472 | thus avoiding the ``training'' phase.) The task: Write a program | |
6473 | that computes Stirling numbers of the first kind, given @cite{n} and | |
6474 | @cite{m} on the stack. Test it with @emph{small} inputs like | |
6475 | @cite{s(4,2)}. (There is a built-in command for Stirling numbers, | |
6476 | @kbd{k s}, which you can use to check your answers.) | |
6477 | @xref{Programming Answer 11, 11}. (@bullet{}) | |
6478 | ||
6479 | The programming commands we've seen in this part of the tutorial | |
6480 | are low-level, general-purpose operations. Often you will find | |
6481 | that a higher-level function, such as vector mapping or rewrite | |
6482 | rules, will do the job much more easily than a detailed, step-by-step | |
6483 | program can: | |
6484 | ||
6485 | (@bullet{}) @strong{Exercise 12.} Write another program for | |
6486 | computing Stirling numbers of the first kind, this time using | |
6487 | rewrite rules. Once again, @cite{n} and @cite{m} should be taken | |
6488 | from the stack. @xref{Programming Answer 12, 12}. (@bullet{}) | |
6489 | ||
6490 | @example | |
6491 | ||
6492 | @end example | |
6493 | This ends the tutorial section of the Calc manual. Now you know enough | |
6494 | about Calc to use it effectively for many kinds of calculations. But | |
6495 | Calc has many features that were not even touched upon in this tutorial. | |
6496 | @c [not-split] | |
6497 | The rest of this manual tells the whole story. | |
6498 | @c [when-split] | |
6499 | @c Volume II of this manual, the @dfn{Calc Reference}, tells the whole story. | |
6500 | ||
6501 | @page | |
6502 | @node Answers to Exercises, , Programming Tutorial, Tutorial | |
6503 | @section Answers to Exercises | |
6504 | ||
6505 | @noindent | |
6506 | This section includes answers to all the exercises in the Calc tutorial. | |
6507 | ||
6508 | @menu | |
5d67986c | 6509 | * RPN Answer 1:: 1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * - |
d7b8e6c6 EZ |
6510 | * RPN Answer 2:: 2*4 + 7*9.5 + 5/4 |
6511 | * RPN Answer 3:: Operating on levels 2 and 3 | |
6512 | * RPN Answer 4:: Joe's complex problems | |
6513 | * Algebraic Answer 1:: Simulating Q command | |
6514 | * Algebraic Answer 2:: Joe's algebraic woes | |
6515 | * Algebraic Answer 3:: 1 / 0 | |
6516 | * Modes Answer 1:: 3#0.1 = 3#0.0222222? | |
6517 | * Modes Answer 2:: 16#f.e8fe15 | |
6518 | * Modes Answer 3:: Joe's rounding bug | |
6519 | * Modes Answer 4:: Why floating point? | |
6520 | * Arithmetic Answer 1:: Why the \ command? | |
6521 | * Arithmetic Answer 2:: Tripping up the B command | |
6522 | * Vector Answer 1:: Normalizing a vector | |
6523 | * Vector Answer 2:: Average position | |
6524 | * Matrix Answer 1:: Row and column sums | |
6525 | * Matrix Answer 2:: Symbolic system of equations | |
6526 | * Matrix Answer 3:: Over-determined system | |
6527 | * List Answer 1:: Powers of two | |
6528 | * List Answer 2:: Least-squares fit with matrices | |
6529 | * List Answer 3:: Geometric mean | |
6530 | * List Answer 4:: Divisor function | |
6531 | * List Answer 5:: Duplicate factors | |
6532 | * List Answer 6:: Triangular list | |
6533 | * List Answer 7:: Another triangular list | |
6534 | * List Answer 8:: Maximum of Bessel function | |
6535 | * List Answer 9:: Integers the hard way | |
6536 | * List Answer 10:: All elements equal | |
6537 | * List Answer 11:: Estimating pi with darts | |
6538 | * List Answer 12:: Estimating pi with matchsticks | |
6539 | * List Answer 13:: Hash codes | |
6540 | * List Answer 14:: Random walk | |
6541 | * Types Answer 1:: Square root of pi times rational | |
6542 | * Types Answer 2:: Infinities | |
6543 | * Types Answer 3:: What can "nan" be? | |
6544 | * Types Answer 4:: Abbey Road | |
6545 | * Types Answer 5:: Friday the 13th | |
6546 | * Types Answer 6:: Leap years | |
6547 | * Types Answer 7:: Erroneous donut | |
6548 | * Types Answer 8:: Dividing intervals | |
6549 | * Types Answer 9:: Squaring intervals | |
6550 | * Types Answer 10:: Fermat's primality test | |
6551 | * Types Answer 11:: pi * 10^7 seconds | |
6552 | * Types Answer 12:: Abbey Road on CD | |
6553 | * Types Answer 13:: Not quite pi * 10^7 seconds | |
6554 | * Types Answer 14:: Supercomputers and c | |
6555 | * Types Answer 15:: Sam the Slug | |
6556 | * Algebra Answer 1:: Squares and square roots | |
6557 | * Algebra Answer 2:: Building polynomial from roots | |
6558 | * Algebra Answer 3:: Integral of x sin(pi x) | |
6559 | * Algebra Answer 4:: Simpson's rule | |
6560 | * Rewrites Answer 1:: Multiplying by conjugate | |
6561 | * Rewrites Answer 2:: Alternative fib rule | |
6562 | * Rewrites Answer 3:: Rewriting opt(a) + opt(b) x | |
6563 | * Rewrites Answer 4:: Sequence of integers | |
6564 | * Rewrites Answer 5:: Number of terms in sum | |
6565 | * Rewrites Answer 6:: Defining 0^0 = 1 | |
6566 | * Rewrites Answer 7:: Truncated Taylor series | |
6567 | * Programming Answer 1:: Fresnel's C(x) | |
6568 | * Programming Answer 2:: Negate third stack element | |
6569 | * Programming Answer 3:: Compute sin(x) / x, etc. | |
6570 | * Programming Answer 4:: Average value of a list | |
6571 | * Programming Answer 5:: Continued fraction phi | |
6572 | * Programming Answer 6:: Matrix Fibonacci numbers | |
6573 | * Programming Answer 7:: Harmonic number greater than 4 | |
6574 | * Programming Answer 8:: Newton's method | |
6575 | * Programming Answer 9:: Digamma function | |
6576 | * Programming Answer 10:: Unpacking a polynomial | |
6577 | * Programming Answer 11:: Recursive Stirling numbers | |
6578 | * Programming Answer 12:: Stirling numbers with rewrites | |
6579 | @end menu | |
6580 | ||
6581 | @c The following kludgery prevents the individual answers from | |
6582 | @c being entered on the table of contents. | |
6583 | @tex | |
6584 | \global\let\oldwrite=\write | |
6585 | \gdef\skipwrite#1#2{\let\write=\oldwrite} | |
6586 | \global\let\oldchapternofonts=\chapternofonts | |
6587 | \gdef\chapternofonts{\let\write=\skipwrite\oldchapternofonts} | |
6588 | @end tex | |
6589 | ||
6590 | @node RPN Answer 1, RPN Answer 2, Answers to Exercises, Answers to Exercises | |
6591 | @subsection RPN Tutorial Exercise 1 | |
6592 | ||
6593 | @noindent | |
6594 | @kbd{1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -} | |
6595 | ||
6596 | The result is @c{$1 - (2 \times (3 + 4)) = -13$} | |
6597 | @cite{1 - (2 * (3 + 4)) = -13}. | |
6598 | ||
6599 | @node RPN Answer 2, RPN Answer 3, RPN Answer 1, Answers to Exercises | |
6600 | @subsection RPN Tutorial Exercise 2 | |
6601 | ||
6602 | @noindent | |
6603 | @c{$2\times4 + 7\times9.5 + {5\over4} = 75.75$} | |
6604 | @cite{2*4 + 7*9.5 + 5/4 = 75.75} | |
6605 | ||
6606 | After computing the intermediate term @c{$2\times4 = 8$} | |
6607 | @cite{2*4 = 8}, you can leave | |
6608 | that result on the stack while you compute the second term. With | |
6609 | both of these results waiting on the stack you can then compute the | |
6610 | final term, then press @kbd{+ +} to add everything up. | |
6611 | ||
d7b8e6c6 | 6612 | @smallexample |
5d67986c | 6613 | @group |
d7b8e6c6 EZ |
6614 | 2: 2 1: 8 3: 8 2: 8 |
6615 | 1: 4 . 2: 7 1: 66.5 | |
6616 | . 1: 9.5 . | |
6617 | . | |
6618 | ||
5d67986c | 6619 | 2 @key{RET} 4 * 7 @key{RET} 9.5 * |
d7b8e6c6 | 6620 | |
d7b8e6c6 | 6621 | @end group |
5d67986c | 6622 | @end smallexample |
d7b8e6c6 | 6623 | @noindent |
d7b8e6c6 | 6624 | @smallexample |
5d67986c | 6625 | @group |
d7b8e6c6 EZ |
6626 | 4: 8 3: 8 2: 8 1: 75.75 |
6627 | 3: 66.5 2: 66.5 1: 67.75 . | |
6628 | 2: 5 1: 1.25 . | |
6629 | 1: 4 . | |
6630 | . | |
6631 | ||
5d67986c | 6632 | 5 @key{RET} 4 / + + |
d7b8e6c6 | 6633 | @end group |
5d67986c | 6634 | @end smallexample |
d7b8e6c6 EZ |
6635 | |
6636 | Alternatively, you could add the first two terms before going on | |
6637 | with the third term. | |
6638 | ||
d7b8e6c6 | 6639 | @smallexample |
5d67986c | 6640 | @group |
d7b8e6c6 EZ |
6641 | 2: 8 1: 74.5 3: 74.5 2: 74.5 1: 75.75 |
6642 | 1: 66.5 . 2: 5 1: 1.25 . | |
6643 | . 1: 4 . | |
6644 | . | |
6645 | ||
5d67986c | 6646 | ... + 5 @key{RET} 4 / + |
d7b8e6c6 | 6647 | @end group |
5d67986c | 6648 | @end smallexample |
d7b8e6c6 EZ |
6649 | |
6650 | On an old-style RPN calculator this second method would have the | |
6651 | advantage of using only three stack levels. But since Calc's stack | |
6652 | can grow arbitrarily large this isn't really an issue. Which method | |
6653 | you choose is purely a matter of taste. | |
6654 | ||
6655 | @node RPN Answer 3, RPN Answer 4, RPN Answer 2, Answers to Exercises | |
6656 | @subsection RPN Tutorial Exercise 3 | |
6657 | ||
6658 | @noindent | |
6659 | The @key{TAB} key provides a way to operate on the number in level 2. | |
6660 | ||
d7b8e6c6 | 6661 | @smallexample |
5d67986c | 6662 | @group |
d7b8e6c6 EZ |
6663 | 3: 10 3: 10 4: 10 3: 10 3: 10 |
6664 | 2: 20 2: 30 3: 30 2: 30 2: 21 | |
6665 | 1: 30 1: 20 2: 20 1: 21 1: 30 | |
6666 | . . 1: 1 . . | |
6667 | . | |
6668 | ||
5d67986c | 6669 | @key{TAB} 1 + @key{TAB} |
d7b8e6c6 | 6670 | @end group |
5d67986c | 6671 | @end smallexample |
d7b8e6c6 | 6672 | |
5d67986c | 6673 | Similarly, @kbd{M-@key{TAB}} gives you access to the number in level 3. |
d7b8e6c6 | 6674 | |
d7b8e6c6 | 6675 | @smallexample |
5d67986c | 6676 | @group |
d7b8e6c6 EZ |
6677 | 3: 10 3: 21 3: 21 3: 30 3: 11 |
6678 | 2: 21 2: 30 2: 30 2: 11 2: 21 | |
6679 | 1: 30 1: 10 1: 11 1: 21 1: 30 | |
6680 | . . . . . | |
6681 | ||
5d67986c | 6682 | M-@key{TAB} 1 + M-@key{TAB} M-@key{TAB} |
d7b8e6c6 | 6683 | @end group |
5d67986c | 6684 | @end smallexample |
d7b8e6c6 EZ |
6685 | |
6686 | @node RPN Answer 4, Algebraic Answer 1, RPN Answer 3, Answers to Exercises | |
6687 | @subsection RPN Tutorial Exercise 4 | |
6688 | ||
6689 | @noindent | |
6690 | Either @kbd{( 2 , 3 )} or @kbd{( 2 @key{SPC} 3 )} would have worked, | |
6691 | but using both the comma and the space at once yields: | |
6692 | ||
d7b8e6c6 | 6693 | @smallexample |
5d67986c | 6694 | @group |
d7b8e6c6 EZ |
6695 | 1: ( ... 2: ( ... 1: (2, ... 2: (2, ... 2: (2, ... |
6696 | . 1: 2 . 1: (2, ... 1: (2, 3) | |
6697 | . . . | |
6698 | ||
5d67986c | 6699 | ( 2 , @key{SPC} 3 ) |
d7b8e6c6 | 6700 | @end group |
5d67986c | 6701 | @end smallexample |
d7b8e6c6 EZ |
6702 | |
6703 | Joe probably tried to type @kbd{@key{TAB} @key{DEL}} to swap the | |
6704 | extra incomplete object to the top of the stack and delete it. | |
6705 | But a feature of Calc is that @key{DEL} on an incomplete object | |
6706 | deletes just one component out of that object, so he had to press | |
6707 | @key{DEL} twice to finish the job. | |
6708 | ||
d7b8e6c6 | 6709 | @smallexample |
5d67986c | 6710 | @group |
d7b8e6c6 EZ |
6711 | 2: (2, ... 2: (2, 3) 2: (2, 3) 1: (2, 3) |
6712 | 1: (2, 3) 1: (2, ... 1: ( ... . | |
6713 | . . . | |
6714 | ||
5d67986c | 6715 | @key{TAB} @key{DEL} @key{DEL} |
d7b8e6c6 | 6716 | @end group |
5d67986c | 6717 | @end smallexample |
d7b8e6c6 EZ |
6718 | |
6719 | (As it turns out, deleting the second-to-top stack entry happens often | |
5d67986c RS |
6720 | enough that Calc provides a special key, @kbd{M-@key{DEL}}, to do just that. |
6721 | @kbd{M-@key{DEL}} is just like @kbd{@key{TAB} @key{DEL}}, except that it doesn't exhibit | |
d7b8e6c6 EZ |
6722 | the ``feature'' that tripped poor Joe.) |
6723 | ||
6724 | @node Algebraic Answer 1, Algebraic Answer 2, RPN Answer 4, Answers to Exercises | |
6725 | @subsection Algebraic Entry Tutorial Exercise 1 | |
6726 | ||
6727 | @noindent | |
6728 | Type @kbd{' sqrt($) @key{RET}}. | |
6729 | ||
6730 | If the @kbd{Q} key is broken, you could use @kbd{' $^0.5 @key{RET}}. | |
6731 | Or, RPN style, @kbd{0.5 ^}. | |
6732 | ||
6733 | (Actually, @samp{$^1:2}, using the fraction one-half as the power, is | |
6734 | a closer equivalent, since @samp{9^0.5} yields @cite{3.0} whereas | |
6735 | @samp{sqrt(9)} and @samp{9^1:2} yield the exact integer @cite{3}.) | |
6736 | ||
6737 | @node Algebraic Answer 2, Algebraic Answer 3, Algebraic Answer 1, Answers to Exercises | |
6738 | @subsection Algebraic Entry Tutorial Exercise 2 | |
6739 | ||
6740 | @noindent | |
6741 | In the formula @samp{2 x (1+y)}, @samp{x} was interpreted as a function | |
6742 | name with @samp{1+y} as its argument. Assigning a value to a variable | |
6743 | has no relation to a function by the same name. Joe needed to use an | |
6744 | explicit @samp{*} symbol here: @samp{2 x*(1+y)}. | |
6745 | ||
6746 | @node Algebraic Answer 3, Modes Answer 1, Algebraic Answer 2, Answers to Exercises | |
6747 | @subsection Algebraic Entry Tutorial Exercise 3 | |
6748 | ||
6749 | @noindent | |
6750 | The result from @kbd{1 @key{RET} 0 /} will be the formula @cite{1 / 0}. | |
6751 | The ``function'' @samp{/} cannot be evaluated when its second argument | |
6752 | is zero, so it is left in symbolic form. When you now type @kbd{0 *}, | |
6753 | the result will be zero because Calc uses the general rule that ``zero | |
6754 | times anything is zero.'' | |
6755 | ||
6756 | @c [fix-ref Infinities] | |
6757 | The @kbd{m i} command enables an @dfn{infinite mode} in which @cite{1 / 0} | |
6758 | results in a special symbol that represents ``infinity.'' If you | |
6759 | multiply infinity by zero, Calc uses another special new symbol to | |
6760 | show that the answer is ``indeterminate.'' @xref{Infinities}, for | |
6761 | further discussion of infinite and indeterminate values. | |
6762 | ||
6763 | @node Modes Answer 1, Modes Answer 2, Algebraic Answer 3, Answers to Exercises | |
6764 | @subsection Modes Tutorial Exercise 1 | |
6765 | ||
6766 | @noindent | |
6767 | Calc always stores its numbers in decimal, so even though one-third has | |
6768 | an exact base-3 representation (@samp{3#0.1}), it is still stored as | |
6769 | 0.3333333 (chopped off after 12 or however many decimal digits) inside | |
6770 | the calculator's memory. When this inexact number is converted back | |
6771 | to base 3 for display, it may still be slightly inexact. When we | |
6772 | multiply this number by 3, we get 0.999999, also an inexact value. | |
6773 | ||
6774 | When Calc displays a number in base 3, it has to decide how many digits | |
6775 | to show. If the current precision is 12 (decimal) digits, that corresponds | |
6776 | to @samp{12 / log10(3) = 25.15} base-3 digits. Because 25.15 is not an | |
6777 | exact integer, Calc shows only 25 digits, with the result that stored | |
6778 | numbers carry a little bit of extra information that may not show up on | |
6779 | the screen. When Joe entered @samp{3#0.2}, the stored number 0.666666 | |
6780 | happened to round to a pleasing value when it lost that last 0.15 of a | |
6781 | digit, but it was still inexact in Calc's memory. When he divided by 2, | |
6782 | he still got the dreaded inexact value 0.333333. (Actually, he divided | |
6783 | 0.666667 by 2 to get 0.333334, which is why he got something a little | |
6784 | higher than @code{3#0.1} instead of a little lower.) | |
6785 | ||
6786 | If Joe didn't want to be bothered with all this, he could have typed | |
6787 | @kbd{M-24 d n} to display with one less digit than the default. (If | |
6788 | you give @kbd{d n} a negative argument, it uses default-minus-that, | |
6789 | so @kbd{M-- d n} would be an easier way to get the same effect.) Those | |
6790 | inexact results would still be lurking there, but they would now be | |
6791 | rounded to nice, natural-looking values for display purposes. (Remember, | |
6792 | @samp{0.022222} in base 3 is like @samp{0.099999} in base 10; rounding | |
6793 | off one digit will round the number up to @samp{0.1}.) Depending on the | |
6794 | nature of your work, this hiding of the inexactness may be a benefit or | |
6795 | a danger. With the @kbd{d n} command, Calc gives you the choice. | |
6796 | ||
6797 | Incidentally, another consequence of all this is that if you type | |
6798 | @kbd{M-30 d n} to display more digits than are ``really there,'' | |
6799 | you'll see garbage digits at the end of the number. (In decimal | |
6800 | display mode, with decimally-stored numbers, these garbage digits are | |
6801 | always zero so they vanish and you don't notice them.) Because Calc | |
6802 | rounds off that 0.15 digit, there is the danger that two numbers could | |
6803 | be slightly different internally but still look the same. If you feel | |
6804 | uneasy about this, set the @kbd{d n} precision to be a little higher | |
6805 | than normal; you'll get ugly garbage digits, but you'll always be able | |
6806 | to tell two distinct numbers apart. | |
6807 | ||
6808 | An interesting side note is that most computers store their | |
6809 | floating-point numbers in binary, and convert to decimal for display. | |
6810 | Thus everyday programs have the same problem: Decimal 0.1 cannot be | |
6811 | represented exactly in binary (try it: @kbd{0.1 d 2}), so @samp{0.1 * 10} | |
6812 | comes out as an inexact approximation to 1 on some machines (though | |
6813 | they generally arrange to hide it from you by rounding off one digit as | |
6814 | we did above). Because Calc works in decimal instead of binary, you can | |
6815 | be sure that numbers that look exact @emph{are} exact as long as you stay | |
6816 | in decimal display mode. | |
6817 | ||
6818 | It's not hard to show that any number that can be represented exactly | |
6819 | in binary, octal, or hexadecimal is also exact in decimal, so the kinds | |
6820 | of problems we saw in this exercise are likely to be severe only when | |
6821 | you use a relatively unusual radix like 3. | |
6822 | ||
6823 | @node Modes Answer 2, Modes Answer 3, Modes Answer 1, Answers to Exercises | |
6824 | @subsection Modes Tutorial Exercise 2 | |
6825 | ||
6826 | If the radix is 15 or higher, we can't use the letter @samp{e} to mark | |
6827 | the exponent because @samp{e} is interpreted as a digit. When Calc | |
6828 | needs to display scientific notation in a high radix, it writes | |
6829 | @samp{16#F.E8F*16.^15}. You can enter a number like this as an | |
6830 | algebraic entry. Also, pressing @kbd{e} without any digits before it | |
6831 | normally types @kbd{1e}, but in a high radix it types @kbd{16.^} and | |
5d67986c | 6832 | puts you in algebraic entry: @kbd{16#f.e8f @key{RET} e 15 @key{RET} *} is another |
d7b8e6c6 EZ |
6833 | way to enter this number. |
6834 | ||
6835 | The reason Calc puts a decimal point in the @samp{16.^} is to prevent | |
6836 | huge integers from being generated if the exponent is large (consider | |
6837 | @samp{16#1.23*16^1000}, where we compute @samp{16^1000} as a giant | |
6838 | exact integer and then throw away most of the digits when we multiply | |
6839 | it by the floating-point @samp{16#1.23}). While this wouldn't normally | |
6840 | matter for display purposes, it could give you a nasty surprise if you | |
6841 | copied that number into a file and later moved it back into Calc. | |
6842 | ||
6843 | @node Modes Answer 3, Modes Answer 4, Modes Answer 2, Answers to Exercises | |
6844 | @subsection Modes Tutorial Exercise 3 | |
6845 | ||
6846 | @noindent | |
6847 | The answer he got was @cite{0.5000000000006399}. | |
6848 | ||
6849 | The problem is not that the square operation is inexact, but that the | |
6850 | sine of 45 that was already on the stack was accurate to only 12 places. | |
6851 | Arbitrary-precision calculations still only give answers as good as | |
6852 | their inputs. | |
6853 | ||
6854 | The real problem is that there is no 12-digit number which, when | |
6855 | squared, comes out to 0.5 exactly. The @kbd{f [} and @kbd{f ]} | |
6856 | commands decrease or increase a number by one unit in the last | |
6857 | place (according to the current precision). They are useful for | |
6858 | determining facts like this. | |
6859 | ||
d7b8e6c6 | 6860 | @smallexample |
5d67986c | 6861 | @group |
d7b8e6c6 EZ |
6862 | 1: 0.707106781187 1: 0.500000000001 |
6863 | . . | |
6864 | ||
6865 | 45 S 2 ^ | |
6866 | ||
d7b8e6c6 | 6867 | @end group |
5d67986c | 6868 | @end smallexample |
d7b8e6c6 | 6869 | @noindent |
d7b8e6c6 | 6870 | @smallexample |
5d67986c | 6871 | @group |
d7b8e6c6 EZ |
6872 | 1: 0.707106781187 1: 0.707106781186 1: 0.499999999999 |
6873 | . . . | |
6874 | ||
5d67986c | 6875 | U @key{DEL} f [ 2 ^ |
d7b8e6c6 | 6876 | @end group |
5d67986c | 6877 | @end smallexample |
d7b8e6c6 EZ |
6878 | |
6879 | A high-precision calculation must be carried out in high precision | |
6880 | all the way. The only number in the original problem which was known | |
6881 | exactly was the quantity 45 degrees, so the precision must be raised | |
6882 | before anything is done after the number 45 has been entered in order | |
6883 | for the higher precision to be meaningful. | |
6884 | ||
6885 | @node Modes Answer 4, Arithmetic Answer 1, Modes Answer 3, Answers to Exercises | |
6886 | @subsection Modes Tutorial Exercise 4 | |
6887 | ||
6888 | @noindent | |
6889 | Many calculations involve real-world quantities, like the width and | |
6890 | height of a piece of wood or the volume of a jar. Such quantities | |
6891 | can't be measured exactly anyway, and if the data that is input to | |
6892 | a calculation is inexact, doing exact arithmetic on it is a waste | |
6893 | of time. | |
6894 | ||
6895 | Fractions become unwieldy after too many calculations have been | |
6896 | done with them. For example, the sum of the reciprocals of the | |
6897 | integers from 1 to 10 is 7381:2520. The sum from 1 to 30 is | |
6898 | 9304682830147:2329089562800. After a point it will take a long | |
6899 | time to add even one more term to this sum, but a floating-point | |
6900 | calculation of the sum will not have this problem. | |
6901 | ||
6902 | Also, rational numbers cannot express the results of all calculations. | |
6903 | There is no fractional form for the square root of two, so if you type | |
6904 | @w{@kbd{2 Q}}, Calc has no choice but to give you a floating-point answer. | |
6905 | ||
6906 | @node Arithmetic Answer 1, Arithmetic Answer 2, Modes Answer 4, Answers to Exercises | |
6907 | @subsection Arithmetic Tutorial Exercise 1 | |
6908 | ||
6909 | @noindent | |
6910 | Dividing two integers that are larger than the current precision may | |
6911 | give a floating-point result that is inaccurate even when rounded | |
6912 | down to an integer. Consider @cite{123456789 / 2} when the current | |
6913 | precision is 6 digits. The true answer is @cite{61728394.5}, but | |
6914 | with a precision of 6 this will be rounded to @c{$12345700.0/2.0 = 61728500.0$} | |
6915 | @cite{12345700.@: / 2.@: = 61728500.}. | |
6916 | The result, when converted to an integer, will be off by 106. | |
6917 | ||
6918 | Here are two solutions: Raise the precision enough that the | |
6919 | floating-point round-off error is strictly to the right of the | |
6920 | decimal point. Or, convert to fraction mode so that @cite{123456789 / 2} | |
6921 | produces the exact fraction @cite{123456789:2}, which can be rounded | |
6922 | down by the @kbd{F} command without ever switching to floating-point | |
6923 | format. | |
6924 | ||
6925 | @node Arithmetic Answer 2, Vector Answer 1, Arithmetic Answer 1, Answers to Exercises | |
6926 | @subsection Arithmetic Tutorial Exercise 2 | |
6927 | ||
6928 | @noindent | |
6929 | @kbd{27 @key{RET} 9 B} could give the exact result @cite{3:2}, but it | |
6930 | does a floating-point calculation instead and produces @cite{1.5}. | |
6931 | ||
6932 | Calc will find an exact result for a logarithm if the result is an integer | |
6933 | or the reciprocal of an integer. But there is no efficient way to search | |
6934 | the space of all possible rational numbers for an exact answer, so Calc | |
6935 | doesn't try. | |
6936 | ||
6937 | @node Vector Answer 1, Vector Answer 2, Arithmetic Answer 2, Answers to Exercises | |
6938 | @subsection Vector Tutorial Exercise 1 | |
6939 | ||
6940 | @noindent | |
6941 | Duplicate the vector, compute its length, then divide the vector | |
6942 | by its length: @kbd{@key{RET} A /}. | |
6943 | ||
d7b8e6c6 | 6944 | @smallexample |
5d67986c | 6945 | @group |
d7b8e6c6 EZ |
6946 | 1: [1, 2, 3] 2: [1, 2, 3] 1: [0.27, 0.53, 0.80] 1: 1. |
6947 | . 1: 3.74165738677 . . | |
6948 | . | |
6949 | ||
5d67986c | 6950 | r 1 @key{RET} A / A |
d7b8e6c6 | 6951 | @end group |
5d67986c | 6952 | @end smallexample |
d7b8e6c6 EZ |
6953 | |
6954 | The final @kbd{A} command shows that the normalized vector does | |
6955 | indeed have unit length. | |
6956 | ||
6957 | @node Vector Answer 2, Matrix Answer 1, Vector Answer 1, Answers to Exercises | |
6958 | @subsection Vector Tutorial Exercise 2 | |
6959 | ||
6960 | @noindent | |
6961 | The average position is equal to the sum of the products of the | |
6962 | positions times their corresponding probabilities. This is the | |
6963 | definition of the dot product operation. So all you need to do | |
6964 | is to put the two vectors on the stack and press @kbd{*}. | |
6965 | ||
6966 | @node Matrix Answer 1, Matrix Answer 2, Vector Answer 2, Answers to Exercises | |
6967 | @subsection Matrix Tutorial Exercise 1 | |
6968 | ||
6969 | @noindent | |
6970 | The trick is to multiply by a vector of ones. Use @kbd{r 4 [1 1 1] *} to | |
6971 | get the row sum. Similarly, use @kbd{[1 1] r 4 *} to get the column sum. | |
6972 | ||
6973 | @node Matrix Answer 2, Matrix Answer 3, Matrix Answer 1, Answers to Exercises | |
6974 | @subsection Matrix Tutorial Exercise 2 | |
6975 | ||
6976 | @ifinfo | |
d7b8e6c6 | 6977 | @example |
5d67986c | 6978 | @group |
d7b8e6c6 EZ |
6979 | x + a y = 6 |
6980 | x + b y = 10 | |
d7b8e6c6 | 6981 | @end group |
5d67986c | 6982 | @end example |
d7b8e6c6 EZ |
6983 | @end ifinfo |
6984 | @tex | |
6985 | \turnoffactive | |
6986 | \beforedisplay | |
6987 | $$ \eqalign{ x &+ a y = 6 \cr | |
6988 | x &+ b y = 10} | |
6989 | $$ | |
6990 | \afterdisplay | |
6991 | @end tex | |
6992 | ||
6993 | Just enter the righthand side vector, then divide by the lefthand side | |
6994 | matrix as usual. | |
6995 | ||
d7b8e6c6 | 6996 | @smallexample |
5d67986c | 6997 | @group |
d7b8e6c6 EZ |
6998 | 1: [6, 10] 2: [6, 10] 1: [6 - 4 a / (b - a), 4 / (b - a) ] |
6999 | . 1: [ [ 1, a ] . | |
7000 | [ 1, b ] ] | |
7001 | . | |
7002 | ||
5d67986c | 7003 | ' [6 10] @key{RET} ' [1 a; 1 b] @key{RET} / |
d7b8e6c6 | 7004 | @end group |
5d67986c | 7005 | @end smallexample |
d7b8e6c6 EZ |
7006 | |
7007 | This can be made more readable using @kbd{d B} to enable ``big'' display | |
7008 | mode: | |
7009 | ||
d7b8e6c6 | 7010 | @smallexample |
5d67986c | 7011 | @group |
d7b8e6c6 EZ |
7012 | 4 a 4 |
7013 | 1: [6 - -----, -----] | |
7014 | b - a b - a | |
d7b8e6c6 | 7015 | @end group |
5d67986c | 7016 | @end smallexample |
d7b8e6c6 EZ |
7017 | |
7018 | Type @kbd{d N} to return to ``normal'' display mode afterwards. | |
7019 | ||
7020 | @node Matrix Answer 3, List Answer 1, Matrix Answer 2, Answers to Exercises | |
7021 | @subsection Matrix Tutorial Exercise 3 | |
7022 | ||
7023 | @noindent | |
7024 | To solve @c{$A^T A \, X = A^T B$} | |
7025 | @cite{trn(A) * A * X = trn(A) * B}, first we compute | |
7026 | @c{$A' = A^T A$} | |
7027 | @cite{A2 = trn(A) * A} and @c{$B' = A^T B$} | |
7028 | @cite{B2 = trn(A) * B}; now, we have a | |
7029 | system @c{$A' X = B'$} | |
7030 | @cite{A2 * X = B2} which we can solve using Calc's @samp{/} | |
7031 | command. | |
7032 | ||
7033 | @ifinfo | |
d7b8e6c6 | 7034 | @example |
5d67986c | 7035 | @group |
d7b8e6c6 EZ |
7036 | a + 2b + 3c = 6 |
7037 | 4a + 5b + 6c = 2 | |
7038 | 7a + 6b = 3 | |
7039 | 2a + 4b + 6c = 11 | |
d7b8e6c6 | 7040 | @end group |
5d67986c | 7041 | @end example |
d7b8e6c6 EZ |
7042 | @end ifinfo |
7043 | @tex | |
7044 | \turnoffactive | |
7045 | \beforedisplayh | |
7046 | $$ \openup1\jot \tabskip=0pt plus1fil | |
7047 | \halign to\displaywidth{\tabskip=0pt | |
7048 | $\hfil#$&$\hfil{}#{}$& | |
7049 | $\hfil#$&$\hfil{}#{}$& | |
7050 | $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr | |
7051 | a&+&2b&+&3c&=6 \cr | |
7052 | 4a&+&5b&+&6c&=2 \cr | |
7053 | 7a&+&6b& & &=3 \cr | |
7054 | 2a&+&4b&+&6c&=11 \cr} | |
7055 | $$ | |
7056 | \afterdisplayh | |
7057 | @end tex | |
7058 | ||
7059 | The first step is to enter the coefficient matrix. We'll store it in | |
7060 | quick variable number 7 for later reference. Next, we compute the | |
7061 | @c{$B'$} | |
7062 | @cite{B2} vector. | |
7063 | ||
d7b8e6c6 | 7064 | @smallexample |
5d67986c | 7065 | @group |
d7b8e6c6 EZ |
7066 | 1: [ [ 1, 2, 3 ] 2: [ [ 1, 4, 7, 2 ] 1: [57, 84, 96] |
7067 | [ 4, 5, 6 ] [ 2, 5, 6, 4 ] . | |
7068 | [ 7, 6, 0 ] [ 3, 6, 0, 6 ] ] | |
7069 | [ 2, 4, 6 ] ] 1: [6, 2, 3, 11] | |
7070 | . . | |
7071 | ||
5d67986c | 7072 | ' [1 2 3; 4 5 6; 7 6 0; 2 4 6] @key{RET} s 7 v t [6 2 3 11] * |
d7b8e6c6 | 7073 | @end group |
5d67986c | 7074 | @end smallexample |
d7b8e6c6 EZ |
7075 | |
7076 | @noindent | |
7077 | Now we compute the matrix @c{$A'$} | |
7078 | @cite{A2} and divide. | |
7079 | ||
d7b8e6c6 | 7080 | @smallexample |
5d67986c | 7081 | @group |
d7b8e6c6 EZ |
7082 | 2: [57, 84, 96] 1: [-11.64, 14.08, -3.64] |
7083 | 1: [ [ 70, 72, 39 ] . | |
7084 | [ 72, 81, 60 ] | |
7085 | [ 39, 60, 81 ] ] | |
7086 | . | |
7087 | ||
7088 | r 7 v t r 7 * / | |
d7b8e6c6 | 7089 | @end group |
5d67986c | 7090 | @end smallexample |
d7b8e6c6 EZ |
7091 | |
7092 | @noindent | |
7093 | (The actual computed answer will be slightly inexact due to | |
7094 | round-off error.) | |
7095 | ||
7096 | Notice that the answers are similar to those for the @c{$3\times3$} | |
7097 | @asis{3x3} system | |
7098 | solved in the text. That's because the fourth equation that was | |
7099 | added to the system is almost identical to the first one multiplied | |
7100 | by two. (If it were identical, we would have gotten the exact same | |
7101 | answer since the @c{$4\times3$} | |
7102 | @asis{4x3} system would be equivalent to the original @c{$3\times3$} | |
7103 | @asis{3x3} | |
7104 | system.) | |
7105 | ||
7106 | Since the first and fourth equations aren't quite equivalent, they | |
7107 | can't both be satisfied at once. Let's plug our answers back into | |
7108 | the original system of equations to see how well they match. | |
7109 | ||
d7b8e6c6 | 7110 | @smallexample |
5d67986c | 7111 | @group |
d7b8e6c6 EZ |
7112 | 2: [-11.64, 14.08, -3.64] 1: [5.6, 2., 3., 11.2] |
7113 | 1: [ [ 1, 2, 3 ] . | |
7114 | [ 4, 5, 6 ] | |
7115 | [ 7, 6, 0 ] | |
7116 | [ 2, 4, 6 ] ] | |
7117 | . | |
7118 | ||
5d67986c | 7119 | r 7 @key{TAB} * |
d7b8e6c6 | 7120 | @end group |
5d67986c | 7121 | @end smallexample |
d7b8e6c6 EZ |
7122 | |
7123 | @noindent | |
7124 | This is reasonably close to our original @cite{B} vector, | |
7125 | @cite{[6, 2, 3, 11]}. | |
7126 | ||
7127 | @node List Answer 1, List Answer 2, Matrix Answer 3, Answers to Exercises | |
7128 | @subsection List Tutorial Exercise 1 | |
7129 | ||
7130 | @noindent | |
7131 | We can use @kbd{v x} to build a vector of integers. This needs to be | |
7132 | adjusted to get the range of integers we desire. Mapping @samp{-} | |
7133 | across the vector will accomplish this, although it turns out the | |
7134 | plain @samp{-} key will work just as well. | |
7135 | ||
d7b8e6c6 | 7136 | @smallexample |
5d67986c | 7137 | @group |
d7b8e6c6 EZ |
7138 | 2: 2 2: 2 |
7139 | 1: [1, 2, 3, 4, 5, 6, 7, 8, 9] 1: [-4, -3, -2, -1, 0, 1, 2, 3, 4] | |
7140 | . . | |
7141 | ||
5d67986c | 7142 | 2 v x 9 @key{RET} 5 V M - or 5 - |
d7b8e6c6 | 7143 | @end group |
5d67986c | 7144 | @end smallexample |
d7b8e6c6 EZ |
7145 | |
7146 | @noindent | |
7147 | Now we use @kbd{V M ^} to map the exponentiation operator across the | |
7148 | vector. | |
7149 | ||
d7b8e6c6 | 7150 | @smallexample |
5d67986c | 7151 | @group |
d7b8e6c6 EZ |
7152 | 1: [0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8, 16] |
7153 | . | |
7154 | ||
7155 | V M ^ | |
d7b8e6c6 | 7156 | @end group |
5d67986c | 7157 | @end smallexample |
d7b8e6c6 EZ |
7158 | |
7159 | @node List Answer 2, List Answer 3, List Answer 1, Answers to Exercises | |
7160 | @subsection List Tutorial Exercise 2 | |
7161 | ||
7162 | @noindent | |
7163 | Given @cite{x} and @cite{y} vectors in quick variables 1 and 2 as before, | |
7164 | the first job is to form the matrix that describes the problem. | |
7165 | ||
7166 | @ifinfo | |
7167 | @example | |
7168 | m*x + b*1 = y | |
7169 | @end example | |
7170 | @end ifinfo | |
7171 | @tex | |
7172 | \turnoffactive | |
7173 | \beforedisplay | |
7174 | $$ m \times x + b \times 1 = y $$ | |
7175 | \afterdisplay | |
7176 | @end tex | |
7177 | ||
7178 | Thus we want a @c{$19\times2$} | |
7179 | @asis{19x2} matrix with our @cite{x} vector as one column and | |
7180 | ones as the other column. So, first we build the column of ones, then | |
7181 | we combine the two columns to form our @cite{A} matrix. | |
7182 | ||
d7b8e6c6 | 7183 | @smallexample |
5d67986c | 7184 | @group |
d7b8e6c6 EZ |
7185 | 2: [1.34, 1.41, 1.49, ... ] 1: [ [ 1.34, 1 ] |
7186 | 1: [1, 1, 1, ...] [ 1.41, 1 ] | |
7187 | . [ 1.49, 1 ] | |
7188 | @dots{} | |
7189 | ||
5d67986c | 7190 | r 1 1 v b 19 @key{RET} M-2 v p v t s 3 |
d7b8e6c6 | 7191 | @end group |
5d67986c | 7192 | @end smallexample |
d7b8e6c6 EZ |
7193 | |
7194 | @noindent | |
7195 | Now we compute @c{$A^T y$} | |
7196 | @cite{trn(A) * y} and @c{$A^T A$} | |
7197 | @cite{trn(A) * A} and divide. | |
7198 | ||
d7b8e6c6 | 7199 | @smallexample |
5d67986c | 7200 | @group |
d7b8e6c6 EZ |
7201 | 1: [33.36554, 13.613] 2: [33.36554, 13.613] |
7202 | . 1: [ [ 98.0003, 41.63 ] | |
7203 | [ 41.63, 19 ] ] | |
7204 | . | |
7205 | ||
7206 | v t r 2 * r 3 v t r 3 * | |
d7b8e6c6 | 7207 | @end group |
5d67986c | 7208 | @end smallexample |
d7b8e6c6 EZ |
7209 | |
7210 | @noindent | |
7211 | (Hey, those numbers look familiar!) | |
7212 | ||
d7b8e6c6 | 7213 | @smallexample |
5d67986c | 7214 | @group |
d7b8e6c6 EZ |
7215 | 1: [0.52141679, -0.425978] |
7216 | . | |
7217 | ||
7218 | / | |
d7b8e6c6 | 7219 | @end group |
5d67986c | 7220 | @end smallexample |
d7b8e6c6 EZ |
7221 | |
7222 | Since we were solving equations of the form @c{$m \times x + b \times 1 = y$} | |
7223 | @cite{m*x + b*1 = y}, these | |
7224 | numbers should be @cite{m} and @cite{b}, respectively. Sure enough, they | |
7225 | agree exactly with the result computed using @kbd{V M} and @kbd{V R}! | |
7226 | ||
7227 | The moral of this story: @kbd{V M} and @kbd{V R} will probably solve | |
7228 | your problem, but there is often an easier way using the higher-level | |
7229 | arithmetic functions! | |
7230 | ||
7231 | @c [fix-ref Curve Fitting] | |
7232 | In fact, there is a built-in @kbd{a F} command that does least-squares | |
7233 | fits. @xref{Curve Fitting}. | |
7234 | ||
7235 | @node List Answer 3, List Answer 4, List Answer 2, Answers to Exercises | |
7236 | @subsection List Tutorial Exercise 3 | |
7237 | ||
7238 | @noindent | |
5d67986c | 7239 | Move to one end of the list and press @kbd{C-@@} (or @kbd{C-@key{SPC}} or |
d7b8e6c6 EZ |
7240 | whatever) to set the mark, then move to the other end of the list |
7241 | and type @w{@kbd{M-# g}}. | |
7242 | ||
d7b8e6c6 | 7243 | @smallexample |
5d67986c | 7244 | @group |
d7b8e6c6 EZ |
7245 | 1: [2.3, 6, 22, 15.1, 7, 15, 14, 7.5, 2.5] |
7246 | . | |
d7b8e6c6 | 7247 | @end group |
5d67986c | 7248 | @end smallexample |
d7b8e6c6 EZ |
7249 | |
7250 | To make things interesting, let's assume we don't know at a glance | |
7251 | how many numbers are in this list. Then we could type: | |
7252 | ||
d7b8e6c6 | 7253 | @smallexample |
5d67986c | 7254 | @group |
d7b8e6c6 EZ |
7255 | 2: [2.3, 6, 22, ... ] 2: [2.3, 6, 22, ... ] |
7256 | 1: [2.3, 6, 22, ... ] 1: 126356422.5 | |
7257 | . . | |
7258 | ||
5d67986c | 7259 | @key{RET} V R * |
d7b8e6c6 | 7260 | |
d7b8e6c6 | 7261 | @end group |
5d67986c | 7262 | @end smallexample |
d7b8e6c6 | 7263 | @noindent |
d7b8e6c6 | 7264 | @smallexample |
5d67986c | 7265 | @group |
d7b8e6c6 EZ |
7266 | 2: 126356422.5 2: 126356422.5 1: 7.94652913734 |
7267 | 1: [2.3, 6, 22, ... ] 1: 9 . | |
7268 | . . | |
7269 | ||
5d67986c | 7270 | @key{TAB} v l I ^ |
d7b8e6c6 | 7271 | @end group |
5d67986c | 7272 | @end smallexample |
d7b8e6c6 EZ |
7273 | |
7274 | @noindent | |
7275 | (The @kbd{I ^} command computes the @var{n}th root of a number. | |
7276 | You could also type @kbd{& ^} to take the reciprocal of 9 and | |
7277 | then raise the number to that power.) | |
7278 | ||
7279 | @node List Answer 4, List Answer 5, List Answer 3, Answers to Exercises | |
7280 | @subsection List Tutorial Exercise 4 | |
7281 | ||
7282 | @noindent | |
7283 | A number @cite{j} is a divisor of @cite{n} if @c{$n \mathbin{\hbox{\code{\%}}} j = 0$} | |
7284 | @samp{n % j = 0}. The first | |
7285 | step is to get a vector that identifies the divisors. | |
7286 | ||
d7b8e6c6 | 7287 | @smallexample |
5d67986c | 7288 | @group |
d7b8e6c6 EZ |
7289 | 2: 30 2: [0, 0, 0, 2, ...] 1: [1, 1, 1, 0, ...] |
7290 | 1: [1, 2, 3, 4, ...] 1: 0 . | |
7291 | . . | |
7292 | ||
5d67986c | 7293 | 30 @key{RET} v x 30 @key{RET} s 1 V M % 0 V M a = s 2 |
d7b8e6c6 | 7294 | @end group |
5d67986c | 7295 | @end smallexample |
d7b8e6c6 EZ |
7296 | |
7297 | @noindent | |
7298 | This vector has 1's marking divisors of 30 and 0's marking non-divisors. | |
7299 | ||
7300 | The zeroth divisor function is just the total number of divisors. | |
7301 | The first divisor function is the sum of the divisors. | |
7302 | ||
d7b8e6c6 | 7303 | @smallexample |
5d67986c | 7304 | @group |
d7b8e6c6 EZ |
7305 | 1: 8 3: 8 2: 8 2: 8 |
7306 | 2: [1, 2, 3, 4, ...] 1: [1, 2, 3, 0, ...] 1: 72 | |
7307 | 1: [1, 1, 1, 0, ...] . . | |
7308 | . | |
7309 | ||
7310 | V R + r 1 r 2 V M * V R + | |
d7b8e6c6 | 7311 | @end group |
5d67986c | 7312 | @end smallexample |
d7b8e6c6 EZ |
7313 | |
7314 | @noindent | |
7315 | Once again, the last two steps just compute a dot product for which | |
7316 | a simple @kbd{*} would have worked equally well. | |
7317 | ||
7318 | @node List Answer 5, List Answer 6, List Answer 4, Answers to Exercises | |
7319 | @subsection List Tutorial Exercise 5 | |
7320 | ||
7321 | @noindent | |
7322 | The obvious first step is to obtain the list of factors with @kbd{k f}. | |
7323 | This list will always be in sorted order, so if there are duplicates | |
7324 | they will be right next to each other. A suitable method is to compare | |
7325 | the list with a copy of itself shifted over by one. | |
7326 | ||
d7b8e6c6 | 7327 | @smallexample |
5d67986c | 7328 | @group |
d7b8e6c6 EZ |
7329 | 1: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19, 0] |
7330 | . 1: [3, 7, 7, 7, 19, 0] 1: [0, 3, 7, 7, 7, 19] | |
7331 | . . | |
7332 | ||
5d67986c | 7333 | 19551 k f @key{RET} 0 | @key{TAB} 0 @key{TAB} | |
d7b8e6c6 | 7334 | |
d7b8e6c6 | 7335 | @end group |
5d67986c | 7336 | @end smallexample |
d7b8e6c6 | 7337 | @noindent |
d7b8e6c6 | 7338 | @smallexample |
5d67986c | 7339 | @group |
d7b8e6c6 EZ |
7340 | 1: [0, 0, 1, 1, 0, 0] 1: 2 1: 0 |
7341 | . . . | |
7342 | ||
7343 | V M a = V R + 0 a = | |
d7b8e6c6 | 7344 | @end group |
5d67986c | 7345 | @end smallexample |
d7b8e6c6 EZ |
7346 | |
7347 | @noindent | |
7348 | Note that we have to arrange for both vectors to have the same length | |
7349 | so that the mapping operation works; no prime factor will ever be | |
7350 | zero, so adding zeros on the left and right is safe. From then on | |
7351 | the job is pretty straightforward. | |
7352 | ||
7353 | Incidentally, Calc provides the @c{\dfn{M\"obius} $\mu$} | |
7354 | @dfn{Moebius mu} function which is | |
7355 | zero if and only if its argument is square-free. It would be a much | |
7356 | more convenient way to do the above test in practice. | |
7357 | ||
7358 | @node List Answer 6, List Answer 7, List Answer 5, Answers to Exercises | |
7359 | @subsection List Tutorial Exercise 6 | |
7360 | ||
7361 | @noindent | |
5d67986c | 7362 | First use @kbd{v x 6 @key{RET}} to get a list of integers, then @kbd{V M v x} |
d7b8e6c6 EZ |
7363 | to get a list of lists of integers! |
7364 | ||
7365 | @node List Answer 7, List Answer 8, List Answer 6, Answers to Exercises | |
7366 | @subsection List Tutorial Exercise 7 | |
7367 | ||
7368 | @noindent | |
7369 | Here's one solution. First, compute the triangular list from the previous | |
7370 | exercise and type @kbd{1 -} to subtract one from all the elements. | |
7371 | ||
d7b8e6c6 | 7372 | @smallexample |
5d67986c | 7373 | @group |
d7b8e6c6 EZ |
7374 | 1: [ [0], |
7375 | [0, 1], | |
7376 | [0, 1, 2], | |
7377 | @dots{} | |
7378 | ||
7379 | 1 - | |
d7b8e6c6 | 7380 | @end group |
5d67986c | 7381 | @end smallexample |
d7b8e6c6 EZ |
7382 | |
7383 | The numbers down the lefthand edge of the list we desire are called | |
7384 | the ``triangular numbers'' (now you know why!). The @cite{n}th | |
7385 | triangular number is the sum of the integers from 1 to @cite{n}, and | |
7386 | can be computed directly by the formula @c{$n (n+1) \over 2$} | |
7387 | @cite{n * (n+1) / 2}. | |
7388 | ||
d7b8e6c6 | 7389 | @smallexample |
5d67986c | 7390 | @group |
d7b8e6c6 EZ |
7391 | 2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ] |
7392 | 1: [0, 1, 2, 3, 4, 5] 1: [0, 1, 3, 6, 10, 15] | |
7393 | . . | |
7394 | ||
5d67986c | 7395 | v x 6 @key{RET} 1 - V M ' $ ($+1)/2 @key{RET} |
d7b8e6c6 | 7396 | @end group |
5d67986c | 7397 | @end smallexample |
d7b8e6c6 EZ |
7398 | |
7399 | @noindent | |
7400 | Adding this list to the above list of lists produces the desired | |
7401 | result: | |
7402 | ||
d7b8e6c6 | 7403 | @smallexample |
5d67986c | 7404 | @group |
d7b8e6c6 EZ |
7405 | 1: [ [0], |
7406 | [1, 2], | |
7407 | [3, 4, 5], | |
7408 | [6, 7, 8, 9], | |
7409 | [10, 11, 12, 13, 14], | |
7410 | [15, 16, 17, 18, 19, 20] ] | |
7411 | . | |
7412 | ||
7413 | V M + | |
d7b8e6c6 | 7414 | @end group |
5d67986c | 7415 | @end smallexample |
d7b8e6c6 EZ |
7416 | |
7417 | If we did not know the formula for triangular numbers, we could have | |
7418 | computed them using a @kbd{V U +} command. We could also have | |
7419 | gotten them the hard way by mapping a reduction across the original | |
7420 | triangular list. | |
7421 | ||
d7b8e6c6 | 7422 | @smallexample |
5d67986c | 7423 | @group |
d7b8e6c6 EZ |
7424 | 2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ] |
7425 | 1: [ [0], [0, 1], ... ] 1: [0, 1, 3, 6, 10, 15] | |
7426 | . . | |
7427 | ||
5d67986c | 7428 | @key{RET} V M V R + |
d7b8e6c6 | 7429 | @end group |
5d67986c | 7430 | @end smallexample |
d7b8e6c6 EZ |
7431 | |
7432 | @noindent | |
7433 | (This means ``map a @kbd{V R +} command across the vector,'' and | |
7434 | since each element of the main vector is itself a small vector, | |
7435 | @kbd{V R +} computes the sum of its elements.) | |
7436 | ||
7437 | @node List Answer 8, List Answer 9, List Answer 7, Answers to Exercises | |
7438 | @subsection List Tutorial Exercise 8 | |
7439 | ||
7440 | @noindent | |
7441 | The first step is to build a list of values of @cite{x}. | |
7442 | ||
d7b8e6c6 | 7443 | @smallexample |
5d67986c | 7444 | @group |
d7b8e6c6 EZ |
7445 | 1: [1, 2, 3, ..., 21] 1: [0, 1, 2, ..., 20] 1: [0, 0.25, 0.5, ..., 5] |
7446 | . . . | |
7447 | ||
5d67986c | 7448 | v x 21 @key{RET} 1 - 4 / s 1 |
d7b8e6c6 | 7449 | @end group |
5d67986c | 7450 | @end smallexample |
d7b8e6c6 EZ |
7451 | |
7452 | Next, we compute the Bessel function values. | |
7453 | ||
d7b8e6c6 | 7454 | @smallexample |
5d67986c | 7455 | @group |
d7b8e6c6 EZ |
7456 | 1: [0., 0.124, 0.242, ..., -0.328] |
7457 | . | |
7458 | ||
5d67986c | 7459 | V M ' besJ(1,$) @key{RET} |
d7b8e6c6 | 7460 | @end group |
5d67986c | 7461 | @end smallexample |
d7b8e6c6 EZ |
7462 | |
7463 | @noindent | |
5d67986c | 7464 | (Another way to do this would be @kbd{1 @key{TAB} V M f j}.) |
d7b8e6c6 EZ |
7465 | |
7466 | A way to isolate the maximum value is to compute the maximum using | |
7467 | @kbd{V R X}, then compare all the Bessel values with that maximum. | |
7468 | ||
d7b8e6c6 | 7469 | @smallexample |
5d67986c | 7470 | @group |
d7b8e6c6 EZ |
7471 | 2: [0., 0.124, 0.242, ... ] 1: [0, 0, 0, ... ] 2: [0, 0, 0, ... ] |
7472 | 1: 0.5801562 . 1: 1 | |
7473 | . . | |
7474 | ||
5d67986c | 7475 | @key{RET} V R X V M a = @key{RET} V R + @key{DEL} |
d7b8e6c6 | 7476 | @end group |
5d67986c | 7477 | @end smallexample |
d7b8e6c6 EZ |
7478 | |
7479 | @noindent | |
7480 | It's a good idea to verify, as in the last step above, that only | |
7481 | one value is equal to the maximum. (After all, a plot of @c{$\sin x$} | |
7482 | @cite{sin(x)} | |
7483 | might have many points all equal to the maximum value, 1.) | |
7484 | ||
7485 | The vector we have now has a single 1 in the position that indicates | |
7486 | the maximum value of @cite{x}. Now it is a simple matter to convert | |
7487 | this back into the corresponding value itself. | |
7488 | ||
d7b8e6c6 | 7489 | @smallexample |
5d67986c | 7490 | @group |
d7b8e6c6 EZ |
7491 | 2: [0, 0, 0, ... ] 1: [0, 0., 0., ... ] 1: 1.75 |
7492 | 1: [0, 0.25, 0.5, ... ] . . | |
7493 | . | |
7494 | ||
7495 | r 1 V M * V R + | |
d7b8e6c6 | 7496 | @end group |
5d67986c | 7497 | @end smallexample |
d7b8e6c6 EZ |
7498 | |
7499 | If @kbd{a =} had produced more than one @cite{1} value, this method | |
7500 | would have given the sum of all maximum @cite{x} values; not very | |
7501 | useful! In this case we could have used @kbd{v m} (@code{calc-mask-vector}) | |
7502 | instead. This command deletes all elements of a ``data'' vector that | |
7503 | correspond to zeros in a ``mask'' vector, leaving us with, in this | |
7504 | example, a vector of maximum @cite{x} values. | |
7505 | ||
7506 | The built-in @kbd{a X} command maximizes a function using more | |
7507 | efficient methods. Just for illustration, let's use @kbd{a X} | |
7508 | to maximize @samp{besJ(1,x)} over this same interval. | |
7509 | ||
d7b8e6c6 | 7510 | @smallexample |
5d67986c | 7511 | @group |
d7b8e6c6 EZ |
7512 | 2: besJ(1, x) 1: [1.84115, 0.581865] |
7513 | 1: [0 .. 5] . | |
7514 | . | |
7515 | ||
5d67986c | 7516 | ' besJ(1,x), [0..5] @key{RET} a X x @key{RET} |
d7b8e6c6 | 7517 | @end group |
5d67986c | 7518 | @end smallexample |
d7b8e6c6 EZ |
7519 | |
7520 | @noindent | |
7521 | The output from @kbd{a X} is a vector containing the value of @cite{x} | |
7522 | that maximizes the function, and the function's value at that maximum. | |
7523 | As you can see, our simple search got quite close to the right answer. | |
7524 | ||
7525 | @node List Answer 9, List Answer 10, List Answer 8, Answers to Exercises | |
7526 | @subsection List Tutorial Exercise 9 | |
7527 | ||
7528 | @noindent | |
7529 | Step one is to convert our integer into vector notation. | |
7530 | ||
d7b8e6c6 | 7531 | @smallexample |
5d67986c | 7532 | @group |
d7b8e6c6 EZ |
7533 | 1: 25129925999 3: 25129925999 |
7534 | . 2: 10 | |
7535 | 1: [11, 10, 9, ..., 1, 0] | |
7536 | . | |
7537 | ||
5d67986c | 7538 | 25129925999 @key{RET} 10 @key{RET} 12 @key{RET} v x 12 @key{RET} - |
d7b8e6c6 | 7539 | |
d7b8e6c6 | 7540 | @end group |
5d67986c | 7541 | @end smallexample |
d7b8e6c6 | 7542 | @noindent |
d7b8e6c6 | 7543 | @smallexample |
5d67986c | 7544 | @group |
d7b8e6c6 EZ |
7545 | 1: 25129925999 1: [0, 2, 25, 251, 2512, ... ] |
7546 | 2: [100000000000, ... ] . | |
7547 | . | |
7548 | ||
7549 | V M ^ s 1 V M \ | |
d7b8e6c6 | 7550 | @end group |
5d67986c | 7551 | @end smallexample |
d7b8e6c6 EZ |
7552 | |
7553 | @noindent | |
7554 | (Recall, the @kbd{\} command computes an integer quotient.) | |
7555 | ||
d7b8e6c6 | 7556 | @smallexample |
5d67986c | 7557 | @group |
d7b8e6c6 EZ |
7558 | 1: [0, 2, 5, 1, 2, 9, 9, 2, 5, 9, 9, 9] |
7559 | . | |
7560 | ||
7561 | 10 V M % s 2 | |
d7b8e6c6 | 7562 | @end group |
5d67986c | 7563 | @end smallexample |
d7b8e6c6 EZ |
7564 | |
7565 | Next we must increment this number. This involves adding one to | |
7566 | the last digit, plus handling carries. There is a carry to the | |
7567 | left out of a digit if that digit is a nine and all the digits to | |
7568 | the right of it are nines. | |
7569 | ||
d7b8e6c6 | 7570 | @smallexample |
5d67986c | 7571 | @group |
d7b8e6c6 EZ |
7572 | 1: [0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1] 1: [1, 1, 1, 0, 0, 1, ... ] |
7573 | . . | |
7574 | ||
7575 | 9 V M a = v v | |
7576 | ||
d7b8e6c6 | 7577 | @end group |
5d67986c | 7578 | @end smallexample |
d7b8e6c6 | 7579 | @noindent |
d7b8e6c6 | 7580 | @smallexample |
5d67986c | 7581 | @group |
d7b8e6c6 EZ |
7582 | 1: [1, 1, 1, 0, 0, 0, ... ] 1: [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1] |
7583 | . . | |
7584 | ||
7585 | V U * v v 1 | | |
d7b8e6c6 | 7586 | @end group |
5d67986c | 7587 | @end smallexample |
d7b8e6c6 EZ |
7588 | |
7589 | @noindent | |
7590 | Accumulating @kbd{*} across a vector of ones and zeros will preserve | |
7591 | only the initial run of ones. These are the carries into all digits | |
7592 | except the rightmost digit. Concatenating a one on the right takes | |
7593 | care of aligning the carries properly, and also adding one to the | |
7594 | rightmost digit. | |
7595 | ||
d7b8e6c6 | 7596 | @smallexample |
5d67986c | 7597 | @group |
d7b8e6c6 EZ |
7598 | 2: [0, 0, 0, 0, ... ] 1: [0, 0, 2, 5, 1, 2, 9, 9, 2, 6, 0, 0, 0] |
7599 | 1: [0, 0, 2, 5, ... ] . | |
7600 | . | |
7601 | ||
7602 | 0 r 2 | V M + 10 V M % | |
d7b8e6c6 | 7603 | @end group |
5d67986c | 7604 | @end smallexample |
d7b8e6c6 EZ |
7605 | |
7606 | @noindent | |
7607 | Here we have concatenated 0 to the @emph{left} of the original number; | |
7608 | this takes care of shifting the carries by one with respect to the | |
7609 | digits that generated them. | |
7610 | ||
7611 | Finally, we must convert this list back into an integer. | |
7612 | ||
d7b8e6c6 | 7613 | @smallexample |
5d67986c | 7614 | @group |
d7b8e6c6 EZ |
7615 | 3: [0, 0, 2, 5, ... ] 2: [0, 0, 2, 5, ... ] |
7616 | 2: 1000000000000 1: [1000000000000, 100000000000, ... ] | |
7617 | 1: [100000000000, ... ] . | |
7618 | . | |
7619 | ||
5d67986c | 7620 | 10 @key{RET} 12 ^ r 1 | |
d7b8e6c6 | 7621 | |
d7b8e6c6 | 7622 | @end group |
5d67986c | 7623 | @end smallexample |
d7b8e6c6 | 7624 | @noindent |
d7b8e6c6 | 7625 | @smallexample |
5d67986c | 7626 | @group |
d7b8e6c6 EZ |
7627 | 1: [0, 0, 20000000000, 5000000000, ... ] 1: 25129926000 |
7628 | . . | |
7629 | ||
7630 | V M * V R + | |
d7b8e6c6 | 7631 | @end group |
5d67986c | 7632 | @end smallexample |
d7b8e6c6 EZ |
7633 | |
7634 | @noindent | |
7635 | Another way to do this final step would be to reduce the formula | |
7636 | @w{@samp{10 $$ + $}} across the vector of digits. | |
7637 | ||
d7b8e6c6 | 7638 | @smallexample |
5d67986c | 7639 | @group |
d7b8e6c6 EZ |
7640 | 1: [0, 0, 2, 5, ... ] 1: 25129926000 |
7641 | . . | |
7642 | ||
5d67986c | 7643 | V R ' 10 $$ + $ @key{RET} |
d7b8e6c6 | 7644 | @end group |
5d67986c | 7645 | @end smallexample |
d7b8e6c6 EZ |
7646 | |
7647 | @node List Answer 10, List Answer 11, List Answer 9, Answers to Exercises | |
7648 | @subsection List Tutorial Exercise 10 | |
7649 | ||
7650 | @noindent | |
7651 | For the list @cite{[a, b, c, d]}, the result is @cite{((a = b) = c) = d}, | |
7652 | which will compare @cite{a} and @cite{b} to produce a 1 or 0, which is | |
7653 | then compared with @cite{c} to produce another 1 or 0, which is then | |
7654 | compared with @cite{d}. This is not at all what Joe wanted. | |
7655 | ||
7656 | Here's a more correct method: | |
7657 | ||
d7b8e6c6 | 7658 | @smallexample |
5d67986c | 7659 | @group |
d7b8e6c6 EZ |
7660 | 1: [7, 7, 7, 8, 7] 2: [7, 7, 7, 8, 7] |
7661 | . 1: 7 | |
7662 | . | |
7663 | ||
5d67986c | 7664 | ' [7,7,7,8,7] @key{RET} @key{RET} v r 1 @key{RET} |
d7b8e6c6 | 7665 | |
d7b8e6c6 | 7666 | @end group |
5d67986c | 7667 | @end smallexample |
d7b8e6c6 | 7668 | @noindent |
d7b8e6c6 | 7669 | @smallexample |
5d67986c | 7670 | @group |
d7b8e6c6 EZ |
7671 | 1: [1, 1, 1, 0, 1] 1: 0 |
7672 | . . | |
7673 | ||
7674 | V M a = V R * | |
d7b8e6c6 | 7675 | @end group |
5d67986c | 7676 | @end smallexample |
d7b8e6c6 EZ |
7677 | |
7678 | @node List Answer 11, List Answer 12, List Answer 10, Answers to Exercises | |
7679 | @subsection List Tutorial Exercise 11 | |
7680 | ||
7681 | @noindent | |
7682 | The circle of unit radius consists of those points @cite{(x,y)} for which | |
7683 | @cite{x^2 + y^2 < 1}. We start by generating a vector of @cite{x^2} | |
7684 | and a vector of @cite{y^2}. | |
7685 | ||
7686 | We can make this go a bit faster by using the @kbd{v .} and @kbd{t .} | |
7687 | commands. | |
7688 | ||
d7b8e6c6 | 7689 | @smallexample |
5d67986c | 7690 | @group |
d7b8e6c6 EZ |
7691 | 2: [2., 2., ..., 2.] 2: [2., 2., ..., 2.] |
7692 | 1: [2., 2., ..., 2.] 1: [1.16, 1.98, ..., 0.81] | |
7693 | . . | |
7694 | ||
5d67986c | 7695 | v . t . 2. v b 100 @key{RET} @key{RET} V M k r |
d7b8e6c6 | 7696 | |
d7b8e6c6 | 7697 | @end group |
5d67986c | 7698 | @end smallexample |
d7b8e6c6 | 7699 | @noindent |
d7b8e6c6 | 7700 | @smallexample |
5d67986c | 7701 | @group |
d7b8e6c6 EZ |
7702 | 2: [2., 2., ..., 2.] 1: [0.026, 0.96, ..., 0.036] |
7703 | 1: [0.026, 0.96, ..., 0.036] 2: [0.53, 0.81, ..., 0.094] | |
7704 | . . | |
7705 | ||
5d67986c | 7706 | 1 - 2 V M ^ @key{TAB} V M k r 1 - 2 V M ^ |
d7b8e6c6 | 7707 | @end group |
5d67986c | 7708 | @end smallexample |
d7b8e6c6 EZ |
7709 | |
7710 | Now we sum the @cite{x^2} and @cite{y^2} values, compare with 1 to | |
7711 | get a vector of 1/0 truth values, then sum the truth values. | |
7712 | ||
d7b8e6c6 | 7713 | @smallexample |
5d67986c | 7714 | @group |
d7b8e6c6 EZ |
7715 | 1: [0.56, 1.78, ..., 0.13] 1: [1, 0, ..., 1] 1: 84 |
7716 | . . . | |
7717 | ||
7718 | + 1 V M a < V R + | |
d7b8e6c6 | 7719 | @end group |
5d67986c | 7720 | @end smallexample |
d7b8e6c6 EZ |
7721 | |
7722 | @noindent | |
7723 | The ratio @cite{84/100} should approximate the ratio @c{$\pi/4$} | |
7724 | @cite{pi/4}. | |
7725 | ||
d7b8e6c6 | 7726 | @smallexample |
5d67986c | 7727 | @group |
d7b8e6c6 EZ |
7728 | 1: 0.84 1: 3.36 2: 3.36 1: 1.0695 |
7729 | . . 1: 3.14159 . | |
7730 | ||
7731 | 100 / 4 * P / | |
d7b8e6c6 | 7732 | @end group |
5d67986c | 7733 | @end smallexample |
d7b8e6c6 EZ |
7734 | |
7735 | @noindent | |
7736 | Our estimate, 3.36, is off by about 7%. We could get a better estimate | |
7737 | by taking more points (say, 1000), but it's clear that this method is | |
7738 | not very efficient! | |
7739 | ||
7740 | (Naturally, since this example uses random numbers your own answer | |
7741 | will be slightly different from the one shown here!) | |
7742 | ||
7743 | If you typed @kbd{v .} and @kbd{t .} before, type them again to | |
7744 | return to full-sized display of vectors. | |
7745 | ||
7746 | @node List Answer 12, List Answer 13, List Answer 11, Answers to Exercises | |
7747 | @subsection List Tutorial Exercise 12 | |
7748 | ||
7749 | @noindent | |
7750 | This problem can be made a lot easier by taking advantage of some | |
7751 | symmetries. First of all, after some thought it's clear that the | |
7752 | @cite{y} axis can be ignored altogether. Just pick a random @cite{x} | |
7753 | component for one end of the match, pick a random direction @c{$\theta$} | |
7754 | @cite{theta}, | |
7755 | and see if @cite{x} and @c{$x + \cos \theta$} | |
7756 | @cite{x + cos(theta)} (which is the @cite{x} | |
7757 | coordinate of the other endpoint) cross a line. The lines are at | |
7758 | integer coordinates, so this happens when the two numbers surround | |
7759 | an integer. | |
7760 | ||
7761 | Since the two endpoints are equivalent, we may as well choose the leftmost | |
7762 | of the two endpoints as @cite{x}. Then @cite{theta} is an angle pointing | |
7763 | to the right, in the range -90 to 90 degrees. (We could use radians, but | |
7764 | it would feel like cheating to refer to @c{$\pi/2$} | |
7765 | @cite{pi/2} radians while trying | |
7766 | to estimate @c{$\pi$} | |
7767 | @cite{pi}!) | |
7768 | ||
7769 | In fact, since the field of lines is infinite we can choose the | |
7770 | coordinates 0 and 1 for the lines on either side of the leftmost | |
7771 | endpoint. The rightmost endpoint will be between 0 and 1 if the | |
7772 | match does not cross a line, or between 1 and 2 if it does. So: | |
7773 | Pick random @cite{x} and @c{$\theta$} | |
7774 | @cite{theta}, compute @c{$x + \cos \theta$} | |
7775 | @cite{x + cos(theta)}, | |
7776 | and count how many of the results are greater than one. Simple! | |
7777 | ||
7778 | We can make this go a bit faster by using the @kbd{v .} and @kbd{t .} | |
7779 | commands. | |
7780 | ||
d7b8e6c6 | 7781 | @smallexample |
5d67986c | 7782 | @group |
d7b8e6c6 EZ |
7783 | 1: [0.52, 0.71, ..., 0.72] 2: [0.52, 0.71, ..., 0.72] |
7784 | . 1: [78.4, 64.5, ..., -42.9] | |
7785 | . | |
7786 | ||
5d67986c | 7787 | v . t . 1. v b 100 @key{RET} V M k r 180. v b 100 @key{RET} V M k r 90 - |
d7b8e6c6 | 7788 | @end group |
5d67986c | 7789 | @end smallexample |
d7b8e6c6 EZ |
7790 | |
7791 | @noindent | |
7792 | (The next step may be slow, depending on the speed of your computer.) | |
7793 | ||
d7b8e6c6 | 7794 | @smallexample |
5d67986c | 7795 | @group |
d7b8e6c6 EZ |
7796 | 2: [0.52, 0.71, ..., 0.72] 1: [0.72, 1.14, ..., 1.45] |
7797 | 1: [0.20, 0.43, ..., 0.73] . | |
7798 | . | |
7799 | ||
7800 | m d V M C + | |
7801 | ||
d7b8e6c6 | 7802 | @end group |
5d67986c | 7803 | @end smallexample |
d7b8e6c6 | 7804 | @noindent |
d7b8e6c6 | 7805 | @smallexample |
5d67986c | 7806 | @group |
d7b8e6c6 EZ |
7807 | 1: [0, 1, ..., 1] 1: 0.64 1: 3.125 |
7808 | . . . | |
7809 | ||
5d67986c | 7810 | 1 V M a > V R + 100 / 2 @key{TAB} / |
d7b8e6c6 | 7811 | @end group |
5d67986c | 7812 | @end smallexample |
d7b8e6c6 EZ |
7813 | |
7814 | Let's try the third method, too. We'll use random integers up to | |
7815 | one million. The @kbd{k r} command with an integer argument picks | |
7816 | a random integer. | |
7817 | ||
d7b8e6c6 | 7818 | @smallexample |
5d67986c | 7819 | @group |
d7b8e6c6 EZ |
7820 | 2: [1000000, 1000000, ..., 1000000] 2: [78489, 527587, ..., 814975] |
7821 | 1: [1000000, 1000000, ..., 1000000] 1: [324014, 358783, ..., 955450] | |
7822 | . . | |
7823 | ||
5d67986c | 7824 | 1000000 v b 100 @key{RET} @key{RET} V M k r @key{TAB} V M k r |
d7b8e6c6 | 7825 | |
d7b8e6c6 | 7826 | @end group |
5d67986c | 7827 | @end smallexample |
d7b8e6c6 | 7828 | @noindent |
d7b8e6c6 | 7829 | @smallexample |
5d67986c | 7830 | @group |
d7b8e6c6 EZ |
7831 | 1: [1, 1, ..., 25] 1: [1, 1, ..., 0] 1: 0.56 |
7832 | . . . | |
7833 | ||
7834 | V M k g 1 V M a = V R + 100 / | |
7835 | ||
d7b8e6c6 | 7836 | @end group |
5d67986c | 7837 | @end smallexample |
d7b8e6c6 | 7838 | @noindent |
d7b8e6c6 | 7839 | @smallexample |
5d67986c | 7840 | @group |
d7b8e6c6 EZ |
7841 | 1: 10.714 1: 3.273 |
7842 | . . | |
7843 | ||
5d67986c | 7844 | 6 @key{TAB} / Q |
d7b8e6c6 | 7845 | @end group |
5d67986c | 7846 | @end smallexample |
d7b8e6c6 EZ |
7847 | |
7848 | For a proof of this property of the GCD function, see section 4.5.2, | |
7849 | exercise 10, of Knuth's @emph{Art of Computer Programming}, volume II. | |
7850 | ||
7851 | If you typed @kbd{v .} and @kbd{t .} before, type them again to | |
7852 | return to full-sized display of vectors. | |
7853 | ||
7854 | @node List Answer 13, List Answer 14, List Answer 12, Answers to Exercises | |
7855 | @subsection List Tutorial Exercise 13 | |
7856 | ||
7857 | @noindent | |
7858 | First, we put the string on the stack as a vector of ASCII codes. | |
7859 | ||
d7b8e6c6 | 7860 | @smallexample |
5d67986c | 7861 | @group |
d7b8e6c6 EZ |
7862 | 1: [84, 101, 115, ..., 51] |
7863 | . | |
7864 | ||
5d67986c | 7865 | "Testing, 1, 2, 3 @key{RET} |
d7b8e6c6 | 7866 | @end group |
5d67986c | 7867 | @end smallexample |
d7b8e6c6 EZ |
7868 | |
7869 | @noindent | |
7870 | Note that the @kbd{"} key, like @kbd{$}, initiates algebraic entry so | |
7871 | there was no need to type an apostrophe. Also, Calc didn't mind that | |
7872 | we omitted the closing @kbd{"}. (The same goes for all closing delimiters | |
7873 | like @kbd{)} and @kbd{]} at the end of a formula. | |
7874 | ||
7875 | We'll show two different approaches here. In the first, we note that | |
7876 | if the input vector is @cite{[a, b, c, d]}, then the hash code is | |
7877 | @cite{3 (3 (3a + b) + c) + d = 27a + 9b + 3c + d}. In other words, | |
7878 | it's a sum of descending powers of three times the ASCII codes. | |
7879 | ||
d7b8e6c6 | 7880 | @smallexample |
5d67986c | 7881 | @group |
d7b8e6c6 EZ |
7882 | 2: [84, 101, 115, ..., 51] 2: [84, 101, 115, ..., 51] |
7883 | 1: 16 1: [15, 14, 13, ..., 0] | |
7884 | . . | |
7885 | ||
5d67986c | 7886 | @key{RET} v l v x 16 @key{RET} - |
d7b8e6c6 | 7887 | |
d7b8e6c6 | 7888 | @end group |
5d67986c | 7889 | @end smallexample |
d7b8e6c6 | 7890 | @noindent |
d7b8e6c6 | 7891 | @smallexample |
5d67986c | 7892 | @group |
d7b8e6c6 EZ |
7893 | 2: [84, 101, 115, ..., 51] 1: 1960915098 1: 121 |
7894 | 1: [14348907, ..., 1] . . | |
7895 | . | |
7896 | ||
5d67986c | 7897 | 3 @key{TAB} V M ^ * 511 % |
d7b8e6c6 | 7898 | @end group |
5d67986c | 7899 | @end smallexample |
d7b8e6c6 EZ |
7900 | |
7901 | @noindent | |
7902 | Once again, @kbd{*} elegantly summarizes most of the computation. | |
7903 | But there's an even more elegant approach: Reduce the formula | |
7904 | @kbd{3 $$ + $} across the vector. Recall that this represents a | |
7905 | function of two arguments that computes its first argument times three | |
7906 | plus its second argument. | |
7907 | ||
d7b8e6c6 | 7908 | @smallexample |
5d67986c | 7909 | @group |
d7b8e6c6 EZ |
7910 | 1: [84, 101, 115, ..., 51] 1: 1960915098 |
7911 | . . | |
7912 | ||
5d67986c | 7913 | "Testing, 1, 2, 3 @key{RET} V R ' 3$$+$ @key{RET} |
d7b8e6c6 | 7914 | @end group |
5d67986c | 7915 | @end smallexample |
d7b8e6c6 EZ |
7916 | |
7917 | @noindent | |
7918 | If you did the decimal arithmetic exercise, this will be familiar. | |
7919 | Basically, we're turning a base-3 vector of digits into an integer, | |
7920 | except that our ``digits'' are much larger than real digits. | |
7921 | ||
7922 | Instead of typing @kbd{511 %} again to reduce the result, we can be | |
7923 | cleverer still and notice that rather than computing a huge integer | |
7924 | and taking the modulo at the end, we can take the modulo at each step | |
7925 | without affecting the result. While this means there are more | |
7926 | arithmetic operations, the numbers we operate on remain small so | |
7927 | the operations are faster. | |
7928 | ||
d7b8e6c6 | 7929 | @smallexample |
5d67986c | 7930 | @group |
d7b8e6c6 EZ |
7931 | 1: [84, 101, 115, ..., 51] 1: 121 |
7932 | . . | |
7933 | ||
5d67986c | 7934 | "Testing, 1, 2, 3 @key{RET} V R ' (3$$+$)%511 @key{RET} |
d7b8e6c6 | 7935 | @end group |
5d67986c | 7936 | @end smallexample |
d7b8e6c6 EZ |
7937 | |
7938 | Why does this work? Think about a two-step computation: | |
7939 | @w{@cite{3 (3a + b) + c}}. Taking a result modulo 511 basically means | |
7940 | subtracting off enough 511's to put the result in the desired range. | |
7941 | So the result when we take the modulo after every step is, | |
7942 | ||
7943 | @ifinfo | |
7944 | @example | |
7945 | 3 (3 a + b - 511 m) + c - 511 n | |
7946 | @end example | |
7947 | @end ifinfo | |
7948 | @tex | |
7949 | \turnoffactive | |
7950 | \beforedisplay | |
7951 | $$ 3 (3 a + b - 511 m) + c - 511 n $$ | |
7952 | \afterdisplay | |
7953 | @end tex | |
7954 | ||
7955 | @noindent | |
7956 | for some suitable integers @cite{m} and @cite{n}. Expanding out by | |
7957 | the distributive law yields | |
7958 | ||
7959 | @ifinfo | |
7960 | @example | |
7961 | 9 a + 3 b + c - 511*3 m - 511 n | |
7962 | @end example | |
7963 | @end ifinfo | |
7964 | @tex | |
7965 | \turnoffactive | |
7966 | \beforedisplay | |
7967 | $$ 9 a + 3 b + c - 511\times3 m - 511 n $$ | |
7968 | \afterdisplay | |
7969 | @end tex | |
7970 | ||
7971 | @noindent | |
7972 | The @cite{m} term in the latter formula is redundant because any | |
7973 | contribution it makes could just as easily be made by the @cite{n} | |
7974 | term. So we can take it out to get an equivalent formula with | |
7975 | @cite{n' = 3m + n}, | |
7976 | ||
7977 | @ifinfo | |
7978 | @example | |
7979 | 9 a + 3 b + c - 511 n' | |
7980 | @end example | |
7981 | @end ifinfo | |
7982 | @tex | |
7983 | \turnoffactive | |
7984 | \beforedisplay | |
7985 | $$ 9 a + 3 b + c - 511 n' $$ | |
7986 | \afterdisplay | |
7987 | @end tex | |
7988 | ||
7989 | @noindent | |
7990 | which is just the formula for taking the modulo only at the end of | |
7991 | the calculation. Therefore the two methods are essentially the same. | |
7992 | ||
7993 | Later in the tutorial we will encounter @dfn{modulo forms}, which | |
7994 | basically automate the idea of reducing every intermediate result | |
5d67986c | 7995 | modulo some value @var{m}. |
d7b8e6c6 EZ |
7996 | |
7997 | @node List Answer 14, Types Answer 1, List Answer 13, Answers to Exercises | |
7998 | @subsection List Tutorial Exercise 14 | |
7999 | ||
8000 | We want to use @kbd{H V U} to nest a function which adds a random | |
8001 | step to an @cite{(x,y)} coordinate. The function is a bit long, but | |
8002 | otherwise the problem is quite straightforward. | |
8003 | ||
d7b8e6c6 | 8004 | @smallexample |
5d67986c | 8005 | @group |
d7b8e6c6 EZ |
8006 | 2: [0, 0] 1: [ [ 0, 0 ] |
8007 | 1: 50 [ 0.4288, -0.1695 ] | |
8008 | . [ -0.4787, -0.9027 ] | |
8009 | ... | |
8010 | ||
5d67986c | 8011 | [0,0] 50 H V U ' <# + [random(2.0)-1, random(2.0)-1]> @key{RET} |
d7b8e6c6 | 8012 | @end group |
5d67986c | 8013 | @end smallexample |
d7b8e6c6 EZ |
8014 | |
8015 | Just as the text recommended, we used @samp{< >} nameless function | |
8016 | notation to keep the two @code{random} calls from being evaluated | |
8017 | before nesting even begins. | |
8018 | ||
8019 | We now have a vector of @cite{[x, y]} sub-vectors, which by Calc's | |
8020 | rules acts like a matrix. We can transpose this matrix and unpack | |
8021 | to get a pair of vectors, @cite{x} and @cite{y}, suitable for graphing. | |
8022 | ||
d7b8e6c6 | 8023 | @smallexample |
5d67986c | 8024 | @group |
d7b8e6c6 EZ |
8025 | 2: [ 0, 0.4288, -0.4787, ... ] |
8026 | 1: [ 0, -0.1696, -0.9027, ... ] | |
8027 | . | |
8028 | ||
8029 | v t v u g f | |
d7b8e6c6 | 8030 | @end group |
5d67986c | 8031 | @end smallexample |
d7b8e6c6 EZ |
8032 | |
8033 | Incidentally, because the @cite{x} and @cite{y} are completely | |
8034 | independent in this case, we could have done two separate commands | |
8035 | to create our @cite{x} and @cite{y} vectors of numbers directly. | |
8036 | ||
8037 | To make a random walk of unit steps, we note that @code{sincos} of | |
8038 | a random direction exactly gives us an @cite{[x, y]} step of unit | |
8039 | length; in fact, the new nesting function is even briefer, though | |
8040 | we might want to lower the precision a bit for it. | |
8041 | ||
d7b8e6c6 | 8042 | @smallexample |
5d67986c | 8043 | @group |
d7b8e6c6 EZ |
8044 | 2: [0, 0] 1: [ [ 0, 0 ] |
8045 | 1: 50 [ 0.1318, 0.9912 ] | |
8046 | . [ -0.5965, 0.3061 ] | |
8047 | ... | |
8048 | ||
5d67986c | 8049 | [0,0] 50 m d p 6 @key{RET} H V U ' <# + sincos(random(360.0))> @key{RET} |
d7b8e6c6 | 8050 | @end group |
5d67986c | 8051 | @end smallexample |
d7b8e6c6 EZ |
8052 | |
8053 | Another @kbd{v t v u g f} sequence will graph this new random walk. | |
8054 | ||
8055 | An interesting twist on these random walk functions would be to use | |
8056 | complex numbers instead of 2-vectors to represent points on the plane. | |
8057 | In the first example, we'd use something like @samp{random + random*(0,1)}, | |
8058 | and in the second we could use polar complex numbers with random phase | |
8059 | angles. (This exercise was first suggested in this form by Randal | |
8060 | Schwartz.) | |
8061 | ||
8062 | @node Types Answer 1, Types Answer 2, List Answer 14, Answers to Exercises | |
8063 | @subsection Types Tutorial Exercise 1 | |
8064 | ||
8065 | @noindent | |
8066 | If the number is the square root of @c{$\pi$} | |
8067 | @cite{pi} times a rational number, | |
8068 | then its square, divided by @c{$\pi$} | |
8069 | @cite{pi}, should be a rational number. | |
8070 | ||
d7b8e6c6 | 8071 | @smallexample |
5d67986c | 8072 | @group |
d7b8e6c6 EZ |
8073 | 1: 1.26508260337 1: 0.509433962268 1: 2486645810:4881193627 |
8074 | . . . | |
8075 | ||
8076 | 2 ^ P / c F | |
d7b8e6c6 | 8077 | @end group |
5d67986c | 8078 | @end smallexample |
d7b8e6c6 EZ |
8079 | |
8080 | @noindent | |
8081 | Technically speaking this is a rational number, but not one that is | |
8082 | likely to have arisen in the original problem. More likely, it just | |
8083 | happens to be the fraction which most closely represents some | |
8084 | irrational number to within 12 digits. | |
8085 | ||
8086 | But perhaps our result was not quite exact. Let's reduce the | |
8087 | precision slightly and try again: | |
8088 | ||
d7b8e6c6 | 8089 | @smallexample |
5d67986c | 8090 | @group |
d7b8e6c6 EZ |
8091 | 1: 0.509433962268 1: 27:53 |
8092 | . . | |
8093 | ||
5d67986c | 8094 | U p 10 @key{RET} c F |
d7b8e6c6 | 8095 | @end group |
5d67986c | 8096 | @end smallexample |
d7b8e6c6 EZ |
8097 | |
8098 | @noindent | |
8099 | Aha! It's unlikely that an irrational number would equal a fraction | |
8100 | this simple to within ten digits, so our original number was probably | |
8101 | @c{$\sqrt{27 \pi / 53}$} | |
8102 | @cite{sqrt(27 pi / 53)}. | |
8103 | ||
8104 | Notice that we didn't need to re-round the number when we reduced the | |
8105 | precision. Remember, arithmetic operations always round their inputs | |
8106 | to the current precision before they begin. | |
8107 | ||
8108 | @node Types Answer 2, Types Answer 3, Types Answer 1, Answers to Exercises | |
8109 | @subsection Types Tutorial Exercise 2 | |
8110 | ||
8111 | @noindent | |
8112 | @samp{inf / inf = nan}. Perhaps @samp{1} is the ``obvious'' answer. | |
8113 | But if @w{@samp{17 inf = inf}}, then @samp{17 inf / inf = inf / inf = 17}, too. | |
8114 | ||
8115 | @samp{exp(inf) = inf}. It's tempting to say that the exponential | |
8116 | of infinity must be ``bigger'' than ``regular'' infinity, but as | |
8117 | far as Calc is concerned all infinities are as just as big. | |
8118 | In other words, as @cite{x} goes to infinity, @cite{e^x} also goes | |
8119 | to infinity, but the fact the @cite{e^x} grows much faster than | |
8120 | @cite{x} is not relevant here. | |
8121 | ||
8122 | @samp{exp(-inf) = 0}. Here we have a finite answer even though | |
8123 | the input is infinite. | |
8124 | ||
8125 | @samp{sqrt(-inf) = (0, 1) inf}. Remember that @cite{(0, 1)} | |
8126 | represents the imaginary number @cite{i}. Here's a derivation: | |
8127 | @samp{sqrt(-inf) = @w{sqrt((-1) * inf)} = sqrt(-1) * sqrt(inf)}. | |
8128 | The first part is, by definition, @cite{i}; the second is @code{inf} | |
8129 | because, once again, all infinities are the same size. | |
8130 | ||
8131 | @samp{sqrt(uinf) = uinf}. In fact, we do know something about the | |
8132 | direction because @code{sqrt} is defined to return a value in the | |
8133 | right half of the complex plane. But Calc has no notation for this, | |
8134 | so it settles for the conservative answer @code{uinf}. | |
8135 | ||
8136 | @samp{abs(uinf) = inf}. No matter which direction @cite{x} points, | |
8137 | @samp{abs(x)} always points along the positive real axis. | |
8138 | ||
8139 | @samp{ln(0) = -inf}. Here we have an infinite answer to a finite | |
8140 | input. As in the @cite{1 / 0} case, Calc will only use infinities | |
8141 | here if you have turned on ``infinite'' mode. Otherwise, it will | |
8142 | treat @samp{ln(0)} as an error. | |
8143 | ||
8144 | @node Types Answer 3, Types Answer 4, Types Answer 2, Answers to Exercises | |
8145 | @subsection Types Tutorial Exercise 3 | |
8146 | ||
8147 | @noindent | |
8148 | We can make @samp{inf - inf} be any real number we like, say, | |
8149 | @cite{a}, just by claiming that we added @cite{a} to the first | |
8150 | infinity but not to the second. This is just as true for complex | |
8151 | values of @cite{a}, so @code{nan} can stand for a complex number. | |
8152 | (And, similarly, @code{uinf} can stand for an infinity that points | |
8153 | in any direction in the complex plane, such as @samp{(0, 1) inf}). | |
8154 | ||
8155 | In fact, we can multiply the first @code{inf} by two. Surely | |
8156 | @w{@samp{2 inf - inf = inf}}, but also @samp{2 inf - inf = inf - inf = nan}. | |
8157 | So @code{nan} can even stand for infinity. Obviously it's just | |
8158 | as easy to make it stand for minus infinity as for plus infinity. | |
8159 | ||
8160 | The moral of this story is that ``infinity'' is a slippery fish | |
8161 | indeed, and Calc tries to handle it by having a very simple model | |
8162 | for infinities (only the direction counts, not the ``size''); but | |
8163 | Calc is careful to write @code{nan} any time this simple model is | |
8164 | unable to tell what the true answer is. | |
8165 | ||
8166 | @node Types Answer 4, Types Answer 5, Types Answer 3, Answers to Exercises | |
8167 | @subsection Types Tutorial Exercise 4 | |
8168 | ||
d7b8e6c6 | 8169 | @smallexample |
5d67986c | 8170 | @group |
d7b8e6c6 EZ |
8171 | 2: 0@@ 47' 26" 1: 0@@ 2' 47.411765" |
8172 | 1: 17 . | |
8173 | . | |
8174 | ||
5d67986c | 8175 | 0@@ 47' 26" @key{RET} 17 / |
d7b8e6c6 | 8176 | @end group |
5d67986c | 8177 | @end smallexample |
d7b8e6c6 EZ |
8178 | |
8179 | @noindent | |
8180 | The average song length is two minutes and 47.4 seconds. | |
8181 | ||
d7b8e6c6 | 8182 | @smallexample |
5d67986c | 8183 | @group |
d7b8e6c6 EZ |
8184 | 2: 0@@ 2' 47.411765" 1: 0@@ 3' 7.411765" 1: 0@@ 53' 6.000005" |
8185 | 1: 0@@ 0' 20" . . | |
8186 | . | |
8187 | ||
8188 | 20" + 17 * | |
d7b8e6c6 | 8189 | @end group |
5d67986c | 8190 | @end smallexample |
d7b8e6c6 EZ |
8191 | |
8192 | @noindent | |
8193 | The album would be 53 minutes and 6 seconds long. | |
8194 | ||
8195 | @node Types Answer 5, Types Answer 6, Types Answer 4, Answers to Exercises | |
8196 | @subsection Types Tutorial Exercise 5 | |
8197 | ||
8198 | @noindent | |
8199 | Let's suppose it's January 14, 1991. The easiest thing to do is | |
8200 | to keep trying 13ths of months until Calc reports a Friday. | |
8201 | We can do this by manually entering dates, or by using @kbd{t I}: | |
8202 | ||
d7b8e6c6 | 8203 | @smallexample |
5d67986c | 8204 | @group |
d7b8e6c6 EZ |
8205 | 1: <Wed Feb 13, 1991> 1: <Wed Mar 13, 1991> 1: <Sat Apr 13, 1991> |
8206 | . . . | |
8207 | ||
5d67986c | 8208 | ' <2/13> @key{RET} @key{DEL} ' <3/13> @key{RET} t I |
d7b8e6c6 | 8209 | @end group |
5d67986c | 8210 | @end smallexample |
d7b8e6c6 EZ |
8211 | |
8212 | @noindent | |
8213 | (Calc assumes the current year if you don't say otherwise.) | |
8214 | ||
8215 | This is getting tedious---we can keep advancing the date by typing | |
8216 | @kbd{t I} over and over again, but let's automate the job by using | |
8217 | vector mapping. The @kbd{t I} command actually takes a second | |
8218 | ``how-many-months'' argument, which defaults to one. This | |
8219 | argument is exactly what we want to map over: | |
8220 | ||
d7b8e6c6 | 8221 | @smallexample |
5d67986c | 8222 | @group |
d7b8e6c6 EZ |
8223 | 2: <Sat Apr 13, 1991> 1: [<Mon May 13, 1991>, <Thu Jun 13, 1991>, |
8224 | 1: [1, 2, 3, 4, 5, 6] <Sat Jul 13, 1991>, <Tue Aug 13, 1991>, | |
8225 | . <Fri Sep 13, 1991>, <Sun Oct 13, 1991>] | |
8226 | . | |
8227 | ||
5d67986c | 8228 | v x 6 @key{RET} V M t I |
d7b8e6c6 | 8229 | @end group |
5d67986c | 8230 | @end smallexample |
d7b8e6c6 | 8231 | |
d7b8e6c6 | 8232 | @noindent |
28665d46 | 8233 | Et voil@`a, September 13, 1991 is a Friday. |
d7b8e6c6 | 8234 | |
d7b8e6c6 | 8235 | @smallexample |
5d67986c | 8236 | @group |
d7b8e6c6 EZ |
8237 | 1: 242 |
8238 | . | |
8239 | ||
5d67986c | 8240 | ' <sep 13> - <jan 14> @key{RET} |
d7b8e6c6 | 8241 | @end group |
5d67986c | 8242 | @end smallexample |
d7b8e6c6 EZ |
8243 | |
8244 | @noindent | |
8245 | And the answer to our original question: 242 days to go. | |
8246 | ||
8247 | @node Types Answer 6, Types Answer 7, Types Answer 5, Answers to Exercises | |
8248 | @subsection Types Tutorial Exercise 6 | |
8249 | ||
8250 | @noindent | |
8251 | The full rule for leap years is that they occur in every year divisible | |
8252 | by four, except that they don't occur in years divisible by 100, except | |
8253 | that they @emph{do} in years divisible by 400. We could work out the | |
8254 | answer by carefully counting the years divisible by four and the | |
8255 | exceptions, but there is a much simpler way that works even if we | |
8256 | don't know the leap year rule. | |
8257 | ||
8258 | Let's assume the present year is 1991. Years have 365 days, except | |
8259 | that leap years (whenever they occur) have 366 days. So let's count | |
8260 | the number of days between now and then, and compare that to the | |
8261 | number of years times 365. The number of extra days we find must be | |
8262 | equal to the number of leap years there were. | |
8263 | ||
d7b8e6c6 | 8264 | @smallexample |
5d67986c | 8265 | @group |
d7b8e6c6 EZ |
8266 | 1: <Mon Jan 1, 10001> 2: <Mon Jan 1, 10001> 1: 2925593 |
8267 | . 1: <Tue Jan 1, 1991> . | |
8268 | . | |
8269 | ||
5d67986c | 8270 | ' <jan 1 10001> @key{RET} ' <jan 1 1991> @key{RET} - |
d7b8e6c6 | 8271 | |
d7b8e6c6 | 8272 | @end group |
5d67986c | 8273 | @end smallexample |
d7b8e6c6 | 8274 | @noindent |
d7b8e6c6 | 8275 | @smallexample |
5d67986c | 8276 | @group |
d7b8e6c6 EZ |
8277 | 3: 2925593 2: 2925593 2: 2925593 1: 1943 |
8278 | 2: 10001 1: 8010 1: 2923650 . | |
8279 | 1: 1991 . . | |
8280 | . | |
8281 | ||
5d67986c | 8282 | 10001 @key{RET} 1991 - 365 * - |
d7b8e6c6 | 8283 | @end group |
5d67986c | 8284 | @end smallexample |
d7b8e6c6 EZ |
8285 | |
8286 | @c [fix-ref Date Forms] | |
8287 | @noindent | |
8288 | There will be 1943 leap years before the year 10001. (Assuming, | |
8289 | of course, that the algorithm for computing leap years remains | |
8290 | unchanged for that long. @xref{Date Forms}, for some interesting | |
8291 | background information in that regard.) | |
8292 | ||
8293 | @node Types Answer 7, Types Answer 8, Types Answer 6, Answers to Exercises | |
8294 | @subsection Types Tutorial Exercise 7 | |
8295 | ||
8296 | @noindent | |
8297 | The relative errors must be converted to absolute errors so that | |
8298 | @samp{+/-} notation may be used. | |
8299 | ||
d7b8e6c6 | 8300 | @smallexample |
5d67986c | 8301 | @group |
d7b8e6c6 EZ |
8302 | 1: 1. 2: 1. |
8303 | . 1: 0.2 | |
8304 | . | |
8305 | ||
5d67986c | 8306 | 20 @key{RET} .05 * 4 @key{RET} .05 * |
d7b8e6c6 | 8307 | @end group |
5d67986c | 8308 | @end smallexample |
d7b8e6c6 EZ |
8309 | |
8310 | Now we simply chug through the formula. | |
8311 | ||
d7b8e6c6 | 8312 | @smallexample |
5d67986c | 8313 | @group |
d7b8e6c6 EZ |
8314 | 1: 19.7392088022 1: 394.78 +/- 19.739 1: 6316.5 +/- 706.21 |
8315 | . . . | |
8316 | ||
5d67986c | 8317 | 2 P 2 ^ * 20 p 1 * 4 p .2 @key{RET} 2 ^ * |
d7b8e6c6 | 8318 | @end group |
5d67986c | 8319 | @end smallexample |
d7b8e6c6 EZ |
8320 | |
8321 | It turns out the @kbd{v u} command will unpack an error form as | |
8322 | well as a vector. This saves us some retyping of numbers. | |
8323 | ||
d7b8e6c6 | 8324 | @smallexample |
5d67986c | 8325 | @group |
d7b8e6c6 EZ |
8326 | 3: 6316.5 +/- 706.21 2: 6316.5 +/- 706.21 |
8327 | 2: 6316.5 1: 0.1118 | |
8328 | 1: 706.21 . | |
8329 | . | |
8330 | ||
5d67986c | 8331 | @key{RET} v u @key{TAB} / |
d7b8e6c6 | 8332 | @end group |
5d67986c | 8333 | @end smallexample |
d7b8e6c6 EZ |
8334 | |
8335 | @noindent | |
8336 | Thus the volume is 6316 cubic centimeters, within about 11 percent. | |
8337 | ||
8338 | @node Types Answer 8, Types Answer 9, Types Answer 7, Answers to Exercises | |
8339 | @subsection Types Tutorial Exercise 8 | |
8340 | ||
8341 | @noindent | |
8342 | The first answer is pretty simple: @samp{1 / (0 .. 10) = (0.1 .. inf)}. | |
8343 | Since a number in the interval @samp{(0 .. 10)} can get arbitrarily | |
8344 | close to zero, its reciprocal can get arbitrarily large, so the answer | |
8345 | is an interval that effectively means, ``any number greater than 0.1'' | |
8346 | but with no upper bound. | |
8347 | ||
8348 | The second answer, similarly, is @samp{1 / (-10 .. 0) = (-inf .. -0.1)}. | |
8349 | ||
8350 | Calc normally treats division by zero as an error, so that the formula | |
8351 | @w{@samp{1 / 0}} is left unsimplified. Our third problem, | |
8352 | @w{@samp{1 / [0 .. 10]}}, also (potentially) divides by zero because zero | |
8353 | is now a member of the interval. So Calc leaves this one unevaluated, too. | |
8354 | ||
8355 | If you turn on ``infinite'' mode by pressing @kbd{m i}, you will | |
8356 | instead get the answer @samp{[0.1 .. inf]}, which includes infinity | |
8357 | as a possible value. | |
8358 | ||
8359 | The fourth calculation, @samp{1 / (-10 .. 10)}, has the same problem. | |
8360 | Zero is buried inside the interval, but it's still a possible value. | |
8361 | It's not hard to see that the actual result of @samp{1 / (-10 .. 10)} | |
8362 | will be either greater than @i{0.1}, or less than @i{-0.1}. Thus | |
8363 | the interval goes from minus infinity to plus infinity, with a ``hole'' | |
8364 | in it from @i{-0.1} to @i{0.1}. Calc doesn't have any way to | |
8365 | represent this, so it just reports @samp{[-inf .. inf]} as the answer. | |
8366 | It may be disappointing to hear ``the answer lies somewhere between | |
8367 | minus infinity and plus infinity, inclusive,'' but that's the best | |
8368 | that interval arithmetic can do in this case. | |
8369 | ||
8370 | @node Types Answer 9, Types Answer 10, Types Answer 8, Answers to Exercises | |
8371 | @subsection Types Tutorial Exercise 9 | |
8372 | ||
d7b8e6c6 | 8373 | @smallexample |
5d67986c | 8374 | @group |
d7b8e6c6 EZ |
8375 | 1: [-3 .. 3] 2: [-3 .. 3] 2: [0 .. 9] |
8376 | . 1: [0 .. 9] 1: [-9 .. 9] | |
8377 | . . | |
8378 | ||
5d67986c | 8379 | [ 3 n .. 3 ] @key{RET} 2 ^ @key{TAB} @key{RET} * |
d7b8e6c6 | 8380 | @end group |
5d67986c | 8381 | @end smallexample |
d7b8e6c6 EZ |
8382 | |
8383 | @noindent | |
8384 | In the first case the result says, ``if a number is between @i{-3} and | |
8385 | 3, its square is between 0 and 9.'' The second case says, ``the product | |
8386 | of two numbers each between @i{-3} and 3 is between @i{-9} and 9.'' | |
8387 | ||
8388 | An interval form is not a number; it is a symbol that can stand for | |
8389 | many different numbers. Two identical-looking interval forms can stand | |
8390 | for different numbers. | |
8391 | ||
8392 | The same issue arises when you try to square an error form. | |
8393 | ||
8394 | @node Types Answer 10, Types Answer 11, Types Answer 9, Answers to Exercises | |
8395 | @subsection Types Tutorial Exercise 10 | |
8396 | ||
8397 | @noindent | |
8398 | Testing the first number, we might arbitrarily choose 17 for @cite{x}. | |
8399 | ||
d7b8e6c6 | 8400 | @smallexample |
5d67986c | 8401 | @group |
d7b8e6c6 EZ |
8402 | 1: 17 mod 811749613 2: 17 mod 811749613 1: 533694123 mod 811749613 |
8403 | . 811749612 . | |
8404 | . | |
8405 | ||
5d67986c | 8406 | 17 M 811749613 @key{RET} 811749612 ^ |
d7b8e6c6 | 8407 | @end group |
5d67986c | 8408 | @end smallexample |
d7b8e6c6 EZ |
8409 | |
8410 | @noindent | |
8411 | Since 533694123 is (considerably) different from 1, the number 811749613 | |
8412 | must not be prime. | |
8413 | ||
8414 | It's awkward to type the number in twice as we did above. There are | |
8415 | various ways to avoid this, and algebraic entry is one. In fact, using | |
8416 | a vector mapping operation we can perform several tests at once. Let's | |
8417 | use this method to test the second number. | |
8418 | ||
d7b8e6c6 | 8419 | @smallexample |
5d67986c | 8420 | @group |
d7b8e6c6 EZ |
8421 | 2: [17, 42, 100000] 1: [1 mod 15485863, 1 mod ... ] |
8422 | 1: 15485863 . | |
8423 | . | |
8424 | ||
5d67986c | 8425 | [17 42 100000] 15485863 @key{RET} V M ' ($$ mod $)^($-1) @key{RET} |
d7b8e6c6 | 8426 | @end group |
5d67986c | 8427 | @end smallexample |
d7b8e6c6 EZ |
8428 | |
8429 | @noindent | |
8430 | The result is three ones (modulo @cite{n}), so it's very probable that | |
8431 | 15485863 is prime. (In fact, this number is the millionth prime.) | |
8432 | ||
8433 | Note that the functions @samp{($$^($-1)) mod $} or @samp{$$^($-1) % $} | |
8434 | would have been hopelessly inefficient, since they would have calculated | |
8435 | the power using full integer arithmetic. | |
8436 | ||
8437 | Calc has a @kbd{k p} command that does primality testing. For small | |
8438 | numbers it does an exact test; for large numbers it uses a variant | |
8439 | of the Fermat test we used here. You can use @kbd{k p} repeatedly | |
8440 | to prove that a large integer is prime with any desired probability. | |
8441 | ||
8442 | @node Types Answer 11, Types Answer 12, Types Answer 10, Answers to Exercises | |
8443 | @subsection Types Tutorial Exercise 11 | |
8444 | ||
8445 | @noindent | |
8446 | There are several ways to insert a calculated number into an HMS form. | |
8447 | One way to convert a number of seconds to an HMS form is simply to | |
8448 | multiply the number by an HMS form representing one second: | |
8449 | ||
d7b8e6c6 | 8450 | @smallexample |
5d67986c | 8451 | @group |
d7b8e6c6 EZ |
8452 | 1: 31415926.5359 2: 31415926.5359 1: 8726@@ 38' 46.5359" |
8453 | . 1: 0@@ 0' 1" . | |
8454 | . | |
8455 | ||
8456 | P 1e7 * 0@@ 0' 1" * | |
8457 | ||
d7b8e6c6 | 8458 | @end group |
5d67986c | 8459 | @end smallexample |
d7b8e6c6 | 8460 | @noindent |
d7b8e6c6 | 8461 | @smallexample |
5d67986c | 8462 | @group |
d7b8e6c6 EZ |
8463 | 2: 8726@@ 38' 46.5359" 1: 6@@ 6' 2.5359" mod 24@@ 0' 0" |
8464 | 1: 15@@ 27' 16" mod 24@@ 0' 0" . | |
8465 | . | |
8466 | ||
5d67986c | 8467 | x time @key{RET} + |
d7b8e6c6 | 8468 | @end group |
5d67986c | 8469 | @end smallexample |
d7b8e6c6 EZ |
8470 | |
8471 | @noindent | |
8472 | It will be just after six in the morning. | |
8473 | ||
8474 | The algebraic @code{hms} function can also be used to build an | |
8475 | HMS form: | |
8476 | ||
d7b8e6c6 | 8477 | @smallexample |
5d67986c | 8478 | @group |
d7b8e6c6 EZ |
8479 | 1: hms(0, 0, 10000000. pi) 1: 8726@@ 38' 46.5359" |
8480 | . . | |
8481 | ||
5d67986c | 8482 | ' hms(0, 0, 1e7 pi) @key{RET} = |
d7b8e6c6 | 8483 | @end group |
5d67986c | 8484 | @end smallexample |
d7b8e6c6 EZ |
8485 | |
8486 | @noindent | |
8487 | The @kbd{=} key is necessary to evaluate the symbol @samp{pi} to | |
8488 | the actual number 3.14159... | |
8489 | ||
8490 | @node Types Answer 12, Types Answer 13, Types Answer 11, Answers to Exercises | |
8491 | @subsection Types Tutorial Exercise 12 | |
8492 | ||
8493 | @noindent | |
8494 | As we recall, there are 17 songs of about 2 minutes and 47 seconds | |
8495 | each. | |
8496 | ||
d7b8e6c6 | 8497 | @smallexample |
5d67986c | 8498 | @group |
d7b8e6c6 EZ |
8499 | 2: 0@@ 2' 47" 1: [0@@ 3' 7" .. 0@@ 3' 47"] |
8500 | 1: [0@@ 0' 20" .. 0@@ 1' 0"] . | |
8501 | . | |
8502 | ||
8503 | [ 0@@ 20" .. 0@@ 1' ] + | |
8504 | ||
d7b8e6c6 | 8505 | @end group |
5d67986c | 8506 | @end smallexample |
d7b8e6c6 | 8507 | @noindent |
d7b8e6c6 | 8508 | @smallexample |
5d67986c | 8509 | @group |
d7b8e6c6 EZ |
8510 | 1: [0@@ 52' 59." .. 1@@ 4' 19."] |
8511 | . | |
8512 | ||
8513 | 17 * | |
d7b8e6c6 | 8514 | @end group |
5d67986c | 8515 | @end smallexample |
d7b8e6c6 EZ |
8516 | |
8517 | @noindent | |
8518 | No matter how long it is, the album will fit nicely on one CD. | |
8519 | ||
8520 | @node Types Answer 13, Types Answer 14, Types Answer 12, Answers to Exercises | |
8521 | @subsection Types Tutorial Exercise 13 | |
8522 | ||
8523 | @noindent | |
5d67986c | 8524 | Type @kbd{' 1 yr @key{RET} u c s @key{RET}}. The answer is 31557600 seconds. |
d7b8e6c6 EZ |
8525 | |
8526 | @node Types Answer 14, Types Answer 15, Types Answer 13, Answers to Exercises | |
8527 | @subsection Types Tutorial Exercise 14 | |
8528 | ||
8529 | @noindent | |
8530 | How long will it take for a signal to get from one end of the computer | |
8531 | to the other? | |
8532 | ||
d7b8e6c6 | 8533 | @smallexample |
5d67986c | 8534 | @group |
d7b8e6c6 EZ |
8535 | 1: m / c 1: 3.3356 ns |
8536 | . . | |
8537 | ||
5d67986c | 8538 | ' 1 m / c @key{RET} u c ns @key{RET} |
d7b8e6c6 | 8539 | @end group |
5d67986c | 8540 | @end smallexample |
d7b8e6c6 EZ |
8541 | |
8542 | @noindent | |
8543 | (Recall, @samp{c} is a ``unit'' corresponding to the speed of light.) | |
8544 | ||
d7b8e6c6 | 8545 | @smallexample |
5d67986c | 8546 | @group |
d7b8e6c6 EZ |
8547 | 1: 3.3356 ns 1: 0.81356 ns / ns 1: 0.81356 |
8548 | 2: 4.1 ns . . | |
8549 | . | |
8550 | ||
5d67986c | 8551 | ' 4.1 ns @key{RET} / u s |
d7b8e6c6 | 8552 | @end group |
5d67986c | 8553 | @end smallexample |
d7b8e6c6 EZ |
8554 | |
8555 | @noindent | |
8556 | Thus a signal could take up to 81 percent of a clock cycle just to | |
8557 | go from one place to another inside the computer, assuming the signal | |
8558 | could actually attain the full speed of light. Pretty tight! | |
8559 | ||
8560 | @node Types Answer 15, Algebra Answer 1, Types Answer 14, Answers to Exercises | |
8561 | @subsection Types Tutorial Exercise 15 | |
8562 | ||
8563 | @noindent | |
8564 | The speed limit is 55 miles per hour on most highways. We want to | |
8565 | find the ratio of Sam's speed to the US speed limit. | |
8566 | ||
d7b8e6c6 | 8567 | @smallexample |
5d67986c | 8568 | @group |
d7b8e6c6 EZ |
8569 | 1: 55 mph 2: 55 mph 3: 11 hr mph / yd |
8570 | . 1: 5 yd / hr . | |
8571 | . | |
8572 | ||
5d67986c | 8573 | ' 55 mph @key{RET} ' 5 yd/hr @key{RET} / |
d7b8e6c6 | 8574 | @end group |
5d67986c | 8575 | @end smallexample |
d7b8e6c6 EZ |
8576 | |
8577 | The @kbd{u s} command cancels out these units to get a plain | |
8578 | number. Now we take the logarithm base two to find the final | |
8579 | answer, assuming that each successive pill doubles his speed. | |
8580 | ||
d7b8e6c6 | 8581 | @smallexample |
5d67986c | 8582 | @group |
d7b8e6c6 EZ |
8583 | 1: 19360. 2: 19360. 1: 14.24 |
8584 | . 1: 2 . | |
8585 | . | |
8586 | ||
8587 | u s 2 B | |
d7b8e6c6 | 8588 | @end group |
5d67986c | 8589 | @end smallexample |
d7b8e6c6 EZ |
8590 | |
8591 | @noindent | |
8592 | Thus Sam can take up to 14 pills without a worry. | |
8593 | ||
8594 | @node Algebra Answer 1, Algebra Answer 2, Types Answer 15, Answers to Exercises | |
8595 | @subsection Algebra Tutorial Exercise 1 | |
8596 | ||
8597 | @noindent | |
8598 | @c [fix-ref Declarations] | |
8599 | The result @samp{sqrt(x)^2} is simplified back to @cite{x} by the | |
8600 | Calculator, but @samp{sqrt(x^2)} is not. (Consider what happens | |
8601 | if @w{@cite{x = -4}}.) If @cite{x} is real, this formula could be | |
8602 | simplified to @samp{abs(x)}, but for general complex arguments even | |
8603 | that is not safe. (@xref{Declarations}, for a way to tell Calc | |
8604 | that @cite{x} is known to be real.) | |
8605 | ||
8606 | @node Algebra Answer 2, Algebra Answer 3, Algebra Answer 1, Answers to Exercises | |
8607 | @subsection Algebra Tutorial Exercise 2 | |
8608 | ||
8609 | @noindent | |
8610 | Suppose our roots are @cite{[a, b, c]}. We want a polynomial which | |
8611 | is zero when @cite{x} is any of these values. The trivial polynomial | |
8612 | @cite{x-a} is zero when @cite{x=a}, so the product @cite{(x-a)(x-b)(x-c)} | |
8613 | will do the job. We can use @kbd{a c x} to write this in a more | |
8614 | familiar form. | |
8615 | ||
d7b8e6c6 | 8616 | @smallexample |
5d67986c | 8617 | @group |
d7b8e6c6 EZ |
8618 | 1: 34 x - 24 x^3 1: [1.19023, -1.19023, 0] |
8619 | . . | |
8620 | ||
5d67986c | 8621 | r 2 a P x @key{RET} |
d7b8e6c6 | 8622 | |
d7b8e6c6 | 8623 | @end group |
5d67986c | 8624 | @end smallexample |
d7b8e6c6 | 8625 | @noindent |
d7b8e6c6 | 8626 | @smallexample |
5d67986c | 8627 | @group |
d7b8e6c6 EZ |
8628 | 1: [x - 1.19023, x + 1.19023, x] 1: (x - 1.19023) (x + 1.19023) x |
8629 | . . | |
8630 | ||
5d67986c | 8631 | V M ' x-$ @key{RET} V R * |
d7b8e6c6 | 8632 | |
d7b8e6c6 | 8633 | @end group |
5d67986c | 8634 | @end smallexample |
d7b8e6c6 | 8635 | @noindent |
d7b8e6c6 | 8636 | @smallexample |
5d67986c | 8637 | @group |
d7b8e6c6 EZ |
8638 | 1: x^3 - 1.41666 x 1: 34 x - 24 x^3 |
8639 | . . | |
8640 | ||
5d67986c | 8641 | a c x @key{RET} 24 n * a x |
d7b8e6c6 | 8642 | @end group |
5d67986c | 8643 | @end smallexample |
d7b8e6c6 EZ |
8644 | |
8645 | @noindent | |
8646 | Sure enough, our answer (multiplied by a suitable constant) is the | |
8647 | same as the original polynomial. | |
8648 | ||
8649 | @node Algebra Answer 3, Algebra Answer 4, Algebra Answer 2, Answers to Exercises | |
8650 | @subsection Algebra Tutorial Exercise 3 | |
8651 | ||
d7b8e6c6 | 8652 | @smallexample |
5d67986c | 8653 | @group |
d7b8e6c6 EZ |
8654 | 1: x sin(pi x) 1: (sin(pi x) - pi x cos(pi x)) / pi^2 |
8655 | . . | |
8656 | ||
5d67986c | 8657 | ' x sin(pi x) @key{RET} m r a i x @key{RET} |
d7b8e6c6 | 8658 | |
d7b8e6c6 | 8659 | @end group |
5d67986c | 8660 | @end smallexample |
d7b8e6c6 | 8661 | @noindent |
d7b8e6c6 | 8662 | @smallexample |
5d67986c | 8663 | @group |
d7b8e6c6 EZ |
8664 | 1: [y, 1] |
8665 | 2: (sin(pi x) - pi x cos(pi x)) / pi^2 | |
8666 | . | |
8667 | ||
5d67986c | 8668 | ' [y,1] @key{RET} @key{TAB} |
d7b8e6c6 | 8669 | |
d7b8e6c6 | 8670 | @end group |
5d67986c | 8671 | @end smallexample |
d7b8e6c6 | 8672 | @noindent |
d7b8e6c6 | 8673 | @smallexample |
5d67986c | 8674 | @group |
d7b8e6c6 EZ |
8675 | 1: [(sin(pi y) - pi y cos(pi y)) / pi^2, (sin(pi) - pi cos(pi)) / pi^2] |
8676 | . | |
8677 | ||
5d67986c | 8678 | V M $ @key{RET} |
d7b8e6c6 | 8679 | |
d7b8e6c6 | 8680 | @end group |
5d67986c | 8681 | @end smallexample |
d7b8e6c6 | 8682 | @noindent |
d7b8e6c6 | 8683 | @smallexample |
5d67986c | 8684 | @group |
d7b8e6c6 EZ |
8685 | 1: (sin(pi y) - pi y cos(pi y)) / pi^2 + (pi cos(pi) - sin(pi)) / pi^2 |
8686 | . | |
8687 | ||
8688 | V R - | |
8689 | ||
d7b8e6c6 | 8690 | @end group |
5d67986c | 8691 | @end smallexample |
d7b8e6c6 | 8692 | @noindent |
d7b8e6c6 | 8693 | @smallexample |
5d67986c | 8694 | @group |
d7b8e6c6 EZ |
8695 | 1: (sin(3.14159 y) - 3.14159 y cos(3.14159 y)) / 9.8696 - 0.3183 |
8696 | . | |
8697 | ||
8698 | = | |
8699 | ||
d7b8e6c6 | 8700 | @end group |
5d67986c | 8701 | @end smallexample |
d7b8e6c6 | 8702 | @noindent |
d7b8e6c6 | 8703 | @smallexample |
5d67986c | 8704 | @group |
d7b8e6c6 EZ |
8705 | 1: [0., -0.95493, 0.63662, -1.5915, 1.2732] |
8706 | . | |
8707 | ||
5d67986c | 8708 | v x 5 @key{RET} @key{TAB} V M $ @key{RET} |
d7b8e6c6 | 8709 | @end group |
5d67986c | 8710 | @end smallexample |
d7b8e6c6 EZ |
8711 | |
8712 | @node Algebra Answer 4, Rewrites Answer 1, Algebra Answer 3, Answers to Exercises | |
8713 | @subsection Algebra Tutorial Exercise 4 | |
8714 | ||
8715 | @noindent | |
8716 | The hard part is that @kbd{V R +} is no longer sufficient to add up all | |
8717 | the contributions from the slices, since the slices have varying | |
8718 | coefficients. So first we must come up with a vector of these | |
8719 | coefficients. Here's one way: | |
8720 | ||
d7b8e6c6 | 8721 | @smallexample |
5d67986c | 8722 | @group |
d7b8e6c6 EZ |
8723 | 2: -1 2: 3 1: [4, 2, ..., 4] |
8724 | 1: [1, 2, ..., 9] 1: [-1, 1, ..., -1] . | |
8725 | . . | |
8726 | ||
5d67986c | 8727 | 1 n v x 9 @key{RET} V M ^ 3 @key{TAB} - |
d7b8e6c6 | 8728 | |
d7b8e6c6 | 8729 | @end group |
5d67986c | 8730 | @end smallexample |
d7b8e6c6 | 8731 | @noindent |
d7b8e6c6 | 8732 | @smallexample |
5d67986c | 8733 | @group |
d7b8e6c6 EZ |
8734 | 1: [4, 2, ..., 4, 1] 1: [1, 4, 2, ..., 4, 1] |
8735 | . . | |
8736 | ||
5d67986c | 8737 | 1 | 1 @key{TAB} | |
d7b8e6c6 | 8738 | @end group |
5d67986c | 8739 | @end smallexample |
d7b8e6c6 EZ |
8740 | |
8741 | @noindent | |
8742 | Now we compute the function values. Note that for this method we need | |
8743 | eleven values, including both endpoints of the desired interval. | |
8744 | ||
d7b8e6c6 | 8745 | @smallexample |
5d67986c | 8746 | @group |
d7b8e6c6 EZ |
8747 | 2: [1, 4, 2, ..., 4, 1] |
8748 | 1: [1, 1.1, 1.2, ... , 1.8, 1.9, 2.] | |
8749 | . | |
8750 | ||
5d67986c | 8751 | 11 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x |
d7b8e6c6 | 8752 | |
d7b8e6c6 | 8753 | @end group |
5d67986c | 8754 | @end smallexample |
d7b8e6c6 | 8755 | @noindent |
d7b8e6c6 | 8756 | @smallexample |
5d67986c | 8757 | @group |
d7b8e6c6 EZ |
8758 | 2: [1, 4, 2, ..., 4, 1] |
8759 | 1: [0., 0.084941, 0.16993, ... ] | |
8760 | . | |
8761 | ||
5d67986c | 8762 | ' sin(x) ln(x) @key{RET} m r p 5 @key{RET} V M $ @key{RET} |
d7b8e6c6 | 8763 | @end group |
5d67986c | 8764 | @end smallexample |
d7b8e6c6 EZ |
8765 | |
8766 | @noindent | |
8767 | Once again this calls for @kbd{V M * V R +}; a simple @kbd{*} does the | |
8768 | same thing. | |
8769 | ||
d7b8e6c6 | 8770 | @smallexample |
5d67986c | 8771 | @group |
d7b8e6c6 EZ |
8772 | 1: 11.22 1: 1.122 1: 0.374 |
8773 | . . . | |
8774 | ||
8775 | * .1 * 3 / | |
d7b8e6c6 | 8776 | @end group |
5d67986c | 8777 | @end smallexample |
d7b8e6c6 EZ |
8778 | |
8779 | @noindent | |
8780 | Wow! That's even better than the result from the Taylor series method. | |
8781 | ||
8782 | @node Rewrites Answer 1, Rewrites Answer 2, Algebra Answer 4, Answers to Exercises | |
8783 | @subsection Rewrites Tutorial Exercise 1 | |
8784 | ||
8785 | @noindent | |
8786 | We'll use Big mode to make the formulas more readable. | |
8787 | ||
d7b8e6c6 | 8788 | @smallexample |
5d67986c | 8789 | @group |
d7b8e6c6 EZ |
8790 | ___ |
8791 | 2 + V 2 | |
8792 | 1: (2 + sqrt(2)) / (1 + sqrt(2)) 1: -------- | |
8793 | . ___ | |
8794 | 1 + V 2 | |
8795 | ||
8796 | . | |
8797 | ||
5d67986c | 8798 | ' (2+sqrt(2)) / (1+sqrt(2)) @key{RET} d B |
d7b8e6c6 | 8799 | @end group |
5d67986c | 8800 | @end smallexample |
d7b8e6c6 EZ |
8801 | |
8802 | @noindent | |
8803 | Multiplying by the conjugate helps because @cite{(a+b) (a-b) = a^2 - b^2}. | |
8804 | ||
d7b8e6c6 | 8805 | @smallexample |
5d67986c | 8806 | @group |
d7b8e6c6 EZ |
8807 | ___ ___ |
8808 | 1: (2 + V 2 ) (V 2 - 1) | |
8809 | . | |
8810 | ||
5d67986c | 8811 | a r a/(b+c) := a*(b-c) / (b^2-c^2) @key{RET} |
d7b8e6c6 | 8812 | |
d7b8e6c6 | 8813 | @end group |
5d67986c | 8814 | @end smallexample |
d7b8e6c6 | 8815 | @noindent |
d7b8e6c6 | 8816 | @smallexample |
5d67986c | 8817 | @group |
d7b8e6c6 EZ |
8818 | ___ ___ |
8819 | 1: 2 + V 2 - 2 1: V 2 | |
8820 | . . | |
8821 | ||
8822 | a r a*(b+c) := a*b + a*c a s | |
d7b8e6c6 | 8823 | @end group |
5d67986c | 8824 | @end smallexample |
d7b8e6c6 EZ |
8825 | |
8826 | @noindent | |
8827 | (We could have used @kbd{a x} instead of a rewrite rule for the | |
8828 | second step.) | |
8829 | ||
8830 | The multiply-by-conjugate rule turns out to be useful in many | |
8831 | different circumstances, such as when the denominator involves | |
8832 | sines and cosines or the imaginary constant @code{i}. | |
8833 | ||
8834 | @node Rewrites Answer 2, Rewrites Answer 3, Rewrites Answer 1, Answers to Exercises | |
8835 | @subsection Rewrites Tutorial Exercise 2 | |
8836 | ||
8837 | @noindent | |
8838 | Here is the rule set: | |
8839 | ||
d7b8e6c6 | 8840 | @smallexample |
5d67986c | 8841 | @group |
d7b8e6c6 EZ |
8842 | [ fib(n) := fib(n, 1, 1) :: integer(n) :: n >= 1, |
8843 | fib(1, x, y) := x, | |
8844 | fib(n, x, y) := fib(n-1, y, x+y) ] | |
d7b8e6c6 | 8845 | @end group |
5d67986c | 8846 | @end smallexample |
d7b8e6c6 EZ |
8847 | |
8848 | @noindent | |
8849 | The first rule turns a one-argument @code{fib} that people like to write | |
8850 | into a three-argument @code{fib} that makes computation easier. The | |
8851 | second rule converts back from three-argument form once the computation | |
8852 | is done. The third rule does the computation itself. It basically | |
8853 | says that if @cite{x} and @cite{y} are two consecutive Fibonacci numbers, | |
8854 | then @cite{y} and @cite{x+y} are the next (overlapping) pair of Fibonacci | |
8855 | numbers. | |
8856 | ||
8857 | Notice that because the number @cite{n} was ``validated'' by the | |
8858 | conditions on the first rule, there is no need to put conditions on | |
8859 | the other rules because the rule set would never get that far unless | |
8860 | the input were valid. That further speeds computation, since no | |
8861 | extra conditions need to be checked at every step. | |
8862 | ||
8863 | Actually, a user with a nasty sense of humor could enter a bad | |
8864 | three-argument @code{fib} call directly, say, @samp{fib(0, 1, 1)}, | |
8865 | which would get the rules into an infinite loop. One thing that would | |
8866 | help keep this from happening by accident would be to use something like | |
8867 | @samp{ZzFib} instead of @code{fib} as the name of the three-argument | |
8868 | function. | |
8869 | ||
8870 | @node Rewrites Answer 3, Rewrites Answer 4, Rewrites Answer 2, Answers to Exercises | |
8871 | @subsection Rewrites Tutorial Exercise 3 | |
8872 | ||
8873 | @noindent | |
8874 | He got an infinite loop. First, Calc did as expected and rewrote | |
8875 | @w{@samp{2 + 3 x}} to @samp{f(2, 3, x)}. Then it looked for ways to | |
8876 | apply the rule again, and found that @samp{f(2, 3, x)} looks like | |
8877 | @samp{a + b x} with @w{@samp{a = 0}} and @samp{b = 1}, so it rewrote to | |
8878 | @samp{f(0, 1, f(2, 3, x))}. It then wrapped another @samp{f(0, 1, ...)} | |
8879 | around that, and so on, ad infinitum. Joe should have used @kbd{M-1 a r} | |
8880 | to make sure the rule applied only once. | |
8881 | ||
8882 | (Actually, even the first step didn't work as he expected. What Calc | |
8883 | really gives for @kbd{M-1 a r} in this situation is @samp{f(3 x, 1, 2)}, | |
8884 | treating 2 as the ``variable,'' and @samp{3 x} as a constant being added | |
8885 | to it. While this may seem odd, it's just as valid a solution as the | |
8886 | ``obvious'' one. One way to fix this would be to add the condition | |
8887 | @samp{:: variable(x)} to the rule, to make sure the thing that matches | |
8888 | @samp{x} is indeed a variable, or to change @samp{x} to @samp{quote(x)} | |
8889 | on the lefthand side, so that the rule matches the actual variable | |
8890 | @samp{x} rather than letting @samp{x} stand for something else.) | |
8891 | ||
8892 | @node Rewrites Answer 4, Rewrites Answer 5, Rewrites Answer 3, Answers to Exercises | |
8893 | @subsection Rewrites Tutorial Exercise 4 | |
8894 | ||
8895 | @noindent | |
5d67986c RS |
8896 | @ignore |
8897 | @starindex | |
8898 | @end ignore | |
d7b8e6c6 EZ |
8899 | @tindex seq |
8900 | Here is a suitable set of rules to solve the first part of the problem: | |
8901 | ||
d7b8e6c6 | 8902 | @smallexample |
5d67986c | 8903 | @group |
d7b8e6c6 EZ |
8904 | [ seq(n, c) := seq(n/2, c+1) :: n%2 = 0, |
8905 | seq(n, c) := seq(3n+1, c+1) :: n%2 = 1 :: n > 1 ] | |
d7b8e6c6 | 8906 | @end group |
5d67986c | 8907 | @end smallexample |
d7b8e6c6 EZ |
8908 | |
8909 | Given the initial formula @samp{seq(6, 0)}, application of these | |
8910 | rules produces the following sequence of formulas: | |
8911 | ||
8912 | @example | |
8913 | seq( 3, 1) | |
8914 | seq(10, 2) | |
8915 | seq( 5, 3) | |
8916 | seq(16, 4) | |
8917 | seq( 8, 5) | |
8918 | seq( 4, 6) | |
8919 | seq( 2, 7) | |
8920 | seq( 1, 8) | |
8921 | @end example | |
8922 | ||
8923 | @noindent | |
8924 | whereupon neither of the rules match, and rewriting stops. | |
8925 | ||
8926 | We can pretty this up a bit with a couple more rules: | |
8927 | ||
d7b8e6c6 | 8928 | @smallexample |
5d67986c | 8929 | @group |
d7b8e6c6 EZ |
8930 | [ seq(n) := seq(n, 0), |
8931 | seq(1, c) := c, | |
8932 | ... ] | |
d7b8e6c6 | 8933 | @end group |
5d67986c | 8934 | @end smallexample |
d7b8e6c6 EZ |
8935 | |
8936 | @noindent | |
8937 | Now, given @samp{seq(6)} as the starting configuration, we get 8 | |
8938 | as the result. | |
8939 | ||
8940 | The change to return a vector is quite simple: | |
8941 | ||
d7b8e6c6 | 8942 | @smallexample |
5d67986c | 8943 | @group |
d7b8e6c6 EZ |
8944 | [ seq(n) := seq(n, []) :: integer(n) :: n > 0, |
8945 | seq(1, v) := v | 1, | |
8946 | seq(n, v) := seq(n/2, v | n) :: n%2 = 0, | |
8947 | seq(n, v) := seq(3n+1, v | n) :: n%2 = 1 ] | |
d7b8e6c6 | 8948 | @end group |
5d67986c | 8949 | @end smallexample |
d7b8e6c6 EZ |
8950 | |
8951 | @noindent | |
8952 | Given @samp{seq(6)}, the result is @samp{[6, 3, 10, 5, 16, 8, 4, 2, 1]}. | |
8953 | ||
8954 | Notice that the @cite{n > 1} guard is no longer necessary on the last | |
8955 | rule since the @cite{n = 1} case is now detected by another rule. | |
8956 | But a guard has been added to the initial rule to make sure the | |
8957 | initial value is suitable before the computation begins. | |
8958 | ||
8959 | While still a good idea, this guard is not as vitally important as it | |
8960 | was for the @code{fib} function, since calling, say, @samp{seq(x, [])} | |
8961 | will not get into an infinite loop. Calc will not be able to prove | |
8962 | the symbol @samp{x} is either even or odd, so none of the rules will | |
8963 | apply and the rewrites will stop right away. | |
8964 | ||
8965 | @node Rewrites Answer 5, Rewrites Answer 6, Rewrites Answer 4, Answers to Exercises | |
8966 | @subsection Rewrites Tutorial Exercise 5 | |
8967 | ||
8968 | @noindent | |
5d67986c RS |
8969 | @ignore |
8970 | @starindex | |
8971 | @end ignore | |
d7b8e6c6 | 8972 | @tindex nterms |
5d67986c RS |
8973 | If @cite{x} is the sum @cite{a + b}, then `@t{nterms(}@var{x}@t{)}' must |
8974 | be `@t{nterms(}@var{a}@t{)}' plus `@t{nterms(}@var{b}@t{)}'. If @cite{x} | |
8975 | is not a sum, then `@t{nterms(}@var{x}@t{)}' = 1. | |
d7b8e6c6 | 8976 | |
d7b8e6c6 | 8977 | @smallexample |
5d67986c | 8978 | @group |
d7b8e6c6 EZ |
8979 | [ nterms(a + b) := nterms(a) + nterms(b), |
8980 | nterms(x) := 1 ] | |
d7b8e6c6 | 8981 | @end group |
5d67986c | 8982 | @end smallexample |
d7b8e6c6 EZ |
8983 | |
8984 | @noindent | |
8985 | Here we have taken advantage of the fact that earlier rules always | |
8986 | match before later rules; @samp{nterms(x)} will only be tried if we | |
8987 | already know that @samp{x} is not a sum. | |
8988 | ||
8989 | @node Rewrites Answer 6, Rewrites Answer 7, Rewrites Answer 5, Answers to Exercises | |
8990 | @subsection Rewrites Tutorial Exercise 6 | |
8991 | ||
8992 | Just put the rule @samp{0^0 := 1} into @code{EvalRules}. For example, | |
8993 | before making this definition we have: | |
8994 | ||
d7b8e6c6 | 8995 | @smallexample |
5d67986c | 8996 | @group |
d7b8e6c6 EZ |
8997 | 2: [-2, -1, 0, 1, 2] 1: [1, 1, 0^0, 1, 1] |
8998 | 1: 0 . | |
8999 | . | |
9000 | ||
5d67986c | 9001 | v x 5 @key{RET} 3 - 0 V M ^ |
d7b8e6c6 | 9002 | @end group |
5d67986c | 9003 | @end smallexample |
d7b8e6c6 EZ |
9004 | |
9005 | @noindent | |
9006 | But then: | |
9007 | ||
d7b8e6c6 | 9008 | @smallexample |
5d67986c | 9009 | @group |
d7b8e6c6 EZ |
9010 | 2: [-2, -1, 0, 1, 2] 1: [1, 1, 1, 1, 1] |
9011 | 1: 0 . | |
9012 | . | |
9013 | ||
5d67986c | 9014 | U ' 0^0:=1 @key{RET} s t EvalRules @key{RET} V M ^ |
d7b8e6c6 | 9015 | @end group |
5d67986c | 9016 | @end smallexample |
d7b8e6c6 EZ |
9017 | |
9018 | Perhaps more surprisingly, this rule still works with infinite mode | |
9019 | turned on. Calc tries @code{EvalRules} before any built-in rules for | |
9020 | a function. This allows you to override the default behavior of any | |
9021 | Calc feature: Even though Calc now wants to evaluate @cite{0^0} to | |
9022 | @code{nan}, your rule gets there first and evaluates it to 1 instead. | |
9023 | ||
9024 | Just for kicks, try adding the rule @code{2+3 := 6} to @code{EvalRules}. | |
9025 | What happens? (Be sure to remove this rule afterward, or you might get | |
9026 | a nasty surprise when you use Calc to balance your checkbook!) | |
9027 | ||
9028 | @node Rewrites Answer 7, Programming Answer 1, Rewrites Answer 6, Answers to Exercises | |
9029 | @subsection Rewrites Tutorial Exercise 7 | |
9030 | ||
9031 | @noindent | |
9032 | Here is a rule set that will do the job: | |
9033 | ||
d7b8e6c6 | 9034 | @smallexample |
5d67986c | 9035 | @group |
d7b8e6c6 EZ |
9036 | [ a*(b + c) := a*b + a*c, |
9037 | opt(a) O(x^n) + opt(b) O(x^m) := O(x^n) :: n <= m | |
9038 | :: constant(a) :: constant(b), | |
9039 | opt(a) O(x^n) + opt(b) x^m := O(x^n) :: n <= m | |
9040 | :: constant(a) :: constant(b), | |
9041 | a O(x^n) := O(x^n) :: constant(a), | |
9042 | x^opt(m) O(x^n) := O(x^(n+m)), | |
9043 | O(x^n) O(x^m) := O(x^(n+m)) ] | |
d7b8e6c6 | 9044 | @end group |
5d67986c | 9045 | @end smallexample |
d7b8e6c6 EZ |
9046 | |
9047 | If we really want the @kbd{+} and @kbd{*} keys to operate naturally | |
9048 | on power series, we should put these rules in @code{EvalRules}. For | |
9049 | testing purposes, it is better to put them in a different variable, | |
9050 | say, @code{O}, first. | |
9051 | ||
9052 | The first rule just expands products of sums so that the rest of the | |
9053 | rules can assume they have an expanded-out polynomial to work with. | |
9054 | Note that this rule does not mention @samp{O} at all, so it will | |
9055 | apply to any product-of-sum it encounters---this rule may surprise | |
9056 | you if you put it into @code{EvalRules}! | |
9057 | ||
9058 | In the second rule, the sum of two O's is changed to the smaller O. | |
9059 | The optional constant coefficients are there mostly so that | |
9060 | @samp{O(x^2) - O(x^3)} and @samp{O(x^3) - O(x^2)} are handled | |
9061 | as well as @samp{O(x^2) + O(x^3)}. | |
9062 | ||
9063 | The third rule absorbs higher powers of @samp{x} into O's. | |
9064 | ||
9065 | The fourth rule says that a constant times a negligible quantity | |
9066 | is still negligible. (This rule will also match @samp{O(x^3) / 4}, | |
9067 | with @samp{a = 1/4}.) | |
9068 | ||
9069 | The fifth rule rewrites, for example, @samp{x^2 O(x^3)} to @samp{O(x^5)}. | |
9070 | (It is easy to see that if one of these forms is negligible, the other | |
9071 | is, too.) Notice the @samp{x^opt(m)} to pick up terms like | |
9072 | @w{@samp{x O(x^3)}}. Optional powers will match @samp{x} as @samp{x^1} | |
9073 | but not 1 as @samp{x^0}. This turns out to be exactly what we want here. | |
9074 | ||
9075 | The sixth rule is the corresponding rule for products of two O's. | |
9076 | ||
9077 | Another way to solve this problem would be to create a new ``data type'' | |
9078 | that represents truncated power series. We might represent these as | |
9079 | function calls @samp{series(@var{coefs}, @var{x})} where @var{coefs} is | |
9080 | a vector of coefficients for @cite{x^0}, @cite{x^1}, @cite{x^2}, and so | |
9081 | on. Rules would exist for sums and products of such @code{series} | |
9082 | objects, and as an optional convenience could also know how to combine a | |
9083 | @code{series} object with a normal polynomial. (With this, and with a | |
9084 | rule that rewrites @samp{O(x^n)} to the equivalent @code{series} form, | |
9085 | you could still enter power series in exactly the same notation as | |
9086 | before.) Operations on such objects would probably be more efficient, | |
9087 | although the objects would be a bit harder to read. | |
9088 | ||
9089 | @c [fix-ref Compositions] | |
9090 | Some other symbolic math programs provide a power series data type | |
9091 | similar to this. Mathematica, for example, has an object that looks | |
9092 | like @samp{PowerSeries[@var{x}, @var{x0}, @var{coefs}, @var{nmin}, | |
9093 | @var{nmax}, @var{den}]}, where @var{x0} is the point about which the | |
9094 | power series is taken (we've been assuming this was always zero), | |
9095 | and @var{nmin}, @var{nmax}, and @var{den} allow pseudo-power-series | |
9096 | with fractional or negative powers. Also, the @code{PowerSeries} | |
9097 | objects have a special display format that makes them look like | |
9098 | @samp{2 x^2 + O(x^4)} when they are printed out. (@xref{Compositions}, | |
9099 | for a way to do this in Calc, although for something as involved as | |
9100 | this it would probably be better to write the formatting routine | |
9101 | in Lisp.) | |
9102 | ||
9103 | @node Programming Answer 1, Programming Answer 2, Rewrites Answer 7, Answers to Exercises | |
9104 | @subsection Programming Tutorial Exercise 1 | |
9105 | ||
9106 | @noindent | |
9107 | Just enter the formula @samp{ninteg(sin(t)/t, t, 0, x)}, type | |
9108 | @kbd{Z F}, and answer the questions. Since this formula contains two | |
9109 | variables, the default argument list will be @samp{(t x)}. We want to | |
9110 | change this to @samp{(x)} since @cite{t} is really a dummy variable | |
9111 | to be used within @code{ninteg}. | |
9112 | ||
5d67986c RS |
9113 | The exact keystrokes are @kbd{Z F s Si @key{RET} @key{RET} C-b C-b @key{DEL} @key{DEL} @key{RET} y}. |
9114 | (The @kbd{C-b C-b @key{DEL} @key{DEL}} are what fix the argument list.) | |
d7b8e6c6 EZ |
9115 | |
9116 | @node Programming Answer 2, Programming Answer 3, Programming Answer 1, Answers to Exercises | |
9117 | @subsection Programming Tutorial Exercise 2 | |
9118 | ||
9119 | @noindent | |
9120 | One way is to move the number to the top of the stack, operate on | |
5d67986c | 9121 | it, then move it back: @kbd{C-x ( M-@key{TAB} n M-@key{TAB} M-@key{TAB} C-x )}. |
d7b8e6c6 EZ |
9122 | |
9123 | Another way is to negate the top three stack entries, then negate | |
9124 | again the top two stack entries: @kbd{C-x ( M-3 n M-2 n C-x )}. | |
9125 | ||
9126 | Finally, it turns out that a negative prefix argument causes a | |
9127 | command like @kbd{n} to operate on the specified stack entry only, | |
9128 | which is just what we want: @kbd{C-x ( M-- 3 n C-x )}. | |
9129 | ||
9130 | Just for kicks, let's also do it algebraically: | |
5d67986c | 9131 | @w{@kbd{C-x ( ' -$$$, $$, $ @key{RET} C-x )}}. |
d7b8e6c6 EZ |
9132 | |
9133 | @node Programming Answer 3, Programming Answer 4, Programming Answer 2, Answers to Exercises | |
9134 | @subsection Programming Tutorial Exercise 3 | |
9135 | ||
9136 | @noindent | |
9137 | Each of these functions can be computed using the stack, or using | |
9138 | algebraic entry, whichever way you prefer: | |
9139 | ||
9140 | @noindent | |
9141 | Computing @c{$\displaystyle{\sin x \over x}$} | |
9142 | @cite{sin(x) / x}: | |
9143 | ||
5d67986c | 9144 | Using the stack: @kbd{C-x ( @key{RET} S @key{TAB} / C-x )}. |
d7b8e6c6 | 9145 | |
5d67986c | 9146 | Using algebraic entry: @kbd{C-x ( ' sin($)/$ @key{RET} C-x )}. |
d7b8e6c6 EZ |
9147 | |
9148 | @noindent | |
9149 | Computing the logarithm: | |
9150 | ||
5d67986c | 9151 | Using the stack: @kbd{C-x ( @key{TAB} B C-x )} |
d7b8e6c6 | 9152 | |
5d67986c | 9153 | Using algebraic entry: @kbd{C-x ( ' log($,$$) @key{RET} C-x )}. |
d7b8e6c6 EZ |
9154 | |
9155 | @noindent | |
9156 | Computing the vector of integers: | |
9157 | ||
5d67986c | 9158 | Using the stack: @kbd{C-x ( 1 @key{RET} 1 C-u v x C-x )}. (Recall that |
d7b8e6c6 EZ |
9159 | @kbd{C-u v x} takes the vector size, starting value, and increment |
9160 | from the stack.) | |
9161 | ||
9162 | Alternatively: @kbd{C-x ( ~ v x C-x )}. (The @kbd{~} key pops a | |
9163 | number from the stack and uses it as the prefix argument for the | |
9164 | next command.) | |
9165 | ||
5d67986c | 9166 | Using algebraic entry: @kbd{C-x ( ' index($) @key{RET} C-x )}. |
d7b8e6c6 EZ |
9167 | |
9168 | @node Programming Answer 4, Programming Answer 5, Programming Answer 3, Answers to Exercises | |
9169 | @subsection Programming Tutorial Exercise 4 | |
9170 | ||
9171 | @noindent | |
5d67986c | 9172 | Here's one way: @kbd{C-x ( @key{RET} V R + @key{TAB} v l / C-x )}. |
d7b8e6c6 EZ |
9173 | |
9174 | @node Programming Answer 5, Programming Answer 6, Programming Answer 4, Answers to Exercises | |
9175 | @subsection Programming Tutorial Exercise 5 | |
9176 | ||
d7b8e6c6 | 9177 | @smallexample |
5d67986c | 9178 | @group |
d7b8e6c6 EZ |
9179 | 2: 1 1: 1.61803398502 2: 1.61803398502 |
9180 | 1: 20 . 1: 1.61803398875 | |
9181 | . . | |
9182 | ||
5d67986c | 9183 | 1 @key{RET} 20 Z < & 1 + Z > I H P |
d7b8e6c6 | 9184 | @end group |
5d67986c | 9185 | @end smallexample |
d7b8e6c6 EZ |
9186 | |
9187 | @noindent | |
9188 | This answer is quite accurate. | |
9189 | ||
9190 | @node Programming Answer 6, Programming Answer 7, Programming Answer 5, Answers to Exercises | |
9191 | @subsection Programming Tutorial Exercise 6 | |
9192 | ||
9193 | @noindent | |
9194 | Here is the matrix: | |
9195 | ||
9196 | @example | |
9197 | [ [ 0, 1 ] * [a, b] = [b, a + b] | |
9198 | [ 1, 1 ] ] | |
9199 | @end example | |
9200 | ||
9201 | @noindent | |
9202 | Thus @samp{[0, 1; 1, 1]^n * [1, 1]} computes Fibonacci numbers @cite{n+1} | |
9203 | and @cite{n+2}. Here's one program that does the job: | |
9204 | ||
9205 | @example | |
5d67986c | 9206 | C-x ( ' [0, 1; 1, 1] ^ ($-1) * [1, 1] @key{RET} v u @key{DEL} C-x ) |
d7b8e6c6 EZ |
9207 | @end example |
9208 | ||
9209 | @noindent | |
9210 | This program is quite efficient because Calc knows how to raise a | |
9211 | matrix (or other value) to the power @cite{n} in only @c{$\log_2 n$} | |
9212 | @cite{log(n,2)} | |
9213 | steps. For example, this program can compute the 1000th Fibonacci | |
9214 | number (a 209-digit integer!) in about 10 steps; even though the | |
9215 | @kbd{Z < ... Z >} solution had much simpler steps, it would have | |
9216 | required so many steps that it would not have been practical. | |
9217 | ||
9218 | @node Programming Answer 7, Programming Answer 8, Programming Answer 6, Answers to Exercises | |
9219 | @subsection Programming Tutorial Exercise 7 | |
9220 | ||
9221 | @noindent | |
9222 | The trick here is to compute the harmonic numbers differently, so that | |
9223 | the loop counter itself accumulates the sum of reciprocals. We use | |
9224 | a separate variable to hold the integer counter. | |
9225 | ||
d7b8e6c6 | 9226 | @smallexample |
5d67986c | 9227 | @group |
d7b8e6c6 EZ |
9228 | 1: 1 2: 1 1: . |
9229 | . 1: 4 | |
9230 | . | |
9231 | ||
5d67986c | 9232 | 1 t 1 1 @key{RET} 4 Z ( t 2 r 1 1 + s 1 & Z ) |
d7b8e6c6 | 9233 | @end group |
5d67986c | 9234 | @end smallexample |
d7b8e6c6 EZ |
9235 | |
9236 | @noindent | |
9237 | The body of the loop goes as follows: First save the harmonic sum | |
9238 | so far in variable 2. Then delete it from the stack; the for loop | |
9239 | itself will take care of remembering it for us. Next, recall the | |
9240 | count from variable 1, add one to it, and feed its reciprocal to | |
9241 | the for loop to use as the step value. The for loop will increase | |
9242 | the ``loop counter'' by that amount and keep going until the | |
9243 | loop counter exceeds 4. | |
9244 | ||
d7b8e6c6 | 9245 | @smallexample |
5d67986c | 9246 | @group |
d7b8e6c6 EZ |
9247 | 2: 31 3: 31 |
9248 | 1: 3.99498713092 2: 3.99498713092 | |
9249 | . 1: 4.02724519544 | |
9250 | . | |
9251 | ||
5d67986c | 9252 | r 1 r 2 @key{RET} 31 & + |
d7b8e6c6 | 9253 | @end group |
5d67986c | 9254 | @end smallexample |
d7b8e6c6 EZ |
9255 | |
9256 | Thus we find that the 30th harmonic number is 3.99, and the 31st | |
9257 | harmonic number is 4.02. | |
9258 | ||
9259 | @node Programming Answer 8, Programming Answer 9, Programming Answer 7, Answers to Exercises | |
9260 | @subsection Programming Tutorial Exercise 8 | |
9261 | ||
9262 | @noindent | |
9263 | The first step is to compute the derivative @cite{f'(x)} and thus | |
9264 | the formula @c{$\displaystyle{x - {f(x) \over f'(x)}}$} | |
9265 | @cite{x - f(x)/f'(x)}. | |
9266 | ||
9267 | (Because this definition is long, it will be repeated in concise form | |
9268 | below. You can use @w{@kbd{M-# m}} to load it from there. While you are | |
9269 | entering a @kbd{Z ` Z '} body in a macro, Calc simply collects | |
9270 | keystrokes without executing them. In the following diagrams we'll | |
9271 | pretend Calc actually executed the keystrokes as you typed them, | |
9272 | just for purposes of illustration.) | |
9273 | ||
d7b8e6c6 | 9274 | @smallexample |
5d67986c | 9275 | @group |
d7b8e6c6 EZ |
9276 | 2: sin(cos(x)) - 0.5 3: 4.5 |
9277 | 1: 4.5 2: sin(cos(x)) - 0.5 | |
9278 | . 1: -(sin(x) cos(cos(x))) | |
9279 | . | |
9280 | ||
5d67986c | 9281 | ' sin(cos(x))-0.5 @key{RET} 4.5 m r C-x ( Z ` @key{TAB} @key{RET} a d x @key{RET} |
d7b8e6c6 | 9282 | |
d7b8e6c6 | 9283 | @end group |
5d67986c | 9284 | @end smallexample |
d7b8e6c6 | 9285 | @noindent |
d7b8e6c6 | 9286 | @smallexample |
5d67986c | 9287 | @group |
d7b8e6c6 EZ |
9288 | 2: 4.5 |
9289 | 1: x + (sin(cos(x)) - 0.5) / sin(x) cos(cos(x)) | |
9290 | . | |
9291 | ||
5d67986c | 9292 | / ' x @key{RET} @key{TAB} - t 1 |
d7b8e6c6 | 9293 | @end group |
5d67986c | 9294 | @end smallexample |
d7b8e6c6 EZ |
9295 | |
9296 | Now, we enter the loop. We'll use a repeat loop with a 20-repetition | |
9297 | limit just in case the method fails to converge for some reason. | |
9298 | (Normally, the @w{@kbd{Z /}} command will stop the loop before all 20 | |
9299 | repetitions are done.) | |
9300 | ||
d7b8e6c6 | 9301 | @smallexample |
5d67986c | 9302 | @group |
d7b8e6c6 EZ |
9303 | 1: 4.5 3: 4.5 2: 4.5 |
9304 | . 2: x + (sin(cos(x)) ... 1: 5.24196456928 | |
9305 | 1: 4.5 . | |
9306 | . | |
9307 | ||
5d67986c | 9308 | 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET} |
d7b8e6c6 | 9309 | @end group |
5d67986c | 9310 | @end smallexample |
d7b8e6c6 EZ |
9311 | |
9312 | This is the new guess for @cite{x}. Now we compare it with the | |
9313 | old one to see if we've converged. | |
9314 | ||
d7b8e6c6 | 9315 | @smallexample |
5d67986c | 9316 | @group |
d7b8e6c6 EZ |
9317 | 3: 5.24196 2: 5.24196 1: 5.24196 1: 5.26345856348 |
9318 | 2: 5.24196 1: 0 . . | |
9319 | 1: 4.5 . | |
9320 | . | |
9321 | ||
5d67986c | 9322 | @key{RET} M-@key{TAB} a = Z / Z > Z ' C-x ) |
d7b8e6c6 | 9323 | @end group |
5d67986c | 9324 | @end smallexample |
d7b8e6c6 EZ |
9325 | |
9326 | The loop converges in just a few steps to this value. To check | |
9327 | the result, we can simply substitute it back into the equation. | |
9328 | ||
d7b8e6c6 | 9329 | @smallexample |
5d67986c | 9330 | @group |
d7b8e6c6 EZ |
9331 | 2: 5.26345856348 |
9332 | 1: 0.499999999997 | |
9333 | . | |
9334 | ||
5d67986c | 9335 | @key{RET} ' sin(cos($)) @key{RET} |
d7b8e6c6 | 9336 | @end group |
5d67986c | 9337 | @end smallexample |
d7b8e6c6 EZ |
9338 | |
9339 | Let's test the new definition again: | |
9340 | ||
d7b8e6c6 | 9341 | @smallexample |
5d67986c | 9342 | @group |
d7b8e6c6 EZ |
9343 | 2: x^2 - 9 1: 3. |
9344 | 1: 1 . | |
9345 | . | |
9346 | ||
5d67986c | 9347 | ' x^2-9 @key{RET} 1 X |
d7b8e6c6 | 9348 | @end group |
5d67986c | 9349 | @end smallexample |
d7b8e6c6 EZ |
9350 | |
9351 | Once again, here's the full Newton's Method definition: | |
9352 | ||
d7b8e6c6 | 9353 | @example |
5d67986c RS |
9354 | @group |
9355 | C-x ( Z ` @key{TAB} @key{RET} a d x @key{RET} / ' x @key{RET} @key{TAB} - t 1 | |
9356 | 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET} | |
9357 | @key{RET} M-@key{TAB} a = Z / | |
d7b8e6c6 EZ |
9358 | Z > |
9359 | Z ' | |
9360 | C-x ) | |
d7b8e6c6 | 9361 | @end group |
5d67986c | 9362 | @end example |
d7b8e6c6 EZ |
9363 | |
9364 | @c [fix-ref Nesting and Fixed Points] | |
9365 | It turns out that Calc has a built-in command for applying a formula | |
9366 | repeatedly until it converges to a number. @xref{Nesting and Fixed Points}, | |
9367 | to see how to use it. | |
9368 | ||
9369 | @c [fix-ref Root Finding] | |
9370 | Also, of course, @kbd{a R} is a built-in command that uses Newton's | |
9371 | method (among others) to look for numerical solutions to any equation. | |
9372 | @xref{Root Finding}. | |
9373 | ||
9374 | @node Programming Answer 9, Programming Answer 10, Programming Answer 8, Answers to Exercises | |
9375 | @subsection Programming Tutorial Exercise 9 | |
9376 | ||
9377 | @noindent | |
9378 | The first step is to adjust @cite{z} to be greater than 5. A simple | |
9379 | ``for'' loop will do the job here. If @cite{z} is less than 5, we | |
9380 | reduce the problem using @c{$\psi(z) = \psi(z+1) - 1/z$} | |
9381 | @cite{psi(z) = psi(z+1) - 1/z}. We go | |
9382 | on to compute @c{$\psi(z+1)$} | |
9383 | @cite{psi(z+1)}, and remember to add back a factor of | |
9384 | @cite{-1/z} when we're done. This step is repeated until @cite{z > 5}. | |
9385 | ||
9386 | (Because this definition is long, it will be repeated in concise form | |
9387 | below. You can use @w{@kbd{M-# m}} to load it from there. While you are | |
9388 | entering a @kbd{Z ` Z '} body in a macro, Calc simply collects | |
9389 | keystrokes without executing them. In the following diagrams we'll | |
9390 | pretend Calc actually executed the keystrokes as you typed them, | |
9391 | just for purposes of illustration.) | |
9392 | ||
d7b8e6c6 | 9393 | @smallexample |
5d67986c | 9394 | @group |
d7b8e6c6 EZ |
9395 | 1: 1. 1: 1. |
9396 | . . | |
9397 | ||
5d67986c | 9398 | 1.0 @key{RET} C-x ( Z ` s 1 0 t 2 |
d7b8e6c6 | 9399 | @end group |
5d67986c | 9400 | @end smallexample |
d7b8e6c6 EZ |
9401 | |
9402 | Here, variable 1 holds @cite{z} and variable 2 holds the adjustment | |
9403 | factor. If @cite{z < 5}, we use a loop to increase it. | |
9404 | ||
9405 | (By the way, we started with @samp{1.0} instead of the integer 1 because | |
9406 | otherwise the calculation below will try to do exact fractional arithmetic, | |
9407 | and will never converge because fractions compare equal only if they | |
9408 | are exactly equal, not just equal to within the current precision.) | |
9409 | ||
d7b8e6c6 | 9410 | @smallexample |
5d67986c | 9411 | @group |
d7b8e6c6 EZ |
9412 | 3: 1. 2: 1. 1: 6. |
9413 | 2: 1. 1: 1 . | |
9414 | 1: 5 . | |
9415 | . | |
9416 | ||
5d67986c | 9417 | @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ] |
d7b8e6c6 | 9418 | @end group |
5d67986c | 9419 | @end smallexample |
d7b8e6c6 EZ |
9420 | |
9421 | Now we compute the initial part of the sum: @c{$\ln z - {1 \over 2z}$} | |
9422 | @cite{ln(z) - 1/2z} | |
9423 | minus the adjustment factor. | |
9424 | ||
d7b8e6c6 | 9425 | @smallexample |
5d67986c | 9426 | @group |
d7b8e6c6 EZ |
9427 | 2: 1.79175946923 2: 1.7084261359 1: -0.57490719743 |
9428 | 1: 0.0833333333333 1: 2.28333333333 . | |
9429 | . . | |
9430 | ||
9431 | L r 1 2 * & - r 2 - | |
d7b8e6c6 | 9432 | @end group |
5d67986c | 9433 | @end smallexample |
d7b8e6c6 EZ |
9434 | |
9435 | Now we evaluate the series. We'll use another ``for'' loop counting | |
9436 | up the value of @cite{2 n}. (Calc does have a summation command, | |
9437 | @kbd{a +}, but we'll use loops just to get more practice with them.) | |
9438 | ||
d7b8e6c6 | 9439 | @smallexample |
5d67986c | 9440 | @group |
d7b8e6c6 EZ |
9441 | 3: -0.5749 3: -0.5749 4: -0.5749 2: -0.5749 |
9442 | 2: 2 2: 1:6 3: 1:6 1: 2.3148e-3 | |
9443 | 1: 40 1: 2 2: 2 . | |
9444 | . . 1: 36. | |
9445 | . | |
9446 | ||
5d67986c | 9447 | 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1 @key{TAB} ^ * / |
d7b8e6c6 | 9448 | |
d7b8e6c6 | 9449 | @end group |
5d67986c | 9450 | @end smallexample |
d7b8e6c6 | 9451 | @noindent |
d7b8e6c6 | 9452 | @smallexample |
5d67986c | 9453 | @group |
d7b8e6c6 EZ |
9454 | 3: -0.5749 3: -0.5772 2: -0.5772 1: -0.577215664892 |
9455 | 2: -0.5749 2: -0.5772 1: 0 . | |
9456 | 1: 2.3148e-3 1: -0.5749 . | |
9457 | . . | |
9458 | ||
5d67986c | 9459 | @key{TAB} @key{RET} M-@key{TAB} - @key{RET} M-@key{TAB} a = Z / 2 Z ) Z ' C-x ) |
d7b8e6c6 | 9460 | @end group |
5d67986c | 9461 | @end smallexample |
d7b8e6c6 EZ |
9462 | |
9463 | This is the value of @c{$-\gamma$} | |
9464 | @cite{- gamma}, with a slight bit of roundoff error. | |
9465 | To get a full 12 digits, let's use a higher precision: | |
9466 | ||
d7b8e6c6 | 9467 | @smallexample |
5d67986c | 9468 | @group |
d7b8e6c6 EZ |
9469 | 2: -0.577215664892 2: -0.577215664892 |
9470 | 1: 1. 1: -0.577215664901532 | |
9471 | ||
5d67986c | 9472 | 1. @key{RET} p 16 @key{RET} X |
d7b8e6c6 | 9473 | @end group |
5d67986c | 9474 | @end smallexample |
d7b8e6c6 EZ |
9475 | |
9476 | Here's the complete sequence of keystrokes: | |
9477 | ||
d7b8e6c6 | 9478 | @example |
5d67986c | 9479 | @group |
d7b8e6c6 | 9480 | C-x ( Z ` s 1 0 t 2 |
5d67986c | 9481 | @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ] |
d7b8e6c6 | 9482 | L r 1 2 * & - r 2 - |
5d67986c RS |
9483 | 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1 @key{TAB} ^ * / |
9484 | @key{TAB} @key{RET} M-@key{TAB} - @key{RET} M-@key{TAB} a = Z / | |
d7b8e6c6 EZ |
9485 | 2 Z ) |
9486 | Z ' | |
9487 | C-x ) | |
d7b8e6c6 | 9488 | @end group |
5d67986c | 9489 | @end example |
d7b8e6c6 EZ |
9490 | |
9491 | @node Programming Answer 10, Programming Answer 11, Programming Answer 9, Answers to Exercises | |
9492 | @subsection Programming Tutorial Exercise 10 | |
9493 | ||
9494 | @noindent | |
9495 | Taking the derivative of a term of the form @cite{x^n} will produce | |
9496 | a term like @c{$n x^{n-1}$} | |
9497 | @cite{n x^(n-1)}. Taking the derivative of a constant | |
9498 | produces zero. From this it is easy to see that the @cite{n}th | |
9499 | derivative of a polynomial, evaluated at @cite{x = 0}, will equal the | |
9500 | coefficient on the @cite{x^n} term times @cite{n!}. | |
9501 | ||
9502 | (Because this definition is long, it will be repeated in concise form | |
9503 | below. You can use @w{@kbd{M-# m}} to load it from there. While you are | |
9504 | entering a @kbd{Z ` Z '} body in a macro, Calc simply collects | |
9505 | keystrokes without executing them. In the following diagrams we'll | |
9506 | pretend Calc actually executed the keystrokes as you typed them, | |
9507 | just for purposes of illustration.) | |
9508 | ||
d7b8e6c6 | 9509 | @smallexample |
5d67986c | 9510 | @group |
d7b8e6c6 EZ |
9511 | 2: 5 x^4 + (x + 1)^2 3: 5 x^4 + (x + 1)^2 |
9512 | 1: 6 2: 0 | |
9513 | . 1: 6 | |
9514 | . | |
9515 | ||
5d67986c | 9516 | ' 5 x^4 + (x+1)^2 @key{RET} 6 C-x ( Z ` [ ] t 1 0 @key{TAB} |
d7b8e6c6 | 9517 | @end group |
5d67986c | 9518 | @end smallexample |
d7b8e6c6 EZ |
9519 | |
9520 | @noindent | |
9521 | Variable 1 will accumulate the vector of coefficients. | |
9522 | ||
d7b8e6c6 | 9523 | @smallexample |
5d67986c | 9524 | @group |
d7b8e6c6 EZ |
9525 | 2: 0 3: 0 2: 5 x^4 + ... |
9526 | 1: 5 x^4 + ... 2: 5 x^4 + ... 1: 1 | |
9527 | . 1: 1 . | |
9528 | . | |
9529 | ||
5d67986c | 9530 | Z ( @key{TAB} @key{RET} 0 s l x @key{RET} M-@key{TAB} ! / s | 1 |
d7b8e6c6 | 9531 | @end group |
5d67986c | 9532 | @end smallexample |
d7b8e6c6 EZ |
9533 | |
9534 | @noindent | |
9535 | Note that @kbd{s | 1} appends the top-of-stack value to the vector | |
9536 | in a variable; it is completely analogous to @kbd{s + 1}. We could | |
5d67986c | 9537 | have written instead, @kbd{r 1 @key{TAB} | t 1}. |
d7b8e6c6 | 9538 | |
d7b8e6c6 | 9539 | @smallexample |
5d67986c | 9540 | @group |
d7b8e6c6 EZ |
9541 | 1: 20 x^3 + 2 x + 2 1: 0 1: [1, 2, 1, 0, 5, 0, 0] |
9542 | . . . | |
9543 | ||
5d67986c | 9544 | a d x @key{RET} 1 Z ) @key{DEL} r 1 Z ' C-x ) |
d7b8e6c6 | 9545 | @end group |
5d67986c | 9546 | @end smallexample |
d7b8e6c6 EZ |
9547 | |
9548 | To convert back, a simple method is just to map the coefficients | |
9549 | against a table of powers of @cite{x}. | |
9550 | ||
d7b8e6c6 | 9551 | @smallexample |
5d67986c | 9552 | @group |
d7b8e6c6 EZ |
9553 | 2: [1, 2, 1, 0, 5, 0, 0] 2: [1, 2, 1, 0, 5, 0, 0] |
9554 | 1: 6 1: [0, 1, 2, 3, 4, 5, 6] | |
9555 | . . | |
9556 | ||
5d67986c | 9557 | 6 @key{RET} 1 + 0 @key{RET} 1 C-u v x |
d7b8e6c6 | 9558 | |
d7b8e6c6 | 9559 | @end group |
5d67986c | 9560 | @end smallexample |
d7b8e6c6 | 9561 | @noindent |
d7b8e6c6 | 9562 | @smallexample |
5d67986c | 9563 | @group |
d7b8e6c6 EZ |
9564 | 2: [1, 2, 1, 0, 5, 0, 0] 2: 1 + 2 x + x^2 + 5 x^4 |
9565 | 1: [1, x, x^2, x^3, ... ] . | |
9566 | . | |
9567 | ||
5d67986c | 9568 | ' x @key{RET} @key{TAB} V M ^ * |
d7b8e6c6 | 9569 | @end group |
5d67986c | 9570 | @end smallexample |
d7b8e6c6 EZ |
9571 | |
9572 | Once again, here are the whole polynomial to/from vector programs: | |
9573 | ||
d7b8e6c6 | 9574 | @example |
5d67986c RS |
9575 | @group |
9576 | C-x ( Z ` [ ] t 1 0 @key{TAB} | |
9577 | Z ( @key{TAB} @key{RET} 0 s l x @key{RET} M-@key{TAB} ! / s | 1 | |
9578 | a d x @key{RET} | |
d7b8e6c6 EZ |
9579 | 1 Z ) r 1 |
9580 | Z ' | |
9581 | C-x ) | |
9582 | ||
5d67986c | 9583 | C-x ( 1 + 0 @key{RET} 1 C-u v x ' x @key{RET} @key{TAB} V M ^ * C-x ) |
d7b8e6c6 | 9584 | @end group |
5d67986c | 9585 | @end example |
d7b8e6c6 EZ |
9586 | |
9587 | @node Programming Answer 11, Programming Answer 12, Programming Answer 10, Answers to Exercises | |
9588 | @subsection Programming Tutorial Exercise 11 | |
9589 | ||
9590 | @noindent | |
9591 | First we define a dummy program to go on the @kbd{z s} key. The true | |
9592 | @w{@kbd{z s}} key is supposed to take two numbers from the stack and | |
5d67986c | 9593 | return one number, so @key{DEL} as a dummy definition will make |
d7b8e6c6 EZ |
9594 | sure the stack comes out right. |
9595 | ||
d7b8e6c6 | 9596 | @smallexample |
5d67986c | 9597 | @group |
d7b8e6c6 EZ |
9598 | 2: 4 1: 4 2: 4 |
9599 | 1: 2 . 1: 2 | |
9600 | . . | |
9601 | ||
5d67986c | 9602 | 4 @key{RET} 2 C-x ( @key{DEL} C-x ) Z K s @key{RET} 2 |
d7b8e6c6 | 9603 | @end group |
5d67986c | 9604 | @end smallexample |
d7b8e6c6 EZ |
9605 | |
9606 | The last step replaces the 2 that was eaten during the creation | |
9607 | of the dummy @kbd{z s} command. Now we move on to the real | |
9608 | definition. The recurrence needs to be rewritten slightly, | |
9609 | to the form @cite{s(n,m) = s(n-1,m-1) - (n-1) s(n-1,m)}. | |
9610 | ||
9611 | (Because this definition is long, it will be repeated in concise form | |
9612 | below. You can use @kbd{M-# m} to load it from there.) | |
9613 | ||
d7b8e6c6 | 9614 | @smallexample |
5d67986c | 9615 | @group |
d7b8e6c6 EZ |
9616 | 2: 4 4: 4 3: 4 2: 4 |
9617 | 1: 2 3: 2 2: 2 1: 2 | |
9618 | . 2: 4 1: 0 . | |
9619 | 1: 2 . | |
9620 | . | |
9621 | ||
5d67986c | 9622 | C-x ( M-2 @key{RET} a = Z [ @key{DEL} @key{DEL} 1 Z : |
d7b8e6c6 | 9623 | |
d7b8e6c6 | 9624 | @end group |
5d67986c | 9625 | @end smallexample |
d7b8e6c6 | 9626 | @noindent |
d7b8e6c6 | 9627 | @smallexample |
5d67986c | 9628 | @group |
d7b8e6c6 EZ |
9629 | 4: 4 2: 4 2: 3 4: 3 4: 3 3: 3 |
9630 | 3: 2 1: 2 1: 2 3: 2 3: 2 2: 2 | |
9631 | 2: 2 . . 2: 3 2: 3 1: 3 | |
9632 | 1: 0 1: 2 1: 1 . | |
9633 | . . . | |
9634 | ||
5d67986c | 9635 | @key{RET} 0 a = Z [ @key{DEL} @key{DEL} 0 Z : @key{TAB} 1 - @key{TAB} M-2 @key{RET} 1 - z s |
d7b8e6c6 | 9636 | @end group |
5d67986c | 9637 | @end smallexample |
d7b8e6c6 EZ |
9638 | |
9639 | @noindent | |
9640 | (Note that the value 3 that our dummy @kbd{z s} produces is not correct; | |
9641 | it is merely a placeholder that will do just as well for now.) | |
9642 | ||
d7b8e6c6 | 9643 | @smallexample |
5d67986c | 9644 | @group |
d7b8e6c6 EZ |
9645 | 3: 3 4: 3 3: 3 2: 3 1: -6 |
9646 | 2: 3 3: 3 2: 3 1: 9 . | |
9647 | 1: 2 2: 3 1: 3 . | |
9648 | . 1: 2 . | |
9649 | . | |
9650 | ||
5d67986c | 9651 | M-@key{TAB} M-@key{TAB} @key{TAB} @key{RET} M-@key{TAB} z s * - |
d7b8e6c6 | 9652 | |
d7b8e6c6 | 9653 | @end group |
5d67986c | 9654 | @end smallexample |
d7b8e6c6 | 9655 | @noindent |
d7b8e6c6 | 9656 | @smallexample |
5d67986c | 9657 | @group |
d7b8e6c6 EZ |
9658 | 1: -6 2: 4 1: 11 2: 11 |
9659 | . 1: 2 . 1: 11 | |
9660 | . . | |
9661 | ||
5d67986c | 9662 | Z ] Z ] C-x ) Z K s @key{RET} @key{DEL} 4 @key{RET} 2 z s M-@key{RET} k s |
d7b8e6c6 | 9663 | @end group |
5d67986c | 9664 | @end smallexample |
d7b8e6c6 EZ |
9665 | |
9666 | Even though the result that we got during the definition was highly | |
9667 | bogus, once the definition is complete the @kbd{z s} command gets | |
9668 | the right answers. | |
9669 | ||
9670 | Here's the full program once again: | |
9671 | ||
d7b8e6c6 | 9672 | @example |
5d67986c RS |
9673 | @group |
9674 | C-x ( M-2 @key{RET} a = | |
9675 | Z [ @key{DEL} @key{DEL} 1 | |
9676 | Z : @key{RET} 0 a = | |
9677 | Z [ @key{DEL} @key{DEL} 0 | |
9678 | Z : @key{TAB} 1 - @key{TAB} M-2 @key{RET} 1 - z s | |
9679 | M-@key{TAB} M-@key{TAB} @key{TAB} @key{RET} M-@key{TAB} z s * - | |
d7b8e6c6 EZ |
9680 | Z ] |
9681 | Z ] | |
9682 | C-x ) | |
d7b8e6c6 | 9683 | @end group |
5d67986c | 9684 | @end example |
d7b8e6c6 EZ |
9685 | |
9686 | You can read this definition using @kbd{M-# m} (@code{read-kbd-macro}) | |
9687 | followed by @kbd{Z K s}, without having to make a dummy definition | |
9688 | first, because @code{read-kbd-macro} doesn't need to execute the | |
9689 | definition as it reads it in. For this reason, @code{M-# m} is often | |
9690 | the easiest way to create recursive programs in Calc. | |
9691 | ||
9692 | @node Programming Answer 12, , Programming Answer 11, Answers to Exercises | |
9693 | @subsection Programming Tutorial Exercise 12 | |
9694 | ||
9695 | @noindent | |
9696 | This turns out to be a much easier way to solve the problem. Let's | |
9697 | denote Stirling numbers as calls of the function @samp{s}. | |
9698 | ||
9699 | First, we store the rewrite rules corresponding to the definition of | |
9700 | Stirling numbers in a convenient variable: | |
9701 | ||
9702 | @smallexample | |
5d67986c | 9703 | s e StirlingRules @key{RET} |
d7b8e6c6 EZ |
9704 | [ s(n,n) := 1 :: n >= 0, |
9705 | s(n,0) := 0 :: n > 0, | |
9706 | s(n,m) := s(n-1,m-1) - (n-1) s(n-1,m) :: n >= m :: m >= 1 ] | |
9707 | C-c C-c | |
9708 | @end smallexample | |
9709 | ||
9710 | Now, it's just a matter of applying the rules: | |
9711 | ||
d7b8e6c6 | 9712 | @smallexample |
5d67986c | 9713 | @group |
d7b8e6c6 EZ |
9714 | 2: 4 1: s(4, 2) 1: 11 |
9715 | 1: 2 . . | |
9716 | . | |
9717 | ||
5d67986c | 9718 | 4 @key{RET} 2 C-x ( ' s($$,$) @key{RET} a r StirlingRules @key{RET} C-x ) |
d7b8e6c6 | 9719 | @end group |
5d67986c | 9720 | @end smallexample |
d7b8e6c6 EZ |
9721 | |
9722 | As in the case of the @code{fib} rules, it would be useful to put these | |
9723 | rules in @code{EvalRules} and to add a @samp{:: remember} condition to | |
9724 | the last rule. | |
9725 | ||
9726 | @c This ends the table-of-contents kludge from above: | |
9727 | @tex | |
9728 | \global\let\chapternofonts=\oldchapternofonts | |
9729 | @end tex | |
9730 | ||
9731 | @c [reference] | |
9732 | ||
9733 | @node Introduction, Data Types, Tutorial, Top | |
9734 | @chapter Introduction | |
9735 | ||
9736 | @noindent | |
9737 | This chapter is the beginning of the Calc reference manual. | |
9738 | It covers basic concepts such as the stack, algebraic and | |
9739 | numeric entry, undo, numeric prefix arguments, etc. | |
9740 | ||
9741 | @c [when-split] | |
9742 | @c (Chapter 2, the Tutorial, has been printed in a separate volume.) | |
9743 | ||
9744 | @menu | |
9745 | * Basic Commands:: | |
9746 | * Help Commands:: | |
9747 | * Stack Basics:: | |
9748 | * Numeric Entry:: | |
9749 | * Algebraic Entry:: | |
9750 | * Quick Calculator:: | |
9751 | * Keypad Mode:: | |
9752 | * Prefix Arguments:: | |
9753 | * Undo:: | |
9754 | * Error Messages:: | |
9755 | * Multiple Calculators:: | |
9756 | * Troubleshooting Commands:: | |
9757 | @end menu | |
9758 | ||
9759 | @node Basic Commands, Help Commands, Introduction, Introduction | |
9760 | @section Basic Commands | |
9761 | ||
9762 | @noindent | |
9763 | @pindex calc | |
9764 | @pindex calc-mode | |
9765 | @cindex Starting the Calculator | |
9766 | @cindex Running the Calculator | |
9767 | To start the Calculator in its standard interface, type @kbd{M-x calc}. | |
9768 | By default this creates a pair of small windows, @samp{*Calculator*} | |
9769 | and @samp{*Calc Trail*}. The former displays the contents of the | |
9770 | Calculator stack and is manipulated exclusively through Calc commands. | |
9771 | It is possible (though not usually necessary) to create several Calc | |
9772 | Mode buffers each of which has an independent stack, undo list, and | |
9773 | mode settings. There is exactly one Calc Trail buffer; it records a | |
9774 | list of the results of all calculations that have been done. The | |
9775 | Calc Trail buffer uses a variant of Calc Mode, so Calculator commands | |
9776 | still work when the trail buffer's window is selected. It is possible | |
9777 | to turn the trail window off, but the @samp{*Calc Trail*} buffer itself | |
9778 | still exists and is updated silently. @xref{Trail Commands}.@refill | |
9779 | ||
9780 | @kindex M-# c | |
9781 | @kindex M-# M-# | |
5d67986c RS |
9782 | @ignore |
9783 | @mindex @null | |
9784 | @end ignore | |
d7b8e6c6 EZ |
9785 | @kindex M-# # |
9786 | In most installations, the @kbd{M-# c} key sequence is a more | |
9787 | convenient way to start the Calculator. Also, @kbd{M-# M-#} and | |
9788 | @kbd{M-# #} are synonyms for @kbd{M-# c} unless you last used Calc | |
9789 | in its ``keypad'' mode. | |
9790 | ||
9791 | @kindex x | |
9792 | @kindex M-x | |
9793 | @pindex calc-execute-extended-command | |
9794 | Most Calc commands use one or two keystrokes. Lower- and upper-case | |
9795 | letters are distinct. Commands may also be entered in full @kbd{M-x} form; | |
9796 | for some commands this is the only form. As a convenience, the @kbd{x} | |
9797 | key (@code{calc-execute-extended-command}) | |
9798 | is like @kbd{M-x} except that it enters the initial string @samp{calc-} | |
9799 | for you. For example, the following key sequences are equivalent: | |
9800 | @kbd{S}, @kbd{M-x calc-sin @key{RET}}, @kbd{x sin @key{RET}}.@refill | |
9801 | ||
9802 | @cindex Extensions module | |
9803 | @cindex @file{calc-ext} module | |
9804 | The Calculator exists in many parts. When you type @kbd{M-# c}, the | |
9805 | Emacs ``auto-load'' mechanism will bring in only the first part, which | |
9806 | contains the basic arithmetic functions. The other parts will be | |
9807 | auto-loaded the first time you use the more advanced commands like trig | |
9808 | functions or matrix operations. This is done to improve the response time | |
9809 | of the Calculator in the common case when all you need to do is a | |
9810 | little arithmetic. If for some reason the Calculator fails to load an | |
9811 | extension module automatically, you can force it to load all the | |
9812 | extensions by using the @kbd{M-# L} (@code{calc-load-everything}) | |
9813 | command. @xref{Mode Settings}.@refill | |
9814 | ||
9815 | If you type @kbd{M-x calc} or @kbd{M-# c} with any numeric prefix argument, | |
9816 | the Calculator is loaded if necessary, but it is not actually started. | |
9817 | If the argument is positive, the @file{calc-ext} extensions are also | |
9818 | loaded if necessary. User-written Lisp code that wishes to make use | |
9819 | of Calc's arithmetic routines can use @samp{(calc 0)} or @samp{(calc 1)} | |
9820 | to auto-load the Calculator.@refill | |
9821 | ||
9822 | @kindex M-# b | |
9823 | @pindex full-calc | |
9824 | If you type @kbd{M-# b}, then next time you use @kbd{M-# c} you | |
9825 | will get a Calculator that uses the full height of the Emacs screen. | |
9826 | When full-screen mode is on, @kbd{M-# c} runs the @code{full-calc} | |
9827 | command instead of @code{calc}. From the Unix shell you can type | |
9828 | @samp{emacs -f full-calc} to start a new Emacs specifically for use | |
9829 | as a calculator. When Calc is started from the Emacs command line | |
9830 | like this, Calc's normal ``quit'' commands actually quit Emacs itself. | |
9831 | ||
9832 | @kindex M-# o | |
9833 | @pindex calc-other-window | |
9834 | The @kbd{M-# o} command is like @kbd{M-# c} except that the Calc | |
9835 | window is not actually selected. If you are already in the Calc | |
9836 | window, @kbd{M-# o} switches you out of it. (The regular Emacs | |
9837 | @kbd{C-x o} command would also work for this, but it has a | |
9838 | tendency to drop you into the Calc Trail window instead, which | |
9839 | @kbd{M-# o} takes care not to do.) | |
9840 | ||
5d67986c RS |
9841 | @ignore |
9842 | @mindex M-# q | |
9843 | @end ignore | |
d7b8e6c6 EZ |
9844 | For one quick calculation, you can type @kbd{M-# q} (@code{quick-calc}) |
9845 | which prompts you for a formula (like @samp{2+3/4}). The result is | |
9846 | displayed at the bottom of the Emacs screen without ever creating | |
9847 | any special Calculator windows. @xref{Quick Calculator}. | |
9848 | ||
5d67986c RS |
9849 | @ignore |
9850 | @mindex M-# k | |
9851 | @end ignore | |
d7b8e6c6 EZ |
9852 | Finally, if you are using the X window system you may want to try |
9853 | @kbd{M-# k} (@code{calc-keypad}) which runs Calc with a | |
9854 | ``calculator keypad'' picture as well as a stack display. Click on | |
9855 | the keys with the mouse to operate the calculator. @xref{Keypad Mode}. | |
9856 | ||
9857 | @kindex q | |
9858 | @pindex calc-quit | |
9859 | @cindex Quitting the Calculator | |
9860 | @cindex Exiting the Calculator | |
9861 | The @kbd{q} key (@code{calc-quit}) exits Calc Mode and closes the | |
9862 | Calculator's window(s). It does not delete the Calculator buffers. | |
9863 | If you type @kbd{M-x calc} again, the Calculator will reappear with the | |
9864 | contents of the stack intact. Typing @kbd{M-# c} or @kbd{M-# M-#} | |
9865 | again from inside the Calculator buffer is equivalent to executing | |
9866 | @code{calc-quit}; you can think of @kbd{M-# M-#} as toggling the | |
9867 | Calculator on and off.@refill | |
9868 | ||
9869 | @kindex M-# x | |
9870 | The @kbd{M-# x} command also turns the Calculator off, no matter which | |
9871 | user interface (standard, Keypad, or Embedded) is currently active. | |
9872 | It also cancels @code{calc-edit} mode if used from there. | |
9873 | ||
5d67986c | 9874 | @kindex d @key{SPC} |
d7b8e6c6 EZ |
9875 | @pindex calc-refresh |
9876 | @cindex Refreshing a garbled display | |
9877 | @cindex Garbled displays, refreshing | |
5d67986c | 9878 | The @kbd{d @key{SPC}} key sequence (@code{calc-refresh}) redraws the contents |
d7b8e6c6 EZ |
9879 | of the Calculator buffer from memory. Use this if the contents of the |
9880 | buffer have been damaged somehow. | |
9881 | ||
5d67986c RS |
9882 | @ignore |
9883 | @mindex o | |
9884 | @end ignore | |
d7b8e6c6 EZ |
9885 | The @kbd{o} key (@code{calc-realign}) moves the cursor back to its |
9886 | ``home'' position at the bottom of the Calculator buffer. | |
9887 | ||
9888 | @kindex < | |
9889 | @kindex > | |
9890 | @pindex calc-scroll-left | |
9891 | @pindex calc-scroll-right | |
9892 | @cindex Horizontal scrolling | |
9893 | @cindex Scrolling | |
9894 | @cindex Wide text, scrolling | |
9895 | The @kbd{<} and @kbd{>} keys are bound to @code{calc-scroll-left} and | |
9896 | @code{calc-scroll-right}. These are just like the normal horizontal | |
9897 | scrolling commands except that they scroll one half-screen at a time by | |
9898 | default. (Calc formats its output to fit within the bounds of the | |
9899 | window whenever it can.)@refill | |
9900 | ||
9901 | @kindex @{ | |
9902 | @kindex @} | |
9903 | @pindex calc-scroll-down | |
9904 | @pindex calc-scroll-up | |
9905 | @cindex Vertical scrolling | |
9906 | The @kbd{@{} and @kbd{@}} keys are bound to @code{calc-scroll-down} | |
9907 | and @code{calc-scroll-up}. They scroll up or down by one-half the | |
9908 | height of the Calc window.@refill | |
9909 | ||
9910 | @kindex M-# 0 | |
9911 | @pindex calc-reset | |
9912 | The @kbd{M-# 0} command (@code{calc-reset}; that's @kbd{M-#} followed | |
9913 | by a zero) resets the Calculator to its default state. This clears | |
9914 | the stack, resets all the modes, clears the caches (@pxref{Caches}), | |
9915 | and so on. (It does @emph{not} erase the values of any variables.) | |
9916 | With a numeric prefix argument, @kbd{M-# 0} preserves the contents | |
9917 | of the stack but resets everything else. | |
9918 | ||
9919 | @pindex calc-version | |
9920 | The @kbd{M-x calc-version} command displays the current version number | |
9921 | of Calc and the name of the person who installed it on your system. | |
9922 | (This information is also present in the @samp{*Calc Trail*} buffer, | |
9923 | and in the output of the @kbd{h h} command.) | |
9924 | ||
9925 | @node Help Commands, Stack Basics, Basic Commands, Introduction | |
9926 | @section Help Commands | |
9927 | ||
9928 | @noindent | |
9929 | @cindex Help commands | |
9930 | @kindex ? | |
9931 | @pindex calc-help | |
9932 | The @kbd{?} key (@code{calc-help}) displays a series of brief help messages. | |
9933 | Some keys (such as @kbd{b} and @kbd{d}) are prefix keys, like Emacs' | |
9934 | @key{ESC} and @kbd{C-x} prefixes. You can type | |
9935 | @kbd{?} after a prefix to see a list of commands beginning with that | |
9936 | prefix. (If the message includes @samp{[MORE]}, press @kbd{?} again | |
9937 | to see additional commands for that prefix.) | |
9938 | ||
9939 | @kindex h h | |
9940 | @pindex calc-full-help | |
9941 | The @kbd{h h} (@code{calc-full-help}) command displays all the @kbd{?} | |
9942 | responses at once. When printed, this makes a nice, compact (three pages) | |
9943 | summary of Calc keystrokes. | |
9944 | ||
9945 | In general, the @kbd{h} key prefix introduces various commands that | |
9946 | provide help within Calc. Many of the @kbd{h} key functions are | |
9947 | Calc-specific analogues to the @kbd{C-h} functions for Emacs help. | |
9948 | ||
9949 | @kindex h i | |
9950 | @kindex M-# i | |
9951 | @kindex i | |
9952 | @pindex calc-info | |
9953 | The @kbd{h i} (@code{calc-info}) command runs the Emacs Info system | |
9954 | to read this manual on-line. This is basically the same as typing | |
9955 | @kbd{C-h i} (the regular way to run the Info system), then, if Info | |
9956 | is not already in the Calc manual, selecting the beginning of the | |
9957 | manual. The @kbd{M-# i} command is another way to read the Calc | |
9958 | manual; it is different from @kbd{h i} in that it works any time, | |
9959 | not just inside Calc. The plain @kbd{i} key is also equivalent to | |
9960 | @kbd{h i}, though this key is obsolete and may be replaced with a | |
9961 | different command in a future version of Calc. | |
9962 | ||
9963 | @kindex h t | |
9964 | @kindex M-# t | |
9965 | @pindex calc-tutorial | |
9966 | The @kbd{h t} (@code{calc-tutorial}) command runs the Info system on | |
9967 | the Tutorial section of the Calc manual. It is like @kbd{h i}, | |
9968 | except that it selects the starting node of the tutorial rather | |
9969 | than the beginning of the whole manual. (It actually selects the | |
9970 | node ``Interactive Tutorial'' which tells a few things about | |
9971 | using the Info system before going on to the actual tutorial.) | |
9972 | The @kbd{M-# t} key is equivalent to @kbd{h t} (but it works at | |
9973 | all times). | |
9974 | ||
9975 | @kindex h s | |
9976 | @kindex M-# s | |
9977 | @pindex calc-info-summary | |
9978 | The @kbd{h s} (@code{calc-info-summary}) command runs the Info system | |
9979 | on the Summary node of the Calc manual. @xref{Summary}. The @kbd{M-# s} | |
9980 | key is equivalent to @kbd{h s}. | |
9981 | ||
9982 | @kindex h k | |
9983 | @pindex calc-describe-key | |
9984 | The @kbd{h k} (@code{calc-describe-key}) command looks up a key | |
9985 | sequence in the Calc manual. For example, @kbd{h k H a S} looks | |
9986 | up the documentation on the @kbd{H a S} (@code{calc-solve-for}) | |
9987 | command. This works by looking up the textual description of | |
9988 | the key(s) in the Key Index of the manual, then jumping to the | |
9989 | node indicated by the index. | |
9990 | ||
9991 | Most Calc commands do not have traditional Emacs documentation | |
9992 | strings, since the @kbd{h k} command is both more convenient and | |
9993 | more instructive. This means the regular Emacs @kbd{C-h k} | |
9994 | (@code{describe-key}) command will not be useful for Calc keystrokes. | |
9995 | ||
9996 | @kindex h c | |
9997 | @pindex calc-describe-key-briefly | |
9998 | The @kbd{h c} (@code{calc-describe-key-briefly}) command reads a | |
9999 | key sequence and displays a brief one-line description of it at | |
10000 | the bottom of the screen. It looks for the key sequence in the | |
10001 | Summary node of the Calc manual; if it doesn't find the sequence | |
10002 | there, it acts just like its regular Emacs counterpart @kbd{C-h c} | |
10003 | (@code{describe-key-briefly}). For example, @kbd{h c H a S} | |
10004 | gives the description: | |
10005 | ||
10006 | @smallexample | |
10007 | H a S runs calc-solve-for: a `H a S' v => fsolve(a,v) (?=notes) | |
10008 | @end smallexample | |
10009 | ||
10010 | @noindent | |
10011 | which means the command @kbd{H a S} or @kbd{H M-x calc-solve-for} | |
10012 | takes a value @cite{a} from the stack, prompts for a value @cite{v}, | |
10013 | then applies the algebraic function @code{fsolve} to these values. | |
10014 | The @samp{?=notes} message means you can now type @kbd{?} to see | |
10015 | additional notes from the summary that apply to this command. | |
10016 | ||
10017 | @kindex h f | |
10018 | @pindex calc-describe-function | |
10019 | The @kbd{h f} (@code{calc-describe-function}) command looks up an | |
10020 | algebraic function or a command name in the Calc manual. The | |
10021 | prompt initially contains @samp{calcFunc-}; follow this with an | |
10022 | algebraic function name to look up that function in the Function | |
10023 | Index. Or, backspace and enter a command name beginning with | |
10024 | @samp{calc-} to look it up in the Command Index. This command | |
10025 | will also look up operator symbols that can appear in algebraic | |
10026 | formulas, like @samp{%} and @samp{=>}. | |
10027 | ||
10028 | @kindex h v | |
10029 | @pindex calc-describe-variable | |
10030 | The @kbd{h v} (@code{calc-describe-variable}) command looks up a | |
10031 | variable in the Calc manual. The prompt initially contains the | |
10032 | @samp{var-} prefix; just add a variable name like @code{pi} or | |
10033 | @code{PlotRejects}. | |
10034 | ||
10035 | @kindex h b | |
10036 | @pindex describe-bindings | |
10037 | The @kbd{h b} (@code{calc-describe-bindings}) command is just like | |
10038 | @kbd{C-h b}, except that only local (Calc-related) key bindings are | |
10039 | listed. | |
10040 | ||
10041 | @kindex h n | |
10042 | The @kbd{h n} or @kbd{h C-n} (@code{calc-view-news}) command displays | |
10043 | the ``news'' or change history of Calc. This is kept in the file | |
10044 | @file{README}, which Calc looks for in the same directory as the Calc | |
10045 | source files. | |
10046 | ||
10047 | @kindex h C-c | |
10048 | @kindex h C-d | |
10049 | @kindex h C-w | |
10050 | The @kbd{h C-c}, @kbd{h C-d}, and @kbd{h C-w} keys display copying, | |
10051 | distribution, and warranty information about Calc. These work by | |
10052 | pulling up the appropriate parts of the ``Copying'' or ``Reporting | |
10053 | Bugs'' sections of the manual. | |
10054 | ||
10055 | @node Stack Basics, Numeric Entry, Help Commands, Introduction | |
10056 | @section Stack Basics | |
10057 | ||
10058 | @noindent | |
10059 | @cindex Stack basics | |
10060 | @c [fix-tut RPN Calculations and the Stack] | |
28665d46 | 10061 | Calc uses RPN notation. If you are not familiar with RPN, @pxref{RPN |
d7b8e6c6 EZ |
10062 | Tutorial}. |
10063 | ||
10064 | To add the numbers 1 and 2 in Calc you would type the keys: | |
10065 | @kbd{1 @key{RET} 2 +}. | |
10066 | (@key{RET} corresponds to the @key{ENTER} key on most calculators.) | |
10067 | The first three keystrokes ``push'' the numbers 1 and 2 onto the stack. The | |
10068 | @kbd{+} key always ``pops'' the top two numbers from the stack, adds them, | |
10069 | and pushes the result (3) back onto the stack. This number is ready for | |
10070 | further calculations: @kbd{5 -} pushes 5 onto the stack, then pops the | |
10071 | 3 and 5, subtracts them, and pushes the result (@i{-2}).@refill | |
10072 | ||
10073 | Note that the ``top'' of the stack actually appears at the @emph{bottom} | |
10074 | of the buffer. A line containing a single @samp{.} character signifies | |
10075 | the end of the buffer; Calculator commands operate on the number(s) | |
10076 | directly above this line. The @kbd{d t} (@code{calc-truncate-stack}) | |
10077 | command allows you to move the @samp{.} marker up and down in the stack; | |
10078 | @pxref{Truncating the Stack}. | |
10079 | ||
10080 | @kindex d l | |
10081 | @pindex calc-line-numbering | |
10082 | Stack elements are numbered consecutively, with number 1 being the top of | |
10083 | the stack. These line numbers are ordinarily displayed on the lefthand side | |
10084 | of the window. The @kbd{d l} (@code{calc-line-numbering}) command controls | |
10085 | whether these numbers appear. (Line numbers may be turned off since they | |
10086 | slow the Calculator down a bit and also clutter the display.) | |
10087 | ||
10088 | @kindex o | |
10089 | @pindex calc-realign | |
10090 | The unshifted letter @kbd{o} (@code{calc-realign}) command repositions | |
10091 | the cursor to its top-of-stack ``home'' position. It also undoes any | |
10092 | horizontal scrolling in the window. If you give it a numeric prefix | |
10093 | argument, it instead moves the cursor to the specified stack element. | |
10094 | ||
10095 | The @key{RET} (or equivalent @key{SPC}) key is only required to separate | |
10096 | two consecutive numbers. | |
10097 | (After all, if you typed @kbd{1 2} by themselves the Calculator | |
5d67986c | 10098 | would enter the number 12.) If you press @key{RET} or @key{SPC} @emph{not} |
d7b8e6c6 EZ |
10099 | right after typing a number, the key duplicates the number on the top of |
10100 | the stack. @kbd{@key{RET} *} is thus a handy way to square a number.@refill | |
10101 | ||
10102 | The @key{DEL} key pops and throws away the top number on the stack. | |
10103 | The @key{TAB} key swaps the top two objects on the stack. | |
10104 | @xref{Stack and Trail}, for descriptions of these and other stack-related | |
10105 | commands.@refill | |
10106 | ||
10107 | @node Numeric Entry, Algebraic Entry, Stack Basics, Introduction | |
10108 | @section Numeric Entry | |
10109 | ||
10110 | @noindent | |
10111 | @kindex 0-9 | |
10112 | @kindex . | |
10113 | @kindex e | |
10114 | @cindex Numeric entry | |
10115 | @cindex Entering numbers | |
10116 | Pressing a digit or other numeric key begins numeric entry using the | |
10117 | minibuffer. The number is pushed on the stack when you press the @key{RET} | |
10118 | or @key{SPC} keys. If you press any other non-numeric key, the number is | |
10119 | pushed onto the stack and the appropriate operation is performed. If | |
10120 | you press a numeric key which is not valid, the key is ignored. | |
10121 | ||
10122 | @cindex Minus signs | |
10123 | @cindex Negative numbers, entering | |
10124 | @kindex _ | |
10125 | There are three different concepts corresponding to the word ``minus,'' | |
10126 | typified by @cite{a-b} (subtraction), @cite{-x} | |
10127 | (change-sign), and @cite{-5} (negative number). Calc uses three | |
10128 | different keys for these operations, respectively: | |
10129 | @kbd{-}, @kbd{n}, and @kbd{_} (the underscore). The @kbd{-} key subtracts | |
10130 | the two numbers on the top of the stack. The @kbd{n} key changes the sign | |
10131 | of the number on the top of the stack or the number currently being entered. | |
10132 | The @kbd{_} key begins entry of a negative number or changes the sign of | |
10133 | the number currently being entered. The following sequences all enter the | |
10134 | number @i{-5} onto the stack: @kbd{0 @key{RET} 5 -}, @kbd{5 n @key{RET}}, | |
10135 | @kbd{5 @key{RET} n}, @kbd{_ 5 @key{RET}}, @kbd{5 _ @key{RET}}.@refill | |
10136 | ||
10137 | Some other keys are active during numeric entry, such as @kbd{#} for | |
10138 | non-decimal numbers, @kbd{:} for fractions, and @kbd{@@} for HMS forms. | |
10139 | These notations are described later in this manual with the corresponding | |
10140 | data types. @xref{Data Types}. | |
10141 | ||
5d67986c | 10142 | During numeric entry, the only editing key available is @key{DEL}. |
d7b8e6c6 EZ |
10143 | |
10144 | @node Algebraic Entry, Quick Calculator, Numeric Entry, Introduction | |
10145 | @section Algebraic Entry | |
10146 | ||
10147 | @noindent | |
10148 | @kindex ' | |
10149 | @pindex calc-algebraic-entry | |
10150 | @cindex Algebraic notation | |
10151 | @cindex Formulas, entering | |
10152 | Calculations can also be entered in algebraic form. This is accomplished | |
10153 | by typing the apostrophe key, @kbd{'}, followed by the expression in | |
10154 | standard format: @kbd{@key{'} 2+3*4 @key{RET}} computes | |
10155 | @c{$2+(3\times4) = 14$} | |
10156 | @cite{2+(3*4) = 14} and pushes that on the stack. If you wish you can | |
10157 | ignore the RPN aspect of Calc altogether and simply enter algebraic | |
10158 | expressions in this way. You may want to use @key{DEL} every so often to | |
10159 | clear previous results off the stack.@refill | |
10160 | ||
10161 | You can press the apostrophe key during normal numeric entry to switch | |
10162 | the half-entered number into algebraic entry mode. One reason to do this | |
10163 | would be to use the full Emacs cursor motion and editing keys, which are | |
10164 | available during algebraic entry but not during numeric entry. | |
10165 | ||
10166 | In the same vein, during either numeric or algebraic entry you can | |
10167 | press @kbd{`} (backquote) to switch to @code{calc-edit} mode, where | |
10168 | you complete your half-finished entry in a separate buffer. | |
10169 | @xref{Editing Stack Entries}. | |
10170 | ||
10171 | @kindex m a | |
10172 | @pindex calc-algebraic-mode | |
10173 | @cindex Algebraic mode | |
10174 | If you prefer algebraic entry, you can use the command @kbd{m a} | |
10175 | (@code{calc-algebraic-mode}) to set Algebraic mode. In this mode, | |
10176 | digits and other keys that would normally start numeric entry instead | |
10177 | start full algebraic entry; as long as your formula begins with a digit | |
10178 | you can omit the apostrophe. Open parentheses and square brackets also | |
10179 | begin algebraic entry. You can still do RPN calculations in this mode, | |
10180 | but you will have to press @key{RET} to terminate every number: | |
10181 | @kbd{2 @key{RET} 3 @key{RET} * 4 @key{RET} +} would accomplish the same | |
10182 | thing as @kbd{2*3+4 @key{RET}}.@refill | |
10183 | ||
10184 | @cindex Incomplete algebraic mode | |
10185 | If you give a numeric prefix argument like @kbd{C-u} to the @kbd{m a} | |
10186 | command, it enables Incomplete Algebraic mode; this is like regular | |
10187 | Algebraic mode except that it applies to the @kbd{(} and @kbd{[} keys | |
10188 | only. Numeric keys still begin a numeric entry in this mode. | |
10189 | ||
10190 | @kindex m t | |
10191 | @pindex calc-total-algebraic-mode | |
10192 | @cindex Total algebraic mode | |
10193 | The @kbd{m t} (@code{calc-total-algebraic-mode}) gives you an even | |
10194 | stronger algebraic-entry mode, in which @emph{all} regular letter and | |
10195 | punctuation keys begin algebraic entry. Use this if you prefer typing | |
10196 | @w{@kbd{sqrt( )}} instead of @kbd{Q}, @w{@kbd{factor( )}} instead of | |
10197 | @kbd{a f}, and so on. To type regular Calc commands when you are in | |
10198 | ``total'' algebraic mode, hold down the @key{META} key. Thus @kbd{M-q} | |
10199 | is the command to quit Calc, @kbd{M-p} sets the precision, and | |
10200 | @kbd{M-m t} (or @kbd{M-m M-t}, if you prefer) turns total algebraic | |
10201 | mode back off again. Meta keys also terminate algebraic entry, so | |
5d67986c | 10202 | that @kbd{2+3 M-S} is equivalent to @kbd{2+3 @key{RET} M-S}. The symbol |
d7b8e6c6 EZ |
10203 | @samp{Alg*} will appear in the mode line whenever you are in this mode. |
10204 | ||
10205 | Pressing @kbd{'} (the apostrophe) a second time re-enters the previous | |
10206 | algebraic formula. You can then use the normal Emacs editing keys to | |
10207 | modify this formula to your liking before pressing @key{RET}. | |
10208 | ||
10209 | @kindex $ | |
10210 | @cindex Formulas, referring to stack | |
10211 | Within a formula entered from the keyboard, the symbol @kbd{$} | |
10212 | represents the number on the top of the stack. If an entered formula | |
10213 | contains any @kbd{$} characters, the Calculator replaces the top of | |
10214 | stack with that formula rather than simply pushing the formula onto the | |
10215 | stack. Thus, @kbd{' 1+2 @key{RET}} pushes 3 on the stack, and @kbd{$*2 | |
10216 | @key{RET}} replaces it with 6. Note that the @kbd{$} key always | |
10217 | initiates algebraic entry; the @kbd{'} is unnecessary if @kbd{$} is the | |
10218 | first character in the new formula.@refill | |
10219 | ||
10220 | Higher stack elements can be accessed from an entered formula with the | |
10221 | symbols @kbd{$$}, @kbd{$$$}, and so on. The number of stack elements | |
10222 | removed (to be replaced by the entered values) equals the number of dollar | |
10223 | signs in the longest such symbol in the formula. For example, @samp{$$+$$$} | |
10224 | adds the second and third stack elements, replacing the top three elements | |
10225 | with the answer. (All information about the top stack element is thus lost | |
10226 | since no single @samp{$} appears in this formula.)@refill | |
10227 | ||
10228 | A slightly different way to refer to stack elements is with a dollar | |
10229 | sign followed by a number: @samp{$1}, @samp{$2}, and so on are much | |
10230 | like @samp{$}, @samp{$$}, etc., except that stack entries referred | |
10231 | to numerically are not replaced by the algebraic entry. That is, while | |
10232 | @samp{$+1} replaces 5 on the stack with 6, @samp{$1+1} leaves the 5 | |
10233 | on the stack and pushes an additional 6. | |
10234 | ||
10235 | If a sequence of formulas are entered separated by commas, each formula | |
10236 | is pushed onto the stack in turn. For example, @samp{1,2,3} pushes | |
10237 | those three numbers onto the stack (leaving the 3 at the top), and | |
10238 | @samp{$+1,$-1} replaces a 5 on the stack with 4 followed by 6. Also, | |
10239 | @samp{$,$$} exchanges the top two elements of the stack, just like the | |
10240 | @key{TAB} key. | |
10241 | ||
5d67986c | 10242 | You can finish an algebraic entry with @kbd{M-=} or @kbd{M-@key{RET}} instead |
d7b8e6c6 EZ |
10243 | of @key{RET}. This uses @kbd{=} to evaluate the variables in each |
10244 | formula that goes onto the stack. (Thus @kbd{' pi @key{RET}} pushes | |
5d67986c | 10245 | the variable @samp{pi}, but @kbd{' pi M-@key{RET}} pushes 3.1415.) |
d7b8e6c6 | 10246 | |
5d67986c | 10247 | If you finish your algebraic entry by pressing @key{LFD} (or @kbd{C-j}) |
d7b8e6c6 EZ |
10248 | instead of @key{RET}, Calc disables the default simplifications |
10249 | (as if by @kbd{m O}; @pxref{Simplification Modes}) while the entry | |
10250 | is being pushed on the stack. Thus @kbd{' 1+2 @key{RET}} pushes 3 | |
10251 | on the stack, but @kbd{' 1+2 @key{LFD}} pushes the formula @cite{1+2}; | |
10252 | you might then press @kbd{=} when it is time to evaluate this formula. | |
10253 | ||
10254 | @node Quick Calculator, Prefix Arguments, Algebraic Entry, Introduction | |
10255 | @section ``Quick Calculator'' Mode | |
10256 | ||
10257 | @noindent | |
10258 | @kindex M-# q | |
10259 | @pindex quick-calc | |
10260 | @cindex Quick Calculator | |
10261 | There is another way to invoke the Calculator if all you need to do | |
10262 | is make one or two quick calculations. Type @kbd{M-# q} (or | |
10263 | @kbd{M-x quick-calc}), then type any formula as an algebraic entry. | |
10264 | The Calculator will compute the result and display it in the echo | |
10265 | area, without ever actually putting up a Calc window. | |
10266 | ||
10267 | You can use the @kbd{$} character in a Quick Calculator formula to | |
10268 | refer to the previous Quick Calculator result. Older results are | |
10269 | not retained; the Quick Calculator has no effect on the full | |
10270 | Calculator's stack or trail. If you compute a result and then | |
10271 | forget what it was, just run @code{M-# q} again and enter | |
10272 | @samp{$} as the formula. | |
10273 | ||
10274 | If this is the first time you have used the Calculator in this Emacs | |
10275 | session, the @kbd{M-# q} command will create the @code{*Calculator*} | |
10276 | buffer and perform all the usual initializations; it simply will | |
10277 | refrain from putting that buffer up in a new window. The Quick | |
10278 | Calculator refers to the @code{*Calculator*} buffer for all mode | |
10279 | settings. Thus, for example, to set the precision that the Quick | |
10280 | Calculator uses, simply run the full Calculator momentarily and use | |
10281 | the regular @kbd{p} command. | |
10282 | ||
10283 | If you use @code{M-# q} from inside the Calculator buffer, the | |
10284 | effect is the same as pressing the apostrophe key (algebraic entry). | |
10285 | ||
10286 | The result of a Quick calculation is placed in the Emacs ``kill ring'' | |
10287 | as well as being displayed. A subsequent @kbd{C-y} command will | |
10288 | yank the result into the editing buffer. You can also use this | |
10289 | to yank the result into the next @kbd{M-# q} input line as a more | |
10290 | explicit alternative to @kbd{$} notation, or to yank the result | |
10291 | into the Calculator stack after typing @kbd{M-# c}. | |
10292 | ||
10293 | If you finish your formula by typing @key{LFD} (or @kbd{C-j}) instead | |
10294 | of @key{RET}, the result is inserted immediately into the current | |
10295 | buffer rather than going into the kill ring. | |
10296 | ||
10297 | Quick Calculator results are actually evaluated as if by the @kbd{=} | |
10298 | key (which replaces variable names by their stored values, if any). | |
10299 | If the formula you enter is an assignment to a variable using the | |
10300 | @samp{:=} operator, say, @samp{foo := 2 + 3} or @samp{foo := foo + 1}, | |
10301 | then the result of the evaluation is stored in that Calc variable. | |
10302 | @xref{Store and Recall}. | |
10303 | ||
10304 | If the result is an integer and the current display radix is decimal, | |
10305 | the number will also be displayed in hex and octal formats. If the | |
10306 | integer is in the range from 1 to 126, it will also be displayed as | |
10307 | an ASCII character. | |
10308 | ||
10309 | For example, the quoted character @samp{"x"} produces the vector | |
10310 | result @samp{[120]} (because 120 is the ASCII code of the lower-case | |
10311 | `x'; @pxref{Strings}). Since this is a vector, not an integer, it | |
10312 | is displayed only according to the current mode settings. But | |
10313 | running Quick Calc again and entering @samp{120} will produce the | |
10314 | result @samp{120 (16#78, 8#170, x)} which shows the number in its | |
10315 | decimal, hexadecimal, octal, and ASCII forms. | |
10316 | ||
10317 | Please note that the Quick Calculator is not any faster at loading | |
10318 | or computing the answer than the full Calculator; the name ``quick'' | |
10319 | merely refers to the fact that it's much less hassle to use for | |
10320 | small calculations. | |
10321 | ||
10322 | @node Prefix Arguments, Undo, Quick Calculator, Introduction | |
10323 | @section Numeric Prefix Arguments | |
10324 | ||
10325 | @noindent | |
10326 | Many Calculator commands use numeric prefix arguments. Some, such as | |
10327 | @kbd{d s} (@code{calc-sci-notation}), set a parameter to the value of | |
10328 | the prefix argument or use a default if you don't use a prefix. | |
10329 | Others (like @kbd{d f} (@code{calc-fix-notation})) require an argument | |
10330 | and prompt for a number if you don't give one as a prefix.@refill | |
10331 | ||
10332 | As a rule, stack-manipulation commands accept a numeric prefix argument | |
10333 | which is interpreted as an index into the stack. A positive argument | |
10334 | operates on the top @var{n} stack entries; a negative argument operates | |
10335 | on the @var{n}th stack entry in isolation; and a zero argument operates | |
10336 | on the entire stack. | |
10337 | ||
10338 | Most commands that perform computations (such as the arithmetic and | |
10339 | scientific functions) accept a numeric prefix argument that allows the | |
10340 | operation to be applied across many stack elements. For unary operations | |
10341 | (that is, functions of one argument like absolute value or complex | |
10342 | conjugate), a positive prefix argument applies that function to the top | |
10343 | @var{n} stack entries simultaneously, and a negative argument applies it | |
10344 | to the @var{n}th stack entry only. For binary operations (functions of | |
10345 | two arguments like addition, GCD, and vector concatenation), a positive | |
10346 | prefix argument ``reduces'' the function across the top @var{n} | |
10347 | stack elements (for example, @kbd{C-u 5 +} sums the top 5 stack entries; | |
10348 | @pxref{Reducing and Mapping}), and a negative argument maps the next-to-top | |
10349 | @var{n} stack elements with the top stack element as a second argument | |
10350 | (for example, @kbd{7 c-u -5 +} adds 7 to the top 5 stack elements). | |
10351 | This feature is not available for operations which use the numeric prefix | |
10352 | argument for some other purpose. | |
10353 | ||
10354 | Numeric prefixes are specified the same way as always in Emacs: Press | |
10355 | a sequence of @key{META}-digits, or press @key{ESC} followed by digits, | |
10356 | or press @kbd{C-u} followed by digits. Some commands treat plain | |
10357 | @kbd{C-u} (without any actual digits) specially.@refill | |
10358 | ||
10359 | @kindex ~ | |
10360 | @pindex calc-num-prefix | |
10361 | You can type @kbd{~} (@code{calc-num-prefix}) to pop an integer from the | |
10362 | top of the stack and enter it as the numeric prefix for the next command. | |
10363 | For example, @kbd{C-u 16 p} sets the precision to 16 digits; an alternate | |
10364 | (silly) way to do this would be @kbd{2 @key{RET} 4 ^ ~ p}, i.e., compute 2 | |
10365 | to the fourth power and set the precision to that value.@refill | |
10366 | ||
10367 | Conversely, if you have typed a numeric prefix argument the @kbd{~} key | |
10368 | pushes it onto the stack in the form of an integer. | |
10369 | ||
10370 | @node Undo, Error Messages, Prefix Arguments, Introduction | |
10371 | @section Undoing Mistakes | |
10372 | ||
10373 | @noindent | |
10374 | @kindex U | |
10375 | @kindex C-_ | |
10376 | @pindex calc-undo | |
10377 | @cindex Mistakes, undoing | |
10378 | @cindex Undoing mistakes | |
10379 | @cindex Errors, undoing | |
10380 | The shift-@kbd{U} key (@code{calc-undo}) undoes the most recent operation. | |
10381 | If that operation added or dropped objects from the stack, those objects | |
10382 | are removed or restored. If it was a ``store'' operation, you are | |
10383 | queried whether or not to restore the variable to its original value. | |
10384 | The @kbd{U} key may be pressed any number of times to undo successively | |
10385 | farther back in time; with a numeric prefix argument it undoes a | |
10386 | specified number of operations. The undo history is cleared only by the | |
10387 | @kbd{q} (@code{calc-quit}) command. (Recall that @kbd{M-# c} is | |
10388 | synonymous with @code{calc-quit} while inside the Calculator; this | |
10389 | also clears the undo history.) | |
10390 | ||
10391 | Currently the mode-setting commands (like @code{calc-precision}) are not | |
10392 | undoable. You can undo past a point where you changed a mode, but you | |
10393 | will need to reset the mode yourself. | |
10394 | ||
10395 | @kindex D | |
10396 | @pindex calc-redo | |
10397 | @cindex Redoing after an Undo | |
10398 | The shift-@kbd{D} key (@code{calc-redo}) redoes an operation that was | |
10399 | mistakenly undone. Pressing @kbd{U} with a negative prefix argument is | |
10400 | equivalent to executing @code{calc-redo}. You can redo any number of | |
10401 | times, up to the number of recent consecutive undo commands. Redo | |
10402 | information is cleared whenever you give any command that adds new undo | |
10403 | information, i.e., if you undo, then enter a number on the stack or make | |
10404 | any other change, then it will be too late to redo. | |
10405 | ||
5d67986c | 10406 | @kindex M-@key{RET} |
d7b8e6c6 EZ |
10407 | @pindex calc-last-args |
10408 | @cindex Last-arguments feature | |
10409 | @cindex Arguments, restoring | |
10410 | The @kbd{M-@key{RET}} key (@code{calc-last-args}) is like undo in that | |
10411 | it restores the arguments of the most recent command onto the stack; | |
10412 | however, it does not remove the result of that command. Given a numeric | |
10413 | prefix argument, this command applies to the @cite{n}th most recent | |
10414 | command which removed items from the stack; it pushes those items back | |
10415 | onto the stack. | |
10416 | ||
10417 | The @kbd{K} (@code{calc-keep-args}) command provides a related function | |
10418 | to @kbd{M-@key{RET}}. @xref{Stack and Trail}. | |
10419 | ||
10420 | It is also possible to recall previous results or inputs using the trail. | |
10421 | @xref{Trail Commands}. | |
10422 | ||
10423 | The standard Emacs @kbd{C-_} undo key is recognized as a synonym for @kbd{U}. | |
10424 | ||
10425 | @node Error Messages, Multiple Calculators, Undo, Introduction | |
10426 | @section Error Messages | |
10427 | ||
10428 | @noindent | |
10429 | @kindex w | |
10430 | @pindex calc-why | |
10431 | @cindex Errors, messages | |
10432 | @cindex Why did an error occur? | |
10433 | Many situations that would produce an error message in other calculators | |
10434 | simply create unsimplified formulas in the Emacs Calculator. For example, | |
10435 | @kbd{1 @key{RET} 0 /} pushes the formula @cite{1 / 0}; @w{@kbd{0 L}} pushes | |
10436 | the formula @samp{ln(0)}. Floating-point overflow and underflow are also | |
10437 | reasons for this to happen. | |
10438 | ||
10439 | When a function call must be left in symbolic form, Calc usually | |
10440 | produces a message explaining why. Messages that are probably | |
10441 | surprising or indicative of user errors are displayed automatically. | |
10442 | Other messages are simply kept in Calc's memory and are displayed only | |
10443 | if you type @kbd{w} (@code{calc-why}). You can also press @kbd{w} if | |
10444 | the same computation results in several messages. (The first message | |
10445 | will end with @samp{[w=more]} in this case.) | |
10446 | ||
10447 | @kindex d w | |
10448 | @pindex calc-auto-why | |
10449 | The @kbd{d w} (@code{calc-auto-why}) command controls when error messages | |
10450 | are displayed automatically. (Calc effectively presses @kbd{w} for you | |
10451 | after your computation finishes.) By default, this occurs only for | |
10452 | ``important'' messages. The other possible modes are to report | |
10453 | @emph{all} messages automatically, or to report none automatically (so | |
10454 | that you must always press @kbd{w} yourself to see the messages). | |
10455 | ||
10456 | @node Multiple Calculators, Troubleshooting Commands, Error Messages, Introduction | |
10457 | @section Multiple Calculators | |
10458 | ||
10459 | @noindent | |
10460 | @pindex another-calc | |
10461 | It is possible to have any number of Calc Mode buffers at once. | |
10462 | Usually this is done by executing @kbd{M-x another-calc}, which | |
10463 | is similar to @kbd{M-# c} except that if a @samp{*Calculator*} | |
10464 | buffer already exists, a new, independent one with a name of the | |
10465 | form @samp{*Calculator*<@var{n}>} is created. You can also use the | |
10466 | command @code{calc-mode} to put any buffer into Calculator mode, but | |
10467 | this would ordinarily never be done. | |
10468 | ||
10469 | The @kbd{q} (@code{calc-quit}) command does not destroy a Calculator buffer; | |
10470 | it only closes its window. Use @kbd{M-x kill-buffer} to destroy a | |
10471 | Calculator buffer. | |
10472 | ||
10473 | Each Calculator buffer keeps its own stack, undo list, and mode settings | |
10474 | such as precision, angular mode, and display formats. In Emacs terms, | |
10475 | variables such as @code{calc-stack} are buffer-local variables. The | |
10476 | global default values of these variables are used only when a new | |
10477 | Calculator buffer is created. The @code{calc-quit} command saves | |
10478 | the stack and mode settings of the buffer being quit as the new defaults. | |
10479 | ||
10480 | There is only one trail buffer, @samp{*Calc Trail*}, used by all | |
10481 | Calculator buffers. | |
10482 | ||
10483 | @node Troubleshooting Commands, , Multiple Calculators, Introduction | |
10484 | @section Troubleshooting Commands | |
10485 | ||
10486 | @noindent | |
10487 | This section describes commands you can use in case a computation | |
10488 | incorrectly fails or gives the wrong answer. | |
10489 | ||
10490 | @xref{Reporting Bugs}, if you find a problem that appears to be due | |
10491 | to a bug or deficiency in Calc. | |
10492 | ||
10493 | @menu | |
10494 | * Autoloading Problems:: | |
10495 | * Recursion Depth:: | |
10496 | * Caches:: | |
10497 | * Debugging Calc:: | |
10498 | @end menu | |
10499 | ||
10500 | @node Autoloading Problems, Recursion Depth, Troubleshooting Commands, Troubleshooting Commands | |
10501 | @subsection Autoloading Problems | |
10502 | ||
10503 | @noindent | |
10504 | The Calc program is split into many component files; components are | |
10505 | loaded automatically as you use various commands that require them. | |
10506 | Occasionally Calc may lose track of when a certain component is | |
10507 | necessary; typically this means you will type a command and it won't | |
10508 | work because some function you've never heard of was undefined. | |
10509 | ||
10510 | @kindex M-# L | |
10511 | @pindex calc-load-everything | |
10512 | If this happens, the easiest workaround is to type @kbd{M-# L} | |
10513 | (@code{calc-load-everything}) to force all the parts of Calc to be | |
10514 | loaded right away. This will cause Emacs to take up a lot more | |
10515 | memory than it would otherwise, but it's guaranteed to fix the problem. | |
10516 | ||
10517 | If you seem to run into this problem no matter what you do, or if | |
10518 | even the @kbd{M-# L} command crashes, Calc may have been improperly | |
10519 | installed. @xref{Installation}, for details of the installation | |
10520 | process. | |
10521 | ||
10522 | @node Recursion Depth, Caches, Autoloading Problems, Troubleshooting Commands | |
10523 | @subsection Recursion Depth | |
10524 | ||
10525 | @noindent | |
10526 | @kindex M | |
10527 | @kindex I M | |
10528 | @pindex calc-more-recursion-depth | |
10529 | @pindex calc-less-recursion-depth | |
10530 | @cindex Recursion depth | |
10531 | @cindex ``Computation got stuck'' message | |
10532 | @cindex @code{max-lisp-eval-depth} | |
10533 | @cindex @code{max-specpdl-size} | |
10534 | Calc uses recursion in many of its calculations. Emacs Lisp keeps a | |
10535 | variable @code{max-lisp-eval-depth} which limits the amount of recursion | |
10536 | possible in an attempt to recover from program bugs. If a calculation | |
10537 | ever halts incorrectly with the message ``Computation got stuck or | |
10538 | ran too long,'' use the @kbd{M} command (@code{calc-more-recursion-depth}) | |
10539 | to increase this limit. (Of course, this will not help if the | |
10540 | calculation really did get stuck due to some problem inside Calc.)@refill | |
10541 | ||
10542 | The limit is always increased (multiplied) by a factor of two. There | |
10543 | is also an @kbd{I M} (@code{calc-less-recursion-depth}) command which | |
10544 | decreases this limit by a factor of two, down to a minimum value of 200. | |
10545 | The default value is 1000. | |
10546 | ||
10547 | These commands also double or halve @code{max-specpdl-size}, another | |
10548 | internal Lisp recursion limit. The minimum value for this limit is 600. | |
10549 | ||
10550 | @node Caches, Debugging Calc, Recursion Depth, Troubleshooting Commands | |
10551 | @subsection Caches | |
10552 | ||
10553 | @noindent | |
10554 | @cindex Caches | |
10555 | @cindex Flushing caches | |
10556 | Calc saves certain values after they have been computed once. For | |
10557 | example, the @kbd{P} (@code{calc-pi}) command initially ``knows'' the | |
10558 | constant @c{$\pi$} | |
10559 | @cite{pi} to about 20 decimal places; if the current precision | |
10560 | is greater than this, it will recompute @c{$\pi$} | |
10561 | @cite{pi} using a series | |
10562 | approximation. This value will not need to be recomputed ever again | |
10563 | unless you raise the precision still further. Many operations such as | |
10564 | logarithms and sines make use of similarly cached values such as | |
10565 | @c{$\pi \over 4$} | |
10566 | @cite{pi/4} and @c{$\ln 2$} | |
10567 | @cite{ln(2)}. The visible effect of caching is that | |
10568 | high-precision computations may seem to do extra work the first time. | |
10569 | Other things cached include powers of two (for the binary arithmetic | |
10570 | functions), matrix inverses and determinants, symbolic integrals, and | |
10571 | data points computed by the graphing commands. | |
10572 | ||
10573 | @pindex calc-flush-caches | |
10574 | If you suspect a Calculator cache has become corrupt, you can use the | |
10575 | @code{calc-flush-caches} command to reset all caches to the empty state. | |
10576 | (This should only be necessary in the event of bugs in the Calculator.) | |
10577 | The @kbd{M-# 0} (with the zero key) command also resets caches along | |
10578 | with all other aspects of the Calculator's state. | |
10579 | ||
10580 | @node Debugging Calc, , Caches, Troubleshooting Commands | |
10581 | @subsection Debugging Calc | |
10582 | ||
10583 | @noindent | |
10584 | A few commands exist to help in the debugging of Calc commands. | |
10585 | @xref{Programming}, to see the various ways that you can write | |
10586 | your own Calc commands. | |
10587 | ||
10588 | @kindex Z T | |
10589 | @pindex calc-timing | |
10590 | The @kbd{Z T} (@code{calc-timing}) command turns on and off a mode | |
10591 | in which the timing of slow commands is reported in the Trail. | |
10592 | Any Calc command that takes two seconds or longer writes a line | |
10593 | to the Trail showing how many seconds it took. This value is | |
10594 | accurate only to within one second. | |
10595 | ||
10596 | All steps of executing a command are included; in particular, time | |
10597 | taken to format the result for display in the stack and trail is | |
10598 | counted. Some prompts also count time taken waiting for them to | |
10599 | be answered, while others do not; this depends on the exact | |
10600 | implementation of the command. For best results, if you are timing | |
10601 | a sequence that includes prompts or multiple commands, define a | |
10602 | keyboard macro to run the whole sequence at once. Calc's @kbd{X} | |
10603 | command (@pxref{Keyboard Macros}) will then report the time taken | |
10604 | to execute the whole macro. | |
10605 | ||
10606 | Another advantage of the @kbd{X} command is that while it is | |
10607 | executing, the stack and trail are not updated from step to step. | |
10608 | So if you expect the output of your test sequence to leave a result | |
10609 | that may take a long time to format and you don't wish to count | |
10610 | this formatting time, end your sequence with a @key{DEL} keystroke | |
10611 | to clear the result from the stack. When you run the sequence with | |
10612 | @kbd{X}, Calc will never bother to format the large result. | |
10613 | ||
10614 | Another thing @kbd{Z T} does is to increase the Emacs variable | |
10615 | @code{gc-cons-threshold} to a much higher value (two million; the | |
10616 | usual default in Calc is 250,000) for the duration of each command. | |
10617 | This generally prevents garbage collection during the timing of | |
10618 | the command, though it may cause your Emacs process to grow | |
10619 | abnormally large. (Garbage collection time is a major unpredictable | |
10620 | factor in the timing of Emacs operations.) | |
10621 | ||
10622 | Another command that is useful when debugging your own Lisp | |
10623 | extensions to Calc is @kbd{M-x calc-pass-errors}, which disables | |
10624 | the error handler that changes the ``@code{max-lisp-eval-depth} | |
10625 | exceeded'' message to the much more friendly ``Computation got | |
10626 | stuck or ran too long.'' This handler interferes with the Emacs | |
10627 | Lisp debugger's @code{debug-on-error} mode. Errors are reported | |
10628 | in the handler itself rather than at the true location of the | |
10629 | error. After you have executed @code{calc-pass-errors}, Lisp | |
10630 | errors will be reported correctly but the user-friendly message | |
10631 | will be lost. | |
10632 | ||
10633 | @node Data Types, Stack and Trail, Introduction, Top | |
10634 | @chapter Data Types | |
10635 | ||
10636 | @noindent | |
10637 | This chapter discusses the various types of objects that can be placed | |
10638 | on the Calculator stack, how they are displayed, and how they are | |
10639 | entered. (@xref{Data Type Formats}, for information on how these data | |
10640 | types are represented as underlying Lisp objects.)@refill | |
10641 | ||
10642 | Integers, fractions, and floats are various ways of describing real | |
10643 | numbers. HMS forms also for many purposes act as real numbers. These | |
10644 | types can be combined to form complex numbers, modulo forms, error forms, | |
10645 | or interval forms. (But these last four types cannot be combined | |
10646 | arbitrarily:@: error forms may not contain modulo forms, for example.) | |
10647 | Finally, all these types of numbers may be combined into vectors, | |
10648 | matrices, or algebraic formulas. | |
10649 | ||
10650 | @menu | |
10651 | * Integers:: The most basic data type. | |
10652 | * Fractions:: This and above are called @dfn{rationals}. | |
10653 | * Floats:: This and above are called @dfn{reals}. | |
10654 | * Complex Numbers:: This and above are called @dfn{numbers}. | |
10655 | * Infinities:: | |
10656 | * Vectors and Matrices:: | |
10657 | * Strings:: | |
10658 | * HMS Forms:: | |
10659 | * Date Forms:: | |
10660 | * Modulo Forms:: | |
10661 | * Error Forms:: | |
10662 | * Interval Forms:: | |
10663 | * Incomplete Objects:: | |
10664 | * Variables:: | |
10665 | * Formulas:: | |
10666 | @end menu | |
10667 | ||
10668 | @node Integers, Fractions, Data Types, Data Types | |
10669 | @section Integers | |
10670 | ||
10671 | @noindent | |
10672 | @cindex Integers | |
10673 | The Calculator stores integers to arbitrary precision. Addition, | |
10674 | subtraction, and multiplication of integers always yields an exact | |
10675 | integer result. (If the result of a division or exponentiation of | |
10676 | integers is not an integer, it is expressed in fractional or | |
10677 | floating-point form according to the current Fraction Mode. | |
10678 | @xref{Fraction Mode}.) | |
10679 | ||
10680 | A decimal integer is represented as an optional sign followed by a | |
10681 | sequence of digits. Grouping (@pxref{Grouping Digits}) can be used to | |
10682 | insert a comma at every third digit for display purposes, but you | |
10683 | must not type commas during the entry of numbers.@refill | |
10684 | ||
10685 | @kindex # | |
10686 | A non-decimal integer is represented as an optional sign, a radix | |
10687 | between 2 and 36, a @samp{#} symbol, and one or more digits. For radix 11 | |
10688 | and above, the letters A through Z (upper- or lower-case) count as | |
10689 | digits and do not terminate numeric entry mode. @xref{Radix Modes}, for how | |
10690 | to set the default radix for display of integers. Numbers of any radix | |
10691 | may be entered at any time. If you press @kbd{#} at the beginning of a | |
10692 | number, the current display radix is used.@refill | |
10693 | ||
10694 | @node Fractions, Floats, Integers, Data Types | |
10695 | @section Fractions | |
10696 | ||
10697 | @noindent | |
10698 | @cindex Fractions | |
10699 | A @dfn{fraction} is a ratio of two integers. Fractions are traditionally | |
10700 | written ``2/3'' but Calc uses the notation @samp{2:3}. (The @kbd{/} key | |
10701 | performs RPN division; the following two sequences push the number | |
10702 | @samp{2:3} on the stack: @kbd{2 :@: 3 @key{RET}}, or @kbd{2 @key{RET} 3 /} | |
10703 | assuming Fraction Mode has been enabled.) | |
10704 | When the Calculator produces a fractional result it always reduces it to | |
10705 | simplest form, which may in fact be an integer.@refill | |
10706 | ||
10707 | Fractions may also be entered in a three-part form, where @samp{2:3:4} | |
10708 | represents two-and-three-quarters. @xref{Fraction Formats}, for fraction | |
10709 | display formats.@refill | |
10710 | ||
10711 | Non-decimal fractions are entered and displayed as | |
10712 | @samp{@var{radix}#@var{num}:@var{denom}} (or in the analogous three-part | |
10713 | form). The numerator and denominator always use the same radix.@refill | |
10714 | ||
10715 | @node Floats, Complex Numbers, Fractions, Data Types | |
10716 | @section Floats | |
10717 | ||
10718 | @noindent | |
10719 | @cindex Floating-point numbers | |
10720 | A floating-point number or @dfn{float} is a number stored in scientific | |
10721 | notation. The number of significant digits in the fractional part is | |
10722 | governed by the current floating precision (@pxref{Precision}). The | |
10723 | range of acceptable values is from @c{$10^{-3999999}$} | |
10724 | @cite{10^-3999999} (inclusive) | |
10725 | to @c{$10^{4000000}$} | |
10726 | @cite{10^4000000} | |
10727 | (exclusive), plus the corresponding negative | |
10728 | values and zero. | |
10729 | ||
10730 | Calculations that would exceed the allowable range of values (such | |
10731 | as @samp{exp(exp(20))}) are left in symbolic form by Calc. The | |
10732 | messages ``floating-point overflow'' or ``floating-point underflow'' | |
10733 | indicate that during the calculation a number would have been produced | |
10734 | that was too large or too close to zero, respectively, to be represented | |
10735 | by Calc. This does not necessarily mean the final result would have | |
10736 | overflowed, just that an overflow occurred while computing the result. | |
10737 | (In fact, it could report an underflow even though the final result | |
10738 | would have overflowed!) | |
10739 | ||
10740 | If a rational number and a float are mixed in a calculation, the result | |
10741 | will in general be expressed as a float. Commands that require an integer | |
10742 | value (such as @kbd{k g} [@code{gcd}]) will also accept integer-valued | |
10743 | floats, i.e., floating-point numbers with nothing after the decimal point. | |
10744 | ||
10745 | Floats are identified by the presence of a decimal point and/or an | |
10746 | exponent. In general a float consists of an optional sign, digits | |
10747 | including an optional decimal point, and an optional exponent consisting | |
10748 | of an @samp{e}, an optional sign, and up to seven exponent digits. | |
10749 | For example, @samp{23.5e-2} is 23.5 times ten to the minus-second power, | |
10750 | or 0.235. | |
10751 | ||
10752 | Floating-point numbers are normally displayed in decimal notation with | |
10753 | all significant figures shown. Exceedingly large or small numbers are | |
10754 | displayed in scientific notation. Various other display options are | |
10755 | available. @xref{Float Formats}. | |
10756 | ||
10757 | @cindex Accuracy of calculations | |
10758 | Floating-point numbers are stored in decimal, not binary. The result | |
10759 | of each operation is rounded to the nearest value representable in the | |
10760 | number of significant digits specified by the current precision, | |
10761 | rounding away from zero in the case of a tie. Thus (in the default | |
10762 | display mode) what you see is exactly what you get. Some operations such | |
10763 | as square roots and transcendental functions are performed with several | |
10764 | digits of extra precision and then rounded down, in an effort to make the | |
10765 | final result accurate to the full requested precision. However, | |
10766 | accuracy is not rigorously guaranteed. If you suspect the validity of a | |
10767 | result, try doing the same calculation in a higher precision. The | |
10768 | Calculator's arithmetic is not intended to be IEEE-conformant in any | |
10769 | way.@refill | |
10770 | ||
10771 | While floats are always @emph{stored} in decimal, they can be entered | |
10772 | and displayed in any radix just like integers and fractions. The | |
10773 | notation @samp{@var{radix}#@var{ddd}.@var{ddd}} is a floating-point | |
10774 | number whose digits are in the specified radix. Note that the @samp{.} | |
10775 | is more aptly referred to as a ``radix point'' than as a decimal | |
10776 | point in this case. The number @samp{8#123.4567} is defined as | |
10777 | @samp{8#1234567 * 8^-4}. If the radix is 14 or less, you can use | |
10778 | @samp{e} notation to write a non-decimal number in scientific notation. | |
10779 | The exponent is written in decimal, and is considered to be a power | |
10780 | of the radix: @samp{8#1234567e-4}. If the radix is 15 or above, the | |
10781 | letter @samp{e} is a digit, so scientific notation must be written | |
10782 | out, e.g., @samp{16#123.4567*16^2}. The first two exercises of the | |
10783 | Modes Tutorial explore some of the properties of non-decimal floats. | |
10784 | ||
10785 | @node Complex Numbers, Infinities, Floats, Data Types | |
10786 | @section Complex Numbers | |
10787 | ||
10788 | @noindent | |
10789 | @cindex Complex numbers | |
10790 | There are two supported formats for complex numbers: rectangular and | |
10791 | polar. The default format is rectangular, displayed in the form | |
10792 | @samp{(@var{real},@var{imag})} where @var{real} is the real part and | |
10793 | @var{imag} is the imaginary part, each of which may be any real number. | |
10794 | Rectangular complex numbers can also be displayed in @samp{@var{a}+@var{b}i} | |
10795 | notation; @pxref{Complex Formats}.@refill | |
10796 | ||
10797 | Polar complex numbers are displayed in the form `@t{(}@var{r}@t{;}@c{$\theta$} | |
10798 | @var{theta}@t{)}' | |
10799 | where @var{r} is the nonnegative magnitude and @c{$\theta$} | |
10800 | @var{theta} is the argument | |
10801 | or phase angle. The range of @c{$\theta$} | |
10802 | @var{theta} depends on the current angular | |
10803 | mode (@pxref{Angular Modes}); it is generally between @i{-180} and | |
10804 | @i{+180} degrees or the equivalent range in radians.@refill | |
10805 | ||
10806 | Complex numbers are entered in stages using incomplete objects. | |
10807 | @xref{Incomplete Objects}. | |
10808 | ||
10809 | Operations on rectangular complex numbers yield rectangular complex | |
10810 | results, and similarly for polar complex numbers. Where the two types | |
10811 | are mixed, or where new complex numbers arise (as for the square root of | |
10812 | a negative real), the current @dfn{Polar Mode} is used to determine the | |
10813 | type. @xref{Polar Mode}. | |
10814 | ||
10815 | A complex result in which the imaginary part is zero (or the phase angle | |
10816 | is 0 or 180 degrees or @c{$\pi$} | |
10817 | @cite{pi} radians) is automatically converted to a real | |
10818 | number. | |
10819 | ||
10820 | @node Infinities, Vectors and Matrices, Complex Numbers, Data Types | |
10821 | @section Infinities | |
10822 | ||
10823 | @noindent | |
10824 | @cindex Infinity | |
10825 | @cindex @code{inf} variable | |
10826 | @cindex @code{uinf} variable | |
10827 | @cindex @code{nan} variable | |
10828 | @vindex inf | |
10829 | @vindex uinf | |
10830 | @vindex nan | |
10831 | The word @code{inf} represents the mathematical concept of @dfn{infinity}. | |
10832 | Calc actually has three slightly different infinity-like values: | |
10833 | @code{inf}, @code{uinf}, and @code{nan}. These are just regular | |
10834 | variable names (@pxref{Variables}); you should avoid using these | |
10835 | names for your own variables because Calc gives them special | |
10836 | treatment. Infinities, like all variable names, are normally | |
10837 | entered using algebraic entry. | |
10838 | ||
10839 | Mathematically speaking, it is not rigorously correct to treat | |
10840 | ``infinity'' as if it were a number, but mathematicians often do | |
10841 | so informally. When they say that @samp{1 / inf = 0}, what they | |
10842 | really mean is that @cite{1 / x}, as @cite{x} becomes larger and | |
10843 | larger, becomes arbitrarily close to zero. So you can imagine | |
10844 | that if @cite{x} got ``all the way to infinity,'' then @cite{1 / x} | |
10845 | would go all the way to zero. Similarly, when they say that | |
10846 | @samp{exp(inf) = inf}, they mean that @c{$e^x$} | |
10847 | @cite{exp(x)} grows without | |
10848 | bound as @cite{x} grows. The symbol @samp{-inf} likewise stands | |
10849 | for an infinitely negative real value; for example, we say that | |
10850 | @samp{exp(-inf) = 0}. You can have an infinity pointing in any | |
10851 | direction on the complex plane: @samp{sqrt(-inf) = i inf}. | |
10852 | ||
10853 | The same concept of limits can be used to define @cite{1 / 0}. We | |
10854 | really want the value that @cite{1 / x} approaches as @cite{x} | |
10855 | approaches zero. But if all we have is @cite{1 / 0}, we can't | |
10856 | tell which direction @cite{x} was coming from. If @cite{x} was | |
10857 | positive and decreasing toward zero, then we should say that | |
10858 | @samp{1 / 0 = inf}. But if @cite{x} was negative and increasing | |
10859 | toward zero, the answer is @samp{1 / 0 = -inf}. In fact, @cite{x} | |
10860 | could be an imaginary number, giving the answer @samp{i inf} or | |
10861 | @samp{-i inf}. Calc uses the special symbol @samp{uinf} to mean | |
10862 | @dfn{undirected infinity}, i.e., a value which is infinitely | |
10863 | large but with an unknown sign (or direction on the complex plane). | |
10864 | ||
10865 | Calc actually has three modes that say how infinities are handled. | |
10866 | Normally, infinities never arise from calculations that didn't | |
10867 | already have them. Thus, @cite{1 / 0} is treated simply as an | |
10868 | error and left unevaluated. The @kbd{m i} (@code{calc-infinite-mode}) | |
10869 | command (@pxref{Infinite Mode}) enables a mode in which | |
10870 | @cite{1 / 0} evaluates to @code{uinf} instead. There is also | |
10871 | an alternative type of infinite mode which says to treat zeros | |
10872 | as if they were positive, so that @samp{1 / 0 = inf}. While this | |
10873 | is less mathematically correct, it may be the answer you want in | |
10874 | some cases. | |
10875 | ||
10876 | Since all infinities are ``as large'' as all others, Calc simplifies, | |
10877 | e.g., @samp{5 inf} to @samp{inf}. Another example is | |
10878 | @samp{5 - inf = -inf}, where the @samp{-inf} is so large that | |
10879 | adding a finite number like five to it does not affect it. | |
10880 | Note that @samp{a - inf} also results in @samp{-inf}; Calc assumes | |
10881 | that variables like @code{a} always stand for finite quantities. | |
10882 | Just to show that infinities really are all the same size, | |
10883 | note that @samp{sqrt(inf) = inf^2 = exp(inf) = inf} in Calc's | |
10884 | notation. | |
10885 | ||
10886 | It's not so easy to define certain formulas like @samp{0 * inf} and | |
10887 | @samp{inf / inf}. Depending on where these zeros and infinities | |
10888 | came from, the answer could be literally anything. The latter | |
10889 | formula could be the limit of @cite{x / x} (giving a result of one), | |
10890 | or @cite{2 x / x} (giving two), or @cite{x^2 / x} (giving @code{inf}), | |
10891 | or @cite{x / x^2} (giving zero). Calc uses the symbol @code{nan} | |
10892 | to represent such an @dfn{indeterminate} value. (The name ``nan'' | |
10893 | comes from analogy with the ``NAN'' concept of IEEE standard | |
10894 | arithmetic; it stands for ``Not A Number.'' This is somewhat of a | |
10895 | misnomer, since @code{nan} @emph{does} stand for some number or | |
10896 | infinity, it's just that @emph{which} number it stands for | |
10897 | cannot be determined.) In Calc's notation, @samp{0 * inf = nan} | |
10898 | and @samp{inf / inf = nan}. A few other common indeterminate | |
10899 | expressions are @samp{inf - inf} and @samp{inf ^ 0}. Also, | |
10900 | @samp{0 / 0 = nan} if you have turned on ``infinite mode'' | |
10901 | (as described above). | |
10902 | ||
10903 | Infinities are especially useful as parts of @dfn{intervals}. | |
10904 | @xref{Interval Forms}. | |
10905 | ||
10906 | @node Vectors and Matrices, Strings, Infinities, Data Types | |
10907 | @section Vectors and Matrices | |
10908 | ||
10909 | @noindent | |
10910 | @cindex Vectors | |
10911 | @cindex Plain vectors | |
10912 | @cindex Matrices | |
10913 | The @dfn{vector} data type is flexible and general. A vector is simply a | |
10914 | list of zero or more data objects. When these objects are numbers, the | |
10915 | whole is a vector in the mathematical sense. When these objects are | |
10916 | themselves vectors of equal (nonzero) length, the whole is a @dfn{matrix}. | |
10917 | A vector which is not a matrix is referred to here as a @dfn{plain vector}. | |
10918 | ||
10919 | A vector is displayed as a list of values separated by commas and enclosed | |
10920 | in square brackets: @samp{[1, 2, 3]}. Thus the following is a 2 row by | |
10921 | 3 column matrix: @samp{[[1, 2, 3], [4, 5, 6]]}. Vectors, like complex | |
10922 | numbers, are entered as incomplete objects. @xref{Incomplete Objects}. | |
10923 | During algebraic entry, vectors are entered all at once in the usual | |
10924 | brackets-and-commas form. Matrices may be entered algebraically as nested | |
10925 | vectors, or using the shortcut notation @w{@samp{[1, 2, 3; 4, 5, 6]}}, | |
10926 | with rows separated by semicolons. The commas may usually be omitted | |
10927 | when entering vectors: @samp{[1 2 3]}. Curly braces may be used in | |
10928 | place of brackets: @samp{@{1, 2, 3@}}, but the commas are required in | |
10929 | this case. | |
10930 | ||
10931 | Traditional vector and matrix arithmetic is also supported; | |
10932 | @pxref{Basic Arithmetic} and @pxref{Matrix Functions}. | |
10933 | Many other operations are applied to vectors element-wise. For example, | |
10934 | the complex conjugate of a vector is a vector of the complex conjugates | |
10935 | of its elements.@refill | |
10936 | ||
5d67986c RS |
10937 | @ignore |
10938 | @starindex | |
10939 | @end ignore | |
d7b8e6c6 EZ |
10940 | @tindex vec |
10941 | Algebraic functions for building vectors include @samp{vec(a, b, c)} | |
10942 | to build @samp{[a, b, c]}, @samp{cvec(a, n, m)} to build an @c{$n\times m$} | |
10943 | @asis{@var{n}x@var{m}} | |
10944 | matrix of @samp{a}s, and @samp{index(n)} to build a vector of integers | |
10945 | from 1 to @samp{n}. | |
10946 | ||
10947 | @node Strings, HMS Forms, Vectors and Matrices, Data Types | |
10948 | @section Strings | |
10949 | ||
10950 | @noindent | |
10951 | @kindex " | |
10952 | @cindex Strings | |
10953 | @cindex Character strings | |
10954 | Character strings are not a special data type in the Calculator. | |
10955 | Rather, a string is represented simply as a vector all of whose | |
10956 | elements are integers in the range 0 to 255 (ASCII codes). You can | |
10957 | enter a string at any time by pressing the @kbd{"} key. Quotation | |
10958 | marks and backslashes are written @samp{\"} and @samp{\\}, respectively, | |
10959 | inside strings. Other notations introduced by backslashes are: | |
10960 | ||
d7b8e6c6 | 10961 | @example |
5d67986c | 10962 | @group |
d7b8e6c6 EZ |
10963 | \a 7 \^@@ 0 |
10964 | \b 8 \^a-z 1-26 | |
10965 | \e 27 \^[ 27 | |
10966 | \f 12 \^\\ 28 | |
10967 | \n 10 \^] 29 | |
10968 | \r 13 \^^ 30 | |
10969 | \t 9 \^_ 31 | |
10970 | \^? 127 | |
d7b8e6c6 | 10971 | @end group |
5d67986c | 10972 | @end example |
d7b8e6c6 EZ |
10973 | |
10974 | @noindent | |
10975 | Finally, a backslash followed by three octal digits produces any | |
10976 | character from its ASCII code. | |
10977 | ||
10978 | @kindex d " | |
10979 | @pindex calc-display-strings | |
10980 | Strings are normally displayed in vector-of-integers form. The | |
10981 | @w{@kbd{d "}} (@code{calc-display-strings}) command toggles a mode in | |
10982 | which any vectors of small integers are displayed as quoted strings | |
10983 | instead. | |
10984 | ||
10985 | The backslash notations shown above are also used for displaying | |
10986 | strings. Characters 128 and above are not translated by Calc; unless | |
10987 | you have an Emacs modified for 8-bit fonts, these will show up in | |
10988 | backslash-octal-digits notation. For characters below 32, and | |
10989 | for character 127, Calc uses the backslash-letter combination if | |
10990 | there is one, or otherwise uses a @samp{\^} sequence. | |
10991 | ||
10992 | The only Calc feature that uses strings is @dfn{compositions}; | |
10993 | @pxref{Compositions}. Strings also provide a convenient | |
10994 | way to do conversions between ASCII characters and integers. | |
10995 | ||
5d67986c RS |
10996 | @ignore |
10997 | @starindex | |
10998 | @end ignore | |
d7b8e6c6 EZ |
10999 | @tindex string |
11000 | There is a @code{string} function which provides a different display | |
11001 | format for strings. Basically, @samp{string(@var{s})}, where @var{s} | |
11002 | is a vector of integers in the proper range, is displayed as the | |
11003 | corresponding string of characters with no surrounding quotation | |
11004 | marks or other modifications. Thus @samp{string("ABC")} (or | |
11005 | @samp{string([65 66 67])}) will look like @samp{ABC} on the stack. | |
11006 | This happens regardless of whether @w{@kbd{d "}} has been used. The | |
11007 | only way to turn it off is to use @kbd{d U} (unformatted language | |
11008 | mode) which will display @samp{string("ABC")} instead. | |
11009 | ||
11010 | Control characters are displayed somewhat differently by @code{string}. | |
11011 | Characters below 32, and character 127, are shown using @samp{^} notation | |
11012 | (same as shown above, but without the backslash). The quote and | |
11013 | backslash characters are left alone, as are characters 128 and above. | |
11014 | ||
5d67986c RS |
11015 | @ignore |
11016 | @starindex | |
11017 | @end ignore | |
d7b8e6c6 EZ |
11018 | @tindex bstring |
11019 | The @code{bstring} function is just like @code{string} except that | |
11020 | the resulting string is breakable across multiple lines if it doesn't | |
11021 | fit all on one line. Potential break points occur at every space | |
11022 | character in the string. | |
11023 | ||
11024 | @node HMS Forms, Date Forms, Strings, Data Types | |
11025 | @section HMS Forms | |
11026 | ||
11027 | @noindent | |
11028 | @cindex Hours-minutes-seconds forms | |
11029 | @cindex Degrees-minutes-seconds forms | |
11030 | @dfn{HMS} stands for Hours-Minutes-Seconds; when used as an angular | |
11031 | argument, the interpretation is Degrees-Minutes-Seconds. All functions | |
11032 | that operate on angles accept HMS forms. These are interpreted as | |
11033 | degrees regardless of the current angular mode. It is also possible to | |
11034 | use HMS as the angular mode so that calculated angles are expressed in | |
11035 | degrees, minutes, and seconds. | |
11036 | ||
11037 | @kindex @@ | |
5d67986c RS |
11038 | @ignore |
11039 | @mindex @null | |
11040 | @end ignore | |
d7b8e6c6 | 11041 | @kindex ' (HMS forms) |
5d67986c RS |
11042 | @ignore |
11043 | @mindex @null | |
11044 | @end ignore | |
d7b8e6c6 | 11045 | @kindex " (HMS forms) |
5d67986c RS |
11046 | @ignore |
11047 | @mindex @null | |
11048 | @end ignore | |
d7b8e6c6 | 11049 | @kindex h (HMS forms) |
5d67986c RS |
11050 | @ignore |
11051 | @mindex @null | |
11052 | @end ignore | |
d7b8e6c6 | 11053 | @kindex o (HMS forms) |
5d67986c RS |
11054 | @ignore |
11055 | @mindex @null | |
11056 | @end ignore | |
d7b8e6c6 | 11057 | @kindex m (HMS forms) |
5d67986c RS |
11058 | @ignore |
11059 | @mindex @null | |
11060 | @end ignore | |
d7b8e6c6 EZ |
11061 | @kindex s (HMS forms) |
11062 | The default format for HMS values is | |
11063 | @samp{@var{hours}@@ @var{mins}' @var{secs}"}. During entry, the letters | |
11064 | @samp{h} (for ``hours'') or | |
11065 | @samp{o} (approximating the ``degrees'' symbol) are accepted as well as | |
11066 | @samp{@@}, @samp{m} is accepted in place of @samp{'}, and @samp{s} is | |
11067 | accepted in place of @samp{"}. | |
11068 | The @var{hours} value is an integer (or integer-valued float). | |
11069 | The @var{mins} value is an integer or integer-valued float between 0 and 59. | |
11070 | The @var{secs} value is a real number between 0 (inclusive) and 60 | |
11071 | (exclusive). A positive HMS form is interpreted as @var{hours} + | |
11072 | @var{mins}/60 + @var{secs}/3600. A negative HMS form is interpreted | |
11073 | as @i{- @var{hours}} @i{-} @var{mins}/60 @i{-} @var{secs}/3600. | |
11074 | Display format for HMS forms is quite flexible. @xref{HMS Formats}.@refill | |
11075 | ||
11076 | HMS forms can be added and subtracted. When they are added to numbers, | |
11077 | the numbers are interpreted according to the current angular mode. HMS | |
11078 | forms can also be multiplied and divided by real numbers. Dividing | |
11079 | two HMS forms produces a real-valued ratio of the two angles. | |
11080 | ||
11081 | @pindex calc-time | |
11082 | @cindex Time of day | |
11083 | Just for kicks, @kbd{M-x calc-time} pushes the current time of day on | |
11084 | the stack as an HMS form. | |
11085 | ||
11086 | @node Date Forms, Modulo Forms, HMS Forms, Data Types | |
11087 | @section Date Forms | |
11088 | ||
11089 | @noindent | |
11090 | @cindex Date forms | |
11091 | A @dfn{date form} represents a date and possibly an associated time. | |
11092 | Simple date arithmetic is supported: Adding a number to a date | |
11093 | produces a new date shifted by that many days; adding an HMS form to | |
11094 | a date shifts it by that many hours. Subtracting two date forms | |
11095 | computes the number of days between them (represented as a simple | |
11096 | number). Many other operations, such as multiplying two date forms, | |
11097 | are nonsensical and are not allowed by Calc. | |
11098 | ||
11099 | Date forms are entered and displayed enclosed in @samp{< >} brackets. | |
11100 | The default format is, e.g., @samp{<Wed Jan 9, 1991>} for dates, | |
11101 | or @samp{<3:32:20pm Wed Jan 9, 1991>} for dates with times. | |
11102 | Input is flexible; date forms can be entered in any of the usual | |
11103 | notations for dates and times. @xref{Date Formats}. | |
11104 | ||
11105 | Date forms are stored internally as numbers, specifically the number | |
11106 | of days since midnight on the morning of January 1 of the year 1 AD. | |
11107 | If the internal number is an integer, the form represents a date only; | |
11108 | if the internal number is a fraction or float, the form represents | |
11109 | a date and time. For example, @samp{<6:00am Wed Jan 9, 1991>} | |
11110 | is represented by the number 726842.25. The standard precision of | |
11111 | 12 decimal digits is enough to ensure that a (reasonable) date and | |
11112 | time can be stored without roundoff error. | |
11113 | ||
11114 | If the current precision is greater than 12, date forms will keep | |
11115 | additional digits in the seconds position. For example, if the | |
11116 | precision is 15, the seconds will keep three digits after the | |
11117 | decimal point. Decreasing the precision below 12 may cause the | |
11118 | time part of a date form to become inaccurate. This can also happen | |
11119 | if astronomically high years are used, though this will not be an | |
28665d46 | 11120 | issue in everyday (or even everymillennium) use. Note that date |
d7b8e6c6 EZ |
11121 | forms without times are stored as exact integers, so roundoff is |
11122 | never an issue for them. | |
11123 | ||
11124 | You can use the @kbd{v p} (@code{calc-pack}) and @kbd{v u} | |
11125 | (@code{calc-unpack}) commands to get at the numerical representation | |
11126 | of a date form. @xref{Packing and Unpacking}. | |
11127 | ||
11128 | Date forms can go arbitrarily far into the future or past. Negative | |
11129 | year numbers represent years BC. Calc uses a combination of the | |
11130 | Gregorian and Julian calendars, following the history of Great | |
11131 | Britain and the British colonies. This is the same calendar that | |
11132 | is used by the @code{cal} program in most Unix implementations. | |
11133 | ||
11134 | @cindex Julian calendar | |
11135 | @cindex Gregorian calendar | |
11136 | Some historical background: The Julian calendar was created by | |
11137 | Julius Caesar in the year 46 BC as an attempt to fix the gradual | |
11138 | drift caused by the lack of leap years in the calendar used | |
11139 | until that time. The Julian calendar introduced an extra day in | |
11140 | all years divisible by four. After some initial confusion, the | |
11141 | calendar was adopted around the year we call 8 AD. Some centuries | |
11142 | later it became apparent that the Julian year of 365.25 days was | |
11143 | itself not quite right. In 1582 Pope Gregory XIII introduced the | |
11144 | Gregorian calendar, which added the new rule that years divisible | |
11145 | by 100, but not by 400, were not to be considered leap years | |
11146 | despite being divisible by four. Many countries delayed adoption | |
11147 | of the Gregorian calendar because of religious differences; | |
11148 | in Britain it was put off until the year 1752, by which time | |
11149 | the Julian calendar had fallen eleven days behind the true | |
11150 | seasons. So the switch to the Gregorian calendar in early | |
11151 | September 1752 introduced a discontinuity: The day after | |
11152 | Sep 2, 1752 is Sep 14, 1752. Calc follows this convention. | |
11153 | To take another example, Russia waited until 1918 before | |
11154 | adopting the new calendar, and thus needed to remove thirteen | |
11155 | days (between Feb 1, 1918 and Feb 14, 1918). This means that | |
11156 | Calc's reckoning will be inconsistent with Russian history between | |
11157 | 1752 and 1918, and similarly for various other countries. | |
11158 | ||
11159 | Today's timekeepers introduce an occasional ``leap second'' as | |
11160 | well, but Calc does not take these minor effects into account. | |
11161 | (If it did, it would have to report a non-integer number of days | |
11162 | between, say, @samp{<12:00am Mon Jan 1, 1900>} and | |
11163 | @samp{<12:00am Sat Jan 1, 2000>}.) | |
11164 | ||
11165 | Calc uses the Julian calendar for all dates before the year 1752, | |
11166 | including dates BC when the Julian calendar technically had not | |
11167 | yet been invented. Thus the claim that day number @i{-10000} is | |
11168 | called ``August 16, 28 BC'' should be taken with a grain of salt. | |
11169 | ||
11170 | Please note that there is no ``year 0''; the day before | |
11171 | @samp{<Sat Jan 1, +1>} is @samp{<Fri Dec 31, -1>}. These are | |
11172 | days 0 and @i{-1} respectively in Calc's internal numbering scheme. | |
11173 | ||
11174 | @cindex Julian day counting | |
11175 | Another day counting system in common use is, confusingly, also | |
11176 | called ``Julian.'' It was invented in 1583 by Joseph Justus | |
11177 | Scaliger, who named it in honor of his father Julius Caesar | |
11178 | Scaliger. For obscure reasons he chose to start his day | |
11179 | numbering on Jan 1, 4713 BC at noon, which in Calc's scheme | |
11180 | is @i{-1721423.5} (recall that Calc starts at midnight instead | |
11181 | of noon). Thus to convert a Calc date code obtained by | |
11182 | unpacking a date form into a Julian day number, simply add | |
11183 | 1721423.5. The Julian code for @samp{6:00am Jan 9, 1991} | |
11184 | is 2448265.75. The built-in @kbd{t J} command performs | |
11185 | this conversion for you. | |
11186 | ||
11187 | @cindex Unix time format | |
11188 | The Unix operating system measures time as an integer number of | |
11189 | seconds since midnight, Jan 1, 1970. To convert a Calc date | |
11190 | value into a Unix time stamp, first subtract 719164 (the code | |
11191 | for @samp{<Jan 1, 1970>}), then multiply by 86400 (the number of | |
11192 | seconds in a day) and press @kbd{R} to round to the nearest | |
11193 | integer. If you have a date form, you can simply subtract the | |
11194 | day @samp{<Jan 1, 1970>} instead of unpacking and subtracting | |
11195 | 719164. Likewise, divide by 86400 and add @samp{<Jan 1, 1970>} | |
11196 | to convert from Unix time to a Calc date form. (Note that | |
11197 | Unix normally maintains the time in the GMT time zone; you may | |
11198 | need to subtract five hours to get New York time, or eight hours | |
11199 | for California time. The same is usually true of Julian day | |
11200 | counts.) The built-in @kbd{t U} command performs these | |
11201 | conversions. | |
11202 | ||
11203 | @node Modulo Forms, Error Forms, Date Forms, Data Types | |
11204 | @section Modulo Forms | |
11205 | ||
11206 | @noindent | |
11207 | @cindex Modulo forms | |
11208 | A @dfn{modulo form} is a real number which is taken modulo (i.e., within | |
5d67986c | 11209 | an integer multiple of) some value @var{M}. Arithmetic modulo @var{M} |
d7b8e6c6 | 11210 | often arises in number theory. Modulo forms are written |
5d67986c RS |
11211 | `@var{a} @t{mod} @var{M}', |
11212 | where @var{a} and @var{M} are real numbers or HMS forms, and | |
d7b8e6c6 EZ |
11213 | @c{$0 \le a < M$} |
11214 | @cite{0 <= a < @var{M}}. | |
11215 | In many applications @cite{a} and @cite{M} will be | |
11216 | integers but this is not required.@refill | |
11217 | ||
11218 | Modulo forms are not to be confused with the modulo operator @samp{%}. | |
11219 | The expression @samp{27 % 10} means to compute 27 modulo 10 to produce | |
11220 | the result 7. Further computations treat this 7 as just a regular integer. | |
11221 | The expression @samp{27 mod 10} produces the result @samp{7 mod 10}; | |
11222 | further computations with this value are again reduced modulo 10 so that | |
11223 | the result always lies in the desired range. | |
11224 | ||
11225 | When two modulo forms with identical @cite{M}'s are added or multiplied, | |
11226 | the Calculator simply adds or multiplies the values, then reduces modulo | |
11227 | @cite{M}. If one argument is a modulo form and the other a plain number, | |
11228 | the plain number is treated like a compatible modulo form. It is also | |
11229 | possible to raise modulo forms to powers; the result is the value raised | |
11230 | to the power, then reduced modulo @cite{M}. (When all values involved | |
11231 | are integers, this calculation is done much more efficiently than | |
11232 | actually computing the power and then reducing.) | |
11233 | ||
11234 | @cindex Modulo division | |
5d67986c | 11235 | Two modulo forms `@var{a} @t{mod} @var{M}' and `@var{b} @t{mod} @var{M}' |
d7b8e6c6 EZ |
11236 | can be divided if @cite{a}, @cite{b}, and @cite{M} are all |
11237 | integers. The result is the modulo form which, when multiplied by | |
5d67986c | 11238 | `@var{b} @t{mod} @var{M}', produces `@var{a} @t{mod} @var{M}'. If |
d7b8e6c6 EZ |
11239 | there is no solution to this equation (which can happen only when |
11240 | @cite{M} is non-prime), or if any of the arguments are non-integers, the | |
11241 | division is left in symbolic form. Other operations, such as square | |
11242 | roots, are not yet supported for modulo forms. (Note that, although | |
5d67986c | 11243 | @w{`@t{(}@var{a} @t{mod} @var{M}@t{)^.5}'} will compute a ``modulo square root'' |
d7b8e6c6 EZ |
11244 | in the sense of reducing @c{$\sqrt a$} |
11245 | @cite{sqrt(a)} modulo @cite{M}, this is not a | |
11246 | useful definition from the number-theoretical point of view.)@refill | |
11247 | ||
5d67986c RS |
11248 | @ignore |
11249 | @mindex M | |
11250 | @end ignore | |
d7b8e6c6 | 11251 | @kindex M (modulo forms) |
5d67986c RS |
11252 | @ignore |
11253 | @mindex mod | |
11254 | @end ignore | |
d7b8e6c6 EZ |
11255 | @tindex mod (operator) |
11256 | To create a modulo form during numeric entry, press the shift-@kbd{M} | |
11257 | key to enter the word @samp{mod}. As a special convenience, pressing | |
11258 | shift-@kbd{M} a second time automatically enters the value of @cite{M} | |
11259 | that was most recently used before. During algebraic entry, either | |
11260 | type @samp{mod} by hand or press @kbd{M-m} (that's @kbd{@key{META}-m}). | |
11261 | Once again, pressing this a second time enters the current modulo.@refill | |
11262 | ||
11263 | You can also use @kbd{v p} and @kbd{%} to modify modulo forms. | |
11264 | @xref{Building Vectors}. @xref{Basic Arithmetic}. | |
11265 | ||
11266 | It is possible to mix HMS forms and modulo forms. For example, an | |
11267 | HMS form modulo 24 could be used to manipulate clock times; an HMS | |
11268 | form modulo 360 would be suitable for angles. Making the modulo @cite{M} | |
11269 | also be an HMS form eliminates troubles that would arise if the angular | |
11270 | mode were inadvertently set to Radians, in which case | |
11271 | @w{@samp{2@@ 0' 0" mod 24}} would be interpreted as two degrees modulo | |
11272 | 24 radians! | |
11273 | ||
11274 | Modulo forms cannot have variables or formulas for components. If you | |
11275 | enter the formula @samp{(x + 2) mod 5}, Calc propagates the modulus | |
11276 | to each of the coefficients: @samp{(1 mod 5) x + (2 mod 5)}. | |
11277 | ||
5d67986c RS |
11278 | @ignore |
11279 | @starindex | |
11280 | @end ignore | |
d7b8e6c6 EZ |
11281 | @tindex makemod |
11282 | The algebraic function @samp{makemod(a, m)} builds the modulo form | |
11283 | @w{@samp{a mod m}}. | |
11284 | ||
11285 | @node Error Forms, Interval Forms, Modulo Forms, Data Types | |
11286 | @section Error Forms | |
11287 | ||
11288 | @noindent | |
11289 | @cindex Error forms | |
11290 | @cindex Standard deviations | |
11291 | An @dfn{error form} is a number with an associated standard | |
11292 | deviation, as in @samp{2.3 +/- 0.12}. The notation | |
5d67986c | 11293 | `@var{x} @t{+/-} @c{$\sigma$} |
d7b8e6c6 EZ |
11294 | @asis{sigma}' stands for an uncertain value which follows a normal or |
11295 | Gaussian distribution of mean @cite{x} and standard deviation or | |
11296 | ``error'' @c{$\sigma$} | |
11297 | @cite{sigma}. Both the mean and the error can be either numbers or | |
11298 | formulas. Generally these are real numbers but the mean may also be | |
11299 | complex. If the error is negative or complex, it is changed to its | |
11300 | absolute value. An error form with zero error is converted to a | |
11301 | regular number by the Calculator.@refill | |
11302 | ||
11303 | All arithmetic and transcendental functions accept error forms as input. | |
11304 | Operations on the mean-value part work just like operations on regular | |
11305 | numbers. The error part for any function @cite{f(x)} (such as @c{$\sin x$} | |
11306 | @cite{sin(x)}) | |
11307 | is defined by the error of @cite{x} times the derivative of @cite{f} | |
11308 | evaluated at the mean value of @cite{x}. For a two-argument function | |
11309 | @cite{f(x,y)} (such as addition) the error is the square root of the sum | |
11310 | of the squares of the errors due to @cite{x} and @cite{y}. | |
11311 | @tex | |
11312 | $$ \eqalign{ | |
11313 | f(x \hbox{\code{ +/- }} \sigma) | |
11314 | &= f(x) \hbox{\code{ +/- }} \sigma \left| {df(x) \over dx} \right| \cr | |
11315 | f(x \hbox{\code{ +/- }} \sigma_x, y \hbox{\code{ +/- }} \sigma_y) | |
11316 | &= f(x,y) \hbox{\code{ +/- }} | |
11317 | \sqrt{\left(\sigma_x \left| {\partial f(x,y) \over \partial x} | |
11318 | \right| \right)^2 | |
11319 | +\left(\sigma_y \left| {\partial f(x,y) \over \partial y} | |
11320 | \right| \right)^2 } \cr | |
11321 | } $$ | |
11322 | @end tex | |
11323 | Note that this | |
11324 | definition assumes the errors in @cite{x} and @cite{y} are uncorrelated. | |
11325 | A side effect of this definition is that @samp{(2 +/- 1) * (2 +/- 1)} | |
11326 | is not the same as @samp{(2 +/- 1)^2}; the former represents the product | |
11327 | of two independent values which happen to have the same probability | |
11328 | distributions, and the latter is the product of one random value with itself. | |
11329 | The former will produce an answer with less error, since on the average | |
11330 | the two independent errors can be expected to cancel out.@refill | |
11331 | ||
11332 | Consult a good text on error analysis for a discussion of the proper use | |
11333 | of standard deviations. Actual errors often are neither Gaussian-distributed | |
11334 | nor uncorrelated, and the above formulas are valid only when errors | |
11335 | are small. As an example, the error arising from | |
5d67986c RS |
11336 | `@t{sin(}@var{x} @t{+/-} @c{$\sigma$} |
11337 | @var{sigma}@t{)}' is | |
d7b8e6c6 | 11338 | `@c{$\sigma$\nobreak} |
5d67986c | 11339 | @var{sigma} @t{abs(cos(}@var{x}@t{))}'. When @cite{x} is close to zero, |
d7b8e6c6 EZ |
11340 | @c{$\cos x$} |
11341 | @cite{cos(x)} is | |
11342 | close to one so the error in the sine is close to @c{$\sigma$} | |
11343 | @cite{sigma}; this makes sense, since @c{$\sin x$} | |
11344 | @cite{sin(x)} is approximately @cite{x} near zero, so a given | |
11345 | error in @cite{x} will produce about the same error in the sine. Likewise, | |
11346 | near 90 degrees @c{$\cos x$} | |
11347 | @cite{cos(x)} is nearly zero and so the computed error is | |
11348 | small: The sine curve is nearly flat in that region, so an error in @cite{x} | |
11349 | has relatively little effect on the value of @c{$\sin x$} | |
11350 | @cite{sin(x)}. However, consider | |
11351 | @samp{sin(90 +/- 1000)}. The cosine of 90 is zero, so Calc will report | |
11352 | zero error! We get an obviously wrong result because we have violated | |
11353 | the small-error approximation underlying the error analysis. If the error | |
11354 | in @cite{x} had been small, the error in @c{$\sin x$} | |
11355 | @cite{sin(x)} would indeed have been negligible.@refill | |
11356 | ||
5d67986c RS |
11357 | @ignore |
11358 | @mindex p | |
11359 | @end ignore | |
d7b8e6c6 EZ |
11360 | @kindex p (error forms) |
11361 | @tindex +/- | |
11362 | To enter an error form during regular numeric entry, use the @kbd{p} | |
11363 | (``plus-or-minus'') key to type the @samp{+/-} symbol. (If you try actually | |
11364 | typing @samp{+/-} the @kbd{+} key will be interpreted as the Calculator's | |
11365 | @kbd{+} command!) Within an algebraic formula, you can press @kbd{M-p} to | |
11366 | type the @samp{+/-} symbol, or type it out by hand. | |
11367 | ||
11368 | Error forms and complex numbers can be mixed; the formulas shown above | |
11369 | are used for complex numbers, too; note that if the error part evaluates | |
11370 | to a complex number its absolute value (or the square root of the sum of | |
11371 | the squares of the absolute values of the two error contributions) is | |
11372 | used. Mathematically, this corresponds to a radially symmetric Gaussian | |
11373 | distribution of numbers on the complex plane. However, note that Calc | |
11374 | considers an error form with real components to represent a real number, | |
11375 | not a complex distribution around a real mean. | |
11376 | ||
11377 | Error forms may also be composed of HMS forms. For best results, both | |
11378 | the mean and the error should be HMS forms if either one is. | |
11379 | ||
5d67986c RS |
11380 | @ignore |
11381 | @starindex | |
11382 | @end ignore | |
d7b8e6c6 EZ |
11383 | @tindex sdev |
11384 | The algebraic function @samp{sdev(a, b)} builds the error form @samp{a +/- b}. | |
11385 | ||
11386 | @node Interval Forms, Incomplete Objects, Error Forms, Data Types | |
11387 | @section Interval Forms | |
11388 | ||
11389 | @noindent | |
11390 | @cindex Interval forms | |
11391 | An @dfn{interval} is a subset of consecutive real numbers. For example, | |
11392 | the interval @samp{[2 ..@: 4]} represents all the numbers from 2 to 4, | |
11393 | inclusive. If you multiply it by the interval @samp{[0.5 ..@: 2]} you | |
11394 | obtain @samp{[1 ..@: 8]}. This calculation represents the fact that if | |
11395 | you multiply some number in the range @samp{[2 ..@: 4]} by some other | |
11396 | number in the range @samp{[0.5 ..@: 2]}, your result will lie in the range | |
11397 | from 1 to 8. Interval arithmetic is used to get a worst-case estimate | |
11398 | of the possible range of values a computation will produce, given the | |
11399 | set of possible values of the input. | |
11400 | ||
11401 | @ifinfo | |
11402 | Calc supports several varieties of intervals, including @dfn{closed} | |
11403 | intervals of the type shown above, @dfn{open} intervals such as | |
11404 | @samp{(2 ..@: 4)}, which represents the range of numbers from 2 to 4 | |
11405 | @emph{exclusive}, and @dfn{semi-open} intervals in which one end | |
11406 | uses a round parenthesis and the other a square bracket. In mathematical | |
11407 | terms, | |
11408 | @samp{[2 ..@: 4]} means @cite{2 <= x <= 4}, whereas | |
11409 | @samp{[2 ..@: 4)} represents @cite{2 <= x < 4}, | |
11410 | @samp{(2 ..@: 4]} represents @cite{2 < x <= 4}, and | |
11411 | @samp{(2 ..@: 4)} represents @cite{2 < x < 4}.@refill | |
11412 | @end ifinfo | |
11413 | @tex | |
11414 | Calc supports several varieties of intervals, including \dfn{closed} | |
11415 | intervals of the type shown above, \dfn{open} intervals such as | |
11416 | \samp{(2 ..\: 4)}, which represents the range of numbers from 2 to 4 | |
11417 | \emph{exclusive}, and \dfn{semi-open} intervals in which one end | |
11418 | uses a round parenthesis and the other a square bracket. In mathematical | |
11419 | terms, | |
11420 | $$ \eqalign{ | |
11421 | [2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 \le x \le 4 \cr | |
11422 | [2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 \le x < 4 \cr | |
11423 | (2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 < x \le 4 \cr | |
11424 | (2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 < x < 4 \cr | |
11425 | } $$ | |
11426 | @end tex | |
11427 | ||
11428 | The lower and upper limits of an interval must be either real numbers | |
11429 | (or HMS or date forms), or symbolic expressions which are assumed to be | |
11430 | real-valued, or @samp{-inf} and @samp{inf}. In general the lower limit | |
11431 | must be less than the upper limit. A closed interval containing only | |
11432 | one value, @samp{[3 ..@: 3]}, is converted to a plain number (3) | |
11433 | automatically. An interval containing no values at all (such as | |
11434 | @samp{[3 ..@: 2]} or @samp{[2 ..@: 2)}) can be represented but is not | |
11435 | guaranteed to behave well when used in arithmetic. Note that the | |
11436 | interval @samp{[3 .. inf)} represents all real numbers greater than | |
11437 | or equal to 3, and @samp{(-inf .. inf)} represents all real numbers. | |
11438 | In fact, @samp{[-inf .. inf]} represents all real numbers including | |
11439 | the real infinities. | |
11440 | ||
11441 | Intervals are entered in the notation shown here, either as algebraic | |
11442 | formulas, or using incomplete forms. (@xref{Incomplete Objects}.) | |
11443 | In algebraic formulas, multiple periods in a row are collected from | |
11444 | left to right, so that @samp{1...1e2} is interpreted as @samp{1.0 ..@: 1e2} | |
11445 | rather than @samp{1 ..@: 0.1e2}. Add spaces or zeros if you want to | |
11446 | get the other interpretation. If you omit the lower or upper limit, | |
11447 | a default of @samp{-inf} or @samp{inf} (respectively) is furnished. | |
11448 | ||
11449 | ``Infinite mode'' also affects operations on intervals | |
11450 | (@pxref{Infinities}). Calc will always introduce an open infinity, | |
11451 | as in @samp{1 / (0 .. 2] = [0.5 .. inf)}. But closed infinities, | |
11452 | @w{@samp{1 / [0 .. 2] = [0.5 .. inf]}}, arise only in infinite mode; | |
11453 | otherwise they are left unevaluated. Note that the ``direction'' of | |
11454 | a zero is not an issue in this case since the zero is always assumed | |
11455 | to be continuous with the rest of the interval. For intervals that | |
11456 | contain zero inside them Calc is forced to give the result, | |
11457 | @samp{1 / (-2 .. 2) = [-inf .. inf]}. | |
11458 | ||
11459 | While it may seem that intervals and error forms are similar, they are | |
11460 | based on entirely different concepts of inexact quantities. An error | |
5d67986c RS |
11461 | form `@var{x} @t{+/-} @c{$\sigma$} |
11462 | @var{sigma}' means a variable is random, and its value could | |
d7b8e6c6 | 11463 | be anything but is ``probably'' within one @c{$\sigma$} |
5d67986c RS |
11464 | @var{sigma} of the mean value @cite{x}. |
11465 | An interval `@t{[}@var{a} @t{..@:} @var{b}@t{]}' means a variable's value | |
d7b8e6c6 EZ |
11466 | is unknown, but guaranteed to lie in the specified range. Error forms |
11467 | are statistical or ``average case'' approximations; interval arithmetic | |
11468 | tends to produce ``worst case'' bounds on an answer.@refill | |
11469 | ||
11470 | Intervals may not contain complex numbers, but they may contain | |
11471 | HMS forms or date forms. | |
11472 | ||
11473 | @xref{Set Operations}, for commands that interpret interval forms | |
11474 | as subsets of the set of real numbers. | |
11475 | ||
5d67986c RS |
11476 | @ignore |
11477 | @starindex | |
11478 | @end ignore | |
d7b8e6c6 EZ |
11479 | @tindex intv |
11480 | The algebraic function @samp{intv(n, a, b)} builds an interval form | |
11481 | from @samp{a} to @samp{b}; @samp{n} is an integer code which must | |
11482 | be 0 for @samp{(..)}, 1 for @samp{(..]}, 2 for @samp{[..)}, or | |
11483 | 3 for @samp{[..]}. | |
11484 | ||
11485 | Please note that in fully rigorous interval arithmetic, care would be | |
11486 | taken to make sure that the computation of the lower bound rounds toward | |
11487 | minus infinity, while upper bound computations round toward plus | |
11488 | infinity. Calc's arithmetic always uses a round-to-nearest mode, | |
11489 | which means that roundoff errors could creep into an interval | |
11490 | calculation to produce intervals slightly smaller than they ought to | |
11491 | be. For example, entering @samp{[1..2]} and pressing @kbd{Q 2 ^} | |
11492 | should yield the interval @samp{[1..2]} again, but in fact it yields the | |
11493 | (slightly too small) interval @samp{[1..1.9999999]} due to roundoff | |
11494 | error. | |
11495 | ||
11496 | @node Incomplete Objects, Variables, Interval Forms, Data Types | |
11497 | @section Incomplete Objects | |
11498 | ||
11499 | @noindent | |
5d67986c RS |
11500 | @ignore |
11501 | @mindex [ ] | |
11502 | @end ignore | |
d7b8e6c6 | 11503 | @kindex [ |
5d67986c RS |
11504 | @ignore |
11505 | @mindex ( ) | |
11506 | @end ignore | |
d7b8e6c6 EZ |
11507 | @kindex ( |
11508 | @kindex , | |
5d67986c RS |
11509 | @ignore |
11510 | @mindex @null | |
11511 | @end ignore | |
d7b8e6c6 | 11512 | @kindex ] |
5d67986c RS |
11513 | @ignore |
11514 | @mindex @null | |
11515 | @end ignore | |
d7b8e6c6 EZ |
11516 | @kindex ) |
11517 | @cindex Incomplete vectors | |
11518 | @cindex Incomplete complex numbers | |
11519 | @cindex Incomplete interval forms | |
11520 | When @kbd{(} or @kbd{[} is typed to begin entering a complex number or | |
11521 | vector, respectively, the effect is to push an @dfn{incomplete} complex | |
11522 | number or vector onto the stack. The @kbd{,} key adds the value(s) at | |
11523 | the top of the stack onto the current incomplete object. The @kbd{)} | |
11524 | and @kbd{]} keys ``close'' the incomplete object after adding any values | |
11525 | on the top of the stack in front of the incomplete object. | |
11526 | ||
11527 | As a result, the sequence of keystrokes @kbd{[ 2 , 3 @key{RET} 2 * , 9 ]} | |
11528 | pushes the vector @samp{[2, 6, 9]} onto the stack. Likewise, @kbd{( 1 , 2 Q )} | |
11529 | pushes the complex number @samp{(1, 1.414)} (approximately). | |
11530 | ||
11531 | If several values lie on the stack in front of the incomplete object, | |
11532 | all are collected and appended to the object. Thus the @kbd{,} key | |
11533 | is redundant: @kbd{[ 2 @key{RET} 3 @key{RET} 2 * 9 ]}. Some people | |
11534 | prefer the equivalent @key{SPC} key to @key{RET}.@refill | |
11535 | ||
11536 | As a special case, typing @kbd{,} immediately after @kbd{(}, @kbd{[}, or | |
11537 | @kbd{,} adds a zero or duplicates the preceding value in the list being | |
11538 | formed. Typing @key{DEL} during incomplete entry removes the last item | |
11539 | from the list. | |
11540 | ||
11541 | @kindex ; | |
11542 | The @kbd{;} key is used in the same way as @kbd{,} to create polar complex | |
11543 | numbers: @kbd{( 1 ; 2 )}. When entering a vector, @kbd{;} is useful for | |
11544 | creating a matrix. In particular, @kbd{[ [ 1 , 2 ; 3 , 4 ; 5 , 6 ] ]} is | |
11545 | equivalent to @kbd{[ [ 1 , 2 ] , [ 3 , 4 ] , [ 5 , 6 ] ]}. | |
11546 | ||
11547 | @kindex .. | |
11548 | @pindex calc-dots | |
11549 | Incomplete entry is also used to enter intervals. For example, | |
11550 | @kbd{[ 2 ..@: 4 )} enters a semi-open interval. Note that when you type | |
11551 | the first period, it will be interpreted as a decimal point, but when | |
11552 | you type a second period immediately afterward, it is re-interpreted as | |
11553 | part of the interval symbol. Typing @kbd{..} corresponds to executing | |
11554 | the @code{calc-dots} command. | |
11555 | ||
11556 | If you find incomplete entry distracting, you may wish to enter vectors | |
11557 | and complex numbers as algebraic formulas by pressing the apostrophe key. | |
11558 | ||
11559 | @node Variables, Formulas, Incomplete Objects, Data Types | |
11560 | @section Variables | |
11561 | ||
11562 | @noindent | |
11563 | @cindex Variables, in formulas | |
11564 | A @dfn{variable} is somewhere between a storage register on a conventional | |
11565 | calculator, and a variable in a programming language. (In fact, a Calc | |
11566 | variable is really just an Emacs Lisp variable that contains a Calc number | |
11567 | or formula.) A variable's name is normally composed of letters and digits. | |
11568 | Calc also allows apostrophes and @code{#} signs in variable names. | |
11569 | The Calc variable @code{foo} corresponds to the Emacs Lisp variable | |
11570 | @code{var-foo}. Commands like @kbd{s s} (@code{calc-store}) that operate | |
11571 | on variables can be made to use any arbitrary Lisp variable simply by | |
11572 | backspacing over the @samp{var-} prefix in the minibuffer.@refill | |
11573 | ||
11574 | In a command that takes a variable name, you can either type the full | |
11575 | name of a variable, or type a single digit to use one of the special | |
11576 | convenience variables @code{var-q0} through @code{var-q9}. For example, | |
11577 | @kbd{3 s s 2} stores the number 3 in variable @code{var-q2}, and | |
11578 | @w{@kbd{3 s s foo @key{RET}}} stores that number in variable | |
11579 | @code{var-foo}.@refill | |
11580 | ||
11581 | To push a variable itself (as opposed to the variable's value) on the | |
11582 | stack, enter its name as an algebraic expression using the apostrophe | |
11583 | (@key{'}) key. Variable names in algebraic formulas implicitly have | |
11584 | @samp{var-} prefixed to their names. The @samp{#} character in variable | |
11585 | names used in algebraic formulas corresponds to a dash @samp{-} in the | |
11586 | Lisp variable name. If the name contains any dashes, the prefix @samp{var-} | |
11587 | is @emph{not} automatically added. Thus the two formulas @samp{foo + 1} | |
11588 | and @samp{var#foo + 1} both refer to the same variable. | |
11589 | ||
11590 | @kindex = | |
11591 | @pindex calc-evaluate | |
11592 | @cindex Evaluation of variables in a formula | |
11593 | @cindex Variables, evaluation | |
11594 | @cindex Formulas, evaluation | |
11595 | The @kbd{=} (@code{calc-evaluate}) key ``evaluates'' a formula by | |
11596 | replacing all variables in the formula which have been given values by a | |
11597 | @code{calc-store} or @code{calc-let} command by their stored values. | |
11598 | Other variables are left alone. Thus a variable that has not been | |
11599 | stored acts like an abstract variable in algebra; a variable that has | |
11600 | been stored acts more like a register in a traditional calculator. | |
11601 | With a positive numeric prefix argument, @kbd{=} evaluates the top | |
11602 | @var{n} stack entries; with a negative argument, @kbd{=} evaluates | |
11603 | the @var{n}th stack entry. | |
11604 | ||
11605 | @cindex @code{e} variable | |
11606 | @cindex @code{pi} variable | |
11607 | @cindex @code{i} variable | |
11608 | @cindex @code{phi} variable | |
11609 | @cindex @code{gamma} variable | |
11610 | @vindex e | |
11611 | @vindex pi | |
11612 | @vindex i | |
11613 | @vindex phi | |
11614 | @vindex gamma | |
11615 | A few variables are called @dfn{special constants}. Their names are | |
11616 | @samp{e}, @samp{pi}, @samp{i}, @samp{phi}, and @samp{gamma}. | |
11617 | (@xref{Scientific Functions}.) When they are evaluated with @kbd{=}, | |
11618 | their values are calculated if necessary according to the current precision | |
11619 | or complex polar mode. If you wish to use these symbols for other purposes, | |
11620 | simply undefine or redefine them using @code{calc-store}.@refill | |
11621 | ||
11622 | The variables @samp{inf}, @samp{uinf}, and @samp{nan} stand for | |
11623 | infinite or indeterminate values. It's best not to use them as | |
11624 | regular variables, since Calc uses special algebraic rules when | |
11625 | it manipulates them. Calc displays a warning message if you store | |
11626 | a value into any of these special variables. | |
11627 | ||
11628 | @xref{Store and Recall}, for a discussion of commands dealing with variables. | |
11629 | ||
11630 | @node Formulas, , Variables, Data Types | |
11631 | @section Formulas | |
11632 | ||
11633 | @noindent | |
11634 | @cindex Formulas | |
11635 | @cindex Expressions | |
11636 | @cindex Operators in formulas | |
11637 | @cindex Precedence of operators | |
11638 | When you press the apostrophe key you may enter any expression or formula | |
11639 | in algebraic form. (Calc uses the terms ``expression'' and ``formula'' | |
11640 | interchangeably.) An expression is built up of numbers, variable names, | |
11641 | and function calls, combined with various arithmetic operators. | |
11642 | Parentheses may | |
11643 | be used to indicate grouping. Spaces are ignored within formulas, except | |
11644 | that spaces are not permitted within variable names or numbers. | |
11645 | Arithmetic operators, in order from highest to lowest precedence, and | |
11646 | with their equivalent function names, are: | |
11647 | ||
11648 | @samp{_} [@code{subscr}] (subscripts); | |
11649 | ||
11650 | postfix @samp{%} [@code{percent}] (as in @samp{25% = 0.25}); | |
11651 | ||
11652 | prefix @samp{+} and @samp{-} [@code{neg}] (as in @samp{-x}) | |
11653 | and prefix @samp{!} [@code{lnot}] (logical ``not,'' as in @samp{!x}); | |
11654 | ||
11655 | @samp{+/-} [@code{sdev}] (the standard deviation symbol) and | |
11656 | @samp{mod} [@code{makemod}] (the symbol for modulo forms); | |
11657 | ||
11658 | postfix @samp{!} [@code{fact}] (factorial, as in @samp{n!}) | |
11659 | and postfix @samp{!!} [@code{dfact}] (double factorial); | |
11660 | ||
11661 | @samp{^} [@code{pow}] (raised-to-the-power-of); | |
11662 | ||
11663 | @samp{*} [@code{mul}]; | |
11664 | ||
11665 | @samp{/} [@code{div}], @samp{%} [@code{mod}] (modulo), and | |
11666 | @samp{\} [@code{idiv}] (integer division); | |
11667 | ||
11668 | infix @samp{+} [@code{add}] and @samp{-} [@code{sub}] (as in @samp{x-y}); | |
11669 | ||
11670 | @samp{|} [@code{vconcat}] (vector concatenation); | |
11671 | ||
11672 | relations @samp{=} [@code{eq}], @samp{!=} [@code{neq}], @samp{<} [@code{lt}], | |
11673 | @samp{>} [@code{gt}], @samp{<=} [@code{leq}], and @samp{>=} [@code{geq}]; | |
11674 | ||
11675 | @samp{&&} [@code{land}] (logical ``and''); | |
11676 | ||
11677 | @samp{||} [@code{lor}] (logical ``or''); | |
11678 | ||
11679 | the C-style ``if'' operator @samp{a?b:c} [@code{if}]; | |
11680 | ||
11681 | @samp{!!!} [@code{pnot}] (rewrite pattern ``not''); | |
11682 | ||
11683 | @samp{&&&} [@code{pand}] (rewrite pattern ``and''); | |
11684 | ||
11685 | @samp{|||} [@code{por}] (rewrite pattern ``or''); | |
11686 | ||
11687 | @samp{:=} [@code{assign}] (for assignments and rewrite rules); | |
11688 | ||
11689 | @samp{::} [@code{condition}] (rewrite pattern condition); | |
11690 | ||
11691 | @samp{=>} [@code{evalto}]. | |
11692 | ||
11693 | Note that, unlike in usual computer notation, multiplication binds more | |
11694 | strongly than division: @samp{a*b/c*d} is equivalent to @c{$a b \over c d$} | |
11695 | @cite{(a*b)/(c*d)}. | |
11696 | ||
11697 | @cindex Multiplication, implicit | |
11698 | @cindex Implicit multiplication | |
11699 | The multiplication sign @samp{*} may be omitted in many cases. In particular, | |
11700 | if the righthand side is a number, variable name, or parenthesized | |
11701 | expression, the @samp{*} may be omitted. Implicit multiplication has the | |
11702 | same precedence as the explicit @samp{*} operator. The one exception to | |
11703 | the rule is that a variable name followed by a parenthesized expression, | |
11704 | as in @samp{f(x)}, | |
11705 | is interpreted as a function call, not an implicit @samp{*}. In many | |
11706 | cases you must use a space if you omit the @samp{*}: @samp{2a} is the | |
11707 | same as @samp{2*a}, and @samp{a b} is the same as @samp{a*b}, but @samp{ab} | |
11708 | is a variable called @code{ab}, @emph{not} the product of @samp{a} and | |
11709 | @samp{b}! Also note that @samp{f (x)} is still a function call.@refill | |
11710 | ||
11711 | @cindex Implicit comma in vectors | |
11712 | The rules are slightly different for vectors written with square brackets. | |
11713 | In vectors, the space character is interpreted (like the comma) as a | |
11714 | separator of elements of the vector. Thus @w{@samp{[ 2a b+c d ]}} is | |
11715 | equivalent to @samp{[2*a, b+c, d]}, whereas @samp{2a b+c d} is equivalent | |
11716 | to @samp{2*a*b + c*d}. | |
11717 | Note that spaces around the brackets, and around explicit commas, are | |
11718 | ignored. To force spaces to be interpreted as multiplication you can | |
11719 | enclose a formula in parentheses as in @samp{[(a b) 2(c d)]}, which is | |
11720 | interpreted as @samp{[a*b, 2*c*d]}. An implicit comma is also inserted | |
11721 | between @samp{][}, as in the matrix @samp{[[1 2][3 4]]}.@refill | |
11722 | ||
11723 | Vectors that contain commas (not embedded within nested parentheses or | |
11724 | brackets) do not treat spaces specially: @samp{[a b, 2 c d]} is a vector | |
11725 | of two elements. Also, if it would be an error to treat spaces as | |
11726 | separators, but not otherwise, then Calc will ignore spaces: | |
11727 | @w{@samp{[a - b]}} is a vector of one element, but @w{@samp{[a -b]}} is | |
11728 | a vector of two elements. Finally, vectors entered with curly braces | |
11729 | instead of square brackets do not give spaces any special treatment. | |
11730 | When Calc displays a vector that does not contain any commas, it will | |
11731 | insert parentheses if necessary to make the meaning clear: | |
11732 | @w{@samp{[(a b)]}}. | |
11733 | ||
11734 | The expression @samp{5%-2} is ambiguous; is this five-percent minus two, | |
11735 | or five modulo minus-two? Calc always interprets the leftmost symbol as | |
11736 | an infix operator preferentially (modulo, in this case), so you would | |
11737 | need to write @samp{(5%)-2} to get the former interpretation. | |
11738 | ||
11739 | @cindex Function call notation | |
11740 | A function call is, e.g., @samp{sin(1+x)}. Function names follow the same | |
11741 | rules as variable names except that the default prefix @samp{calcFunc-} is | |
11742 | used (instead of @samp{var-}) for the internal Lisp form. | |
11743 | Most mathematical Calculator commands like | |
11744 | @code{calc-sin} have function equivalents like @code{sin}. | |
11745 | If no Lisp function is defined for a function called by a formula, the | |
11746 | call is left as it is during algebraic manipulation: @samp{f(x+y)} is | |
11747 | left alone. Beware that many innocent-looking short names like @code{in} | |
11748 | and @code{re} have predefined meanings which could surprise you; however, | |
11749 | single letters or single letters followed by digits are always safe to | |
11750 | use for your own function names. @xref{Function Index}.@refill | |
11751 | ||
11752 | In the documentation for particular commands, the notation @kbd{H S} | |
11753 | (@code{calc-sinh}) [@code{sinh}] means that the key sequence @kbd{H S}, the | |
11754 | command @kbd{M-x calc-sinh}, and the algebraic function @code{sinh(x)} all | |
11755 | represent the same operation.@refill | |
11756 | ||
11757 | Commands that interpret (``parse'') text as algebraic formulas include | |
11758 | algebraic entry (@kbd{'}), editing commands like @kbd{`} which parse | |
11759 | the contents of the editing buffer when you finish, the @kbd{M-# g} | |
11760 | and @w{@kbd{M-# r}} commands, the @kbd{C-y} command, the X window system | |
11761 | ``paste'' mouse operation, and Embedded Mode. All of these operations | |
11762 | use the same rules for parsing formulas; in particular, language modes | |
11763 | (@pxref{Language Modes}) affect them all in the same way. | |
11764 | ||
11765 | When you read a large amount of text into the Calculator (say a vector | |
11766 | which represents a big set of rewrite rules; @pxref{Rewrite Rules}), | |
11767 | you may wish to include comments in the text. Calc's formula parser | |
11768 | ignores the symbol @samp{%%} and anything following it on a line: | |
11769 | ||
11770 | @example | |
11771 | [ a + b, %% the sum of "a" and "b" | |
11772 | c + d, | |
11773 | %% last line is coming up: | |
11774 | e + f ] | |
11775 | @end example | |
11776 | ||
11777 | @noindent | |
11778 | This is parsed exactly the same as @samp{[ a + b, c + d, e + f ]}. | |
11779 | ||
11780 | @xref{Syntax Tables}, for a way to create your own operators and other | |
11781 | input notations. @xref{Compositions}, for a way to create new display | |
11782 | formats. | |
11783 | ||
11784 | @xref{Algebra}, for commands for manipulating formulas symbolically. | |
11785 | ||
11786 | @node Stack and Trail, Mode Settings, Data Types, Top | |
11787 | @chapter Stack and Trail Commands | |
11788 | ||
11789 | @noindent | |
11790 | This chapter describes the Calc commands for manipulating objects on the | |
11791 | stack and in the trail buffer. (These commands operate on objects of any | |
11792 | type, such as numbers, vectors, formulas, and incomplete objects.) | |
11793 | ||
11794 | @menu | |
11795 | * Stack Manipulation:: | |
11796 | * Editing Stack Entries:: | |
11797 | * Trail Commands:: | |
11798 | * Keep Arguments:: | |
11799 | @end menu | |
11800 | ||
11801 | @node Stack Manipulation, Editing Stack Entries, Stack and Trail, Stack and Trail | |
11802 | @section Stack Manipulation Commands | |
11803 | ||
11804 | @noindent | |
5d67986c RS |
11805 | @kindex @key{RET} |
11806 | @kindex @key{SPC} | |
d7b8e6c6 EZ |
11807 | @pindex calc-enter |
11808 | @cindex Duplicating stack entries | |
11809 | To duplicate the top object on the stack, press @key{RET} or @key{SPC} | |
11810 | (two equivalent keys for the @code{calc-enter} command). | |
11811 | Given a positive numeric prefix argument, these commands duplicate | |
11812 | several elements at the top of the stack. | |
11813 | Given a negative argument, | |
11814 | these commands duplicate the specified element of the stack. | |
11815 | Given an argument of zero, they duplicate the entire stack. | |
11816 | For example, with @samp{10 20 30} on the stack, | |
11817 | @key{RET} creates @samp{10 20 30 30}, | |
11818 | @kbd{C-u 2 @key{RET}} creates @samp{10 20 30 20 30}, | |
11819 | @kbd{C-u - 2 @key{RET}} creates @samp{10 20 30 20}, and | |
11820 | @kbd{C-u 0 @key{RET}} creates @samp{10 20 30 10 20 30}.@refill | |
11821 | ||
5d67986c | 11822 | @kindex @key{LFD} |
d7b8e6c6 EZ |
11823 | @pindex calc-over |
11824 | The @key{LFD} (@code{calc-over}) command (on a key marked Line-Feed if you | |
11825 | have it, else on @kbd{C-j}) is like @code{calc-enter} | |
11826 | except that the sign of the numeric prefix argument is interpreted | |
11827 | oppositely. Also, with no prefix argument the default argument is 2. | |
11828 | Thus with @samp{10 20 30} on the stack, @key{LFD} and @kbd{C-u 2 @key{LFD}} | |
11829 | are both equivalent to @kbd{C-u - 2 @key{RET}}, producing | |
11830 | @samp{10 20 30 20}.@refill | |
11831 | ||
5d67986c | 11832 | @kindex @key{DEL} |
d7b8e6c6 EZ |
11833 | @kindex C-d |
11834 | @pindex calc-pop | |
11835 | @cindex Removing stack entries | |
11836 | @cindex Deleting stack entries | |
11837 | To remove the top element from the stack, press @key{DEL} (@code{calc-pop}). | |
11838 | The @kbd{C-d} key is a synonym for @key{DEL}. | |
11839 | (If the top element is an incomplete object with at least one element, the | |
11840 | last element is removed from it.) Given a positive numeric prefix argument, | |
11841 | several elements are removed. Given a negative argument, the specified | |
11842 | element of the stack is deleted. Given an argument of zero, the entire | |
11843 | stack is emptied. | |
11844 | For example, with @samp{10 20 30} on the stack, | |
11845 | @key{DEL} leaves @samp{10 20}, | |
11846 | @kbd{C-u 2 @key{DEL}} leaves @samp{10}, | |
11847 | @kbd{C-u - 2 @key{DEL}} leaves @samp{10 30}, and | |
11848 | @kbd{C-u 0 @key{DEL}} leaves an empty stack.@refill | |
11849 | ||
5d67986c | 11850 | @kindex M-@key{DEL} |
d7b8e6c6 | 11851 | @pindex calc-pop-above |
0d48e8aa | 11852 | The @kbd{M-@key{DEL}} (@code{calc-pop-above}) command is to @key{DEL} what |
d7b8e6c6 EZ |
11853 | @key{LFD} is to @key{RET}: It interprets the sign of the numeric |
11854 | prefix argument in the opposite way, and the default argument is 2. | |
0d48e8aa | 11855 | Thus @kbd{M-@key{DEL}} by itself removes the second-from-top stack element, |
5d67986c | 11856 | leaving the first, third, fourth, and so on; @kbd{M-3 M-@key{DEL}} deletes |
d7b8e6c6 EZ |
11857 | the third stack element. |
11858 | ||
5d67986c | 11859 | @kindex @key{TAB} |
d7b8e6c6 EZ |
11860 | @pindex calc-roll-down |
11861 | To exchange the top two elements of the stack, press @key{TAB} | |
11862 | (@code{calc-roll-down}). Given a positive numeric prefix argument, the | |
11863 | specified number of elements at the top of the stack are rotated downward. | |
11864 | Given a negative argument, the entire stack is rotated downward the specified | |
11865 | number of times. Given an argument of zero, the entire stack is reversed | |
11866 | top-for-bottom. | |
11867 | For example, with @samp{10 20 30 40 50} on the stack, | |
11868 | @key{TAB} creates @samp{10 20 30 50 40}, | |
11869 | @kbd{C-u 3 @key{TAB}} creates @samp{10 20 50 30 40}, | |
11870 | @kbd{C-u - 2 @key{TAB}} creates @samp{40 50 10 20 30}, and | |
11871 | @kbd{C-u 0 @key{TAB}} creates @samp{50 40 30 20 10}.@refill | |
11872 | ||
5d67986c | 11873 | @kindex M-@key{TAB} |
d7b8e6c6 | 11874 | @pindex calc-roll-up |
5d67986c | 11875 | The command @kbd{M-@key{TAB}} (@code{calc-roll-up}) is analogous to @key{TAB} |
d7b8e6c6 EZ |
11876 | except that it rotates upward instead of downward. Also, the default |
11877 | with no prefix argument is to rotate the top 3 elements. | |
11878 | For example, with @samp{10 20 30 40 50} on the stack, | |
5d67986c RS |
11879 | @kbd{M-@key{TAB}} creates @samp{10 20 40 50 30}, |
11880 | @kbd{C-u 4 M-@key{TAB}} creates @samp{10 30 40 50 20}, | |
11881 | @kbd{C-u - 2 M-@key{TAB}} creates @samp{30 40 50 10 20}, and | |
11882 | @kbd{C-u 0 M-@key{TAB}} creates @samp{50 40 30 20 10}.@refill | |
d7b8e6c6 | 11883 | |
5d67986c | 11884 | A good way to view the operation of @key{TAB} and @kbd{M-@key{TAB}} is in |
d7b8e6c6 | 11885 | terms of moving a particular element to a new position in the stack. |
5d67986c RS |
11886 | With a positive argument @var{n}, @key{TAB} moves the top stack |
11887 | element down to level @var{n}, making room for it by pulling all the | |
11888 | intervening stack elements toward the top. @kbd{M-@key{TAB}} moves the | |
11889 | element at level @var{n} up to the top. (Compare with @key{LFD}, | |
11890 | which copies instead of moving the element in level @var{n}.) | |
11891 | ||
11892 | With a negative argument @i{-@var{n}}, @key{TAB} rotates the stack | |
11893 | to move the object in level @var{n} to the deepest place in the | |
11894 | stack, and the object in level @i{@var{n}+1} to the top. @kbd{M-@key{TAB}} | |
d7b8e6c6 | 11895 | rotates the deepest stack element to be in level @i{n}, also |
5d67986c | 11896 | putting the top stack element in level @i{@var{n}+1}. |
d7b8e6c6 EZ |
11897 | |
11898 | @xref{Selecting Subformulas}, for a way to apply these commands to | |
11899 | any portion of a vector or formula on the stack. | |
11900 | ||
11901 | @node Editing Stack Entries, Trail Commands, Stack Manipulation, Stack and Trail | |
11902 | @section Editing Stack Entries | |
11903 | ||
11904 | @noindent | |
11905 | @kindex ` | |
11906 | @pindex calc-edit | |
11907 | @pindex calc-edit-finish | |
11908 | @cindex Editing the stack with Emacs | |
11909 | The backquote, @kbd{`} (@code{calc-edit}) command creates a temporary | |
11910 | buffer (@samp{*Calc Edit*}) for editing the top-of-stack value using | |
11911 | regular Emacs commands. With a numeric prefix argument, it edits the | |
11912 | specified number of stack entries at once. (An argument of zero edits | |
11913 | the entire stack; a negative argument edits one specific stack entry.) | |
11914 | ||
11915 | When you are done editing, press @kbd{M-# M-#} to finish and return | |
11916 | to Calc. The @key{RET} and @key{LFD} keys also work to finish most | |
11917 | sorts of editing, though in some cases Calc leaves @key{RET} with its | |
11918 | usual meaning (``insert a newline'') if it's a situation where you | |
11919 | might want to insert new lines into the editing buffer. The traditional | |
11920 | Emacs ``finish'' key sequence, @kbd{C-c C-c}, also works to finish | |
11921 | editing and may be easier to type, depending on your keyboard. | |
11922 | ||
11923 | When you finish editing, the Calculator parses the lines of text in | |
11924 | the @samp{*Calc Edit*} buffer as numbers or formulas, replaces the | |
11925 | original stack elements in the original buffer with these new values, | |
11926 | then kills the @samp{*Calc Edit*} buffer. The original Calculator buffer | |
11927 | continues to exist during editing, but for best results you should be | |
11928 | careful not to change it until you have finished the edit. You can | |
11929 | also cancel the edit by pressing @kbd{M-# x}. | |
11930 | ||
11931 | The formula is normally reevaluated as it is put onto the stack. | |
11932 | For example, editing @samp{a + 2} to @samp{3 + 2} and pressing | |
11933 | @kbd{M-# M-#} will push 5 on the stack. If you use @key{LFD} to | |
11934 | finish, Calc will put the result on the stack without evaluating it. | |
11935 | ||
11936 | If you give a prefix argument to @kbd{M-# M-#} (or @kbd{C-c C-c}), | |
11937 | Calc will not kill the @samp{*Calc Edit*} buffer. You can switch | |
11938 | back to that buffer and continue editing if you wish. However, you | |
11939 | should understand that if you initiated the edit with @kbd{`}, the | |
11940 | @kbd{M-# M-#} operation will be programmed to replace the top of the | |
11941 | stack with the new edited value, and it will do this even if you have | |
11942 | rearranged the stack in the meanwhile. This is not so much of a problem | |
11943 | with other editing commands, though, such as @kbd{s e} | |
11944 | (@code{calc-edit-variable}; @pxref{Operations on Variables}). | |
11945 | ||
11946 | If the @code{calc-edit} command involves more than one stack entry, | |
11947 | each line of the @samp{*Calc Edit*} buffer is interpreted as a | |
11948 | separate formula. Otherwise, the entire buffer is interpreted as | |
11949 | one formula, with line breaks ignored. (You can use @kbd{C-o} or | |
11950 | @kbd{C-q C-j} to insert a newline in the buffer without pressing @key{RET}.) | |
11951 | ||
11952 | The @kbd{`} key also works during numeric or algebraic entry. The | |
11953 | text entered so far is moved to the @code{*Calc Edit*} buffer for | |
11954 | more extensive editing than is convenient in the minibuffer. | |
11955 | ||
11956 | @node Trail Commands, Keep Arguments, Editing Stack Entries, Stack and Trail | |
11957 | @section Trail Commands | |
11958 | ||
11959 | @noindent | |
11960 | @cindex Trail buffer | |
11961 | The commands for manipulating the Calc Trail buffer are two-key sequences | |
11962 | beginning with the @kbd{t} prefix. | |
11963 | ||
11964 | @kindex t d | |
11965 | @pindex calc-trail-display | |
11966 | The @kbd{t d} (@code{calc-trail-display}) command turns display of the | |
11967 | trail on and off. Normally the trail display is toggled on if it was off, | |
11968 | off if it was on. With a numeric prefix of zero, this command always | |
11969 | turns the trail off; with a prefix of one, it always turns the trail on. | |
11970 | The other trail-manipulation commands described here automatically turn | |
11971 | the trail on. Note that when the trail is off values are still recorded | |
11972 | there; they are simply not displayed. To set Emacs to turn the trail | |
11973 | off by default, type @kbd{t d} and then save the mode settings with | |
11974 | @kbd{m m} (@code{calc-save-modes}). | |
11975 | ||
11976 | @kindex t i | |
11977 | @pindex calc-trail-in | |
11978 | @kindex t o | |
11979 | @pindex calc-trail-out | |
11980 | The @kbd{t i} (@code{calc-trail-in}) and @kbd{t o} | |
11981 | (@code{calc-trail-out}) commands switch the cursor into and out of the | |
11982 | Calc Trail window. In practice they are rarely used, since the commands | |
11983 | shown below are a more convenient way to move around in the | |
11984 | trail, and they work ``by remote control'' when the cursor is still | |
11985 | in the Calculator window.@refill | |
11986 | ||
11987 | @cindex Trail pointer | |
11988 | There is a @dfn{trail pointer} which selects some entry of the trail at | |
11989 | any given time. The trail pointer looks like a @samp{>} symbol right | |
11990 | before the selected number. The following commands operate on the | |
11991 | trail pointer in various ways. | |
11992 | ||
11993 | @kindex t y | |
11994 | @pindex calc-trail-yank | |
11995 | @cindex Retrieving previous results | |
11996 | The @kbd{t y} (@code{calc-trail-yank}) command reads the selected value in | |
11997 | the trail and pushes it onto the Calculator stack. It allows you to | |
11998 | re-use any previously computed value without retyping. With a numeric | |
11999 | prefix argument @var{n}, it yanks the value @var{n} lines above the current | |
12000 | trail pointer. | |
12001 | ||
12002 | @kindex t < | |
12003 | @pindex calc-trail-scroll-left | |
12004 | @kindex t > | |
12005 | @pindex calc-trail-scroll-right | |
12006 | The @kbd{t <} (@code{calc-trail-scroll-left}) and @kbd{t >} | |
12007 | (@code{calc-trail-scroll-right}) commands horizontally scroll the trail | |
12008 | window left or right by one half of its width.@refill | |
12009 | ||
12010 | @kindex t n | |
12011 | @pindex calc-trail-next | |
12012 | @kindex t p | |
12013 | @pindex calc-trail-previous | |
12014 | @kindex t f | |
12015 | @pindex calc-trail-forward | |
12016 | @kindex t b | |
12017 | @pindex calc-trail-backward | |
12018 | The @kbd{t n} (@code{calc-trail-next}) and @kbd{t p} | |
12019 | (@code{calc-trail-previous)} commands move the trail pointer down or up | |
12020 | one line. The @kbd{t f} (@code{calc-trail-forward}) and @kbd{t b} | |
12021 | (@code{calc-trail-backward}) commands move the trail pointer down or up | |
12022 | one screenful at a time. All of these commands accept numeric prefix | |
12023 | arguments to move several lines or screenfuls at a time.@refill | |
12024 | ||
12025 | @kindex t [ | |
12026 | @pindex calc-trail-first | |
12027 | @kindex t ] | |
12028 | @pindex calc-trail-last | |
12029 | @kindex t h | |
12030 | @pindex calc-trail-here | |
12031 | The @kbd{t [} (@code{calc-trail-first}) and @kbd{t ]} | |
12032 | (@code{calc-trail-last}) commands move the trail pointer to the first or | |
12033 | last line of the trail. The @kbd{t h} (@code{calc-trail-here}) command | |
12034 | moves the trail pointer to the cursor position; unlike the other trail | |
12035 | commands, @kbd{t h} works only when Calc Trail is the selected window.@refill | |
12036 | ||
12037 | @kindex t s | |
12038 | @pindex calc-trail-isearch-forward | |
12039 | @kindex t r | |
12040 | @pindex calc-trail-isearch-backward | |
12041 | @ifinfo | |
12042 | The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r} | |
12043 | (@code{calc-trail-isearch-backward}) commands perform an incremental | |
12044 | search forward or backward through the trail. You can press @key{RET} | |
12045 | to terminate the search; the trail pointer moves to the current line. | |
12046 | If you cancel the search with @kbd{C-g}, the trail pointer stays where | |
12047 | it was when the search began.@refill | |
12048 | @end ifinfo | |
12049 | @tex | |
12050 | The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r} | |
12051 | (@code{calc-trail-isearch-backward}) com\-mands perform an incremental | |
12052 | search forward or backward through the trail. You can press @key{RET} | |
12053 | to terminate the search; the trail pointer moves to the current line. | |
12054 | If you cancel the search with @kbd{C-g}, the trail pointer stays where | |
12055 | it was when the search began. | |
12056 | @end tex | |
12057 | ||
12058 | @kindex t m | |
12059 | @pindex calc-trail-marker | |
12060 | The @kbd{t m} (@code{calc-trail-marker}) command allows you to enter a | |
12061 | line of text of your own choosing into the trail. The text is inserted | |
12062 | after the line containing the trail pointer; this usually means it is | |
12063 | added to the end of the trail. Trail markers are useful mainly as the | |
12064 | targets for later incremental searches in the trail. | |
12065 | ||
12066 | @kindex t k | |
12067 | @pindex calc-trail-kill | |
12068 | The @kbd{t k} (@code{calc-trail-kill}) command removes the selected line | |
12069 | from the trail. The line is saved in the Emacs kill ring suitable for | |
12070 | yanking into another buffer, but it is not easy to yank the text back | |
12071 | into the trail buffer. With a numeric prefix argument, this command | |
12072 | kills the @var{n} lines below or above the selected one. | |
12073 | ||
12074 | The @kbd{t .} (@code{calc-full-trail-vectors}) command is described | |
12075 | elsewhere; @pxref{Vector and Matrix Formats}. | |
12076 | ||
12077 | @node Keep Arguments, , Trail Commands, Stack and Trail | |
12078 | @section Keep Arguments | |
12079 | ||
12080 | @noindent | |
12081 | @kindex K | |
12082 | @pindex calc-keep-args | |
12083 | The @kbd{K} (@code{calc-keep-args}) command acts like a prefix for | |
12084 | the following command. It prevents that command from removing its | |
12085 | arguments from the stack. For example, after @kbd{2 @key{RET} 3 +}, | |
12086 | the stack contains the sole number 5, but after @kbd{2 @key{RET} 3 K +}, | |
12087 | the stack contains the arguments and the result: @samp{2 3 5}. | |
12088 | ||
12089 | This works for all commands that take arguments off the stack. As | |
12090 | another example, @kbd{K a s} simplifies a formula, pushing the | |
12091 | simplified version of the formula onto the stack after the original | |
12092 | formula (rather than replacing the original formula). | |
12093 | ||
5d67986c | 12094 | Note that you could get the same effect by typing @kbd{@key{RET} a s}, |
d7b8e6c6 EZ |
12095 | copying the formula and then simplifying the copy. One difference |
12096 | is that for a very large formula the time taken to format the | |
5d67986c | 12097 | intermediate copy in @kbd{@key{RET} a s} could be noticeable; @kbd{K a s} |
d7b8e6c6 EZ |
12098 | would avoid this extra work. |
12099 | ||
12100 | Even stack manipulation commands are affected. @key{TAB} works by | |
12101 | popping two values and pushing them back in the opposite order, | |
12102 | so @kbd{2 @key{RET} 3 K @key{TAB}} produces @samp{2 3 3 2}. | |
12103 | ||
12104 | A few Calc commands provide other ways of doing the same thing. | |
12105 | For example, @kbd{' sin($)} replaces the number on the stack with | |
12106 | its sine using algebraic entry; to push the sine and keep the | |
12107 | original argument you could use either @kbd{' sin($1)} or | |
12108 | @kbd{K ' sin($)}. @xref{Algebraic Entry}. Also, the @kbd{s s} | |
12109 | command is effectively the same as @kbd{K s t}. @xref{Storing Variables}. | |
12110 | ||
12111 | Keyboard macros may interact surprisingly with the @kbd{K} prefix. | |
12112 | If you have defined a keyboard macro to be, say, @samp{Q +} to add | |
12113 | one number to the square root of another, then typing @kbd{K X} will | |
12114 | execute @kbd{K Q +}, probably not what you expected. The @kbd{K} | |
12115 | prefix will apply to just the first command in the macro rather than | |
12116 | the whole macro. | |
12117 | ||
12118 | If you execute a command and then decide you really wanted to keep | |
12119 | the argument, you can press @kbd{M-@key{RET}} (@code{calc-last-args}). | |
12120 | This command pushes the last arguments that were popped by any command | |
12121 | onto the stack. Note that the order of things on the stack will be | |
12122 | different than with @kbd{K}: @kbd{2 @key{RET} 3 + M-@key{RET}} leaves | |
12123 | @samp{5 2 3} on the stack instead of @samp{2 3 5}. @xref{Undo}. | |
12124 | ||
12125 | @node Mode Settings, Arithmetic, Stack and Trail, Top | |
12126 | @chapter Mode Settings | |
12127 | ||
12128 | @noindent | |
12129 | This chapter describes commands that set modes in the Calculator. | |
12130 | They do not affect the contents of the stack, although they may change | |
12131 | the @emph{appearance} or @emph{interpretation} of the stack's contents. | |
12132 | ||
12133 | @menu | |
12134 | * General Mode Commands:: | |
12135 | * Precision:: | |
12136 | * Inverse and Hyperbolic:: | |
12137 | * Calculation Modes:: | |
12138 | * Simplification Modes:: | |
12139 | * Declarations:: | |
12140 | * Display Modes:: | |
12141 | * Language Modes:: | |
12142 | * Modes Variable:: | |
12143 | * Calc Mode Line:: | |
12144 | @end menu | |
12145 | ||
12146 | @node General Mode Commands, Precision, Mode Settings, Mode Settings | |
12147 | @section General Mode Commands | |
12148 | ||
12149 | @noindent | |
12150 | @kindex m m | |
12151 | @pindex calc-save-modes | |
12152 | @cindex Continuous memory | |
12153 | @cindex Saving mode settings | |
12154 | @cindex Permanent mode settings | |
12155 | @cindex @file{.emacs} file, mode settings | |
12156 | You can save all of the current mode settings in your @file{.emacs} file | |
12157 | with the @kbd{m m} (@code{calc-save-modes}) command. This will cause | |
12158 | Emacs to reestablish these modes each time it starts up. The modes saved | |
12159 | in the file include everything controlled by the @kbd{m} and @kbd{d} | |
12160 | prefix keys, the current precision and binary word size, whether or not | |
12161 | the trail is displayed, the current height of the Calc window, and more. | |
12162 | The current interface (used when you type @kbd{M-# M-#}) is also saved. | |
12163 | If there were already saved mode settings in the file, they are replaced. | |
12164 | Otherwise, the new mode information is appended to the end of the file. | |
12165 | ||
12166 | @kindex m R | |
12167 | @pindex calc-mode-record-mode | |
12168 | The @kbd{m R} (@code{calc-mode-record-mode}) command tells Calc to | |
12169 | record the new mode settings (as if by pressing @kbd{m m}) every | |
12170 | time a mode setting changes. If Embedded Mode is enabled, other | |
12171 | options are available; @pxref{Mode Settings in Embedded Mode}. | |
12172 | ||
12173 | @kindex m F | |
12174 | @pindex calc-settings-file-name | |
12175 | The @kbd{m F} (@code{calc-settings-file-name}) command allows you to | |
12176 | choose a different place than your @file{.emacs} file for @kbd{m m}, | |
12177 | @kbd{Z P}, and similar commands to save permanent information. | |
12178 | You are prompted for a file name. All Calc modes are then reset to | |
12179 | their default values, then settings from the file you named are loaded | |
12180 | if this file exists, and this file becomes the one that Calc will | |
12181 | use in the future for commands like @kbd{m m}. The default settings | |
12182 | file name is @file{~/.emacs}. You can see the current file name by | |
12183 | giving a blank response to the @kbd{m F} prompt. See also the | |
12184 | discussion of the @code{calc-settings-file} variable; @pxref{Installation}. | |
12185 | ||
12186 | If the file name you give contains the string @samp{.emacs} anywhere | |
12187 | inside it, @kbd{m F} will not automatically load the new file. This | |
12188 | is because you are presumably switching to your @file{~/.emacs} file, | |
12189 | which may contain other things you don't want to reread. You can give | |
12190 | a numeric prefix argument of 1 to @kbd{m F} to force it to read the | |
12191 | file no matter what its name. Conversely, an argument of @i{-1} tells | |
12192 | @kbd{m F} @emph{not} to read the new file. An argument of 2 or @i{-2} | |
12193 | tells @kbd{m F} not to reset the modes to their defaults beforehand, | |
12194 | which is useful if you intend your new file to have a variant of the | |
12195 | modes present in the file you were using before. | |
12196 | ||
12197 | @kindex m x | |
12198 | @pindex calc-always-load-extensions | |
12199 | The @kbd{m x} (@code{calc-always-load-extensions}) command enables a mode | |
12200 | in which the first use of Calc loads the entire program, including all | |
12201 | extensions modules. Otherwise, the extensions modules will not be loaded | |
12202 | until the various advanced Calc features are used. Since this mode only | |
12203 | has effect when Calc is first loaded, @kbd{m x} is usually followed by | |
12204 | @kbd{m m} to make the mode-setting permanent. To load all of Calc just | |
12205 | once, rather than always in the future, you can press @kbd{M-# L}. | |
12206 | ||
12207 | @kindex m S | |
12208 | @pindex calc-shift-prefix | |
12209 | The @kbd{m S} (@code{calc-shift-prefix}) command enables a mode in which | |
12210 | all of Calc's letter prefix keys may be typed shifted as well as unshifted. | |
12211 | If you are typing, say, @kbd{a S} (@code{calc-solve-for}) quite often | |
12212 | you might find it easier to turn this mode on so that you can type | |
12213 | @kbd{A S} instead. When this mode is enabled, the commands that used to | |
12214 | be on those single shifted letters (e.g., @kbd{A} (@code{calc-abs})) can | |
12215 | now be invoked by pressing the shifted letter twice: @kbd{A A}. Note | |
12216 | that the @kbd{v} prefix key always works both shifted and unshifted, and | |
12217 | the @kbd{z} and @kbd{Z} prefix keys are always distinct. Also, the @kbd{h} | |
12218 | prefix is not affected by this mode. Press @kbd{m S} again to disable | |
12219 | shifted-prefix mode. | |
12220 | ||
12221 | @node Precision, Inverse and Hyperbolic, General Mode Commands, Mode Settings | |
12222 | @section Precision | |
12223 | ||
12224 | @noindent | |
12225 | @kindex p | |
12226 | @pindex calc-precision | |
12227 | @cindex Precision of calculations | |
12228 | The @kbd{p} (@code{calc-precision}) command controls the precision to | |
12229 | which floating-point calculations are carried. The precision must be | |
12230 | at least 3 digits and may be arbitrarily high, within the limits of | |
12231 | memory and time. This affects only floats: Integer and rational | |
12232 | calculations are always carried out with as many digits as necessary. | |
12233 | ||
12234 | The @kbd{p} key prompts for the current precision. If you wish you | |
12235 | can instead give the precision as a numeric prefix argument. | |
12236 | ||
12237 | Many internal calculations are carried to one or two digits higher | |
12238 | precision than normal. Results are rounded down afterward to the | |
12239 | current precision. Unless a special display mode has been selected, | |
12240 | floats are always displayed with their full stored precision, i.e., | |
12241 | what you see is what you get. Reducing the current precision does not | |
12242 | round values already on the stack, but those values will be rounded | |
12243 | down before being used in any calculation. The @kbd{c 0} through | |
12244 | @kbd{c 9} commands (@pxref{Conversions}) can be used to round an | |
12245 | existing value to a new precision.@refill | |
12246 | ||
12247 | @cindex Accuracy of calculations | |
12248 | It is important to distinguish the concepts of @dfn{precision} and | |
12249 | @dfn{accuracy}. In the normal usage of these words, the number | |
12250 | 123.4567 has a precision of 7 digits but an accuracy of 4 digits. | |
12251 | The precision is the total number of digits not counting leading | |
12252 | or trailing zeros (regardless of the position of the decimal point). | |
12253 | The accuracy is simply the number of digits after the decimal point | |
12254 | (again not counting trailing zeros). In Calc you control the precision, | |
12255 | not the accuracy of computations. If you were to set the accuracy | |
12256 | instead, then calculations like @samp{exp(100)} would generate many | |
12257 | more digits than you would typically need, while @samp{exp(-100)} would | |
12258 | probably round to zero! In Calc, both these computations give you | |
12259 | exactly 12 (or the requested number of) significant digits. | |
12260 | ||
12261 | The only Calc features that deal with accuracy instead of precision | |
12262 | are fixed-point display mode for floats (@kbd{d f}; @pxref{Float Formats}), | |
12263 | and the rounding functions like @code{floor} and @code{round} | |
12264 | (@pxref{Integer Truncation}). Also, @kbd{c 0} through @kbd{c 9} | |
12265 | deal with both precision and accuracy depending on the magnitudes | |
12266 | of the numbers involved. | |
12267 | ||
12268 | If you need to work with a particular fixed accuracy (say, dollars and | |
12269 | cents with two digits after the decimal point), one solution is to work | |
12270 | with integers and an ``implied'' decimal point. For example, $8.99 | |
5d67986c | 12271 | divided by 6 would be entered @kbd{899 @key{RET} 6 /}, yielding 149.833 |
d7b8e6c6 EZ |
12272 | (actually $1.49833 with our implied decimal point); pressing @kbd{R} |
12273 | would round this to 150 cents, i.e., $1.50. | |
12274 | ||
12275 | @xref{Floats}, for still more on floating-point precision and related | |
12276 | issues. | |
12277 | ||
12278 | @node Inverse and Hyperbolic, Calculation Modes, Precision, Mode Settings | |
12279 | @section Inverse and Hyperbolic Flags | |
12280 | ||
12281 | @noindent | |
12282 | @kindex I | |
12283 | @pindex calc-inverse | |
12284 | There is no single-key equivalent to the @code{calc-arcsin} function. | |
12285 | Instead, you must first press @kbd{I} (@code{calc-inverse}) to set | |
12286 | the @dfn{Inverse Flag}, then press @kbd{S} (@code{calc-sin}). | |
12287 | The @kbd{I} key actually toggles the Inverse Flag. When this flag | |
12288 | is set, the word @samp{Inv} appears in the mode line.@refill | |
12289 | ||
12290 | @kindex H | |
12291 | @pindex calc-hyperbolic | |
12292 | Likewise, the @kbd{H} key (@code{calc-hyperbolic}) sets or clears the | |
12293 | Hyperbolic Flag, which transforms @code{calc-sin} into @code{calc-sinh}. | |
12294 | If both of these flags are set at once, the effect will be | |
12295 | @code{calc-arcsinh}. (The Hyperbolic flag is also used by some | |
12296 | non-trigonometric commands; for example @kbd{H L} computes a base-10, | |
12297 | instead of base-@i{e}, logarithm.)@refill | |
12298 | ||
12299 | Command names like @code{calc-arcsin} are provided for completeness, and | |
12300 | may be executed with @kbd{x} or @kbd{M-x}. Their effect is simply to | |
12301 | toggle the Inverse and/or Hyperbolic flags and then execute the | |
12302 | corresponding base command (@code{calc-sin} in this case). | |
12303 | ||
12304 | The Inverse and Hyperbolic flags apply only to the next Calculator | |
12305 | command, after which they are automatically cleared. (They are also | |
12306 | cleared if the next keystroke is not a Calc command.) Digits you | |
12307 | type after @kbd{I} or @kbd{H} (or @kbd{K}) are treated as prefix | |
12308 | arguments for the next command, not as numeric entries. The same | |
12309 | is true of @kbd{C-u}, but not of the minus sign (@kbd{K -} means to | |
12310 | subtract and keep arguments). | |
12311 | ||
12312 | The third Calc prefix flag, @kbd{K} (keep-arguments), is discussed | |
12313 | elsewhere. @xref{Keep Arguments}. | |
12314 | ||
12315 | @node Calculation Modes, Simplification Modes, Inverse and Hyperbolic, Mode Settings | |
12316 | @section Calculation Modes | |
12317 | ||
12318 | @noindent | |
12319 | The commands in this section are two-key sequences beginning with | |
12320 | the @kbd{m} prefix. (That's the letter @kbd{m}, not the @key{META} key.) | |
12321 | The @samp{m a} (@code{calc-algebraic-mode}) command is described elsewhere | |
12322 | (@pxref{Algebraic Entry}). | |
12323 | ||
12324 | @menu | |
12325 | * Angular Modes:: | |
12326 | * Polar Mode:: | |
12327 | * Fraction Mode:: | |
12328 | * Infinite Mode:: | |
12329 | * Symbolic Mode:: | |
12330 | * Matrix Mode:: | |
12331 | * Automatic Recomputation:: | |
12332 | * Working Message:: | |
12333 | @end menu | |
12334 | ||
12335 | @node Angular Modes, Polar Mode, Calculation Modes, Calculation Modes | |
12336 | @subsection Angular Modes | |
12337 | ||
12338 | @noindent | |
12339 | @cindex Angular mode | |
12340 | The Calculator supports three notations for angles: radians, degrees, | |
12341 | and degrees-minutes-seconds. When a number is presented to a function | |
12342 | like @code{sin} that requires an angle, the current angular mode is | |
12343 | used to interpret the number as either radians or degrees. If an HMS | |
12344 | form is presented to @code{sin}, it is always interpreted as | |
12345 | degrees-minutes-seconds. | |
12346 | ||
12347 | Functions that compute angles produce a number in radians, a number in | |
12348 | degrees, or an HMS form depending on the current angular mode. If the | |
12349 | result is a complex number and the current mode is HMS, the number is | |
12350 | instead expressed in degrees. (Complex-number calculations would | |
12351 | normally be done in radians mode, though. Complex numbers are converted | |
12352 | to degrees by calculating the complex result in radians and then | |
12353 | multiplying by 180 over @c{$\pi$} | |
12354 | @cite{pi}.) | |
12355 | ||
12356 | @kindex m r | |
12357 | @pindex calc-radians-mode | |
12358 | @kindex m d | |
12359 | @pindex calc-degrees-mode | |
12360 | @kindex m h | |
12361 | @pindex calc-hms-mode | |
12362 | The @kbd{m r} (@code{calc-radians-mode}), @kbd{m d} (@code{calc-degrees-mode}), | |
12363 | and @kbd{m h} (@code{calc-hms-mode}) commands control the angular mode. | |
12364 | The current angular mode is displayed on the Emacs mode line. | |
12365 | The default angular mode is degrees.@refill | |
12366 | ||
12367 | @node Polar Mode, Fraction Mode, Angular Modes, Calculation Modes | |
12368 | @subsection Polar Mode | |
12369 | ||
12370 | @noindent | |
12371 | @cindex Polar mode | |
12372 | The Calculator normally ``prefers'' rectangular complex numbers in the | |
12373 | sense that rectangular form is used when the proper form can not be | |
12374 | decided from the input. This might happen by multiplying a rectangular | |
12375 | number by a polar one, by taking the square root of a negative real | |
12376 | number, or by entering @kbd{( 2 @key{SPC} 3 )}. | |
12377 | ||
12378 | @kindex m p | |
12379 | @pindex calc-polar-mode | |
12380 | The @kbd{m p} (@code{calc-polar-mode}) command toggles complex-number | |
12381 | preference between rectangular and polar forms. In polar mode, all | |
12382 | of the above example situations would produce polar complex numbers. | |
12383 | ||
12384 | @node Fraction Mode, Infinite Mode, Polar Mode, Calculation Modes | |
12385 | @subsection Fraction Mode | |
12386 | ||
12387 | @noindent | |
12388 | @cindex Fraction mode | |
12389 | @cindex Division of integers | |
12390 | Division of two integers normally yields a floating-point number if the | |
12391 | result cannot be expressed as an integer. In some cases you would | |
12392 | rather get an exact fractional answer. One way to accomplish this is | |
12393 | to multiply fractions instead: @kbd{6 @key{RET} 1:4 *} produces @cite{3:2} | |
12394 | even though @kbd{6 @key{RET} 4 /} produces @cite{1.5}. | |
12395 | ||
12396 | @kindex m f | |
12397 | @pindex calc-frac-mode | |
12398 | To set the Calculator to produce fractional results for normal integer | |
12399 | divisions, use the @kbd{m f} (@code{calc-frac-mode}) command. | |
12400 | For example, @cite{8/4} produces @cite{2} in either mode, | |
12401 | but @cite{6/4} produces @cite{3:2} in Fraction Mode, @cite{1.5} in | |
12402 | Float Mode.@refill | |
12403 | ||
12404 | At any time you can use @kbd{c f} (@code{calc-float}) to convert a | |
12405 | fraction to a float, or @kbd{c F} (@code{calc-fraction}) to convert a | |
12406 | float to a fraction. @xref{Conversions}. | |
12407 | ||
12408 | @node Infinite Mode, Symbolic Mode, Fraction Mode, Calculation Modes | |
12409 | @subsection Infinite Mode | |
12410 | ||
12411 | @noindent | |
12412 | @cindex Infinite mode | |
12413 | The Calculator normally treats results like @cite{1 / 0} as errors; | |
12414 | formulas like this are left in unsimplified form. But Calc can be | |
12415 | put into a mode where such calculations instead produce ``infinite'' | |
12416 | results. | |
12417 | ||
12418 | @kindex m i | |
12419 | @pindex calc-infinite-mode | |
12420 | The @kbd{m i} (@code{calc-infinite-mode}) command turns this mode | |
12421 | on and off. When the mode is off, infinities do not arise except | |
12422 | in calculations that already had infinities as inputs. (One exception | |
12423 | is that infinite open intervals like @samp{[0 .. inf)} can be | |
12424 | generated; however, intervals closed at infinity (@samp{[0 .. inf]}) | |
12425 | will not be generated when infinite mode is off.) | |
12426 | ||
12427 | With infinite mode turned on, @samp{1 / 0} will generate @code{uinf}, | |
12428 | an undirected infinity. @xref{Infinities}, for a discussion of the | |
12429 | difference between @code{inf} and @code{uinf}. Also, @cite{0 / 0} | |
12430 | evaluates to @code{nan}, the ``indeterminate'' symbol. Various other | |
12431 | functions can also return infinities in this mode; for example, | |
12432 | @samp{ln(0) = -inf}, and @samp{gamma(-7) = uinf}. Once again, | |
12433 | note that @samp{exp(inf) = inf} regardless of infinite mode because | |
12434 | this calculation has infinity as an input. | |
12435 | ||
12436 | @cindex Positive infinite mode | |
12437 | The @kbd{m i} command with a numeric prefix argument of zero, | |
12438 | i.e., @kbd{C-u 0 m i}, turns on a ``positive infinite mode'' in | |
177c0ea7 | 12439 | which zero is treated as positive instead of being directionless. |
d7b8e6c6 EZ |
12440 | Thus, @samp{1 / 0 = inf} and @samp{-1 / 0 = -inf} in this mode. |
12441 | Note that zero never actually has a sign in Calc; there are no | |
12442 | separate representations for @i{+0} and @i{-0}. Positive | |
12443 | infinite mode merely changes the interpretation given to the | |
12444 | single symbol, @samp{0}. One consequence of this is that, while | |
12445 | you might expect @samp{1 / -0 = -inf}, actually @samp{1 / -0} | |
12446 | is equivalent to @samp{1 / 0}, which is equal to positive @code{inf}. | |
12447 | ||
12448 | @node Symbolic Mode, Matrix Mode, Infinite Mode, Calculation Modes | |
12449 | @subsection Symbolic Mode | |
12450 | ||
12451 | @noindent | |
12452 | @cindex Symbolic mode | |
12453 | @cindex Inexact results | |
12454 | Calculations are normally performed numerically wherever possible. | |
12455 | For example, the @code{calc-sqrt} command, or @code{sqrt} function in an | |
12456 | algebraic expression, produces a numeric answer if the argument is a | |
12457 | number or a symbolic expression if the argument is an expression: | |
12458 | @kbd{2 Q} pushes 1.4142 but @kbd{@key{'} x+1 @key{RET} Q} pushes @samp{sqrt(x+1)}. | |
12459 | ||
12460 | @kindex m s | |
12461 | @pindex calc-symbolic-mode | |
12462 | In @dfn{symbolic mode}, controlled by the @kbd{m s} (@code{calc-symbolic-mode}) | |
12463 | command, functions which would produce inexact, irrational results are | |
12464 | left in symbolic form. Thus @kbd{16 Q} pushes 4, but @kbd{2 Q} pushes | |
12465 | @samp{sqrt(2)}. | |
12466 | ||
12467 | @kindex N | |
12468 | @pindex calc-eval-num | |
12469 | The shift-@kbd{N} (@code{calc-eval-num}) command evaluates numerically | |
12470 | the expression at the top of the stack, by temporarily disabling | |
12471 | @code{calc-symbolic-mode} and executing @kbd{=} (@code{calc-evaluate}). | |
12472 | Given a numeric prefix argument, it also | |
12473 | sets the floating-point precision to the specified value for the duration | |
12474 | of the command.@refill | |
12475 | ||
12476 | To evaluate a formula numerically without expanding the variables it | |
12477 | contains, you can use the key sequence @kbd{m s a v m s} (this uses | |
12478 | @code{calc-alg-evaluate}, which resimplifies but doesn't evaluate | |
12479 | variables.) | |
12480 | ||
12481 | @node Matrix Mode, Automatic Recomputation, Symbolic Mode, Calculation Modes | |
12482 | @subsection Matrix and Scalar Modes | |
12483 | ||
12484 | @noindent | |
12485 | @cindex Matrix mode | |
12486 | @cindex Scalar mode | |
12487 | Calc sometimes makes assumptions during algebraic manipulation that | |
12488 | are awkward or incorrect when vectors and matrices are involved. | |
12489 | Calc has two modes, @dfn{matrix mode} and @dfn{scalar mode}, which | |
12490 | modify its behavior around vectors in useful ways. | |
12491 | ||
12492 | @kindex m v | |
12493 | @pindex calc-matrix-mode | |
12494 | Press @kbd{m v} (@code{calc-matrix-mode}) once to enter matrix mode. | |
12495 | In this mode, all objects are assumed to be matrices unless provably | |
12496 | otherwise. One major effect is that Calc will no longer consider | |
12497 | multiplication to be commutative. (Recall that in matrix arithmetic, | |
12498 | @samp{A*B} is not the same as @samp{B*A}.) This assumption affects | |
12499 | rewrite rules and algebraic simplification. Another effect of this | |
12500 | mode is that calculations that would normally produce constants like | |
12501 | 0 and 1 (e.g., @cite{a - a} and @cite{a / a}, respectively) will now | |
12502 | produce function calls that represent ``generic'' zero or identity | |
12503 | matrices: @samp{idn(0)}, @samp{idn(1)}. The @code{idn} function | |
12504 | @samp{idn(@var{a},@var{n})} returns @var{a} times an @var{n}x@var{n} | |
12505 | identity matrix; if @var{n} is omitted, it doesn't know what | |
12506 | dimension to use and so the @code{idn} call remains in symbolic | |
12507 | form. However, if this generic identity matrix is later combined | |
12508 | with a matrix whose size is known, it will be converted into | |
12509 | a true identity matrix of the appropriate size. On the other hand, | |
12510 | if it is combined with a scalar (as in @samp{idn(1) + 2}), Calc | |
12511 | will assume it really was a scalar after all and produce, e.g., 3. | |
12512 | ||
12513 | Press @kbd{m v} a second time to get scalar mode. Here, objects are | |
12514 | assumed @emph{not} to be vectors or matrices unless provably so. | |
12515 | For example, normally adding a variable to a vector, as in | |
12516 | @samp{[x, y, z] + a}, will leave the sum in symbolic form because | |
12517 | as far as Calc knows, @samp{a} could represent either a number or | |
12518 | another 3-vector. In scalar mode, @samp{a} is assumed to be a | |
12519 | non-vector, and the addition is evaluated to @samp{[x+a, y+a, z+a]}. | |
12520 | ||
12521 | Press @kbd{m v} a third time to return to the normal mode of operation. | |
12522 | ||
12523 | If you press @kbd{m v} with a numeric prefix argument @var{n}, you | |
12524 | get a special ``dimensioned matrix mode'' in which matrices of | |
12525 | unknown size are assumed to be @var{n}x@var{n} square matrices. | |
12526 | Then, the function call @samp{idn(1)} will expand into an actual | |
12527 | matrix rather than representing a ``generic'' matrix. | |
12528 | ||
12529 | @cindex Declaring scalar variables | |
12530 | Of course these modes are approximations to the true state of | |
12531 | affairs, which is probably that some quantities will be matrices | |
12532 | and others will be scalars. One solution is to ``declare'' | |
12533 | certain variables or functions to be scalar-valued. | |
12534 | @xref{Declarations}, to see how to make declarations in Calc. | |
12535 | ||
12536 | There is nothing stopping you from declaring a variable to be | |
12537 | scalar and then storing a matrix in it; however, if you do, the | |
12538 | results you get from Calc may not be valid. Suppose you let Calc | |
12539 | get the result @samp{[x+a, y+a, z+a]} shown above, and then stored | |
12540 | @samp{[1, 2, 3]} in @samp{a}. The result would not be the same as | |
12541 | for @samp{[x, y, z] + [1, 2, 3]}, but that's because you have broken | |
12542 | your earlier promise to Calc that @samp{a} would be scalar. | |
12543 | ||
12544 | Another way to mix scalars and matrices is to use selections | |
12545 | (@pxref{Selecting Subformulas}). Use matrix mode when operating on | |
12546 | your formula normally; then, to apply scalar mode to a certain part | |
12547 | of the formula without affecting the rest just select that part, | |
12548 | change into scalar mode and press @kbd{=} to resimplify the part | |
12549 | under this mode, then change back to matrix mode before deselecting. | |
12550 | ||
12551 | @node Automatic Recomputation, Working Message, Matrix Mode, Calculation Modes | |
12552 | @subsection Automatic Recomputation | |
12553 | ||
12554 | @noindent | |
12555 | The @dfn{evaluates-to} operator, @samp{=>}, has the special | |
12556 | property that any @samp{=>} formulas on the stack are recomputed | |
12557 | whenever variable values or mode settings that might affect them | |
12558 | are changed. @xref{Evaluates-To Operator}. | |
12559 | ||
12560 | @kindex m C | |
12561 | @pindex calc-auto-recompute | |
12562 | The @kbd{m C} (@code{calc-auto-recompute}) command turns this | |
12563 | automatic recomputation on and off. If you turn it off, Calc will | |
12564 | not update @samp{=>} operators on the stack (nor those in the | |
12565 | attached Embedded Mode buffer, if there is one). They will not | |
12566 | be updated unless you explicitly do so by pressing @kbd{=} or until | |
12567 | you press @kbd{m C} to turn recomputation back on. (While automatic | |
12568 | recomputation is off, you can think of @kbd{m C m C} as a command | |
12569 | to update all @samp{=>} operators while leaving recomputation off.) | |
12570 | ||
12571 | To update @samp{=>} operators in an Embedded buffer while | |
12572 | automatic recomputation is off, use @w{@kbd{M-# u}}. | |
12573 | @xref{Embedded Mode}. | |
12574 | ||
12575 | @node Working Message, , Automatic Recomputation, Calculation Modes | |
12576 | @subsection Working Messages | |
12577 | ||
12578 | @noindent | |
12579 | @cindex Performance | |
12580 | @cindex Working messages | |
12581 | Since the Calculator is written entirely in Emacs Lisp, which is not | |
12582 | designed for heavy numerical work, many operations are quite slow. | |
12583 | The Calculator normally displays the message @samp{Working...} in the | |
12584 | echo area during any command that may be slow. In addition, iterative | |
12585 | operations such as square roots and trigonometric functions display the | |
12586 | intermediate result at each step. Both of these types of messages can | |
12587 | be disabled if you find them distracting. | |
12588 | ||
12589 | @kindex m w | |
12590 | @pindex calc-working | |
12591 | Type @kbd{m w} (@code{calc-working}) with a numeric prefix of 0 to | |
12592 | disable all ``working'' messages. Use a numeric prefix of 1 to enable | |
12593 | only the plain @samp{Working...} message. Use a numeric prefix of 2 to | |
12594 | see intermediate results as well. With no numeric prefix this displays | |
12595 | the current mode.@refill | |
12596 | ||
12597 | While it may seem that the ``working'' messages will slow Calc down | |
12598 | considerably, experiments have shown that their impact is actually | |
12599 | quite small. But if your terminal is slow you may find that it helps | |
12600 | to turn the messages off. | |
12601 | ||
12602 | @node Simplification Modes, Declarations, Calculation Modes, Mode Settings | |
12603 | @section Simplification Modes | |
12604 | ||
12605 | @noindent | |
12606 | The current @dfn{simplification mode} controls how numbers and formulas | |
12607 | are ``normalized'' when being taken from or pushed onto the stack. | |
12608 | Some normalizations are unavoidable, such as rounding floating-point | |
12609 | results to the current precision, and reducing fractions to simplest | |
12610 | form. Others, such as simplifying a formula like @cite{a+a} (or @cite{2+3}), | |
12611 | are done by default but can be turned off when necessary. | |
12612 | ||
12613 | When you press a key like @kbd{+} when @cite{2} and @cite{3} are on the | |
12614 | stack, Calc pops these numbers, normalizes them, creates the formula | |
12615 | @cite{2+3}, normalizes it, and pushes the result. Of course the standard | |
12616 | rules for normalizing @cite{2+3} will produce the result @cite{5}. | |
12617 | ||
12618 | Simplification mode commands consist of the lower-case @kbd{m} prefix key | |
12619 | followed by a shifted letter. | |
12620 | ||
12621 | @kindex m O | |
12622 | @pindex calc-no-simplify-mode | |
12623 | The @kbd{m O} (@code{calc-no-simplify-mode}) command turns off all optional | |
12624 | simplifications. These would leave a formula like @cite{2+3} alone. In | |
12625 | fact, nothing except simple numbers are ever affected by normalization | |
12626 | in this mode. | |
12627 | ||
12628 | @kindex m N | |
12629 | @pindex calc-num-simplify-mode | |
12630 | The @kbd{m N} (@code{calc-num-simplify-mode}) command turns off simplification | |
12631 | of any formulas except those for which all arguments are constants. For | |
12632 | example, @cite{1+2} is simplified to @cite{3}, and @cite{a+(2-2)} is | |
12633 | simplified to @cite{a+0} but no further, since one argument of the sum | |
12634 | is not a constant. Unfortunately, @cite{(a+2)-2} is @emph{not} simplified | |
12635 | because the top-level @samp{-} operator's arguments are not both | |
12636 | constant numbers (one of them is the formula @cite{a+2}). | |
12637 | A constant is a number or other numeric object (such as a constant | |
12638 | error form or modulo form), or a vector all of whose | |
12639 | elements are constant.@refill | |
12640 | ||
12641 | @kindex m D | |
12642 | @pindex calc-default-simplify-mode | |
12643 | The @kbd{m D} (@code{calc-default-simplify-mode}) command restores the | |
12644 | default simplifications for all formulas. This includes many easy and | |
12645 | fast algebraic simplifications such as @cite{a+0} to @cite{a}, and | |
12646 | @cite{a + 2 a} to @cite{3 a}, as well as evaluating functions like | |
12647 | @cite{@t{deriv}(x^2, x)} to @cite{2 x}. | |
12648 | ||
12649 | @kindex m B | |
12650 | @pindex calc-bin-simplify-mode | |
12651 | The @kbd{m B} (@code{calc-bin-simplify-mode}) mode applies the default | |
12652 | simplifications to a result and then, if the result is an integer, | |
12653 | uses the @kbd{b c} (@code{calc-clip}) command to clip the integer according | |
12654 | to the current binary word size. @xref{Binary Functions}. Real numbers | |
12655 | are rounded to the nearest integer and then clipped; other kinds of | |
12656 | results (after the default simplifications) are left alone. | |
12657 | ||
12658 | @kindex m A | |
12659 | @pindex calc-alg-simplify-mode | |
12660 | The @kbd{m A} (@code{calc-alg-simplify-mode}) mode does algebraic | |
12661 | simplification; it applies all the default simplifications, and also | |
12662 | the more powerful (and slower) simplifications made by @kbd{a s} | |
12663 | (@code{calc-simplify}). @xref{Algebraic Simplifications}. | |
12664 | ||
12665 | @kindex m E | |
12666 | @pindex calc-ext-simplify-mode | |
12667 | The @kbd{m E} (@code{calc-ext-simplify-mode}) mode does ``extended'' | |
12668 | algebraic simplification, as by the @kbd{a e} (@code{calc-simplify-extended}) | |
12669 | command. @xref{Unsafe Simplifications}. | |
12670 | ||
12671 | @kindex m U | |
12672 | @pindex calc-units-simplify-mode | |
12673 | The @kbd{m U} (@code{calc-units-simplify-mode}) mode does units | |
12674 | simplification; it applies the command @kbd{u s} | |
12675 | (@code{calc-simplify-units}), which in turn | |
12676 | is a superset of @kbd{a s}. In this mode, variable names which | |
12677 | are identifiable as unit names (like @samp{mm} for ``millimeters'') | |
12678 | are simplified with their unit definitions in mind.@refill | |
12679 | ||
12680 | A common technique is to set the simplification mode down to the lowest | |
12681 | amount of simplification you will allow to be applied automatically, then | |
12682 | use manual commands like @kbd{a s} and @kbd{c c} (@code{calc-clean}) to | |
12683 | perform higher types of simplifications on demand. @xref{Algebraic | |
12684 | Definitions}, for another sample use of no-simplification mode.@refill | |
12685 | ||
12686 | @node Declarations, Display Modes, Simplification Modes, Mode Settings | |
12687 | @section Declarations | |
12688 | ||
12689 | @noindent | |
12690 | A @dfn{declaration} is a statement you make that promises you will | |
12691 | use a certain variable or function in a restricted way. This may | |
12692 | give Calc the freedom to do things that it couldn't do if it had to | |
12693 | take the fully general situation into account. | |
12694 | ||
12695 | @menu | |
12696 | * Declaration Basics:: | |
12697 | * Kinds of Declarations:: | |
12698 | * Functions for Declarations:: | |
12699 | @end menu | |
12700 | ||
12701 | @node Declaration Basics, Kinds of Declarations, Declarations, Declarations | |
12702 | @subsection Declaration Basics | |
12703 | ||
12704 | @noindent | |
12705 | @kindex s d | |
12706 | @pindex calc-declare-variable | |
12707 | The @kbd{s d} (@code{calc-declare-variable}) command is the easiest | |
12708 | way to make a declaration for a variable. This command prompts for | |
12709 | the variable name, then prompts for the declaration. The default | |
12710 | at the declaration prompt is the previous declaration, if any. | |
12711 | You can edit this declaration, or press @kbd{C-k} to erase it and | |
12712 | type a new declaration. (Or, erase it and press @key{RET} to clear | |
12713 | the declaration, effectively ``undeclaring'' the variable.) | |
12714 | ||
12715 | A declaration is in general a vector of @dfn{type symbols} and | |
12716 | @dfn{range} values. If there is only one type symbol or range value, | |
12717 | you can write it directly rather than enclosing it in a vector. | |
5d67986c RS |
12718 | For example, @kbd{s d foo @key{RET} real @key{RET}} declares @code{foo} to |
12719 | be a real number, and @kbd{s d bar @key{RET} [int, const, [1..6]] @key{RET}} | |
d7b8e6c6 EZ |
12720 | declares @code{bar} to be a constant integer between 1 and 6. |
12721 | (Actually, you can omit the outermost brackets and Calc will | |
5d67986c | 12722 | provide them for you: @kbd{s d bar @key{RET} int, const, [1..6] @key{RET}}.) |
d7b8e6c6 EZ |
12723 | |
12724 | @cindex @code{Decls} variable | |
12725 | @vindex Decls | |
12726 | Declarations in Calc are kept in a special variable called @code{Decls}. | |
12727 | This variable encodes the set of all outstanding declarations in | |
12728 | the form of a matrix. Each row has two elements: A variable or | |
12729 | vector of variables declared by that row, and the declaration | |
12730 | specifier as described above. You can use the @kbd{s D} command to | |
12731 | edit this variable if you wish to see all the declarations at once. | |
12732 | @xref{Operations on Variables}, for a description of this command | |
12733 | and the @kbd{s p} command that allows you to save your declarations | |
12734 | permanently if you wish. | |
12735 | ||
12736 | Items being declared can also be function calls. The arguments in | |
12737 | the call are ignored; the effect is to say that this function returns | |
12738 | values of the declared type for any valid arguments. The @kbd{s d} | |
12739 | command declares only variables, so if you wish to make a function | |
12740 | declaration you will have to edit the @code{Decls} matrix yourself. | |
12741 | ||
12742 | For example, the declaration matrix | |
12743 | ||
d7b8e6c6 | 12744 | @smallexample |
5d67986c | 12745 | @group |
d7b8e6c6 EZ |
12746 | [ [ foo, real ] |
12747 | [ [j, k, n], int ] | |
12748 | [ f(1,2,3), [0 .. inf) ] ] | |
d7b8e6c6 | 12749 | @end group |
5d67986c | 12750 | @end smallexample |
d7b8e6c6 EZ |
12751 | |
12752 | @noindent | |
12753 | declares that @code{foo} represents a real number, @code{j}, @code{k} | |
12754 | and @code{n} represent integers, and the function @code{f} always | |
12755 | returns a real number in the interval shown. | |
12756 | ||
12757 | @vindex All | |
12758 | If there is a declaration for the variable @code{All}, then that | |
12759 | declaration applies to all variables that are not otherwise declared. | |
12760 | It does not apply to function names. For example, using the row | |
12761 | @samp{[All, real]} says that all your variables are real unless they | |
12762 | are explicitly declared without @code{real} in some other row. | |
12763 | The @kbd{s d} command declares @code{All} if you give a blank | |
12764 | response to the variable-name prompt. | |
12765 | ||
12766 | @node Kinds of Declarations, Functions for Declarations, Declaration Basics, Declarations | |
12767 | @subsection Kinds of Declarations | |
12768 | ||
12769 | @noindent | |
12770 | The type-specifier part of a declaration (that is, the second prompt | |
12771 | in the @kbd{s d} command) can be a type symbol, an interval, or a | |
12772 | vector consisting of zero or more type symbols followed by zero or | |
12773 | more intervals or numbers that represent the set of possible values | |
12774 | for the variable. | |
12775 | ||
d7b8e6c6 | 12776 | @smallexample |
5d67986c | 12777 | @group |
d7b8e6c6 EZ |
12778 | [ [ a, [1, 2, 3, 4, 5] ] |
12779 | [ b, [1 .. 5] ] | |
12780 | [ c, [int, 1 .. 5] ] ] | |
d7b8e6c6 | 12781 | @end group |
5d67986c | 12782 | @end smallexample |
d7b8e6c6 EZ |
12783 | |
12784 | Here @code{a} is declared to contain one of the five integers shown; | |
12785 | @code{b} is any number in the interval from 1 to 5 (any real number | |
12786 | since we haven't specified), and @code{c} is any integer in that | |
12787 | interval. Thus the declarations for @code{a} and @code{c} are | |
12788 | nearly equivalent (see below). | |
12789 | ||
12790 | The type-specifier can be the empty vector @samp{[]} to say that | |
12791 | nothing is known about a given variable's value. This is the same | |
12792 | as not declaring the variable at all except that it overrides any | |
12793 | @code{All} declaration which would otherwise apply. | |
12794 | ||
12795 | The initial value of @code{Decls} is the empty vector @samp{[]}. | |
12796 | If @code{Decls} has no stored value or if the value stored in it | |
12797 | is not valid, it is ignored and there are no declarations as far | |
12798 | as Calc is concerned. (The @kbd{s d} command will replace such a | |
12799 | malformed value with a fresh empty matrix, @samp{[]}, before recording | |
12800 | the new declaration.) Unrecognized type symbols are ignored. | |
12801 | ||
12802 | The following type symbols describe what sorts of numbers will be | |
12803 | stored in a variable: | |
12804 | ||
12805 | @table @code | |
12806 | @item int | |
12807 | Integers. | |
12808 | @item numint | |
12809 | Numerical integers. (Integers or integer-valued floats.) | |
12810 | @item frac | |
12811 | Fractions. (Rational numbers which are not integers.) | |
12812 | @item rat | |
12813 | Rational numbers. (Either integers or fractions.) | |
12814 | @item float | |
12815 | Floating-point numbers. | |
12816 | @item real | |
12817 | Real numbers. (Integers, fractions, or floats. Actually, | |
12818 | intervals and error forms with real components also count as | |
12819 | reals here.) | |
12820 | @item pos | |
12821 | Positive real numbers. (Strictly greater than zero.) | |
12822 | @item nonneg | |
12823 | Nonnegative real numbers. (Greater than or equal to zero.) | |
12824 | @item number | |
12825 | Numbers. (Real or complex.) | |
12826 | @end table | |
12827 | ||
12828 | Calc uses this information to determine when certain simplifications | |
12829 | of formulas are safe. For example, @samp{(x^y)^z} cannot be | |
12830 | simplified to @samp{x^(y z)} in general; for example, | |
12831 | @samp{((-3)^2)^1:2} is 3, but @samp{(-3)^(2*1:2) = (-3)^1} is @i{-3}. | |
12832 | However, this simplification @emph{is} safe if @code{z} is known | |
12833 | to be an integer, or if @code{x} is known to be a nonnegative | |
12834 | real number. If you have given declarations that allow Calc to | |
12835 | deduce either of these facts, Calc will perform this simplification | |
12836 | of the formula. | |
12837 | ||
12838 | Calc can apply a certain amount of logic when using declarations. | |
12839 | For example, @samp{(x^y)^(2n+1)} will be simplified if @code{n} | |
12840 | has been declared @code{int}; Calc knows that an integer times an | |
12841 | integer, plus an integer, must always be an integer. (In fact, | |
12842 | Calc would simplify @samp{(-x)^(2n+1)} to @samp{-(x^(2n+1))} since | |
12843 | it is able to determine that @samp{2n+1} must be an odd integer.) | |
12844 | ||
12845 | Similarly, @samp{(abs(x)^y)^z} will be simplified to @samp{abs(x)^(y z)} | |
12846 | because Calc knows that the @code{abs} function always returns a | |
12847 | nonnegative real. If you had a @code{myabs} function that also had | |
12848 | this property, you could get Calc to recognize it by adding the row | |
12849 | @samp{[myabs(), nonneg]} to the @code{Decls} matrix. | |
12850 | ||
12851 | One instance of this simplification is @samp{sqrt(x^2)} (since the | |
12852 | @code{sqrt} function is effectively a one-half power). Normally | |
12853 | Calc leaves this formula alone. After the command | |
5d67986c RS |
12854 | @kbd{s d x @key{RET} real @key{RET}}, however, it can simplify the formula to |
12855 | @samp{abs(x)}. And after @kbd{s d x @key{RET} nonneg @key{RET}}, Calc can | |
d7b8e6c6 EZ |
12856 | simplify this formula all the way to @samp{x}. |
12857 | ||
12858 | If there are any intervals or real numbers in the type specifier, | |
12859 | they comprise the set of possible values that the variable or | |
12860 | function being declared can have. In particular, the type symbol | |
12861 | @code{real} is effectively the same as the range @samp{[-inf .. inf]} | |
12862 | (note that infinity is included in the range of possible values); | |
12863 | @code{pos} is the same as @samp{(0 .. inf]}, and @code{nonneg} is | |
12864 | the same as @samp{[0 .. inf]}. Saying @samp{[real, [-5 .. 5]]} is | |
12865 | redundant because the fact that the variable is real can be | |
12866 | deduced just from the interval, but @samp{[int, [-5 .. 5]]} and | |
12867 | @samp{[rat, [-5 .. 5]]} are useful combinations. | |
12868 | ||
12869 | Note that the vector of intervals or numbers is in the same format | |
12870 | used by Calc's set-manipulation commands. @xref{Set Operations}. | |
12871 | ||
12872 | The type specifier @samp{[1, 2, 3]} is equivalent to | |
12873 | @samp{[numint, 1, 2, 3]}, @emph{not} to @samp{[int, 1, 2, 3]}. | |
12874 | In other words, the range of possible values means only that | |
12875 | the variable's value must be numerically equal to a number in | |
12876 | that range, but not that it must be equal in type as well. | |
12877 | Calc's set operations act the same way; @samp{in(2, [1., 2., 3.])} | |
12878 | and @samp{in(1.5, [1:2, 3:2, 5:2])} both report ``true.'' | |
12879 | ||
12880 | If you use a conflicting combination of type specifiers, the | |
12881 | results are unpredictable. An example is @samp{[pos, [0 .. 5]]}, | |
12882 | where the interval does not lie in the range described by the | |
12883 | type symbol. | |
12884 | ||
12885 | ``Real'' declarations mostly affect simplifications involving powers | |
12886 | like the one described above. Another case where they are used | |
12887 | is in the @kbd{a P} command which returns a list of all roots of a | |
12888 | polynomial; if the variable has been declared real, only the real | |
12889 | roots (if any) will be included in the list. | |
12890 | ||
12891 | ``Integer'' declarations are used for simplifications which are valid | |
12892 | only when certain values are integers (such as @samp{(x^y)^z} | |
12893 | shown above). | |
12894 | ||
12895 | Another command that makes use of declarations is @kbd{a s}, when | |
12896 | simplifying equations and inequalities. It will cancel @code{x} | |
12897 | from both sides of @samp{a x = b x} only if it is sure @code{x} | |
12898 | is non-zero, say, because it has a @code{pos} declaration. | |
12899 | To declare specifically that @code{x} is real and non-zero, | |
12900 | use @samp{[[-inf .. 0), (0 .. inf]]}. (There is no way in the | |
12901 | current notation to say that @code{x} is nonzero but not necessarily | |
12902 | real.) The @kbd{a e} command does ``unsafe'' simplifications, | |
12903 | including cancelling @samp{x} from the equation when @samp{x} is | |
12904 | not known to be nonzero. | |
12905 | ||
12906 | Another set of type symbols distinguish between scalars and vectors. | |
12907 | ||
12908 | @table @code | |
12909 | @item scalar | |
12910 | The value is not a vector. | |
12911 | @item vector | |
12912 | The value is a vector. | |
12913 | @item matrix | |
12914 | The value is a matrix (a rectangular vector of vectors). | |
12915 | @end table | |
12916 | ||
12917 | These type symbols can be combined with the other type symbols | |
12918 | described above; @samp{[int, matrix]} describes an object which | |
12919 | is a matrix of integers. | |
12920 | ||
12921 | Scalar/vector declarations are used to determine whether certain | |
12922 | algebraic operations are safe. For example, @samp{[a, b, c] + x} | |
12923 | is normally not simplified to @samp{[a + x, b + x, c + x]}, but | |
12924 | it will be if @code{x} has been declared @code{scalar}. On the | |
12925 | other hand, multiplication is usually assumed to be commutative, | |
12926 | but the terms in @samp{x y} will never be exchanged if both @code{x} | |
12927 | and @code{y} are known to be vectors or matrices. (Calc currently | |
12928 | never distinguishes between @code{vector} and @code{matrix} | |
12929 | declarations.) | |
12930 | ||
12931 | @xref{Matrix Mode}, for a discussion of ``matrix mode'' and | |
12932 | ``scalar mode,'' which are similar to declaring @samp{[All, matrix]} | |
12933 | or @samp{[All, scalar]} but much more convenient. | |
12934 | ||
12935 | One more type symbol that is recognized is used with the @kbd{H a d} | |
12936 | command for taking total derivatives of a formula. @xref{Calculus}. | |
12937 | ||
12938 | @table @code | |
12939 | @item const | |
12940 | The value is a constant with respect to other variables. | |
12941 | @end table | |
12942 | ||
12943 | Calc does not check the declarations for a variable when you store | |
12944 | a value in it. However, storing @i{-3.5} in a variable that has | |
12945 | been declared @code{pos}, @code{int}, or @code{matrix} may have | |
12946 | unexpected effects; Calc may evaluate @samp{sqrt(x^2)} to @cite{3.5} | |
12947 | if it substitutes the value first, or to @cite{-3.5} if @code{x} | |
12948 | was declared @code{pos} and the formula @samp{sqrt(x^2)} is | |
12949 | simplified to @samp{x} before the value is substituted. Before | |
12950 | using a variable for a new purpose, it is best to use @kbd{s d} | |
12951 | or @kbd{s D} to check to make sure you don't still have an old | |
12952 | declaration for the variable that will conflict with its new meaning. | |
12953 | ||
12954 | @node Functions for Declarations, , Kinds of Declarations, Declarations | |
12955 | @subsection Functions for Declarations | |
12956 | ||
12957 | @noindent | |
12958 | Calc has a set of functions for accessing the current declarations | |
12959 | in a convenient manner. These functions return 1 if the argument | |
12960 | can be shown to have the specified property, or 0 if the argument | |
12961 | can be shown @emph{not} to have that property; otherwise they are | |
12962 | left unevaluated. These functions are suitable for use with rewrite | |
12963 | rules (@pxref{Conditional Rewrite Rules}) or programming constructs | |
12964 | (@pxref{Conditionals in Macros}). They can be entered only using | |
12965 | algebraic notation. @xref{Logical Operations}, for functions | |
12966 | that perform other tests not related to declarations. | |
12967 | ||
12968 | For example, @samp{dint(17)} returns 1 because 17 is an integer, as | |
12969 | do @samp{dint(n)} and @samp{dint(2 n - 3)} if @code{n} has been declared | |
12970 | @code{int}, but @samp{dint(2.5)} and @samp{dint(n + 0.5)} return 0. | |
12971 | Calc consults knowledge of its own built-in functions as well as your | |
12972 | own declarations: @samp{dint(floor(x))} returns 1. | |
12973 | ||
5d67986c RS |
12974 | @ignore |
12975 | @starindex | |
12976 | @end ignore | |
d7b8e6c6 | 12977 | @tindex dint |
5d67986c RS |
12978 | @ignore |
12979 | @starindex | |
12980 | @end ignore | |
d7b8e6c6 | 12981 | @tindex dnumint |
5d67986c RS |
12982 | @ignore |
12983 | @starindex | |
12984 | @end ignore | |
d7b8e6c6 EZ |
12985 | @tindex dnatnum |
12986 | The @code{dint} function checks if its argument is an integer. | |
12987 | The @code{dnatnum} function checks if its argument is a natural | |
12988 | number, i.e., a nonnegative integer. The @code{dnumint} function | |
12989 | checks if its argument is numerically an integer, i.e., either an | |
12990 | integer or an integer-valued float. Note that these and the other | |
12991 | data type functions also accept vectors or matrices composed of | |
12992 | suitable elements, and that real infinities @samp{inf} and @samp{-inf} | |
12993 | are considered to be integers for the purposes of these functions. | |
12994 | ||
5d67986c RS |
12995 | @ignore |
12996 | @starindex | |
12997 | @end ignore | |
d7b8e6c6 EZ |
12998 | @tindex drat |
12999 | The @code{drat} function checks if its argument is rational, i.e., | |
13000 | an integer or fraction. Infinities count as rational, but intervals | |
13001 | and error forms do not. | |
13002 | ||
5d67986c RS |
13003 | @ignore |
13004 | @starindex | |
13005 | @end ignore | |
d7b8e6c6 EZ |
13006 | @tindex dreal |
13007 | The @code{dreal} function checks if its argument is real. This | |
13008 | includes integers, fractions, floats, real error forms, and intervals. | |
13009 | ||
5d67986c RS |
13010 | @ignore |
13011 | @starindex | |
13012 | @end ignore | |
d7b8e6c6 EZ |
13013 | @tindex dimag |
13014 | The @code{dimag} function checks if its argument is imaginary, | |
13015 | i.e., is mathematically equal to a real number times @cite{i}. | |
13016 | ||
5d67986c RS |
13017 | @ignore |
13018 | @starindex | |
13019 | @end ignore | |
d7b8e6c6 | 13020 | @tindex dpos |
5d67986c RS |
13021 | @ignore |
13022 | @starindex | |
13023 | @end ignore | |
d7b8e6c6 | 13024 | @tindex dneg |
5d67986c RS |
13025 | @ignore |
13026 | @starindex | |
13027 | @end ignore | |
d7b8e6c6 EZ |
13028 | @tindex dnonneg |
13029 | The @code{dpos} function checks for positive (but nonzero) reals. | |
13030 | The @code{dneg} function checks for negative reals. The @code{dnonneg} | |
13031 | function checks for nonnegative reals, i.e., reals greater than or | |
13032 | equal to zero. Note that the @kbd{a s} command can simplify an | |
13033 | expression like @cite{x > 0} to 1 or 0 using @code{dpos}, and that | |
13034 | @kbd{a s} is effectively applied to all conditions in rewrite rules, | |
13035 | so the actual functions @code{dpos}, @code{dneg}, and @code{dnonneg} | |
13036 | are rarely necessary. | |
13037 | ||
5d67986c RS |
13038 | @ignore |
13039 | @starindex | |
13040 | @end ignore | |
d7b8e6c6 EZ |
13041 | @tindex dnonzero |
13042 | The @code{dnonzero} function checks that its argument is nonzero. | |
13043 | This includes all nonzero real or complex numbers, all intervals that | |
13044 | do not include zero, all nonzero modulo forms, vectors all of whose | |
13045 | elements are nonzero, and variables or formulas whose values can be | |
13046 | deduced to be nonzero. It does not include error forms, since they | |
13047 | represent values which could be anything including zero. (This is | |
13048 | also the set of objects considered ``true'' in conditional contexts.) | |
13049 | ||
5d67986c RS |
13050 | @ignore |
13051 | @starindex | |
13052 | @end ignore | |
d7b8e6c6 | 13053 | @tindex deven |
5d67986c RS |
13054 | @ignore |
13055 | @starindex | |
13056 | @end ignore | |
d7b8e6c6 EZ |
13057 | @tindex dodd |
13058 | The @code{deven} function returns 1 if its argument is known to be | |
13059 | an even integer (or integer-valued float); it returns 0 if its argument | |
13060 | is known not to be even (because it is known to be odd or a non-integer). | |
13061 | The @kbd{a s} command uses this to simplify a test of the form | |
13062 | @samp{x % 2 = 0}. There is also an analogous @code{dodd} function. | |
13063 | ||
5d67986c RS |
13064 | @ignore |
13065 | @starindex | |
13066 | @end ignore | |
d7b8e6c6 EZ |
13067 | @tindex drange |
13068 | The @code{drange} function returns a set (an interval or a vector | |
13069 | of intervals and/or numbers; @pxref{Set Operations}) that describes | |
13070 | the set of possible values of its argument. If the argument is | |
13071 | a variable or a function with a declaration, the range is copied | |
13072 | from the declaration. Otherwise, the possible signs of the | |
13073 | expression are determined using a method similar to @code{dpos}, | |
13074 | etc., and a suitable set like @samp{[0 .. inf]} is returned. If | |
13075 | the expression is not provably real, the @code{drange} function | |
13076 | remains unevaluated. | |
13077 | ||
5d67986c RS |
13078 | @ignore |
13079 | @starindex | |
13080 | @end ignore | |
d7b8e6c6 EZ |
13081 | @tindex dscalar |
13082 | The @code{dscalar} function returns 1 if its argument is provably | |
13083 | scalar, or 0 if its argument is provably non-scalar. It is left | |
13084 | unevaluated if this cannot be determined. (If matrix mode or scalar | |
13085 | mode are in effect, this function returns 1 or 0, respectively, | |
13086 | if it has no other information.) When Calc interprets a condition | |
13087 | (say, in a rewrite rule) it considers an unevaluated formula to be | |
13088 | ``false.'' Thus, @samp{dscalar(a)} is ``true'' only if @code{a} is | |
13089 | provably scalar, and @samp{!dscalar(a)} is ``true'' only if @code{a} | |
13090 | is provably non-scalar; both are ``false'' if there is insufficient | |
13091 | information to tell. | |
13092 | ||
13093 | @node Display Modes, Language Modes, Declarations, Mode Settings | |
13094 | @section Display Modes | |
13095 | ||
13096 | @noindent | |
13097 | The commands in this section are two-key sequences beginning with the | |
13098 | @kbd{d} prefix. The @kbd{d l} (@code{calc-line-numbering}) and @kbd{d b} | |
13099 | (@code{calc-line-breaking}) commands are described elsewhere; | |
13100 | @pxref{Stack Basics} and @pxref{Normal Language Modes}, respectively. | |
13101 | Display formats for vectors and matrices are also covered elsewhere; | |
13102 | @pxref{Vector and Matrix Formats}.@refill | |
13103 | ||
13104 | One thing all display modes have in common is their treatment of the | |
13105 | @kbd{H} prefix. This prefix causes any mode command that would normally | |
13106 | refresh the stack to leave the stack display alone. The word ``Dirty'' | |
13107 | will appear in the mode line when Calc thinks the stack display may not | |
13108 | reflect the latest mode settings. | |
13109 | ||
5d67986c | 13110 | @kindex d @key{RET} |
d7b8e6c6 | 13111 | @pindex calc-refresh-top |
5d67986c | 13112 | The @kbd{d @key{RET}} (@code{calc-refresh-top}) command reformats the |
d7b8e6c6 EZ |
13113 | top stack entry according to all the current modes. Positive prefix |
13114 | arguments reformat the top @var{n} entries; negative prefix arguments | |
13115 | reformat the specified entry, and a prefix of zero is equivalent to | |
5d67986c RS |
13116 | @kbd{d @key{SPC}} (@code{calc-refresh}), which reformats the entire stack. |
13117 | For example, @kbd{H d s M-2 d @key{RET}} changes to scientific notation | |
d7b8e6c6 EZ |
13118 | but reformats only the top two stack entries in the new mode. |
13119 | ||
13120 | The @kbd{I} prefix has another effect on the display modes. The mode | |
13121 | is set only temporarily; the top stack entry is reformatted according | |
13122 | to that mode, then the original mode setting is restored. In other | |
5d67986c | 13123 | words, @kbd{I d s} is equivalent to @kbd{H d s d @key{RET} H d (@var{old mode})}. |
d7b8e6c6 EZ |
13124 | |
13125 | @menu | |
13126 | * Radix Modes:: | |
13127 | * Grouping Digits:: | |
13128 | * Float Formats:: | |
13129 | * Complex Formats:: | |
13130 | * Fraction Formats:: | |
13131 | * HMS Formats:: | |
13132 | * Date Formats:: | |
13133 | * Truncating the Stack:: | |
13134 | * Justification:: | |
13135 | * Labels:: | |
13136 | @end menu | |
13137 | ||
13138 | @node Radix Modes, Grouping Digits, Display Modes, Display Modes | |
13139 | @subsection Radix Modes | |
13140 | ||
13141 | @noindent | |
13142 | @cindex Radix display | |
13143 | @cindex Non-decimal numbers | |
13144 | @cindex Decimal and non-decimal numbers | |
13145 | Calc normally displays numbers in decimal (@dfn{base-10} or @dfn{radix-10}) | |
13146 | notation. Calc can actually display in any radix from two (binary) to 36. | |
13147 | When the radix is above 10, the letters @code{A} to @code{Z} are used as | |
13148 | digits. When entering such a number, letter keys are interpreted as | |
13149 | potential digits rather than terminating numeric entry mode. | |
13150 | ||
13151 | @kindex d 2 | |
13152 | @kindex d 8 | |
13153 | @kindex d 6 | |
13154 | @kindex d 0 | |
13155 | @cindex Hexadecimal integers | |
13156 | @cindex Octal integers | |
13157 | The key sequences @kbd{d 2}, @kbd{d 8}, @kbd{d 6}, and @kbd{d 0} select | |
13158 | binary, octal, hexadecimal, and decimal as the current display radix, | |
13159 | respectively. Numbers can always be entered in any radix, though the | |
13160 | current radix is used as a default if you press @kbd{#} without any initial | |
13161 | digits. A number entered without a @kbd{#} is @emph{always} interpreted | |
13162 | as decimal.@refill | |
13163 | ||
13164 | @kindex d r | |
13165 | @pindex calc-radix | |
13166 | To set the radix generally, use @kbd{d r} (@code{calc-radix}) and enter | |
13167 | an integer from 2 to 36. You can specify the radix as a numeric prefix | |
13168 | argument; otherwise you will be prompted for it. | |
13169 | ||
13170 | @kindex d z | |
13171 | @pindex calc-leading-zeros | |
13172 | @cindex Leading zeros | |
13173 | Integers normally are displayed with however many digits are necessary to | |
13174 | represent the integer and no more. The @kbd{d z} (@code{calc-leading-zeros}) | |
13175 | command causes integers to be padded out with leading zeros according to the | |
13176 | current binary word size. (@xref{Binary Functions}, for a discussion of | |
13177 | word size.) If the absolute value of the word size is @cite{w}, all integers | |
13178 | are displayed with at least enough digits to represent @c{$2^w-1$} | |
13179 | @cite{(2^w)-1} in the | |
13180 | current radix. (Larger integers will still be displayed in their entirety.) | |
13181 | ||
13182 | @node Grouping Digits, Float Formats, Radix Modes, Display Modes | |
13183 | @subsection Grouping Digits | |
13184 | ||
13185 | @noindent | |
13186 | @kindex d g | |
13187 | @pindex calc-group-digits | |
13188 | @cindex Grouping digits | |
13189 | @cindex Digit grouping | |
13190 | Long numbers can be hard to read if they have too many digits. For | |
13191 | example, the factorial of 30 is 33 digits long! Press @kbd{d g} | |
13192 | (@code{calc-group-digits}) to enable @dfn{grouping} mode, in which digits | |
13193 | are displayed in clumps of 3 or 4 (depending on the current radix) | |
13194 | separated by commas. | |
13195 | ||
13196 | The @kbd{d g} command toggles grouping on and off. | |
13197 | With a numerix prefix of 0, this command displays the current state of | |
13198 | the grouping flag; with an argument of minus one it disables grouping; | |
13199 | with a positive argument @cite{N} it enables grouping on every @cite{N} | |
13200 | digits. For floating-point numbers, grouping normally occurs only | |
13201 | before the decimal point. A negative prefix argument @cite{-N} enables | |
13202 | grouping every @cite{N} digits both before and after the decimal point.@refill | |
13203 | ||
13204 | @kindex d , | |
13205 | @pindex calc-group-char | |
13206 | The @kbd{d ,} (@code{calc-group-char}) command allows you to choose any | |
13207 | character as the grouping separator. The default is the comma character. | |
13208 | If you find it difficult to read vectors of large integers grouped with | |
13209 | commas, you may wish to use spaces or some other character instead. | |
13210 | This command takes the next character you type, whatever it is, and | |
13211 | uses it as the digit separator. As a special case, @kbd{d , \} selects | |
13212 | @samp{\,} (@TeX{}'s thin-space symbol) as the digit separator. | |
13213 | ||
13214 | Please note that grouped numbers will not generally be parsed correctly | |
13215 | if re-read in textual form, say by the use of @kbd{M-# y} and @kbd{M-# g}. | |
13216 | (@xref{Kill and Yank}, for details on these commands.) One exception is | |
13217 | the @samp{\,} separator, which doesn't interfere with parsing because it | |
13218 | is ignored by @TeX{} language mode. | |
13219 | ||
13220 | @node Float Formats, Complex Formats, Grouping Digits, Display Modes | |
13221 | @subsection Float Formats | |
13222 | ||
13223 | @noindent | |
13224 | Floating-point quantities are normally displayed in standard decimal | |
13225 | form, with scientific notation used if the exponent is especially high | |
13226 | or low. All significant digits are normally displayed. The commands | |
13227 | in this section allow you to choose among several alternative display | |
13228 | formats for floats. | |
13229 | ||
13230 | @kindex d n | |
13231 | @pindex calc-normal-notation | |
13232 | The @kbd{d n} (@code{calc-normal-notation}) command selects the normal | |
13233 | display format. All significant figures in a number are displayed. | |
13234 | With a positive numeric prefix, numbers are rounded if necessary to | |
13235 | that number of significant digits. With a negative numerix prefix, | |
13236 | the specified number of significant digits less than the current | |
13237 | precision is used. (Thus @kbd{C-u -2 d n} displays 10 digits if the | |
13238 | current precision is 12.) | |
13239 | ||
13240 | @kindex d f | |
13241 | @pindex calc-fix-notation | |
13242 | The @kbd{d f} (@code{calc-fix-notation}) command selects fixed-point | |
13243 | notation. The numeric argument is the number of digits after the | |
13244 | decimal point, zero or more. This format will relax into scientific | |
13245 | notation if a nonzero number would otherwise have been rounded all the | |
13246 | way to zero. Specifying a negative number of digits is the same as | |
13247 | for a positive number, except that small nonzero numbers will be rounded | |
13248 | to zero rather than switching to scientific notation. | |
13249 | ||
13250 | @kindex d s | |
13251 | @pindex calc-sci-notation | |
13252 | @cindex Scientific notation, display of | |
13253 | The @kbd{d s} (@code{calc-sci-notation}) command selects scientific | |
13254 | notation. A positive argument sets the number of significant figures | |
13255 | displayed, of which one will be before and the rest after the decimal | |
13256 | point. A negative argument works the same as for @kbd{d n} format. | |
13257 | The default is to display all significant digits. | |
13258 | ||
13259 | @kindex d e | |
13260 | @pindex calc-eng-notation | |
13261 | @cindex Engineering notation, display of | |
13262 | The @kbd{d e} (@code{calc-eng-notation}) command selects engineering | |
13263 | notation. This is similar to scientific notation except that the | |
13264 | exponent is rounded down to a multiple of three, with from one to three | |
13265 | digits before the decimal point. An optional numeric prefix sets the | |
13266 | number of significant digits to display, as for @kbd{d s}. | |
13267 | ||
13268 | It is important to distinguish between the current @emph{precision} and | |
13269 | the current @emph{display format}. After the commands @kbd{C-u 10 p} | |
13270 | and @kbd{C-u 6 d n} the Calculator computes all results to ten | |
13271 | significant figures but displays only six. (In fact, intermediate | |
13272 | calculations are often carried to one or two more significant figures, | |
13273 | but values placed on the stack will be rounded down to ten figures.) | |
13274 | Numbers are never actually rounded to the display precision for storage, | |
13275 | except by commands like @kbd{C-k} and @kbd{M-# y} which operate on the | |
13276 | actual displayed text in the Calculator buffer. | |
13277 | ||
13278 | @kindex d . | |
13279 | @pindex calc-point-char | |
13280 | The @kbd{d .} (@code{calc-point-char}) command selects the character used | |
13281 | as a decimal point. Normally this is a period; users in some countries | |
13282 | may wish to change this to a comma. Note that this is only a display | |
13283 | style; on entry, periods must always be used to denote floating-point | |
13284 | numbers, and commas to separate elements in a list. | |
13285 | ||
13286 | @node Complex Formats, Fraction Formats, Float Formats, Display Modes | |
13287 | @subsection Complex Formats | |
13288 | ||
13289 | @noindent | |
13290 | @kindex d c | |
13291 | @pindex calc-complex-notation | |
13292 | There are three supported notations for complex numbers in rectangular | |
13293 | form. The default is as a pair of real numbers enclosed in parentheses | |
13294 | and separated by a comma: @samp{(a,b)}. The @kbd{d c} | |
13295 | (@code{calc-complex-notation}) command selects this style.@refill | |
13296 | ||
13297 | @kindex d i | |
13298 | @pindex calc-i-notation | |
13299 | @kindex d j | |
13300 | @pindex calc-j-notation | |
13301 | The other notations are @kbd{d i} (@code{calc-i-notation}), in which | |
13302 | numbers are displayed in @samp{a+bi} form, and @kbd{d j} | |
13303 | (@code{calc-j-notation}) which displays the form @samp{a+bj} preferred | |
13304 | in some disciplines.@refill | |
13305 | ||
13306 | @cindex @code{i} variable | |
13307 | @vindex i | |
13308 | Complex numbers are normally entered in @samp{(a,b)} format. | |
13309 | If you enter @samp{2+3i} as an algebraic formula, it will be stored as | |
13310 | the formula @samp{2 + 3 * i}. However, if you use @kbd{=} to evaluate | |
13311 | this formula and you have not changed the variable @samp{i}, the @samp{i} | |
13312 | will be interpreted as @samp{(0,1)} and the formula will be simplified | |
13313 | to @samp{(2,3)}. Other commands (like @code{calc-sin}) will @emph{not} | |
13314 | interpret the formula @samp{2 + 3 * i} as a complex number. | |
13315 | @xref{Variables}, under ``special constants.''@refill | |
13316 | ||
13317 | @node Fraction Formats, HMS Formats, Complex Formats, Display Modes | |
13318 | @subsection Fraction Formats | |
13319 | ||
13320 | @noindent | |
13321 | @kindex d o | |
13322 | @pindex calc-over-notation | |
13323 | Display of fractional numbers is controlled by the @kbd{d o} | |
13324 | (@code{calc-over-notation}) command. By default, a number like | |
13325 | eight thirds is displayed in the form @samp{8:3}. The @kbd{d o} command | |
13326 | prompts for a one- or two-character format. If you give one character, | |
13327 | that character is used as the fraction separator. Common separators are | |
13328 | @samp{:} and @samp{/}. (During input of numbers, the @kbd{:} key must be | |
13329 | used regardless of the display format; in particular, the @kbd{/} is used | |
13330 | for RPN-style division, @emph{not} for entering fractions.) | |
13331 | ||
13332 | If you give two characters, fractions use ``integer-plus-fractional-part'' | |
13333 | notation. For example, the format @samp{+/} would display eight thirds | |
13334 | as @samp{2+2/3}. If two colons are present in a number being entered, | |
13335 | the number is interpreted in this form (so that the entries @kbd{2:2:3} | |
13336 | and @kbd{8:3} are equivalent). | |
13337 | ||
13338 | It is also possible to follow the one- or two-character format with | |
13339 | a number. For example: @samp{:10} or @samp{+/3}. In this case, | |
13340 | Calc adjusts all fractions that are displayed to have the specified | |
13341 | denominator, if possible. Otherwise it adjusts the denominator to | |
13342 | be a multiple of the specified value. For example, in @samp{:6} mode | |
13343 | the fraction @cite{1:6} will be unaffected, but @cite{2:3} will be | |
13344 | displayed as @cite{4:6}, @cite{1:2} will be displayed as @cite{3:6}, | |
13345 | and @cite{1:8} will be displayed as @cite{3:24}. Integers are also | |
13346 | affected by this mode: 3 is displayed as @cite{18:6}. Note that the | |
13347 | format @samp{:1} writes fractions the same as @samp{:}, but it writes | |
13348 | integers as @cite{n:1}. | |
13349 | ||
13350 | The fraction format does not affect the way fractions or integers are | |
13351 | stored, only the way they appear on the screen. The fraction format | |
13352 | never affects floats. | |
13353 | ||
13354 | @node HMS Formats, Date Formats, Fraction Formats, Display Modes | |
13355 | @subsection HMS Formats | |
13356 | ||
13357 | @noindent | |
13358 | @kindex d h | |
13359 | @pindex calc-hms-notation | |
13360 | The @kbd{d h} (@code{calc-hms-notation}) command controls the display of | |
13361 | HMS (hours-minutes-seconds) forms. It prompts for a string which | |
13362 | consists basically of an ``hours'' marker, optional punctuation, a | |
13363 | ``minutes'' marker, more optional punctuation, and a ``seconds'' marker. | |
13364 | Punctuation is zero or more spaces, commas, or semicolons. The hours | |
13365 | marker is one or more non-punctuation characters. The minutes and | |
13366 | seconds markers must be single non-punctuation characters. | |
13367 | ||
13368 | The default HMS format is @samp{@@ ' "}, producing HMS values of the form | |
13369 | @samp{23@@ 30' 15.75"}. The format @samp{deg, ms} would display this same | |
13370 | value as @samp{23deg, 30m15.75s}. During numeric entry, the @kbd{h} or @kbd{o} | |
13371 | keys are recognized as synonyms for @kbd{@@} regardless of display format. | |
13372 | The @kbd{m} and @kbd{s} keys are recognized as synonyms for @kbd{'} and | |
13373 | @kbd{"}, respectively, but only if an @kbd{@@} (or @kbd{h} or @kbd{o}) has | |
13374 | already been typed; otherwise, they have their usual meanings | |
13375 | (@kbd{m-} prefix and @kbd{s-} prefix). Thus, @kbd{5 "}, @kbd{0 @@ 5 "}, and | |
13376 | @kbd{0 h 5 s} are some of the ways to enter the quantity ``five seconds.'' | |
13377 | The @kbd{'} key is recognized as ``minutes'' only if @kbd{@@} (or @kbd{h} or | |
13378 | @kbd{o}) has already been pressed; otherwise it means to switch to algebraic | |
13379 | entry. | |
13380 | ||
13381 | @node Date Formats, Truncating the Stack, HMS Formats, Display Modes | |
13382 | @subsection Date Formats | |
13383 | ||
13384 | @noindent | |
13385 | @kindex d d | |
13386 | @pindex calc-date-notation | |
13387 | The @kbd{d d} (@code{calc-date-notation}) command controls the display | |
13388 | of date forms (@pxref{Date Forms}). It prompts for a string which | |
13389 | contains letters that represent the various parts of a date and time. | |
13390 | To show which parts should be omitted when the form represents a pure | |
13391 | date with no time, parts of the string can be enclosed in @samp{< >} | |
13392 | marks. If you don't include @samp{< >} markers in the format, Calc | |
13393 | guesses at which parts, if any, should be omitted when formatting | |
13394 | pure dates. | |
13395 | ||
13396 | The default format is: @samp{<H:mm:SSpp >Www Mmm D, YYYY}. | |
13397 | An example string in this format is @samp{3:32pm Wed Jan 9, 1991}. | |
13398 | If you enter a blank format string, this default format is | |
13399 | reestablished. | |
13400 | ||
13401 | Calc uses @samp{< >} notation for nameless functions as well as for | |
13402 | dates. @xref{Specifying Operators}. To avoid confusion with nameless | |
13403 | functions, your date formats should avoid using the @samp{#} character. | |
13404 | ||
13405 | @menu | |
13406 | * Date Formatting Codes:: | |
13407 | * Free-Form Dates:: | |
13408 | * Standard Date Formats:: | |
13409 | @end menu | |
13410 | ||
13411 | @node Date Formatting Codes, Free-Form Dates, Date Formats, Date Formats | |
13412 | @subsubsection Date Formatting Codes | |
13413 | ||
13414 | @noindent | |
13415 | When displaying a date, the current date format is used. All | |
13416 | characters except for letters and @samp{<} and @samp{>} are | |
13417 | copied literally when dates are formatted. The portion between | |
13418 | @samp{< >} markers is omitted for pure dates, or included for | |
13419 | date/time forms. Letters are interpreted according to the table | |
13420 | below. | |
13421 | ||
13422 | When dates are read in during algebraic entry, Calc first tries to | |
13423 | match the input string to the current format either with or without | |
13424 | the time part. The punctuation characters (including spaces) must | |
13425 | match exactly; letter fields must correspond to suitable text in | |
13426 | the input. If this doesn't work, Calc checks if the input is a | |
13427 | simple number; if so, the number is interpreted as a number of days | |
13428 | since Jan 1, 1 AD. Otherwise, Calc tries a much more relaxed and | |
13429 | flexible algorithm which is described in the next section. | |
13430 | ||
13431 | Weekday names are ignored during reading. | |
13432 | ||
13433 | Two-digit year numbers are interpreted as lying in the range | |
13434 | from 1941 to 2039. Years outside that range are always | |
13435 | entered and displayed in full. Year numbers with a leading | |
13436 | @samp{+} sign are always interpreted exactly, allowing the | |
13437 | entry and display of the years 1 through 99 AD. | |
13438 | ||
13439 | Here is a complete list of the formatting codes for dates: | |
13440 | ||
13441 | @table @asis | |
13442 | @item Y | |
13443 | Year: ``91'' for 1991, ``7'' for 2007, ``+23'' for 23 AD. | |
13444 | @item YY | |
13445 | Year: ``91'' for 1991, ``07'' for 2007, ``+23'' for 23 AD. | |
13446 | @item BY | |
13447 | Year: ``91'' for 1991, `` 7'' for 2007, ``+23'' for 23 AD. | |
13448 | @item YYY | |
13449 | Year: ``1991'' for 1991, ``23'' for 23 AD. | |
13450 | @item YYYY | |
13451 | Year: ``1991'' for 1991, ``+23'' for 23 AD. | |
13452 | @item aa | |
13453 | Year: ``ad'' or blank. | |
13454 | @item AA | |
13455 | Year: ``AD'' or blank. | |
13456 | @item aaa | |
13457 | Year: ``ad '' or blank. (Note trailing space.) | |
13458 | @item AAA | |
13459 | Year: ``AD '' or blank. | |
13460 | @item aaaa | |
13461 | Year: ``a.d.'' or blank. | |
13462 | @item AAAA | |
13463 | Year: ``A.D.'' or blank. | |
13464 | @item bb | |
13465 | Year: ``bc'' or blank. | |
13466 | @item BB | |
13467 | Year: ``BC'' or blank. | |
13468 | @item bbb | |
13469 | Year: `` bc'' or blank. (Note leading space.) | |
13470 | @item BBB | |
13471 | Year: `` BC'' or blank. | |
13472 | @item bbbb | |
13473 | Year: ``b.c.'' or blank. | |
13474 | @item BBBB | |
13475 | Year: ``B.C.'' or blank. | |
13476 | @item M | |
13477 | Month: ``8'' for August. | |
13478 | @item MM | |
13479 | Month: ``08'' for August. | |
13480 | @item BM | |
13481 | Month: `` 8'' for August. | |
13482 | @item MMM | |
13483 | Month: ``AUG'' for August. | |
13484 | @item Mmm | |
13485 | Month: ``Aug'' for August. | |
13486 | @item mmm | |
13487 | Month: ``aug'' for August. | |
13488 | @item MMMM | |
13489 | Month: ``AUGUST'' for August. | |
13490 | @item Mmmm | |
13491 | Month: ``August'' for August. | |
13492 | @item D | |
13493 | Day: ``7'' for 7th day of month. | |
13494 | @item DD | |
13495 | Day: ``07'' for 7th day of month. | |
13496 | @item BD | |
13497 | Day: `` 7'' for 7th day of month. | |
13498 | @item W | |
13499 | Weekday: ``0'' for Sunday, ``6'' for Saturday. | |
13500 | @item WWW | |
13501 | Weekday: ``SUN'' for Sunday. | |
13502 | @item Www | |
13503 | Weekday: ``Sun'' for Sunday. | |
13504 | @item www | |
13505 | Weekday: ``sun'' for Sunday. | |
13506 | @item WWWW | |
13507 | Weekday: ``SUNDAY'' for Sunday. | |
13508 | @item Wwww | |
13509 | Weekday: ``Sunday'' for Sunday. | |
13510 | @item d | |
13511 | Day of year: ``34'' for Feb. 3. | |
13512 | @item ddd | |
13513 | Day of year: ``034'' for Feb. 3. | |
13514 | @item bdd | |
13515 | Day of year: `` 34'' for Feb. 3. | |
13516 | @item h | |
13517 | Hour: ``5'' for 5 AM; ``17'' for 5 PM. | |
13518 | @item hh | |
13519 | Hour: ``05'' for 5 AM; ``17'' for 5 PM. | |
13520 | @item bh | |
13521 | Hour: `` 5'' for 5 AM; ``17'' for 5 PM. | |
13522 | @item H | |
13523 | Hour: ``5'' for 5 AM and 5 PM. | |
13524 | @item HH | |
13525 | Hour: ``05'' for 5 AM and 5 PM. | |
13526 | @item BH | |
13527 | Hour: `` 5'' for 5 AM and 5 PM. | |
13528 | @item p | |
13529 | AM/PM: ``a'' or ``p''. | |
13530 | @item P | |
13531 | AM/PM: ``A'' or ``P''. | |
13532 | @item pp | |
13533 | AM/PM: ``am'' or ``pm''. | |
13534 | @item PP | |
13535 | AM/PM: ``AM'' or ``PM''. | |
13536 | @item pppp | |
13537 | AM/PM: ``a.m.'' or ``p.m.''. | |
13538 | @item PPPP | |
13539 | AM/PM: ``A.M.'' or ``P.M.''. | |
13540 | @item m | |
13541 | Minutes: ``7'' for 7. | |
13542 | @item mm | |
13543 | Minutes: ``07'' for 7. | |
13544 | @item bm | |
13545 | Minutes: `` 7'' for 7. | |
13546 | @item s | |
13547 | Seconds: ``7'' for 7; ``7.23'' for 7.23. | |
13548 | @item ss | |
13549 | Seconds: ``07'' for 7; ``07.23'' for 7.23. | |
13550 | @item bs | |
13551 | Seconds: `` 7'' for 7; `` 7.23'' for 7.23. | |
13552 | @item SS | |
13553 | Optional seconds: ``07'' for 7; blank for 0. | |
13554 | @item BS | |
13555 | Optional seconds: `` 7'' for 7; blank for 0. | |
13556 | @item N | |
13557 | Numeric date/time: ``726842.25'' for 6:00am Wed Jan 9, 1991. | |
13558 | @item n | |
13559 | Numeric date: ``726842'' for any time on Wed Jan 9, 1991. | |
13560 | @item J | |
13561 | Julian date/time: ``2448265.75'' for 6:00am Wed Jan 9, 1991. | |
13562 | @item j | |
13563 | Julian date: ``2448266'' for any time on Wed Jan 9, 1991. | |
13564 | @item U | |
13565 | Unix time: ``663400800'' for 6:00am Wed Jan 9, 1991. | |
13566 | @item X | |
13567 | Brackets suppression. An ``X'' at the front of the format | |
13568 | causes the surrounding @w{@samp{< >}} delimiters to be omitted | |
13569 | when formatting dates. Note that the brackets are still | |
13570 | required for algebraic entry. | |
13571 | @end table | |
13572 | ||
13573 | If ``SS'' or ``BS'' (optional seconds) is preceded by a colon, the | |
13574 | colon is also omitted if the seconds part is zero. | |
13575 | ||
13576 | If ``bb,'' ``bbb'' or ``bbbb'' or their upper-case equivalents | |
13577 | appear in the format, then negative year numbers are displayed | |
13578 | without a minus sign. Note that ``aa'' and ``bb'' are mutually | |
13579 | exclusive. Some typical usages would be @samp{YYYY AABB}; | |
13580 | @samp{AAAYYYYBBB}; @samp{YYYYBBB}. | |
13581 | ||
13582 | The formats ``YY,'' ``YYYY,'' ``MM,'' ``DD,'' ``ddd,'' ``hh,'' ``HH,'' | |
13583 | ``mm,'' ``ss,'' and ``SS'' actually match any number of digits during | |
13584 | reading unless several of these codes are strung together with no | |
13585 | punctuation in between, in which case the input must have exactly as | |
13586 | many digits as there are letters in the format. | |
13587 | ||
13588 | The ``j,'' ``J,'' and ``U'' formats do not make any time zone | |
13589 | adjustment. They effectively use @samp{julian(x,0)} and | |
13590 | @samp{unixtime(x,0)} to make the conversion; @pxref{Date Arithmetic}. | |
13591 | ||
13592 | @node Free-Form Dates, Standard Date Formats, Date Formatting Codes, Date Formats | |
13593 | @subsubsection Free-Form Dates | |
13594 | ||
13595 | @noindent | |
13596 | When reading a date form during algebraic entry, Calc falls back | |
13597 | on the algorithm described here if the input does not exactly | |
13598 | match the current date format. This algorithm generally | |
13599 | ``does the right thing'' and you don't have to worry about it, | |
13600 | but it is described here in full detail for the curious. | |
13601 | ||
13602 | Calc does not distinguish between upper- and lower-case letters | |
13603 | while interpreting dates. | |
13604 | ||
13605 | First, the time portion, if present, is located somewhere in the | |
13606 | text and then removed. The remaining text is then interpreted as | |
13607 | the date. | |
13608 | ||
13609 | A time is of the form @samp{hh:mm:ss}, possibly with the seconds | |
13610 | part omitted and possibly with an AM/PM indicator added to indicate | |
13611 | 12-hour time. If the AM/PM is present, the minutes may also be | |
13612 | omitted. The AM/PM part may be any of the words @samp{am}, | |
13613 | @samp{pm}, @samp{noon}, or @samp{midnight}; each of these may be | |
13614 | abbreviated to one letter, and the alternate forms @samp{a.m.}, | |
13615 | @samp{p.m.}, and @samp{mid} are also understood. Obviously | |
13616 | @samp{noon} and @samp{midnight} are allowed only on 12:00:00. | |
13617 | The words @samp{noon}, @samp{mid}, and @samp{midnight} are also | |
13618 | recognized with no number attached. | |
13619 | ||
13620 | If there is no AM/PM indicator, the time is interpreted in 24-hour | |
13621 | format. | |
13622 | ||
13623 | To read the date portion, all words and numbers are isolated | |
13624 | from the string; other characters are ignored. All words must | |
13625 | be either month names or day-of-week names (the latter of which | |
13626 | are ignored). Names can be written in full or as three-letter | |
13627 | abbreviations. | |
13628 | ||
13629 | Large numbers, or numbers with @samp{+} or @samp{-} signs, | |
13630 | are interpreted as years. If one of the other numbers is | |
13631 | greater than 12, then that must be the day and the remaining | |
13632 | number in the input is therefore the month. Otherwise, Calc | |
13633 | assumes the month, day and year are in the same order that they | |
13634 | appear in the current date format. If the year is omitted, the | |
13635 | current year is taken from the system clock. | |
13636 | ||
13637 | If there are too many or too few numbers, or any unrecognizable | |
13638 | words, then the input is rejected. | |
13639 | ||
13640 | If there are any large numbers (of five digits or more) other than | |
13641 | the year, they are ignored on the assumption that they are something | |
13642 | like Julian dates that were included along with the traditional | |
13643 | date components when the date was formatted. | |
13644 | ||
13645 | One of the words @samp{ad}, @samp{a.d.}, @samp{bc}, or @samp{b.c.} | |
13646 | may optionally be used; the latter two are equivalent to a | |
13647 | minus sign on the year value. | |
13648 | ||
13649 | If you always enter a four-digit year, and use a name instead | |
13650 | of a number for the month, there is no danger of ambiguity. | |
13651 | ||
13652 | @node Standard Date Formats, , Free-Form Dates, Date Formats | |
13653 | @subsubsection Standard Date Formats | |
13654 | ||
13655 | @noindent | |
13656 | There are actually ten standard date formats, numbered 0 through 9. | |
13657 | Entering a blank line at the @kbd{d d} command's prompt gives | |
13658 | you format number 1, Calc's usual format. You can enter any digit | |
13659 | to select the other formats. | |
13660 | ||
13661 | To create your own standard date formats, give a numeric prefix | |
13662 | argument from 0 to 9 to the @w{@kbd{d d}} command. The format you | |
13663 | enter will be recorded as the new standard format of that | |
13664 | number, as well as becoming the new current date format. | |
13665 | You can save your formats permanently with the @w{@kbd{m m}} | |
13666 | command (@pxref{Mode Settings}). | |
13667 | ||
13668 | @table @asis | |
13669 | @item 0 | |
13670 | @samp{N} (Numerical format) | |
13671 | @item 1 | |
13672 | @samp{<H:mm:SSpp >Www Mmm D, YYYY} (American format) | |
13673 | @item 2 | |
13674 | @samp{D Mmm YYYY<, h:mm:SS>} (European format) | |
13675 | @item 3 | |
13676 | @samp{Www Mmm BD< hh:mm:ss> YYYY} (Unix written date format) | |
13677 | @item 4 | |
13678 | @samp{M/D/Y< H:mm:SSpp>} (American slashed format) | |
13679 | @item 5 | |
13680 | @samp{D.M.Y< h:mm:SS>} (European dotted format) | |
13681 | @item 6 | |
13682 | @samp{M-D-Y< H:mm:SSpp>} (American dashed format) | |
13683 | @item 7 | |
13684 | @samp{D-M-Y< h:mm:SS>} (European dashed format) | |
13685 | @item 8 | |
13686 | @samp{j<, h:mm:ss>} (Julian day plus time) | |
13687 | @item 9 | |
13688 | @samp{YYddd< hh:mm:ss>} (Year-day format) | |
13689 | @end table | |
13690 | ||
13691 | @node Truncating the Stack, Justification, Date Formats, Display Modes | |
13692 | @subsection Truncating the Stack | |
13693 | ||
13694 | @noindent | |
13695 | @kindex d t | |
13696 | @pindex calc-truncate-stack | |
13697 | @cindex Truncating the stack | |
13698 | @cindex Narrowing the stack | |
13699 | The @kbd{d t} (@code{calc-truncate-stack}) command moves the @samp{.}@: | |
13700 | line that marks the top-of-stack up or down in the Calculator buffer. | |
13701 | The number right above that line is considered to the be at the top of | |
13702 | the stack. Any numbers below that line are ``hidden'' from all stack | |
13703 | operations. This is similar to the Emacs ``narrowing'' feature, except | |
13704 | that the values below the @samp{.} are @emph{visible}, just temporarily | |
13705 | frozen. This feature allows you to keep several independent calculations | |
13706 | running at once in different parts of the stack, or to apply a certain | |
13707 | command to an element buried deep in the stack.@refill | |
13708 | ||
13709 | Pressing @kbd{d t} by itself moves the @samp{.} to the line the cursor | |
13710 | is on. Thus, this line and all those below it become hidden. To un-hide | |
13711 | these lines, move down to the end of the buffer and press @w{@kbd{d t}}. | |
13712 | With a positive numeric prefix argument @cite{n}, @kbd{d t} hides the | |
13713 | bottom @cite{n} values in the buffer. With a negative argument, it hides | |
13714 | all but the top @cite{n} values. With an argument of zero, it hides zero | |
13715 | values, i.e., moves the @samp{.} all the way down to the bottom.@refill | |
13716 | ||
13717 | @kindex d [ | |
13718 | @pindex calc-truncate-up | |
13719 | @kindex d ] | |
13720 | @pindex calc-truncate-down | |
13721 | The @kbd{d [} (@code{calc-truncate-up}) and @kbd{d ]} | |
13722 | (@code{calc-truncate-down}) commands move the @samp{.} up or down one | |
13723 | line at a time (or several lines with a prefix argument).@refill | |
13724 | ||
13725 | @node Justification, Labels, Truncating the Stack, Display Modes | |
13726 | @subsection Justification | |
13727 | ||
13728 | @noindent | |
13729 | @kindex d < | |
13730 | @pindex calc-left-justify | |
13731 | @kindex d = | |
13732 | @pindex calc-center-justify | |
13733 | @kindex d > | |
13734 | @pindex calc-right-justify | |
13735 | Values on the stack are normally left-justified in the window. You can | |
13736 | control this arrangement by typing @kbd{d <} (@code{calc-left-justify}), | |
13737 | @kbd{d >} (@code{calc-right-justify}), or @kbd{d =} | |
13738 | (@code{calc-center-justify}). For example, in right-justification mode, | |
13739 | stack entries are displayed flush-right against the right edge of the | |
13740 | window.@refill | |
13741 | ||
13742 | If you change the width of the Calculator window you may have to type | |
5d67986c | 13743 | @kbd{d @key{SPC}} (@code{calc-refresh}) to re-align right-justified or centered |
d7b8e6c6 EZ |
13744 | text. |
13745 | ||
13746 | Right-justification is especially useful together with fixed-point | |
13747 | notation (see @code{d f}; @code{calc-fix-notation}). With these modes | |
13748 | together, the decimal points on numbers will always line up. | |
13749 | ||
13750 | With a numeric prefix argument, the justification commands give you | |
13751 | a little extra control over the display. The argument specifies the | |
13752 | horizontal ``origin'' of a display line. It is also possible to | |
13753 | specify a maximum line width using the @kbd{d b} command (@pxref{Normal | |
13754 | Language Modes}). For reference, the precise rules for formatting and | |
13755 | breaking lines are given below. Notice that the interaction between | |
13756 | origin and line width is slightly different in each justification | |
13757 | mode. | |
13758 | ||
13759 | In left-justified mode, the line is indented by a number of spaces | |
13760 | given by the origin (default zero). If the result is longer than the | |
13761 | maximum line width, if given, or too wide to fit in the Calc window | |
13762 | otherwise, then it is broken into lines which will fit; each broken | |
13763 | line is indented to the origin. | |
13764 | ||
13765 | In right-justified mode, lines are shifted right so that the rightmost | |
13766 | character is just before the origin, or just before the current | |
13767 | window width if no origin was specified. If the line is too long | |
13768 | for this, then it is broken; the current line width is used, if | |
13769 | specified, or else the origin is used as a width if that is | |
13770 | specified, or else the line is broken to fit in the window. | |
13771 | ||
13772 | In centering mode, the origin is the column number of the center of | |
13773 | each stack entry. If a line width is specified, lines will not be | |
13774 | allowed to go past that width; Calc will either indent less or | |
13775 | break the lines if necessary. If no origin is specified, half the | |
13776 | line width or Calc window width is used. | |
13777 | ||
13778 | Note that, in each case, if line numbering is enabled the display | |
13779 | is indented an additional four spaces to make room for the line | |
13780 | number. The width of the line number is taken into account when | |
13781 | positioning according to the current Calc window width, but not | |
13782 | when positioning by explicit origins and widths. In the latter | |
13783 | case, the display is formatted as specified, and then uniformly | |
13784 | shifted over four spaces to fit the line numbers. | |
13785 | ||
13786 | @node Labels, , Justification, Display Modes | |
13787 | @subsection Labels | |
13788 | ||
13789 | @noindent | |
13790 | @kindex d @{ | |
13791 | @pindex calc-left-label | |
13792 | The @kbd{d @{} (@code{calc-left-label}) command prompts for a string, | |
13793 | then displays that string to the left of every stack entry. If the | |
13794 | entries are left-justified (@pxref{Justification}), then they will | |
13795 | appear immediately after the label (unless you specified an origin | |
13796 | greater than the length of the label). If the entries are centered | |
13797 | or right-justified, the label appears on the far left and does not | |
13798 | affect the horizontal position of the stack entry. | |
13799 | ||
13800 | Give a blank string (with @kbd{d @{ @key{RET}}) to turn the label off. | |
13801 | ||
13802 | @kindex d @} | |
13803 | @pindex calc-right-label | |
13804 | The @kbd{d @}} (@code{calc-right-label}) command similarly adds a | |
13805 | label on the righthand side. It does not affect positioning of | |
13806 | the stack entries unless they are right-justified. Also, if both | |
13807 | a line width and an origin are given in right-justified mode, the | |
13808 | stack entry is justified to the origin and the righthand label is | |
13809 | justified to the line width. | |
13810 | ||
13811 | One application of labels would be to add equation numbers to | |
13812 | formulas you are manipulating in Calc and then copying into a | |
13813 | document (possibly using Embedded Mode). The equations would | |
13814 | typically be centered, and the equation numbers would be on the | |
13815 | left or right as you prefer. | |
13816 | ||
13817 | @node Language Modes, Modes Variable, Display Modes, Mode Settings | |
13818 | @section Language Modes | |
13819 | ||
13820 | @noindent | |
13821 | The commands in this section change Calc to use a different notation for | |
13822 | entry and display of formulas, corresponding to the conventions of some | |
13823 | other common language such as Pascal or @TeX{}. Objects displayed on the | |
13824 | stack or yanked from the Calculator to an editing buffer will be formatted | |
13825 | in the current language; objects entered in algebraic entry or yanked from | |
13826 | another buffer will be interpreted according to the current language. | |
13827 | ||
13828 | The current language has no effect on things written to or read from the | |
13829 | trail buffer, nor does it affect numeric entry. Only algebraic entry is | |
13830 | affected. You can make even algebraic entry ignore the current language | |
13831 | and use the standard notation by giving a numeric prefix, e.g., @kbd{C-u '}. | |
13832 | ||
13833 | For example, suppose the formula @samp{2*a[1] + atan(a[2])} occurs in a C | |
13834 | program; elsewhere in the program you need the derivatives of this formula | |
13835 | with respect to @samp{a[1]} and @samp{a[2]}. First, type @kbd{d C} | |
13836 | to switch to C notation. Now use @code{C-u M-# g} to grab the formula | |
13837 | into the Calculator, @kbd{a d a[1] @key{RET}} to differentiate with respect | |
13838 | to the first variable, and @kbd{M-# y} to yank the formula for the derivative | |
13839 | back into your C program. Press @kbd{U} to undo the differentiation and | |
13840 | repeat with @kbd{a d a[2] @key{RET}} for the other derivative. | |
13841 | ||
13842 | Without being switched into C mode first, Calc would have misinterpreted | |
13843 | the brackets in @samp{a[1]} and @samp{a[2]}, would not have known that | |
13844 | @code{atan} was equivalent to Calc's built-in @code{arctan} function, | |
13845 | and would have written the formula back with notations (like implicit | |
13846 | multiplication) which would not have been legal for a C program. | |
13847 | ||
13848 | As another example, suppose you are maintaining a C program and a @TeX{} | |
13849 | document, each of which needs a copy of the same formula. You can grab the | |
13850 | formula from the program in C mode, switch to @TeX{} mode, and yank the | |
13851 | formula into the document in @TeX{} math-mode format. | |
13852 | ||
13853 | Language modes are selected by typing the letter @kbd{d} followed by a | |
13854 | shifted letter key. | |
13855 | ||
13856 | @menu | |
13857 | * Normal Language Modes:: | |
13858 | * C FORTRAN Pascal:: | |
13859 | * TeX Language Mode:: | |
13860 | * Eqn Language Mode:: | |
13861 | * Mathematica Language Mode:: | |
13862 | * Maple Language Mode:: | |
13863 | * Compositions:: | |
13864 | * Syntax Tables:: | |
13865 | @end menu | |
13866 | ||
13867 | @node Normal Language Modes, C FORTRAN Pascal, Language Modes, Language Modes | |
13868 | @subsection Normal Language Modes | |
13869 | ||
13870 | @noindent | |
13871 | @kindex d N | |
13872 | @pindex calc-normal-language | |
13873 | The @kbd{d N} (@code{calc-normal-language}) command selects the usual | |
13874 | notation for Calc formulas, as described in the rest of this manual. | |
13875 | Matrices are displayed in a multi-line tabular format, but all other | |
13876 | objects are written in linear form, as they would be typed from the | |
13877 | keyboard. | |
13878 | ||
13879 | @kindex d O | |
13880 | @pindex calc-flat-language | |
13881 | @cindex Matrix display | |
13882 | The @kbd{d O} (@code{calc-flat-language}) command selects a language | |
13883 | identical with the normal one, except that matrices are written in | |
13884 | one-line form along with everything else. In some applications this | |
13885 | form may be more suitable for yanking data into other buffers. | |
13886 | ||
13887 | @kindex d b | |
13888 | @pindex calc-line-breaking | |
13889 | @cindex Line breaking | |
13890 | @cindex Breaking up long lines | |
13891 | Even in one-line mode, long formulas or vectors will still be split | |
13892 | across multiple lines if they exceed the width of the Calculator window. | |
13893 | The @kbd{d b} (@code{calc-line-breaking}) command turns this line-breaking | |
13894 | feature on and off. (It works independently of the current language.) | |
13895 | If you give a numeric prefix argument of five or greater to the @kbd{d b} | |
13896 | command, that argument will specify the line width used when breaking | |
13897 | long lines. | |
13898 | ||
13899 | @kindex d B | |
13900 | @pindex calc-big-language | |
13901 | The @kbd{d B} (@code{calc-big-language}) command selects a language | |
13902 | which uses textual approximations to various mathematical notations, | |
13903 | such as powers, quotients, and square roots: | |
13904 | ||
13905 | @example | |
13906 | ____________ | |
13907 | | a + 1 2 | |
13908 | | ----- + c | |
13909 | \| b | |
13910 | @end example | |
13911 | ||
13912 | @noindent | |
13913 | in place of @samp{sqrt((a+1)/b + c^2)}. | |
13914 | ||
13915 | Subscripts like @samp{a_i} are displayed as actual subscripts in ``big'' | |
13916 | mode. Double subscripts, @samp{a_i_j} (@samp{subscr(subscr(a, i), j)}) | |
13917 | are displayed as @samp{a} with subscripts separated by commas: | |
13918 | @samp{i, j}. They must still be entered in the usual underscore | |
13919 | notation. | |
13920 | ||
13921 | One slight ambiguity of Big notation is that | |
13922 | ||
13923 | @example | |
13924 | 3 | |
13925 | - - | |
13926 | 4 | |
13927 | @end example | |
13928 | ||
13929 | @noindent | |
13930 | can represent either the negative rational number @cite{-3:4}, or the | |
13931 | actual expression @samp{-(3/4)}; but the latter formula would normally | |
13932 | never be displayed because it would immediately be evaluated to | |
13933 | @cite{-3:4} or @cite{-0.75}, so this ambiguity is not a problem in | |
13934 | typical use. | |
13935 | ||
13936 | Non-decimal numbers are displayed with subscripts. Thus there is no | |
13937 | way to tell the difference between @samp{16#C2} and @samp{C2_16}, | |
13938 | though generally you will know which interpretation is correct. | |
13939 | Logarithms @samp{log(x,b)} and @samp{log10(x)} also use subscripts | |
13940 | in Big mode. | |
13941 | ||
13942 | In Big mode, stack entries often take up several lines. To aid | |
13943 | readability, stack entries are separated by a blank line in this mode. | |
13944 | You may find it useful to expand the Calc window's height using | |
13945 | @kbd{C-x ^} (@code{enlarge-window}) or to make the Calc window the only | |
13946 | one on the screen with @kbd{C-x 1} (@code{delete-other-windows}). | |
13947 | ||
13948 | Long lines are currently not rearranged to fit the window width in | |
13949 | Big mode, so you may need to use the @kbd{<} and @kbd{>} keys | |
13950 | to scroll across a wide formula. For really big formulas, you may | |
13951 | even need to use @kbd{@{} and @kbd{@}} to scroll up and down. | |
13952 | ||
13953 | @kindex d U | |
13954 | @pindex calc-unformatted-language | |
13955 | The @kbd{d U} (@code{calc-unformatted-language}) command altogether disables | |
13956 | the use of operator notation in formulas. In this mode, the formula | |
13957 | shown above would be displayed: | |
13958 | ||
13959 | @example | |
13960 | sqrt(add(div(add(a, 1), b), pow(c, 2))) | |
13961 | @end example | |
13962 | ||
13963 | These four modes differ only in display format, not in the format | |
13964 | expected for algebraic entry. The standard Calc operators work in | |
13965 | all four modes, and unformatted notation works in any language mode | |
13966 | (except that Mathematica mode expects square brackets instead of | |
13967 | parentheses). | |
13968 | ||
13969 | @node C FORTRAN Pascal, TeX Language Mode, Normal Language Modes, Language Modes | |
13970 | @subsection C, FORTRAN, and Pascal Modes | |
13971 | ||
13972 | @noindent | |
13973 | @kindex d C | |
13974 | @pindex calc-c-language | |
13975 | @cindex C language | |
13976 | The @kbd{d C} (@code{calc-c-language}) command selects the conventions | |
13977 | of the C language for display and entry of formulas. This differs from | |
13978 | the normal language mode in a variety of (mostly minor) ways. In | |
13979 | particular, C language operators and operator precedences are used in | |
13980 | place of Calc's usual ones. For example, @samp{a^b} means @samp{xor(a,b)} | |
13981 | in C mode; a value raised to a power is written as a function call, | |
13982 | @samp{pow(a,b)}. | |
13983 | ||
13984 | In C mode, vectors and matrices use curly braces instead of brackets. | |
13985 | Octal and hexadecimal values are written with leading @samp{0} or @samp{0x} | |
13986 | rather than using the @samp{#} symbol. Array subscripting is | |
13987 | translated into @code{subscr} calls, so that @samp{a[i]} in C | |
13988 | mode is the same as @samp{a_i} in normal mode. Assignments | |
13989 | turn into the @code{assign} function, which Calc normally displays | |
13990 | using the @samp{:=} symbol. | |
13991 | ||
13992 | The variables @code{var-pi} and @code{var-e} would be displayed @samp{pi} | |
13993 | and @samp{e} in normal mode, but in C mode they are displayed as | |
13994 | @samp{M_PI} and @samp{M_E}, corresponding to the names of constants | |
13995 | typically provided in the @file{<math.h>} header. Functions whose | |
13996 | names are different in C are translated automatically for entry and | |
13997 | display purposes. For example, entering @samp{asin(x)} will push the | |
13998 | formula @samp{arcsin(x)} onto the stack; this formula will be displayed | |
13999 | as @samp{asin(x)} as long as C mode is in effect. | |
14000 | ||
14001 | @kindex d P | |
14002 | @pindex calc-pascal-language | |
14003 | @cindex Pascal language | |
14004 | The @kbd{d P} (@code{calc-pascal-language}) command selects Pascal | |
14005 | conventions. Like C mode, Pascal mode interprets array brackets and uses | |
14006 | a different table of operators. Hexadecimal numbers are entered and | |
14007 | displayed with a preceding dollar sign. (Thus the regular meaning of | |
14008 | @kbd{$2} during algebraic entry does not work in Pascal mode, though | |
14009 | @kbd{$} (and @kbd{$$}, etc.) not followed by digits works the same as | |
14010 | always.) No special provisions are made for other non-decimal numbers, | |
14011 | vectors, and so on, since there is no universally accepted standard way | |
14012 | of handling these in Pascal. | |
14013 | ||
14014 | @kindex d F | |
14015 | @pindex calc-fortran-language | |
14016 | @cindex FORTRAN language | |
14017 | The @kbd{d F} (@code{calc-fortran-language}) command selects FORTRAN | |
14018 | conventions. Various function names are transformed into FORTRAN | |
14019 | equivalents. Vectors are written as @samp{/1, 2, 3/}, and may be | |
14020 | entered this way or using square brackets. Since FORTRAN uses round | |
14021 | parentheses for both function calls and array subscripts, Calc displays | |
14022 | both in the same way; @samp{a(i)} is interpreted as a function call | |
14023 | upon reading, and subscripts must be entered as @samp{subscr(a, i)}. | |
14024 | Also, if the variable @code{a} has been declared to have type | |
14025 | @code{vector} or @code{matrix} then @samp{a(i)} will be parsed as a | |
14026 | subscript. (@xref{Declarations}.) Usually it doesn't matter, though; | |
14027 | if you enter the subscript expression @samp{a(i)} and Calc interprets | |
14028 | it as a function call, you'll never know the difference unless you | |
14029 | switch to another language mode or replace @code{a} with an actual | |
14030 | vector (or unless @code{a} happens to be the name of a built-in | |
14031 | function!). | |
14032 | ||
14033 | Underscores are allowed in variable and function names in all of these | |
14034 | language modes. The underscore here is equivalent to the @samp{#} in | |
14035 | normal mode, or to hyphens in the underlying Emacs Lisp variable names. | |
14036 | ||
14037 | FORTRAN and Pascal modes normally do not adjust the case of letters in | |
14038 | formulas. Most built-in Calc names use lower-case letters. If you use a | |
14039 | positive numeric prefix argument with @kbd{d P} or @kbd{d F}, these | |
14040 | modes will use upper-case letters exclusively for display, and will | |
14041 | convert to lower-case on input. With a negative prefix, these modes | |
14042 | convert to lower-case for display and input. | |
14043 | ||
14044 | @node TeX Language Mode, Eqn Language Mode, C FORTRAN Pascal, Language Modes | |
14045 | @subsection @TeX{} Language Mode | |
14046 | ||
14047 | @noindent | |
14048 | @kindex d T | |
14049 | @pindex calc-tex-language | |
14050 | @cindex TeX language | |
14051 | The @kbd{d T} (@code{calc-tex-language}) command selects the conventions | |
14052 | of ``math mode'' in the @TeX{} typesetting language, by Donald Knuth. | |
14053 | Formulas are entered | |
14054 | and displayed in @TeX{} notation, as in @samp{\sin\left( a \over b \right)}. | |
14055 | Math formulas are usually enclosed by @samp{$ $} signs in @TeX{}; these | |
14056 | should be omitted when interfacing with Calc. To Calc, the @samp{$} sign | |
14057 | has the same meaning it always does in algebraic formulas (a reference to | |
14058 | an existing entry on the stack).@refill | |
14059 | ||
14060 | Complex numbers are displayed as in @samp{3 + 4i}. Fractions and | |
14061 | quotients are written using @code{\over}; | |
14062 | binomial coefficients are written with @code{\choose}. | |
14063 | Interval forms are written with @code{\ldots}, and | |
14064 | error forms are written with @code{\pm}. | |
14065 | Absolute values are written as in @samp{|x + 1|}, and the floor and | |
14066 | ceiling functions are written with @code{\lfloor}, @code{\rfloor}, etc. | |
14067 | The words @code{\left} and @code{\right} are ignored when reading | |
14068 | formulas in @TeX{} mode. Both @code{inf} and @code{uinf} are written | |
14069 | as @code{\infty}; when read, @code{\infty} always translates to | |
14070 | @code{inf}.@refill | |
14071 | ||
14072 | Function calls are written the usual way, with the function name followed | |
14073 | by the arguments in parentheses. However, functions for which @TeX{} has | |
14074 | special names (like @code{\sin}) will use curly braces instead of | |
14075 | parentheses for very simple arguments. During input, curly braces and | |
14076 | parentheses work equally well for grouping, but when the document is | |
14077 | formatted the curly braces will be invisible. Thus the printed result is | |
14078 | @c{$\sin{2 x}$} | |
14079 | @cite{sin 2x} but @c{$\sin(2 + x)$} | |
14080 | @cite{sin(2 + x)}. | |
14081 | ||
14082 | Function and variable names not treated specially by @TeX{} are simply | |
14083 | written out as-is, which will cause them to come out in italic letters | |
14084 | in the printed document. If you invoke @kbd{d T} with a positive numeric | |
14085 | prefix argument, names of more than one character will instead be written | |
14086 | @samp{\hbox@{@var{name}@}}. The @samp{\hbox@{ @}} notation is ignored | |
14087 | during reading. If you use a negative prefix argument, such function | |
14088 | names are written @samp{\@var{name}}, and function names that begin | |
14089 | with @code{\} during reading have the @code{\} removed. (Note that | |
14090 | in this mode, long variable names are still written with @code{\hbox}. | |
14091 | However, you can always make an actual variable name like @code{\bar} | |
14092 | in any @TeX{} mode.) | |
14093 | ||
14094 | During reading, text of the form @samp{\matrix@{ ...@: @}} is replaced | |
14095 | by @samp{[ ...@: ]}. The same also applies to @code{\pmatrix} and | |
14096 | @code{\bmatrix}. The symbol @samp{&} is interpreted as a comma, | |
14097 | and the symbols @samp{\cr} and @samp{\\} are interpreted as semicolons. | |
14098 | During output, matrices are displayed in @samp{\matrix@{ a & b \\ c & d@}} | |
14099 | format; you may need to edit this afterwards to change @code{\matrix} | |
14100 | to @code{\pmatrix} or @code{\\} to @code{\cr}. | |
14101 | ||
14102 | Accents like @code{\tilde} and @code{\bar} translate into function | |
14103 | calls internally (@samp{tilde(x)}, @samp{bar(x)}). The @code{\underline} | |
14104 | sequence is treated as an accent. The @code{\vec} accent corresponds | |
14105 | to the function name @code{Vec}, because @code{vec} is the name of | |
14106 | a built-in Calc function. The following table shows the accents | |
14107 | in Calc, @TeX{}, and @dfn{eqn} (described in the next section): | |
14108 | ||
14109 | @iftex | |
14110 | @begingroup | |
14111 | @let@calcindexershow=@calcindexernoshow @c Suppress marginal notes | |
14112 | @let@calcindexersh=@calcindexernoshow | |
14113 | @end iftex | |
5d67986c RS |
14114 | @ignore |
14115 | @starindex | |
14116 | @end ignore | |
d7b8e6c6 | 14117 | @tindex acute |
5d67986c RS |
14118 | @ignore |
14119 | @starindex | |
14120 | @end ignore | |
d7b8e6c6 | 14121 | @tindex bar |
5d67986c RS |
14122 | @ignore |
14123 | @starindex | |
14124 | @end ignore | |
d7b8e6c6 | 14125 | @tindex breve |
5d67986c RS |
14126 | @ignore |
14127 | @starindex | |
14128 | @end ignore | |
d7b8e6c6 | 14129 | @tindex check |
5d67986c RS |
14130 | @ignore |
14131 | @starindex | |
14132 | @end ignore | |
d7b8e6c6 | 14133 | @tindex dot |
5d67986c RS |
14134 | @ignore |
14135 | @starindex | |
14136 | @end ignore | |
d7b8e6c6 | 14137 | @tindex dotdot |
5d67986c RS |
14138 | @ignore |
14139 | @starindex | |
14140 | @end ignore | |
d7b8e6c6 | 14141 | @tindex dyad |
5d67986c RS |
14142 | @ignore |
14143 | @starindex | |
14144 | @end ignore | |
d7b8e6c6 | 14145 | @tindex grave |
5d67986c RS |
14146 | @ignore |
14147 | @starindex | |
14148 | @end ignore | |
d7b8e6c6 | 14149 | @tindex hat |
5d67986c RS |
14150 | @ignore |
14151 | @starindex | |
14152 | @end ignore | |
d7b8e6c6 | 14153 | @tindex Prime |
5d67986c RS |
14154 | @ignore |
14155 | @starindex | |
14156 | @end ignore | |
d7b8e6c6 | 14157 | @tindex tilde |
5d67986c RS |
14158 | @ignore |
14159 | @starindex | |
14160 | @end ignore | |
d7b8e6c6 | 14161 | @tindex under |
5d67986c RS |
14162 | @ignore |
14163 | @starindex | |
14164 | @end ignore | |
d7b8e6c6 EZ |
14165 | @tindex Vec |
14166 | @iftex | |
14167 | @endgroup | |
14168 | @end iftex | |
14169 | @example | |
14170 | Calc TeX eqn | |
14171 | ---- --- --- | |
14172 | acute \acute | |
14173 | bar \bar bar | |
177c0ea7 | 14174 | breve \breve |
d7b8e6c6 EZ |
14175 | check \check |
14176 | dot \dot dot | |
14177 | dotdot \ddot dotdot | |
14178 | dyad dyad | |
14179 | grave \grave | |
14180 | hat \hat hat | |
14181 | Prime prime | |
14182 | tilde \tilde tilde | |
14183 | under \underline under | |
14184 | Vec \vec vec | |
14185 | @end example | |
14186 | ||
14187 | The @samp{=>} (evaluates-to) operator appears as a @code{\to} symbol: | |
14188 | @samp{@{@var{a} \to @var{b}@}}. @TeX{} defines @code{\to} as an | |
14189 | alias for @code{\rightarrow}. However, if the @samp{=>} is the | |
14190 | top-level expression being formatted, a slightly different notation | |
14191 | is used: @samp{\evalto @var{a} \to @var{b}}. The @code{\evalto} | |
14192 | word is ignored by Calc's input routines, and is undefined in @TeX{}. | |
14193 | You will typically want to include one of the following definitions | |
14194 | at the top of a @TeX{} file that uses @code{\evalto}: | |
14195 | ||
14196 | @example | |
14197 | \def\evalto@{@} | |
14198 | \def\evalto#1\to@{@} | |
14199 | @end example | |
14200 | ||
14201 | The first definition formats evaluates-to operators in the usual | |
14202 | way. The second causes only the @var{b} part to appear in the | |
14203 | printed document; the @var{a} part and the arrow are hidden. | |
14204 | Another definition you may wish to use is @samp{\let\to=\Rightarrow} | |
14205 | which causes @code{\to} to appear more like Calc's @samp{=>} symbol. | |
14206 | @xref{Evaluates-To Operator}, for a discussion of @code{evalto}. | |
14207 | ||
14208 | The complete set of @TeX{} control sequences that are ignored during | |
14209 | reading is: | |
14210 | ||
14211 | @example | |
14212 | \hbox \mbox \text \left \right | |
14213 | \, \> \: \; \! \quad \qquad \hfil \hfill | |
14214 | \displaystyle \textstyle \dsize \tsize | |
14215 | \scriptstyle \scriptscriptstyle \ssize \ssize | |
14216 | \rm \bf \it \sl \roman \bold \italic \slanted | |
14217 | \cal \mit \Cal \Bbb \frak \goth | |
14218 | \evalto | |
14219 | @end example | |
14220 | ||
14221 | Note that, because these symbols are ignored, reading a @TeX{} formula | |
14222 | into Calc and writing it back out may lose spacing and font information. | |
14223 | ||
14224 | Also, the ``discretionary multiplication sign'' @samp{\*} is read | |
14225 | the same as @samp{*}. | |
14226 | ||
14227 | @ifinfo | |
14228 | The @TeX{} version of this manual includes some printed examples at the | |
14229 | end of this section. | |
14230 | @end ifinfo | |
14231 | @iftex | |
14232 | Here are some examples of how various Calc formulas are formatted in @TeX{}: | |
14233 | ||
d7b8e6c6 | 14234 | @example |
5d67986c | 14235 | @group |
d7b8e6c6 EZ |
14236 | sin(a^2 / b_i) |
14237 | \sin\left( {a^2 \over b_i} \right) | |
5d67986c | 14238 | @end group |
d7b8e6c6 EZ |
14239 | @end example |
14240 | @tex | |
14241 | \let\rm\goodrm | |
14242 | $$ \sin\left( a^2 \over b_i \right) $$ | |
14243 | @end tex | |
14244 | @sp 1 | |
d7b8e6c6 | 14245 | |
d7b8e6c6 | 14246 | @example |
5d67986c | 14247 | @group |
d7b8e6c6 EZ |
14248 | [(3, 4), 3:4, 3 +/- 4, [3 .. inf)] |
14249 | [3 + 4i, @{3 \over 4@}, 3 \pm 4, [3 \ldots \infty)] | |
5d67986c | 14250 | @end group |
d7b8e6c6 EZ |
14251 | @end example |
14252 | @tex | |
14253 | \turnoffactive | |
14254 | $$ [3 + 4i, {3 \over 4}, 3 \pm 4, [ 3 \ldots \infty)] $$ | |
14255 | @end tex | |
14256 | @sp 1 | |
d7b8e6c6 | 14257 | |
d7b8e6c6 | 14258 | @example |
5d67986c | 14259 | @group |
d7b8e6c6 EZ |
14260 | [abs(a), abs(a / b), floor(a), ceil(a / b)] |
14261 | [|a|, \left| a \over b \right|, | |
14262 | \lfloor a \rfloor, \left\lceil a \over b \right\rceil] | |
5d67986c | 14263 | @end group |
d7b8e6c6 EZ |
14264 | @end example |
14265 | @tex | |
14266 | $$ [|a|, \left| a \over b \right|, | |
14267 | \lfloor a \rfloor, \left\lceil a \over b \right\rceil] $$ | |
14268 | @end tex | |
14269 | @sp 1 | |
d7b8e6c6 | 14270 | |
d7b8e6c6 | 14271 | @example |
5d67986c | 14272 | @group |
d7b8e6c6 EZ |
14273 | [sin(a), sin(2 a), sin(2 + a), sin(a / b)] |
14274 | [\sin@{a@}, \sin@{2 a@}, \sin(2 + a), | |
14275 | \sin\left( @{a \over b@} \right)] | |
5d67986c | 14276 | @end group |
d7b8e6c6 EZ |
14277 | @end example |
14278 | @tex | |
14279 | \turnoffactive\let\rm\goodrm | |
14280 | $$ [\sin{a}, \sin{2 a}, \sin(2 + a), \sin\left( {a \over b} \right)] $$ | |
14281 | @end tex | |
14282 | @sp 2 | |
d7b8e6c6 | 14283 | |
d7b8e6c6 EZ |
14284 | First with plain @kbd{d T}, then with @kbd{C-u d T}, then finally with |
14285 | @kbd{C-u - d T} (using the example definition | |
14286 | @samp{\def\foo#1@{\tilde F(#1)@}}: | |
14287 | ||
14288 | @example | |
5d67986c | 14289 | @group |
d7b8e6c6 EZ |
14290 | [f(a), foo(bar), sin(pi)] |
14291 | [f(a), foo(bar), \sin{\pi}] | |
14292 | [f(a), \hbox@{foo@}(\hbox@{bar@}), \sin@{\pi@}] | |
14293 | [f(a), \foo@{\hbox@{bar@}@}, \sin@{\pi@}] | |
5d67986c | 14294 | @end group |
d7b8e6c6 EZ |
14295 | @end example |
14296 | @tex | |
14297 | \let\rm\goodrm | |
14298 | $$ [f(a), foo(bar), \sin{\pi}] $$ | |
14299 | $$ [f(a), \hbox{foo}(\hbox{bar}), \sin{\pi}] $$ | |
14300 | $$ [f(a), \tilde F(\hbox{bar}), \sin{\pi}] $$ | |
14301 | @end tex | |
14302 | @sp 2 | |
d7b8e6c6 | 14303 | |
d7b8e6c6 EZ |
14304 | First with @samp{\def\evalto@{@}}, then with @samp{\def\evalto#1\to@{@}}: |
14305 | ||
14306 | @example | |
5d67986c | 14307 | @group |
d7b8e6c6 EZ |
14308 | 2 + 3 => 5 |
14309 | \evalto 2 + 3 \to 5 | |
5d67986c | 14310 | @end group |
d7b8e6c6 EZ |
14311 | @end example |
14312 | @tex | |
14313 | \turnoffactive | |
14314 | $$ 2 + 3 \to 5 $$ | |
14315 | $$ 5 $$ | |
14316 | @end tex | |
14317 | @sp 2 | |
d7b8e6c6 | 14318 | |
d7b8e6c6 EZ |
14319 | First with standard @code{\to}, then with @samp{\let\to\Rightarrow}: |
14320 | ||
14321 | @example | |
5d67986c | 14322 | @group |
d7b8e6c6 EZ |
14323 | [2 + 3 => 5, a / 2 => (b + c) / 2] |
14324 | [@{2 + 3 \to 5@}, @{@{a \over 2@} \to @{b + c \over 2@}@}] | |
5d67986c | 14325 | @end group |
d7b8e6c6 EZ |
14326 | @end example |
14327 | @tex | |
14328 | \turnoffactive | |
14329 | $$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$ | |
14330 | {\let\to\Rightarrow | |
14331 | $$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$} | |
14332 | @end tex | |
14333 | @sp 2 | |
d7b8e6c6 | 14334 | |
d7b8e6c6 EZ |
14335 | Matrices normally, then changing @code{\matrix} to @code{\pmatrix}: |
14336 | ||
14337 | @example | |
5d67986c | 14338 | @group |
d7b8e6c6 EZ |
14339 | [ [ a / b, 0 ], [ 0, 2^(x + 1) ] ] |
14340 | \matrix@{ @{a \over b@} & 0 \\ 0 & 2^@{(x + 1)@} @} | |
14341 | \pmatrix@{ @{a \over b@} & 0 \\ 0 & 2^@{(x + 1)@} @} | |
5d67986c | 14342 | @end group |
d7b8e6c6 EZ |
14343 | @end example |
14344 | @tex | |
14345 | \turnoffactive | |
14346 | $$ \matrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$ | |
14347 | $$ \pmatrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$ | |
14348 | @end tex | |
14349 | @sp 2 | |
d7b8e6c6 EZ |
14350 | @end iftex |
14351 | ||
14352 | @node Eqn Language Mode, Mathematica Language Mode, TeX Language Mode, Language Modes | |
14353 | @subsection Eqn Language Mode | |
14354 | ||
14355 | @noindent | |
14356 | @kindex d E | |
14357 | @pindex calc-eqn-language | |
14358 | @dfn{Eqn} is another popular formatter for math formulas. It is | |
14359 | designed for use with the TROFF text formatter, and comes standard | |
14360 | with many versions of Unix. The @kbd{d E} (@code{calc-eqn-language}) | |
14361 | command selects @dfn{eqn} notation. | |
14362 | ||
14363 | The @dfn{eqn} language's main idiosyncrasy is that whitespace plays | |
14364 | a significant part in the parsing of the language. For example, | |
14365 | @samp{sqrt x+1 + y} treats @samp{x+1} as the argument of the | |
14366 | @code{sqrt} operator. @dfn{Eqn} also understands more conventional | |
14367 | grouping using curly braces: @samp{sqrt@{x+1@} + y}. Braces are | |
14368 | required only when the argument contains spaces. | |
14369 | ||
14370 | In Calc's @dfn{eqn} mode, however, curly braces are required to | |
14371 | delimit arguments of operators like @code{sqrt}. The first of the | |
14372 | above examples would treat only the @samp{x} as the argument of | |
14373 | @code{sqrt}, and in fact @samp{sin x+1} would be interpreted as | |
14374 | @samp{sin * x + 1}, because @code{sin} is not a special operator | |
14375 | in the @dfn{eqn} language. If you always surround the argument | |
14376 | with curly braces, Calc will never misunderstand. | |
14377 | ||
14378 | Calc also understands parentheses as grouping characters. Another | |
14379 | peculiarity of @dfn{eqn}'s syntax makes it advisable to separate | |
14380 | words with spaces from any surrounding characters that aren't curly | |
14381 | braces, so Calc writes @samp{sin ( x + y )} in @dfn{eqn} mode. | |
14382 | (The spaces around @code{sin} are important to make @dfn{eqn} | |
14383 | recognize that @code{sin} should be typeset in a roman font, and | |
14384 | the spaces around @code{x} and @code{y} are a good idea just in | |
14385 | case the @dfn{eqn} document has defined special meanings for these | |
14386 | names, too.) | |
14387 | ||
14388 | Powers and subscripts are written with the @code{sub} and @code{sup} | |
14389 | operators, respectively. Note that the caret symbol @samp{^} is | |
14390 | treated the same as a space in @dfn{eqn} mode, as is the @samp{~} | |
14391 | symbol (these are used to introduce spaces of various widths into | |
14392 | the typeset output of @dfn{eqn}). | |
14393 | ||
14394 | As in @TeX{} mode, Calc's formatter omits parentheses around the | |
14395 | arguments of functions like @code{ln} and @code{sin} if they are | |
14396 | ``simple-looking''; in this case Calc surrounds the argument with | |
14397 | braces, separated by a @samp{~} from the function name: @samp{sin~@{x@}}. | |
14398 | ||
14399 | Font change codes (like @samp{roman @var{x}}) and positioning codes | |
14400 | (like @samp{~} and @samp{down @var{n} @var{x}}) are ignored by the | |
14401 | @dfn{eqn} reader. Also ignored are the words @code{left}, @code{right}, | |
14402 | @code{mark}, and @code{lineup}. Quotation marks in @dfn{eqn} mode input | |
14403 | are treated the same as curly braces: @samp{sqrt "1+x"} is equivalent to | |
14404 | @samp{sqrt @{1+x@}}; this is only an approximation to the true meaning | |
14405 | of quotes in @dfn{eqn}, but it is good enough for most uses. | |
14406 | ||
14407 | Accent codes (@samp{@var{x} dot}) are handled by treating them as | |
14408 | function calls (@samp{dot(@var{x})}) internally. @xref{TeX Language | |
b275eac7 | 14409 | Mode}, for a table of these accent functions. The @code{prime} accent |
d7b8e6c6 EZ |
14410 | is treated specially if it occurs on a variable or function name: |
14411 | @samp{f prime prime @w{( x prime )}} is stored internally as | |
14412 | @samp{f'@w{'}(x')}. For example, taking the derivative of @samp{f(2 x)} | |
14413 | with @kbd{a d x} will produce @samp{2 f'(2 x)}, which @dfn{eqn} mode | |
14414 | will display as @samp{2 f prime ( 2 x )}. | |
14415 | ||
14416 | Assignments are written with the @samp{<-} (left-arrow) symbol, | |
14417 | and @code{evalto} operators are written with @samp{->} or | |
14418 | @samp{evalto ... ->} (@pxref{TeX Language Mode}, for a discussion | |
14419 | of this). The regular Calc symbols @samp{:=} and @samp{=>} are also | |
14420 | recognized for these operators during reading. | |
14421 | ||
14422 | Vectors in @dfn{eqn} mode use regular Calc square brackets, but | |
14423 | matrices are formatted as @samp{matrix @{ ccol @{ a above b @} ... @}}. | |
14424 | The words @code{lcol} and @code{rcol} are recognized as synonyms | |
14425 | for @code{ccol} during input, and are generated instead of @code{ccol} | |
14426 | if the matrix justification mode so specifies. | |
14427 | ||
14428 | @node Mathematica Language Mode, Maple Language Mode, Eqn Language Mode, Language Modes | |
14429 | @subsection Mathematica Language Mode | |
14430 | ||
14431 | @noindent | |
14432 | @kindex d M | |
14433 | @pindex calc-mathematica-language | |
14434 | @cindex Mathematica language | |
14435 | The @kbd{d M} (@code{calc-mathematica-language}) command selects the | |
14436 | conventions of Mathematica, a powerful and popular mathematical tool | |
14437 | from Wolfram Research, Inc. Notable differences in Mathematica mode | |
14438 | are that the names of built-in functions are capitalized, and function | |
14439 | calls use square brackets instead of parentheses. Thus the Calc | |
14440 | formula @samp{sin(2 x)} is entered and displayed @w{@samp{Sin[2 x]}} in | |
14441 | Mathematica mode. | |
14442 | ||
14443 | Vectors and matrices use curly braces in Mathematica. Complex numbers | |
14444 | are written @samp{3 + 4 I}. The standard special constants in Calc are | |
14445 | written @code{Pi}, @code{E}, @code{I}, @code{GoldenRatio}, @code{EulerGamma}, | |
14446 | @code{Infinity}, @code{ComplexInfinity}, and @code{Indeterminate} in | |
14447 | Mathematica mode. | |
14448 | Non-decimal numbers are written, e.g., @samp{16^^7fff}. Floating-point | |
14449 | numbers in scientific notation are written @samp{1.23*10.^3}. | |
14450 | Subscripts use double square brackets: @samp{a[[i]]}.@refill | |
14451 | ||
14452 | @node Maple Language Mode, Compositions, Mathematica Language Mode, Language Modes | |
14453 | @subsection Maple Language Mode | |
14454 | ||
14455 | @noindent | |
14456 | @kindex d W | |
14457 | @pindex calc-maple-language | |
14458 | @cindex Maple language | |
14459 | The @kbd{d W} (@code{calc-maple-language}) command selects the | |
14460 | conventions of Maple, another mathematical tool from the University | |
177c0ea7 | 14461 | of Waterloo. |
d7b8e6c6 EZ |
14462 | |
14463 | Maple's language is much like C. Underscores are allowed in symbol | |
14464 | names; square brackets are used for subscripts; explicit @samp{*}s for | |
14465 | multiplications are required. Use either @samp{^} or @samp{**} to | |
14466 | denote powers. | |
14467 | ||
14468 | Maple uses square brackets for lists and curly braces for sets. Calc | |
14469 | interprets both notations as vectors, and displays vectors with square | |
14470 | brackets. This means Maple sets will be converted to lists when they | |
14471 | pass through Calc. As a special case, matrices are written as calls | |
14472 | to the function @code{matrix}, given a list of lists as the argument, | |
14473 | and can be read in this form or with all-capitals @code{MATRIX}. | |
14474 | ||
14475 | The Maple interval notation @samp{2 .. 3} has no surrounding brackets; | |
14476 | Calc reads @samp{2 .. 3} as the closed interval @samp{[2 .. 3]}, and | |
14477 | writes any kind of interval as @samp{2 .. 3}. This means you cannot | |
14478 | see the difference between an open and a closed interval while in | |
14479 | Maple display mode. | |
14480 | ||
14481 | Maple writes complex numbers as @samp{3 + 4*I}. Its special constants | |
14482 | are @code{Pi}, @code{E}, @code{I}, and @code{infinity} (all three of | |
14483 | @code{inf}, @code{uinf}, and @code{nan} display as @code{infinity}). | |
14484 | Floating-point numbers are written @samp{1.23*10.^3}. | |
14485 | ||
14486 | Among things not currently handled by Calc's Maple mode are the | |
14487 | various quote symbols, procedures and functional operators, and | |
14488 | inert (@samp{&}) operators. | |
14489 | ||
14490 | @node Compositions, Syntax Tables, Maple Language Mode, Language Modes | |
14491 | @subsection Compositions | |
14492 | ||
14493 | @noindent | |
14494 | @cindex Compositions | |
14495 | There are several @dfn{composition functions} which allow you to get | |
14496 | displays in a variety of formats similar to those in Big language | |
14497 | mode. Most of these functions do not evaluate to anything; they are | |
14498 | placeholders which are left in symbolic form by Calc's evaluator but | |
14499 | are recognized by Calc's display formatting routines. | |
14500 | ||
14501 | Two of these, @code{string} and @code{bstring}, are described elsewhere. | |
14502 | @xref{Strings}. For example, @samp{string("ABC")} is displayed as | |
14503 | @samp{ABC}. When viewed on the stack it will be indistinguishable from | |
14504 | the variable @code{ABC}, but internally it will be stored as | |
14505 | @samp{string([65, 66, 67])} and can still be manipulated this way; for | |
14506 | example, the selection and vector commands @kbd{j 1 v v j u} would | |
14507 | select the vector portion of this object and reverse the elements, then | |
14508 | deselect to reveal a string whose characters had been reversed. | |
14509 | ||
14510 | The composition functions do the same thing in all language modes | |
14511 | (although their components will of course be formatted in the current | |
14512 | language mode). The one exception is Unformatted mode (@kbd{d U}), | |
14513 | which does not give the composition functions any special treatment. | |
14514 | The functions are discussed here because of their relationship to | |
14515 | the language modes. | |
14516 | ||
14517 | @menu | |
14518 | * Composition Basics:: | |
14519 | * Horizontal Compositions:: | |
14520 | * Vertical Compositions:: | |
14521 | * Other Compositions:: | |
14522 | * Information about Compositions:: | |
14523 | * User-Defined Compositions:: | |
14524 | @end menu | |
14525 | ||
14526 | @node Composition Basics, Horizontal Compositions, Compositions, Compositions | |
14527 | @subsubsection Composition Basics | |
14528 | ||
14529 | @noindent | |
14530 | Compositions are generally formed by stacking formulas together | |
14531 | horizontally or vertically in various ways. Those formulas are | |
14532 | themselves compositions. @TeX{} users will find this analogous | |
14533 | to @TeX{}'s ``boxes.'' Each multi-line composition has a | |
14534 | @dfn{baseline}; horizontal compositions use the baselines to | |
14535 | decide how formulas should be positioned relative to one another. | |
14536 | For example, in the Big mode formula | |
14537 | ||
d7b8e6c6 | 14538 | @example |
5d67986c | 14539 | @group |
d7b8e6c6 EZ |
14540 | 2 |
14541 | a + b | |
14542 | 17 + ------ | |
14543 | c | |
d7b8e6c6 | 14544 | @end group |
5d67986c | 14545 | @end example |
d7b8e6c6 EZ |
14546 | |
14547 | @noindent | |
14548 | the second term of the sum is four lines tall and has line three as | |
14549 | its baseline. Thus when the term is combined with 17, line three | |
14550 | is placed on the same level as the baseline of 17. | |
14551 | ||
14552 | @tex | |
14553 | \bigskip | |
14554 | @end tex | |
14555 | ||
14556 | Another important composition concept is @dfn{precedence}. This is | |
14557 | an integer that represents the binding strength of various operators. | |
14558 | For example, @samp{*} has higher precedence (195) than @samp{+} (180), | |
14559 | which means that @samp{(a * b) + c} will be formatted without the | |
14560 | parentheses, but @samp{a * (b + c)} will keep the parentheses. | |
14561 | ||
14562 | The operator table used by normal and Big language modes has the | |
14563 | following precedences: | |
14564 | ||
14565 | @example | |
14566 | _ 1200 @r{(subscripts)} | |
14567 | % 1100 @r{(as in n}%@r{)} | |
14568 | - 1000 @r{(as in }-@r{n)} | |
14569 | ! 1000 @r{(as in }!@r{n)} | |
14570 | mod 400 | |
14571 | +/- 300 | |
14572 | !! 210 @r{(as in n}!!@r{)} | |
14573 | ! 210 @r{(as in n}!@r{)} | |
14574 | ^ 200 | |
14575 | * 195 @r{(or implicit multiplication)} | |
14576 | / % \ 190 | |
14577 | + - 180 @r{(as in a}+@r{b)} | |
14578 | | 170 | |
14579 | < = 160 @r{(and other relations)} | |
14580 | && 110 | |
14581 | || 100 | |
14582 | ? : 90 | |
14583 | !!! 85 | |
14584 | &&& 80 | |
14585 | ||| 75 | |
14586 | := 50 | |
14587 | :: 45 | |
14588 | => 40 | |
14589 | @end example | |
14590 | ||
14591 | The general rule is that if an operator with precedence @cite{n} | |
14592 | occurs as an argument to an operator with precedence @cite{m}, then | |
14593 | the argument is enclosed in parentheses if @cite{n < m}. Top-level | |
14594 | expressions and expressions which are function arguments, vector | |
14595 | components, etc., are formatted with precedence zero (so that they | |
14596 | normally never get additional parentheses). | |
14597 | ||
14598 | For binary left-associative operators like @samp{+}, the righthand | |
14599 | argument is actually formatted with one-higher precedence than shown | |
14600 | in the table. This makes sure @samp{(a + b) + c} omits the parentheses, | |
14601 | but the unnatural form @samp{a + (b + c)} keeps its parentheses. | |
14602 | Right-associative operators like @samp{^} format the lefthand argument | |
14603 | with one-higher precedence. | |
14604 | ||
5d67986c RS |
14605 | @ignore |
14606 | @starindex | |
14607 | @end ignore | |
d7b8e6c6 EZ |
14608 | @tindex cprec |
14609 | The @code{cprec} function formats an expression with an arbitrary | |
14610 | precedence. For example, @samp{cprec(abc, 185)} will combine into | |
14611 | sums and products as follows: @samp{7 + abc}, @samp{7 (abc)} (because | |
14612 | this @code{cprec} form has higher precedence than addition, but lower | |
14613 | precedence than multiplication). | |
14614 | ||
14615 | @tex | |
14616 | \bigskip | |
14617 | @end tex | |
14618 | ||
14619 | A final composition issue is @dfn{line breaking}. Calc uses two | |
14620 | different strategies for ``flat'' and ``non-flat'' compositions. | |
14621 | A non-flat composition is anything that appears on multiple lines | |
14622 | (not counting line breaking). Examples would be matrices and Big | |
14623 | mode powers and quotients. Non-flat compositions are displayed | |
14624 | exactly as specified. If they come out wider than the current | |
14625 | window, you must use horizontal scrolling (@kbd{<} and @kbd{>}) to | |
14626 | view them. | |
14627 | ||
14628 | Flat compositions, on the other hand, will be broken across several | |
14629 | lines if they are too wide to fit the window. Certain points in a | |
14630 | composition are noted internally as @dfn{break points}. Calc's | |
14631 | general strategy is to fill each line as much as possible, then to | |
14632 | move down to the next line starting at the first break point that | |
14633 | didn't fit. However, the line breaker understands the hierarchical | |
14634 | structure of formulas. It will not break an ``inner'' formula if | |
14635 | it can use an earlier break point from an ``outer'' formula instead. | |
14636 | For example, a vector of sums might be formatted as: | |
14637 | ||
d7b8e6c6 | 14638 | @example |
5d67986c | 14639 | @group |
d7b8e6c6 EZ |
14640 | [ a + b + c, d + e + f, |
14641 | g + h + i, j + k + l, m ] | |
d7b8e6c6 | 14642 | @end group |
5d67986c | 14643 | @end example |
d7b8e6c6 EZ |
14644 | |
14645 | @noindent | |
14646 | If the @samp{m} can fit, then so, it seems, could the @samp{g}. | |
14647 | But Calc prefers to break at the comma since the comma is part | |
14648 | of a ``more outer'' formula. Calc would break at a plus sign | |
14649 | only if it had to, say, if the very first sum in the vector had | |
14650 | itself been too large to fit. | |
14651 | ||
14652 | Of the composition functions described below, only @code{choriz} | |
14653 | generates break points. The @code{bstring} function (@pxref{Strings}) | |
14654 | also generates breakable items: A break point is added after every | |
14655 | space (or group of spaces) except for spaces at the very beginning or | |
14656 | end of the string. | |
14657 | ||
14658 | Composition functions themselves count as levels in the formula | |
14659 | hierarchy, so a @code{choriz} that is a component of a larger | |
14660 | @code{choriz} will be less likely to be broken. As a special case, | |
14661 | if a @code{bstring} occurs as a component of a @code{choriz} or | |
14662 | @code{choriz}-like object (such as a vector or a list of arguments | |
14663 | in a function call), then the break points in that @code{bstring} | |
14664 | will be on the same level as the break points of the surrounding | |
14665 | object. | |
14666 | ||
14667 | @node Horizontal Compositions, Vertical Compositions, Composition Basics, Compositions | |
14668 | @subsubsection Horizontal Compositions | |
14669 | ||
14670 | @noindent | |
5d67986c RS |
14671 | @ignore |
14672 | @starindex | |
14673 | @end ignore | |
d7b8e6c6 EZ |
14674 | @tindex choriz |
14675 | The @code{choriz} function takes a vector of objects and composes | |
14676 | them horizontally. For example, @samp{choriz([17, a b/c, d])} formats | |
14677 | as @w{@samp{17a b / cd}} in normal language mode, or as | |
14678 | ||
d7b8e6c6 | 14679 | @example |
5d67986c | 14680 | @group |
d7b8e6c6 EZ |
14681 | a b |
14682 | 17---d | |
14683 | c | |
d7b8e6c6 | 14684 | @end group |
5d67986c | 14685 | @end example |
d7b8e6c6 EZ |
14686 | |
14687 | @noindent | |
14688 | in Big language mode. This is actually one case of the general | |
14689 | function @samp{choriz(@var{vec}, @var{sep}, @var{prec})}, where | |
14690 | either or both of @var{sep} and @var{prec} may be omitted. | |
14691 | @var{Prec} gives the @dfn{precedence} to use when formatting | |
14692 | each of the components of @var{vec}. The default precedence is | |
14693 | the precedence from the surrounding environment. | |
14694 | ||
14695 | @var{Sep} is a string (i.e., a vector of character codes as might | |
14696 | be entered with @code{" "} notation) which should separate components | |
14697 | of the composition. Also, if @var{sep} is given, the line breaker | |
14698 | will allow lines to be broken after each occurrence of @var{sep}. | |
14699 | If @var{sep} is omitted, the composition will not be breakable | |
14700 | (unless any of its component compositions are breakable). | |
14701 | ||
14702 | For example, @samp{2 choriz([a, b c, d = e], " + ", 180)} is | |
14703 | formatted as @samp{2 a + b c + (d = e)}. To get the @code{choriz} | |
14704 | to have precedence 180 ``outwards'' as well as ``inwards,'' | |
14705 | enclose it in a @code{cprec} form: @samp{2 cprec(choriz(...), 180)} | |
14706 | formats as @samp{2 (a + b c + (d = e))}. | |
14707 | ||
14708 | The baseline of a horizontal composition is the same as the | |
14709 | baselines of the component compositions, which are all aligned. | |
14710 | ||
14711 | @node Vertical Compositions, Other Compositions, Horizontal Compositions, Compositions | |
14712 | @subsubsection Vertical Compositions | |
14713 | ||
14714 | @noindent | |
5d67986c RS |
14715 | @ignore |
14716 | @starindex | |
14717 | @end ignore | |
d7b8e6c6 EZ |
14718 | @tindex cvert |
14719 | The @code{cvert} function makes a vertical composition. Each | |
14720 | component of the vector is centered in a column. The baseline of | |
14721 | the result is by default the top line of the resulting composition. | |
14722 | For example, @samp{f(cvert([a, bb, ccc]), cvert([a^2 + 1, b^2]))} | |
14723 | formats in Big mode as | |
14724 | ||
d7b8e6c6 | 14725 | @example |
5d67986c | 14726 | @group |
d7b8e6c6 EZ |
14727 | f( a , 2 ) |
14728 | bb a + 1 | |
14729 | ccc 2 | |
14730 | b | |
d7b8e6c6 | 14731 | @end group |
5d67986c | 14732 | @end example |
d7b8e6c6 | 14733 | |
5d67986c RS |
14734 | @ignore |
14735 | @starindex | |
14736 | @end ignore | |
d7b8e6c6 EZ |
14737 | @tindex cbase |
14738 | There are several special composition functions that work only as | |
14739 | components of a vertical composition. The @code{cbase} function | |
14740 | controls the baseline of the vertical composition; the baseline | |
14741 | will be the same as the baseline of whatever component is enclosed | |
14742 | in @code{cbase}. Thus @samp{f(cvert([a, cbase(bb), ccc]), | |
14743 | cvert([a^2 + 1, cbase(b^2)]))} displays as | |
14744 | ||
d7b8e6c6 | 14745 | @example |
5d67986c | 14746 | @group |
d7b8e6c6 EZ |
14747 | 2 |
14748 | a + 1 | |
14749 | a 2 | |
14750 | f(bb , b ) | |
14751 | ccc | |
d7b8e6c6 | 14752 | @end group |
5d67986c | 14753 | @end example |
d7b8e6c6 | 14754 | |
5d67986c RS |
14755 | @ignore |
14756 | @starindex | |
14757 | @end ignore | |
d7b8e6c6 | 14758 | @tindex ctbase |
5d67986c RS |
14759 | @ignore |
14760 | @starindex | |
14761 | @end ignore | |
d7b8e6c6 EZ |
14762 | @tindex cbbase |
14763 | There are also @code{ctbase} and @code{cbbase} functions which | |
14764 | make the baseline of the vertical composition equal to the top | |
14765 | or bottom line (rather than the baseline) of that component. | |
14766 | Thus @samp{cvert([cbase(a / b)]) + cvert([ctbase(a / b)]) + | |
14767 | cvert([cbbase(a / b)])} gives | |
14768 | ||
d7b8e6c6 | 14769 | @example |
5d67986c | 14770 | @group |
d7b8e6c6 EZ |
14771 | a |
14772 | a - | |
14773 | - + a + b | |
14774 | b - | |
14775 | b | |
d7b8e6c6 | 14776 | @end group |
5d67986c | 14777 | @end example |
d7b8e6c6 EZ |
14778 | |
14779 | There should be only one @code{cbase}, @code{ctbase}, or @code{cbbase} | |
14780 | function in a given vertical composition. These functions can also | |
14781 | be written with no arguments: @samp{ctbase()} is a zero-height object | |
14782 | which means the baseline is the top line of the following item, and | |
14783 | @samp{cbbase()} means the baseline is the bottom line of the preceding | |
14784 | item. | |
14785 | ||
5d67986c RS |
14786 | @ignore |
14787 | @starindex | |
14788 | @end ignore | |
d7b8e6c6 EZ |
14789 | @tindex crule |
14790 | The @code{crule} function builds a ``rule,'' or horizontal line, | |
14791 | across a vertical composition. By itself @samp{crule()} uses @samp{-} | |
14792 | characters to build the rule. You can specify any other character, | |
14793 | e.g., @samp{crule("=")}. The argument must be a character code or | |
14794 | vector of exactly one character code. It is repeated to match the | |
14795 | width of the widest item in the stack. For example, a quotient | |
14796 | with a thick line is @samp{cvert([a + 1, cbase(crule("=")), b^2])}: | |
14797 | ||
d7b8e6c6 | 14798 | @example |
5d67986c | 14799 | @group |
d7b8e6c6 EZ |
14800 | a + 1 |
14801 | ===== | |
14802 | 2 | |
14803 | b | |
d7b8e6c6 | 14804 | @end group |
5d67986c | 14805 | @end example |
d7b8e6c6 | 14806 | |
5d67986c RS |
14807 | @ignore |
14808 | @starindex | |
14809 | @end ignore | |
d7b8e6c6 | 14810 | @tindex clvert |
5d67986c RS |
14811 | @ignore |
14812 | @starindex | |
14813 | @end ignore | |
d7b8e6c6 EZ |
14814 | @tindex crvert |
14815 | Finally, the functions @code{clvert} and @code{crvert} act exactly | |
14816 | like @code{cvert} except that the items are left- or right-justified | |
14817 | in the stack. Thus @samp{clvert([a, bb, ccc]) + crvert([a, bb, ccc])} | |
14818 | gives: | |
14819 | ||
d7b8e6c6 | 14820 | @example |
5d67986c | 14821 | @group |
d7b8e6c6 EZ |
14822 | a + a |
14823 | bb bb | |
14824 | ccc ccc | |
d7b8e6c6 | 14825 | @end group |
5d67986c | 14826 | @end example |
d7b8e6c6 EZ |
14827 | |
14828 | Like @code{choriz}, the vertical compositions accept a second argument | |
14829 | which gives the precedence to use when formatting the components. | |
14830 | Vertical compositions do not support separator strings. | |
14831 | ||
14832 | @node Other Compositions, Information about Compositions, Vertical Compositions, Compositions | |
14833 | @subsubsection Other Compositions | |
14834 | ||
14835 | @noindent | |
5d67986c RS |
14836 | @ignore |
14837 | @starindex | |
14838 | @end ignore | |
d7b8e6c6 EZ |
14839 | @tindex csup |
14840 | The @code{csup} function builds a superscripted expression. For | |
14841 | example, @samp{csup(a, b)} looks the same as @samp{a^b} does in Big | |
14842 | language mode. This is essentially a horizontal composition of | |
14843 | @samp{a} and @samp{b}, where @samp{b} is shifted up so that its | |
14844 | bottom line is one above the baseline. | |
14845 | ||
5d67986c RS |
14846 | @ignore |
14847 | @starindex | |
14848 | @end ignore | |
d7b8e6c6 EZ |
14849 | @tindex csub |
14850 | Likewise, the @code{csub} function builds a subscripted expression. | |
14851 | This shifts @samp{b} down so that its top line is one below the | |
14852 | bottom line of @samp{a} (note that this is not quite analogous to | |
14853 | @code{csup}). Other arrangements can be obtained by using | |
14854 | @code{choriz} and @code{cvert} directly. | |
14855 | ||
5d67986c RS |
14856 | @ignore |
14857 | @starindex | |
14858 | @end ignore | |
d7b8e6c6 EZ |
14859 | @tindex cflat |
14860 | The @code{cflat} function formats its argument in ``flat'' mode, | |
14861 | as obtained by @samp{d O}, if the current language mode is normal | |
14862 | or Big. It has no effect in other language modes. For example, | |
14863 | @samp{a^(b/c)} is formatted by Big mode like @samp{csup(a, cflat(b/c))} | |
14864 | to improve its readability. | |
14865 | ||
5d67986c RS |
14866 | @ignore |
14867 | @starindex | |
14868 | @end ignore | |
d7b8e6c6 EZ |
14869 | @tindex cspace |
14870 | The @code{cspace} function creates horizontal space. For example, | |
14871 | @samp{cspace(4)} is effectively the same as @samp{string(" ")}. | |
14872 | A second string (i.e., vector of characters) argument is repeated | |
14873 | instead of the space character. For example, @samp{cspace(4, "ab")} | |
14874 | looks like @samp{abababab}. If the second argument is not a string, | |
14875 | it is formatted in the normal way and then several copies of that | |
14876 | are composed together: @samp{cspace(4, a^2)} yields | |
14877 | ||
d7b8e6c6 | 14878 | @example |
5d67986c | 14879 | @group |
d7b8e6c6 EZ |
14880 | 2 2 2 2 |
14881 | a a a a | |
d7b8e6c6 | 14882 | @end group |
5d67986c | 14883 | @end example |
d7b8e6c6 EZ |
14884 | |
14885 | @noindent | |
14886 | If the number argument is zero, this is a zero-width object. | |
14887 | ||
5d67986c RS |
14888 | @ignore |
14889 | @starindex | |
14890 | @end ignore | |
d7b8e6c6 EZ |
14891 | @tindex cvspace |
14892 | The @code{cvspace} function creates vertical space, or a vertical | |
14893 | stack of copies of a certain string or formatted object. The | |
14894 | baseline is the center line of the resulting stack. A numerical | |
14895 | argument of zero will produce an object which contributes zero | |
14896 | height if used in a vertical composition. | |
14897 | ||
5d67986c RS |
14898 | @ignore |
14899 | @starindex | |
14900 | @end ignore | |
d7b8e6c6 | 14901 | @tindex ctspace |
5d67986c RS |
14902 | @ignore |
14903 | @starindex | |
14904 | @end ignore | |
d7b8e6c6 EZ |
14905 | @tindex cbspace |
14906 | There are also @code{ctspace} and @code{cbspace} functions which | |
14907 | create vertical space with the baseline the same as the baseline | |
14908 | of the top or bottom copy, respectively, of the second argument. | |
14909 | Thus @samp{cvspace(2, a/b) + ctspace(2, a/b) + cbspace(2, a/b)} | |
14910 | displays as: | |
14911 | ||
d7b8e6c6 | 14912 | @example |
5d67986c | 14913 | @group |
d7b8e6c6 EZ |
14914 | a |
14915 | - | |
14916 | a b | |
14917 | - a a | |
14918 | b + - + - | |
14919 | a b b | |
14920 | - a | |
14921 | b - | |
14922 | b | |
d7b8e6c6 | 14923 | @end group |
5d67986c | 14924 | @end example |
d7b8e6c6 EZ |
14925 | |
14926 | @node Information about Compositions, User-Defined Compositions, Other Compositions, Compositions | |
14927 | @subsubsection Information about Compositions | |
14928 | ||
14929 | @noindent | |
14930 | The functions in this section are actual functions; they compose their | |
14931 | arguments according to the current language and other display modes, | |
14932 | then return a certain measurement of the composition as an integer. | |
14933 | ||
5d67986c RS |
14934 | @ignore |
14935 | @starindex | |
14936 | @end ignore | |
d7b8e6c6 EZ |
14937 | @tindex cwidth |
14938 | The @code{cwidth} function measures the width, in characters, of a | |
14939 | composition. For example, @samp{cwidth(a + b)} is 5, and | |
14940 | @samp{cwidth(a / b)} is 5 in normal mode, 1 in Big mode, and 11 in | |
14941 | @TeX{} mode (for @samp{@{a \over b@}}). The argument may involve | |
14942 | the composition functions described in this section. | |
14943 | ||
5d67986c RS |
14944 | @ignore |
14945 | @starindex | |
14946 | @end ignore | |
d7b8e6c6 EZ |
14947 | @tindex cheight |
14948 | The @code{cheight} function measures the height of a composition. | |
14949 | This is the total number of lines in the argument's printed form. | |
14950 | ||
5d67986c RS |
14951 | @ignore |
14952 | @starindex | |
14953 | @end ignore | |
d7b8e6c6 | 14954 | @tindex cascent |
5d67986c RS |
14955 | @ignore |
14956 | @starindex | |
14957 | @end ignore | |
d7b8e6c6 EZ |
14958 | @tindex cdescent |
14959 | The functions @code{cascent} and @code{cdescent} measure the amount | |
14960 | of the height that is above (and including) the baseline, or below | |
14961 | the baseline, respectively. Thus @samp{cascent(@var{x}) + cdescent(@var{x})} | |
14962 | always equals @samp{cheight(@var{x})}. For a one-line formula like | |
14963 | @samp{a + b}, @code{cascent} returns 1 and @code{cdescent} returns 0. | |
14964 | For @samp{a / b} in Big mode, @code{cascent} returns 2 and @code{cdescent} | |
14965 | returns 1. The only formula for which @code{cascent} will return zero | |
14966 | is @samp{cvspace(0)} or equivalents. | |
14967 | ||
14968 | @node User-Defined Compositions, , Information about Compositions, Compositions | |
14969 | @subsubsection User-Defined Compositions | |
14970 | ||
14971 | @noindent | |
14972 | @kindex Z C | |
14973 | @pindex calc-user-define-composition | |
14974 | The @kbd{Z C} (@code{calc-user-define-composition}) command lets you | |
14975 | define the display format for any algebraic function. You provide a | |
14976 | formula containing a certain number of argument variables on the stack. | |
14977 | Any time Calc formats a call to the specified function in the current | |
14978 | language mode and with that number of arguments, Calc effectively | |
14979 | replaces the function call with that formula with the arguments | |
14980 | replaced. | |
14981 | ||
14982 | Calc builds the default argument list by sorting all the variable names | |
14983 | that appear in the formula into alphabetical order. You can edit this | |
14984 | argument list before pressing @key{RET} if you wish. Any variables in | |
14985 | the formula that do not appear in the argument list will be displayed | |
14986 | literally; any arguments that do not appear in the formula will not | |
14987 | affect the display at all. | |
14988 | ||
14989 | You can define formats for built-in functions, for functions you have | |
14990 | defined with @kbd{Z F} (@pxref{Algebraic Definitions}), or for functions | |
14991 | which have no definitions but are being used as purely syntactic objects. | |
14992 | You can define different formats for each language mode, and for each | |
14993 | number of arguments, using a succession of @kbd{Z C} commands. When | |
14994 | Calc formats a function call, it first searches for a format defined | |
14995 | for the current language mode (and number of arguments); if there is | |
14996 | none, it uses the format defined for the Normal language mode. If | |
14997 | neither format exists, Calc uses its built-in standard format for that | |
14998 | function (usually just @samp{@var{func}(@var{args})}). | |
14999 | ||
15000 | If you execute @kbd{Z C} with the number 0 on the stack instead of a | |
15001 | formula, any defined formats for the function in the current language | |
15002 | mode will be removed. The function will revert to its standard format. | |
15003 | ||
15004 | For example, the default format for the binomial coefficient function | |
15005 | @samp{choose(n, m)} in the Big language mode is | |
15006 | ||
d7b8e6c6 | 15007 | @example |
5d67986c | 15008 | @group |
d7b8e6c6 EZ |
15009 | n |
15010 | ( ) | |
15011 | m | |
d7b8e6c6 | 15012 | @end group |
5d67986c | 15013 | @end example |
d7b8e6c6 EZ |
15014 | |
15015 | @noindent | |
15016 | You might prefer the notation, | |
15017 | ||
d7b8e6c6 | 15018 | @example |
5d67986c | 15019 | @group |
d7b8e6c6 EZ |
15020 | C |
15021 | n m | |
d7b8e6c6 | 15022 | @end group |
5d67986c | 15023 | @end example |
d7b8e6c6 EZ |
15024 | |
15025 | @noindent | |
15026 | To define this notation, first make sure you are in Big mode, | |
15027 | then put the formula | |
15028 | ||
15029 | @smallexample | |
15030 | choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])]) | |
15031 | @end smallexample | |
15032 | ||
15033 | @noindent | |
15034 | on the stack and type @kbd{Z C}. Answer the first prompt with | |
15035 | @code{choose}. The second prompt will be the default argument list | |
15036 | of @samp{(C m n)}. Edit this list to be @samp{(n m)} and press | |
15037 | @key{RET}. Now, try it out: For example, turn simplification | |
15038 | off with @kbd{m O} and enter @samp{choose(a,b) + choose(7,3)} | |
15039 | as an algebraic entry. | |
15040 | ||
d7b8e6c6 | 15041 | @example |
5d67986c | 15042 | @group |
177c0ea7 | 15043 | C + C |
d7b8e6c6 | 15044 | a b 7 3 |
d7b8e6c6 | 15045 | @end group |
5d67986c | 15046 | @end example |
d7b8e6c6 EZ |
15047 | |
15048 | As another example, let's define the usual notation for Stirling | |
15049 | numbers of the first kind, @samp{stir1(n, m)}. This is just like | |
15050 | the regular format for binomial coefficients but with square brackets | |
15051 | instead of parentheses. | |
15052 | ||
15053 | @smallexample | |
15054 | choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")]) | |
15055 | @end smallexample | |
15056 | ||
15057 | Now type @kbd{Z C stir1 @key{RET}}, edit the argument list to | |
15058 | @samp{(n m)}, and type @key{RET}. | |
15059 | ||
15060 | The formula provided to @kbd{Z C} usually will involve composition | |
15061 | functions, but it doesn't have to. Putting the formula @samp{a + b + c} | |
15062 | onto the stack and typing @kbd{Z C foo @key{RET} @key{RET}} would define | |
15063 | the function @samp{foo(x,y,z)} to display like @samp{x + y + z}. | |
15064 | This ``sum'' will act exactly like a real sum for all formatting | |
15065 | purposes (it will be parenthesized the same, and so on). However | |
15066 | it will be computationally unrelated to a sum. For example, the | |
15067 | formula @samp{2 * foo(1, 2, 3)} will display as @samp{2 (1 + 2 + 3)}. | |
15068 | Operator precedences have caused the ``sum'' to be written in | |
15069 | parentheses, but the arguments have not actually been summed. | |
15070 | (Generally a display format like this would be undesirable, since | |
15071 | it can easily be confused with a real sum.) | |
15072 | ||
15073 | The special function @code{eval} can be used inside a @kbd{Z C} | |
15074 | composition formula to cause all or part of the formula to be | |
15075 | evaluated at display time. For example, if the formula is | |
15076 | @samp{a + eval(b + c)}, then @samp{foo(1, 2, 3)} will be displayed | |
15077 | as @samp{1 + 5}. Evaluation will use the default simplifications, | |
15078 | regardless of the current simplification mode. There are also | |
15079 | @code{evalsimp} and @code{evalextsimp} which simplify as if by | |
15080 | @kbd{a s} and @kbd{a e} (respectively). Note that these ``functions'' | |
15081 | operate only in the context of composition formulas (and also in | |
15082 | rewrite rules, where they serve a similar purpose; @pxref{Rewrite | |
15083 | Rules}). On the stack, a call to @code{eval} will be left in | |
15084 | symbolic form. | |
15085 | ||
15086 | It is not a good idea to use @code{eval} except as a last resort. | |
15087 | It can cause the display of formulas to be extremely slow. For | |
15088 | example, while @samp{eval(a + b)} might seem quite fast and simple, | |
15089 | there are several situations where it could be slow. For example, | |
15090 | @samp{a} and/or @samp{b} could be polar complex numbers, in which | |
15091 | case doing the sum requires trigonometry. Or, @samp{a} could be | |
15092 | the factorial @samp{fact(100)} which is unevaluated because you | |
15093 | have typed @kbd{m O}; @code{eval} will evaluate it anyway to | |
15094 | produce a large, unwieldy integer. | |
15095 | ||
15096 | You can save your display formats permanently using the @kbd{Z P} | |
15097 | command (@pxref{Creating User Keys}). | |
15098 | ||
15099 | @node Syntax Tables, , Compositions, Language Modes | |
15100 | @subsection Syntax Tables | |
15101 | ||
15102 | @noindent | |
15103 | @cindex Syntax tables | |
15104 | @cindex Parsing formulas, customized | |
15105 | Syntax tables do for input what compositions do for output: They | |
15106 | allow you to teach custom notations to Calc's formula parser. | |
15107 | Calc keeps a separate syntax table for each language mode. | |
15108 | ||
15109 | (Note that the Calc ``syntax tables'' discussed here are completely | |
15110 | unrelated to the syntax tables described in the Emacs manual.) | |
15111 | ||
15112 | @kindex Z S | |
15113 | @pindex calc-edit-user-syntax | |
15114 | The @kbd{Z S} (@code{calc-edit-user-syntax}) command edits the | |
15115 | syntax table for the current language mode. If you want your | |
15116 | syntax to work in any language, define it in the normal language | |
15117 | mode. Type @kbd{M-# M-#} to finish editing the syntax table, or | |
15118 | @kbd{M-# x} to cancel the edit. The @kbd{m m} command saves all | |
15119 | the syntax tables along with the other mode settings; | |
15120 | @pxref{General Mode Commands}. | |
15121 | ||
15122 | @menu | |
15123 | * Syntax Table Basics:: | |
15124 | * Precedence in Syntax Tables:: | |
15125 | * Advanced Syntax Patterns:: | |
15126 | * Conditional Syntax Rules:: | |
15127 | @end menu | |
15128 | ||
15129 | @node Syntax Table Basics, Precedence in Syntax Tables, Syntax Tables, Syntax Tables | |
15130 | @subsubsection Syntax Table Basics | |
15131 | ||
15132 | @noindent | |
15133 | @dfn{Parsing} is the process of converting a raw string of characters, | |
15134 | such as you would type in during algebraic entry, into a Calc formula. | |
15135 | Calc's parser works in two stages. First, the input is broken down | |
15136 | into @dfn{tokens}, such as words, numbers, and punctuation symbols | |
15137 | like @samp{+}, @samp{:=}, and @samp{+/-}. Space between tokens is | |
15138 | ignored (except when it serves to separate adjacent words). Next, | |
15139 | the parser matches this string of tokens against various built-in | |
15140 | syntactic patterns, such as ``an expression followed by @samp{+} | |
15141 | followed by another expression'' or ``a name followed by @samp{(}, | |
15142 | zero or more expressions separated by commas, and @samp{)}.'' | |
15143 | ||
15144 | A @dfn{syntax table} is a list of user-defined @dfn{syntax rules}, | |
15145 | which allow you to specify new patterns to define your own | |
15146 | favorite input notations. Calc's parser always checks the syntax | |
15147 | table for the current language mode, then the table for the normal | |
15148 | language mode, before it uses its built-in rules to parse an | |
15149 | algebraic formula you have entered. Each syntax rule should go on | |
15150 | its own line; it consists of a @dfn{pattern}, a @samp{:=} symbol, | |
15151 | and a Calc formula with an optional @dfn{condition}. (Syntax rules | |
15152 | resemble algebraic rewrite rules, but the notation for patterns is | |
15153 | completely different.) | |
15154 | ||
15155 | A syntax pattern is a list of tokens, separated by spaces. | |
15156 | Except for a few special symbols, tokens in syntax patterns are | |
15157 | matched literally, from left to right. For example, the rule, | |
15158 | ||
15159 | @example | |
15160 | foo ( ) := 2+3 | |
15161 | @end example | |
15162 | ||
15163 | @noindent | |
15164 | would cause Calc to parse the formula @samp{4+foo()*5} as if it | |
15165 | were @samp{4+(2+3)*5}. Notice that the parentheses were written | |
15166 | as two separate tokens in the rule. As a result, the rule works | |
15167 | for both @samp{foo()} and @w{@samp{foo ( )}}. If we had written | |
15168 | the rule as @samp{foo () := 2+3}, then Calc would treat @samp{()} | |
15169 | as a single, indivisible token, so that @w{@samp{foo( )}} would | |
15170 | not be recognized by the rule. (It would be parsed as a regular | |
15171 | zero-argument function call instead.) In fact, this rule would | |
15172 | also make trouble for the rest of Calc's parser: An unrelated | |
15173 | formula like @samp{bar()} would now be tokenized into @samp{bar ()} | |
15174 | instead of @samp{bar ( )}, so that the standard parser for function | |
15175 | calls would no longer recognize it! | |
15176 | ||
15177 | While it is possible to make a token with a mixture of letters | |
15178 | and punctuation symbols, this is not recommended. It is better to | |
15179 | break it into several tokens, as we did with @samp{foo()} above. | |
15180 | ||
15181 | The symbol @samp{#} in a syntax pattern matches any Calc expression. | |
15182 | On the righthand side, the things that matched the @samp{#}s can | |
15183 | be referred to as @samp{#1}, @samp{#2}, and so on (where @samp{#1} | |
15184 | matches the leftmost @samp{#} in the pattern). For example, these | |
15185 | rules match a user-defined function, prefix operator, infix operator, | |
15186 | and postfix operator, respectively: | |
15187 | ||
15188 | @example | |
15189 | foo ( # ) := myfunc(#1) | |
15190 | foo # := myprefix(#1) | |
15191 | # foo # := myinfix(#1,#2) | |
15192 | # foo := mypostfix(#1) | |
15193 | @end example | |
15194 | ||
15195 | Thus @samp{foo(3)} will parse as @samp{myfunc(3)}, and @samp{2+3 foo} | |
15196 | will parse as @samp{mypostfix(2+3)}. | |
15197 | ||
15198 | It is important to write the first two rules in the order shown, | |
15199 | because Calc tries rules in order from first to last. If the | |
15200 | pattern @samp{foo #} came first, it would match anything that could | |
15201 | match the @samp{foo ( # )} rule, since an expression in parentheses | |
15202 | is itself a valid expression. Thus the @w{@samp{foo ( # )}} rule would | |
15203 | never get to match anything. Likewise, the last two rules must be | |
15204 | written in the order shown or else @samp{3 foo 4} will be parsed as | |
15205 | @samp{mypostfix(3) * 4}. (Of course, the best way to avoid these | |
15206 | ambiguities is not to use the same symbol in more than one way at | |
15207 | the same time! In case you're not convinced, try the following | |
15208 | exercise: How will the above rules parse the input @samp{foo(3,4)}, | |
15209 | if at all? Work it out for yourself, then try it in Calc and see.) | |
15210 | ||
15211 | Calc is quite flexible about what sorts of patterns are allowed. | |
15212 | The only rule is that every pattern must begin with a literal | |
15213 | token (like @samp{foo} in the first two patterns above), or with | |
15214 | a @samp{#} followed by a literal token (as in the last two | |
15215 | patterns). After that, any mixture is allowed, although putting | |
15216 | two @samp{#}s in a row will not be very useful since two | |
15217 | expressions with nothing between them will be parsed as one | |
15218 | expression that uses implicit multiplication. | |
15219 | ||
15220 | As a more practical example, Maple uses the notation | |
15221 | @samp{sum(a(i), i=1..10)} for sums, which Calc's Maple mode doesn't | |
15222 | recognize at present. To handle this syntax, we simply add the | |
15223 | rule, | |
15224 | ||
15225 | @example | |
15226 | sum ( # , # = # .. # ) := sum(#1,#2,#3,#4) | |
15227 | @end example | |
15228 | ||
15229 | @noindent | |
15230 | to the Maple mode syntax table. As another example, C mode can't | |
15231 | read assignment operators like @samp{++} and @samp{*=}. We can | |
15232 | define these operators quite easily: | |
15233 | ||
15234 | @example | |
15235 | # *= # := muleq(#1,#2) | |
15236 | # ++ := postinc(#1) | |
15237 | ++ # := preinc(#1) | |
15238 | @end example | |
15239 | ||
15240 | @noindent | |
15241 | To complete the job, we would use corresponding composition functions | |
15242 | and @kbd{Z C} to cause these functions to display in their respective | |
15243 | Maple and C notations. (Note that the C example ignores issues of | |
15244 | operator precedence, which are discussed in the next section.) | |
15245 | ||
15246 | You can enclose any token in quotes to prevent its usual | |
15247 | interpretation in syntax patterns: | |
15248 | ||
15249 | @example | |
15250 | # ":=" # := becomes(#1,#2) | |
15251 | @end example | |
15252 | ||
15253 | Quotes also allow you to include spaces in a token, although once | |
15254 | again it is generally better to use two tokens than one token with | |
15255 | an embedded space. To include an actual quotation mark in a quoted | |
15256 | token, precede it with a backslash. (This also works to include | |
15257 | backslashes in tokens.) | |
15258 | ||
15259 | @example | |
15260 | # "bad token" # "/\"\\" # := silly(#1,#2,#3) | |
15261 | @end example | |
15262 | ||
15263 | @noindent | |
15264 | This will parse @samp{3 bad token 4 /"\ 5} to @samp{silly(3,4,5)}. | |
15265 | ||
15266 | The token @kbd{#} has a predefined meaning in Calc's formula parser; | |
15267 | it is not legal to use @samp{"#"} in a syntax rule. However, longer | |
15268 | tokens that include the @samp{#} character are allowed. Also, while | |
15269 | @samp{"$"} and @samp{"\""} are allowed as tokens, their presence in | |
15270 | the syntax table will prevent those characters from working in their | |
15271 | usual ways (referring to stack entries and quoting strings, | |
15272 | respectively). | |
15273 | ||
15274 | Finally, the notation @samp{%%} anywhere in a syntax table causes | |
15275 | the rest of the line to be ignored as a comment. | |
15276 | ||
15277 | @node Precedence in Syntax Tables, Advanced Syntax Patterns, Syntax Table Basics, Syntax Tables | |
15278 | @subsubsection Precedence | |
15279 | ||
15280 | @noindent | |
15281 | Different operators are generally assigned different @dfn{precedences}. | |
15282 | By default, an operator defined by a rule like | |
15283 | ||
15284 | @example | |
15285 | # foo # := foo(#1,#2) | |
15286 | @end example | |
15287 | ||
15288 | @noindent | |
15289 | will have an extremely low precedence, so that @samp{2*3+4 foo 5 == 6} | |
15290 | will be parsed as @samp{(2*3+4) foo (5 == 6)}. To change the | |
15291 | precedence of an operator, use the notation @samp{#/@var{p}} in | |
15292 | place of @samp{#}, where @var{p} is an integer precedence level. | |
15293 | For example, 185 lies between the precedences for @samp{+} and | |
15294 | @samp{*}, so if we change this rule to | |
15295 | ||
15296 | @example | |
15297 | #/185 foo #/186 := foo(#1,#2) | |
15298 | @end example | |
15299 | ||
15300 | @noindent | |
15301 | then @samp{2+3 foo 4*5} will be parsed as @samp{2+(3 foo (4*5))}. | |
15302 | Also, because we've given the righthand expression slightly higher | |
15303 | precedence, our new operator will be left-associative: | |
15304 | @samp{1 foo 2 foo 3} will be parsed as @samp{(1 foo 2) foo 3}. | |
15305 | By raising the precedence of the lefthand expression instead, we | |
15306 | can create a right-associative operator. | |
15307 | ||
15308 | @xref{Composition Basics}, for a table of precedences of the | |
15309 | standard Calc operators. For the precedences of operators in other | |
15310 | language modes, look in the Calc source file @file{calc-lang.el}. | |
15311 | ||
15312 | @node Advanced Syntax Patterns, Conditional Syntax Rules, Precedence in Syntax Tables, Syntax Tables | |
15313 | @subsubsection Advanced Syntax Patterns | |
15314 | ||
15315 | @noindent | |
15316 | To match a function with a variable number of arguments, you could | |
15317 | write | |
15318 | ||
15319 | @example | |
15320 | foo ( # ) := myfunc(#1) | |
15321 | foo ( # , # ) := myfunc(#1,#2) | |
15322 | foo ( # , # , # ) := myfunc(#1,#2,#3) | |
15323 | @end example | |
15324 | ||
15325 | @noindent | |
15326 | but this isn't very elegant. To match variable numbers of items, | |
15327 | Calc uses some notations inspired regular expressions and the | |
15328 | ``extended BNF'' style used by some language designers. | |
15329 | ||
15330 | @example | |
15331 | foo ( @{ # @}*, ) := apply(myfunc,#1) | |
15332 | @end example | |
15333 | ||
15334 | The token @samp{@{} introduces a repeated or optional portion. | |
15335 | One of the three tokens @samp{@}*}, @samp{@}+}, or @samp{@}?} | |
15336 | ends the portion. These will match zero or more, one or more, | |
15337 | or zero or one copies of the enclosed pattern, respectively. | |
15338 | In addition, @samp{@}*} and @samp{@}+} can be followed by a | |
15339 | separator token (with no space in between, as shown above). | |
15340 | Thus @samp{@{ # @}*,} matches nothing, or one expression, or | |
15341 | several expressions separated by commas. | |
15342 | ||
15343 | A complete @samp{@{ ... @}} item matches as a vector of the | |
15344 | items that matched inside it. For example, the above rule will | |
15345 | match @samp{foo(1,2,3)} to get @samp{apply(myfunc,[1,2,3])}. | |
15346 | The Calc @code{apply} function takes a function name and a vector | |
15347 | of arguments and builds a call to the function with those | |
15348 | arguments, so the net result is the formula @samp{myfunc(1,2,3)}. | |
15349 | ||
15350 | If the body of a @samp{@{ ... @}} contains several @samp{#}s | |
15351 | (or nested @samp{@{ ... @}} constructs), then the items will be | |
15352 | strung together into the resulting vector. If the body | |
15353 | does not contain anything but literal tokens, the result will | |
15354 | always be an empty vector. | |
15355 | ||
15356 | @example | |
15357 | foo ( @{ # , # @}+, ) := bar(#1) | |
15358 | foo ( @{ @{ # @}*, @}*; ) := matrix(#1) | |
15359 | @end example | |
15360 | ||
15361 | @noindent | |
5d67986c RS |
15362 | will parse @samp{foo(1, 2, 3, 4)} as @samp{bar([1, 2, 3, 4])}, and |
15363 | @samp{foo(1, 2; 3, 4)} as @samp{matrix([[1, 2], [3, 4]])}. Also, after | |
d7b8e6c6 | 15364 | some thought it's easy to see how this pair of rules will parse |
5d67986c | 15365 | @samp{foo(1, 2, 3)} as @samp{matrix([[1, 2, 3]])}, since the first |
d7b8e6c6 EZ |
15366 | rule will only match an even number of arguments. The rule |
15367 | ||
15368 | @example | |
15369 | foo ( # @{ , # , # @}? ) := bar(#1,#2) | |
15370 | @end example | |
15371 | ||
15372 | @noindent | |
15373 | will parse @samp{foo(2,3,4)} as @samp{bar(2,[3,4])}, and | |
15374 | @samp{foo(2)} as @samp{bar(2,[])}. | |
15375 | ||
15376 | The notation @samp{@{ ... @}?.} (note the trailing period) works | |
15377 | just the same as regular @samp{@{ ... @}?}, except that it does not | |
15378 | count as an argument; the following two rules are equivalent: | |
15379 | ||
15380 | @example | |
15381 | foo ( # , @{ also @}? # ) := bar(#1,#3) | |
15382 | foo ( # , @{ also @}?. # ) := bar(#1,#2) | |
15383 | @end example | |
15384 | ||
15385 | @noindent | |
15386 | Note that in the first case the optional text counts as @samp{#2}, | |
15387 | which will always be an empty vector, but in the second case no | |
15388 | empty vector is produced. | |
15389 | ||
15390 | Another variant is @samp{@{ ... @}?$}, which means the body is | |
15391 | optional only at the end of the input formula. All built-in syntax | |
15392 | rules in Calc use this for closing delimiters, so that during | |
5d67986c | 15393 | algebraic entry you can type @kbd{[sqrt(2), sqrt(3 @key{RET}}, omitting |
d7b8e6c6 EZ |
15394 | the closing parenthesis and bracket. Calc does this automatically |
15395 | for trailing @samp{)}, @samp{]}, and @samp{>} tokens in syntax | |
15396 | rules, but you can use @samp{@{ ... @}?$} explicitly to get | |
15397 | this effect with any token (such as @samp{"@}"} or @samp{end}). | |
15398 | Like @samp{@{ ... @}?.}, this notation does not count as an | |
15399 | argument. Conversely, you can use quotes, as in @samp{")"}, to | |
15400 | prevent a closing-delimiter token from being automatically treated | |
15401 | as optional. | |
15402 | ||
15403 | Calc's parser does not have full backtracking, which means some | |
15404 | patterns will not work as you might expect: | |
15405 | ||
15406 | @example | |
15407 | foo ( @{ # , @}? # , # ) := bar(#1,#2,#3) | |
15408 | @end example | |
15409 | ||
15410 | @noindent | |
15411 | Here we are trying to make the first argument optional, so that | |
15412 | @samp{foo(2,3)} parses as @samp{bar([],2,3)}. Unfortunately, Calc | |
15413 | first tries to match @samp{2,} against the optional part of the | |
15414 | pattern, finds a match, and so goes ahead to match the rest of the | |
15415 | pattern. Later on it will fail to match the second comma, but it | |
15416 | doesn't know how to go back and try the other alternative at that | |
15417 | point. One way to get around this would be to use two rules: | |
15418 | ||
15419 | @example | |
15420 | foo ( # , # , # ) := bar([#1],#2,#3) | |
15421 | foo ( # , # ) := bar([],#1,#2) | |
15422 | @end example | |
15423 | ||
15424 | More precisely, when Calc wants to match an optional or repeated | |
15425 | part of a pattern, it scans forward attempting to match that part. | |
15426 | If it reaches the end of the optional part without failing, it | |
15427 | ``finalizes'' its choice and proceeds. If it fails, though, it | |
15428 | backs up and tries the other alternative. Thus Calc has ``partial'' | |
15429 | backtracking. A fully backtracking parser would go on to make sure | |
15430 | the rest of the pattern matched before finalizing the choice. | |
15431 | ||
15432 | @node Conditional Syntax Rules, , Advanced Syntax Patterns, Syntax Tables | |
15433 | @subsubsection Conditional Syntax Rules | |
15434 | ||
15435 | @noindent | |
15436 | It is possible to attach a @dfn{condition} to a syntax rule. For | |
15437 | example, the rules | |
15438 | ||
15439 | @example | |
15440 | foo ( # ) := ifoo(#1) :: integer(#1) | |
15441 | foo ( # ) := gfoo(#1) | |
15442 | @end example | |
15443 | ||
15444 | @noindent | |
15445 | will parse @samp{foo(3)} as @samp{ifoo(3)}, but will parse | |
15446 | @samp{foo(3.5)} and @samp{foo(x)} as calls to @code{gfoo}. Any | |
15447 | number of conditions may be attached; all must be true for the | |
15448 | rule to succeed. A condition is ``true'' if it evaluates to a | |
15449 | nonzero number. @xref{Logical Operations}, for a list of Calc | |
15450 | functions like @code{integer} that perform logical tests. | |
15451 | ||
15452 | The exact sequence of events is as follows: When Calc tries a | |
15453 | rule, it first matches the pattern as usual. It then substitutes | |
15454 | @samp{#1}, @samp{#2}, etc., in the conditions, if any. Next, the | |
15455 | conditions are simplified and evaluated in order from left to right, | |
15456 | as if by the @w{@kbd{a s}} algebra command (@pxref{Simplifying Formulas}). | |
15457 | Each result is true if it is a nonzero number, or an expression | |
15458 | that can be proven to be nonzero (@pxref{Declarations}). If the | |
15459 | results of all conditions are true, the expression (such as | |
15460 | @samp{ifoo(#1)}) has its @samp{#}s substituted, and that is the | |
15461 | result of the parse. If the result of any condition is false, Calc | |
15462 | goes on to try the next rule in the syntax table. | |
15463 | ||
15464 | Syntax rules also support @code{let} conditions, which operate in | |
15465 | exactly the same way as they do in algebraic rewrite rules. | |
15466 | @xref{Other Features of Rewrite Rules}, for details. A @code{let} | |
15467 | condition is always true, but as a side effect it defines a | |
15468 | variable which can be used in later conditions, and also in the | |
15469 | expression after the @samp{:=} sign: | |
15470 | ||
15471 | @example | |
15472 | foo ( # ) := hifoo(x) :: let(x := #1 + 0.5) :: dnumint(x) | |
15473 | @end example | |
15474 | ||
15475 | @noindent | |
15476 | The @code{dnumint} function tests if a value is numerically an | |
15477 | integer, i.e., either a true integer or an integer-valued float. | |
15478 | This rule will parse @code{foo} with a half-integer argument, | |
15479 | like @samp{foo(3.5)}, to a call like @samp{hifoo(4.)}. | |
15480 | ||
15481 | The lefthand side of a syntax rule @code{let} must be a simple | |
15482 | variable, not the arbitrary pattern that is allowed in rewrite | |
15483 | rules. | |
15484 | ||
15485 | The @code{matches} function is also treated specially in syntax | |
15486 | rule conditions (again, in the same way as in rewrite rules). | |
15487 | @xref{Matching Commands}. If the matching pattern contains | |
15488 | meta-variables, then those meta-variables may be used in later | |
15489 | conditions and in the result expression. The arguments to | |
15490 | @code{matches} are not evaluated in this situation. | |
15491 | ||
15492 | @example | |
15493 | sum ( # , # ) := sum(#1,a,b,c) :: matches(#2, a=[b..c]) | |
15494 | @end example | |
15495 | ||
15496 | @noindent | |
15497 | This is another way to implement the Maple mode @code{sum} notation. | |
15498 | In this approach, we allow @samp{#2} to equal the whole expression | |
15499 | @samp{i=1..10}. Then, we use @code{matches} to break it apart into | |
15500 | its components. If the expression turns out not to match the pattern, | |
15501 | the syntax rule will fail. Note that @kbd{Z S} always uses Calc's | |
15502 | normal language mode for editing expressions in syntax rules, so we | |
15503 | must use regular Calc notation for the interval @samp{[b..c]} that | |
15504 | will correspond to the Maple mode interval @samp{1..10}. | |
15505 | ||
15506 | @node Modes Variable, Calc Mode Line, Language Modes, Mode Settings | |
15507 | @section The @code{Modes} Variable | |
15508 | ||
15509 | @noindent | |
15510 | @kindex m g | |
15511 | @pindex calc-get-modes | |
15512 | The @kbd{m g} (@code{calc-get-modes}) command pushes onto the stack | |
15513 | a vector of numbers that describes the various mode settings that | |
15514 | are in effect. With a numeric prefix argument, it pushes only the | |
15515 | @var{n}th mode, i.e., the @var{n}th element of this vector. Keyboard | |
15516 | macros can use the @kbd{m g} command to modify their behavior based | |
15517 | on the current mode settings. | |
15518 | ||
15519 | @cindex @code{Modes} variable | |
15520 | @vindex Modes | |
15521 | The modes vector is also available in the special variable | |
5d67986c | 15522 | @code{Modes}. In other words, @kbd{m g} is like @kbd{s r Modes @key{RET}}. |
d7b8e6c6 EZ |
15523 | It will not work to store into this variable; in fact, if you do, |
15524 | @code{Modes} will cease to track the current modes. (The @kbd{m g} | |
15525 | command will continue to work, however.) | |
15526 | ||
15527 | In general, each number in this vector is suitable as a numeric | |
15528 | prefix argument to the associated mode-setting command. (Recall | |
15529 | that the @kbd{~} key takes a number from the stack and gives it as | |
15530 | a numeric prefix to the next command.) | |
15531 | ||
15532 | The elements of the modes vector are as follows: | |
15533 | ||
15534 | @enumerate | |
15535 | @item | |
15536 | Current precision. Default is 12; associated command is @kbd{p}. | |
15537 | ||
15538 | @item | |
15539 | Binary word size. Default is 32; associated command is @kbd{b w}. | |
15540 | ||
15541 | @item | |
15542 | Stack size (not counting the value about to be pushed by @kbd{m g}). | |
15543 | This is zero if @kbd{m g} is executed with an empty stack. | |
15544 | ||
15545 | @item | |
15546 | Number radix. Default is 10; command is @kbd{d r}. | |
15547 | ||
15548 | @item | |
15549 | Floating-point format. This is the number of digits, plus the | |
15550 | constant 0 for normal notation, 10000 for scientific notation, | |
15551 | 20000 for engineering notation, or 30000 for fixed-point notation. | |
15552 | These codes are acceptable as prefix arguments to the @kbd{d n} | |
15553 | command, but note that this may lose information: For example, | |
15554 | @kbd{d s} and @kbd{C-u 12 d s} have similar (but not quite | |
15555 | identical) effects if the current precision is 12, but they both | |
15556 | produce a code of 10012, which will be treated by @kbd{d n} as | |
15557 | @kbd{C-u 12 d s}. If the precision then changes, the float format | |
15558 | will still be frozen at 12 significant figures. | |
15559 | ||
15560 | @item | |
15561 | Angular mode. Default is 1 (degrees). Other values are 2 (radians) | |
15562 | and 3 (HMS). The @kbd{m d} command accepts these prefixes. | |
15563 | ||
15564 | @item | |
15565 | Symbolic mode. Value is 0 or 1; default is 0. Command is @kbd{m s}. | |
15566 | ||
177c0ea7 | 15567 | @item |
d7b8e6c6 EZ |
15568 | Fraction mode. Value is 0 or 1; default is 0. Command is @kbd{m f}. |
15569 | ||
15570 | @item | |
15571 | Polar mode. Value is 0 (rectangular) or 1 (polar); default is 0. | |
15572 | Command is @kbd{m p}. | |
15573 | ||
15574 | @item | |
15575 | Matrix/scalar mode. Default value is @i{-1}. Value is 0 for scalar | |
5d67986c RS |
15576 | mode, @i{-2} for matrix mode, or @var{N} for @c{$N\times N$} |
15577 | @var{N}x@var{N} matrix mode. Command is @kbd{m v}. | |
d7b8e6c6 EZ |
15578 | |
15579 | @item | |
15580 | Simplification mode. Default is 1. Value is @i{-1} for off (@kbd{m O}), | |
15581 | 0 for @kbd{m N}, 2 for @kbd{m B}, 3 for @kbd{m A}, 4 for @kbd{m E}, | |
15582 | or 5 for @w{@kbd{m U}}. The @kbd{m D} command accepts these prefixes. | |
15583 | ||
15584 | @item | |
15585 | Infinite mode. Default is @i{-1} (off). Value is 1 if the mode is on, | |
15586 | or 0 if the mode is on with positive zeros. Command is @kbd{m i}. | |
15587 | @end enumerate | |
15588 | ||
5d67986c | 15589 | For example, the sequence @kbd{M-1 m g @key{RET} 2 + ~ p} increases the |
d7b8e6c6 EZ |
15590 | precision by two, leaving a copy of the old precision on the stack. |
15591 | Later, @kbd{~ p} will restore the original precision using that | |
15592 | stack value. (This sequence might be especially useful inside a | |
15593 | keyboard macro.) | |
15594 | ||
5d67986c | 15595 | As another example, @kbd{M-3 m g 1 - ~ @key{DEL}} deletes all but the |
d7b8e6c6 EZ |
15596 | oldest (bottommost) stack entry. |
15597 | ||
15598 | Yet another example: The HP-48 ``round'' command rounds a number | |
15599 | to the current displayed precision. You could roughly emulate this | |
15600 | in Calc with the sequence @kbd{M-5 m g 10000 % ~ c c}. (This | |
15601 | would not work for fixed-point mode, but it wouldn't be hard to | |
15602 | do a full emulation with the help of the @kbd{Z [} and @kbd{Z ]} | |
15603 | programming commands. @xref{Conditionals in Macros}.) | |
15604 | ||
15605 | @node Calc Mode Line, , Modes Variable, Mode Settings | |
15606 | @section The Calc Mode Line | |
15607 | ||
15608 | @noindent | |
15609 | @cindex Mode line indicators | |
15610 | This section is a summary of all symbols that can appear on the | |
15611 | Calc mode line, the highlighted bar that appears under the Calc | |
15612 | stack window (or under an editing window in Embedded Mode). | |
15613 | ||
15614 | The basic mode line format is: | |
15615 | ||
15616 | @example | |
15617 | --%%-Calc: 12 Deg @var{other modes} (Calculator) | |
15618 | @end example | |
15619 | ||
15620 | The @samp{%%} is the Emacs symbol for ``read-only''; it shows that | |
15621 | regular Emacs commands are not allowed to edit the stack buffer | |
15622 | as if it were text. | |
15623 | ||
15624 | The word @samp{Calc:} changes to @samp{CalcEmbed:} if Embedded Mode | |
15625 | is enabled. The words after this describe the various Calc modes | |
15626 | that are in effect. | |
15627 | ||
15628 | The first mode is always the current precision, an integer. | |
15629 | The second mode is always the angular mode, either @code{Deg}, | |
15630 | @code{Rad}, or @code{Hms}. | |
15631 | ||
15632 | Here is a complete list of the remaining symbols that can appear | |
15633 | on the mode line: | |
15634 | ||
15635 | @table @code | |
15636 | @item Alg | |
15637 | Algebraic mode (@kbd{m a}; @pxref{Algebraic Entry}). | |
15638 | ||
15639 | @item Alg[( | |
15640 | Incomplete algebraic mode (@kbd{C-u m a}). | |
15641 | ||
15642 | @item Alg* | |
15643 | Total algebraic mode (@kbd{m t}). | |
15644 | ||
15645 | @item Symb | |
15646 | Symbolic mode (@kbd{m s}; @pxref{Symbolic Mode}). | |
15647 | ||
15648 | @item Matrix | |
15649 | Matrix mode (@kbd{m v}; @pxref{Matrix Mode}). | |
15650 | ||
15651 | @item Matrix@var{n} | |
15652 | Dimensioned matrix mode (@kbd{C-u @var{n} m v}). | |
15653 | ||
15654 | @item Scalar | |
15655 | Scalar mode (@kbd{m v}; @pxref{Matrix Mode}). | |
15656 | ||
15657 | @item Polar | |
15658 | Polar complex mode (@kbd{m p}; @pxref{Polar Mode}). | |
15659 | ||
15660 | @item Frac | |
15661 | Fraction mode (@kbd{m f}; @pxref{Fraction Mode}). | |
15662 | ||
15663 | @item Inf | |
15664 | Infinite mode (@kbd{m i}; @pxref{Infinite Mode}). | |
15665 | ||
15666 | @item +Inf | |
15667 | Positive infinite mode (@kbd{C-u 0 m i}). | |
15668 | ||
15669 | @item NoSimp | |
15670 | Default simplifications off (@kbd{m O}; @pxref{Simplification Modes}). | |
15671 | ||
15672 | @item NumSimp | |
15673 | Default simplifications for numeric arguments only (@kbd{m N}). | |
15674 | ||
15675 | @item BinSimp@var{w} | |
15676 | Binary-integer simplification mode; word size @var{w} (@kbd{m B}, @kbd{b w}). | |
15677 | ||
15678 | @item AlgSimp | |
15679 | Algebraic simplification mode (@kbd{m A}). | |
15680 | ||
15681 | @item ExtSimp | |
15682 | Extended algebraic simplification mode (@kbd{m E}). | |
15683 | ||
15684 | @item UnitSimp | |
15685 | Units simplification mode (@kbd{m U}). | |
15686 | ||
15687 | @item Bin | |
15688 | Current radix is 2 (@kbd{d 2}; @pxref{Radix Modes}). | |
15689 | ||
15690 | @item Oct | |
15691 | Current radix is 8 (@kbd{d 8}). | |
15692 | ||
15693 | @item Hex | |
15694 | Current radix is 16 (@kbd{d 6}). | |
15695 | ||
15696 | @item Radix@var{n} | |
15697 | Current radix is @var{n} (@kbd{d r}). | |
15698 | ||
15699 | @item Zero | |
15700 | Leading zeros (@kbd{d z}; @pxref{Radix Modes}). | |
15701 | ||
15702 | @item Big | |
15703 | Big language mode (@kbd{d B}; @pxref{Normal Language Modes}). | |
15704 | ||
15705 | @item Flat | |
15706 | One-line normal language mode (@kbd{d O}). | |
15707 | ||
15708 | @item Unform | |
15709 | Unformatted language mode (@kbd{d U}). | |
15710 | ||
15711 | @item C | |
15712 | C language mode (@kbd{d C}; @pxref{C FORTRAN Pascal}). | |
15713 | ||
15714 | @item Pascal | |
15715 | Pascal language mode (@kbd{d P}). | |
15716 | ||
15717 | @item Fortran | |
15718 | FORTRAN language mode (@kbd{d F}). | |
15719 | ||
15720 | @item TeX | |
15721 | @TeX{} language mode (@kbd{d T}; @pxref{TeX Language Mode}). | |
15722 | ||
15723 | @item Eqn | |
15724 | @dfn{Eqn} language mode (@kbd{d E}; @pxref{Eqn Language Mode}). | |
15725 | ||
15726 | @item Math | |
15727 | Mathematica language mode (@kbd{d M}; @pxref{Mathematica Language Mode}). | |
15728 | ||
15729 | @item Maple | |
15730 | Maple language mode (@kbd{d W}; @pxref{Maple Language Mode}). | |
15731 | ||
15732 | @item Norm@var{n} | |
15733 | Normal float mode with @var{n} digits (@kbd{d n}; @pxref{Float Formats}). | |
15734 | ||
15735 | @item Fix@var{n} | |
15736 | Fixed point mode with @var{n} digits after the point (@kbd{d f}). | |
15737 | ||
15738 | @item Sci | |
15739 | Scientific notation mode (@kbd{d s}). | |
15740 | ||
15741 | @item Sci@var{n} | |
15742 | Scientific notation with @var{n} digits (@kbd{d s}). | |
15743 | ||
15744 | @item Eng | |
15745 | Engineering notation mode (@kbd{d e}). | |
15746 | ||
15747 | @item Eng@var{n} | |
15748 | Engineering notation with @var{n} digits (@kbd{d e}). | |
15749 | ||
15750 | @item Left@var{n} | |
15751 | Left-justified display indented by @var{n} (@kbd{d <}; @pxref{Justification}). | |
15752 | ||
15753 | @item Right | |
15754 | Right-justified display (@kbd{d >}). | |
15755 | ||
15756 | @item Right@var{n} | |
15757 | Right-justified display with width @var{n} (@kbd{d >}). | |
15758 | ||
15759 | @item Center | |
15760 | Centered display (@kbd{d =}). | |
15761 | ||
15762 | @item Center@var{n} | |
15763 | Centered display with center column @var{n} (@kbd{d =}). | |
15764 | ||
15765 | @item Wid@var{n} | |
15766 | Line breaking with width @var{n} (@kbd{d b}; @pxref{Normal Language Modes}). | |
15767 | ||
15768 | @item Wide | |
15769 | No line breaking (@kbd{d b}). | |
15770 | ||
15771 | @item Break | |
15772 | Selections show deep structure (@kbd{j b}; @pxref{Making Selections}). | |
15773 | ||
15774 | @item Save | |
15775 | Record modes in @file{~/.emacs} (@kbd{m R}; @pxref{General Mode Commands}). | |
15776 | ||
15777 | @item Local | |
15778 | Record modes in Embedded buffer (@kbd{m R}). | |
15779 | ||
15780 | @item LocEdit | |
15781 | Record modes as editing-only in Embedded buffer (@kbd{m R}). | |
15782 | ||
15783 | @item LocPerm | |
15784 | Record modes as permanent-only in Embedded buffer (@kbd{m R}). | |
15785 | ||
15786 | @item Global | |
15787 | Record modes as global in Embedded buffer (@kbd{m R}). | |
15788 | ||
15789 | @item Manual | |
15790 | Automatic recomputation turned off (@kbd{m C}; @pxref{Automatic | |
15791 | Recomputation}). | |
15792 | ||
15793 | @item Graph | |
15794 | GNUPLOT process is alive in background (@pxref{Graphics}). | |
15795 | ||
15796 | @item Sel | |
15797 | Top-of-stack has a selection (Embedded only; @pxref{Making Selections}). | |
15798 | ||
15799 | @item Dirty | |
15800 | The stack display may not be up-to-date (@pxref{Display Modes}). | |
15801 | ||
15802 | @item Inv | |
15803 | ``Inverse'' prefix was pressed (@kbd{I}; @pxref{Inverse and Hyperbolic}). | |
15804 | ||
15805 | @item Hyp | |
15806 | ``Hyperbolic'' prefix was pressed (@kbd{H}). | |
15807 | ||
15808 | @item Keep | |
15809 | ``Keep-arguments'' prefix was pressed (@kbd{K}). | |
15810 | ||
15811 | @item Narrow | |
15812 | Stack is truncated (@kbd{d t}; @pxref{Truncating the Stack}). | |
15813 | @end table | |
15814 | ||
15815 | In addition, the symbols @code{Active} and @code{~Active} can appear | |
15816 | as minor modes on an Embedded buffer's mode line. @xref{Embedded Mode}. | |
15817 | ||
15818 | @node Arithmetic, Scientific Functions, Mode Settings, Top | |
15819 | @chapter Arithmetic Functions | |
15820 | ||
15821 | @noindent | |
15822 | This chapter describes the Calc commands for doing simple calculations | |
15823 | on numbers, such as addition, absolute value, and square roots. These | |
15824 | commands work by removing the top one or two values from the stack, | |
15825 | performing the desired operation, and pushing the result back onto the | |
15826 | stack. If the operation cannot be performed, the result pushed is a | |
15827 | formula instead of a number, such as @samp{2/0} (because division by zero | |
15828 | is illegal) or @samp{sqrt(x)} (because the argument @samp{x} is a formula). | |
15829 | ||
15830 | Most of the commands described here can be invoked by a single keystroke. | |
15831 | Some of the more obscure ones are two-letter sequences beginning with | |
15832 | the @kbd{f} (``functions'') prefix key. | |
15833 | ||
15834 | @xref{Prefix Arguments}, for a discussion of the effect of numeric | |
15835 | prefix arguments on commands in this chapter which do not otherwise | |
15836 | interpret a prefix argument. | |
15837 | ||
15838 | @menu | |
15839 | * Basic Arithmetic:: | |
15840 | * Integer Truncation:: | |
15841 | * Complex Number Functions:: | |
15842 | * Conversions:: | |
15843 | * Date Arithmetic:: | |
15844 | * Financial Functions:: | |
15845 | * Binary Functions:: | |
15846 | @end menu | |
15847 | ||
15848 | @node Basic Arithmetic, Integer Truncation, Arithmetic, Arithmetic | |
15849 | @section Basic Arithmetic | |
15850 | ||
15851 | @noindent | |
15852 | @kindex + | |
15853 | @pindex calc-plus | |
5d67986c RS |
15854 | @ignore |
15855 | @mindex @null | |
15856 | @end ignore | |
d7b8e6c6 EZ |
15857 | @tindex + |
15858 | The @kbd{+} (@code{calc-plus}) command adds two numbers. The numbers may | |
15859 | be any of the standard Calc data types. The resulting sum is pushed back | |
15860 | onto the stack. | |
15861 | ||
15862 | If both arguments of @kbd{+} are vectors or matrices (of matching dimensions), | |
15863 | the result is a vector or matrix sum. If one argument is a vector and the | |
15864 | other a scalar (i.e., a non-vector), the scalar is added to each of the | |
15865 | elements of the vector to form a new vector. If the scalar is not a | |
15866 | number, the operation is left in symbolic form: Suppose you added @samp{x} | |
15867 | to the vector @samp{[1,2]}. You may want the result @samp{[1+x,2+x]}, or | |
15868 | you may plan to substitute a 2-vector for @samp{x} in the future. Since | |
15869 | the Calculator can't tell which interpretation you want, it makes the | |
15870 | safest assumption. @xref{Reducing and Mapping}, for a way to add @samp{x} | |
15871 | to every element of a vector. | |
15872 | ||
15873 | If either argument of @kbd{+} is a complex number, the result will in general | |
15874 | be complex. If one argument is in rectangular form and the other polar, | |
15875 | the current Polar Mode determines the form of the result. If Symbolic | |
15876 | Mode is enabled, the sum may be left as a formula if the necessary | |
15877 | conversions for polar addition are non-trivial. | |
15878 | ||
15879 | If both arguments of @kbd{+} are HMS forms, the forms are added according to | |
15880 | the usual conventions of hours-minutes-seconds notation. If one argument | |
15881 | is an HMS form and the other is a number, that number is converted from | |
15882 | degrees or radians (depending on the current Angular Mode) to HMS format | |
15883 | and then the two HMS forms are added. | |
15884 | ||
15885 | If one argument of @kbd{+} is a date form, the other can be either a | |
15886 | real number, which advances the date by a certain number of days, or | |
15887 | an HMS form, which advances the date by a certain amount of time. | |
15888 | Subtracting two date forms yields the number of days between them. | |
15889 | Adding two date forms is meaningless, but Calc interprets it as the | |
15890 | subtraction of one date form and the negative of the other. (The | |
15891 | negative of a date form can be understood by remembering that dates | |
15892 | are stored as the number of days before or after Jan 1, 1 AD.) | |
15893 | ||
15894 | If both arguments of @kbd{+} are error forms, the result is an error form | |
15895 | with an appropriately computed standard deviation. If one argument is an | |
15896 | error form and the other is a number, the number is taken to have zero error. | |
15897 | Error forms may have symbolic formulas as their mean and/or error parts; | |
15898 | adding these will produce a symbolic error form result. However, adding an | |
15899 | error form to a plain symbolic formula (as in @samp{(a +/- b) + c}) will not | |
15900 | work, for the same reasons just mentioned for vectors. Instead you must | |
15901 | write @samp{(a +/- b) + (c +/- 0)}. | |
15902 | ||
15903 | If both arguments of @kbd{+} are modulo forms with equal values of @cite{M}, | |
15904 | or if one argument is a modulo form and the other a plain number, the | |
15905 | result is a modulo form which represents the sum, modulo @cite{M}, of | |
15906 | the two values. | |
15907 | ||
15908 | If both arguments of @kbd{+} are intervals, the result is an interval | |
15909 | which describes all possible sums of the possible input values. If | |
15910 | one argument is a plain number, it is treated as the interval | |
15911 | @w{@samp{[x ..@: x]}}. | |
15912 | ||
15913 | If one argument of @kbd{+} is an infinity and the other is not, the | |
15914 | result is that same infinity. If both arguments are infinite and in | |
15915 | the same direction, the result is the same infinity, but if they are | |
15916 | infinite in different directions the result is @code{nan}. | |
15917 | ||
15918 | @kindex - | |
15919 | @pindex calc-minus | |
5d67986c RS |
15920 | @ignore |
15921 | @mindex @null | |
15922 | @end ignore | |
d7b8e6c6 EZ |
15923 | @tindex - |
15924 | The @kbd{-} (@code{calc-minus}) command subtracts two values. The top | |
15925 | number on the stack is subtracted from the one behind it, so that the | |
15926 | computation @kbd{5 @key{RET} 2 -} produces 3, not @i{-3}. All options | |
15927 | available for @kbd{+} are available for @kbd{-} as well. | |
15928 | ||
15929 | @kindex * | |
15930 | @pindex calc-times | |
5d67986c RS |
15931 | @ignore |
15932 | @mindex @null | |
15933 | @end ignore | |
d7b8e6c6 EZ |
15934 | @tindex * |
15935 | The @kbd{*} (@code{calc-times}) command multiplies two numbers. If one | |
15936 | argument is a vector and the other a scalar, the scalar is multiplied by | |
15937 | the elements of the vector to produce a new vector. If both arguments | |
15938 | are vectors, the interpretation depends on the dimensions of the | |
15939 | vectors: If both arguments are matrices, a matrix multiplication is | |
15940 | done. If one argument is a matrix and the other a plain vector, the | |
15941 | vector is interpreted as a row vector or column vector, whichever is | |
15942 | dimensionally correct. If both arguments are plain vectors, the result | |
15943 | is a single scalar number which is the dot product of the two vectors. | |
15944 | ||
15945 | If one argument of @kbd{*} is an HMS form and the other a number, the | |
15946 | HMS form is multiplied by that amount. It is an error to multiply two | |
15947 | HMS forms together, or to attempt any multiplication involving date | |
15948 | forms. Error forms, modulo forms, and intervals can be multiplied; | |
15949 | see the comments for addition of those forms. When two error forms | |
15950 | or intervals are multiplied they are considered to be statistically | |
15951 | independent; thus, @samp{[-2 ..@: 3] * [-2 ..@: 3]} is @samp{[-6 ..@: 9]}, | |
15952 | whereas @w{@samp{[-2 ..@: 3] ^ 2}} is @samp{[0 ..@: 9]}. | |
15953 | ||
15954 | @kindex / | |
15955 | @pindex calc-divide | |
5d67986c RS |
15956 | @ignore |
15957 | @mindex @null | |
15958 | @end ignore | |
d7b8e6c6 EZ |
15959 | @tindex / |
15960 | The @kbd{/} (@code{calc-divide}) command divides two numbers. When | |
15961 | dividing a scalar @cite{B} by a square matrix @cite{A}, the computation | |
15962 | performed is @cite{B} times the inverse of @cite{A}. This also occurs | |
15963 | if @cite{B} is itself a vector or matrix, in which case the effect is | |
15964 | to solve the set of linear equations represented by @cite{B}. If @cite{B} | |
15965 | is a matrix with the same number of rows as @cite{A}, or a plain vector | |
15966 | (which is interpreted here as a column vector), then the equation | |
15967 | @cite{A X = B} is solved for the vector or matrix @cite{X}. Otherwise, | |
15968 | if @cite{B} is a non-square matrix with the same number of @emph{columns} | |
15969 | as @cite{A}, the equation @cite{X A = B} is solved. If you wish a vector | |
15970 | @cite{B} to be interpreted as a row vector to be solved as @cite{X A = B}, | |
15971 | make it into a one-row matrix with @kbd{C-u 1 v p} first. To force a | |
15972 | left-handed solution with a square matrix @cite{B}, transpose @cite{A} and | |
15973 | @cite{B} before dividing, then transpose the result. | |
15974 | ||
15975 | HMS forms can be divided by real numbers or by other HMS forms. Error | |
15976 | forms can be divided in any combination of ways. Modulo forms where both | |
15977 | values and the modulo are integers can be divided to get an integer modulo | |
15978 | form result. Intervals can be divided; dividing by an interval that | |
15979 | encompasses zero or has zero as a limit will result in an infinite | |
15980 | interval. | |
15981 | ||
15982 | @kindex ^ | |
15983 | @pindex calc-power | |
5d67986c RS |
15984 | @ignore |
15985 | @mindex @null | |
15986 | @end ignore | |
d7b8e6c6 EZ |
15987 | @tindex ^ |
15988 | The @kbd{^} (@code{calc-power}) command raises a number to a power. If | |
15989 | the power is an integer, an exact result is computed using repeated | |
15990 | multiplications. For non-integer powers, Calc uses Newton's method or | |
15991 | logarithms and exponentials. Square matrices can be raised to integer | |
15992 | powers. If either argument is an error (or interval or modulo) form, | |
15993 | the result is also an error (or interval or modulo) form. | |
15994 | ||
15995 | @kindex I ^ | |
15996 | @tindex nroot | |
15997 | If you press the @kbd{I} (inverse) key first, the @kbd{I ^} command | |
5d67986c RS |
15998 | computes an Nth root: @kbd{125 @key{RET} 3 I ^} computes the number 5. |
15999 | (This is entirely equivalent to @kbd{125 @key{RET} 1:3 ^}.) | |
d7b8e6c6 EZ |
16000 | |
16001 | @kindex \ | |
16002 | @pindex calc-idiv | |
16003 | @tindex idiv | |
5d67986c RS |
16004 | @ignore |
16005 | @mindex @null | |
16006 | @end ignore | |
d7b8e6c6 EZ |
16007 | @tindex \ |
16008 | The @kbd{\} (@code{calc-idiv}) command divides two numbers on the stack | |
16009 | to produce an integer result. It is equivalent to dividing with | |
16010 | @key{/}, then rounding down with @kbd{F} (@code{calc-floor}), only a bit | |
16011 | more convenient and efficient. Also, since it is an all-integer | |
16012 | operation when the arguments are integers, it avoids problems that | |
16013 | @kbd{/ F} would have with floating-point roundoff. | |
16014 | ||
16015 | @kindex % | |
16016 | @pindex calc-mod | |
5d67986c RS |
16017 | @ignore |
16018 | @mindex @null | |
16019 | @end ignore | |
d7b8e6c6 EZ |
16020 | @tindex % |
16021 | The @kbd{%} (@code{calc-mod}) command performs a ``modulo'' (or ``remainder'') | |
16022 | operation. Mathematically, @samp{a%b = a - (a\b)*b}, and is defined | |
16023 | for all real numbers @cite{a} and @cite{b} (except @cite{b=0}). For | |
16024 | positive @cite{b}, the result will always be between 0 (inclusive) and | |
16025 | @cite{b} (exclusive). Modulo does not work for HMS forms and error forms. | |
16026 | If @cite{a} is a modulo form, its modulo is changed to @cite{b}, which | |
16027 | must be positive real number. | |
16028 | ||
16029 | @kindex : | |
16030 | @pindex calc-fdiv | |
16031 | @tindex fdiv | |
16032 | The @kbd{:} (@code{calc-fdiv}) command [@code{fdiv} function in a formula] | |
16033 | divides the two integers on the top of the stack to produce a fractional | |
16034 | result. This is a convenient shorthand for enabling Fraction Mode (with | |
16035 | @kbd{m f}) temporarily and using @samp{/}. Note that during numeric entry | |
16036 | the @kbd{:} key is interpreted as a fraction separator, so to divide 8 by 6 | |
16037 | you would have to type @kbd{8 @key{RET} 6 @key{RET} :}. (Of course, in | |
16038 | this case, it would be much easier simply to enter the fraction directly | |
16039 | as @kbd{8:6 @key{RET}}!) | |
16040 | ||
16041 | @kindex n | |
16042 | @pindex calc-change-sign | |
16043 | The @kbd{n} (@code{calc-change-sign}) command negates the number on the top | |
16044 | of the stack. It works on numbers, vectors and matrices, HMS forms, date | |
16045 | forms, error forms, intervals, and modulo forms. | |
16046 | ||
16047 | @kindex A | |
16048 | @pindex calc-abs | |
16049 | @tindex abs | |
16050 | The @kbd{A} (@code{calc-abs}) [@code{abs}] command computes the absolute | |
16051 | value of a number. The result of @code{abs} is always a nonnegative | |
16052 | real number: With a complex argument, it computes the complex magnitude. | |
16053 | With a vector or matrix argument, it computes the Frobenius norm, i.e., | |
16054 | the square root of the sum of the squares of the absolute values of the | |
16055 | elements. The absolute value of an error form is defined by replacing | |
16056 | the mean part with its absolute value and leaving the error part the same. | |
16057 | The absolute value of a modulo form is undefined. The absolute value of | |
16058 | an interval is defined in the obvious way. | |
16059 | ||
16060 | @kindex f A | |
16061 | @pindex calc-abssqr | |
16062 | @tindex abssqr | |
16063 | The @kbd{f A} (@code{calc-abssqr}) [@code{abssqr}] command computes the | |
16064 | absolute value squared of a number, vector or matrix, or error form. | |
16065 | ||
16066 | @kindex f s | |
16067 | @pindex calc-sign | |
16068 | @tindex sign | |
16069 | The @kbd{f s} (@code{calc-sign}) [@code{sign}] command returns 1 if its | |
16070 | argument is positive, @i{-1} if its argument is negative, or 0 if its | |
16071 | argument is zero. In algebraic form, you can also write @samp{sign(a,x)} | |
16072 | which evaluates to @samp{x * sign(a)}, i.e., either @samp{x}, @samp{-x}, or | |
16073 | zero depending on the sign of @samp{a}. | |
16074 | ||
16075 | @kindex & | |
16076 | @pindex calc-inv | |
16077 | @tindex inv | |
16078 | @cindex Reciprocal | |
16079 | The @kbd{&} (@code{calc-inv}) [@code{inv}] command computes the | |
16080 | reciprocal of a number, i.e., @cite{1 / x}. Operating on a square | |
16081 | matrix, it computes the inverse of that matrix. | |
16082 | ||
16083 | @kindex Q | |
16084 | @pindex calc-sqrt | |
16085 | @tindex sqrt | |
16086 | The @kbd{Q} (@code{calc-sqrt}) [@code{sqrt}] command computes the square | |
16087 | root of a number. For a negative real argument, the result will be a | |
16088 | complex number whose form is determined by the current Polar Mode. | |
16089 | ||
16090 | @kindex f h | |
16091 | @pindex calc-hypot | |
16092 | @tindex hypot | |
16093 | The @kbd{f h} (@code{calc-hypot}) [@code{hypot}] command computes the square | |
16094 | root of the sum of the squares of two numbers. That is, @samp{hypot(a,b)} | |
16095 | is the length of the hypotenuse of a right triangle with sides @cite{a} | |
16096 | and @cite{b}. If the arguments are complex numbers, their squared | |
16097 | magnitudes are used. | |
16098 | ||
16099 | @kindex f Q | |
16100 | @pindex calc-isqrt | |
16101 | @tindex isqrt | |
16102 | The @kbd{f Q} (@code{calc-isqrt}) [@code{isqrt}] command computes the | |
16103 | integer square root of an integer. This is the true square root of the | |
16104 | number, rounded down to an integer. For example, @samp{isqrt(10)} | |
16105 | produces 3. Note that, like @kbd{\} [@code{idiv}], this uses exact | |
16106 | integer arithmetic throughout to avoid roundoff problems. If the input | |
16107 | is a floating-point number or other non-integer value, this is exactly | |
16108 | the same as @samp{floor(sqrt(x))}. | |
16109 | ||
16110 | @kindex f n | |
16111 | @kindex f x | |
16112 | @pindex calc-min | |
16113 | @tindex min | |
16114 | @pindex calc-max | |
16115 | @tindex max | |
16116 | The @kbd{f n} (@code{calc-min}) [@code{min}] and @kbd{f x} (@code{calc-max}) | |
16117 | [@code{max}] commands take the minimum or maximum of two real numbers, | |
16118 | respectively. These commands also work on HMS forms, date forms, | |
16119 | intervals, and infinities. (In algebraic expressions, these functions | |
16120 | take any number of arguments and return the maximum or minimum among | |
16121 | all the arguments.)@refill | |
16122 | ||
16123 | @kindex f M | |
16124 | @kindex f X | |
16125 | @pindex calc-mant-part | |
16126 | @tindex mant | |
16127 | @pindex calc-xpon-part | |
16128 | @tindex xpon | |
16129 | The @kbd{f M} (@code{calc-mant-part}) [@code{mant}] function extracts | |
16130 | the ``mantissa'' part @cite{m} of its floating-point argument; @kbd{f X} | |
16131 | (@code{calc-xpon-part}) [@code{xpon}] extracts the ``exponent'' part | |
16132 | @cite{e}. The original number is equal to @c{$m \times 10^e$} | |
16133 | @cite{m * 10^e}, | |
16134 | where @cite{m} is in the interval @samp{[1.0 ..@: 10.0)} except that | |
16135 | @cite{m=e=0} if the original number is zero. For integers | |
16136 | and fractions, @code{mant} returns the number unchanged and @code{xpon} | |
16137 | returns zero. The @kbd{v u} (@code{calc-unpack}) command can also be | |
16138 | used to ``unpack'' a floating-point number; this produces an integer | |
16139 | mantissa and exponent, with the constraint that the mantissa is not | |
16140 | a multiple of ten (again except for the @cite{m=e=0} case).@refill | |
16141 | ||
16142 | @kindex f S | |
16143 | @pindex calc-scale-float | |
16144 | @tindex scf | |
16145 | The @kbd{f S} (@code{calc-scale-float}) [@code{scf}] function scales a number | |
16146 | by a given power of ten. Thus, @samp{scf(mant(x), xpon(x)) = x} for any | |
16147 | real @samp{x}. The second argument must be an integer, but the first | |
16148 | may actually be any numeric value. For example, @samp{scf(5,-2) = 0.05} | |
16149 | or @samp{1:20} depending on the current Fraction Mode.@refill | |
16150 | ||
16151 | @kindex f [ | |
16152 | @kindex f ] | |
16153 | @pindex calc-decrement | |
16154 | @pindex calc-increment | |
16155 | @tindex decr | |
16156 | @tindex incr | |
16157 | The @kbd{f [} (@code{calc-decrement}) [@code{decr}] and @kbd{f ]} | |
16158 | (@code{calc-increment}) [@code{incr}] functions decrease or increase | |
16159 | a number by one unit. For integers, the effect is obvious. For | |
16160 | floating-point numbers, the change is by one unit in the last place. | |
16161 | For example, incrementing @samp{12.3456} when the current precision | |
16162 | is 6 digits yields @samp{12.3457}. If the current precision had been | |
16163 | 8 digits, the result would have been @samp{12.345601}. Incrementing | |
16164 | @samp{0.0} produces @c{$10^{-p}$} | |
16165 | @cite{10^-p}, where @cite{p} is the current | |
16166 | precision. These operations are defined only on integers and floats. | |
16167 | With numeric prefix arguments, they change the number by @cite{n} units. | |
16168 | ||
16169 | Note that incrementing followed by decrementing, or vice-versa, will | |
16170 | almost but not quite always cancel out. Suppose the precision is | |
16171 | 6 digits and the number @samp{9.99999} is on the stack. Incrementing | |
16172 | will produce @samp{10.0000}; decrementing will produce @samp{9.9999}. | |
16173 | One digit has been dropped. This is an unavoidable consequence of the | |
16174 | way floating-point numbers work. | |
16175 | ||
16176 | Incrementing a date/time form adjusts it by a certain number of seconds. | |
16177 | Incrementing a pure date form adjusts it by a certain number of days. | |
16178 | ||
16179 | @node Integer Truncation, Complex Number Functions, Basic Arithmetic, Arithmetic | |
16180 | @section Integer Truncation | |
16181 | ||
16182 | @noindent | |
16183 | There are four commands for truncating a real number to an integer, | |
16184 | differing mainly in their treatment of negative numbers. All of these | |
16185 | commands have the property that if the argument is an integer, the result | |
16186 | is the same integer. An integer-valued floating-point argument is converted | |
16187 | to integer form. | |
16188 | ||
16189 | If you press @kbd{H} (@code{calc-hyperbolic}) first, the result will be | |
16190 | expressed as an integer-valued floating-point number. | |
16191 | ||
16192 | @cindex Integer part of a number | |
16193 | @kindex F | |
16194 | @pindex calc-floor | |
16195 | @tindex floor | |
16196 | @tindex ffloor | |
5d67986c RS |
16197 | @ignore |
16198 | @mindex @null | |
16199 | @end ignore | |
d7b8e6c6 EZ |
16200 | @kindex H F |
16201 | The @kbd{F} (@code{calc-floor}) [@code{floor} or @code{ffloor}] command | |
16202 | truncates a real number to the next lower integer, i.e., toward minus | |
16203 | infinity. Thus @kbd{3.6 F} produces 3, but @kbd{_3.6 F} produces | |
16204 | @i{-4}.@refill | |
16205 | ||
16206 | @kindex I F | |
16207 | @pindex calc-ceiling | |
16208 | @tindex ceil | |
16209 | @tindex fceil | |
5d67986c RS |
16210 | @ignore |
16211 | @mindex @null | |
16212 | @end ignore | |
d7b8e6c6 EZ |
16213 | @kindex H I F |
16214 | The @kbd{I F} (@code{calc-ceiling}) [@code{ceil} or @code{fceil}] | |
16215 | command truncates toward positive infinity. Thus @kbd{3.6 I F} produces | |
16216 | 4, and @kbd{_3.6 I F} produces @i{-3}.@refill | |
16217 | ||
16218 | @kindex R | |
16219 | @pindex calc-round | |
16220 | @tindex round | |
16221 | @tindex fround | |
5d67986c RS |
16222 | @ignore |
16223 | @mindex @null | |
16224 | @end ignore | |
d7b8e6c6 EZ |
16225 | @kindex H R |
16226 | The @kbd{R} (@code{calc-round}) [@code{round} or @code{fround}] command | |
16227 | rounds to the nearest integer. When the fractional part is .5 exactly, | |
16228 | this command rounds away from zero. (All other rounding in the | |
16229 | Calculator uses this convention as well.) Thus @kbd{3.5 R} produces 4 | |
16230 | but @kbd{3.4 R} produces 3; @kbd{_3.5 R} produces @i{-4}.@refill | |
16231 | ||
16232 | @kindex I R | |
16233 | @pindex calc-trunc | |
16234 | @tindex trunc | |
16235 | @tindex ftrunc | |
5d67986c RS |
16236 | @ignore |
16237 | @mindex @null | |
16238 | @end ignore | |
d7b8e6c6 EZ |
16239 | @kindex H I R |
16240 | The @kbd{I R} (@code{calc-trunc}) [@code{trunc} or @code{ftrunc}] | |
16241 | command truncates toward zero. In other words, it ``chops off'' | |
16242 | everything after the decimal point. Thus @kbd{3.6 I R} produces 3 and | |
16243 | @kbd{_3.6 I R} produces @i{-3}.@refill | |
16244 | ||
16245 | These functions may not be applied meaningfully to error forms, but they | |
16246 | do work for intervals. As a convenience, applying @code{floor} to a | |
16247 | modulo form floors the value part of the form. Applied to a vector, | |
16248 | these functions operate on all elements of the vector one by one. | |
16249 | Applied to a date form, they operate on the internal numerical | |
16250 | representation of dates, converting a date/time form into a pure date. | |
16251 | ||
5d67986c RS |
16252 | @ignore |
16253 | @starindex | |
16254 | @end ignore | |
d7b8e6c6 | 16255 | @tindex rounde |
5d67986c RS |
16256 | @ignore |
16257 | @starindex | |
16258 | @end ignore | |
d7b8e6c6 | 16259 | @tindex roundu |
5d67986c RS |
16260 | @ignore |
16261 | @starindex | |
16262 | @end ignore | |
d7b8e6c6 | 16263 | @tindex frounde |
5d67986c RS |
16264 | @ignore |
16265 | @starindex | |
16266 | @end ignore | |
d7b8e6c6 EZ |
16267 | @tindex froundu |
16268 | There are two more rounding functions which can only be entered in | |
16269 | algebraic notation. The @code{roundu} function is like @code{round} | |
16270 | except that it rounds up, toward plus infinity, when the fractional | |
16271 | part is .5. This distinction matters only for negative arguments. | |
16272 | Also, @code{rounde} rounds to an even number in the case of a tie, | |
16273 | rounding up or down as necessary. For example, @samp{rounde(3.5)} and | |
16274 | @samp{rounde(4.5)} both return 4, but @samp{rounde(5.5)} returns 6. | |
16275 | The advantage of round-to-even is that the net error due to rounding | |
16276 | after a long calculation tends to cancel out to zero. An important | |
16277 | subtle point here is that the number being fed to @code{rounde} will | |
16278 | already have been rounded to the current precision before @code{rounde} | |
16279 | begins. For example, @samp{rounde(2.500001)} with a current precision | |
16280 | of 6 will incorrectly, or at least surprisingly, yield 2 because the | |
16281 | argument will first have been rounded down to @cite{2.5} (which | |
16282 | @code{rounde} sees as an exact tie between 2 and 3). | |
16283 | ||
16284 | Each of these functions, when written in algebraic formulas, allows | |
16285 | a second argument which specifies the number of digits after the | |
16286 | decimal point to keep. For example, @samp{round(123.4567, 2)} will | |
16287 | produce the answer 123.46, and @samp{round(123.4567, -1)} will | |
16288 | produce 120 (i.e., the cutoff is one digit to the @emph{left} of | |
16289 | the decimal point). A second argument of zero is equivalent to | |
16290 | no second argument at all. | |
16291 | ||
16292 | @cindex Fractional part of a number | |
16293 | To compute the fractional part of a number (i.e., the amount which, when | |
5d67986c | 16294 | added to `@t{floor(}@var{n}@t{)}', will produce @var{n}) just take @var{n} |
d7b8e6c6 EZ |
16295 | modulo 1 using the @code{%} command.@refill |
16296 | ||
16297 | Note also the @kbd{\} (integer quotient), @kbd{f I} (integer logarithm), | |
16298 | and @kbd{f Q} (integer square root) commands, which are analogous to | |
16299 | @kbd{/}, @kbd{B}, and @kbd{Q}, respectively, except that they take integer | |
16300 | arguments and return the result rounded down to an integer. | |
16301 | ||
16302 | @node Complex Number Functions, Conversions, Integer Truncation, Arithmetic | |
16303 | @section Complex Number Functions | |
16304 | ||
16305 | @noindent | |
16306 | @kindex J | |
16307 | @pindex calc-conj | |
16308 | @tindex conj | |
16309 | The @kbd{J} (@code{calc-conj}) [@code{conj}] command computes the | |
16310 | complex conjugate of a number. For complex number @cite{a+bi}, the | |
16311 | complex conjugate is @cite{a-bi}. If the argument is a real number, | |
16312 | this command leaves it the same. If the argument is a vector or matrix, | |
16313 | this command replaces each element by its complex conjugate. | |
16314 | ||
16315 | @kindex G | |
16316 | @pindex calc-argument | |
16317 | @tindex arg | |
16318 | The @kbd{G} (@code{calc-argument}) [@code{arg}] command computes the | |
16319 | ``argument'' or polar angle of a complex number. For a number in polar | |
16320 | notation, this is simply the second component of the pair | |
5d67986c RS |
16321 | `@t{(}@var{r}@t{;}@c{$\theta$} |
16322 | @var{theta}@t{)}'. | |
d7b8e6c6 EZ |
16323 | The result is expressed according to the current angular mode and will |
16324 | be in the range @i{-180} degrees (exclusive) to @i{+180} degrees | |
16325 | (inclusive), or the equivalent range in radians.@refill | |
16326 | ||
16327 | @pindex calc-imaginary | |
16328 | The @code{calc-imaginary} command multiplies the number on the | |
16329 | top of the stack by the imaginary number @cite{i = (0,1)}. This | |
16330 | command is not normally bound to a key in Calc, but it is available | |
16331 | on the @key{IMAG} button in Keypad Mode. | |
16332 | ||
16333 | @kindex f r | |
16334 | @pindex calc-re | |
16335 | @tindex re | |
16336 | The @kbd{f r} (@code{calc-re}) [@code{re}] command replaces a complex number | |
16337 | by its real part. This command has no effect on real numbers. (As an | |
16338 | added convenience, @code{re} applied to a modulo form extracts | |
16339 | the value part.)@refill | |
16340 | ||
16341 | @kindex f i | |
16342 | @pindex calc-im | |
16343 | @tindex im | |
16344 | The @kbd{f i} (@code{calc-im}) [@code{im}] command replaces a complex number | |
16345 | by its imaginary part; real numbers are converted to zero. With a vector | |
16346 | or matrix argument, these functions operate element-wise.@refill | |
16347 | ||
5d67986c RS |
16348 | @ignore |
16349 | @mindex v p | |
16350 | @end ignore | |
d7b8e6c6 EZ |
16351 | @kindex v p (complex) |
16352 | @pindex calc-pack | |
16353 | The @kbd{v p} (@code{calc-pack}) command can pack the top two numbers on | |
269b7745 | 16354 | the stack into a composite object such as a complex number. With |
d7b8e6c6 EZ |
16355 | a prefix argument of @i{-1}, it produces a rectangular complex number; |
16356 | with an argument of @i{-2}, it produces a polar complex number. | |
16357 | (Also, @pxref{Building Vectors}.) | |
16358 | ||
5d67986c RS |
16359 | @ignore |
16360 | @mindex v u | |
16361 | @end ignore | |
d7b8e6c6 EZ |
16362 | @kindex v u (complex) |
16363 | @pindex calc-unpack | |
16364 | The @kbd{v u} (@code{calc-unpack}) command takes the complex number | |
16365 | (or other composite object) on the top of the stack and unpacks it | |
16366 | into its separate components. | |
16367 | ||
16368 | @node Conversions, Date Arithmetic, Complex Number Functions, Arithmetic | |
16369 | @section Conversions | |
16370 | ||
16371 | @noindent | |
16372 | The commands described in this section convert numbers from one form | |
16373 | to another; they are two-key sequences beginning with the letter @kbd{c}. | |
16374 | ||
16375 | @kindex c f | |
16376 | @pindex calc-float | |
16377 | @tindex pfloat | |
16378 | The @kbd{c f} (@code{calc-float}) [@code{pfloat}] command converts the | |
16379 | number on the top of the stack to floating-point form. For example, | |
16380 | @cite{23} is converted to @cite{23.0}, @cite{3:2} is converted to | |
16381 | @cite{1.5}, and @cite{2.3} is left the same. If the value is a composite | |
16382 | object such as a complex number or vector, each of the components is | |
16383 | converted to floating-point. If the value is a formula, all numbers | |
16384 | in the formula are converted to floating-point. Note that depending | |
16385 | on the current floating-point precision, conversion to floating-point | |
16386 | format may lose information.@refill | |
16387 | ||
16388 | As a special exception, integers which appear as powers or subscripts | |
16389 | are not floated by @kbd{c f}. If you really want to float a power, | |
16390 | you can use a @kbd{j s} command to select the power followed by @kbd{c f}. | |
16391 | Because @kbd{c f} cannot examine the formula outside of the selection, | |
16392 | it does not notice that the thing being floated is a power. | |
16393 | @xref{Selecting Subformulas}. | |
16394 | ||
16395 | The normal @kbd{c f} command is ``pervasive'' in the sense that it | |
16396 | applies to all numbers throughout the formula. The @code{pfloat} | |
16397 | algebraic function never stays around in a formula; @samp{pfloat(a + 1)} | |
16398 | changes to @samp{a + 1.0} as soon as it is evaluated. | |
16399 | ||
16400 | @kindex H c f | |
16401 | @tindex float | |
16402 | With the Hyperbolic flag, @kbd{H c f} [@code{float}] operates | |
16403 | only on the number or vector of numbers at the top level of its | |
16404 | argument. Thus, @samp{float(1)} is 1.0, but @samp{float(a + 1)} | |
16405 | is left unevaluated because its argument is not a number. | |
16406 | ||
16407 | You should use @kbd{H c f} if you wish to guarantee that the final | |
16408 | value, once all the variables have been assigned, is a float; you | |
16409 | would use @kbd{c f} if you wish to do the conversion on the numbers | |
16410 | that appear right now. | |
16411 | ||
16412 | @kindex c F | |
16413 | @pindex calc-fraction | |
16414 | @tindex pfrac | |
16415 | The @kbd{c F} (@code{calc-fraction}) [@code{pfrac}] command converts a | |
16416 | floating-point number into a fractional approximation. By default, it | |
16417 | produces a fraction whose decimal representation is the same as the | |
16418 | input number, to within the current precision. You can also give a | |
16419 | numeric prefix argument to specify a tolerance, either directly, or, | |
16420 | if the prefix argument is zero, by using the number on top of the stack | |
16421 | as the tolerance. If the tolerance is a positive integer, the fraction | |
16422 | is correct to within that many significant figures. If the tolerance is | |
16423 | a non-positive integer, it specifies how many digits fewer than the current | |
16424 | precision to use. If the tolerance is a floating-point number, the | |
16425 | fraction is correct to within that absolute amount. | |
16426 | ||
16427 | @kindex H c F | |
16428 | @tindex frac | |
16429 | The @code{pfrac} function is pervasive, like @code{pfloat}. | |
16430 | There is also a non-pervasive version, @kbd{H c F} [@code{frac}], | |
16431 | which is analogous to @kbd{H c f} discussed above. | |
16432 | ||
16433 | @kindex c d | |
16434 | @pindex calc-to-degrees | |
16435 | @tindex deg | |
16436 | The @kbd{c d} (@code{calc-to-degrees}) [@code{deg}] command converts a | |
16437 | number into degrees form. The value on the top of the stack may be an | |
16438 | HMS form (interpreted as degrees-minutes-seconds), or a real number which | |
16439 | will be interpreted in radians regardless of the current angular mode.@refill | |
16440 | ||
16441 | @kindex c r | |
16442 | @pindex calc-to-radians | |
16443 | @tindex rad | |
16444 | The @kbd{c r} (@code{calc-to-radians}) [@code{rad}] command converts an | |
16445 | HMS form or angle in degrees into an angle in radians. | |
16446 | ||
16447 | @kindex c h | |
16448 | @pindex calc-to-hms | |
16449 | @tindex hms | |
16450 | The @kbd{c h} (@code{calc-to-hms}) [@code{hms}] command converts a real | |
16451 | number, interpreted according to the current angular mode, to an HMS | |
16452 | form describing the same angle. In algebraic notation, the @code{hms} | |
16453 | function also accepts three arguments: @samp{hms(@var{h}, @var{m}, @var{s})}. | |
16454 | (The three-argument version is independent of the current angular mode.) | |
16455 | ||
16456 | @pindex calc-from-hms | |
16457 | The @code{calc-from-hms} command converts the HMS form on the top of the | |
16458 | stack into a real number according to the current angular mode. | |
16459 | ||
16460 | @kindex c p | |
16461 | @kindex I c p | |
16462 | @pindex calc-polar | |
16463 | @tindex polar | |
16464 | @tindex rect | |
16465 | The @kbd{c p} (@code{calc-polar}) command converts the complex number on | |
16466 | the top of the stack from polar to rectangular form, or from rectangular | |
16467 | to polar form, whichever is appropriate. Real numbers are left the same. | |
16468 | This command is equivalent to the @code{rect} or @code{polar} | |
16469 | functions in algebraic formulas, depending on the direction of | |
16470 | conversion. (It uses @code{polar}, except that if the argument is | |
16471 | already a polar complex number, it uses @code{rect} instead. The | |
16472 | @kbd{I c p} command always uses @code{rect}.)@refill | |
16473 | ||
16474 | @kindex c c | |
16475 | @pindex calc-clean | |
16476 | @tindex pclean | |
16477 | The @kbd{c c} (@code{calc-clean}) [@code{pclean}] command ``cleans'' the | |
16478 | number on the top of the stack. Floating point numbers are re-rounded | |
16479 | according to the current precision. Polar numbers whose angular | |
16480 | components have strayed from the @i{-180} to @i{+180} degree range | |
16481 | are normalized. (Note that results will be undesirable if the current | |
16482 | angular mode is different from the one under which the number was | |
16483 | produced!) Integers and fractions are generally unaffected by this | |
16484 | operation. Vectors and formulas are cleaned by cleaning each component | |
16485 | number (i.e., pervasively).@refill | |
16486 | ||
16487 | If the simplification mode is set below the default level, it is raised | |
16488 | to the default level for the purposes of this command. Thus, @kbd{c c} | |
16489 | applies the default simplifications even if their automatic application | |
16490 | is disabled. @xref{Simplification Modes}. | |
16491 | ||
16492 | @cindex Roundoff errors, correcting | |
16493 | A numeric prefix argument to @kbd{c c} sets the floating-point precision | |
16494 | to that value for the duration of the command. A positive prefix (of at | |
16495 | least 3) sets the precision to the specified value; a negative or zero | |
16496 | prefix decreases the precision by the specified amount. | |
16497 | ||
16498 | @kindex c 0-9 | |
16499 | @pindex calc-clean-num | |
16500 | The keystroke sequences @kbd{c 0} through @kbd{c 9} are equivalent | |
16501 | to @kbd{c c} with the corresponding negative prefix argument. If roundoff | |
16502 | errors have changed 2.0 into 1.999999, typing @kbd{c 1} to clip off one | |
16503 | decimal place often conveniently does the trick. | |
16504 | ||
16505 | The @kbd{c c} command with a numeric prefix argument, and the @kbd{c 0} | |
16506 | through @kbd{c 9} commands, also ``clip'' very small floating-point | |
16507 | numbers to zero. If the exponent is less than or equal to the negative | |
16508 | of the specified precision, the number is changed to 0.0. For example, | |
16509 | if the current precision is 12, then @kbd{c 2} changes the vector | |
16510 | @samp{[1e-8, 1e-9, 1e-10, 1e-11]} to @samp{[1e-8, 1e-9, 0, 0]}. | |
16511 | Numbers this small generally arise from roundoff noise. | |
16512 | ||
16513 | If the numbers you are using really are legitimately this small, | |
16514 | you should avoid using the @kbd{c 0} through @kbd{c 9} commands. | |
16515 | (The plain @kbd{c c} command rounds to the current precision but | |
16516 | does not clip small numbers.) | |
16517 | ||
16518 | One more property of @kbd{c 0} through @kbd{c 9}, and of @kbd{c c} with | |
16519 | a prefix argument, is that integer-valued floats are converted to | |
16520 | plain integers, so that @kbd{c 1} on @samp{[1., 1.5, 2., 2.5, 3.]} | |
16521 | produces @samp{[1, 1.5, 2, 2.5, 3]}. This is not done for huge | |
16522 | numbers (@samp{1e100} is technically an integer-valued float, but | |
16523 | you wouldn't want it automatically converted to a 100-digit integer). | |
16524 | ||
16525 | @kindex H c 0-9 | |
16526 | @kindex H c c | |
16527 | @tindex clean | |
16528 | With the Hyperbolic flag, @kbd{H c c} and @kbd{H c 0} through @kbd{H c 9} | |
16529 | operate non-pervasively [@code{clean}]. | |
16530 | ||
16531 | @node Date Arithmetic, Financial Functions, Conversions, Arithmetic | |
16532 | @section Date Arithmetic | |
16533 | ||
16534 | @noindent | |
16535 | @cindex Date arithmetic, additional functions | |
16536 | The commands described in this section perform various conversions | |
16537 | and calculations involving date forms (@pxref{Date Forms}). They | |
16538 | use the @kbd{t} (for time/date) prefix key followed by shifted | |
16539 | letters. | |
16540 | ||
16541 | The simplest date arithmetic is done using the regular @kbd{+} and @kbd{-} | |
16542 | commands. In particular, adding a number to a date form advances the | |
16543 | date form by a certain number of days; adding an HMS form to a date | |
16544 | form advances the date by a certain amount of time; and subtracting two | |
16545 | date forms produces a difference measured in days. The commands | |
16546 | described here provide additional, more specialized operations on dates. | |
16547 | ||
16548 | Many of these commands accept a numeric prefix argument; if you give | |
16549 | plain @kbd{C-u} as the prefix, these commands will instead take the | |
16550 | additional argument from the top of the stack. | |
16551 | ||
16552 | @menu | |
16553 | * Date Conversions:: | |
16554 | * Date Functions:: | |
16555 | * Time Zones:: | |
16556 | * Business Days:: | |
16557 | @end menu | |
16558 | ||
16559 | @node Date Conversions, Date Functions, Date Arithmetic, Date Arithmetic | |
16560 | @subsection Date Conversions | |
16561 | ||
16562 | @noindent | |
16563 | @kindex t D | |
16564 | @pindex calc-date | |
16565 | @tindex date | |
16566 | The @kbd{t D} (@code{calc-date}) [@code{date}] command converts a | |
16567 | date form into a number, measured in days since Jan 1, 1 AD. The | |
16568 | result will be an integer if @var{date} is a pure date form, or a | |
16569 | fraction or float if @var{date} is a date/time form. Or, if its | |
16570 | argument is a number, it converts this number into a date form. | |
16571 | ||
16572 | With a numeric prefix argument, @kbd{t D} takes that many objects | |
16573 | (up to six) from the top of the stack and interprets them in one | |
16574 | of the following ways: | |
16575 | ||
16576 | The @samp{date(@var{year}, @var{month}, @var{day})} function | |
16577 | builds a pure date form out of the specified year, month, and | |
16578 | day, which must all be integers. @var{Year} is a year number, | |
16579 | such as 1991 (@emph{not} the same as 91!). @var{Month} must be | |
16580 | an integer in the range 1 to 12; @var{day} must be in the range | |
16581 | 1 to 31. If the specified month has fewer than 31 days and | |
16582 | @var{day} is too large, the equivalent day in the following | |
16583 | month will be used. | |
16584 | ||
16585 | The @samp{date(@var{month}, @var{day})} function builds a | |
16586 | pure date form using the current year, as determined by the | |
16587 | real-time clock. | |
16588 | ||
16589 | The @samp{date(@var{year}, @var{month}, @var{day}, @var{hms})} | |
16590 | function builds a date/time form using an @var{hms} form. | |
16591 | ||
16592 | The @samp{date(@var{year}, @var{month}, @var{day}, @var{hour}, | |
16593 | @var{minute}, @var{second})} function builds a date/time form. | |
16594 | @var{hour} should be an integer in the range 0 to 23; | |
16595 | @var{minute} should be an integer in the range 0 to 59; | |
16596 | @var{second} should be any real number in the range @samp{[0 .. 60)}. | |
16597 | The last two arguments default to zero if omitted. | |
16598 | ||
16599 | @kindex t J | |
16600 | @pindex calc-julian | |
16601 | @tindex julian | |
16602 | @cindex Julian day counts, conversions | |
16603 | The @kbd{t J} (@code{calc-julian}) [@code{julian}] command converts | |
16604 | a date form into a Julian day count, which is the number of days | |
16605 | since noon on Jan 1, 4713 BC. A pure date is converted to an integer | |
16606 | Julian count representing noon of that day. A date/time form is | |
16607 | converted to an exact floating-point Julian count, adjusted to | |
16608 | interpret the date form in the current time zone but the Julian | |
16609 | day count in Greenwich Mean Time. A numeric prefix argument allows | |
16610 | you to specify the time zone; @pxref{Time Zones}. Use a prefix of | |
16611 | zero to suppress the time zone adjustment. Note that pure date forms | |
16612 | are never time-zone adjusted. | |
16613 | ||
16614 | This command can also do the opposite conversion, from a Julian day | |
16615 | count (either an integer day, or a floating-point day and time in | |
16616 | the GMT zone), into a pure date form or a date/time form in the | |
16617 | current or specified time zone. | |
16618 | ||
16619 | @kindex t U | |
16620 | @pindex calc-unix-time | |
16621 | @tindex unixtime | |
16622 | @cindex Unix time format, conversions | |
16623 | The @kbd{t U} (@code{calc-unix-time}) [@code{unixtime}] command | |
16624 | converts a date form into a Unix time value, which is the number of | |
16625 | seconds since midnight on Jan 1, 1970, or vice-versa. The numeric result | |
16626 | will be an integer if the current precision is 12 or less; for higher | |
5d67986c | 16627 | precisions, the result may be a float with (@var{precision}@minus{}12) |
d7b8e6c6 EZ |
16628 | digits after the decimal. Just as for @kbd{t J}, the numeric time |
16629 | is interpreted in the GMT time zone and the date form is interpreted | |
16630 | in the current or specified zone. Some systems use Unix-like | |
16631 | numbering but with the local time zone; give a prefix of zero to | |
16632 | suppress the adjustment if so. | |
16633 | ||
16634 | @kindex t C | |
16635 | @pindex calc-convert-time-zones | |
16636 | @tindex tzconv | |
16637 | @cindex Time Zones, converting between | |
16638 | The @kbd{t C} (@code{calc-convert-time-zones}) [@code{tzconv}] | |
16639 | command converts a date form from one time zone to another. You | |
16640 | are prompted for each time zone name in turn; you can answer with | |
16641 | any suitable Calc time zone expression (@pxref{Time Zones}). | |
16642 | If you answer either prompt with a blank line, the local time | |
16643 | zone is used for that prompt. You can also answer the first | |
16644 | prompt with @kbd{$} to take the two time zone names from the | |
16645 | stack (and the date to be converted from the third stack level). | |
16646 | ||
16647 | @node Date Functions, Business Days, Date Conversions, Date Arithmetic | |
16648 | @subsection Date Functions | |
16649 | ||
16650 | @noindent | |
16651 | @kindex t N | |
16652 | @pindex calc-now | |
16653 | @tindex now | |
16654 | The @kbd{t N} (@code{calc-now}) [@code{now}] command pushes the | |
16655 | current date and time on the stack as a date form. The time is | |
16656 | reported in terms of the specified time zone; with no numeric prefix | |
16657 | argument, @kbd{t N} reports for the current time zone. | |
16658 | ||
16659 | @kindex t P | |
16660 | @pindex calc-date-part | |
16661 | The @kbd{t P} (@code{calc-date-part}) command extracts one part | |
16662 | of a date form. The prefix argument specifies the part; with no | |
16663 | argument, this command prompts for a part code from 1 to 9. | |
16664 | The various part codes are described in the following paragraphs. | |
16665 | ||
16666 | @tindex year | |
16667 | The @kbd{M-1 t P} [@code{year}] function extracts the year number | |
16668 | from a date form as an integer, e.g., 1991. This and the | |
16669 | following functions will also accept a real number for an | |
16670 | argument, which is interpreted as a standard Calc day number. | |
16671 | Note that this function will never return zero, since the year | |
16672 | 1 BC immediately precedes the year 1 AD. | |
16673 | ||
16674 | @tindex month | |
16675 | The @kbd{M-2 t P} [@code{month}] function extracts the month number | |
16676 | from a date form as an integer in the range 1 to 12. | |
16677 | ||
16678 | @tindex day | |
16679 | The @kbd{M-3 t P} [@code{day}] function extracts the day number | |
16680 | from a date form as an integer in the range 1 to 31. | |
16681 | ||
16682 | @tindex hour | |
16683 | The @kbd{M-4 t P} [@code{hour}] function extracts the hour from | |
16684 | a date form as an integer in the range 0 (midnight) to 23. Note | |
16685 | that 24-hour time is always used. This returns zero for a pure | |
16686 | date form. This function (and the following two) also accept | |
16687 | HMS forms as input. | |
16688 | ||
16689 | @tindex minute | |
16690 | The @kbd{M-5 t P} [@code{minute}] function extracts the minute | |
16691 | from a date form as an integer in the range 0 to 59. | |
16692 | ||
16693 | @tindex second | |
16694 | The @kbd{M-6 t P} [@code{second}] function extracts the second | |
16695 | from a date form. If the current precision is 12 or less, | |
16696 | the result is an integer in the range 0 to 59. For higher | |
16697 | precisions, the result may instead be a floating-point number. | |
16698 | ||
16699 | @tindex weekday | |
16700 | The @kbd{M-7 t P} [@code{weekday}] function extracts the weekday | |
16701 | number from a date form as an integer in the range 0 (Sunday) | |
16702 | to 6 (Saturday). | |
16703 | ||
16704 | @tindex yearday | |
16705 | The @kbd{M-8 t P} [@code{yearday}] function extracts the day-of-year | |
16706 | number from a date form as an integer in the range 1 (January 1) | |
16707 | to 366 (December 31 of a leap year). | |
16708 | ||
16709 | @tindex time | |
16710 | The @kbd{M-9 t P} [@code{time}] function extracts the time portion | |
16711 | of a date form as an HMS form. This returns @samp{0@@ 0' 0"} | |
16712 | for a pure date form. | |
16713 | ||
16714 | @kindex t M | |
16715 | @pindex calc-new-month | |
16716 | @tindex newmonth | |
16717 | The @kbd{t M} (@code{calc-new-month}) [@code{newmonth}] command | |
16718 | computes a new date form that represents the first day of the month | |
16719 | specified by the input date. The result is always a pure date | |
16720 | form; only the year and month numbers of the input are retained. | |
16721 | With a numeric prefix argument @var{n} in the range from 1 to 31, | |
16722 | @kbd{t M} computes the @var{n}th day of the month. (If @var{n} | |
16723 | is greater than the actual number of days in the month, or if | |
16724 | @var{n} is zero, the last day of the month is used.) | |
16725 | ||
16726 | @kindex t Y | |
16727 | @pindex calc-new-year | |
16728 | @tindex newyear | |
16729 | The @kbd{t Y} (@code{calc-new-year}) [@code{newyear}] command | |
16730 | computes a new pure date form that represents the first day of | |
16731 | the year specified by the input. The month, day, and time | |
16732 | of the input date form are lost. With a numeric prefix argument | |
16733 | @var{n} in the range from 1 to 366, @kbd{t Y} computes the | |
16734 | @var{n}th day of the year (366 is treated as 365 in non-leap | |
16735 | years). A prefix argument of 0 computes the last day of the | |
16736 | year (December 31). A negative prefix argument from @i{-1} to | |
16737 | @i{-12} computes the first day of the @var{n}th month of the year. | |
16738 | ||
16739 | @kindex t W | |
16740 | @pindex calc-new-week | |
16741 | @tindex newweek | |
16742 | The @kbd{t W} (@code{calc-new-week}) [@code{newweek}] command | |
16743 | computes a new pure date form that represents the Sunday on or before | |
16744 | the input date. With a numeric prefix argument, it can be made to | |
16745 | use any day of the week as the starting day; the argument must be in | |
16746 | the range from 0 (Sunday) to 6 (Saturday). This function always | |
16747 | subtracts between 0 and 6 days from the input date. | |
16748 | ||
16749 | Here's an example use of @code{newweek}: Find the date of the next | |
16750 | Wednesday after a given date. Using @kbd{M-3 t W} or @samp{newweek(d, 3)} | |
16751 | will give you the @emph{preceding} Wednesday, so @samp{newweek(d+7, 3)} | |
16752 | will give you the following Wednesday. A further look at the definition | |
16753 | of @code{newweek} shows that if the input date is itself a Wednesday, | |
16754 | this formula will return the Wednesday one week in the future. An | |
16755 | exercise for the reader is to modify this formula to yield the same day | |
16756 | if the input is already a Wednesday. Another interesting exercise is | |
16757 | to preserve the time-of-day portion of the input (@code{newweek} resets | |
16758 | the time to midnight; hint:@: how can @code{newweek} be defined in terms | |
16759 | of the @code{weekday} function?). | |
16760 | ||
5d67986c RS |
16761 | @ignore |
16762 | @starindex | |
16763 | @end ignore | |
d7b8e6c6 EZ |
16764 | @tindex pwday |
16765 | The @samp{pwday(@var{date})} function (not on any key) computes the | |
16766 | day-of-month number of the Sunday on or before @var{date}. With | |
16767 | two arguments, @samp{pwday(@var{date}, @var{day})} computes the day | |
16768 | number of the Sunday on or before day number @var{day} of the month | |
16769 | specified by @var{date}. The @var{day} must be in the range from | |
16770 | 7 to 31; if the day number is greater than the actual number of days | |
16771 | in the month, the true number of days is used instead. Thus | |
16772 | @samp{pwday(@var{date}, 7)} finds the first Sunday of the month, and | |
16773 | @samp{pwday(@var{date}, 31)} finds the last Sunday of the month. | |
16774 | With a third @var{weekday} argument, @code{pwday} can be made to look | |
16775 | for any day of the week instead of Sunday. | |
16776 | ||
16777 | @kindex t I | |
16778 | @pindex calc-inc-month | |
16779 | @tindex incmonth | |
16780 | The @kbd{t I} (@code{calc-inc-month}) [@code{incmonth}] command | |
16781 | increases a date form by one month, or by an arbitrary number of | |
16782 | months specified by a numeric prefix argument. The time portion, | |
16783 | if any, of the date form stays the same. The day also stays the | |
16784 | same, except that if the new month has fewer days the day | |
16785 | number may be reduced to lie in the valid range. For example, | |
16786 | @samp{incmonth(<Jan 31, 1991>)} produces @samp{<Feb 28, 1991>}. | |
16787 | Because of this, @kbd{t I t I} and @kbd{M-2 t I} do not always give | |
16788 | the same results (@samp{<Mar 28, 1991>} versus @samp{<Mar 31, 1991>} | |
16789 | in this case). | |
16790 | ||
5d67986c RS |
16791 | @ignore |
16792 | @starindex | |
16793 | @end ignore | |
d7b8e6c6 EZ |
16794 | @tindex incyear |
16795 | The @samp{incyear(@var{date}, @var{step})} function increases | |
16796 | a date form by the specified number of years, which may be | |
16797 | any positive or negative integer. Note that @samp{incyear(d, n)} | |
16798 | is equivalent to @w{@samp{incmonth(d, 12*n)}}, but these do not have | |
16799 | simple equivalents in terms of day arithmetic because | |
16800 | months and years have varying lengths. If the @var{step} | |
16801 | argument is omitted, 1 year is assumed. There is no keyboard | |
16802 | command for this function; use @kbd{C-u 12 t I} instead. | |
16803 | ||
16804 | There is no @code{newday} function at all because @kbd{F} [@code{floor}] | |
16805 | serves this purpose. Similarly, instead of @code{incday} and | |
16806 | @code{incweek} simply use @cite{d + n} or @cite{d + 7 n}. | |
16807 | ||
16808 | @xref{Basic Arithmetic}, for the @kbd{f ]} [@code{incr}] command | |
16809 | which can adjust a date/time form by a certain number of seconds. | |
16810 | ||
16811 | @node Business Days, Time Zones, Date Functions, Date Arithmetic | |
16812 | @subsection Business Days | |
16813 | ||
16814 | @noindent | |
16815 | Often time is measured in ``business days'' or ``working days,'' | |
16816 | where weekends and holidays are skipped. Calc's normal date | |
16817 | arithmetic functions use calendar days, so that subtracting two | |
16818 | consecutive Mondays will yield a difference of 7 days. By contrast, | |
16819 | subtracting two consecutive Mondays would yield 5 business days | |
16820 | (assuming two-day weekends and the absence of holidays). | |
16821 | ||
16822 | @kindex t + | |
16823 | @kindex t - | |
16824 | @tindex badd | |
16825 | @tindex bsub | |
16826 | @pindex calc-business-days-plus | |
16827 | @pindex calc-business-days-minus | |
16828 | The @kbd{t +} (@code{calc-business-days-plus}) [@code{badd}] | |
16829 | and @kbd{t -} (@code{calc-business-days-minus}) [@code{bsub}] | |
16830 | commands perform arithmetic using business days. For @kbd{t +}, | |
16831 | one argument must be a date form and the other must be a real | |
16832 | number (positive or negative). If the number is not an integer, | |
16833 | then a certain amount of time is added as well as a number of | |
16834 | days; for example, adding 0.5 business days to a time in Friday | |
16835 | evening will produce a time in Monday morning. It is also | |
16836 | possible to add an HMS form; adding @samp{12@@ 0' 0"} also adds | |
16837 | half a business day. For @kbd{t -}, the arguments are either a | |
16838 | date form and a number or HMS form, or two date forms, in which | |
16839 | case the result is the number of business days between the two | |
16840 | dates. | |
16841 | ||
16842 | @cindex @code{Holidays} variable | |
16843 | @vindex Holidays | |
16844 | By default, Calc considers any day that is not a Saturday or | |
16845 | Sunday to be a business day. You can define any number of | |
16846 | additional holidays by editing the variable @code{Holidays}. | |
16847 | (There is an @w{@kbd{s H}} convenience command for editing this | |
16848 | variable.) Initially, @code{Holidays} contains the vector | |
16849 | @samp{[sat, sun]}. Entries in the @code{Holidays} vector may | |
16850 | be any of the following kinds of objects: | |
16851 | ||
16852 | @itemize @bullet | |
16853 | @item | |
16854 | Date forms (pure dates, not date/time forms). These specify | |
16855 | particular days which are to be treated as holidays. | |
16856 | ||
16857 | @item | |
16858 | Intervals of date forms. These specify a range of days, all of | |
16859 | which are holidays (e.g., Christmas week). @xref{Interval Forms}. | |
16860 | ||
16861 | @item | |
16862 | Nested vectors of date forms. Each date form in the vector is | |
16863 | considered to be a holiday. | |
16864 | ||
16865 | @item | |
16866 | Any Calc formula which evaluates to one of the above three things. | |
16867 | If the formula involves the variable @cite{y}, it stands for a | |
16868 | yearly repeating holiday; @cite{y} will take on various year | |
16869 | numbers like 1992. For example, @samp{date(y, 12, 25)} specifies | |
16870 | Christmas day, and @samp{newweek(date(y, 11, 7), 4) + 21} specifies | |
16871 | Thanksgiving (which is held on the fourth Thursday of November). | |
16872 | If the formula involves the variable @cite{m}, that variable | |
16873 | takes on month numbers from 1 to 12: @samp{date(y, m, 15)} is | |
16874 | a holiday that takes place on the 15th of every month. | |
16875 | ||
16876 | @item | |
16877 | A weekday name, such as @code{sat} or @code{sun}. This is really | |
16878 | a variable whose name is a three-letter, lower-case day name. | |
16879 | ||
16880 | @item | |
16881 | An interval of year numbers (integers). This specifies the span of | |
16882 | years over which this holiday list is to be considered valid. Any | |
16883 | business-day arithmetic that goes outside this range will result | |
16884 | in an error message. Use this if you are including an explicit | |
16885 | list of holidays, rather than a formula to generate them, and you | |
16886 | want to make sure you don't accidentally go beyond the last point | |
16887 | where the holidays you entered are complete. If there is no | |
16888 | limiting interval in the @code{Holidays} vector, the default | |
16889 | @samp{[1 .. 2737]} is used. (This is the absolute range of years | |
16890 | for which Calc's business-day algorithms will operate.) | |
16891 | ||
16892 | @item | |
16893 | An interval of HMS forms. This specifies the span of hours that | |
16894 | are to be considered one business day. For example, if this | |
16895 | range is @samp{[9@@ 0' 0" .. 17@@ 0' 0"]} (i.e., 9am to 5pm), then | |
16896 | the business day is only eight hours long, so that @kbd{1.5 t +} | |
16897 | on @samp{<4:00pm Fri Dec 13, 1991>} will add one business day and | |
16898 | four business hours to produce @samp{<12:00pm Tue Dec 17, 1991>}. | |
16899 | Likewise, @kbd{t -} will now express differences in time as | |
16900 | fractions of an eight-hour day. Times before 9am will be treated | |
16901 | as 9am by business date arithmetic, and times at or after 5pm will | |
16902 | be treated as 4:59:59pm. If there is no HMS interval in @code{Holidays}, | |
16903 | the full 24-hour day @samp{[0@ 0' 0" .. 24@ 0' 0"]} is assumed. | |
16904 | (Regardless of the type of bounds you specify, the interval is | |
16905 | treated as inclusive on the low end and exclusive on the high end, | |
16906 | so that the work day goes from 9am up to, but not including, 5pm.) | |
16907 | @end itemize | |
16908 | ||
16909 | If the @code{Holidays} vector is empty, then @kbd{t +} and | |
16910 | @kbd{t -} will act just like @kbd{+} and @kbd{-} because there will | |
16911 | then be no difference between business days and calendar days. | |
16912 | ||
16913 | Calc expands the intervals and formulas you give into a complete | |
16914 | list of holidays for internal use. This is done mainly to make | |
16915 | sure it can detect multiple holidays. (For example, | |
16916 | @samp{<Jan 1, 1989>} is both New Year's Day and a Sunday, but | |
16917 | Calc's algorithms take care to count it only once when figuring | |
16918 | the number of holidays between two dates.) | |
16919 | ||
16920 | Since the complete list of holidays for all the years from 1 to | |
16921 | 2737 would be huge, Calc actually computes only the part of the | |
16922 | list between the smallest and largest years that have been involved | |
16923 | in business-day calculations so far. Normally, you won't have to | |
16924 | worry about this. Keep in mind, however, that if you do one | |
16925 | calculation for 1992, and another for 1792, even if both involve | |
16926 | only a small range of years, Calc will still work out all the | |
16927 | holidays that fall in that 200-year span. | |
16928 | ||
16929 | If you add a (positive) number of days to a date form that falls on a | |
16930 | weekend or holiday, the date form is treated as if it were the most | |
16931 | recent business day. (Thus adding one business day to a Friday, | |
16932 | Saturday, or Sunday will all yield the following Monday.) If you | |
16933 | subtract a number of days from a weekend or holiday, the date is | |
16934 | effectively on the following business day. (So subtracting one business | |
16935 | day from Saturday, Sunday, or Monday yields the preceding Friday.) The | |
16936 | difference between two dates one or both of which fall on holidays | |
16937 | equals the number of actual business days between them. These | |
16938 | conventions are consistent in the sense that, if you add @var{n} | |
16939 | business days to any date, the difference between the result and the | |
16940 | original date will come out to @var{n} business days. (It can't be | |
16941 | completely consistent though; a subtraction followed by an addition | |
16942 | might come out a bit differently, since @kbd{t +} is incapable of | |
16943 | producing a date that falls on a weekend or holiday.) | |
16944 | ||
5d67986c RS |
16945 | @ignore |
16946 | @starindex | |
16947 | @end ignore | |
d7b8e6c6 EZ |
16948 | @tindex holiday |
16949 | There is a @code{holiday} function, not on any keys, that takes | |
16950 | any date form and returns 1 if that date falls on a weekend or | |
16951 | holiday, as defined in @code{Holidays}, or 0 if the date is a | |
16952 | business day. | |
16953 | ||
16954 | @node Time Zones, , Business Days, Date Arithmetic | |
16955 | @subsection Time Zones | |
16956 | ||
16957 | @noindent | |
16958 | @cindex Time zones | |
16959 | @cindex Daylight savings time | |
16960 | Time zones and daylight savings time are a complicated business. | |
16961 | The conversions to and from Julian and Unix-style dates automatically | |
16962 | compute the correct time zone and daylight savings adjustment to use, | |
16963 | provided they can figure out this information. This section describes | |
16964 | Calc's time zone adjustment algorithm in detail, in case you want to | |
16965 | do conversions in different time zones or in case Calc's algorithms | |
16966 | can't determine the right correction to use. | |
16967 | ||
16968 | Adjustments for time zones and daylight savings time are done by | |
16969 | @kbd{t U}, @kbd{t J}, @kbd{t N}, and @kbd{t C}, but not by any other | |
16970 | commands. In particular, @samp{<may 1 1991> - <apr 1 1991>} evaluates | |
16971 | to exactly 30 days even though there is a daylight-savings | |
16972 | transition in between. This is also true for Julian pure dates: | |
16973 | @samp{julian(<may 1 1991>) - julian(<apr 1 1991>)}. But Julian | |
16974 | and Unix date/times will adjust for daylight savings time: | |
16975 | @samp{julian(<12am may 1 1991>) - julian(<12am apr 1 1991>)} | |
16976 | evaluates to @samp{29.95834} (that's 29 days and 23 hours) | |
16977 | because one hour was lost when daylight savings commenced on | |
16978 | April 7, 1991. | |
16979 | ||
16980 | In brief, the idiom @samp{julian(@var{date1}) - julian(@var{date2})} | |
16981 | computes the actual number of 24-hour periods between two dates, whereas | |
16982 | @samp{@var{date1} - @var{date2}} computes the number of calendar | |
16983 | days between two dates without taking daylight savings into account. | |
16984 | ||
16985 | @pindex calc-time-zone | |
5d67986c RS |
16986 | @ignore |
16987 | @starindex | |
16988 | @end ignore | |
d7b8e6c6 EZ |
16989 | @tindex tzone |
16990 | The @code{calc-time-zone} [@code{tzone}] command converts the time | |
16991 | zone specified by its numeric prefix argument into a number of | |
16992 | seconds difference from Greenwich mean time (GMT). If the argument | |
16993 | is a number, the result is simply that value multiplied by 3600. | |
16994 | Typical arguments for North America are 5 (Eastern) or 8 (Pacific). If | |
16995 | Daylight Savings time is in effect, one hour should be subtracted from | |
16996 | the normal difference. | |
16997 | ||
16998 | If you give a prefix of plain @kbd{C-u}, @code{calc-time-zone} (like other | |
16999 | date arithmetic commands that include a time zone argument) takes the | |
17000 | zone argument from the top of the stack. (In the case of @kbd{t J} | |
17001 | and @kbd{t U}, the normal argument is then taken from the second-to-top | |
17002 | stack position.) This allows you to give a non-integer time zone | |
17003 | adjustment. The time-zone argument can also be an HMS form, or | |
17004 | it can be a variable which is a time zone name in upper- or lower-case. | |
17005 | For example @samp{tzone(PST) = tzone(8)} and @samp{tzone(pdt) = tzone(7)} | |
17006 | (for Pacific standard and daylight savings times, respectively). | |
17007 | ||
17008 | North American and European time zone names are defined as follows; | |
17009 | note that for each time zone there is one name for standard time, | |
17010 | another for daylight savings time, and a third for ``generalized'' time | |
17011 | in which the daylight savings adjustment is computed from context. | |
17012 | ||
d7b8e6c6 | 17013 | @smallexample |
5d67986c | 17014 | @group |
d7b8e6c6 EZ |
17015 | YST PST MST CST EST AST NST GMT WET MET MEZ |
17016 | 9 8 7 6 5 4 3.5 0 -1 -2 -2 | |
17017 | ||
17018 | YDT PDT MDT CDT EDT ADT NDT BST WETDST METDST MESZ | |
17019 | 8 7 6 5 4 3 2.5 -1 -2 -3 -3 | |
17020 | ||
17021 | YGT PGT MGT CGT EGT AGT NGT BGT WEGT MEGT MEGZ | |
17022 | 9/8 8/7 7/6 6/5 5/4 4/3 3.5/2.5 0/-1 -1/-2 -2/-3 -2/-3 | |
d7b8e6c6 | 17023 | @end group |
5d67986c | 17024 | @end smallexample |
d7b8e6c6 EZ |
17025 | |
17026 | @vindex math-tzone-names | |
17027 | To define time zone names that do not appear in the above table, | |
17028 | you must modify the Lisp variable @code{math-tzone-names}. This | |
17029 | is a list of lists describing the different time zone names; its | |
17030 | structure is best explained by an example. The three entries for | |
17031 | Pacific Time look like this: | |
17032 | ||
d7b8e6c6 | 17033 | @smallexample |
5d67986c | 17034 | @group |
d7b8e6c6 EZ |
17035 | ( ( "PST" 8 0 ) ; Name as an upper-case string, then standard |
17036 | ( "PDT" 8 -1 ) ; adjustment, then daylight savings adjustment. | |
17037 | ( "PGT" 8 "PST" "PDT" ) ) ; Generalized time zone. | |
d7b8e6c6 | 17038 | @end group |
5d67986c | 17039 | @end smallexample |
d7b8e6c6 EZ |
17040 | |
17041 | @cindex @code{TimeZone} variable | |
17042 | @vindex TimeZone | |
17043 | With no arguments, @code{calc-time-zone} or @samp{tzone()} obtains an | |
17044 | argument from the Calc variable @code{TimeZone} if a value has been | |
17045 | stored for that variable. If not, Calc runs the Unix @samp{date} | |
17046 | command and looks for one of the above time zone names in the output; | |
17047 | if this does not succeed, @samp{tzone()} leaves itself unevaluated. | |
17048 | The time zone name in the @samp{date} output may be followed by a signed | |
17049 | adjustment, e.g., @samp{GMT+5} or @samp{GMT+0500} which specifies a | |
17050 | number of hours and minutes to be added to the base time zone. | |
17051 | Calc stores the time zone it finds into @code{TimeZone} to speed | |
17052 | later calls to @samp{tzone()}. | |
17053 | ||
17054 | The special time zone name @code{local} is equivalent to no argument, | |
17055 | i.e., it uses the local time zone as obtained from the @code{date} | |
17056 | command. | |
17057 | ||
17058 | If the time zone name found is one of the standard or daylight | |
17059 | savings zone names from the above table, and Calc's internal | |
17060 | daylight savings algorithm says that time and zone are consistent | |
17061 | (e.g., @code{PDT} accompanies a date that Calc's algorithm would also | |
17062 | consider to be daylight savings, or @code{PST} accompanies a date | |
17063 | that Calc would consider to be standard time), then Calc substitutes | |
17064 | the corresponding generalized time zone (like @code{PGT}). | |
17065 | ||
17066 | If your system does not have a suitable @samp{date} command, you | |
17067 | may wish to put a @samp{(setq var-TimeZone ...)} in your Emacs | |
17068 | initialization file to set the time zone. The easiest way to do | |
17069 | this is to edit the @code{TimeZone} variable using Calc's @kbd{s T} | |
17070 | command, then use the @kbd{s p} (@code{calc-permanent-variable}) | |
17071 | command to save the value of @code{TimeZone} permanently. | |
17072 | ||
17073 | The @kbd{t J} and @code{t U} commands with no numeric prefix | |
17074 | arguments do the same thing as @samp{tzone()}. If the current | |
17075 | time zone is a generalized time zone, e.g., @code{EGT}, Calc | |
17076 | examines the date being converted to tell whether to use standard | |
17077 | or daylight savings time. But if the current time zone is explicit, | |
17078 | e.g., @code{EST} or @code{EDT}, then that adjustment is used exactly | |
17079 | and Calc's daylight savings algorithm is not consulted. | |
17080 | ||
17081 | Some places don't follow the usual rules for daylight savings time. | |
17082 | The state of Arizona, for example, does not observe daylight savings | |
17083 | time. If you run Calc during the winter season in Arizona, the | |
17084 | Unix @code{date} command will report @code{MST} time zone, which | |
17085 | Calc will change to @code{MGT}. If you then convert a time that | |
17086 | lies in the summer months, Calc will apply an incorrect daylight | |
17087 | savings time adjustment. To avoid this, set your @code{TimeZone} | |
17088 | variable explicitly to @code{MST} to force the use of standard, | |
17089 | non-daylight-savings time. | |
17090 | ||
17091 | @vindex math-daylight-savings-hook | |
17092 | @findex math-std-daylight-savings | |
17093 | By default Calc always considers daylight savings time to begin at | |
17094 | 2 a.m.@: on the first Sunday of April, and to end at 2 a.m.@: on the | |
17095 | last Sunday of October. This is the rule that has been in effect | |
17096 | in North America since 1987. If you are in a country that uses | |
17097 | different rules for computing daylight savings time, you have two | |
17098 | choices: Write your own daylight savings hook, or control time | |
17099 | zones explicitly by setting the @code{TimeZone} variable and/or | |
17100 | always giving a time-zone argument for the conversion functions. | |
17101 | ||
17102 | The Lisp variable @code{math-daylight-savings-hook} holds the | |
17103 | name of a function that is used to compute the daylight savings | |
17104 | adjustment for a given date. The default is | |
17105 | @code{math-std-daylight-savings}, which computes an adjustment | |
17106 | (either 0 or @i{-1}) using the North American rules given above. | |
17107 | ||
17108 | The daylight savings hook function is called with four arguments: | |
17109 | The date, as a floating-point number in standard Calc format; | |
17110 | a six-element list of the date decomposed into year, month, day, | |
17111 | hour, minute, and second, respectively; a string which contains | |
17112 | the generalized time zone name in upper-case, e.g., @code{"WEGT"}; | |
17113 | and a special adjustment to be applied to the hour value when | |
17114 | converting into a generalized time zone (see below). | |
17115 | ||
17116 | @findex math-prev-weekday-in-month | |
17117 | The Lisp function @code{math-prev-weekday-in-month} is useful for | |
17118 | daylight savings computations. This is an internal version of | |
17119 | the user-level @code{pwday} function described in the previous | |
17120 | section. It takes four arguments: The floating-point date value, | |
17121 | the corresponding six-element date list, the day-of-month number, | |
17122 | and the weekday number (0-6). | |
17123 | ||
17124 | The default daylight savings hook ignores the time zone name, but a | |
17125 | more sophisticated hook could use different algorithms for different | |
17126 | time zones. It would also be possible to use different algorithms | |
17127 | depending on the year number, but the default hook always uses the | |
17128 | algorithm for 1987 and later. Here is a listing of the default | |
17129 | daylight savings hook: | |
17130 | ||
17131 | @smallexample | |
17132 | (defun math-std-daylight-savings (date dt zone bump) | |
17133 | (cond ((< (nth 1 dt) 4) 0) | |
17134 | ((= (nth 1 dt) 4) | |
17135 | (let ((sunday (math-prev-weekday-in-month date dt 7 0))) | |
17136 | (cond ((< (nth 2 dt) sunday) 0) | |
17137 | ((= (nth 2 dt) sunday) | |
17138 | (if (>= (nth 3 dt) (+ 3 bump)) -1 0)) | |
17139 | (t -1)))) | |
17140 | ((< (nth 1 dt) 10) -1) | |
17141 | ((= (nth 1 dt) 10) | |
17142 | (let ((sunday (math-prev-weekday-in-month date dt 31 0))) | |
17143 | (cond ((< (nth 2 dt) sunday) -1) | |
17144 | ((= (nth 2 dt) sunday) | |
17145 | (if (>= (nth 3 dt) (+ 2 bump)) 0 -1)) | |
17146 | (t 0)))) | |
17147 | (t 0)) | |
17148 | ) | |
17149 | @end smallexample | |
17150 | ||
17151 | @noindent | |
17152 | The @code{bump} parameter is equal to zero when Calc is converting | |
17153 | from a date form in a generalized time zone into a GMT date value. | |
17154 | It is @i{-1} when Calc is converting in the other direction. The | |
17155 | adjustments shown above ensure that the conversion behaves correctly | |
17156 | and reasonably around the 2 a.m.@: transition in each direction. | |
17157 | ||
17158 | There is a ``missing'' hour between 2 a.m.@: and 3 a.m.@: at the | |
17159 | beginning of daylight savings time; converting a date/time form that | |
17160 | falls in this hour results in a time value for the following hour, | |
17161 | from 3 a.m.@: to 4 a.m. At the end of daylight savings time, the | |
17162 | hour from 1 a.m.@: to 2 a.m.@: repeats itself; converting a date/time | |
17163 | form that falls in in this hour results in a time value for the first | |
28665d46 | 17164 | manifestation of that time (@emph{not} the one that occurs one hour later). |
d7b8e6c6 EZ |
17165 | |
17166 | If @code{math-daylight-savings-hook} is @code{nil}, then the | |
17167 | daylight savings adjustment is always taken to be zero. | |
17168 | ||
17169 | In algebraic formulas, @samp{tzone(@var{zone}, @var{date})} | |
17170 | computes the time zone adjustment for a given zone name at a | |
17171 | given date. The @var{date} is ignored unless @var{zone} is a | |
17172 | generalized time zone. If @var{date} is a date form, the | |
17173 | daylight savings computation is applied to it as it appears. | |
17174 | If @var{date} is a numeric date value, it is adjusted for the | |
17175 | daylight-savings version of @var{zone} before being given to | |
17176 | the daylight savings hook. This odd-sounding rule ensures | |
17177 | that the daylight-savings computation is always done in | |
17178 | local time, not in the GMT time that a numeric @var{date} | |
17179 | is typically represented in. | |
17180 | ||
5d67986c RS |
17181 | @ignore |
17182 | @starindex | |
17183 | @end ignore | |
d7b8e6c6 EZ |
17184 | @tindex dsadj |
17185 | The @samp{dsadj(@var{date}, @var{zone})} function computes the | |
17186 | daylight savings adjustment that is appropriate for @var{date} in | |
17187 | time zone @var{zone}. If @var{zone} is explicitly in or not in | |
17188 | daylight savings time (e.g., @code{PDT} or @code{PST}) the | |
17189 | @var{date} is ignored. If @var{zone} is a generalized time zone, | |
17190 | the algorithms described above are used. If @var{zone} is omitted, | |
17191 | the computation is done for the current time zone. | |
17192 | ||
17193 | @xref{Reporting Bugs}, for the address of Calc's author, if you | |
17194 | should wish to contribute your improved versions of | |
17195 | @code{math-tzone-names} and @code{math-daylight-savings-hook} | |
17196 | to the Calc distribution. | |
17197 | ||
17198 | @node Financial Functions, Binary Functions, Date Arithmetic, Arithmetic | |
17199 | @section Financial Functions | |
17200 | ||
17201 | @noindent | |
17202 | Calc's financial or business functions use the @kbd{b} prefix | |
17203 | key followed by a shifted letter. (The @kbd{b} prefix followed by | |
17204 | a lower-case letter is used for operations on binary numbers.) | |
17205 | ||
17206 | Note that the rate and the number of intervals given to these | |
17207 | functions must be on the same time scale, e.g., both months or | |
17208 | both years. Mixing an annual interest rate with a time expressed | |
17209 | in months will give you very wrong answers! | |
17210 | ||
17211 | It is wise to compute these functions to a higher precision than | |
17212 | you really need, just to make sure your answer is correct to the | |
17213 | last penny; also, you may wish to check the definitions at the end | |
17214 | of this section to make sure the functions have the meaning you expect. | |
17215 | ||
17216 | @menu | |
17217 | * Percentages:: | |
17218 | * Future Value:: | |
17219 | * Present Value:: | |
17220 | * Related Financial Functions:: | |
17221 | * Depreciation Functions:: | |
17222 | * Definitions of Financial Functions:: | |
17223 | @end menu | |
17224 | ||
17225 | @node Percentages, Future Value, Financial Functions, Financial Functions | |
17226 | @subsection Percentages | |
17227 | ||
17228 | @kindex M-% | |
17229 | @pindex calc-percent | |
17230 | @tindex % | |
17231 | @tindex percent | |
17232 | The @kbd{M-%} (@code{calc-percent}) command takes a percentage value, | |
17233 | say 5.4, and converts it to an equivalent actual number. For example, | |
17234 | @kbd{5.4 M-%} enters 0.054 on the stack. (That's the @key{META} or | |
17235 | @key{ESC} key combined with @kbd{%}.) | |
17236 | ||
17237 | Actually, @kbd{M-%} creates a formula of the form @samp{5.4%}. | |
17238 | You can enter @samp{5.4%} yourself during algebraic entry. The | |
17239 | @samp{%} operator simply means, ``the preceding value divided by | |
17240 | 100.'' The @samp{%} operator has very high precedence, so that | |
17241 | @samp{1+8%} is interpreted as @samp{1+(8%)}, not as @samp{(1+8)%}. | |
17242 | (The @samp{%} operator is just a postfix notation for the | |
17243 | @code{percent} function, just like @samp{20!} is the notation for | |
17244 | @samp{fact(20)}, or twenty-factorial.) | |
17245 | ||
17246 | The formula @samp{5.4%} would normally evaluate immediately to | |
17247 | 0.054, but the @kbd{M-%} command suppresses evaluation as it puts | |
17248 | the formula onto the stack. However, the next Calc command that | |
17249 | uses the formula @samp{5.4%} will evaluate it as its first step. | |
17250 | The net effect is that you get to look at @samp{5.4%} on the stack, | |
17251 | but Calc commands see it as @samp{0.054}, which is what they expect. | |
17252 | ||
17253 | In particular, @samp{5.4%} and @samp{0.054} are suitable values | |
17254 | for the @var{rate} arguments of the various financial functions, | |
17255 | but the number @samp{5.4} is probably @emph{not} suitable---it | |
17256 | represents a rate of 540 percent! | |
17257 | ||
17258 | The key sequence @kbd{M-% *} effectively means ``percent-of.'' | |
5d67986c | 17259 | For example, @kbd{68 @key{RET} 25 M-% *} computes 17, which is 25% of |
d7b8e6c6 EZ |
17260 | 68 (and also 68% of 25, which comes out to the same thing). |
17261 | ||
17262 | @kindex c % | |
17263 | @pindex calc-convert-percent | |
17264 | The @kbd{c %} (@code{calc-convert-percent}) command converts the | |
17265 | value on the top of the stack from numeric to percentage form. | |
17266 | For example, if 0.08 is on the stack, @kbd{c %} converts it to | |
17267 | @samp{8%}. The quantity is the same, it's just represented | |
17268 | differently. (Contrast this with @kbd{M-%}, which would convert | |
17269 | this number to @samp{0.08%}.) The @kbd{=} key is a convenient way | |
17270 | to convert a formula like @samp{8%} back to numeric form, 0.08. | |
17271 | ||
17272 | To compute what percentage one quantity is of another quantity, | |
5d67986c | 17273 | use @kbd{/ c %}. For example, @w{@kbd{17 @key{RET} 68 / c %}} displays |
d7b8e6c6 EZ |
17274 | @samp{25%}. |
17275 | ||
17276 | @kindex b % | |
17277 | @pindex calc-percent-change | |
17278 | @tindex relch | |
17279 | The @kbd{b %} (@code{calc-percent-change}) [@code{relch}] command | |
17280 | calculates the percentage change from one number to another. | |
5d67986c | 17281 | For example, @kbd{40 @key{RET} 50 b %} produces the answer @samp{25%}, |
d7b8e6c6 | 17282 | since 50 is 25% larger than 40. A negative result represents a |
5d67986c | 17283 | decrease: @kbd{50 @key{RET} 40 b %} produces @samp{-20%}, since 40 is |
d7b8e6c6 EZ |
17284 | 20% smaller than 50. (The answers are different in magnitude |
17285 | because, in the first case, we're increasing by 25% of 40, but | |
17286 | in the second case, we're decreasing by 20% of 50.) The effect | |
5d67986c | 17287 | of @kbd{40 @key{RET} 50 b %} is to compute @cite{(50-40)/40}, converting |
d7b8e6c6 EZ |
17288 | the answer to percentage form as if by @kbd{c %}. |
17289 | ||
17290 | @node Future Value, Present Value, Percentages, Financial Functions | |
17291 | @subsection Future Value | |
17292 | ||
17293 | @noindent | |
17294 | @kindex b F | |
17295 | @pindex calc-fin-fv | |
17296 | @tindex fv | |
17297 | The @kbd{b F} (@code{calc-fin-fv}) [@code{fv}] command computes | |
17298 | the future value of an investment. It takes three arguments | |
17299 | from the stack: @samp{fv(@var{rate}, @var{n}, @var{payment})}. | |
17300 | If you give payments of @var{payment} every year for @var{n} | |
17301 | years, and the money you have paid earns interest at @var{rate} per | |
17302 | year, then this function tells you what your investment would be | |
17303 | worth at the end of the period. (The actual interval doesn't | |
17304 | have to be years, as long as @var{n} and @var{rate} are expressed | |
17305 | in terms of the same intervals.) This function assumes payments | |
17306 | occur at the @emph{end} of each interval. | |
17307 | ||
17308 | @kindex I b F | |
17309 | @tindex fvb | |
17310 | The @kbd{I b F} [@code{fvb}] command does the same computation, | |
17311 | but assuming your payments are at the beginning of each interval. | |
17312 | Suppose you plan to deposit $1000 per year in a savings account | |
17313 | earning 5.4% interest, starting right now. How much will be | |
17314 | in the account after five years? @code{fvb(5.4%, 5, 1000) = 5870.73}. | |
17315 | Thus you will have earned $870 worth of interest over the years. | |
17316 | Using the stack, this calculation would have been | |
5d67986c | 17317 | @kbd{5.4 M-% 5 @key{RET} 1000 I b F}. Note that the rate is expressed |
d7b8e6c6 EZ |
17318 | as a number between 0 and 1, @emph{not} as a percentage. |
17319 | ||
17320 | @kindex H b F | |
17321 | @tindex fvl | |
17322 | The @kbd{H b F} [@code{fvl}] command computes the future value | |
17323 | of an initial lump sum investment. Suppose you could deposit | |
17324 | those five thousand dollars in the bank right now; how much would | |
17325 | they be worth in five years? @code{fvl(5.4%, 5, 5000) = 6503.89}. | |
17326 | ||
17327 | The algebraic functions @code{fv} and @code{fvb} accept an optional | |
17328 | fourth argument, which is used as an initial lump sum in the sense | |
17329 | of @code{fvl}. In other words, @code{fv(@var{rate}, @var{n}, | |
17330 | @var{payment}, @var{initial}) = fv(@var{rate}, @var{n}, @var{payment}) | |
17331 | + fvl(@var{rate}, @var{n}, @var{initial})}.@refill | |
17332 | ||
17333 | To illustrate the relationships between these functions, we could | |
17334 | do the @code{fvb} calculation ``by hand'' using @code{fvl}. The | |
17335 | final balance will be the sum of the contributions of our five | |
17336 | deposits at various times. The first deposit earns interest for | |
17337 | five years: @code{fvl(5.4%, 5, 1000) = 1300.78}. The second | |
17338 | deposit only earns interest for four years: @code{fvl(5.4%, 4, 1000) = | |
17339 | 1234.13}. And so on down to the last deposit, which earns one | |
17340 | year's interest: @code{fvl(5.4%, 1, 1000) = 1054.00}. The sum of | |
17341 | these five values is, sure enough, $5870.73, just as was computed | |
17342 | by @code{fvb} directly. | |
17343 | ||
17344 | What does @code{fv(5.4%, 5, 1000) = 5569.96} mean? The payments | |
17345 | are now at the ends of the periods. The end of one year is the same | |
17346 | as the beginning of the next, so what this really means is that we've | |
17347 | lost the payment at year zero (which contributed $1300.78), but we're | |
17348 | now counting the payment at year five (which, since it didn't have | |
17349 | a chance to earn interest, counts as $1000). Indeed, @cite{5569.96 = | |
17350 | 5870.73 - 1300.78 + 1000} (give or take a bit of roundoff error). | |
17351 | ||
17352 | @node Present Value, Related Financial Functions, Future Value, Financial Functions | |
17353 | @subsection Present Value | |
17354 | ||
17355 | @noindent | |
17356 | @kindex b P | |
17357 | @pindex calc-fin-pv | |
17358 | @tindex pv | |
17359 | The @kbd{b P} (@code{calc-fin-pv}) [@code{pv}] command computes | |
17360 | the present value of an investment. Like @code{fv}, it takes | |
17361 | three arguments: @code{pv(@var{rate}, @var{n}, @var{payment})}. | |
17362 | It computes the present value of a series of regular payments. | |
17363 | Suppose you have the chance to make an investment that will | |
17364 | pay $2000 per year over the next four years; as you receive | |
17365 | these payments you can put them in the bank at 9% interest. | |
17366 | You want to know whether it is better to make the investment, or | |
17367 | to keep the money in the bank where it earns 9% interest right | |
17368 | from the start. The calculation @code{pv(9%, 4, 2000)} gives the | |
17369 | result 6479.44. If your initial investment must be less than this, | |
17370 | say, $6000, then the investment is worthwhile. But if you had to | |
17371 | put up $7000, then it would be better just to leave it in the bank. | |
17372 | ||
17373 | Here is the interpretation of the result of @code{pv}: You are | |
17374 | trying to compare the return from the investment you are | |
17375 | considering, which is @code{fv(9%, 4, 2000) = 9146.26}, with | |
17376 | the return from leaving the money in the bank, which is | |
17377 | @code{fvl(9%, 4, @var{x})} where @var{x} is the amount of money | |
17378 | you would have to put up in advance. The @code{pv} function | |
17379 | finds the break-even point, @cite{x = 6479.44}, at which | |
17380 | @code{fvl(9%, 4, 6479.44)} is also equal to 9146.26. This is | |
17381 | the largest amount you should be willing to invest. | |
17382 | ||
17383 | @kindex I b P | |
17384 | @tindex pvb | |
17385 | The @kbd{I b P} [@code{pvb}] command solves the same problem, | |
17386 | but with payments occurring at the beginning of each interval. | |
17387 | It has the same relationship to @code{fvb} as @code{pv} has | |
17388 | to @code{fv}. For example @code{pvb(9%, 4, 2000) = 7062.59}, | |
17389 | a larger number than @code{pv} produced because we get to start | |
17390 | earning interest on the return from our investment sooner. | |
17391 | ||
17392 | @kindex H b P | |
17393 | @tindex pvl | |
17394 | The @kbd{H b P} [@code{pvl}] command computes the present value of | |
17395 | an investment that will pay off in one lump sum at the end of the | |
17396 | period. For example, if we get our $8000 all at the end of the | |
17397 | four years, @code{pvl(9%, 4, 8000) = 5667.40}. This is much | |
17398 | less than @code{pv} reported, because we don't earn any interest | |
17399 | on the return from this investment. Note that @code{pvl} and | |
17400 | @code{fvl} are simple inverses: @code{fvl(9%, 4, 5667.40) = 8000}. | |
17401 | ||
17402 | You can give an optional fourth lump-sum argument to @code{pv} | |
17403 | and @code{pvb}; this is handled in exactly the same way as the | |
17404 | fourth argument for @code{fv} and @code{fvb}. | |
17405 | ||
17406 | @kindex b N | |
17407 | @pindex calc-fin-npv | |
17408 | @tindex npv | |
17409 | The @kbd{b N} (@code{calc-fin-npv}) [@code{npv}] command computes | |
17410 | the net present value of a series of irregular investments. | |
17411 | The first argument is the interest rate. The second argument is | |
17412 | a vector which represents the expected return from the investment | |
17413 | at the end of each interval. For example, if the rate represents | |
17414 | a yearly interest rate, then the vector elements are the return | |
17415 | from the first year, second year, and so on. | |
17416 | ||
17417 | Thus, @code{npv(9%, [2000,2000,2000,2000]) = pv(9%, 4, 2000) = 6479.44}. | |
17418 | Obviously this function is more interesting when the payments are | |
17419 | not all the same! | |
17420 | ||
17421 | The @code{npv} function can actually have two or more arguments. | |
17422 | Multiple arguments are interpreted in the same way as for the | |
17423 | vector statistical functions like @code{vsum}. | |
17424 | @xref{Single-Variable Statistics}. Basically, if there are several | |
17425 | payment arguments, each either a vector or a plain number, all these | |
17426 | values are collected left-to-right into the complete list of payments. | |
17427 | A numeric prefix argument on the @kbd{b N} command says how many | |
17428 | payment values or vectors to take from the stack.@refill | |
17429 | ||
17430 | @kindex I b N | |
17431 | @tindex npvb | |
17432 | The @kbd{I b N} [@code{npvb}] command computes the net present | |
17433 | value where payments occur at the beginning of each interval | |
17434 | rather than at the end. | |
17435 | ||
17436 | @node Related Financial Functions, Depreciation Functions, Present Value, Financial Functions | |
17437 | @subsection Related Financial Functions | |
17438 | ||
17439 | @noindent | |
17440 | The functions in this section are basically inverses of the | |
17441 | present value functions with respect to the various arguments. | |
17442 | ||
17443 | @kindex b M | |
17444 | @pindex calc-fin-pmt | |
17445 | @tindex pmt | |
17446 | The @kbd{b M} (@code{calc-fin-pmt}) [@code{pmt}] command computes | |
17447 | the amount of periodic payment necessary to amortize a loan. | |
17448 | Thus @code{pmt(@var{rate}, @var{n}, @var{amount})} equals the | |
17449 | value of @var{payment} such that @code{pv(@var{rate}, @var{n}, | |
17450 | @var{payment}) = @var{amount}}.@refill | |
17451 | ||
17452 | @kindex I b M | |
17453 | @tindex pmtb | |
17454 | The @kbd{I b M} [@code{pmtb}] command does the same computation | |
17455 | but using @code{pvb} instead of @code{pv}. Like @code{pv} and | |
17456 | @code{pvb}, these functions can also take a fourth argument which | |
17457 | represents an initial lump-sum investment. | |
17458 | ||
17459 | @kindex H b M | |
17460 | The @kbd{H b M} key just invokes the @code{fvl} function, which is | |
17461 | the inverse of @code{pvl}. There is no explicit @code{pmtl} function. | |
17462 | ||
17463 | @kindex b # | |
17464 | @pindex calc-fin-nper | |
17465 | @tindex nper | |
17466 | The @kbd{b #} (@code{calc-fin-nper}) [@code{nper}] command computes | |
17467 | the number of regular payments necessary to amortize a loan. | |
17468 | Thus @code{nper(@var{rate}, @var{payment}, @var{amount})} equals | |
17469 | the value of @var{n} such that @code{pv(@var{rate}, @var{n}, | |
17470 | @var{payment}) = @var{amount}}. If @var{payment} is too small | |
17471 | ever to amortize a loan for @var{amount} at interest rate @var{rate}, | |
17472 | the @code{nper} function is left in symbolic form.@refill | |
17473 | ||
17474 | @kindex I b # | |
17475 | @tindex nperb | |
17476 | The @kbd{I b #} [@code{nperb}] command does the same computation | |
17477 | but using @code{pvb} instead of @code{pv}. You can give a fourth | |
17478 | lump-sum argument to these functions, but the computation will be | |
17479 | rather slow in the four-argument case.@refill | |
17480 | ||
17481 | @kindex H b # | |
17482 | @tindex nperl | |
17483 | The @kbd{H b #} [@code{nperl}] command does the same computation | |
17484 | using @code{pvl}. By exchanging @var{payment} and @var{amount} you | |
17485 | can also get the solution for @code{fvl}. For example, | |
17486 | @code{nperl(8%, 2000, 1000) = 9.006}, so if you place $1000 in a | |
17487 | bank account earning 8%, it will take nine years to grow to $2000.@refill | |
17488 | ||
17489 | @kindex b T | |
17490 | @pindex calc-fin-rate | |
17491 | @tindex rate | |
17492 | The @kbd{b T} (@code{calc-fin-rate}) [@code{rate}] command computes | |
17493 | the rate of return on an investment. This is also an inverse of @code{pv}: | |
17494 | @code{rate(@var{n}, @var{payment}, @var{amount})} computes the value of | |
17495 | @var{rate} such that @code{pv(@var{rate}, @var{n}, @var{payment}) = | |
17496 | @var{amount}}. The result is expressed as a formula like @samp{6.3%}.@refill | |
17497 | ||
17498 | @kindex I b T | |
17499 | @kindex H b T | |
17500 | @tindex rateb | |
17501 | @tindex ratel | |
17502 | The @kbd{I b T} [@code{rateb}] and @kbd{H b T} [@code{ratel}] | |
17503 | commands solve the analogous equations with @code{pvb} or @code{pvl} | |
17504 | in place of @code{pv}. Also, @code{rate} and @code{rateb} can | |
17505 | accept an optional fourth argument just like @code{pv} and @code{pvb}. | |
17506 | To redo the above example from a different perspective, | |
17507 | @code{ratel(9, 2000, 1000) = 8.00597%}, which says you will need an | |
17508 | interest rate of 8% in order to double your account in nine years.@refill | |
17509 | ||
17510 | @kindex b I | |
17511 | @pindex calc-fin-irr | |
17512 | @tindex irr | |
17513 | The @kbd{b I} (@code{calc-fin-irr}) [@code{irr}] command is the | |
17514 | analogous function to @code{rate} but for net present value. | |
17515 | Its argument is a vector of payments. Thus @code{irr(@var{payments})} | |
17516 | computes the @var{rate} such that @code{npv(@var{rate}, @var{payments}) = 0}; | |
17517 | this rate is known as the @dfn{internal rate of return}. | |
17518 | ||
17519 | @kindex I b I | |
17520 | @tindex irrb | |
17521 | The @kbd{I b I} [@code{irrb}] command computes the internal rate of | |
17522 | return assuming payments occur at the beginning of each period. | |
17523 | ||
17524 | @node Depreciation Functions, Definitions of Financial Functions, Related Financial Functions, Financial Functions | |
17525 | @subsection Depreciation Functions | |
17526 | ||
17527 | @noindent | |
17528 | The functions in this section calculate @dfn{depreciation}, which is | |
17529 | the amount of value that a possession loses over time. These functions | |
17530 | are characterized by three parameters: @var{cost}, the original cost | |
17531 | of the asset; @var{salvage}, the value the asset will have at the end | |
17532 | of its expected ``useful life''; and @var{life}, the number of years | |
17533 | (or other periods) of the expected useful life. | |
17534 | ||
17535 | There are several methods for calculating depreciation that differ in | |
17536 | the way they spread the depreciation over the lifetime of the asset. | |
17537 | ||
17538 | @kindex b S | |
17539 | @pindex calc-fin-sln | |
17540 | @tindex sln | |
17541 | The @kbd{b S} (@code{calc-fin-sln}) [@code{sln}] command computes the | |
17542 | ``straight-line'' depreciation. In this method, the asset depreciates | |
17543 | by the same amount every year (or period). For example, | |
17544 | @samp{sln(12000, 2000, 5)} returns 2000. The asset costs $12000 | |
17545 | initially and will be worth $2000 after five years; it loses $2000 | |
17546 | per year. | |
17547 | ||
17548 | @kindex b Y | |
17549 | @pindex calc-fin-syd | |
17550 | @tindex syd | |
17551 | The @kbd{b Y} (@code{calc-fin-syd}) [@code{syd}] command computes the | |
17552 | accelerated ``sum-of-years'-digits'' depreciation. Here the depreciation | |
17553 | is higher during the early years of the asset's life. Since the | |
17554 | depreciation is different each year, @kbd{b Y} takes a fourth @var{period} | |
17555 | parameter which specifies which year is requested, from 1 to @var{life}. | |
17556 | If @var{period} is outside this range, the @code{syd} function will | |
17557 | return zero. | |
17558 | ||
17559 | @kindex b D | |
17560 | @pindex calc-fin-ddb | |
17561 | @tindex ddb | |
17562 | The @kbd{b D} (@code{calc-fin-ddb}) [@code{ddb}] command computes an | |
17563 | accelerated depreciation using the double-declining balance method. | |
17564 | It also takes a fourth @var{period} parameter. | |
17565 | ||
17566 | For symmetry, the @code{sln} function will accept a @var{period} | |
17567 | parameter as well, although it will ignore its value except that the | |
17568 | return value will as usual be zero if @var{period} is out of range. | |
17569 | ||
17570 | For example, pushing the vector @cite{[1,2,3,4,5]} (perhaps with @kbd{v x 5}) | |
17571 | and then mapping @kbd{V M ' [sln(12000,2000,5,$), syd(12000,2000,5,$), | |
5d67986c | 17572 | ddb(12000,2000,5,$)] @key{RET}} produces a matrix that allows us to compare |
d7b8e6c6 EZ |
17573 | the three depreciation methods: |
17574 | ||
d7b8e6c6 | 17575 | @example |
5d67986c | 17576 | @group |
d7b8e6c6 EZ |
17577 | [ [ 2000, 3333, 4800 ] |
17578 | [ 2000, 2667, 2880 ] | |
17579 | [ 2000, 2000, 1728 ] | |
17580 | [ 2000, 1333, 592 ] | |
17581 | [ 2000, 667, 0 ] ] | |
d7b8e6c6 | 17582 | @end group |
5d67986c | 17583 | @end example |
d7b8e6c6 EZ |
17584 | |
17585 | @noindent | |
17586 | (Values have been rounded to nearest integers in this figure.) | |
17587 | We see that @code{sln} depreciates by the same amount each year, | |
17588 | @kbd{syd} depreciates more at the beginning and less at the end, | |
17589 | and @kbd{ddb} weights the depreciation even more toward the beginning. | |
17590 | ||
17591 | Summing columns with @kbd{V R : +} yields @cite{[10000, 10000, 10000]}; | |
17592 | the total depreciation in any method is (by definition) the | |
17593 | difference between the cost and the salvage value. | |
17594 | ||
17595 | @node Definitions of Financial Functions, , Depreciation Functions, Financial Functions | |
17596 | @subsection Definitions | |
17597 | ||
17598 | @noindent | |
17599 | For your reference, here are the actual formulas used to compute | |
17600 | Calc's financial functions. | |
17601 | ||
17602 | Calc will not evaluate a financial function unless the @var{rate} or | |
17603 | @var{n} argument is known. However, @var{payment} or @var{amount} can | |
17604 | be a variable. Calc expands these functions according to the | |
17605 | formulas below for symbolic arguments only when you use the @kbd{a "} | |
17606 | (@code{calc-expand-formula}) command, or when taking derivatives or | |
17607 | integrals or solving equations involving the functions. | |
17608 | ||
17609 | @ifinfo | |
17610 | These formulas are shown using the conventions of ``Big'' display | |
17611 | mode (@kbd{d B}); for example, the formula for @code{fv} written | |
17612 | linearly is @samp{pmt * ((1 + rate)^n) - 1) / rate}. | |
17613 | ||
17614 | @example | |
17615 | n | |
17616 | (1 + rate) - 1 | |
17617 | fv(rate, n, pmt) = pmt * --------------- | |
17618 | rate | |
17619 | ||
17620 | n | |
17621 | ((1 + rate) - 1) (1 + rate) | |
17622 | fvb(rate, n, pmt) = pmt * ---------------------------- | |
17623 | rate | |
17624 | ||
17625 | n | |
17626 | fvl(rate, n, pmt) = pmt * (1 + rate) | |
17627 | ||
17628 | -n | |
177c0ea7 | 17629 | 1 - (1 + rate) |
d7b8e6c6 EZ |
17630 | pv(rate, n, pmt) = pmt * ---------------- |
17631 | rate | |
17632 | ||
17633 | -n | |
17634 | (1 - (1 + rate) ) (1 + rate) | |
17635 | pvb(rate, n, pmt) = pmt * ----------------------------- | |
17636 | rate | |
17637 | ||
17638 | -n | |
17639 | pvl(rate, n, pmt) = pmt * (1 + rate) | |
17640 | ||
17641 | -1 -2 -3 | |
17642 | npv(rate, [a, b, c]) = a*(1 + rate) + b*(1 + rate) + c*(1 + rate) | |
17643 | ||
17644 | -1 -2 | |
17645 | npvb(rate, [a, b, c]) = a + b*(1 + rate) + c*(1 + rate) | |
17646 | ||
17647 | -n | |
17648 | (amt - x * (1 + rate) ) * rate | |
17649 | pmt(rate, n, amt, x) = ------------------------------- | |
17650 | -n | |
17651 | 1 - (1 + rate) | |
17652 | ||
17653 | -n | |
17654 | (amt - x * (1 + rate) ) * rate | |
17655 | pmtb(rate, n, amt, x) = ------------------------------- | |
17656 | -n | |
17657 | (1 - (1 + rate) ) (1 + rate) | |
17658 | ||
17659 | amt * rate | |
17660 | nper(rate, pmt, amt) = - log(1 - ------------, 1 + rate) | |
17661 | pmt | |
17662 | ||
17663 | amt * rate | |
17664 | nperb(rate, pmt, amt) = - log(1 - ---------------, 1 + rate) | |
17665 | pmt * (1 + rate) | |
17666 | ||
17667 | amt | |
17668 | nperl(rate, pmt, amt) = - log(---, 1 + rate) | |
17669 | pmt | |
17670 | ||
17671 | 1/n | |
17672 | pmt | |
17673 | ratel(n, pmt, amt) = ------ - 1 | |
17674 | 1/n | |
17675 | amt | |
17676 | ||
17677 | cost - salv | |
17678 | sln(cost, salv, life) = ----------- | |
17679 | life | |
17680 | ||
17681 | (cost - salv) * (life - per + 1) | |
17682 | syd(cost, salv, life, per) = -------------------------------- | |
17683 | life * (life + 1) / 2 | |
17684 | ||
17685 | book * 2 | |
17686 | ddb(cost, salv, life, per) = --------, book = cost - depreciation so far | |
17687 | life | |
17688 | @end example | |
17689 | @end ifinfo | |
17690 | @tex | |
17691 | \turnoffactive | |
17692 | $$ \code{fv}(r, n, p) = p { (1 + r)^n - 1 \over r } $$ | |
17693 | $$ \code{fvb}(r, n, p) = p { ((1 + r)^n - 1) (1 + r) \over r } $$ | |
17694 | $$ \code{fvl}(r, n, p) = p (1 + r)^n $$ | |
17695 | $$ \code{pv}(r, n, p) = p { 1 - (1 + r)^{-n} \over r } $$ | |
17696 | $$ \code{pvb}(r, n, p) = p { (1 - (1 + r)^{-n}) (1 + r) \over r } $$ | |
17697 | $$ \code{pvl}(r, n, p) = p (1 + r)^{-n} $$ | |
17698 | $$ \code{npv}(r, [a,b,c]) = a (1 + r)^{-1} + b (1 + r)^{-2} + c (1 + r)^{-3} $$ | |
17699 | $$ \code{npvb}(r, [a,b,c]) = a + b (1 + r)^{-1} + c (1 + r)^{-2} $$ | |
17700 | $$ \code{pmt}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over 1 - (1 + r)^{-n} }$$ | |
17701 | $$ \code{pmtb}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over | |
17702 | (1 - (1 + r)^{-n}) (1 + r) } $$ | |
17703 | $$ \code{nper}(r, p, a) = -\code{log}(1 - { a r \over p }, 1 + r) $$ | |
17704 | $$ \code{nperb}(r, p, a) = -\code{log}(1 - { a r \over p (1 + r) }, 1 + r) $$ | |
17705 | $$ \code{nperl}(r, p, a) = -\code{log}({a \over p}, 1 + r) $$ | |
17706 | $$ \code{ratel}(n, p, a) = { p^{1/n} \over a^{1/n} } - 1 $$ | |
17707 | $$ \code{sln}(c, s, l) = { c - s \over l } $$ | |
17708 | $$ \code{syd}(c, s, l, p) = { (c - s) (l - p + 1) \over l (l+1) / 2 } $$ | |
17709 | $$ \code{ddb}(c, s, l, p) = { 2 (c - \hbox{depreciation so far}) \over l } $$ | |
17710 | @end tex | |
17711 | ||
17712 | @noindent | |
17713 | In @code{pmt} and @code{pmtb}, @cite{x=0} if omitted. | |
17714 | ||
17715 | These functions accept any numeric objects, including error forms, | |
17716 | intervals, and even (though not very usefully) complex numbers. The | |
17717 | above formulas specify exactly the behavior of these functions with | |
17718 | all sorts of inputs. | |
17719 | ||
17720 | Note that if the first argument to the @code{log} in @code{nper} is | |
17721 | negative, @code{nper} leaves itself in symbolic form rather than | |
17722 | returning a (financially meaningless) complex number. | |
17723 | ||
17724 | @samp{rate(num, pmt, amt)} solves the equation | |
17725 | @samp{pv(rate, num, pmt) = amt} for @samp{rate} using @kbd{H a R} | |
17726 | (@code{calc-find-root}), with the interval @samp{[.01% .. 100%]} | |
17727 | for an initial guess. The @code{rateb} function is the same except | |
17728 | that it uses @code{pvb}. Note that @code{ratel} can be solved | |
17729 | directly; its formula is shown in the above list. | |
17730 | ||
17731 | Similarly, @samp{irr(pmts)} solves the equation @samp{npv(rate, pmts) = 0} | |
17732 | for @samp{rate}. | |
17733 | ||
17734 | If you give a fourth argument to @code{nper} or @code{nperb}, Calc | |
17735 | will also use @kbd{H a R} to solve the equation using an initial | |
17736 | guess interval of @samp{[0 .. 100]}. | |
17737 | ||
17738 | A fourth argument to @code{fv} simply sums the two components | |
17739 | calculated from the above formulas for @code{fv} and @code{fvl}. | |
17740 | The same is true of @code{fvb}, @code{pv}, and @code{pvb}. | |
17741 | ||
17742 | The @kbd{ddb} function is computed iteratively; the ``book'' value | |
17743 | starts out equal to @var{cost}, and decreases according to the above | |
17744 | formula for the specified number of periods. If the book value | |
17745 | would decrease below @var{salvage}, it only decreases to @var{salvage} | |
17746 | and the depreciation is zero for all subsequent periods. The @code{ddb} | |
17747 | function returns the amount the book value decreased in the specified | |
17748 | period. | |
17749 | ||
17750 | The Calc financial function names were borrowed mostly from Microsoft | |
17751 | Excel and Borland's Quattro. The @code{ratel} function corresponds to | |
17752 | @samp{@@CGR} in Borland's Reflex. The @code{nper} and @code{nperl} | |
17753 | functions correspond to @samp{@@TERM} and @samp{@@CTERM} in Quattro, | |
17754 | respectively. Beware that the Calc functions may take their arguments | |
17755 | in a different order than the corresponding functions in your favorite | |
17756 | spreadsheet. | |
17757 | ||
17758 | @node Binary Functions, , Financial Functions, Arithmetic | |
17759 | @section Binary Number Functions | |
17760 | ||
17761 | @noindent | |
17762 | The commands in this chapter all use two-letter sequences beginning with | |
17763 | the @kbd{b} prefix. | |
17764 | ||
17765 | @cindex Binary numbers | |
17766 | The ``binary'' operations actually work regardless of the currently | |
17767 | displayed radix, although their results make the most sense in a radix | |
17768 | like 2, 8, or 16 (as obtained by the @kbd{d 2}, @kbd{d 8}, or @w{@kbd{d 6}} | |
17769 | commands, respectively). You may also wish to enable display of leading | |
17770 | zeros with @kbd{d z}. @xref{Radix Modes}. | |
17771 | ||
17772 | @cindex Word size for binary operations | |
17773 | The Calculator maintains a current @dfn{word size} @cite{w}, an | |
17774 | arbitrary positive or negative integer. For a positive word size, all | |
17775 | of the binary operations described here operate modulo @cite{2^w}. In | |
17776 | particular, negative arguments are converted to positive integers modulo | |
17777 | @cite{2^w} by all binary functions.@refill | |
17778 | ||
17779 | If the word size is negative, binary operations produce 2's complement | |
17780 | integers from @c{$-2^{-w-1}$} | |
17781 | @cite{-(2^(-w-1))} to @c{$2^{-w-1}-1$} | |
17782 | @cite{2^(-w-1)-1} inclusive. Either | |
17783 | mode accepts inputs in any range; the sign of @cite{w} affects only | |
17784 | the results produced. | |
17785 | ||
17786 | @kindex b c | |
17787 | @pindex calc-clip | |
17788 | @tindex clip | |
17789 | The @kbd{b c} (@code{calc-clip}) | |
17790 | [@code{clip}] command can be used to clip a number by reducing it modulo | |
17791 | @cite{2^w}. The commands described in this chapter automatically clip | |
17792 | their results to the current word size. Note that other operations like | |
17793 | addition do not use the current word size, since integer addition | |
17794 | generally is not ``binary.'' (However, @pxref{Simplification Modes}, | |
17795 | @code{calc-bin-simplify-mode}.) For example, with a word size of 8 | |
17796 | bits @kbd{b c} converts a number to the range 0 to 255; with a word | |
17797 | size of @i{-8} @kbd{b c} converts to the range @i{-128} to 127.@refill | |
17798 | ||
17799 | @kindex b w | |
17800 | @pindex calc-word-size | |
17801 | The default word size is 32 bits. All operations except the shifts and | |
17802 | rotates allow you to specify a different word size for that one | |
17803 | operation by giving a numeric prefix argument: @kbd{C-u 8 b c} clips the | |
17804 | top of stack to the range 0 to 255 regardless of the current word size. | |
17805 | To set the word size permanently, use @kbd{b w} (@code{calc-word-size}). | |
17806 | This command displays a prompt with the current word size; press @key{RET} | |
17807 | immediately to keep this word size, or type a new word size at the prompt. | |
17808 | ||
17809 | When the binary operations are written in symbolic form, they take an | |
17810 | optional second (or third) word-size parameter. When a formula like | |
17811 | @samp{and(a,b)} is finally evaluated, the word size current at that time | |
17812 | will be used, but when @samp{and(a,b,-8)} is evaluated, a word size of | |
17813 | @i{-8} will always be used. A symbolic binary function will be left | |
17814 | in symbolic form unless the all of its argument(s) are integers or | |
17815 | integer-valued floats. | |
17816 | ||
17817 | If either or both arguments are modulo forms for which @cite{M} is a | |
17818 | power of two, that power of two is taken as the word size unless a | |
17819 | numeric prefix argument overrides it. The current word size is never | |
17820 | consulted when modulo-power-of-two forms are involved. | |
17821 | ||
17822 | @kindex b a | |
17823 | @pindex calc-and | |
17824 | @tindex and | |
17825 | The @kbd{b a} (@code{calc-and}) [@code{and}] command computes the bitwise | |
17826 | AND of the two numbers on the top of the stack. In other words, for each | |
17827 | of the @cite{w} binary digits of the two numbers (pairwise), the corresponding | |
17828 | bit of the result is 1 if and only if both input bits are 1: | |
17829 | @samp{and(2#1100, 2#1010) = 2#1000}. | |
17830 | ||
17831 | @kindex b o | |
17832 | @pindex calc-or | |
17833 | @tindex or | |
17834 | The @kbd{b o} (@code{calc-or}) [@code{or}] command computes the bitwise | |
17835 | inclusive OR of two numbers. A bit is 1 if either of the input bits, or | |
17836 | both, are 1: @samp{or(2#1100, 2#1010) = 2#1110}. | |
17837 | ||
17838 | @kindex b x | |
17839 | @pindex calc-xor | |
17840 | @tindex xor | |
17841 | The @kbd{b x} (@code{calc-xor}) [@code{xor}] command computes the bitwise | |
17842 | exclusive OR of two numbers. A bit is 1 if exactly one of the input bits | |
17843 | is 1: @samp{xor(2#1100, 2#1010) = 2#0110}. | |
17844 | ||
17845 | @kindex b d | |
17846 | @pindex calc-diff | |
17847 | @tindex diff | |
17848 | The @kbd{b d} (@code{calc-diff}) [@code{diff}] command computes the bitwise | |
17849 | difference of two numbers; this is defined by @samp{diff(a,b) = and(a,not(b))}, | |
17850 | so that @samp{diff(2#1100, 2#1010) = 2#0100}. | |
17851 | ||
17852 | @kindex b n | |
17853 | @pindex calc-not | |
17854 | @tindex not | |
17855 | The @kbd{b n} (@code{calc-not}) [@code{not}] command computes the bitwise | |
17856 | NOT of a number. A bit is 1 if the input bit is 0 and vice-versa. | |
17857 | ||
17858 | @kindex b l | |
17859 | @pindex calc-lshift-binary | |
17860 | @tindex lsh | |
17861 | The @kbd{b l} (@code{calc-lshift-binary}) [@code{lsh}] command shifts a | |
17862 | number left by one bit, or by the number of bits specified in the numeric | |
17863 | prefix argument. A negative prefix argument performs a logical right shift, | |
17864 | in which zeros are shifted in on the left. In symbolic form, @samp{lsh(a)} | |
17865 | is short for @samp{lsh(a,1)}, which in turn is short for @samp{lsh(a,n,w)}. | |
17866 | Bits shifted ``off the end,'' according to the current word size, are lost. | |
17867 | ||
17868 | @kindex H b l | |
17869 | @kindex H b r | |
5d67986c RS |
17870 | @ignore |
17871 | @mindex @idots | |
17872 | @end ignore | |
d7b8e6c6 | 17873 | @kindex H b L |
5d67986c RS |
17874 | @ignore |
17875 | @mindex @null | |
17876 | @end ignore | |
d7b8e6c6 | 17877 | @kindex H b R |
5d67986c RS |
17878 | @ignore |
17879 | @mindex @null | |
17880 | @end ignore | |
d7b8e6c6 EZ |
17881 | @kindex H b t |
17882 | The @kbd{H b l} command also does a left shift, but it takes two arguments | |
17883 | from the stack (the value to shift, and, at top-of-stack, the number of | |
17884 | bits to shift). This version interprets the prefix argument just like | |
17885 | the regular binary operations, i.e., as a word size. The Hyperbolic flag | |
17886 | has a similar effect on the rest of the binary shift and rotate commands. | |
17887 | ||
17888 | @kindex b r | |
17889 | @pindex calc-rshift-binary | |
17890 | @tindex rsh | |
17891 | The @kbd{b r} (@code{calc-rshift-binary}) [@code{rsh}] command shifts a | |
17892 | number right by one bit, or by the number of bits specified in the numeric | |
17893 | prefix argument: @samp{rsh(a,n) = lsh(a,-n)}. | |
17894 | ||
17895 | @kindex b L | |
17896 | @pindex calc-lshift-arith | |
17897 | @tindex ash | |
17898 | The @kbd{b L} (@code{calc-lshift-arith}) [@code{ash}] command shifts a | |
17899 | number left. It is analogous to @code{lsh}, except that if the shift | |
17900 | is rightward (the prefix argument is negative), an arithmetic shift | |
17901 | is performed as described below. | |
17902 | ||
17903 | @kindex b R | |
17904 | @pindex calc-rshift-arith | |
17905 | @tindex rash | |
17906 | The @kbd{b R} (@code{calc-rshift-arith}) [@code{rash}] command performs | |
17907 | an ``arithmetic'' shift to the right, in which the leftmost bit (according | |
17908 | to the current word size) is duplicated rather than shifting in zeros. | |
17909 | This corresponds to dividing by a power of two where the input is interpreted | |
17910 | as a signed, twos-complement number. (The distinction between the @samp{rsh} | |
17911 | and @samp{rash} operations is totally independent from whether the word | |
17912 | size is positive or negative.) With a negative prefix argument, this | |
17913 | performs a standard left shift. | |
17914 | ||
17915 | @kindex b t | |
17916 | @pindex calc-rotate-binary | |
17917 | @tindex rot | |
17918 | The @kbd{b t} (@code{calc-rotate-binary}) [@code{rot}] command rotates a | |
17919 | number one bit to the left. The leftmost bit (according to the current | |
17920 | word size) is dropped off the left and shifted in on the right. With a | |
17921 | numeric prefix argument, the number is rotated that many bits to the left | |
17922 | or right. | |
17923 | ||
17924 | @xref{Set Operations}, for the @kbd{b p} and @kbd{b u} commands that | |
17925 | pack and unpack binary integers into sets. (For example, @kbd{b u} | |
17926 | unpacks the number @samp{2#11001} to the set of bit-numbers | |
17927 | @samp{[0, 3, 4]}.) Type @kbd{b u V #} to count the number of ``1'' | |
17928 | bits in a binary integer. | |
17929 | ||
17930 | Another interesting use of the set representation of binary integers | |
17931 | is to reverse the bits in, say, a 32-bit integer. Type @kbd{b u} to | |
5d67986c | 17932 | unpack; type @kbd{31 @key{TAB} -} to replace each bit-number in the set |
d7b8e6c6 EZ |
17933 | with 31 minus that bit-number; type @kbd{b p} to pack the set back |
17934 | into a binary integer. | |
17935 | ||
17936 | @node Scientific Functions, Matrix Functions, Arithmetic, Top | |
17937 | @chapter Scientific Functions | |
17938 | ||
17939 | @noindent | |
17940 | The functions described here perform trigonometric and other transcendental | |
17941 | calculations. They generally produce floating-point answers correct to the | |
17942 | full current precision. The @kbd{H} (Hyperbolic) and @kbd{I} (Inverse) | |
17943 | flag keys must be used to get some of these functions from the keyboard. | |
17944 | ||
17945 | @kindex P | |
17946 | @pindex calc-pi | |
17947 | @cindex @code{pi} variable | |
17948 | @vindex pi | |
17949 | @kindex H P | |
17950 | @cindex @code{e} variable | |
17951 | @vindex e | |
17952 | @kindex I P | |
17953 | @cindex @code{gamma} variable | |
17954 | @vindex gamma | |
17955 | @cindex Gamma constant, Euler's | |
17956 | @cindex Euler's gamma constant | |
17957 | @kindex H I P | |
17958 | @cindex @code{phi} variable | |
17959 | @cindex Phi, golden ratio | |
17960 | @cindex Golden ratio | |
28665d46 | 17961 | One miscellaneous command is shift-@kbd{P} (@code{calc-pi}), which pushes |
d7b8e6c6 EZ |
17962 | the value of @c{$\pi$} |
17963 | @cite{pi} (at the current precision) onto the stack. With the | |
17964 | Hyperbolic flag, it pushes the value @cite{e}, the base of natural logarithms. | |
17965 | With the Inverse flag, it pushes Euler's constant @c{$\gamma$} | |
17966 | @cite{gamma} (about 0.5772). With both Inverse and Hyperbolic, it | |
17967 | pushes the ``golden ratio'' @c{$\phi$} | |
17968 | @cite{phi} (about 1.618). (At present, Euler's constant is not available | |
17969 | to unlimited precision; Calc knows only the first 100 digits.) | |
17970 | In Symbolic mode, these commands push the | |
17971 | actual variables @samp{pi}, @samp{e}, @samp{gamma}, and @samp{phi}, | |
17972 | respectively, instead of their values; @pxref{Symbolic Mode}.@refill | |
17973 | ||
5d67986c RS |
17974 | @ignore |
17975 | @mindex Q | |
17976 | @end ignore | |
17977 | @ignore | |
17978 | @mindex I Q | |
17979 | @end ignore | |
d7b8e6c6 EZ |
17980 | @kindex I Q |
17981 | @tindex sqr | |
17982 | The @kbd{Q} (@code{calc-sqrt}) [@code{sqrt}] function is described elsewhere; | |
17983 | @pxref{Basic Arithmetic}. With the Inverse flag [@code{sqr}], this command | |
17984 | computes the square of the argument. | |
17985 | ||
17986 | @xref{Prefix Arguments}, for a discussion of the effect of numeric | |
17987 | prefix arguments on commands in this chapter which do not otherwise | |
17988 | interpret a prefix argument. | |
17989 | ||
17990 | @menu | |
17991 | * Logarithmic Functions:: | |
17992 | * Trigonometric and Hyperbolic Functions:: | |
17993 | * Advanced Math Functions:: | |
17994 | * Branch Cuts:: | |
17995 | * Random Numbers:: | |
17996 | * Combinatorial Functions:: | |
17997 | * Probability Distribution Functions:: | |
17998 | @end menu | |
17999 | ||
18000 | @node Logarithmic Functions, Trigonometric and Hyperbolic Functions, Scientific Functions, Scientific Functions | |
18001 | @section Logarithmic Functions | |
18002 | ||
18003 | @noindent | |
18004 | @kindex L | |
18005 | @pindex calc-ln | |
18006 | @tindex ln | |
5d67986c RS |
18007 | @ignore |
18008 | @mindex @null | |
18009 | @end ignore | |
d7b8e6c6 EZ |
18010 | @kindex I E |
18011 | The shift-@kbd{L} (@code{calc-ln}) [@code{ln}] command computes the natural | |
18012 | logarithm of the real or complex number on the top of the stack. With | |
18013 | the Inverse flag it computes the exponential function instead, although | |
18014 | this is redundant with the @kbd{E} command. | |
18015 | ||
18016 | @kindex E | |
18017 | @pindex calc-exp | |
18018 | @tindex exp | |
5d67986c RS |
18019 | @ignore |
18020 | @mindex @null | |
18021 | @end ignore | |
d7b8e6c6 EZ |
18022 | @kindex I L |
18023 | The shift-@kbd{E} (@code{calc-exp}) [@code{exp}] command computes the | |
18024 | exponential, i.e., @cite{e} raised to the power of the number on the stack. | |
18025 | The meanings of the Inverse and Hyperbolic flags follow from those for | |
18026 | the @code{calc-ln} command. | |
18027 | ||
18028 | @kindex H L | |
18029 | @kindex H E | |
18030 | @pindex calc-log10 | |
18031 | @tindex log10 | |
18032 | @tindex exp10 | |
5d67986c RS |
18033 | @ignore |
18034 | @mindex @null | |
18035 | @end ignore | |
d7b8e6c6 | 18036 | @kindex H I L |
5d67986c RS |
18037 | @ignore |
18038 | @mindex @null | |
18039 | @end ignore | |
d7b8e6c6 EZ |
18040 | @kindex H I E |
18041 | The @kbd{H L} (@code{calc-log10}) [@code{log10}] command computes the common | |
18042 | (base-10) logarithm of a number. (With the Inverse flag [@code{exp10}], | |
18043 | it raises ten to a given power.) Note that the common logarithm of a | |
18044 | complex number is computed by taking the natural logarithm and dividing | |
18045 | by @c{$\ln10$} | |
18046 | @cite{ln(10)}. | |
18047 | ||
18048 | @kindex B | |
18049 | @kindex I B | |
18050 | @pindex calc-log | |
18051 | @tindex log | |
18052 | @tindex alog | |
18053 | The @kbd{B} (@code{calc-log}) [@code{log}] command computes a logarithm | |
18054 | to any base. For example, @kbd{1024 @key{RET} 2 B} produces 10, since | |
18055 | @c{$2^{10} = 1024$} | |
18056 | @cite{2^10 = 1024}. In certain cases like @samp{log(3,9)}, the result | |
18057 | will be either @cite{1:2} or @cite{0.5} depending on the current Fraction | |
18058 | Mode setting. With the Inverse flag [@code{alog}], this command is | |
18059 | similar to @kbd{^} except that the order of the arguments is reversed. | |
18060 | ||
18061 | @kindex f I | |
18062 | @pindex calc-ilog | |
18063 | @tindex ilog | |
18064 | The @kbd{f I} (@code{calc-ilog}) [@code{ilog}] command computes the | |
18065 | integer logarithm of a number to any base. The number and the base must | |
18066 | themselves be positive integers. This is the true logarithm, rounded | |
18067 | down to an integer. Thus @kbd{ilog(x,10)} is 3 for all @cite{x} in the | |
18068 | range from 1000 to 9999. If both arguments are positive integers, exact | |
18069 | integer arithmetic is used; otherwise, this is equivalent to | |
18070 | @samp{floor(log(x,b))}. | |
18071 | ||
18072 | @kindex f E | |
18073 | @pindex calc-expm1 | |
18074 | @tindex expm1 | |
18075 | The @kbd{f E} (@code{calc-expm1}) [@code{expm1}] command computes | |
18076 | @c{$e^x - 1$} | |
18077 | @cite{exp(x)-1}, but using an algorithm that produces a more accurate | |
18078 | answer when the result is close to zero, i.e., when @c{$e^x$} | |
18079 | @cite{exp(x)} is close | |
18080 | to one. | |
18081 | ||
18082 | @kindex f L | |
18083 | @pindex calc-lnp1 | |
18084 | @tindex lnp1 | |
18085 | The @kbd{f L} (@code{calc-lnp1}) [@code{lnp1}] command computes | |
18086 | @c{$\ln(x+1)$} | |
18087 | @cite{ln(x+1)}, producing a more accurate answer when @cite{x} is close | |
18088 | to zero. | |
18089 | ||
18090 | @node Trigonometric and Hyperbolic Functions, Advanced Math Functions, Logarithmic Functions, Scientific Functions | |
18091 | @section Trigonometric/Hyperbolic Functions | |
18092 | ||
18093 | @noindent | |
18094 | @kindex S | |
18095 | @pindex calc-sin | |
18096 | @tindex sin | |
18097 | The shift-@kbd{S} (@code{calc-sin}) [@code{sin}] command computes the sine | |
18098 | of an angle or complex number. If the input is an HMS form, it is interpreted | |
18099 | as degrees-minutes-seconds; otherwise, the input is interpreted according | |
18100 | to the current angular mode. It is best to use Radians mode when operating | |
18101 | on complex numbers.@refill | |
18102 | ||
18103 | Calc's ``units'' mechanism includes angular units like @code{deg}, | |
18104 | @code{rad}, and @code{grad}. While @samp{sin(45 deg)} is not evaluated | |
18105 | all the time, the @kbd{u s} (@code{calc-simplify-units}) command will | |
18106 | simplify @samp{sin(45 deg)} by taking the sine of 45 degrees, regardless | |
18107 | of the current angular mode. @xref{Basic Operations on Units}. | |
18108 | ||
18109 | Also, the symbolic variable @code{pi} is not ordinarily recognized in | |
18110 | arguments to trigonometric functions, as in @samp{sin(3 pi / 4)}, but | |
18111 | the @kbd{a s} (@code{calc-simplify}) command recognizes many such | |
18112 | formulas when the current angular mode is radians @emph{and} symbolic | |
18113 | mode is enabled; this example would be replaced by @samp{sqrt(2) / 2}. | |
18114 | @xref{Symbolic Mode}. Beware, this simplification occurs even if you | |
18115 | have stored a different value in the variable @samp{pi}; this is one | |
18116 | reason why changing built-in variables is a bad idea. Arguments of | |
18117 | the form @cite{x} plus a multiple of @c{$\pi/2$} | |
18118 | @cite{pi/2} are also simplified. | |
18119 | Calc includes similar formulas for @code{cos} and @code{tan}.@refill | |
18120 | ||
18121 | The @kbd{a s} command knows all angles which are integer multiples of | |
18122 | @c{$\pi/12$} | |
18123 | @cite{pi/12}, @c{$\pi/10$} | |
18124 | @cite{pi/10}, or @c{$\pi/8$} | |
18125 | @cite{pi/8} radians. In degrees mode, | |
18126 | analogous simplifications occur for integer multiples of 15 or 18 | |
18127 | degrees, and for arguments plus multiples of 90 degrees. | |
18128 | ||
18129 | @kindex I S | |
18130 | @pindex calc-arcsin | |
18131 | @tindex arcsin | |
18132 | With the Inverse flag, @code{calc-sin} computes an arcsine. This is also | |
18133 | available as the @code{calc-arcsin} command or @code{arcsin} algebraic | |
18134 | function. The returned argument is converted to degrees, radians, or HMS | |
18135 | notation depending on the current angular mode. | |
18136 | ||
18137 | @kindex H S | |
18138 | @pindex calc-sinh | |
18139 | @tindex sinh | |
18140 | @kindex H I S | |
18141 | @pindex calc-arcsinh | |
18142 | @tindex arcsinh | |
18143 | With the Hyperbolic flag, @code{calc-sin} computes the hyperbolic | |
18144 | sine, also available as @code{calc-sinh} [@code{sinh}]. With the | |
18145 | Hyperbolic and Inverse flags, it computes the hyperbolic arcsine | |
18146 | (@code{calc-arcsinh}) [@code{arcsinh}]. | |
18147 | ||
18148 | @kindex C | |
18149 | @pindex calc-cos | |
18150 | @tindex cos | |
5d67986c RS |
18151 | @ignore |
18152 | @mindex @idots | |
18153 | @end ignore | |
d7b8e6c6 EZ |
18154 | @kindex I C |
18155 | @pindex calc-arccos | |
5d67986c RS |
18156 | @ignore |
18157 | @mindex @null | |
18158 | @end ignore | |
d7b8e6c6 | 18159 | @tindex arccos |
5d67986c RS |
18160 | @ignore |
18161 | @mindex @null | |
18162 | @end ignore | |
d7b8e6c6 EZ |
18163 | @kindex H C |
18164 | @pindex calc-cosh | |
5d67986c RS |
18165 | @ignore |
18166 | @mindex @null | |
18167 | @end ignore | |
d7b8e6c6 | 18168 | @tindex cosh |
5d67986c RS |
18169 | @ignore |
18170 | @mindex @null | |
18171 | @end ignore | |
d7b8e6c6 EZ |
18172 | @kindex H I C |
18173 | @pindex calc-arccosh | |
5d67986c RS |
18174 | @ignore |
18175 | @mindex @null | |
18176 | @end ignore | |
d7b8e6c6 | 18177 | @tindex arccosh |
5d67986c RS |
18178 | @ignore |
18179 | @mindex @null | |
18180 | @end ignore | |
d7b8e6c6 EZ |
18181 | @kindex T |
18182 | @pindex calc-tan | |
5d67986c RS |
18183 | @ignore |
18184 | @mindex @null | |
18185 | @end ignore | |
d7b8e6c6 | 18186 | @tindex tan |
5d67986c RS |
18187 | @ignore |
18188 | @mindex @null | |
18189 | @end ignore | |
d7b8e6c6 EZ |
18190 | @kindex I T |
18191 | @pindex calc-arctan | |
5d67986c RS |
18192 | @ignore |
18193 | @mindex @null | |
18194 | @end ignore | |
d7b8e6c6 | 18195 | @tindex arctan |
5d67986c RS |
18196 | @ignore |
18197 | @mindex @null | |
18198 | @end ignore | |
d7b8e6c6 EZ |
18199 | @kindex H T |
18200 | @pindex calc-tanh | |
5d67986c RS |
18201 | @ignore |
18202 | @mindex @null | |
18203 | @end ignore | |
d7b8e6c6 | 18204 | @tindex tanh |
5d67986c RS |
18205 | @ignore |
18206 | @mindex @null | |
18207 | @end ignore | |
d7b8e6c6 EZ |
18208 | @kindex H I T |
18209 | @pindex calc-arctanh | |
5d67986c RS |
18210 | @ignore |
18211 | @mindex @null | |
18212 | @end ignore | |
d7b8e6c6 EZ |
18213 | @tindex arctanh |
18214 | The shift-@kbd{C} (@code{calc-cos}) [@code{cos}] command computes the cosine | |
18215 | of an angle or complex number, and shift-@kbd{T} (@code{calc-tan}) [@code{tan}] | |
18216 | computes the tangent, along with all the various inverse and hyperbolic | |
18217 | variants of these functions. | |
18218 | ||
18219 | @kindex f T | |
18220 | @pindex calc-arctan2 | |
18221 | @tindex arctan2 | |
18222 | The @kbd{f T} (@code{calc-arctan2}) [@code{arctan2}] command takes two | |
18223 | numbers from the stack and computes the arc tangent of their ratio. The | |
18224 | result is in the full range from @i{-180} (exclusive) to @i{+180} | |
18225 | (inclusive) degrees, or the analogous range in radians. A similar | |
18226 | result would be obtained with @kbd{/} followed by @kbd{I T}, but the | |
18227 | value would only be in the range from @i{-90} to @i{+90} degrees | |
18228 | since the division loses information about the signs of the two | |
18229 | components, and an error might result from an explicit division by zero | |
18230 | which @code{arctan2} would avoid. By (arbitrary) definition, | |
18231 | @samp{arctan2(0,0)=0}. | |
18232 | ||
18233 | @pindex calc-sincos | |
5d67986c RS |
18234 | @ignore |
18235 | @starindex | |
18236 | @end ignore | |
d7b8e6c6 | 18237 | @tindex sincos |
5d67986c RS |
18238 | @ignore |
18239 | @starindex | |
18240 | @end ignore | |
18241 | @ignore | |
18242 | @mindex arc@idots | |
18243 | @end ignore | |
d7b8e6c6 EZ |
18244 | @tindex arcsincos |
18245 | The @code{calc-sincos} [@code{sincos}] command computes the sine and | |
18246 | cosine of a number, returning them as a vector of the form | |
18247 | @samp{[@var{cos}, @var{sin}]}. | |
18248 | With the Inverse flag [@code{arcsincos}], this command takes a two-element | |
18249 | vector as an argument and computes @code{arctan2} of the elements. | |
18250 | (This command does not accept the Hyperbolic flag.)@refill | |
18251 | ||
18252 | @node Advanced Math Functions, Branch Cuts, Trigonometric and Hyperbolic Functions, Scientific Functions | |
18253 | @section Advanced Mathematical Functions | |
18254 | ||
18255 | @noindent | |
18256 | Calc can compute a variety of less common functions that arise in | |
18257 | various branches of mathematics. All of the functions described in | |
18258 | this section allow arbitrary complex arguments and, except as noted, | |
18259 | will work to arbitrarily large precisions. They can not at present | |
18260 | handle error forms or intervals as arguments. | |
18261 | ||
18262 | NOTE: These functions are still experimental. In particular, their | |
18263 | accuracy is not guaranteed in all domains. It is advisable to set the | |
18264 | current precision comfortably higher than you actually need when | |
18265 | using these functions. Also, these functions may be impractically | |
18266 | slow for some values of the arguments. | |
18267 | ||
18268 | @kindex f g | |
18269 | @pindex calc-gamma | |
18270 | @tindex gamma | |
18271 | The @kbd{f g} (@code{calc-gamma}) [@code{gamma}] command computes the Euler | |
18272 | gamma function. For positive integer arguments, this is related to the | |
18273 | factorial function: @samp{gamma(n+1) = fact(n)}. For general complex | |
18274 | arguments the gamma function can be defined by the following definite | |
18275 | integral: @c{$\Gamma(a) = \int_0^\infty t^{a-1} e^t dt$} | |
18276 | @cite{gamma(a) = integ(t^(a-1) exp(t), t, 0, inf)}. | |
18277 | (The actual implementation uses far more efficient computational methods.) | |
18278 | ||
18279 | @kindex f G | |
18280 | @tindex gammaP | |
5d67986c RS |
18281 | @ignore |
18282 | @mindex @idots | |
18283 | @end ignore | |
d7b8e6c6 | 18284 | @kindex I f G |
5d67986c RS |
18285 | @ignore |
18286 | @mindex @null | |
18287 | @end ignore | |
d7b8e6c6 | 18288 | @kindex H f G |
5d67986c RS |
18289 | @ignore |
18290 | @mindex @null | |
18291 | @end ignore | |
d7b8e6c6 EZ |
18292 | @kindex H I f G |
18293 | @pindex calc-inc-gamma | |
5d67986c RS |
18294 | @ignore |
18295 | @mindex @null | |
18296 | @end ignore | |
d7b8e6c6 | 18297 | @tindex gammaQ |
5d67986c RS |
18298 | @ignore |
18299 | @mindex @null | |
18300 | @end ignore | |
d7b8e6c6 | 18301 | @tindex gammag |
5d67986c RS |
18302 | @ignore |
18303 | @mindex @null | |
18304 | @end ignore | |
d7b8e6c6 EZ |
18305 | @tindex gammaG |
18306 | The @kbd{f G} (@code{calc-inc-gamma}) [@code{gammaP}] command computes | |
18307 | the incomplete gamma function, denoted @samp{P(a,x)}. This is defined by | |
18308 | the integral, @c{$P(a,x) = \left( \int_0^x t^{a-1} e^t dt \right) / \Gamma(a)$} | |
18309 | @cite{gammaP(a,x) = integ(t^(a-1) exp(t), t, 0, x) / gamma(a)}. | |
18310 | This implies that @samp{gammaP(a,inf) = 1} for any @cite{a} (see the | |
18311 | definition of the normal gamma function). | |
18312 | ||
18313 | Several other varieties of incomplete gamma function are defined. | |
18314 | The complement of @cite{P(a,x)}, called @cite{Q(a,x) = 1-P(a,x)} by | |
18315 | some authors, is computed by the @kbd{I f G} [@code{gammaQ}] command. | |
18316 | You can think of this as taking the other half of the integral, from | |
18317 | @cite{x} to infinity. | |
18318 | ||
18319 | @ifinfo | |
18320 | The functions corresponding to the integrals that define @cite{P(a,x)} | |
18321 | and @cite{Q(a,x)} but without the normalizing @cite{1/gamma(a)} | |
18322 | factor are called @cite{g(a,x)} and @cite{G(a,x)}, respectively | |
18323 | (where @cite{g} and @cite{G} represent the lower- and upper-case Greek | |
18324 | letter gamma). You can obtain these using the @kbd{H f G} [@code{gammag}] | |
18325 | and @kbd{H I f G} [@code{gammaG}] commands. | |
18326 | @end ifinfo | |
18327 | @tex | |
18328 | \turnoffactive | |
18329 | The functions corresponding to the integrals that define $P(a,x)$ | |
18330 | and $Q(a,x)$ but without the normalizing $1/\Gamma(a)$ | |
18331 | factor are called $\gamma(a,x)$ and $\Gamma(a,x)$, respectively. | |
18332 | You can obtain these using the \kbd{H f G} [\code{gammag}] and | |
18333 | \kbd{I H f G} [\code{gammaG}] commands. | |
18334 | @end tex | |
18335 | ||
18336 | @kindex f b | |
18337 | @pindex calc-beta | |
18338 | @tindex beta | |
18339 | The @kbd{f b} (@code{calc-beta}) [@code{beta}] command computes the | |
18340 | Euler beta function, which is defined in terms of the gamma function as | |
18341 | @c{$B(a,b) = \Gamma(a) \Gamma(b) / \Gamma(a+b)$} | |
18342 | @cite{beta(a,b) = gamma(a) gamma(b) / gamma(a+b)}, or by | |
18343 | @c{$B(a,b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt$} | |
18344 | @cite{beta(a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, 1)}. | |
18345 | ||
18346 | @kindex f B | |
18347 | @kindex H f B | |
18348 | @pindex calc-inc-beta | |
18349 | @tindex betaI | |
18350 | @tindex betaB | |
18351 | The @kbd{f B} (@code{calc-inc-beta}) [@code{betaI}] command computes | |
18352 | the incomplete beta function @cite{I(x,a,b)}. It is defined by | |
18353 | @c{$I(x,a,b) = \left( \int_0^x t^{a-1} (1-t)^{b-1} dt \right) / B(a,b)$} | |
18354 | @cite{betaI(x,a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, x) / beta(a,b)}. | |
18355 | Once again, the @kbd{H} (hyperbolic) prefix gives the corresponding | |
18356 | un-normalized version [@code{betaB}]. | |
18357 | ||
18358 | @kindex f e | |
18359 | @kindex I f e | |
18360 | @pindex calc-erf | |
18361 | @tindex erf | |
18362 | @tindex erfc | |
18363 | The @kbd{f e} (@code{calc-erf}) [@code{erf}] command computes the | |
18364 | error function @c{$\hbox{erf}(x) = {2 \over \sqrt{\pi}} \int_0^x e^{-t^2} dt$} | |
18365 | @cite{erf(x) = 2 integ(exp(-(t^2)), t, 0, x) / sqrt(pi)}. | |
18366 | The complementary error function @kbd{I f e} (@code{calc-erfc}) [@code{erfc}] | |
18367 | is the corresponding integral from @samp{x} to infinity; the sum | |
18368 | @c{$\hbox{erf}(x) + \hbox{erfc}(x) = 1$} | |
18369 | @cite{erf(x) + erfc(x) = 1}. | |
18370 | ||
18371 | @kindex f j | |
18372 | @kindex f y | |
18373 | @pindex calc-bessel-J | |
18374 | @pindex calc-bessel-Y | |
18375 | @tindex besJ | |
18376 | @tindex besY | |
18377 | The @kbd{f j} (@code{calc-bessel-J}) [@code{besJ}] and @kbd{f y} | |
18378 | (@code{calc-bessel-Y}) [@code{besY}] commands compute the Bessel | |
18379 | functions of the first and second kinds, respectively. | |
18380 | In @samp{besJ(n,x)} and @samp{besY(n,x)} the ``order'' parameter | |
18381 | @cite{n} is often an integer, but is not required to be one. | |
18382 | Calc's implementation of the Bessel functions currently limits the | |
18383 | precision to 8 digits, and may not be exact even to that precision. | |
18384 | Use with care!@refill | |
18385 | ||
18386 | @node Branch Cuts, Random Numbers, Advanced Math Functions, Scientific Functions | |
18387 | @section Branch Cuts and Principal Values | |
18388 | ||
18389 | @noindent | |
18390 | @cindex Branch cuts | |
18391 | @cindex Principal values | |
18392 | All of the logarithmic, trigonometric, and other scientific functions are | |
18393 | defined for complex numbers as well as for reals. | |
18394 | This section describes the values | |
18395 | returned in cases where the general result is a family of possible values. | |
18396 | Calc follows section 12.5.3 of Steele's @dfn{Common Lisp, the Language}, | |
18397 | second edition, in these matters. This section will describe each | |
18398 | function briefly; for a more detailed discussion (including some nifty | |
18399 | diagrams), consult Steele's book. | |
18400 | ||
18401 | Note that the branch cuts for @code{arctan} and @code{arctanh} were | |
18402 | changed between the first and second editions of Steele. Versions of | |
18403 | Calc starting with 2.00 follow the second edition. | |
18404 | ||
18405 | The new branch cuts exactly match those of the HP-28/48 calculators. | |
18406 | They also match those of Mathematica 1.2, except that Mathematica's | |
18407 | @code{arctan} cut is always in the right half of the complex plane, | |
18408 | and its @code{arctanh} cut is always in the top half of the plane. | |
18409 | Calc's cuts are continuous with quadrants I and III for @code{arctan}, | |
18410 | or II and IV for @code{arctanh}. | |
18411 | ||
18412 | Note: The current implementations of these functions with complex arguments | |
18413 | are designed with proper behavior around the branch cuts in mind, @emph{not} | |
18414 | efficiency or accuracy. You may need to increase the floating precision | |
18415 | and wait a while to get suitable answers from them. | |
18416 | ||
18417 | For @samp{sqrt(a+bi)}: When @cite{a<0} and @cite{b} is small but positive | |
18418 | or zero, the result is close to the @cite{+i} axis. For @cite{b} small and | |
18419 | negative, the result is close to the @cite{-i} axis. The result always lies | |
18420 | in the right half of the complex plane. | |
18421 | ||
18422 | For @samp{ln(a+bi)}: The real part is defined as @samp{ln(abs(a+bi))}. | |
18423 | The imaginary part is defined as @samp{arg(a+bi) = arctan2(b,a)}. | |
18424 | Thus the branch cuts for @code{sqrt} and @code{ln} both lie on the | |
18425 | negative real axis. | |
18426 | ||
18427 | The following table describes these branch cuts in another way. | |
18428 | If the real and imaginary parts of @cite{z} are as shown, then | |
18429 | the real and imaginary parts of @cite{f(z)} will be as shown. | |
18430 | Here @code{eps} stands for a small positive value; each | |
18431 | occurrence of @code{eps} may stand for a different small value. | |
18432 | ||
18433 | @smallexample | |
18434 | z sqrt(z) ln(z) | |
18435 | ---------------------------------------- | |
18436 | +, 0 +, 0 any, 0 | |
18437 | -, 0 0, + any, pi | |
18438 | -, +eps +eps, + +eps, + | |
18439 | -, -eps +eps, - +eps, - | |
18440 | @end smallexample | |
18441 | ||
18442 | For @samp{z1^z2}: This is defined by @samp{exp(ln(z1)*z2)}. | |
18443 | One interesting consequence of this is that @samp{(-8)^1:3} does | |
18444 | not evaluate to @i{-2} as you might expect, but to the complex | |
18445 | number @cite{(1., 1.732)}. Both of these are valid cube roots | |
18446 | of @i{-8} (as is @cite{(1., -1.732)}); Calc chooses a perhaps | |
18447 | less-obvious root for the sake of mathematical consistency. | |
18448 | ||
18449 | For @samp{arcsin(z)}: This is defined by @samp{-i*ln(i*z + sqrt(1-z^2))}. | |
18450 | The branch cuts are on the real axis, less than @i{-1} and greater than 1. | |
18451 | ||
18452 | For @samp{arccos(z)}: This is defined by @samp{-i*ln(z + i*sqrt(1-z^2))}, | |
18453 | or equivalently by @samp{pi/2 - arcsin(z)}. The branch cuts are on | |
18454 | the real axis, less than @i{-1} and greater than 1. | |
18455 | ||
18456 | For @samp{arctan(z)}: This is defined by | |
18457 | @samp{(ln(1+i*z) - ln(1-i*z)) / (2*i)}. The branch cuts are on the | |
18458 | imaginary axis, below @cite{-i} and above @cite{i}. | |
18459 | ||
18460 | For @samp{arcsinh(z)}: This is defined by @samp{ln(z + sqrt(1+z^2))}. | |
18461 | The branch cuts are on the imaginary axis, below @cite{-i} and | |
18462 | above @cite{i}. | |
18463 | ||
18464 | For @samp{arccosh(z)}: This is defined by | |
18465 | @samp{ln(z + (z+1)*sqrt((z-1)/(z+1)))}. The branch cut is on the | |
18466 | real axis less than 1. | |
18467 | ||
18468 | For @samp{arctanh(z)}: This is defined by @samp{(ln(1+z) - ln(1-z)) / 2}. | |
18469 | The branch cuts are on the real axis, less than @i{-1} and greater than 1. | |
18470 | ||
18471 | The following tables for @code{arcsin}, @code{arccos}, and | |
18472 | @code{arctan} assume the current angular mode is radians. The | |
18473 | hyperbolic functions operate independently of the angular mode. | |
18474 | ||
18475 | @smallexample | |
18476 | z arcsin(z) arccos(z) | |
18477 | ------------------------------------------------------- | |
18478 | (-1..1), 0 (-pi/2..pi/2), 0 (0..pi), 0 | |
18479 | (-1..1), +eps (-pi/2..pi/2), +eps (0..pi), -eps | |
18480 | (-1..1), -eps (-pi/2..pi/2), -eps (0..pi), +eps | |
18481 | <-1, 0 -pi/2, + pi, - | |
18482 | <-1, +eps -pi/2 + eps, + pi - eps, - | |
18483 | <-1, -eps -pi/2 + eps, - pi - eps, + | |
18484 | >1, 0 pi/2, - 0, + | |
18485 | >1, +eps pi/2 - eps, + +eps, - | |
18486 | >1, -eps pi/2 - eps, - +eps, + | |
18487 | @end smallexample | |
18488 | ||
18489 | @smallexample | |
18490 | z arccosh(z) arctanh(z) | |
18491 | ----------------------------------------------------- | |
18492 | (-1..1), 0 0, (0..pi) any, 0 | |
18493 | (-1..1), +eps +eps, (0..pi) any, +eps | |
18494 | (-1..1), -eps +eps, (-pi..0) any, -eps | |
18495 | <-1, 0 +, pi -, pi/2 | |
18496 | <-1, +eps +, pi - eps -, pi/2 - eps | |
18497 | <-1, -eps +, -pi + eps -, -pi/2 + eps | |
18498 | >1, 0 +, 0 +, -pi/2 | |
18499 | >1, +eps +, +eps +, pi/2 - eps | |
18500 | >1, -eps +, -eps +, -pi/2 + eps | |
18501 | @end smallexample | |
18502 | ||
18503 | @smallexample | |
18504 | z arcsinh(z) arctan(z) | |
18505 | ----------------------------------------------------- | |
18506 | 0, (-1..1) 0, (-pi/2..pi/2) 0, any | |
18507 | 0, <-1 -, -pi/2 -pi/2, - | |
18508 | +eps, <-1 +, -pi/2 + eps pi/2 - eps, - | |
18509 | -eps, <-1 -, -pi/2 + eps -pi/2 + eps, - | |
18510 | 0, >1 +, pi/2 pi/2, + | |
18511 | +eps, >1 +, pi/2 - eps pi/2 - eps, + | |
18512 | -eps, >1 -, pi/2 - eps -pi/2 + eps, + | |
18513 | @end smallexample | |
18514 | ||
18515 | Finally, the following identities help to illustrate the relationship | |
18516 | between the complex trigonometric and hyperbolic functions. They | |
18517 | are valid everywhere, including on the branch cuts. | |
18518 | ||
18519 | @smallexample | |
18520 | sin(i*z) = i*sinh(z) arcsin(i*z) = i*arcsinh(z) | |
18521 | cos(i*z) = cosh(z) arcsinh(i*z) = i*arcsin(z) | |
18522 | tan(i*z) = i*tanh(z) arctan(i*z) = i*arctanh(z) | |
18523 | sinh(i*z) = i*sin(z) cosh(i*z) = cos(z) | |
18524 | @end smallexample | |
18525 | ||
18526 | The ``advanced math'' functions (gamma, Bessel, etc.@:) are also defined | |
18527 | for general complex arguments, but their branch cuts and principal values | |
18528 | are not rigorously specified at present. | |
18529 | ||
18530 | @node Random Numbers, Combinatorial Functions, Branch Cuts, Scientific Functions | |
18531 | @section Random Numbers | |
18532 | ||
18533 | @noindent | |
18534 | @kindex k r | |
18535 | @pindex calc-random | |
18536 | @tindex random | |
18537 | The @kbd{k r} (@code{calc-random}) [@code{random}] command produces | |
18538 | random numbers of various sorts. | |
18539 | ||
18540 | Given a positive numeric prefix argument @cite{M}, it produces a random | |
18541 | integer @cite{N} in the range @c{$0 \le N < M$} | |
18542 | @cite{0 <= N < M}. Each of the @cite{M} | |
18543 | values appears with equal probability.@refill | |
18544 | ||
18545 | With no numeric prefix argument, the @kbd{k r} command takes its argument | |
18546 | from the stack instead. Once again, if this is a positive integer @cite{M} | |
18547 | the result is a random integer less than @cite{M}. However, note that | |
18548 | while numeric prefix arguments are limited to six digits or so, an @cite{M} | |
18549 | taken from the stack can be arbitrarily large. If @cite{M} is negative, | |
18550 | the result is a random integer in the range @c{$M < N \le 0$} | |
18551 | @cite{M < N <= 0}. | |
18552 | ||
18553 | If the value on the stack is a floating-point number @cite{M}, the result | |
18554 | is a random floating-point number @cite{N} in the range @c{$0 \le N < M$} | |
18555 | @cite{0 <= N < M} | |
18556 | or @c{$M < N \le 0$} | |
18557 | @cite{M < N <= 0}, according to the sign of @cite{M}. | |
18558 | ||
18559 | If @cite{M} is zero, the result is a Gaussian-distributed random real | |
18560 | number; the distribution has a mean of zero and a standard deviation | |
18561 | of one. The algorithm used generates random numbers in pairs; thus, | |
18562 | every other call to this function will be especially fast. | |
18563 | ||
18564 | If @cite{M} is an error form @c{$m$ @code{+/-} $\sigma$} | |
5d67986c | 18565 | @samp{m +/- s} where @var{m} |
d7b8e6c6 | 18566 | and @c{$\sigma$} |
5d67986c RS |
18567 | @var{s} are both real numbers, the result uses a Gaussian |
18568 | distribution with mean @var{m} and standard deviation @c{$\sigma$} | |
18569 | @var{s}. | |
d7b8e6c6 EZ |
18570 | |
18571 | If @cite{M} is an interval form, the lower and upper bounds specify the | |
18572 | acceptable limits of the random numbers. If both bounds are integers, | |
18573 | the result is a random integer in the specified range. If either bound | |
18574 | is floating-point, the result is a random real number in the specified | |
18575 | range. If the interval is open at either end, the result will be sure | |
18576 | not to equal that end value. (This makes a big difference for integer | |
18577 | intervals, but for floating-point intervals it's relatively minor: | |
18578 | with a precision of 6, @samp{random([1.0..2.0))} will return any of one | |
18579 | million numbers from 1.00000 to 1.99999; @samp{random([1.0..2.0])} may | |
18580 | additionally return 2.00000, but the probability of this happening is | |
18581 | extremely small.) | |
18582 | ||
18583 | If @cite{M} is a vector, the result is one element taken at random from | |
18584 | the vector. All elements of the vector are given equal probabilities. | |
18585 | ||
18586 | @vindex RandSeed | |
18587 | The sequence of numbers produced by @kbd{k r} is completely random by | |
18588 | default, i.e., the sequence is seeded each time you start Calc using | |
18589 | the current time and other information. You can get a reproducible | |
18590 | sequence by storing a particular ``seed value'' in the Calc variable | |
18591 | @code{RandSeed}. Any integer will do for a seed; integers of from 1 | |
18592 | to 12 digits are good. If you later store a different integer into | |
18593 | @code{RandSeed}, Calc will switch to a different pseudo-random | |
18594 | sequence. If you ``unstore'' @code{RandSeed}, Calc will re-seed itself | |
18595 | from the current time. If you store the same integer that you used | |
18596 | before back into @code{RandSeed}, you will get the exact same sequence | |
18597 | of random numbers as before. | |
18598 | ||
18599 | @pindex calc-rrandom | |
18600 | The @code{calc-rrandom} command (not on any key) produces a random real | |
18601 | number between zero and one. It is equivalent to @samp{random(1.0)}. | |
18602 | ||
18603 | @kindex k a | |
18604 | @pindex calc-random-again | |
18605 | The @kbd{k a} (@code{calc-random-again}) command produces another random | |
18606 | number, re-using the most recent value of @cite{M}. With a numeric | |
18607 | prefix argument @var{n}, it produces @var{n} more random numbers using | |
18608 | that value of @cite{M}. | |
18609 | ||
18610 | @kindex k h | |
18611 | @pindex calc-shuffle | |
18612 | @tindex shuffle | |
18613 | The @kbd{k h} (@code{calc-shuffle}) command produces a vector of several | |
18614 | random values with no duplicates. The value on the top of the stack | |
18615 | specifies the set from which the random values are drawn, and may be any | |
18616 | of the @cite{M} formats described above. The numeric prefix argument | |
18617 | gives the length of the desired list. (If you do not provide a numeric | |
18618 | prefix argument, the length of the list is taken from the top of the | |
18619 | stack, and @cite{M} from second-to-top.) | |
18620 | ||
18621 | If @cite{M} is a floating-point number, zero, or an error form (so | |
18622 | that the random values are being drawn from the set of real numbers) | |
18623 | there is little practical difference between using @kbd{k h} and using | |
18624 | @kbd{k r} several times. But if the set of possible values consists | |
18625 | of just a few integers, or the elements of a vector, then there is | |
18626 | a very real chance that multiple @kbd{k r}'s will produce the same | |
18627 | number more than once. The @kbd{k h} command produces a vector whose | |
18628 | elements are always distinct. (Actually, there is a slight exception: | |
18629 | If @cite{M} is a vector, no given vector element will be drawn more | |
18630 | than once, but if several elements of @cite{M} are equal, they may | |
18631 | each make it into the result vector.) | |
18632 | ||
18633 | One use of @kbd{k h} is to rearrange a list at random. This happens | |
18634 | if the prefix argument is equal to the number of values in the list: | |
18635 | @kbd{[1, 1.5, 2, 2.5, 3] 5 k h} might produce the permuted list | |
18636 | @samp{[2.5, 1, 1.5, 3, 2]}. As a convenient feature, if the argument | |
18637 | @var{n} is negative it is replaced by the size of the set represented | |
18638 | by @cite{M}. Naturally, this is allowed only when @cite{M} specifies | |
18639 | a small discrete set of possibilities. | |
18640 | ||
18641 | To do the equivalent of @kbd{k h} but with duplications allowed, | |
18642 | given @cite{M} on the stack and with @var{n} just entered as a numeric | |
18643 | prefix, use @kbd{v b} to build a vector of copies of @cite{M}, then use | |
18644 | @kbd{V M k r} to ``map'' the normal @kbd{k r} function over the | |
18645 | elements of this vector. @xref{Matrix Functions}. | |
18646 | ||
18647 | @menu | |
18648 | * Random Number Generator:: (Complete description of Calc's algorithm) | |
18649 | @end menu | |
18650 | ||
18651 | @node Random Number Generator, , Random Numbers, Random Numbers | |
18652 | @subsection Random Number Generator | |
18653 | ||
18654 | Calc's random number generator uses several methods to ensure that | |
18655 | the numbers it produces are highly random. Knuth's @emph{Art of | |
18656 | Computer Programming}, Volume II, contains a thorough description | |
18657 | of the theory of random number generators and their measurement and | |
18658 | characterization. | |
18659 | ||
18660 | If @code{RandSeed} has no stored value, Calc calls Emacs' built-in | |
18661 | @code{random} function to get a stream of random numbers, which it | |
18662 | then treats in various ways to avoid problems inherent in the simple | |
18663 | random number generators that many systems use to implement @code{random}. | |
18664 | ||
18665 | When Calc's random number generator is first invoked, it ``seeds'' | |
18666 | the low-level random sequence using the time of day, so that the | |
18667 | random number sequence will be different every time you use Calc. | |
18668 | ||
18669 | Since Emacs Lisp doesn't specify the range of values that will be | |
18670 | returned by its @code{random} function, Calc exercises the function | |
18671 | several times to estimate the range. When Calc subsequently uses | |
18672 | the @code{random} function, it takes only 10 bits of the result | |
18673 | near the most-significant end. (It avoids at least the bottom | |
18674 | four bits, preferably more, and also tries to avoid the top two | |
18675 | bits.) This strategy works well with the linear congruential | |
18676 | generators that are typically used to implement @code{random}. | |
18677 | ||
18678 | If @code{RandSeed} contains an integer, Calc uses this integer to | |
18679 | seed an ``additive congruential'' method (Knuth's algorithm 3.2.2A, | |
18680 | computing @c{$X_{n-55} - X_{n-24}$} | |
18681 | @cite{X_n-55 - X_n-24}). This method expands the seed | |
18682 | value into a large table which is maintained internally; the variable | |
18683 | @code{RandSeed} is changed from, e.g., 42 to the vector @cite{[42]} | |
18684 | to indicate that the seed has been absorbed into this table. When | |
18685 | @code{RandSeed} contains a vector, @kbd{k r} and related commands | |
18686 | continue to use the same internal table as last time. There is no | |
18687 | way to extract the complete state of the random number generator | |
18688 | so that you can restart it from any point; you can only restart it | |
18689 | from the same initial seed value. A simple way to restart from the | |
18690 | same seed is to type @kbd{s r RandSeed} to get the seed vector, | |
18691 | @kbd{v u} to unpack it back into a number, then @kbd{s t RandSeed} | |
18692 | to reseed the generator with that number. | |
18693 | ||
18694 | Calc uses a ``shuffling'' method as described in algorithm 3.2.2B | |
18695 | of Knuth. It fills a table with 13 random 10-bit numbers. Then, | |
18696 | to generate a new random number, it uses the previous number to | |
18697 | index into the table, picks the value it finds there as the new | |
18698 | random number, then replaces that table entry with a new value | |
18699 | obtained from a call to the base random number generator (either | |
18700 | the additive congruential generator or the @code{random} function | |
18701 | supplied by the system). If there are any flaws in the base | |
18702 | generator, shuffling will tend to even them out. But if the system | |
18703 | provides an excellent @code{random} function, shuffling will not | |
18704 | damage its randomness. | |
18705 | ||
18706 | To create a random integer of a certain number of digits, Calc | |
18707 | builds the integer three decimal digits at a time. For each group | |
18708 | of three digits, Calc calls its 10-bit shuffling random number generator | |
18709 | (which returns a value from 0 to 1023); if the random value is 1000 | |
18710 | or more, Calc throws it out and tries again until it gets a suitable | |
18711 | value. | |
18712 | ||
18713 | To create a random floating-point number with precision @var{p}, Calc | |
18714 | simply creates a random @var{p}-digit integer and multiplies by | |
18715 | @c{$10^{-p}$} | |
18716 | @cite{10^-p}. The resulting random numbers should be very clean, but note | |
18717 | that relatively small numbers will have few significant random digits. | |
18718 | In other words, with a precision of 12, you will occasionally get | |
18719 | numbers on the order of @c{$10^{-9}$} | |
18720 | @cite{10^-9} or @c{$10^{-10}$} | |
18721 | @cite{10^-10}, but those numbers | |
18722 | will only have two or three random digits since they correspond to small | |
18723 | integers times @c{$10^{-12}$} | |
18724 | @cite{10^-12}. | |
18725 | ||
18726 | To create a random integer in the interval @samp{[0 .. @var{m})}, Calc | |
18727 | counts the digits in @var{m}, creates a random integer with three | |
18728 | additional digits, then reduces modulo @var{m}. Unless @var{m} is a | |
18729 | power of ten the resulting values will be very slightly biased toward | |
18730 | the lower numbers, but this bias will be less than 0.1%. (For example, | |
18731 | if @var{m} is 42, Calc will reduce a random integer less than 100000 | |
18732 | modulo 42 to get a result less than 42. It is easy to show that the | |
18733 | numbers 40 and 41 will be only 2380/2381 as likely to result from this | |
18734 | modulo operation as numbers 39 and below.) If @var{m} is a power of | |
18735 | ten, however, the numbers should be completely unbiased. | |
18736 | ||
18737 | The Gaussian random numbers generated by @samp{random(0.0)} use the | |
18738 | ``polar'' method described in Knuth section 3.4.1C. This method | |
18739 | generates a pair of Gaussian random numbers at a time, so only every | |
18740 | other call to @samp{random(0.0)} will require significant calculations. | |
18741 | ||
18742 | @node Combinatorial Functions, Probability Distribution Functions, Random Numbers, Scientific Functions | |
18743 | @section Combinatorial Functions | |
18744 | ||
18745 | @noindent | |
18746 | Commands relating to combinatorics and number theory begin with the | |
18747 | @kbd{k} key prefix. | |
18748 | ||
18749 | @kindex k g | |
18750 | @pindex calc-gcd | |
18751 | @tindex gcd | |
18752 | The @kbd{k g} (@code{calc-gcd}) [@code{gcd}] command computes the | |
18753 | Greatest Common Divisor of two integers. It also accepts fractions; | |
18754 | the GCD of two fractions is defined by taking the GCD of the | |
18755 | numerators, and the LCM of the denominators. This definition is | |
18756 | consistent with the idea that @samp{a / gcd(a,x)} should yield an | |
18757 | integer for any @samp{a} and @samp{x}. For other types of arguments, | |
18758 | the operation is left in symbolic form.@refill | |
18759 | ||
18760 | @kindex k l | |
18761 | @pindex calc-lcm | |
18762 | @tindex lcm | |
18763 | The @kbd{k l} (@code{calc-lcm}) [@code{lcm}] command computes the | |
18764 | Least Common Multiple of two integers or fractions. The product of | |
18765 | the LCM and GCD of two numbers is equal to the product of the | |
18766 | numbers.@refill | |
18767 | ||
18768 | @kindex k E | |
18769 | @pindex calc-extended-gcd | |
18770 | @tindex egcd | |
18771 | The @kbd{k E} (@code{calc-extended-gcd}) [@code{egcd}] command computes | |
18772 | the GCD of two integers @cite{x} and @cite{y} and returns a vector | |
18773 | @cite{[g, a, b]} where @c{$g = \gcd(x,y) = a x + b y$} | |
18774 | @cite{g = gcd(x,y) = a x + b y}. | |
18775 | ||
18776 | @kindex ! | |
18777 | @pindex calc-factorial | |
18778 | @tindex fact | |
5d67986c RS |
18779 | @ignore |
18780 | @mindex @null | |
18781 | @end ignore | |
d7b8e6c6 EZ |
18782 | @tindex ! |
18783 | The @kbd{!} (@code{calc-factorial}) [@code{fact}] command computes the | |
18784 | factorial of the number at the top of the stack. If the number is an | |
18785 | integer, the result is an exact integer. If the number is an | |
18786 | integer-valued float, the result is a floating-point approximation. If | |
18787 | the number is a non-integral real number, the generalized factorial is used, | |
18788 | as defined by the Euler Gamma function. Please note that computation of | |
18789 | large factorials can be slow; using floating-point format will help | |
18790 | since fewer digits must be maintained. The same is true of many of | |
18791 | the commands in this section.@refill | |
18792 | ||
18793 | @kindex k d | |
18794 | @pindex calc-double-factorial | |
18795 | @tindex dfact | |
5d67986c RS |
18796 | @ignore |
18797 | @mindex @null | |
18798 | @end ignore | |
d7b8e6c6 EZ |
18799 | @tindex !! |
18800 | The @kbd{k d} (@code{calc-double-factorial}) [@code{dfact}] command | |
18801 | computes the ``double factorial'' of an integer. For an even integer, | |
18802 | this is the product of even integers from 2 to @cite{N}. For an odd | |
18803 | integer, this is the product of odd integers from 3 to @cite{N}. If | |
18804 | the argument is an integer-valued float, the result is a floating-point | |
18805 | approximation. This function is undefined for negative even integers. | |
18806 | The notation @cite{N!!} is also recognized for double factorials.@refill | |
18807 | ||
18808 | @kindex k c | |
18809 | @pindex calc-choose | |
18810 | @tindex choose | |
18811 | The @kbd{k c} (@code{calc-choose}) [@code{choose}] command computes the | |
18812 | binomial coefficient @cite{N}-choose-@cite{M}, where @cite{M} is the number | |
18813 | on the top of the stack and @cite{N} is second-to-top. If both arguments | |
18814 | are integers, the result is an exact integer. Otherwise, the result is a | |
18815 | floating-point approximation. The binomial coefficient is defined for all | |
18816 | real numbers by @c{$N! \over M! (N-M)!\,$} | |
18817 | @cite{N! / M! (N-M)!}. | |
18818 | ||
18819 | @kindex H k c | |
18820 | @pindex calc-perm | |
18821 | @tindex perm | |
18822 | @ifinfo | |
18823 | The @kbd{H k c} (@code{calc-perm}) [@code{perm}] command computes the | |
18824 | number-of-permutations function @cite{N! / (N-M)!}. | |
18825 | @end ifinfo | |
18826 | @tex | |
18827 | The \kbd{H k c} (\code{calc-perm}) [\code{perm}] command computes the | |
18828 | number-of-perm\-utations function $N! \over (N-M)!\,$. | |
18829 | @end tex | |
18830 | ||
18831 | @kindex k b | |
18832 | @kindex H k b | |
18833 | @pindex calc-bernoulli-number | |
18834 | @tindex bern | |
18835 | The @kbd{k b} (@code{calc-bernoulli-number}) [@code{bern}] command | |
18836 | computes a given Bernoulli number. The value at the top of the stack | |
18837 | is a nonnegative integer @cite{n} that specifies which Bernoulli number | |
18838 | is desired. The @kbd{H k b} command computes a Bernoulli polynomial, | |
18839 | taking @cite{n} from the second-to-top position and @cite{x} from the | |
18840 | top of the stack. If @cite{x} is a variable or formula the result is | |
18841 | a polynomial in @cite{x}; if @cite{x} is a number the result is a number. | |
18842 | ||
18843 | @kindex k e | |
18844 | @kindex H k e | |
18845 | @pindex calc-euler-number | |
18846 | @tindex euler | |
18847 | The @kbd{k e} (@code{calc-euler-number}) [@code{euler}] command similarly | |
18848 | computes an Euler number, and @w{@kbd{H k e}} computes an Euler polynomial. | |
18849 | Bernoulli and Euler numbers occur in the Taylor expansions of several | |
18850 | functions. | |
18851 | ||
18852 | @kindex k s | |
18853 | @kindex H k s | |
18854 | @pindex calc-stirling-number | |
18855 | @tindex stir1 | |
18856 | @tindex stir2 | |
18857 | The @kbd{k s} (@code{calc-stirling-number}) [@code{stir1}] command | |
18858 | computes a Stirling number of the first kind@c{ $n \brack m$} | |
18859 | @asis{}, given two integers | |
18860 | @cite{n} and @cite{m} on the stack. The @kbd{H k s} [@code{stir2}] | |
18861 | command computes a Stirling number of the second kind@c{ $n \brace m$} | |
18862 | @asis{}. These are | |
18863 | the number of @cite{m}-cycle permutations of @cite{n} objects, and | |
18864 | the number of ways to partition @cite{n} objects into @cite{m} | |
18865 | non-empty sets, respectively. | |
18866 | ||
18867 | @kindex k p | |
18868 | @pindex calc-prime-test | |
18869 | @cindex Primes | |
18870 | The @kbd{k p} (@code{calc-prime-test}) command checks if the integer on | |
18871 | the top of the stack is prime. For integers less than eight million, the | |
18872 | answer is always exact and reasonably fast. For larger integers, a | |
18873 | probabilistic method is used (see Knuth vol. II, section 4.5.4, algorithm P). | |
18874 | The number is first checked against small prime factors (up to 13). Then, | |
18875 | any number of iterations of the algorithm are performed. Each step either | |
18876 | discovers that the number is non-prime, or substantially increases the | |
18877 | certainty that the number is prime. After a few steps, the chance that | |
18878 | a number was mistakenly described as prime will be less than one percent. | |
18879 | (Indeed, this is a worst-case estimate of the probability; in practice | |
18880 | even a single iteration is quite reliable.) After the @kbd{k p} command, | |
18881 | the number will be reported as definitely prime or non-prime if possible, | |
18882 | or otherwise ``probably'' prime with a certain probability of error. | |
18883 | ||
5d67986c RS |
18884 | @ignore |
18885 | @starindex | |
18886 | @end ignore | |
d7b8e6c6 EZ |
18887 | @tindex prime |
18888 | The normal @kbd{k p} command performs one iteration of the primality | |
18889 | test. Pressing @kbd{k p} repeatedly for the same integer will perform | |
18890 | additional iterations. Also, @kbd{k p} with a numeric prefix performs | |
18891 | the specified number of iterations. There is also an algebraic function | |
18892 | @samp{prime(n)} or @samp{prime(n,iters)} which returns 1 if @cite{n} | |
18893 | is (probably) prime and 0 if not. | |
18894 | ||
18895 | @kindex k f | |
18896 | @pindex calc-prime-factors | |
18897 | @tindex prfac | |
18898 | The @kbd{k f} (@code{calc-prime-factors}) [@code{prfac}] command | |
18899 | attempts to decompose an integer into its prime factors. For numbers up | |
18900 | to 25 million, the answer is exact although it may take some time. The | |
18901 | result is a vector of the prime factors in increasing order. For larger | |
18902 | inputs, prime factors above 5000 may not be found, in which case the | |
18903 | last number in the vector will be an unfactored integer greater than 25 | |
18904 | million (with a warning message). For negative integers, the first | |
18905 | element of the list will be @i{-1}. For inputs @i{-1}, @i{0}, and | |
18906 | @i{1}, the result is a list of the same number. | |
18907 | ||
18908 | @kindex k n | |
18909 | @pindex calc-next-prime | |
5d67986c RS |
18910 | @ignore |
18911 | @mindex nextpr@idots | |
18912 | @end ignore | |
d7b8e6c6 EZ |
18913 | @tindex nextprime |
18914 | The @kbd{k n} (@code{calc-next-prime}) [@code{nextprime}] command finds | |
18915 | the next prime above a given number. Essentially, it searches by calling | |
18916 | @code{calc-prime-test} on successive integers until it finds one that | |
18917 | passes the test. This is quite fast for integers less than eight million, | |
18918 | but once the probabilistic test comes into play the search may be rather | |
18919 | slow. Ordinarily this command stops for any prime that passes one iteration | |
18920 | of the primality test. With a numeric prefix argument, a number must pass | |
18921 | the specified number of iterations before the search stops. (This only | |
18922 | matters when searching above eight million.) You can always use additional | |
18923 | @kbd{k p} commands to increase your certainty that the number is indeed | |
18924 | prime. | |
18925 | ||
18926 | @kindex I k n | |
18927 | @pindex calc-prev-prime | |
5d67986c RS |
18928 | @ignore |
18929 | @mindex prevpr@idots | |
18930 | @end ignore | |
d7b8e6c6 EZ |
18931 | @tindex prevprime |
18932 | The @kbd{I k n} (@code{calc-prev-prime}) [@code{prevprime}] command | |
18933 | analogously finds the next prime less than a given number. | |
18934 | ||
18935 | @kindex k t | |
18936 | @pindex calc-totient | |
18937 | @tindex totient | |
18938 | The @kbd{k t} (@code{calc-totient}) [@code{totient}] command computes the | |
18939 | Euler ``totient'' function@c{ $\phi(n)$} | |
18940 | @asis{}, the number of integers less than @cite{n} which | |
18941 | are relatively prime to @cite{n}. | |
18942 | ||
18943 | @kindex k m | |
18944 | @pindex calc-moebius | |
18945 | @tindex moebius | |
18946 | The @kbd{k m} (@code{calc-moebius}) [@code{moebius}] command computes the | |
18947 | @c{M\"obius $\mu$} | |
18948 | @asis{Moebius ``mu''} function. If the input number is a product of @cite{k} | |
18949 | distinct factors, this is @cite{(-1)^k}. If the input number has any | |
18950 | duplicate factors (i.e., can be divided by the same prime more than once), | |
18951 | the result is zero. | |
18952 | ||
18953 | @node Probability Distribution Functions, , Combinatorial Functions, Scientific Functions | |
18954 | @section Probability Distribution Functions | |
18955 | ||
18956 | @noindent | |
18957 | The functions in this section compute various probability distributions. | |
18958 | For continuous distributions, this is the integral of the probability | |
18959 | density function from @cite{x} to infinity. (These are the ``upper | |
18960 | tail'' distribution functions; there are also corresponding ``lower | |
18961 | tail'' functions which integrate from minus infinity to @cite{x}.) | |
18962 | For discrete distributions, the upper tail function gives the sum | |
18963 | from @cite{x} to infinity; the lower tail function gives the sum | |
18964 | from minus infinity up to, but not including,@w{ }@cite{x}. | |
18965 | ||
18966 | To integrate from @cite{x} to @cite{y}, just use the distribution | |
18967 | function twice and subtract. For example, the probability that a | |
18968 | Gaussian random variable with mean 2 and standard deviation 1 will | |
18969 | lie in the range from 2.5 to 2.8 is @samp{utpn(2.5,2,1) - utpn(2.8,2,1)} | |
18970 | (``the probability that it is greater than 2.5, but not greater than 2.8''), | |
18971 | or equivalently @samp{ltpn(2.8,2,1) - ltpn(2.5,2,1)}. | |
18972 | ||
18973 | @kindex k B | |
18974 | @kindex I k B | |
18975 | @pindex calc-utpb | |
18976 | @tindex utpb | |
18977 | @tindex ltpb | |
18978 | The @kbd{k B} (@code{calc-utpb}) [@code{utpb}] function uses the | |
18979 | binomial distribution. Push the parameters @var{n}, @var{p}, and | |
18980 | then @var{x} onto the stack; the result (@samp{utpb(x,n,p)}) is the | |
18981 | probability that an event will occur @var{x} or more times out | |
18982 | of @var{n} trials, if its probability of occurring in any given | |
18983 | trial is @var{p}. The @kbd{I k B} [@code{ltpb}] function is | |
18984 | the probability that the event will occur fewer than @var{x} times. | |
18985 | ||
18986 | The other probability distribution functions similarly take the | |
18987 | form @kbd{k @var{X}} (@code{calc-utp@var{x}}) [@code{utp@var{x}}] | |
18988 | and @kbd{I k @var{X}} [@code{ltp@var{x}}], for various letters | |
18989 | @var{x}. The arguments to the algebraic functions are the value of | |
18990 | the random variable first, then whatever other parameters define the | |
18991 | distribution. Note these are among the few Calc functions where the | |
18992 | order of the arguments in algebraic form differs from the order of | |
18993 | arguments as found on the stack. (The random variable comes last on | |
18994 | the stack, so that you can type, e.g., @kbd{2 @key{RET} 1 @key{RET} 2.5 | |
18995 | k N M-@key{RET} @key{DEL} 2.8 k N -}, using @kbd{M-@key{RET} @key{DEL}} to | |
18996 | recover the original arguments but substitute a new value for @cite{x}.) | |
18997 | ||
18998 | @kindex k C | |
18999 | @pindex calc-utpc | |
19000 | @tindex utpc | |
5d67986c RS |
19001 | @ignore |
19002 | @mindex @idots | |
19003 | @end ignore | |
d7b8e6c6 | 19004 | @kindex I k C |
5d67986c RS |
19005 | @ignore |
19006 | @mindex @null | |
19007 | @end ignore | |
d7b8e6c6 EZ |
19008 | @tindex ltpc |
19009 | The @samp{utpc(x,v)} function uses the chi-square distribution with | |
19010 | @c{$\nu$} | |
19011 | @cite{v} degrees of freedom. It is the probability that a model is | |
19012 | correct if its chi-square statistic is @cite{x}. | |
19013 | ||
19014 | @kindex k F | |
19015 | @pindex calc-utpf | |
19016 | @tindex utpf | |
5d67986c RS |
19017 | @ignore |
19018 | @mindex @idots | |
19019 | @end ignore | |
d7b8e6c6 | 19020 | @kindex I k F |
5d67986c RS |
19021 | @ignore |
19022 | @mindex @null | |
19023 | @end ignore | |
d7b8e6c6 EZ |
19024 | @tindex ltpf |
19025 | The @samp{utpf(F,v1,v2)} function uses the F distribution, used in | |
19026 | various statistical tests. The parameters @c{$\nu_1$} | |
19027 | @cite{v1} and @c{$\nu_2$} | |
19028 | @cite{v2} | |
19029 | are the degrees of freedom in the numerator and denominator, | |
19030 | respectively, used in computing the statistic @cite{F}. | |
19031 | ||
19032 | @kindex k N | |
19033 | @pindex calc-utpn | |
19034 | @tindex utpn | |
5d67986c RS |
19035 | @ignore |
19036 | @mindex @idots | |
19037 | @end ignore | |
d7b8e6c6 | 19038 | @kindex I k N |
5d67986c RS |
19039 | @ignore |
19040 | @mindex @null | |
19041 | @end ignore | |
d7b8e6c6 EZ |
19042 | @tindex ltpn |
19043 | The @samp{utpn(x,m,s)} function uses a normal (Gaussian) distribution | |
19044 | with mean @cite{m} and standard deviation @c{$\sigma$} | |
19045 | @cite{s}. It is the | |
19046 | probability that such a normal-distributed random variable would | |
19047 | exceed @cite{x}. | |
19048 | ||
19049 | @kindex k P | |
19050 | @pindex calc-utpp | |
19051 | @tindex utpp | |
5d67986c RS |
19052 | @ignore |
19053 | @mindex @idots | |
19054 | @end ignore | |
d7b8e6c6 | 19055 | @kindex I k P |
5d67986c RS |
19056 | @ignore |
19057 | @mindex @null | |
19058 | @end ignore | |
d7b8e6c6 EZ |
19059 | @tindex ltpp |
19060 | The @samp{utpp(n,x)} function uses a Poisson distribution with | |
19061 | mean @cite{x}. It is the probability that @cite{n} or more such | |
19062 | Poisson random events will occur. | |
19063 | ||
19064 | @kindex k T | |
19065 | @pindex calc-ltpt | |
19066 | @tindex utpt | |
5d67986c RS |
19067 | @ignore |
19068 | @mindex @idots | |
19069 | @end ignore | |
d7b8e6c6 | 19070 | @kindex I k T |
5d67986c RS |
19071 | @ignore |
19072 | @mindex @null | |
19073 | @end ignore | |
d7b8e6c6 EZ |
19074 | @tindex ltpt |
19075 | The @samp{utpt(t,v)} function uses the Student's ``t'' distribution | |
19076 | with @c{$\nu$} | |
19077 | @cite{v} degrees of freedom. It is the probability that a | |
19078 | t-distributed random variable will be greater than @cite{t}. | |
19079 | (Note: This computes the distribution function @c{$A(t|\nu)$} | |
19080 | @cite{A(t|v)} | |
19081 | where @c{$A(0|\nu) = 1$} | |
19082 | @cite{A(0|v) = 1} and @c{$A(\infty|\nu) \to 0$} | |
19083 | @cite{A(inf|v) -> 0}. The | |
19084 | @code{UTPT} operation on the HP-48 uses a different definition | |
19085 | which returns half of Calc's value: @samp{UTPT(t,v) = .5*utpt(t,v)}.) | |
19086 | ||
19087 | While Calc does not provide inverses of the probability distribution | |
19088 | functions, the @kbd{a R} command can be used to solve for the inverse. | |
19089 | Since the distribution functions are monotonic, @kbd{a R} is guaranteed | |
19090 | to be able to find a solution given any initial guess. | |
19091 | @xref{Numerical Solutions}. | |
19092 | ||
19093 | @node Matrix Functions, Algebra, Scientific Functions, Top | |
19094 | @chapter Vector/Matrix Functions | |
19095 | ||
19096 | @noindent | |
19097 | Many of the commands described here begin with the @kbd{v} prefix. | |
19098 | (For convenience, the shift-@kbd{V} prefix is equivalent to @kbd{v}.) | |
19099 | The commands usually apply to both plain vectors and matrices; some | |
19100 | apply only to matrices or only to square matrices. If the argument | |
19101 | has the wrong dimensions the operation is left in symbolic form. | |
19102 | ||
19103 | Vectors are entered and displayed using @samp{[a,b,c]} notation. | |
19104 | Matrices are vectors of which all elements are vectors of equal length. | |
19105 | (Though none of the standard Calc commands use this concept, a | |
19106 | three-dimensional matrix or rank-3 tensor could be defined as a | |
19107 | vector of matrices, and so on.) | |
19108 | ||
19109 | @menu | |
19110 | * Packing and Unpacking:: | |
19111 | * Building Vectors:: | |
19112 | * Extracting Elements:: | |
19113 | * Manipulating Vectors:: | |
19114 | * Vector and Matrix Arithmetic:: | |
19115 | * Set Operations:: | |
19116 | * Statistical Operations:: | |
19117 | * Reducing and Mapping:: | |
19118 | * Vector and Matrix Formats:: | |
19119 | @end menu | |
19120 | ||
19121 | @node Packing and Unpacking, Building Vectors, Matrix Functions, Matrix Functions | |
19122 | @section Packing and Unpacking | |
19123 | ||
19124 | @noindent | |
19125 | Calc's ``pack'' and ``unpack'' commands collect stack entries to build | |
19126 | composite objects such as vectors and complex numbers. They are | |
19127 | described in this chapter because they are most often used to build | |
19128 | vectors. | |
19129 | ||
19130 | @kindex v p | |
19131 | @pindex calc-pack | |
19132 | The @kbd{v p} (@code{calc-pack}) [@code{pack}] command collects several | |
19133 | elements from the stack into a matrix, complex number, HMS form, error | |
19134 | form, etc. It uses a numeric prefix argument to specify the kind of | |
19135 | object to be built; this argument is referred to as the ``packing mode.'' | |
19136 | If the packing mode is a nonnegative integer, a vector of that | |
19137 | length is created. For example, @kbd{C-u 5 v p} will pop the top | |
19138 | five stack elements and push back a single vector of those five | |
19139 | elements. (@kbd{C-u 0 v p} simply creates an empty vector.) | |
19140 | ||
19141 | The same effect can be had by pressing @kbd{[} to push an incomplete | |
19142 | vector on the stack, using @key{TAB} (@code{calc-roll-down}) to sneak | |
19143 | the incomplete object up past a certain number of elements, and | |
19144 | then pressing @kbd{]} to complete the vector. | |
19145 | ||
19146 | Negative packing modes create other kinds of composite objects: | |
19147 | ||
19148 | @table @cite | |
19149 | @item -1 | |
19150 | Two values are collected to build a complex number. For example, | |
19151 | @kbd{5 @key{RET} 7 C-u -1 v p} creates the complex number | |
19152 | @cite{(5, 7)}. The result is always a rectangular complex | |
19153 | number. The two input values must both be real numbers, | |
19154 | i.e., integers, fractions, or floats. If they are not, Calc | |
19155 | will instead build a formula like @samp{a + (0, 1) b}. (The | |
19156 | other packing modes also create a symbolic answer if the | |
19157 | components are not suitable.) | |
19158 | ||
19159 | @item -2 | |
19160 | Two values are collected to build a polar complex number. | |
19161 | The first is the magnitude; the second is the phase expressed | |
19162 | in either degrees or radians according to the current angular | |
19163 | mode. | |
19164 | ||
19165 | @item -3 | |
19166 | Three values are collected into an HMS form. The first | |
19167 | two values (hours and minutes) must be integers or | |
19168 | integer-valued floats. The third value may be any real | |
19169 | number. | |
19170 | ||
19171 | @item -4 | |
19172 | Two values are collected into an error form. The inputs | |
19173 | may be real numbers or formulas. | |
19174 | ||
19175 | @item -5 | |
19176 | Two values are collected into a modulo form. The inputs | |
19177 | must be real numbers. | |
19178 | ||
19179 | @item -6 | |
19180 | Two values are collected into the interval @samp{[a .. b]}. | |
19181 | The inputs may be real numbers, HMS or date forms, or formulas. | |
19182 | ||
19183 | @item -7 | |
19184 | Two values are collected into the interval @samp{[a .. b)}. | |
19185 | ||
19186 | @item -8 | |
19187 | Two values are collected into the interval @samp{(a .. b]}. | |
19188 | ||
19189 | @item -9 | |
19190 | Two values are collected into the interval @samp{(a .. b)}. | |
19191 | ||
19192 | @item -10 | |
19193 | Two integer values are collected into a fraction. | |
19194 | ||
19195 | @item -11 | |
19196 | Two values are collected into a floating-point number. | |
19197 | The first is the mantissa; the second, which must be an | |
19198 | integer, is the exponent. The result is the mantissa | |
19199 | times ten to the power of the exponent. | |
19200 | ||
19201 | @item -12 | |
19202 | This is treated the same as @i{-11} by the @kbd{v p} command. | |
19203 | When unpacking, @i{-12} specifies that a floating-point mantissa | |
19204 | is desired. | |
19205 | ||
19206 | @item -13 | |
19207 | A real number is converted into a date form. | |
19208 | ||
19209 | @item -14 | |
19210 | Three numbers (year, month, day) are packed into a pure date form. | |
19211 | ||
19212 | @item -15 | |
19213 | Six numbers are packed into a date/time form. | |
19214 | @end table | |
19215 | ||
19216 | With any of the two-input negative packing modes, either or both | |
19217 | of the inputs may be vectors. If both are vectors of the same | |
19218 | length, the result is another vector made by packing corresponding | |
19219 | elements of the input vectors. If one input is a vector and the | |
19220 | other is a plain number, the number is packed along with each vector | |
19221 | element to produce a new vector. For example, @kbd{C-u -4 v p} | |
19222 | could be used to convert a vector of numbers and a vector of errors | |
19223 | into a single vector of error forms; @kbd{C-u -5 v p} could convert | |
19224 | a vector of numbers and a single number @var{M} into a vector of | |
19225 | numbers modulo @var{M}. | |
19226 | ||
19227 | If you don't give a prefix argument to @kbd{v p}, it takes | |
19228 | the packing mode from the top of the stack. The elements to | |
19229 | be packed then begin at stack level 2. Thus | |
19230 | @kbd{1 @key{RET} 2 @key{RET} 4 n v p} is another way to | |
19231 | enter the error form @samp{1 +/- 2}. | |
19232 | ||
19233 | If the packing mode taken from the stack is a vector, the result is a | |
19234 | matrix with the dimensions specified by the elements of the vector, | |
19235 | which must each be integers. For example, if the packing mode is | |
19236 | @samp{[2, 3]}, then six numbers will be taken from the stack and | |
19237 | returned in the form @samp{[@w{[a, b, c]}, [d, e, f]]}. | |
19238 | ||
19239 | If any elements of the vector are negative, other kinds of | |
19240 | packing are done at that level as described above. For | |
19241 | example, @samp{[2, 3, -4]} takes 12 objects and creates a | |
19242 | @c{$2\times3$} | |
19243 | @asis{2x3} matrix of error forms: @samp{[[a +/- b, c +/- d ... ]]}. | |
19244 | Also, @samp{[-4, -10]} will convert four integers into an | |
19245 | error form consisting of two fractions: @samp{a:b +/- c:d}. | |
19246 | ||
5d67986c RS |
19247 | @ignore |
19248 | @starindex | |
19249 | @end ignore | |
d7b8e6c6 EZ |
19250 | @tindex pack |
19251 | There is an equivalent algebraic function, | |
19252 | @samp{pack(@var{mode}, @var{items})} where @var{mode} is a | |
19253 | packing mode (an integer or a vector of integers) and @var{items} | |
19254 | is a vector of objects to be packed (re-packed, really) according | |
19255 | to that mode. For example, @samp{pack([3, -4], [a,b,c,d,e,f])} | |
19256 | yields @samp{[a +/- b, @w{c +/- d}, e +/- f]}. The function is | |
19257 | left in symbolic form if the packing mode is illegal, or if the | |
19258 | number of data items does not match the number of items required | |
19259 | by the mode. | |
19260 | ||
19261 | @kindex v u | |
19262 | @pindex calc-unpack | |
19263 | The @kbd{v u} (@code{calc-unpack}) command takes the vector, complex | |
19264 | number, HMS form, or other composite object on the top of the stack and | |
19265 | ``unpacks'' it, pushing each of its elements onto the stack as separate | |
19266 | objects. Thus, it is the ``inverse'' of @kbd{v p}. If the value | |
19267 | at the top of the stack is a formula, @kbd{v u} unpacks it by pushing | |
19268 | each of the arguments of the top-level operator onto the stack. | |
19269 | ||
19270 | You can optionally give a numeric prefix argument to @kbd{v u} | |
19271 | to specify an explicit (un)packing mode. If the packing mode is | |
19272 | negative and the input is actually a vector or matrix, the result | |
19273 | will be two or more similar vectors or matrices of the elements. | |
19274 | For example, given the vector @samp{[@w{a +/- b}, c^2, d +/- 7]}, | |
19275 | the result of @kbd{C-u -4 v u} will be the two vectors | |
19276 | @samp{[a, c^2, d]} and @w{@samp{[b, 0, 7]}}. | |
19277 | ||
19278 | Note that the prefix argument can have an effect even when the input is | |
19279 | not a vector. For example, if the input is the number @i{-5}, then | |
19280 | @kbd{c-u -1 v u} yields @i{-5} and 0 (the components of @i{-5} | |
19281 | when viewed as a rectangular complex number); @kbd{C-u -2 v u} yields 5 | |
19282 | and 180 (assuming degrees mode); and @kbd{C-u -10 v u} yields @i{-5} | |
19283 | and 1 (the numerator and denominator of @i{-5}, viewed as a rational | |
19284 | number). Plain @kbd{v u} with this input would complain that the input | |
19285 | is not a composite object. | |
19286 | ||
19287 | Unpacking mode @i{-11} converts a float into an integer mantissa and | |
19288 | an integer exponent, where the mantissa is not divisible by 10 | |
19289 | (except that 0.0 is represented by a mantissa and exponent of 0). | |
19290 | Unpacking mode @i{-12} converts a float into a floating-point mantissa | |
19291 | and integer exponent, where the mantissa (for non-zero numbers) | |
19292 | is guaranteed to lie in the range [1 .. 10). In both cases, | |
19293 | the mantissa is shifted left or right (and the exponent adjusted | |
19294 | to compensate) in order to satisfy these constraints. | |
19295 | ||
19296 | Positive unpacking modes are treated differently than for @kbd{v p}. | |
19297 | A mode of 1 is much like plain @kbd{v u} with no prefix argument, | |
19298 | except that in addition to the components of the input object, | |
19299 | a suitable packing mode to re-pack the object is also pushed. | |
19300 | Thus, @kbd{C-u 1 v u} followed by @kbd{v p} will re-build the | |
19301 | original object. | |
19302 | ||
19303 | A mode of 2 unpacks two levels of the object; the resulting | |
19304 | re-packing mode will be a vector of length 2. This might be used | |
19305 | to unpack a matrix, say, or a vector of error forms. Higher | |
19306 | unpacking modes unpack the input even more deeply. | |
19307 | ||
5d67986c RS |
19308 | @ignore |
19309 | @starindex | |
19310 | @end ignore | |
d7b8e6c6 EZ |
19311 | @tindex unpack |
19312 | There are two algebraic functions analogous to @kbd{v u}. | |
19313 | The @samp{unpack(@var{mode}, @var{item})} function unpacks the | |
19314 | @var{item} using the given @var{mode}, returning the result as | |
19315 | a vector of components. Here the @var{mode} must be an | |
19316 | integer, not a vector. For example, @samp{unpack(-4, a +/- b)} | |
19317 | returns @samp{[a, b]}, as does @samp{unpack(1, a +/- b)}. | |
19318 | ||
5d67986c RS |
19319 | @ignore |
19320 | @starindex | |
19321 | @end ignore | |
d7b8e6c6 EZ |
19322 | @tindex unpackt |
19323 | The @code{unpackt} function is like @code{unpack} but instead | |
19324 | of returning a simple vector of items, it returns a vector of | |
19325 | two things: The mode, and the vector of items. For example, | |
19326 | @samp{unpackt(1, 2:3 +/- 1:4)} returns @samp{[-4, [2:3, 1:4]]}, | |
19327 | and @samp{unpackt(2, 2:3 +/- 1:4)} returns @samp{[[-4, -10], [2, 3, 1, 4]]}. | |
19328 | The identity for re-building the original object is | |
19329 | @samp{apply(pack, unpackt(@var{n}, @var{x})) = @var{x}}. (The | |
19330 | @code{apply} function builds a function call given the function | |
19331 | name and a vector of arguments.) | |
19332 | ||
19333 | @cindex Numerator of a fraction, extracting | |
19334 | Subscript notation is a useful way to extract a particular part | |
19335 | of an object. For example, to get the numerator of a rational | |
19336 | number, you can use @samp{unpack(-10, @var{x})_1}. | |
19337 | ||
19338 | @node Building Vectors, Extracting Elements, Packing and Unpacking, Matrix Functions | |
19339 | @section Building Vectors | |
19340 | ||
19341 | @noindent | |
19342 | Vectors and matrices can be added, | |
19343 | subtracted, multiplied, and divided; @pxref{Basic Arithmetic}.@refill | |
19344 | ||
19345 | @kindex | | |
19346 | @pindex calc-concat | |
5d67986c RS |
19347 | @ignore |
19348 | @mindex @null | |
19349 | @end ignore | |
d7b8e6c6 EZ |
19350 | @tindex | |
19351 | The @kbd{|} (@code{calc-concat}) command ``concatenates'' two vectors | |
19352 | into one. For example, after @kbd{@w{[ 1 , 2 ]} [ 3 , 4 ] |}, the stack | |
19353 | will contain the single vector @samp{[1, 2, 3, 4]}. If the arguments | |
19354 | are matrices, the rows of the first matrix are concatenated with the | |
19355 | rows of the second. (In other words, two matrices are just two vectors | |
19356 | of row-vectors as far as @kbd{|} is concerned.) | |
19357 | ||
19358 | If either argument to @kbd{|} is a scalar (a non-vector), it is treated | |
19359 | like a one-element vector for purposes of concatenation: @kbd{1 [ 2 , 3 ] |} | |
19360 | produces the vector @samp{[1, 2, 3]}. Likewise, if one argument is a | |
19361 | matrix and the other is a plain vector, the vector is treated as a | |
19362 | one-row matrix. | |
19363 | ||
19364 | @kindex H | | |
19365 | @tindex append | |
19366 | The @kbd{H |} (@code{calc-append}) [@code{append}] command concatenates | |
19367 | two vectors without any special cases. Both inputs must be vectors. | |
19368 | Whether or not they are matrices is not taken into account. If either | |
19369 | argument is a scalar, the @code{append} function is left in symbolic form. | |
19370 | See also @code{cons} and @code{rcons} below. | |
19371 | ||
19372 | @kindex I | | |
19373 | @kindex H I | | |
19374 | The @kbd{I |} and @kbd{H I |} commands are similar, but they use their | |
19375 | two stack arguments in the opposite order. Thus @kbd{I |} is equivalent | |
5d67986c | 19376 | to @kbd{@key{TAB} |}, but possibly more convenient and also a bit faster. |
d7b8e6c6 EZ |
19377 | |
19378 | @kindex v d | |
19379 | @pindex calc-diag | |
19380 | @tindex diag | |
19381 | The @kbd{v d} (@code{calc-diag}) [@code{diag}] function builds a diagonal | |
19382 | square matrix. The optional numeric prefix gives the number of rows | |
19383 | and columns in the matrix. If the value at the top of the stack is a | |
19384 | vector, the elements of the vector are used as the diagonal elements; the | |
19385 | prefix, if specified, must match the size of the vector. If the value on | |
19386 | the stack is a scalar, it is used for each element on the diagonal, and | |
19387 | the prefix argument is required. | |
19388 | ||
19389 | To build a constant square matrix, e.g., a @c{$3\times3$} | |
19390 | @asis{3x3} matrix filled with ones, | |
19391 | use @kbd{0 M-3 v d 1 +}, i.e., build a zero matrix first and then add a | |
19392 | constant value to that matrix. (Another alternative would be to use | |
19393 | @kbd{v b} and @kbd{v a}; see below.) | |
19394 | ||
19395 | @kindex v i | |
19396 | @pindex calc-ident | |
19397 | @tindex idn | |
19398 | The @kbd{v i} (@code{calc-ident}) [@code{idn}] function builds an identity | |
19399 | matrix of the specified size. It is a convenient form of @kbd{v d} | |
19400 | where the diagonal element is always one. If no prefix argument is given, | |
19401 | this command prompts for one. | |
19402 | ||
19403 | In algebraic notation, @samp{idn(a,n)} acts much like @samp{diag(a,n)}, | |
19404 | except that @cite{a} is required to be a scalar (non-vector) quantity. | |
19405 | If @cite{n} is omitted, @samp{idn(a)} represents @cite{a} times an | |
19406 | identity matrix of unknown size. Calc can operate algebraically on | |
19407 | such generic identity matrices, and if one is combined with a matrix | |
19408 | whose size is known, it is converted automatically to an identity | |
19409 | matrix of a suitable matching size. The @kbd{v i} command with an | |
19410 | argument of zero creates a generic identity matrix, @samp{idn(1)}. | |
19411 | Note that in dimensioned matrix mode (@pxref{Matrix Mode}), generic | |
19412 | identity matrices are immediately expanded to the current default | |
19413 | dimensions. | |
19414 | ||
19415 | @kindex v x | |
19416 | @pindex calc-index | |
19417 | @tindex index | |
19418 | The @kbd{v x} (@code{calc-index}) [@code{index}] function builds a vector | |
19419 | of consecutive integers from 1 to @var{n}, where @var{n} is the numeric | |
19420 | prefix argument. If you do not provide a prefix argument, you will be | |
19421 | prompted to enter a suitable number. If @var{n} is negative, the result | |
19422 | is a vector of negative integers from @var{n} to @i{-1}. | |
19423 | ||
19424 | With a prefix argument of just @kbd{C-u}, the @kbd{v x} command takes | |
19425 | three values from the stack: @var{n}, @var{start}, and @var{incr} (with | |
19426 | @var{incr} at top-of-stack). Counting starts at @var{start} and increases | |
19427 | by @var{incr} for successive vector elements. If @var{start} or @var{n} | |
19428 | is in floating-point format, the resulting vector elements will also be | |
19429 | floats. Note that @var{start} and @var{incr} may in fact be any kind | |
19430 | of numbers or formulas. | |
19431 | ||
19432 | When @var{start} and @var{incr} are specified, a negative @var{n} has a | |
19433 | different interpretation: It causes a geometric instead of arithmetic | |
19434 | sequence to be generated. For example, @samp{index(-3, a, b)} produces | |
19435 | @samp{[a, a b, a b^2]}. If you omit @var{incr} in the algebraic form, | |
19436 | @samp{index(@var{n}, @var{start})}, the default value for @var{incr} | |
19437 | is one for positive @var{n} or two for negative @var{n}. | |
19438 | ||
19439 | @kindex v b | |
19440 | @pindex calc-build-vector | |
19441 | @tindex cvec | |
19442 | The @kbd{v b} (@code{calc-build-vector}) [@code{cvec}] function builds a | |
19443 | vector of @var{n} copies of the value on the top of the stack, where @var{n} | |
19444 | is the numeric prefix argument. In algebraic formulas, @samp{cvec(x,n,m)} | |
19445 | can also be used to build an @var{n}-by-@var{m} matrix of copies of @var{x}. | |
19446 | (Interactively, just use @kbd{v b} twice: once to build a row, then again | |
19447 | to build a matrix of copies of that row.) | |
19448 | ||
19449 | @kindex v h | |
19450 | @kindex I v h | |
19451 | @pindex calc-head | |
19452 | @pindex calc-tail | |
19453 | @tindex head | |
19454 | @tindex tail | |
19455 | The @kbd{v h} (@code{calc-head}) [@code{head}] function returns the first | |
19456 | element of a vector. The @kbd{I v h} (@code{calc-tail}) [@code{tail}] | |
19457 | function returns the vector with its first element removed. In both | |
19458 | cases, the argument must be a non-empty vector. | |
19459 | ||
19460 | @kindex v k | |
19461 | @pindex calc-cons | |
19462 | @tindex cons | |
19463 | The @kbd{v k} (@code{calc-cons}) [@code{cons}] function takes a value @var{h} | |
19464 | and a vector @var{t} from the stack, and produces the vector whose head is | |
19465 | @var{h} and whose tail is @var{t}. This is similar to @kbd{|}, except | |
19466 | if @var{h} is itself a vector, @kbd{|} will concatenate the two vectors | |
19467 | whereas @code{cons} will insert @var{h} at the front of the vector @var{t}. | |
19468 | ||
19469 | @kindex H v h | |
19470 | @tindex rhead | |
5d67986c RS |
19471 | @ignore |
19472 | @mindex @idots | |
19473 | @end ignore | |
d7b8e6c6 | 19474 | @kindex H I v h |
5d67986c RS |
19475 | @ignore |
19476 | @mindex @null | |
19477 | @end ignore | |
d7b8e6c6 | 19478 | @kindex H v k |
5d67986c RS |
19479 | @ignore |
19480 | @mindex @null | |
19481 | @end ignore | |
d7b8e6c6 | 19482 | @tindex rtail |
5d67986c RS |
19483 | @ignore |
19484 | @mindex @null | |
19485 | @end ignore | |
d7b8e6c6 EZ |
19486 | @tindex rcons |
19487 | Each of these three functions also accepts the Hyperbolic flag [@code{rhead}, | |
19488 | @code{rtail}, @code{rcons}] in which case @var{t} instead represents | |
19489 | the @emph{last} single element of the vector, with @var{h} | |
19490 | representing the remainder of the vector. Thus the vector | |
19491 | @samp{[a, b, c, d] = cons(a, [b, c, d]) = rcons([a, b, c], d)}. | |
19492 | Also, @samp{head([a, b, c, d]) = a}, @samp{tail([a, b, c, d]) = [b, c, d]}, | |
19493 | @samp{rhead([a, b, c, d]) = [a, b, c]}, and @samp{rtail([a, b, c, d]) = d}. | |
19494 | ||
19495 | @node Extracting Elements, Manipulating Vectors, Building Vectors, Matrix Functions | |
19496 | @section Extracting Vector Elements | |
19497 | ||
19498 | @noindent | |
19499 | @kindex v r | |
19500 | @pindex calc-mrow | |
19501 | @tindex mrow | |
19502 | The @kbd{v r} (@code{calc-mrow}) [@code{mrow}] command extracts one row of | |
19503 | the matrix on the top of the stack, or one element of the plain vector on | |
19504 | the top of the stack. The row or element is specified by the numeric | |
19505 | prefix argument; the default is to prompt for the row or element number. | |
19506 | The matrix or vector is replaced by the specified row or element in the | |
19507 | form of a vector or scalar, respectively. | |
19508 | ||
19509 | @cindex Permutations, applying | |
19510 | With a prefix argument of @kbd{C-u} only, @kbd{v r} takes the index of | |
19511 | the element or row from the top of the stack, and the vector or matrix | |
19512 | from the second-to-top position. If the index is itself a vector of | |
19513 | integers, the result is a vector of the corresponding elements of the | |
19514 | input vector, or a matrix of the corresponding rows of the input matrix. | |
19515 | This command can be used to obtain any permutation of a vector. | |
19516 | ||
19517 | With @kbd{C-u}, if the index is an interval form with integer components, | |
19518 | it is interpreted as a range of indices and the corresponding subvector or | |
19519 | submatrix is returned. | |
19520 | ||
19521 | @cindex Subscript notation | |
19522 | @kindex a _ | |
19523 | @pindex calc-subscript | |
19524 | @tindex subscr | |
19525 | @tindex _ | |
19526 | Subscript notation in algebraic formulas (@samp{a_b}) stands for the | |
19527 | Calc function @code{subscr}, which is synonymous with @code{mrow}. | |
19528 | Thus, @samp{[x, y, z]_k} produces @cite{x}, @cite{y}, or @cite{z} if | |
19529 | @cite{k} is one, two, or three, respectively. A double subscript | |
19530 | (@samp{M_i_j}, equivalent to @samp{subscr(subscr(M, i), j)}) will | |
19531 | access the element at row @cite{i}, column @cite{j} of a matrix. | |
19532 | The @kbd{a _} (@code{calc-subscript}) command creates a subscript | |
19533 | formula @samp{a_b} out of two stack entries. (It is on the @kbd{a} | |
19534 | ``algebra'' prefix because subscripted variables are often used | |
19535 | purely as an algebraic notation.) | |
19536 | ||
19537 | @tindex mrrow | |
19538 | Given a negative prefix argument, @kbd{v r} instead deletes one row or | |
19539 | element from the matrix or vector on the top of the stack. Thus | |
19540 | @kbd{C-u 2 v r} replaces a matrix with its second row, but @kbd{C-u -2 v r} | |
19541 | replaces the matrix with the same matrix with its second row removed. | |
19542 | In algebraic form this function is called @code{mrrow}. | |
19543 | ||
19544 | @tindex getdiag | |
19545 | Given a prefix argument of zero, @kbd{v r} extracts the diagonal elements | |
19546 | of a square matrix in the form of a vector. In algebraic form this | |
19547 | function is called @code{getdiag}. | |
19548 | ||
19549 | @kindex v c | |
19550 | @pindex calc-mcol | |
19551 | @tindex mcol | |
19552 | @tindex mrcol | |
19553 | The @kbd{v c} (@code{calc-mcol}) [@code{mcol} or @code{mrcol}] command is | |
19554 | the analogous operation on columns of a matrix. Given a plain vector | |
19555 | it extracts (or removes) one element, just like @kbd{v r}. If the | |
19556 | index in @kbd{C-u v c} is an interval or vector and the argument is a | |
19557 | matrix, the result is a submatrix with only the specified columns | |
19558 | retained (and possibly permuted in the case of a vector index).@refill | |
19559 | ||
19560 | To extract a matrix element at a given row and column, use @kbd{v r} to | |
19561 | extract the row as a vector, then @kbd{v c} to extract the column element | |
19562 | from that vector. In algebraic formulas, it is often more convenient to | |
19563 | use subscript notation: @samp{m_i_j} gives row @cite{i}, column @cite{j} | |
19564 | of matrix @cite{m}. | |
19565 | ||
19566 | @kindex v s | |
19567 | @pindex calc-subvector | |
19568 | @tindex subvec | |
19569 | The @kbd{v s} (@code{calc-subvector}) [@code{subvec}] command extracts | |
19570 | a subvector of a vector. The arguments are the vector, the starting | |
19571 | index, and the ending index, with the ending index in the top-of-stack | |
19572 | position. The starting index indicates the first element of the vector | |
19573 | to take. The ending index indicates the first element @emph{past} the | |
19574 | range to be taken. Thus, @samp{subvec([a, b, c, d, e], 2, 4)} produces | |
19575 | the subvector @samp{[b, c]}. You could get the same result using | |
19576 | @samp{mrow([a, b, c, d, e], @w{[2 .. 4)})}. | |
19577 | ||
19578 | If either the start or the end index is zero or negative, it is | |
19579 | interpreted as relative to the end of the vector. Thus | |
19580 | @samp{subvec([a, b, c, d, e], 2, -2)} also produces @samp{[b, c]}. In | |
19581 | the algebraic form, the end index can be omitted in which case it | |
19582 | is taken as zero, i.e., elements from the starting element to the | |
19583 | end of the vector are used. The infinity symbol, @code{inf}, also | |
19584 | has this effect when used as the ending index. | |
19585 | ||
19586 | @kindex I v s | |
19587 | @tindex rsubvec | |
19588 | With the Inverse flag, @kbd{I v s} [@code{rsubvec}] removes a subvector | |
19589 | from a vector. The arguments are interpreted the same as for the | |
19590 | normal @kbd{v s} command. Thus, @samp{rsubvec([a, b, c, d, e], 2, 4)} | |
19591 | produces @samp{[a, d, e]}. It is always true that @code{subvec} and | |
19592 | @code{rsubvec} return complementary parts of the input vector. | |
19593 | ||
19594 | @xref{Selecting Subformulas}, for an alternative way to operate on | |
19595 | vectors one element at a time. | |
19596 | ||
19597 | @node Manipulating Vectors, Vector and Matrix Arithmetic, Extracting Elements, Matrix Functions | |
19598 | @section Manipulating Vectors | |
19599 | ||
19600 | @noindent | |
19601 | @kindex v l | |
19602 | @pindex calc-vlength | |
19603 | @tindex vlen | |
19604 | The @kbd{v l} (@code{calc-vlength}) [@code{vlen}] command computes the | |
19605 | length of a vector. The length of a non-vector is considered to be zero. | |
19606 | Note that matrices are just vectors of vectors for the purposes of this | |
19607 | command.@refill | |
19608 | ||
19609 | @kindex H v l | |
19610 | @tindex mdims | |
19611 | With the Hyperbolic flag, @kbd{H v l} [@code{mdims}] computes a vector | |
19612 | of the dimensions of a vector, matrix, or higher-order object. For | |
19613 | example, @samp{mdims([[a,b,c],[d,e,f]])} returns @samp{[2, 3]} since | |
19614 | its argument is a @c{$2\times3$} | |
19615 | @asis{2x3} matrix. | |
19616 | ||
19617 | @kindex v f | |
19618 | @pindex calc-vector-find | |
19619 | @tindex find | |
19620 | The @kbd{v f} (@code{calc-vector-find}) [@code{find}] command searches | |
19621 | along a vector for the first element equal to a given target. The target | |
19622 | is on the top of the stack; the vector is in the second-to-top position. | |
19623 | If a match is found, the result is the index of the matching element. | |
19624 | Otherwise, the result is zero. The numeric prefix argument, if given, | |
19625 | allows you to select any starting index for the search. | |
19626 | ||
19627 | @kindex v a | |
19628 | @pindex calc-arrange-vector | |
19629 | @tindex arrange | |
19630 | @cindex Arranging a matrix | |
19631 | @cindex Reshaping a matrix | |
19632 | @cindex Flattening a matrix | |
19633 | The @kbd{v a} (@code{calc-arrange-vector}) [@code{arrange}] command | |
19634 | rearranges a vector to have a certain number of columns and rows. The | |
19635 | numeric prefix argument specifies the number of columns; if you do not | |
19636 | provide an argument, you will be prompted for the number of columns. | |
19637 | The vector or matrix on the top of the stack is @dfn{flattened} into a | |
19638 | plain vector. If the number of columns is nonzero, this vector is | |
19639 | then formed into a matrix by taking successive groups of @var{n} elements. | |
19640 | If the number of columns does not evenly divide the number of elements | |
19641 | in the vector, the last row will be short and the result will not be | |
19642 | suitable for use as a matrix. For example, with the matrix | |
19643 | @samp{[[1, 2], @w{[3, 4]}]} on the stack, @kbd{v a 4} produces | |
19644 | @samp{[[1, 2, 3, 4]]} (a @c{$1\times4$} | |
19645 | @asis{1x4} matrix), @kbd{v a 1} produces | |
19646 | @samp{[[1], [2], [3], [4]]} (a @c{$4\times1$} | |
19647 | @asis{4x1} matrix), @kbd{v a 2} produces | |
19648 | @samp{[[1, 2], [3, 4]]} (the original @c{$2\times2$} | |
19649 | @asis{2x2} matrix), @w{@kbd{v a 3}} produces | |
19650 | @samp{[[1, 2, 3], [4]]} (not a matrix), and @kbd{v a 0} produces | |
19651 | the flattened list @samp{[1, 2, @w{3, 4}]}. | |
19652 | ||
19653 | @cindex Sorting data | |
19654 | @kindex V S | |
19655 | @kindex I V S | |
19656 | @pindex calc-sort | |
19657 | @tindex sort | |
19658 | @tindex rsort | |
19659 | The @kbd{V S} (@code{calc-sort}) [@code{sort}] command sorts the elements of | |
19660 | a vector into increasing order. Real numbers, real infinities, and | |
19661 | constant interval forms come first in this ordering; next come other | |
19662 | kinds of numbers, then variables (in alphabetical order), then finally | |
19663 | come formulas and other kinds of objects; these are sorted according | |
19664 | to a kind of lexicographic ordering with the useful property that | |
19665 | one vector is less or greater than another if the first corresponding | |
19666 | unequal elements are less or greater, respectively. Since quoted strings | |
19667 | are stored by Calc internally as vectors of ASCII character codes | |
19668 | (@pxref{Strings}), this means vectors of strings are also sorted into | |
19669 | alphabetical order by this command. | |
19670 | ||
19671 | The @kbd{I V S} [@code{rsort}] command sorts a vector into decreasing order. | |
19672 | ||
19673 | @cindex Permutation, inverse of | |
19674 | @cindex Inverse of permutation | |
19675 | @cindex Index tables | |
19676 | @cindex Rank tables | |
19677 | @kindex V G | |
19678 | @kindex I V G | |
19679 | @pindex calc-grade | |
19680 | @tindex grade | |
19681 | @tindex rgrade | |
19682 | The @kbd{V G} (@code{calc-grade}) [@code{grade}, @code{rgrade}] command | |
19683 | produces an index table or permutation vector which, if applied to the | |
19684 | input vector (as the index of @kbd{C-u v r}, say), would sort the vector. | |
19685 | A permutation vector is just a vector of integers from 1 to @var{n}, where | |
19686 | each integer occurs exactly once. One application of this is to sort a | |
19687 | matrix of data rows using one column as the sort key; extract that column, | |
19688 | grade it with @kbd{V G}, then use the result to reorder the original matrix | |
19689 | with @kbd{C-u v r}. Another interesting property of the @code{V G} command | |
19690 | is that, if the input is itself a permutation vector, the result will | |
19691 | be the inverse of the permutation. The inverse of an index table is | |
19692 | a rank table, whose @var{k}th element says where the @var{k}th original | |
19693 | vector element will rest when the vector is sorted. To get a rank | |
19694 | table, just use @kbd{V G V G}. | |
19695 | ||
19696 | With the Inverse flag, @kbd{I V G} produces an index table that would | |
19697 | sort the input into decreasing order. Note that @kbd{V S} and @kbd{V G} | |
19698 | use a ``stable'' sorting algorithm, i.e., any two elements which are equal | |
19699 | will not be moved out of their original order. Generally there is no way | |
19700 | to tell with @kbd{V S}, since two elements which are equal look the same, | |
19701 | but with @kbd{V G} this can be an important issue. In the matrix-of-rows | |
19702 | example, suppose you have names and telephone numbers as two columns and | |
19703 | you wish to sort by phone number primarily, and by name when the numbers | |
19704 | are equal. You can sort the data matrix by names first, and then again | |
19705 | by phone numbers. Because the sort is stable, any two rows with equal | |
19706 | phone numbers will remain sorted by name even after the second sort. | |
19707 | ||
19708 | @cindex Histograms | |
19709 | @kindex V H | |
19710 | @pindex calc-histogram | |
5d67986c RS |
19711 | @ignore |
19712 | @mindex histo@idots | |
19713 | @end ignore | |
d7b8e6c6 EZ |
19714 | @tindex histogram |
19715 | The @kbd{V H} (@code{calc-histogram}) [@code{histogram}] command builds a | |
19716 | histogram of a vector of numbers. Vector elements are assumed to be | |
19717 | integers or real numbers in the range [0..@var{n}) for some ``number of | |
19718 | bins'' @var{n}, which is the numeric prefix argument given to the | |
19719 | command. The result is a vector of @var{n} counts of how many times | |
19720 | each value appeared in the original vector. Non-integers in the input | |
19721 | are rounded down to integers. Any vector elements outside the specified | |
19722 | range are ignored. (You can tell if elements have been ignored by noting | |
19723 | that the counts in the result vector don't add up to the length of the | |
19724 | input vector.) | |
19725 | ||
19726 | @kindex H V H | |
19727 | With the Hyperbolic flag, @kbd{H V H} pulls two vectors from the stack. | |
19728 | The second-to-top vector is the list of numbers as before. The top | |
19729 | vector is an equal-sized list of ``weights'' to attach to the elements | |
19730 | of the data vector. For example, if the first data element is 4.2 and | |
19731 | the first weight is 10, then 10 will be added to bin 4 of the result | |
19732 | vector. Without the hyperbolic flag, every element has a weight of one. | |
19733 | ||
19734 | @kindex v t | |
19735 | @pindex calc-transpose | |
19736 | @tindex trn | |
19737 | The @kbd{v t} (@code{calc-transpose}) [@code{trn}] command computes | |
19738 | the transpose of the matrix at the top of the stack. If the argument | |
19739 | is a plain vector, it is treated as a row vector and transposed into | |
19740 | a one-column matrix. | |
19741 | ||
19742 | @kindex v v | |
19743 | @pindex calc-reverse-vector | |
19744 | @tindex rev | |
19745 | The @kbd{v v} (@code{calc-reverse-vector}) [@code{vec}] command reverses | |
19746 | a vector end-for-end. Given a matrix, it reverses the order of the rows. | |
19747 | (To reverse the columns instead, just use @kbd{v t v v v t}. The same | |
19748 | principle can be used to apply other vector commands to the columns of | |
19749 | a matrix.) | |
19750 | ||
19751 | @kindex v m | |
19752 | @pindex calc-mask-vector | |
19753 | @tindex vmask | |
19754 | The @kbd{v m} (@code{calc-mask-vector}) [@code{vmask}] command uses | |
19755 | one vector as a mask to extract elements of another vector. The mask | |
19756 | is in the second-to-top position; the target vector is on the top of | |
19757 | the stack. These vectors must have the same length. The result is | |
19758 | the same as the target vector, but with all elements which correspond | |
19759 | to zeros in the mask vector deleted. Thus, for example, | |
19760 | @samp{vmask([1, 0, 1, 0, 1], [a, b, c, d, e])} produces @samp{[a, c, e]}. | |
19761 | @xref{Logical Operations}. | |
19762 | ||
19763 | @kindex v e | |
19764 | @pindex calc-expand-vector | |
19765 | @tindex vexp | |
19766 | The @kbd{v e} (@code{calc-expand-vector}) [@code{vexp}] command | |
19767 | expands a vector according to another mask vector. The result is a | |
19768 | vector the same length as the mask, but with nonzero elements replaced | |
19769 | by successive elements from the target vector. The length of the target | |
19770 | vector is normally the number of nonzero elements in the mask. If the | |
19771 | target vector is longer, its last few elements are lost. If the target | |
19772 | vector is shorter, the last few nonzero mask elements are left | |
19773 | unreplaced in the result. Thus @samp{vexp([2, 0, 3, 0, 7], [a, b])} | |
19774 | produces @samp{[a, 0, b, 0, 7]}. | |
19775 | ||
19776 | @kindex H v e | |
19777 | With the Hyperbolic flag, @kbd{H v e} takes a filler value from the | |
19778 | top of the stack; the mask and target vectors come from the third and | |
19779 | second elements of the stack. This filler is used where the mask is | |
19780 | zero: @samp{vexp([2, 0, 3, 0, 7], [a, b], z)} produces | |
19781 | @samp{[a, z, c, z, 7]}. If the filler value is itself a vector, | |
19782 | then successive values are taken from it, so that the effect is to | |
19783 | interleave two vectors according to the mask: | |
19784 | @samp{vexp([2, 0, 3, 7, 0, 0], [a, b], [x, y])} produces | |
19785 | @samp{[a, x, b, 7, y, 0]}. | |
19786 | ||
19787 | Another variation on the masking idea is to combine @samp{[a, b, c, d, e]} | |
19788 | with the mask @samp{[1, 0, 1, 0, 1]} to produce @samp{[a, 0, c, 0, e]}. | |
19789 | You can accomplish this with @kbd{V M a &}, mapping the logical ``and'' | |
19790 | operation across the two vectors. @xref{Logical Operations}. Note that | |
19791 | the @code{? :} operation also discussed there allows other types of | |
19792 | masking using vectors. | |
19793 | ||
19794 | @node Vector and Matrix Arithmetic, Set Operations, Manipulating Vectors, Matrix Functions | |
19795 | @section Vector and Matrix Arithmetic | |
19796 | ||
19797 | @noindent | |
19798 | Basic arithmetic operations like addition and multiplication are defined | |
19799 | for vectors and matrices as well as for numbers. Division of matrices, in | |
19800 | the sense of multiplying by the inverse, is supported. (Division by a | |
19801 | matrix actually uses LU-decomposition for greater accuracy and speed.) | |
19802 | @xref{Basic Arithmetic}. | |
19803 | ||
19804 | The following functions are applied element-wise if their arguments are | |
19805 | vectors or matrices: @code{change-sign}, @code{conj}, @code{arg}, | |
19806 | @code{re}, @code{im}, @code{polar}, @code{rect}, @code{clean}, | |
19807 | @code{float}, @code{frac}. @xref{Function Index}.@refill | |
19808 | ||
19809 | @kindex V J | |
19810 | @pindex calc-conj-transpose | |
19811 | @tindex ctrn | |
19812 | The @kbd{V J} (@code{calc-conj-transpose}) [@code{ctrn}] command computes | |
19813 | the conjugate transpose of its argument, i.e., @samp{conj(trn(x))}. | |
19814 | ||
5d67986c RS |
19815 | @ignore |
19816 | @mindex A | |
19817 | @end ignore | |
d7b8e6c6 EZ |
19818 | @kindex A (vectors) |
19819 | @pindex calc-abs (vectors) | |
5d67986c RS |
19820 | @ignore |
19821 | @mindex abs | |
19822 | @end ignore | |
d7b8e6c6 EZ |
19823 | @tindex abs (vectors) |
19824 | The @kbd{A} (@code{calc-abs}) [@code{abs}] command computes the | |
19825 | Frobenius norm of a vector or matrix argument. This is the square | |
19826 | root of the sum of the squares of the absolute values of the | |
19827 | elements of the vector or matrix. If the vector is interpreted as | |
19828 | a point in two- or three-dimensional space, this is the distance | |
19829 | from that point to the origin.@refill | |
19830 | ||
19831 | @kindex v n | |
19832 | @pindex calc-rnorm | |
19833 | @tindex rnorm | |
19834 | The @kbd{v n} (@code{calc-rnorm}) [@code{rnorm}] command computes | |
19835 | the row norm, or infinity-norm, of a vector or matrix. For a plain | |
19836 | vector, this is the maximum of the absolute values of the elements. | |
19837 | For a matrix, this is the maximum of the row-absolute-value-sums, | |
19838 | i.e., of the sums of the absolute values of the elements along the | |
19839 | various rows. | |
19840 | ||
19841 | @kindex V N | |
19842 | @pindex calc-cnorm | |
19843 | @tindex cnorm | |
19844 | The @kbd{V N} (@code{calc-cnorm}) [@code{cnorm}] command computes | |
19845 | the column norm, or one-norm, of a vector or matrix. For a plain | |
19846 | vector, this is the sum of the absolute values of the elements. | |
19847 | For a matrix, this is the maximum of the column-absolute-value-sums. | |
19848 | General @cite{k}-norms for @cite{k} other than one or infinity are | |
19849 | not provided. | |
19850 | ||
19851 | @kindex V C | |
19852 | @pindex calc-cross | |
19853 | @tindex cross | |
19854 | The @kbd{V C} (@code{calc-cross}) [@code{cross}] command computes the | |
19855 | right-handed cross product of two vectors, each of which must have | |
19856 | exactly three elements. | |
19857 | ||
5d67986c RS |
19858 | @ignore |
19859 | @mindex & | |
19860 | @end ignore | |
d7b8e6c6 EZ |
19861 | @kindex & (matrices) |
19862 | @pindex calc-inv (matrices) | |
5d67986c RS |
19863 | @ignore |
19864 | @mindex inv | |
19865 | @end ignore | |
d7b8e6c6 EZ |
19866 | @tindex inv (matrices) |
19867 | The @kbd{&} (@code{calc-inv}) [@code{inv}] command computes the | |
19868 | inverse of a square matrix. If the matrix is singular, the inverse | |
19869 | operation is left in symbolic form. Matrix inverses are recorded so | |
19870 | that once an inverse (or determinant) of a particular matrix has been | |
19871 | computed, the inverse and determinant of the matrix can be recomputed | |
19872 | quickly in the future. | |
19873 | ||
19874 | If the argument to @kbd{&} is a plain number @cite{x}, this | |
19875 | command simply computes @cite{1/x}. This is okay, because the | |
19876 | @samp{/} operator also does a matrix inversion when dividing one | |
19877 | by a matrix. | |
19878 | ||
19879 | @kindex V D | |
19880 | @pindex calc-mdet | |
19881 | @tindex det | |
19882 | The @kbd{V D} (@code{calc-mdet}) [@code{det}] command computes the | |
19883 | determinant of a square matrix. | |
19884 | ||
19885 | @kindex V L | |
19886 | @pindex calc-mlud | |
19887 | @tindex lud | |
19888 | The @kbd{V L} (@code{calc-mlud}) [@code{lud}] command computes the | |
19889 | LU decomposition of a matrix. The result is a list of three matrices | |
19890 | which, when multiplied together left-to-right, form the original matrix. | |
19891 | The first is a permutation matrix that arises from pivoting in the | |
19892 | algorithm, the second is lower-triangular with ones on the diagonal, | |
19893 | and the third is upper-triangular. | |
19894 | ||
19895 | @kindex V T | |
19896 | @pindex calc-mtrace | |
19897 | @tindex tr | |
19898 | The @kbd{V T} (@code{calc-mtrace}) [@code{tr}] command computes the | |
19899 | trace of a square matrix. This is defined as the sum of the diagonal | |
19900 | elements of the matrix. | |
19901 | ||
19902 | @node Set Operations, Statistical Operations, Vector and Matrix Arithmetic, Matrix Functions | |
19903 | @section Set Operations using Vectors | |
19904 | ||
19905 | @noindent | |
19906 | @cindex Sets, as vectors | |
19907 | Calc includes several commands which interpret vectors as @dfn{sets} of | |
19908 | objects. A set is a collection of objects; any given object can appear | |
19909 | only once in the set. Calc stores sets as vectors of objects in | |
19910 | sorted order. Objects in a Calc set can be any of the usual things, | |
19911 | such as numbers, variables, or formulas. Two set elements are considered | |
19912 | equal if they are identical, except that numerically equal numbers like | |
19913 | the integer 4 and the float 4.0 are considered equal even though they | |
19914 | are not ``identical.'' Variables are treated like plain symbols without | |
19915 | attached values by the set operations; subtracting the set @samp{[b]} | |
19916 | from @samp{[a, b]} always yields the set @samp{[a]} even though if | |
28665d46 | 19917 | the variables @samp{a} and @samp{b} both equaled 17, you might |
d7b8e6c6 EZ |
19918 | expect the answer @samp{[]}. |
19919 | ||
19920 | If a set contains interval forms, then it is assumed to be a set of | |
19921 | real numbers. In this case, all set operations require the elements | |
19922 | of the set to be only things that are allowed in intervals: Real | |
19923 | numbers, plus and minus infinity, HMS forms, and date forms. If | |
19924 | there are variables or other non-real objects present in a real set, | |
19925 | all set operations on it will be left in unevaluated form. | |
19926 | ||
19927 | If the input to a set operation is a plain number or interval form | |
19928 | @var{a}, it is treated like the one-element vector @samp{[@var{a}]}. | |
19929 | The result is always a vector, except that if the set consists of a | |
19930 | single interval, the interval itself is returned instead. | |
19931 | ||
19932 | @xref{Logical Operations}, for the @code{in} function which tests if | |
19933 | a certain value is a member of a given set. To test if the set @cite{A} | |
19934 | is a subset of the set @cite{B}, use @samp{vdiff(A, B) = []}. | |
19935 | ||
19936 | @kindex V + | |
19937 | @pindex calc-remove-duplicates | |
19938 | @tindex rdup | |
19939 | The @kbd{V +} (@code{calc-remove-duplicates}) [@code{rdup}] command | |
19940 | converts an arbitrary vector into set notation. It works by sorting | |
19941 | the vector as if by @kbd{V S}, then removing duplicates. (For example, | |
19942 | @kbd{[a, 5, 4, a, 4.0]} is sorted to @samp{[4, 4.0, 5, a, a]} and then | |
19943 | reduced to @samp{[4, 5, a]}). Overlapping intervals are merged as | |
19944 | necessary. You rarely need to use @kbd{V +} explicitly, since all the | |
19945 | other set-based commands apply @kbd{V +} to their inputs before using | |
19946 | them. | |
19947 | ||
19948 | @kindex V V | |
19949 | @pindex calc-set-union | |
19950 | @tindex vunion | |
19951 | The @kbd{V V} (@code{calc-set-union}) [@code{vunion}] command computes | |
19952 | the union of two sets. An object is in the union of two sets if and | |
19953 | only if it is in either (or both) of the input sets. (You could | |
19954 | accomplish the same thing by concatenating the sets with @kbd{|}, | |
19955 | then using @kbd{V +}.) | |
19956 | ||
19957 | @kindex V ^ | |
19958 | @pindex calc-set-intersect | |
19959 | @tindex vint | |
19960 | The @kbd{V ^} (@code{calc-set-intersect}) [@code{vint}] command computes | |
19961 | the intersection of two sets. An object is in the intersection if | |
19962 | and only if it is in both of the input sets. Thus if the input | |
19963 | sets are disjoint, i.e., if they share no common elements, the result | |
19964 | will be the empty vector @samp{[]}. Note that the characters @kbd{V} | |
19965 | and @kbd{^} were chosen to be close to the conventional mathematical | |
19966 | notation for set union@c{ ($A \cup B$)} | |
19967 | @asis{} and intersection@c{ ($A \cap B$)} | |
19968 | @asis{}. | |
19969 | ||
19970 | @kindex V - | |
19971 | @pindex calc-set-difference | |
19972 | @tindex vdiff | |
19973 | The @kbd{V -} (@code{calc-set-difference}) [@code{vdiff}] command computes | |
19974 | the difference between two sets. An object is in the difference | |
19975 | @cite{A - B} if and only if it is in @cite{A} but not in @cite{B}. | |
19976 | Thus subtracting @samp{[y,z]} from a set will remove the elements | |
19977 | @samp{y} and @samp{z} if they are present. You can also think of this | |
19978 | as a general @dfn{set complement} operator; if @cite{A} is the set of | |
19979 | all possible values, then @cite{A - B} is the ``complement'' of @cite{B}. | |
19980 | Obviously this is only practical if the set of all possible values in | |
19981 | your problem is small enough to list in a Calc vector (or simple | |
19982 | enough to express in a few intervals). | |
19983 | ||
19984 | @kindex V X | |
19985 | @pindex calc-set-xor | |
19986 | @tindex vxor | |
19987 | The @kbd{V X} (@code{calc-set-xor}) [@code{vxor}] command computes | |
19988 | the ``exclusive-or,'' or ``symmetric difference'' of two sets. | |
19989 | An object is in the symmetric difference of two sets if and only | |
19990 | if it is in one, but @emph{not} both, of the sets. Objects that | |
19991 | occur in both sets ``cancel out.'' | |
19992 | ||
19993 | @kindex V ~ | |
19994 | @pindex calc-set-complement | |
19995 | @tindex vcompl | |
19996 | The @kbd{V ~} (@code{calc-set-complement}) [@code{vcompl}] command | |
19997 | computes the complement of a set with respect to the real numbers. | |
19998 | Thus @samp{vcompl(x)} is equivalent to @samp{vdiff([-inf .. inf], x)}. | |
19999 | For example, @samp{vcompl([2, (3 .. 4]])} evaluates to | |
20000 | @samp{[[-inf .. 2), (2 .. 3], (4 .. inf]]}. | |
20001 | ||
20002 | @kindex V F | |
20003 | @pindex calc-set-floor | |
20004 | @tindex vfloor | |
20005 | The @kbd{V F} (@code{calc-set-floor}) [@code{vfloor}] command | |
20006 | reinterprets a set as a set of integers. Any non-integer values, | |
20007 | and intervals that do not enclose any integers, are removed. Open | |
20008 | intervals are converted to equivalent closed intervals. Successive | |
20009 | integers are converted into intervals of integers. For example, the | |
20010 | complement of the set @samp{[2, 6, 7, 8]} is messy, but if you wanted | |
20011 | the complement with respect to the set of integers you could type | |
20012 | @kbd{V ~ V F} to get @samp{[[-inf .. 1], [3 .. 5], [9 .. inf]]}. | |
20013 | ||
20014 | @kindex V E | |
20015 | @pindex calc-set-enumerate | |
20016 | @tindex venum | |
20017 | The @kbd{V E} (@code{calc-set-enumerate}) [@code{venum}] command | |
20018 | converts a set of integers into an explicit vector. Intervals in | |
20019 | the set are expanded out to lists of all integers encompassed by | |
20020 | the intervals. This only works for finite sets (i.e., sets which | |
20021 | do not involve @samp{-inf} or @samp{inf}). | |
20022 | ||
20023 | @kindex V : | |
20024 | @pindex calc-set-span | |
20025 | @tindex vspan | |
20026 | The @kbd{V :} (@code{calc-set-span}) [@code{vspan}] command converts any | |
20027 | set of reals into an interval form that encompasses all its elements. | |
20028 | The lower limit will be the smallest element in the set; the upper | |
20029 | limit will be the largest element. For an empty set, @samp{vspan([])} | |
20030 | returns the empty interval @w{@samp{[0 .. 0)}}. | |
20031 | ||
20032 | @kindex V # | |
20033 | @pindex calc-set-cardinality | |
20034 | @tindex vcard | |
20035 | The @kbd{V #} (@code{calc-set-cardinality}) [@code{vcard}] command counts | |
20036 | the number of integers in a set. The result is the length of the vector | |
20037 | that would be produced by @kbd{V E}, although the computation is much | |
20038 | more efficient than actually producing that vector. | |
20039 | ||
20040 | @cindex Sets, as binary numbers | |
20041 | Another representation for sets that may be more appropriate in some | |
20042 | cases is binary numbers. If you are dealing with sets of integers | |
20043 | in the range 0 to 49, you can use a 50-bit binary number where a | |
20044 | particular bit is 1 if the corresponding element is in the set. | |
20045 | @xref{Binary Functions}, for a list of commands that operate on | |
20046 | binary numbers. Note that many of the above set operations have | |
20047 | direct equivalents in binary arithmetic: @kbd{b o} (@code{calc-or}), | |
20048 | @kbd{b a} (@code{calc-and}), @kbd{b d} (@code{calc-diff}), | |
20049 | @kbd{b x} (@code{calc-xor}), and @kbd{b n} (@code{calc-not}), | |
20050 | respectively. You can use whatever representation for sets is most | |
20051 | convenient to you. | |
20052 | ||
20053 | @kindex b p | |
20054 | @kindex b u | |
20055 | @pindex calc-pack-bits | |
20056 | @pindex calc-unpack-bits | |
20057 | @tindex vpack | |
20058 | @tindex vunpack | |
20059 | The @kbd{b u} (@code{calc-unpack-bits}) [@code{vunpack}] command | |
20060 | converts an integer that represents a set in binary into a set | |
20061 | in vector/interval notation. For example, @samp{vunpack(67)} | |
20062 | returns @samp{[[0 .. 1], 6]}. If the input is negative, the set | |
20063 | it represents is semi-infinite: @samp{vunpack(-4) = [2 .. inf)}. | |
20064 | Use @kbd{V E} afterwards to expand intervals to individual | |
20065 | values if you wish. Note that this command uses the @kbd{b} | |
20066 | (binary) prefix key. | |
20067 | ||
20068 | The @kbd{b p} (@code{calc-pack-bits}) [@code{vpack}] command | |
20069 | converts the other way, from a vector or interval representing | |
20070 | a set of nonnegative integers into a binary integer describing | |
20071 | the same set. The set may include positive infinity, but must | |
20072 | not include any negative numbers. The input is interpreted as a | |
20073 | set of integers in the sense of @kbd{V F} (@code{vfloor}). Beware | |
20074 | that a simple input like @samp{[100]} can result in a huge integer | |
20075 | representation (@c{$2^{100}$} | |
20076 | @cite{2^100}, a 31-digit integer, in this case). | |
20077 | ||
20078 | @node Statistical Operations, Reducing and Mapping, Set Operations, Matrix Functions | |
20079 | @section Statistical Operations on Vectors | |
20080 | ||
20081 | @noindent | |
20082 | @cindex Statistical functions | |
20083 | The commands in this section take vectors as arguments and compute | |
20084 | various statistical measures on the data stored in the vectors. The | |
20085 | references used in the definitions of these functions are Bevington's | |
20086 | @emph{Data Reduction and Error Analysis for the Physical Sciences}, | |
20087 | and @emph{Numerical Recipes} by Press, Flannery, Teukolsky and | |
20088 | Vetterling. | |
20089 | ||
20090 | The statistical commands use the @kbd{u} prefix key followed by | |
20091 | a shifted letter or other character. | |
20092 | ||
20093 | @xref{Manipulating Vectors}, for a description of @kbd{V H} | |
20094 | (@code{calc-histogram}). | |
20095 | ||
20096 | @xref{Curve Fitting}, for the @kbd{a F} command for doing | |
20097 | least-squares fits to statistical data. | |
20098 | ||
20099 | @xref{Probability Distribution Functions}, for several common | |
20100 | probability distribution functions. | |
20101 | ||
20102 | @menu | |
20103 | * Single-Variable Statistics:: | |
20104 | * Paired-Sample Statistics:: | |
20105 | @end menu | |
20106 | ||
20107 | @node Single-Variable Statistics, Paired-Sample Statistics, Statistical Operations, Statistical Operations | |
20108 | @subsection Single-Variable Statistics | |
20109 | ||
20110 | @noindent | |
20111 | These functions do various statistical computations on single | |
20112 | vectors. Given a numeric prefix argument, they actually pop | |
20113 | @var{n} objects from the stack and combine them into a data | |
20114 | vector. Each object may be either a number or a vector; if a | |
20115 | vector, any sub-vectors inside it are ``flattened'' as if by | |
20116 | @kbd{v a 0}; @pxref{Manipulating Vectors}. By default one object | |
20117 | is popped, which (in order to be useful) is usually a vector. | |
20118 | ||
20119 | If an argument is a variable name, and the value stored in that | |
20120 | variable is a vector, then the stored vector is used. This method | |
20121 | has the advantage that if your data vector is large, you can avoid | |
20122 | the slow process of manipulating it directly on the stack. | |
20123 | ||
20124 | These functions are left in symbolic form if any of their arguments | |
20125 | are not numbers or vectors, e.g., if an argument is a formula, or | |
20126 | a non-vector variable. However, formulas embedded within vector | |
20127 | arguments are accepted; the result is a symbolic representation | |
20128 | of the computation, based on the assumption that the formula does | |
20129 | not itself represent a vector. All varieties of numbers such as | |
20130 | error forms and interval forms are acceptable. | |
20131 | ||
20132 | Some of the functions in this section also accept a single error form | |
20133 | or interval as an argument. They then describe a property of the | |
20134 | normal or uniform (respectively) statistical distribution described | |
20135 | by the argument. The arguments are interpreted in the same way as | |
20136 | the @var{M} argument of the random number function @kbd{k r}. In | |
20137 | particular, an interval with integer limits is considered an integer | |
20138 | distribution, so that @samp{[2 .. 6)} is the same as @samp{[2 .. 5]}. | |
20139 | An interval with at least one floating-point limit is a continuous | |
20140 | distribution: @samp{[2.0 .. 6.0)} is @emph{not} the same as | |
20141 | @samp{[2.0 .. 5.0]}! | |
20142 | ||
20143 | @kindex u # | |
20144 | @pindex calc-vector-count | |
20145 | @tindex vcount | |
20146 | The @kbd{u #} (@code{calc-vector-count}) [@code{vcount}] command | |
20147 | computes the number of data values represented by the inputs. | |
20148 | For example, @samp{vcount(1, [2, 3], [[4, 5], [], x, y])} returns 7. | |
20149 | If the argument is a single vector with no sub-vectors, this | |
20150 | simply computes the length of the vector. | |
20151 | ||
20152 | @kindex u + | |
20153 | @kindex u * | |
20154 | @pindex calc-vector-sum | |
20155 | @pindex calc-vector-prod | |
20156 | @tindex vsum | |
20157 | @tindex vprod | |
20158 | @cindex Summations (statistical) | |
20159 | The @kbd{u +} (@code{calc-vector-sum}) [@code{vsum}] command | |
20160 | computes the sum of the data values. The @kbd{u *} | |
20161 | (@code{calc-vector-prod}) [@code{vprod}] command computes the | |
20162 | product of the data values. If the input is a single flat vector, | |
20163 | these are the same as @kbd{V R +} and @kbd{V R *} | |
20164 | (@pxref{Reducing and Mapping}).@refill | |
20165 | ||
20166 | @kindex u X | |
20167 | @kindex u N | |
20168 | @pindex calc-vector-max | |
20169 | @pindex calc-vector-min | |
20170 | @tindex vmax | |
20171 | @tindex vmin | |
20172 | The @kbd{u X} (@code{calc-vector-max}) [@code{vmax}] command | |
20173 | computes the maximum of the data values, and the @kbd{u N} | |
20174 | (@code{calc-vector-min}) [@code{vmin}] command computes the minimum. | |
20175 | If the argument is an interval, this finds the minimum or maximum | |
20176 | value in the interval. (Note that @samp{vmax([2..6)) = 5} as | |
20177 | described above.) If the argument is an error form, this returns | |
20178 | plus or minus infinity. | |
20179 | ||
20180 | @kindex u M | |
20181 | @pindex calc-vector-mean | |
20182 | @tindex vmean | |
20183 | @cindex Mean of data values | |
20184 | The @kbd{u M} (@code{calc-vector-mean}) [@code{vmean}] command | |
20185 | computes the average (arithmetic mean) of the data values. | |
20186 | If the inputs are error forms @c{$x$ @code{+/-} $\sigma$} | |
20187 | @samp{x +/- s}, this is the weighted | |
20188 | mean of the @cite{x} values with weights @c{$1 / \sigma^2$} | |
20189 | @cite{1 / s^2}. | |
20190 | @tex | |
20191 | \turnoffactive | |
20192 | $$ \mu = { \displaystyle \sum { x_i \over \sigma_i^2 } \over | |
20193 | \displaystyle \sum { 1 \over \sigma_i^2 } } $$ | |
20194 | @end tex | |
20195 | If the inputs are not error forms, this is simply the sum of the | |
20196 | values divided by the count of the values.@refill | |
20197 | ||
20198 | Note that a plain number can be considered an error form with | |
20199 | error @c{$\sigma = 0$} | |
20200 | @cite{s = 0}. If the input to @kbd{u M} is a mixture of | |
20201 | plain numbers and error forms, the result is the mean of the | |
20202 | plain numbers, ignoring all values with non-zero errors. (By the | |
20203 | above definitions it's clear that a plain number effectively | |
20204 | has an infinite weight, next to which an error form with a finite | |
20205 | weight is completely negligible.) | |
20206 | ||
20207 | This function also works for distributions (error forms or | |
5d67986c | 20208 | intervals). The mean of an error form `@var{a} @t{+/-} @var{b}' is simply |
d7b8e6c6 EZ |
20209 | @cite{a}. The mean of an interval is the mean of the minimum |
20210 | and maximum values of the interval. | |
20211 | ||
20212 | @kindex I u M | |
20213 | @pindex calc-vector-mean-error | |
20214 | @tindex vmeane | |
20215 | The @kbd{I u M} (@code{calc-vector-mean-error}) [@code{vmeane}] | |
20216 | command computes the mean of the data points expressed as an | |
20217 | error form. This includes the estimated error associated with | |
20218 | the mean. If the inputs are error forms, the error is the square | |
20219 | root of the reciprocal of the sum of the reciprocals of the squares | |
20220 | of the input errors. (I.e., the variance is the reciprocal of the | |
20221 | sum of the reciprocals of the variances.) | |
20222 | @tex | |
20223 | \turnoffactive | |
20224 | $$ \sigma_\mu^2 = {1 \over \displaystyle \sum {1 \over \sigma_i^2}} $$ | |
20225 | @end tex | |
20226 | If the inputs are plain | |
20227 | numbers, the error is equal to the standard deviation of the values | |
20228 | divided by the square root of the number of values. (This works | |
20229 | out to be equivalent to calculating the standard deviation and | |
20230 | then assuming each value's error is equal to this standard | |
20231 | deviation.)@refill | |
20232 | @tex | |
20233 | \turnoffactive | |
20234 | $$ \sigma_\mu^2 = {\sigma^2 \over N} $$ | |
20235 | @end tex | |
20236 | ||
20237 | @kindex H u M | |
20238 | @pindex calc-vector-median | |
20239 | @tindex vmedian | |
20240 | @cindex Median of data values | |
20241 | The @kbd{H u M} (@code{calc-vector-median}) [@code{vmedian}] | |
20242 | command computes the median of the data values. The values are | |
20243 | first sorted into numerical order; the median is the middle | |
20244 | value after sorting. (If the number of data values is even, | |
20245 | the median is taken to be the average of the two middle values.) | |
20246 | The median function is different from the other functions in | |
20247 | this section in that the arguments must all be real numbers; | |
20248 | variables are not accepted even when nested inside vectors. | |
20249 | (Otherwise it is not possible to sort the data values.) If | |
20250 | any of the input values are error forms, their error parts are | |
20251 | ignored. | |
20252 | ||
20253 | The median function also accepts distributions. For both normal | |
20254 | (error form) and uniform (interval) distributions, the median is | |
20255 | the same as the mean. | |
20256 | ||
20257 | @kindex H I u M | |
20258 | @pindex calc-vector-harmonic-mean | |
20259 | @tindex vhmean | |
20260 | @cindex Harmonic mean | |
20261 | The @kbd{H I u M} (@code{calc-vector-harmonic-mean}) [@code{vhmean}] | |
20262 | command computes the harmonic mean of the data values. This is | |
20263 | defined as the reciprocal of the arithmetic mean of the reciprocals | |
20264 | of the values. | |
20265 | @tex | |
20266 | \turnoffactive | |
20267 | $$ { N \over \displaystyle \sum {1 \over x_i} } $$ | |
20268 | @end tex | |
20269 | ||
20270 | @kindex u G | |
20271 | @pindex calc-vector-geometric-mean | |
20272 | @tindex vgmean | |
20273 | @cindex Geometric mean | |
20274 | The @kbd{u G} (@code{calc-vector-geometric-mean}) [@code{vgmean}] | |
20275 | command computes the geometric mean of the data values. This | |
5d67986c | 20276 | is the @var{n}th root of the product of the values. This is also |
d7b8e6c6 EZ |
20277 | equal to the @code{exp} of the arithmetic mean of the logarithms |
20278 | of the data values. | |
20279 | @tex | |
20280 | \turnoffactive | |
20281 | $$ \exp \left ( \sum { \ln x_i } \right ) = | |
20282 | \left ( \prod { x_i } \right)^{1 / N} $$ | |
20283 | @end tex | |
20284 | ||
20285 | @kindex H u G | |
20286 | @tindex agmean | |
20287 | The @kbd{H u G} [@code{agmean}] command computes the ``arithmetic-geometric | |
20288 | mean'' of two numbers taken from the stack. This is computed by | |
20289 | replacing the two numbers with their arithmetic mean and geometric | |
20290 | mean, then repeating until the two values converge. | |
20291 | @tex | |
20292 | \turnoffactive | |
20293 | $$ a_{i+1} = { a_i + b_i \over 2 } , \qquad b_{i+1} = \sqrt{a_i b_i} $$ | |
20294 | @end tex | |
20295 | ||
20296 | @cindex Root-mean-square | |
20297 | Another commonly used mean, the RMS (root-mean-square), can be computed | |
20298 | for a vector of numbers simply by using the @kbd{A} command. | |
20299 | ||
20300 | @kindex u S | |
20301 | @pindex calc-vector-sdev | |
20302 | @tindex vsdev | |
20303 | @cindex Standard deviation | |
20304 | @cindex Sample statistics | |
20305 | The @kbd{u S} (@code{calc-vector-sdev}) [@code{vsdev}] command | |
20306 | computes the standard deviation@c{ $\sigma$} | |
20307 | @asis{} of the data values. If the | |
20308 | values are error forms, the errors are used as weights just | |
20309 | as for @kbd{u M}. This is the @emph{sample} standard deviation, | |
20310 | whose value is the square root of the sum of the squares of the | |
20311 | differences between the values and the mean of the @cite{N} values, | |
20312 | divided by @cite{N-1}. | |
20313 | @tex | |
20314 | \turnoffactive | |
20315 | $$ \sigma^2 = {1 \over N - 1} \sum (x_i - \mu)^2 $$ | |
20316 | @end tex | |
20317 | ||
20318 | This function also applies to distributions. The standard deviation | |
20319 | of a single error form is simply the error part. The standard deviation | |
20320 | of a continuous interval happens to equal the difference between the | |
20321 | limits, divided by @c{$\sqrt{12}$} | |
20322 | @cite{sqrt(12)}. The standard deviation of an | |
20323 | integer interval is the same as the standard deviation of a vector | |
20324 | of those integers. | |
20325 | ||
20326 | @kindex I u S | |
20327 | @pindex calc-vector-pop-sdev | |
20328 | @tindex vpsdev | |
20329 | @cindex Population statistics | |
20330 | The @kbd{I u S} (@code{calc-vector-pop-sdev}) [@code{vpsdev}] | |
20331 | command computes the @emph{population} standard deviation. | |
20332 | It is defined by the same formula as above but dividing | |
20333 | by @cite{N} instead of by @cite{N-1}. The population standard | |
20334 | deviation is used when the input represents the entire set of | |
20335 | data values in the distribution; the sample standard deviation | |
20336 | is used when the input represents a sample of the set of all | |
20337 | data values, so that the mean computed from the input is itself | |
20338 | only an estimate of the true mean. | |
20339 | @tex | |
20340 | \turnoffactive | |
20341 | $$ \sigma^2 = {1 \over N} \sum (x_i - \mu)^2 $$ | |
20342 | @end tex | |
20343 | ||
20344 | For error forms and continuous intervals, @code{vpsdev} works | |
20345 | exactly like @code{vsdev}. For integer intervals, it computes the | |
20346 | population standard deviation of the equivalent vector of integers. | |
20347 | ||
20348 | @kindex H u S | |
20349 | @kindex H I u S | |
20350 | @pindex calc-vector-variance | |
20351 | @pindex calc-vector-pop-variance | |
20352 | @tindex vvar | |
20353 | @tindex vpvar | |
20354 | @cindex Variance of data values | |
20355 | The @kbd{H u S} (@code{calc-vector-variance}) [@code{vvar}] and | |
20356 | @kbd{H I u S} (@code{calc-vector-pop-variance}) [@code{vpvar}] | |
20357 | commands compute the variance of the data values. The variance | |
20358 | is the square@c{ $\sigma^2$} | |
20359 | @asis{} of the standard deviation, i.e., the sum of the | |
20360 | squares of the deviations of the data values from the mean. | |
20361 | (This definition also applies when the argument is a distribution.) | |
20362 | ||
5d67986c RS |
20363 | @ignore |
20364 | @starindex | |
20365 | @end ignore | |
d7b8e6c6 EZ |
20366 | @tindex vflat |
20367 | The @code{vflat} algebraic function returns a vector of its | |
20368 | arguments, interpreted in the same way as the other functions | |
20369 | in this section. For example, @samp{vflat(1, [2, [3, 4]], 5)} | |
20370 | returns @samp{[1, 2, 3, 4, 5]}. | |
20371 | ||
20372 | @node Paired-Sample Statistics, , Single-Variable Statistics, Statistical Operations | |
20373 | @subsection Paired-Sample Statistics | |
20374 | ||
20375 | @noindent | |
20376 | The functions in this section take two arguments, which must be | |
20377 | vectors of equal size. The vectors are each flattened in the same | |
20378 | way as by the single-variable statistical functions. Given a numeric | |
20379 | prefix argument of 1, these functions instead take one object from | |
20380 | the stack, which must be an @c{$N\times2$} | |
20381 | @asis{Nx2} matrix of data values. Once | |
20382 | again, variable names can be used in place of actual vectors and | |
20383 | matrices. | |
20384 | ||
20385 | @kindex u C | |
20386 | @pindex calc-vector-covariance | |
20387 | @tindex vcov | |
20388 | @cindex Covariance | |
20389 | The @kbd{u C} (@code{calc-vector-covariance}) [@code{vcov}] command | |
20390 | computes the sample covariance of two vectors. The covariance | |
20391 | of vectors @var{x} and @var{y} is the sum of the products of the | |
20392 | differences between the elements of @var{x} and the mean of @var{x} | |
20393 | times the differences between the corresponding elements of @var{y} | |
20394 | and the mean of @var{y}, all divided by @cite{N-1}. Note that | |
20395 | the variance of a vector is just the covariance of the vector | |
20396 | with itself. Once again, if the inputs are error forms the | |
20397 | errors are used as weight factors. If both @var{x} and @var{y} | |
20398 | are composed of error forms, the error for a given data point | |
20399 | is taken as the square root of the sum of the squares of the two | |
20400 | input errors. | |
20401 | @tex | |
20402 | \turnoffactive | |
20403 | $$ \sigma_{x\!y}^2 = {1 \over N-1} \sum (x_i - \mu_x) (y_i - \mu_y) $$ | |
20404 | $$ \sigma_{x\!y}^2 = | |
20405 | {\displaystyle {1 \over N-1} | |
20406 | \sum {(x_i - \mu_x) (y_i - \mu_y) \over \sigma_i^2} | |
20407 | \over \displaystyle {1 \over N} \sum {1 \over \sigma_i^2}} | |
20408 | $$ | |
20409 | @end tex | |
20410 | ||
20411 | @kindex I u C | |
20412 | @pindex calc-vector-pop-covariance | |
20413 | @tindex vpcov | |
20414 | The @kbd{I u C} (@code{calc-vector-pop-covariance}) [@code{vpcov}] | |
20415 | command computes the population covariance, which is the same as the | |
20416 | sample covariance computed by @kbd{u C} except dividing by @cite{N} | |
20417 | instead of @cite{N-1}. | |
20418 | ||
20419 | @kindex H u C | |
20420 | @pindex calc-vector-correlation | |
20421 | @tindex vcorr | |
20422 | @cindex Correlation coefficient | |
20423 | @cindex Linear correlation | |
20424 | The @kbd{H u C} (@code{calc-vector-correlation}) [@code{vcorr}] | |
20425 | command computes the linear correlation coefficient of two vectors. | |
20426 | This is defined by the covariance of the vectors divided by the | |
20427 | product of their standard deviations. (There is no difference | |
20428 | between sample or population statistics here.) | |
20429 | @tex | |
20430 | \turnoffactive | |
20431 | $$ r_{x\!y} = { \sigma_{x\!y}^2 \over \sigma_x^2 \sigma_y^2 } $$ | |
20432 | @end tex | |
20433 | ||
20434 | @node Reducing and Mapping, Vector and Matrix Formats, Statistical Operations, Matrix Functions | |
20435 | @section Reducing and Mapping Vectors | |
20436 | ||
20437 | @noindent | |
20438 | The commands in this section allow for more general operations on the | |
20439 | elements of vectors. | |
20440 | ||
20441 | @kindex V A | |
20442 | @pindex calc-apply | |
20443 | @tindex apply | |
20444 | The simplest of these operations is @kbd{V A} (@code{calc-apply}) | |
20445 | [@code{apply}], which applies a given operator to the elements of a vector. | |
20446 | For example, applying the hypothetical function @code{f} to the vector | |
20447 | @w{@samp{[1, 2, 3]}} would produce the function call @samp{f(1, 2, 3)}. | |
20448 | Applying the @code{+} function to the vector @samp{[a, b]} gives | |
20449 | @samp{a + b}. Applying @code{+} to the vector @samp{[a, b, c]} is an | |
20450 | error, since the @code{+} function expects exactly two arguments. | |
20451 | ||
20452 | While @kbd{V A} is useful in some cases, you will usually find that either | |
20453 | @kbd{V R} or @kbd{V M}, described below, is closer to what you want. | |
20454 | ||
20455 | @menu | |
20456 | * Specifying Operators:: | |
20457 | * Mapping:: | |
20458 | * Reducing:: | |
20459 | * Nesting and Fixed Points:: | |
20460 | * Generalized Products:: | |
20461 | @end menu | |
20462 | ||
20463 | @node Specifying Operators, Mapping, Reducing and Mapping, Reducing and Mapping | |
20464 | @subsection Specifying Operators | |
20465 | ||
20466 | @noindent | |
20467 | Commands in this section (like @kbd{V A}) prompt you to press the key | |
20468 | corresponding to the desired operator. Press @kbd{?} for a partial | |
20469 | list of the available operators. Generally, an operator is any key or | |
20470 | sequence of keys that would normally take one or more arguments from | |
20471 | the stack and replace them with a result. For example, @kbd{V A H C} | |
20472 | uses the hyperbolic cosine operator, @code{cosh}. (Since @code{cosh} | |
20473 | expects one argument, @kbd{V A H C} requires a vector with a single | |
20474 | element as its argument.) | |
20475 | ||
20476 | You can press @kbd{x} at the operator prompt to select any algebraic | |
20477 | function by name to use as the operator. This includes functions you | |
20478 | have defined yourself using the @kbd{Z F} command. (@xref{Algebraic | |
20479 | Definitions}.) If you give a name for which no function has been | |
20480 | defined, the result is left in symbolic form, as in @samp{f(1, 2, 3)}. | |
20481 | Calc will prompt for the number of arguments the function takes if it | |
20482 | can't figure it out on its own (say, because you named a function that | |
20483 | is currently undefined). It is also possible to type a digit key before | |
20484 | the function name to specify the number of arguments, e.g., | |
5d67986c | 20485 | @kbd{V M 3 x f @key{RET}} calls @code{f} with three arguments even if it |
d7b8e6c6 EZ |
20486 | looks like it ought to have only two. This technique may be necessary |
20487 | if the function allows a variable number of arguments. For example, | |
20488 | the @kbd{v e} [@code{vexp}] function accepts two or three arguments; | |
20489 | if you want to map with the three-argument version, you will have to | |
20490 | type @kbd{V M 3 v e}. | |
20491 | ||
20492 | It is also possible to apply any formula to a vector by treating that | |
20493 | formula as a function. When prompted for the operator to use, press | |
20494 | @kbd{'} (the apostrophe) and type your formula as an algebraic entry. | |
20495 | You will then be prompted for the argument list, which defaults to a | |
20496 | list of all variables that appear in the formula, sorted into alphabetic | |
20497 | order. For example, suppose you enter the formula @w{@samp{x + 2y^x}}. | |
20498 | The default argument list would be @samp{(x y)}, which means that if | |
20499 | this function is applied to the arguments @samp{[3, 10]} the result will | |
20500 | be @samp{3 + 2*10^3}. (If you plan to use a certain formula in this | |
20501 | way often, you might consider defining it as a function with @kbd{Z F}.) | |
20502 | ||
20503 | Another way to specify the arguments to the formula you enter is with | |
20504 | @kbd{$}, @kbd{$$}, and so on. For example, @kbd{V A ' $$ + 2$^$$} | |
20505 | has the same effect as the previous example. The argument list is | |
20506 | automatically taken to be @samp{($$ $)}. (The order of the arguments | |
20507 | may seem backwards, but it is analogous to the way normal algebraic | |
20508 | entry interacts with the stack.) | |
20509 | ||
20510 | If you press @kbd{$} at the operator prompt, the effect is similar to | |
20511 | the apostrophe except that the relevant formula is taken from top-of-stack | |
20512 | instead. The actual vector arguments of the @kbd{V A $} or related command | |
20513 | then start at the second-to-top stack position. You will still be | |
20514 | prompted for an argument list. | |
20515 | ||
20516 | @cindex Nameless functions | |
20517 | @cindex Generic functions | |
20518 | A function can be written without a name using the notation @samp{<#1 - #2>}, | |
20519 | which means ``a function of two arguments that computes the first | |
20520 | argument minus the second argument.'' The symbols @samp{#1} and @samp{#2} | |
20521 | are placeholders for the arguments. You can use any names for these | |
20522 | placeholders if you wish, by including an argument list followed by a | |
5d67986c | 20523 | colon: @samp{<x, y : x - y>}. When you type @kbd{V A ' $$ + 2$^$$ @key{RET}}, |
d7b8e6c6 | 20524 | Calc builds the nameless function @samp{<#1 + 2 #2^#1>} as the function |
5d67986c | 20525 | to map across the vectors. When you type @kbd{V A ' x + 2y^x @key{RET} @key{RET}}, |
d7b8e6c6 EZ |
20526 | Calc builds the nameless function @w{@samp{<x, y : x + 2 y^x>}}. In both |
20527 | cases, Calc also writes the nameless function to the Trail so that you | |
20528 | can get it back later if you wish. | |
20529 | ||
20530 | If there is only one argument, you can write @samp{#} in place of @samp{#1}. | |
20531 | (Note that @samp{< >} notation is also used for date forms. Calc tells | |
20532 | that @samp{<@var{stuff}>} is a nameless function by the presence of | |
20533 | @samp{#} signs inside @var{stuff}, or by the fact that @var{stuff} | |
20534 | begins with a list of variables followed by a colon.) | |
20535 | ||
20536 | You can type a nameless function directly to @kbd{V A '}, or put one on | |
20537 | the stack and use it with @w{@kbd{V A $}}. Calc will not prompt for an | |
20538 | argument list in this case, since the nameless function specifies the | |
20539 | argument list as well as the function itself. In @kbd{V A '}, you can | |
20540 | omit the @samp{< >} marks if you use @samp{#} notation for the arguments, | |
5d67986c RS |
20541 | so that @kbd{V A ' #1+#2 @key{RET}} is the same as @kbd{V A ' <#1+#2> @key{RET}}, |
20542 | which in turn is the same as @kbd{V A ' $$+$ @key{RET}}. | |
d7b8e6c6 EZ |
20543 | |
20544 | @cindex Lambda expressions | |
5d67986c RS |
20545 | @ignore |
20546 | @starindex | |
20547 | @end ignore | |
d7b8e6c6 EZ |
20548 | @tindex lambda |
20549 | The internal format for @samp{<x, y : x + y>} is @samp{lambda(x, y, x + y)}. | |
20550 | (The word @code{lambda} derives from Lisp notation and the theory of | |
20551 | functions.) The internal format for @samp{<#1 + #2>} is @samp{lambda(ArgA, | |
20552 | ArgB, ArgA + ArgB)}. Note that there is no actual Calc function called | |
20553 | @code{lambda}; the whole point is that the @code{lambda} expression is | |
20554 | used in its symbolic form, not evaluated for an answer until it is applied | |
20555 | to specific arguments by a command like @kbd{V A} or @kbd{V M}. | |
20556 | ||
20557 | (Actually, @code{lambda} does have one special property: Its arguments | |
20558 | are never evaluated; for example, putting @samp{<(2/3) #>} on the stack | |
20559 | will not simplify the @samp{2/3} until the nameless function is actually | |
20560 | called.) | |
20561 | ||
20562 | @tindex add | |
20563 | @tindex sub | |
5d67986c RS |
20564 | @ignore |
20565 | @mindex @idots | |
20566 | @end ignore | |
d7b8e6c6 | 20567 | @tindex mul |
5d67986c RS |
20568 | @ignore |
20569 | @mindex @null | |
20570 | @end ignore | |
d7b8e6c6 | 20571 | @tindex div |
5d67986c RS |
20572 | @ignore |
20573 | @mindex @null | |
20574 | @end ignore | |
d7b8e6c6 | 20575 | @tindex pow |
5d67986c RS |
20576 | @ignore |
20577 | @mindex @null | |
20578 | @end ignore | |
d7b8e6c6 | 20579 | @tindex neg |
5d67986c RS |
20580 | @ignore |
20581 | @mindex @null | |
20582 | @end ignore | |
d7b8e6c6 | 20583 | @tindex mod |
5d67986c RS |
20584 | @ignore |
20585 | @mindex @null | |
20586 | @end ignore | |
d7b8e6c6 EZ |
20587 | @tindex vconcat |
20588 | As usual, commands like @kbd{V A} have algebraic function name equivalents. | |
20589 | For example, @kbd{V A k g} with an argument of @samp{v} is equivalent to | |
20590 | @samp{apply(gcd, v)}. The first argument specifies the operator name, | |
20591 | and is either a variable whose name is the same as the function name, | |
20592 | or a nameless function like @samp{<#^3+1>}. Operators that are normally | |
20593 | written as algebraic symbols have the names @code{add}, @code{sub}, | |
20594 | @code{mul}, @code{div}, @code{pow}, @code{neg}, @code{mod}, and | |
20595 | @code{vconcat}.@refill | |
20596 | ||
5d67986c RS |
20597 | @ignore |
20598 | @starindex | |
20599 | @end ignore | |
d7b8e6c6 EZ |
20600 | @tindex call |
20601 | The @code{call} function builds a function call out of several arguments: | |
20602 | @samp{call(gcd, x, y)} is the same as @samp{apply(gcd, [x, y])}, which | |
20603 | in turn is the same as @samp{gcd(x, y)}. The first argument of @code{call}, | |
20604 | like the other functions described here, may be either a variable naming a | |
20605 | function, or a nameless function (@samp{call(<#1+2#2>, x, y)} is the same | |
20606 | as @samp{x + 2y}). | |
20607 | ||
20608 | (Experts will notice that it's not quite proper to use a variable to name | |
20609 | a function, since the name @code{gcd} corresponds to the Lisp variable | |
20610 | @code{var-gcd} but to the Lisp function @code{calcFunc-gcd}. Calc | |
20611 | automatically makes this translation, so you don't have to worry | |
20612 | about it.) | |
20613 | ||
20614 | @node Mapping, Reducing, Specifying Operators, Reducing and Mapping | |
20615 | @subsection Mapping | |
20616 | ||
20617 | @noindent | |
20618 | @kindex V M | |
20619 | @pindex calc-map | |
20620 | @tindex map | |
20621 | The @kbd{V M} (@code{calc-map}) [@code{map}] command applies a given | |
20622 | operator elementwise to one or more vectors. For example, mapping | |
20623 | @code{A} [@code{abs}] produces a vector of the absolute values of the | |
20624 | elements in the input vector. Mapping @code{+} pops two vectors from | |
20625 | the stack, which must be of equal length, and produces a vector of the | |
20626 | pairwise sums of the elements. If either argument is a non-vector, it | |
20627 | is duplicated for each element of the other vector. For example, | |
20628 | @kbd{[1,2,3] 2 V M ^} squares the elements of the specified vector. | |
20629 | With the 2 listed first, it would have computed a vector of powers of | |
20630 | two. Mapping a user-defined function pops as many arguments from the | |
20631 | stack as the function requires. If you give an undefined name, you will | |
20632 | be prompted for the number of arguments to use.@refill | |
20633 | ||
20634 | If any argument to @kbd{V M} is a matrix, the operator is normally mapped | |
20635 | across all elements of the matrix. For example, given the matrix | |
20636 | @cite{[[1, -2, 3], [-4, 5, -6]]}, @kbd{V M A} takes six absolute values to | |
20637 | produce another @c{$3\times2$} | |
20638 | @asis{3x2} matrix, @cite{[[1, 2, 3], [4, 5, 6]]}. | |
20639 | ||
20640 | @tindex mapr | |
20641 | The command @kbd{V M _} [@code{mapr}] (i.e., type an underscore at the | |
20642 | operator prompt) maps by rows instead. For example, @kbd{V M _ A} views | |
20643 | the above matrix as a vector of two 3-element row vectors. It produces | |
20644 | a new vector which contains the absolute values of those row vectors, | |
20645 | namely @cite{[3.74, 8.77]}. (Recall, the absolute value of a vector is | |
20646 | defined as the square root of the sum of the squares of the elements.) | |
20647 | Some operators accept vectors and return new vectors; for example, | |
20648 | @kbd{v v} reverses a vector, so @kbd{V M _ v v} would reverse each row | |
20649 | of the matrix to get a new matrix, @cite{[[3, -2, 1], [-6, 5, -4]]}. | |
20650 | ||
20651 | Sometimes a vector of vectors (representing, say, strings, sets, or lists) | |
20652 | happens to look like a matrix. If so, remember to use @kbd{V M _} if you | |
20653 | want to map a function across the whole strings or sets rather than across | |
20654 | their individual elements. | |
20655 | ||
20656 | @tindex mapc | |
20657 | The command @kbd{V M :} [@code{mapc}] maps by columns. Basically, it | |
20658 | transposes the input matrix, maps by rows, and then, if the result is a | |
20659 | matrix, transposes again. For example, @kbd{V M : A} takes the absolute | |
20660 | values of the three columns of the matrix, treating each as a 2-vector, | |
20661 | and @kbd{V M : v v} reverses the columns to get the matrix | |
20662 | @cite{[[-4, 5, -6], [1, -2, 3]]}. | |
20663 | ||
20664 | (The symbols @kbd{_} and @kbd{:} were chosen because they had row-like | |
20665 | and column-like appearances, and were not already taken by useful | |
20666 | operators. Also, they appear shifted on most keyboards so they are easy | |
20667 | to type after @kbd{V M}.) | |
20668 | ||
20669 | The @kbd{_} and @kbd{:} modifiers have no effect on arguments that are | |
20670 | not matrices (so if none of the arguments are matrices, they have no | |
20671 | effect at all). If some of the arguments are matrices and others are | |
20672 | plain numbers, the plain numbers are held constant for all rows of the | |
20673 | matrix (so that @kbd{2 V M _ ^} squares every row of a matrix; squaring | |
20674 | a vector takes a dot product of the vector with itself). | |
20675 | ||
20676 | If some of the arguments are vectors with the same lengths as the | |
20677 | rows (for @kbd{V M _}) or columns (for @kbd{V M :}) of the matrix | |
20678 | arguments, those vectors are also held constant for every row or | |
20679 | column. | |
20680 | ||
20681 | Sometimes it is useful to specify another mapping command as the operator | |
20682 | to use with @kbd{V M}. For example, @kbd{V M _ V A +} applies @kbd{V A +} | |
20683 | to each row of the input matrix, which in turn adds the two values on that | |
20684 | row. If you give another vector-operator command as the operator for | |
20685 | @kbd{V M}, it automatically uses map-by-rows mode if you don't specify | |
20686 | otherwise; thus @kbd{V M V A +} is equivalent to @kbd{V M _ V A +}. (If | |
20687 | you really want to map-by-elements another mapping command, you can use | |
20688 | a triple-nested mapping command: @kbd{V M V M V A +} means to map | |
20689 | @kbd{V M V A +} over the rows of the matrix; in turn, @kbd{V A +} is | |
20690 | mapped over the elements of each row.) | |
20691 | ||
20692 | @tindex mapa | |
20693 | @tindex mapd | |
20694 | Previous versions of Calc had ``map across'' and ``map down'' modes | |
20695 | that are now considered obsolete; the old ``map across'' is now simply | |
20696 | @kbd{V M V A}, and ``map down'' is now @kbd{V M : V A}. The algebraic | |
20697 | functions @code{mapa} and @code{mapd} are still supported, though. | |
20698 | Note also that, while the old mapping modes were persistent (once you | |
20699 | set the mode, it would apply to later mapping commands until you reset | |
20700 | it), the new @kbd{:} and @kbd{_} modifiers apply only to the current | |
20701 | mapping command. The default @kbd{V M} always means map-by-elements. | |
20702 | ||
20703 | @xref{Algebraic Manipulation}, for the @kbd{a M} command, which is like | |
20704 | @kbd{V M} but for equations and inequalities instead of vectors. | |
20705 | @xref{Storing Variables}, for the @kbd{s m} command which modifies a | |
20706 | variable's stored value using a @kbd{V M}-like operator. | |
20707 | ||
20708 | @node Reducing, Nesting and Fixed Points, Mapping, Reducing and Mapping | |
20709 | @subsection Reducing | |
20710 | ||
20711 | @noindent | |
20712 | @kindex V R | |
20713 | @pindex calc-reduce | |
20714 | @tindex reduce | |
20715 | The @kbd{V R} (@code{calc-reduce}) [@code{reduce}] command applies a given | |
20716 | binary operator across all the elements of a vector. A binary operator is | |
20717 | a function such as @code{+} or @code{max} which takes two arguments. For | |
20718 | example, reducing @code{+} over a vector computes the sum of the elements | |
20719 | of the vector. Reducing @code{-} computes the first element minus each of | |
20720 | the remaining elements. Reducing @code{max} computes the maximum element | |
20721 | and so on. In general, reducing @code{f} over the vector @samp{[a, b, c, d]} | |
20722 | produces @samp{f(f(f(a, b), c), d)}. | |
20723 | ||
20724 | @kindex I V R | |
20725 | @tindex rreduce | |
20726 | The @kbd{I V R} [@code{rreduce}] command is similar to @kbd{V R} except | |
20727 | that works from right to left through the vector. For example, plain | |
20728 | @kbd{V R -} on the vector @samp{[a, b, c, d]} produces @samp{a - b - c - d} | |
20729 | but @kbd{I V R -} on the same vector produces @samp{a - (b - (c - d))}, | |
20730 | or @samp{a - b + c - d}. This ``alternating sum'' occurs frequently | |
20731 | in power series expansions. | |
20732 | ||
20733 | @kindex V U | |
20734 | @tindex accum | |
20735 | The @kbd{V U} (@code{calc-accumulate}) [@code{accum}] command does an | |
20736 | accumulation operation. Here Calc does the corresponding reduction | |
20737 | operation, but instead of producing only the final result, it produces | |
20738 | a vector of all the intermediate results. Accumulating @code{+} over | |
20739 | the vector @samp{[a, b, c, d]} produces the vector | |
20740 | @samp{[a, a + b, a + b + c, a + b + c + d]}. | |
20741 | ||
20742 | @kindex I V U | |
20743 | @tindex raccum | |
20744 | The @kbd{I V U} [@code{raccum}] command does a right-to-left accumulation. | |
20745 | For example, @kbd{I V U -} on the vector @samp{[a, b, c, d]} produces the | |
20746 | vector @samp{[a - b + c - d, b - c + d, c - d, d]}. | |
20747 | ||
20748 | @tindex reducea | |
20749 | @tindex rreducea | |
20750 | @tindex reduced | |
20751 | @tindex rreduced | |
20752 | As for @kbd{V M}, @kbd{V R} normally reduces a matrix elementwise. For | |
20753 | example, given the matrix @cite{[[a, b, c], [d, e, f]]}, @kbd{V R +} will | |
20754 | compute @cite{a + b + c + d + e + f}. You can type @kbd{V R _} or | |
20755 | @kbd{V R :} to modify this behavior. The @kbd{V R _} [@code{reducea}] | |
20756 | command reduces ``across'' the matrix; it reduces each row of the matrix | |
20757 | as a vector, then collects the results. Thus @kbd{V R _ +} of this | |
20758 | matrix would produce @cite{[a + b + c, d + e + f]}. Similarly, @kbd{V R :} | |
20759 | [@code{reduced}] reduces down; @kbd{V R : +} would produce @cite{[a + d, | |
20760 | b + e, c + f]}. | |
20761 | ||
20762 | @tindex reducer | |
20763 | @tindex rreducer | |
20764 | There is a third ``by rows'' mode for reduction that is occasionally | |
20765 | useful; @kbd{V R =} [@code{reducer}] simply reduces the operator over | |
20766 | the rows of the matrix themselves. Thus @kbd{V R = +} on the above | |
20767 | matrix would get the same result as @kbd{V R : +}, since adding two | |
20768 | row vectors is equivalent to adding their elements. But @kbd{V R = *} | |
20769 | would multiply the two rows (to get a single number, their dot product), | |
20770 | while @kbd{V R : *} would produce a vector of the products of the columns. | |
20771 | ||
20772 | These three matrix reduction modes work with @kbd{V R} and @kbd{I V R}, | |
20773 | but they are not currently supported with @kbd{V U} or @kbd{I V U}. | |
20774 | ||
20775 | @tindex reducec | |
20776 | @tindex rreducec | |
20777 | The obsolete reduce-by-columns function, @code{reducec}, is still | |
20778 | supported but there is no way to get it through the @kbd{V R} command. | |
20779 | ||
20780 | The commands @kbd{M-# :} and @kbd{M-# _} are equivalent to typing | |
20781 | @kbd{M-# r} to grab a rectangle of data into Calc, and then typing | |
20782 | @kbd{V R : +} or @kbd{V R _ +}, respectively, to sum the columns or | |
20783 | rows of the matrix. @xref{Grabbing From Buffers}. | |
20784 | ||
20785 | @node Nesting and Fixed Points, Generalized Products, Reducing, Reducing and Mapping | |
20786 | @subsection Nesting and Fixed Points | |
20787 | ||
20788 | @noindent | |
20789 | @kindex H V R | |
20790 | @tindex nest | |
20791 | The @kbd{H V R} [@code{nest}] command applies a function to a given | |
20792 | argument repeatedly. It takes two values, @samp{a} and @samp{n}, from | |
20793 | the stack, where @samp{n} must be an integer. It then applies the | |
20794 | function nested @samp{n} times; if the function is @samp{f} and @samp{n} | |
20795 | is 3, the result is @samp{f(f(f(a)))}. The number @samp{n} may be | |
20796 | negative if Calc knows an inverse for the function @samp{f}; for | |
20797 | example, @samp{nest(sin, a, -2)} returns @samp{arcsin(arcsin(a))}. | |
20798 | ||
20799 | @kindex H V U | |
20800 | @tindex anest | |
20801 | The @kbd{H V U} [@code{anest}] command is an accumulating version of | |
20802 | @code{nest}: It returns a vector of @samp{n+1} values, e.g., | |
20803 | @samp{[a, f(a), f(f(a)), f(f(f(a)))]}. If @samp{n} is negative and | |
20804 | @samp{F} is the inverse of @samp{f}, then the result is of the | |
20805 | form @samp{[a, F(a), F(F(a)), F(F(F(a)))]}. | |
20806 | ||
20807 | @kindex H I V R | |
20808 | @tindex fixp | |
20809 | @cindex Fixed points | |
20810 | The @kbd{H I V R} [@code{fixp}] command is like @kbd{H V R}, except | |
20811 | that it takes only an @samp{a} value from the stack; the function is | |
20812 | applied until it reaches a ``fixed point,'' i.e., until the result | |
20813 | no longer changes. | |
20814 | ||
20815 | @kindex H I V U | |
20816 | @tindex afixp | |
20817 | The @kbd{H I V U} [@code{afixp}] command is an accumulating @code{fixp}. | |
20818 | The first element of the return vector will be the initial value @samp{a}; | |
20819 | the last element will be the final result that would have been returned | |
20820 | by @code{fixp}. | |
20821 | ||
20822 | For example, 0.739085 is a fixed point of the cosine function (in radians): | |
20823 | @samp{cos(0.739085) = 0.739085}. You can find this value by putting, say, | |
20824 | 1.0 on the stack and typing @kbd{H I V U C}. (We use the accumulating | |
20825 | version so we can see the intermediate results: @samp{[1, 0.540302, 0.857553, | |
20826 | 0.65329, ...]}. With a precision of six, this command will take 36 steps | |
20827 | to converge to 0.739085.) | |
20828 | ||
20829 | Newton's method for finding roots is a classic example of iteration | |
20830 | to a fixed point. To find the square root of five starting with an | |
20831 | initial guess, Newton's method would look for a fixed point of the | |
20832 | function @samp{(x + 5/x) / 2}. Putting a guess of 1 on the stack | |
5d67986c | 20833 | and typing @kbd{H I V R ' ($ + 5/$)/2 @key{RET}} quickly yields the result |
d7b8e6c6 EZ |
20834 | 2.23607. This is equivalent to using the @kbd{a R} (@code{calc-find-root}) |
20835 | command to find a root of the equation @samp{x^2 = 5}. | |
20836 | ||
20837 | These examples used numbers for @samp{a} values. Calc keeps applying | |
20838 | the function until two successive results are equal to within the | |
20839 | current precision. For complex numbers, both the real parts and the | |
20840 | imaginary parts must be equal to within the current precision. If | |
20841 | @samp{a} is a formula (say, a variable name), then the function is | |
20842 | applied until two successive results are exactly the same formula. | |
20843 | It is up to you to ensure that the function will eventually converge; | |
20844 | if it doesn't, you may have to press @kbd{C-g} to stop the Calculator. | |
20845 | ||
20846 | The algebraic @code{fixp} function takes two optional arguments, @samp{n} | |
20847 | and @samp{tol}. The first is the maximum number of steps to be allowed, | |
20848 | and must be either an integer or the symbol @samp{inf} (infinity, the | |
20849 | default). The second is a convergence tolerance. If a tolerance is | |
20850 | specified, all results during the calculation must be numbers, not | |
20851 | formulas, and the iteration stops when the magnitude of the difference | |
20852 | between two successive results is less than or equal to the tolerance. | |
20853 | (This implies that a tolerance of zero iterates until the results are | |
20854 | exactly equal.) | |
20855 | ||
20856 | Putting it all together, @samp{fixp(<(# + A/#)/2>, B, 20, 1e-10)} | |
20857 | computes the square root of @samp{A} given the initial guess @samp{B}, | |
20858 | stopping when the result is correct within the specified tolerance, or | |
20859 | when 20 steps have been taken, whichever is sooner. | |
20860 | ||
20861 | @node Generalized Products, , Nesting and Fixed Points, Reducing and Mapping | |
20862 | @subsection Generalized Products | |
20863 | ||
20864 | @kindex V O | |
20865 | @pindex calc-outer-product | |
20866 | @tindex outer | |
20867 | The @kbd{V O} (@code{calc-outer-product}) [@code{outer}] command applies | |
20868 | a given binary operator to all possible pairs of elements from two | |
20869 | vectors, to produce a matrix. For example, @kbd{V O *} with @samp{[a, b]} | |
20870 | and @samp{[x, y, z]} on the stack produces a multiplication table: | |
20871 | @samp{[[a x, a y, a z], [b x, b y, b z]]}. Element @var{r},@var{c} of | |
20872 | the result matrix is obtained by applying the operator to element @var{r} | |
20873 | of the lefthand vector and element @var{c} of the righthand vector. | |
20874 | ||
20875 | @kindex V I | |
20876 | @pindex calc-inner-product | |
20877 | @tindex inner | |
20878 | The @kbd{V I} (@code{calc-inner-product}) [@code{inner}] command computes | |
20879 | the generalized inner product of two vectors or matrices, given a | |
20880 | ``multiplicative'' operator and an ``additive'' operator. These can each | |
20881 | actually be any binary operators; if they are @samp{*} and @samp{+}, | |
20882 | respectively, the result is a standard matrix multiplication. Element | |
20883 | @var{r},@var{c} of the result matrix is obtained by mapping the | |
20884 | multiplicative operator across row @var{r} of the lefthand matrix and | |
20885 | column @var{c} of the righthand matrix, and then reducing with the additive | |
20886 | operator. Just as for the standard @kbd{*} command, this can also do a | |
20887 | vector-matrix or matrix-vector inner product, or a vector-vector | |
20888 | generalized dot product. | |
20889 | ||
20890 | Since @kbd{V I} requires two operators, it prompts twice. In each case, | |
20891 | you can use any of the usual methods for entering the operator. If you | |
20892 | use @kbd{$} twice to take both operator formulas from the stack, the | |
20893 | first (multiplicative) operator is taken from the top of the stack | |
20894 | and the second (additive) operator is taken from second-to-top. | |
20895 | ||
20896 | @node Vector and Matrix Formats, , Reducing and Mapping, Matrix Functions | |
20897 | @section Vector and Matrix Display Formats | |
20898 | ||
20899 | @noindent | |
20900 | Commands for controlling vector and matrix display use the @kbd{v} prefix | |
20901 | instead of the usual @kbd{d} prefix. But they are display modes; in | |
20902 | particular, they are influenced by the @kbd{I} and @kbd{H} prefix keys | |
20903 | in the same way (@pxref{Display Modes}). Matrix display is also | |
20904 | influenced by the @kbd{d O} (@code{calc-flat-language}) mode; | |
20905 | @pxref{Normal Language Modes}. | |
20906 | ||
20907 | @kindex V < | |
20908 | @pindex calc-matrix-left-justify | |
20909 | @kindex V = | |
20910 | @pindex calc-matrix-center-justify | |
20911 | @kindex V > | |
20912 | @pindex calc-matrix-right-justify | |
20913 | The commands @kbd{v <} (@code{calc-matrix-left-justify}), @kbd{v >} | |
20914 | (@code{calc-matrix-right-justify}), and @w{@kbd{v =}} | |
20915 | (@code{calc-matrix-center-justify}) control whether matrix elements | |
20916 | are justified to the left, right, or center of their columns.@refill | |
20917 | ||
20918 | @kindex V [ | |
20919 | @pindex calc-vector-brackets | |
20920 | @kindex V @{ | |
20921 | @pindex calc-vector-braces | |
20922 | @kindex V ( | |
20923 | @pindex calc-vector-parens | |
20924 | The @kbd{v [} (@code{calc-vector-brackets}) command turns the square | |
20925 | brackets that surround vectors and matrices displayed in the stack on | |
20926 | and off. The @kbd{v @{} (@code{calc-vector-braces}) and @kbd{v (} | |
20927 | (@code{calc-vector-parens}) commands use curly braces or parentheses, | |
20928 | respectively, instead of square brackets. For example, @kbd{v @{} might | |
20929 | be used in preparation for yanking a matrix into a buffer running | |
20930 | Mathematica. (In fact, the Mathematica language mode uses this mode; | |
20931 | @pxref{Mathematica Language Mode}.) Note that, regardless of the | |
20932 | display mode, either brackets or braces may be used to enter vectors, | |
20933 | and parentheses may never be used for this purpose.@refill | |
20934 | ||
20935 | @kindex V ] | |
20936 | @pindex calc-matrix-brackets | |
20937 | The @kbd{v ]} (@code{calc-matrix-brackets}) command controls the | |
20938 | ``big'' style display of matrices. It prompts for a string of code | |
20939 | letters; currently implemented letters are @code{R}, which enables | |
20940 | brackets on each row of the matrix; @code{O}, which enables outer | |
20941 | brackets in opposite corners of the matrix; and @code{C}, which | |
20942 | enables commas or semicolons at the ends of all rows but the last. | |
20943 | The default format is @samp{RO}. (Before Calc 2.00, the format | |
20944 | was fixed at @samp{ROC}.) Here are some example matrices: | |
20945 | ||
d7b8e6c6 | 20946 | @example |
5d67986c | 20947 | @group |
d7b8e6c6 EZ |
20948 | [ [ 123, 0, 0 ] [ [ 123, 0, 0 ], |
20949 | [ 0, 123, 0 ] [ 0, 123, 0 ], | |
20950 | [ 0, 0, 123 ] ] [ 0, 0, 123 ] ] | |
20951 | ||
20952 | RO ROC | |
20953 | ||
d7b8e6c6 | 20954 | @end group |
5d67986c | 20955 | @end example |
d7b8e6c6 | 20956 | @noindent |
d7b8e6c6 | 20957 | @example |
5d67986c | 20958 | @group |
d7b8e6c6 EZ |
20959 | [ 123, 0, 0 [ 123, 0, 0 ; |
20960 | 0, 123, 0 0, 123, 0 ; | |
20961 | 0, 0, 123 ] 0, 0, 123 ] | |
20962 | ||
20963 | O OC | |
20964 | ||
d7b8e6c6 | 20965 | @end group |
5d67986c | 20966 | @end example |
d7b8e6c6 | 20967 | @noindent |
d7b8e6c6 | 20968 | @example |
5d67986c | 20969 | @group |
d7b8e6c6 EZ |
20970 | [ 123, 0, 0 ] 123, 0, 0 |
20971 | [ 0, 123, 0 ] 0, 123, 0 | |
20972 | [ 0, 0, 123 ] 0, 0, 123 | |
20973 | ||
20974 | R @r{blank} | |
d7b8e6c6 | 20975 | @end group |
5d67986c | 20976 | @end example |
d7b8e6c6 EZ |
20977 | |
20978 | @noindent | |
20979 | Note that of the formats shown here, @samp{RO}, @samp{ROC}, and | |
20980 | @samp{OC} are all recognized as matrices during reading, while | |
20981 | the others are useful for display only. | |
20982 | ||
20983 | @kindex V , | |
20984 | @pindex calc-vector-commas | |
20985 | The @kbd{v ,} (@code{calc-vector-commas}) command turns commas on and | |
20986 | off in vector and matrix display.@refill | |
20987 | ||
20988 | In vectors of length one, and in all vectors when commas have been | |
20989 | turned off, Calc adds extra parentheses around formulas that might | |
20990 | otherwise be ambiguous. For example, @samp{[a b]} could be a vector | |
20991 | of the one formula @samp{a b}, or it could be a vector of two | |
20992 | variables with commas turned off. Calc will display the former | |
20993 | case as @samp{[(a b)]}. You can disable these extra parentheses | |
20994 | (to make the output less cluttered at the expense of allowing some | |
20995 | ambiguity) by adding the letter @code{P} to the control string you | |
20996 | give to @kbd{v ]} (as described above). | |
20997 | ||
20998 | @kindex V . | |
20999 | @pindex calc-full-vectors | |
21000 | The @kbd{v .} (@code{calc-full-vectors}) command turns abbreviated | |
21001 | display of long vectors on and off. In this mode, vectors of six | |
21002 | or more elements, or matrices of six or more rows or columns, will | |
21003 | be displayed in an abbreviated form that displays only the first | |
21004 | three elements and the last element: @samp{[a, b, c, ..., z]}. | |
21005 | When very large vectors are involved this will substantially | |
21006 | improve Calc's display speed. | |
21007 | ||
21008 | @kindex t . | |
21009 | @pindex calc-full-trail-vectors | |
21010 | The @kbd{t .} (@code{calc-full-trail-vectors}) command controls a | |
21011 | similar mode for recording vectors in the Trail. If you turn on | |
21012 | this mode, vectors of six or more elements and matrices of six or | |
21013 | more rows or columns will be abbreviated when they are put in the | |
21014 | Trail. The @kbd{t y} (@code{calc-trail-yank}) command will be | |
21015 | unable to recover those vectors. If you are working with very | |
21016 | large vectors, this mode will improve the speed of all operations | |
21017 | that involve the trail. | |
21018 | ||
21019 | @kindex V / | |
21020 | @pindex calc-break-vectors | |
21021 | The @kbd{v /} (@code{calc-break-vectors}) command turns multi-line | |
21022 | vector display on and off. Normally, matrices are displayed with one | |
21023 | row per line but all other types of vectors are displayed in a single | |
21024 | line. This mode causes all vectors, whether matrices or not, to be | |
21025 | displayed with a single element per line. Sub-vectors within the | |
21026 | vectors will still use the normal linear form. | |
21027 | ||
21028 | @node Algebra, Units, Matrix Functions, Top | |
21029 | @chapter Algebra | |
21030 | ||
21031 | @noindent | |
21032 | This section covers the Calc features that help you work with | |
21033 | algebraic formulas. First, the general sub-formula selection | |
21034 | mechanism is described; this works in conjunction with any Calc | |
21035 | commands. Then, commands for specific algebraic operations are | |
21036 | described. Finally, the flexible @dfn{rewrite rule} mechanism | |
21037 | is discussed. | |
21038 | ||
21039 | The algebraic commands use the @kbd{a} key prefix; selection | |
21040 | commands use the @kbd{j} (for ``just a letter that wasn't used | |
21041 | for anything else'') prefix. | |
21042 | ||
21043 | @xref{Editing Stack Entries}, to see how to manipulate formulas | |
21044 | using regular Emacs editing commands.@refill | |
21045 | ||
21046 | When doing algebraic work, you may find several of the Calculator's | |
21047 | modes to be helpful, including algebraic-simplification mode (@kbd{m A}) | |
21048 | or no-simplification mode (@kbd{m O}), | |
21049 | algebraic-entry mode (@kbd{m a}), fraction mode (@kbd{m f}), and | |
21050 | symbolic mode (@kbd{m s}). @xref{Mode Settings}, for discussions | |
21051 | of these modes. You may also wish to select ``big'' display mode (@kbd{d B}). | |
21052 | @xref{Normal Language Modes}.@refill | |
21053 | ||
21054 | @menu | |
21055 | * Selecting Subformulas:: | |
21056 | * Algebraic Manipulation:: | |
21057 | * Simplifying Formulas:: | |
21058 | * Polynomials:: | |
21059 | * Calculus:: | |
21060 | * Solving Equations:: | |
21061 | * Numerical Solutions:: | |
21062 | * Curve Fitting:: | |
21063 | * Summations:: | |
21064 | * Logical Operations:: | |
21065 | * Rewrite Rules:: | |
21066 | @end menu | |
21067 | ||
21068 | @node Selecting Subformulas, Algebraic Manipulation, Algebra, Algebra | |
21069 | @section Selecting Sub-Formulas | |
21070 | ||
21071 | @noindent | |
21072 | @cindex Selections | |
21073 | @cindex Sub-formulas | |
21074 | @cindex Parts of formulas | |
21075 | When working with an algebraic formula it is often necessary to | |
21076 | manipulate a portion of the formula rather than the formula as a | |
21077 | whole. Calc allows you to ``select'' a portion of any formula on | |
21078 | the stack. Commands which would normally operate on that stack | |
21079 | entry will now operate only on the sub-formula, leaving the | |
21080 | surrounding part of the stack entry alone. | |
21081 | ||
21082 | One common non-algebraic use for selection involves vectors. To work | |
21083 | on one element of a vector in-place, simply select that element as a | |
21084 | ``sub-formula'' of the vector. | |
21085 | ||
21086 | @menu | |
21087 | * Making Selections:: | |
21088 | * Changing Selections:: | |
21089 | * Displaying Selections:: | |
21090 | * Operating on Selections:: | |
21091 | * Rearranging with Selections:: | |
21092 | @end menu | |
21093 | ||
21094 | @node Making Selections, Changing Selections, Selecting Subformulas, Selecting Subformulas | |
21095 | @subsection Making Selections | |
21096 | ||
21097 | @noindent | |
21098 | @kindex j s | |
21099 | @pindex calc-select-here | |
21100 | To select a sub-formula, move the Emacs cursor to any character in that | |
21101 | sub-formula, and press @w{@kbd{j s}} (@code{calc-select-here}). Calc will | |
21102 | highlight the smallest portion of the formula that contains that | |
21103 | character. By default the sub-formula is highlighted by blanking out | |
21104 | all of the rest of the formula with dots. Selection works in any | |
21105 | display mode but is perhaps easiest in ``big'' (@kbd{d B}) mode. | |
21106 | Suppose you enter the following formula: | |
21107 | ||
d7b8e6c6 | 21108 | @smallexample |
5d67986c | 21109 | @group |
d7b8e6c6 EZ |
21110 | 3 ___ |
21111 | (a + b) + V c | |
21112 | 1: --------------- | |
21113 | 2 x + 1 | |
d7b8e6c6 | 21114 | @end group |
5d67986c | 21115 | @end smallexample |
d7b8e6c6 EZ |
21116 | |
21117 | @noindent | |
21118 | (by typing @kbd{' ((a+b)^3 + sqrt(c)) / (2x+1)}). If you move the | |
21119 | cursor to the letter @samp{b} and press @w{@kbd{j s}}, the display changes | |
21120 | to | |
21121 | ||
d7b8e6c6 | 21122 | @smallexample |
5d67986c | 21123 | @group |
d7b8e6c6 EZ |
21124 | . ... |
21125 | .. . b. . . . | |
21126 | 1* ............... | |
21127 | . . . . | |
d7b8e6c6 | 21128 | @end group |
5d67986c | 21129 | @end smallexample |
d7b8e6c6 EZ |
21130 | |
21131 | @noindent | |
21132 | Every character not part of the sub-formula @samp{b} has been changed | |
21133 | to a dot. The @samp{*} next to the line number is to remind you that | |
21134 | the formula has a portion of it selected. (In this case, it's very | |
21135 | obvious, but it might not always be. If Embedded Mode is enabled, | |
21136 | the word @samp{Sel} also appears in the mode line because the stack | |
21137 | may not be visible. @pxref{Embedded Mode}.) | |
21138 | ||
21139 | If you had instead placed the cursor on the parenthesis immediately to | |
21140 | the right of the @samp{b}, the selection would have been: | |
21141 | ||
d7b8e6c6 | 21142 | @smallexample |
5d67986c | 21143 | @group |
d7b8e6c6 EZ |
21144 | . ... |
21145 | (a + b) . . . | |
21146 | 1* ............... | |
21147 | . . . . | |
d7b8e6c6 | 21148 | @end group |
5d67986c | 21149 | @end smallexample |
d7b8e6c6 EZ |
21150 | |
21151 | @noindent | |
21152 | The portion selected is always large enough to be considered a complete | |
21153 | formula all by itself, so selecting the parenthesis selects the whole | |
269b7745 | 21154 | formula that it encloses. Putting the cursor on the @samp{+} sign |
d7b8e6c6 EZ |
21155 | would have had the same effect. |
21156 | ||
21157 | (Strictly speaking, the Emacs cursor is really the manifestation of | |
21158 | the Emacs ``point,'' which is a position @emph{between} two characters | |
21159 | in the buffer. So purists would say that Calc selects the smallest | |
21160 | sub-formula which contains the character to the right of ``point.'') | |
21161 | ||
21162 | If you supply a numeric prefix argument @var{n}, the selection is | |
21163 | expanded to the @var{n}th enclosing sub-formula. Thus, positioning | |
21164 | the cursor on the @samp{b} and typing @kbd{C-u 1 j s} will select | |
21165 | @samp{a + b}; typing @kbd{C-u 2 j s} will select @samp{(a + b)^3}, | |
21166 | and so on. | |
21167 | ||
21168 | If the cursor is not on any part of the formula, or if you give a | |
21169 | numeric prefix that is too large, the entire formula is selected. | |
21170 | ||
21171 | If the cursor is on the @samp{.} line that marks the top of the stack | |
21172 | (i.e., its normal ``rest position''), this command selects the entire | |
21173 | formula at stack level 1. Most selection commands similarly operate | |
21174 | on the formula at the top of the stack if you haven't positioned the | |
21175 | cursor on any stack entry. | |
21176 | ||
21177 | @kindex j a | |
21178 | @pindex calc-select-additional | |
21179 | The @kbd{j a} (@code{calc-select-additional}) command enlarges the | |
21180 | current selection to encompass the cursor. To select the smallest | |
21181 | sub-formula defined by two different points, move to the first and | |
21182 | press @kbd{j s}, then move to the other and press @kbd{j a}. This | |
21183 | is roughly analogous to using @kbd{C-@@} (@code{set-mark-command}) to | |
21184 | select the two ends of a region of text during normal Emacs editing. | |
21185 | ||
21186 | @kindex j o | |
21187 | @pindex calc-select-once | |
21188 | The @kbd{j o} (@code{calc-select-once}) command selects a formula in | |
21189 | exactly the same way as @kbd{j s}, except that the selection will | |
21190 | last only as long as the next command that uses it. For example, | |
21191 | @kbd{j o 1 +} is a handy way to add one to the sub-formula indicated | |
21192 | by the cursor. | |
21193 | ||
21194 | (A somewhat more precise definition: The @kbd{j o} command sets a flag | |
21195 | such that the next command involving selected stack entries will clear | |
21196 | the selections on those stack entries afterwards. All other selection | |
21197 | commands except @kbd{j a} and @kbd{j O} clear this flag.) | |
21198 | ||
21199 | @kindex j S | |
21200 | @kindex j O | |
21201 | @pindex calc-select-here-maybe | |
21202 | @pindex calc-select-once-maybe | |
21203 | The @kbd{j S} (@code{calc-select-here-maybe}) and @kbd{j O} | |
21204 | (@code{calc-select-once-maybe}) commands are equivalent to @kbd{j s} | |
21205 | and @kbd{j o}, respectively, except that if the formula already | |
21206 | has a selection they have no effect. This is analogous to the | |
21207 | behavior of some commands such as @kbd{j r} (@code{calc-rewrite-selection}; | |
21208 | @pxref{Selections with Rewrite Rules}) and is mainly intended to be | |
21209 | used in keyboard macros that implement your own selection-oriented | |
21210 | commands.@refill | |
21211 | ||
21212 | Selection of sub-formulas normally treats associative terms like | |
21213 | @samp{a + b - c + d} and @samp{x * y * z} as single levels of the formula. | |
21214 | If you place the cursor anywhere inside @samp{a + b - c + d} except | |
21215 | on one of the variable names and use @kbd{j s}, you will select the | |
21216 | entire four-term sum. | |
21217 | ||
21218 | @kindex j b | |
21219 | @pindex calc-break-selections | |
21220 | The @kbd{j b} (@code{calc-break-selections}) command controls a mode | |
21221 | in which the ``deep structure'' of these associative formulas shows | |
21222 | through. Calc actually stores the above formulas as @samp{((a + b) - c) + d} | |
21223 | and @samp{x * (y * z)}. (Note that for certain obscure reasons, Calc | |
21224 | treats multiplication as right-associative.) Once you have enabled | |
21225 | @kbd{j b} mode, selecting with the cursor on the @samp{-} sign would | |
21226 | only select the @samp{a + b - c} portion, which makes sense when the | |
21227 | deep structure of the sum is considered. There is no way to select | |
21228 | the @samp{b - c + d} portion; although this might initially look | |
21229 | like just as legitimate a sub-formula as @samp{a + b - c}, the deep | |
21230 | structure shows that it isn't. The @kbd{d U} command can be used | |
21231 | to view the deep structure of any formula (@pxref{Normal Language Modes}). | |
21232 | ||
21233 | When @kbd{j b} mode has not been enabled, the deep structure is | |
21234 | generally hidden by the selection commands---what you see is what | |
21235 | you get. | |
21236 | ||
21237 | @kindex j u | |
21238 | @pindex calc-unselect | |
21239 | The @kbd{j u} (@code{calc-unselect}) command unselects the formula | |
21240 | that the cursor is on. If there was no selection in the formula, | |
21241 | this command has no effect. With a numeric prefix argument, it | |
21242 | unselects the @var{n}th stack element rather than using the cursor | |
21243 | position. | |
21244 | ||
21245 | @kindex j c | |
21246 | @pindex calc-clear-selections | |
21247 | The @kbd{j c} (@code{calc-clear-selections}) command unselects all | |
21248 | stack elements. | |
21249 | ||
21250 | @node Changing Selections, Displaying Selections, Making Selections, Selecting Subformulas | |
21251 | @subsection Changing Selections | |
21252 | ||
21253 | @noindent | |
21254 | @kindex j m | |
21255 | @pindex calc-select-more | |
21256 | Once you have selected a sub-formula, you can expand it using the | |
21257 | @w{@kbd{j m}} (@code{calc-select-more}) command. If @samp{a + b} is | |
21258 | selected, pressing @w{@kbd{j m}} repeatedly works as follows: | |
21259 | ||
d7b8e6c6 | 21260 | @smallexample |
5d67986c | 21261 | @group |
d7b8e6c6 EZ |
21262 | 3 ... 3 ___ 3 ___ |
21263 | (a + b) . . . (a + b) + V c (a + b) + V c | |
21264 | 1* ............... 1* ............... 1* --------------- | |
21265 | . . . . . . . . 2 x + 1 | |
d7b8e6c6 | 21266 | @end group |
5d67986c | 21267 | @end smallexample |
d7b8e6c6 EZ |
21268 | |
21269 | @noindent | |
21270 | In the last example, the entire formula is selected. This is roughly | |
21271 | the same as having no selection at all, but because there are subtle | |
21272 | differences the @samp{*} character is still there on the line number. | |
21273 | ||
21274 | With a numeric prefix argument @var{n}, @kbd{j m} expands @var{n} | |
21275 | times (or until the entire formula is selected). Note that @kbd{j s} | |
21276 | with argument @var{n} is equivalent to plain @kbd{j s} followed by | |
21277 | @kbd{j m} with argument @var{n}. If @w{@kbd{j m}} is used when there | |
21278 | is no current selection, it is equivalent to @w{@kbd{j s}}. | |
21279 | ||
21280 | Even though @kbd{j m} does not explicitly use the location of the | |
21281 | cursor within the formula, it nevertheless uses the cursor to determine | |
21282 | which stack element to operate on. As usual, @kbd{j m} when the cursor | |
21283 | is not on any stack element operates on the top stack element. | |
21284 | ||
21285 | @kindex j l | |
21286 | @pindex calc-select-less | |
21287 | The @kbd{j l} (@code{calc-select-less}) command reduces the current | |
21288 | selection around the cursor position. That is, it selects the | |
21289 | immediate sub-formula of the current selection which contains the | |
21290 | cursor, the opposite of @kbd{j m}. If the cursor is not inside the | |
21291 | current selection, the command de-selects the formula. | |
21292 | ||
21293 | @kindex j 1-9 | |
21294 | @pindex calc-select-part | |
21295 | The @kbd{j 1} through @kbd{j 9} (@code{calc-select-part}) commands | |
21296 | select the @var{n}th sub-formula of the current selection. They are | |
21297 | like @kbd{j l} (@code{calc-select-less}) except they use counting | |
21298 | rather than the cursor position to decide which sub-formula to select. | |
21299 | For example, if the current selection is @kbd{a + b + c} or | |
21300 | @kbd{f(a, b, c)} or @kbd{[a, b, c]}, then @kbd{j 1} selects @samp{a}, | |
21301 | @kbd{j 2} selects @samp{b}, and @kbd{j 3} selects @samp{c}; in each of | |
21302 | these cases, @kbd{j 4} through @kbd{j 9} would be errors. | |
21303 | ||
21304 | If there is no current selection, @kbd{j 1} through @kbd{j 9} select | |
21305 | the @var{n}th top-level sub-formula. (In other words, they act as if | |
21306 | the entire stack entry were selected first.) To select the @var{n}th | |
21307 | sub-formula where @var{n} is greater than nine, you must instead invoke | |
21308 | @w{@kbd{j 1}} with @var{n} as a numeric prefix argument.@refill | |
21309 | ||
21310 | @kindex j n | |
21311 | @kindex j p | |
21312 | @pindex calc-select-next | |
21313 | @pindex calc-select-previous | |
21314 | The @kbd{j n} (@code{calc-select-next}) and @kbd{j p} | |
21315 | (@code{calc-select-previous}) commands change the current selection | |
21316 | to the next or previous sub-formula at the same level. For example, | |
21317 | if @samp{b} is selected in @w{@samp{2 + a*b*c + x}}, then @kbd{j n} | |
21318 | selects @samp{c}. Further @kbd{j n} commands would be in error because, | |
21319 | even though there is something to the right of @samp{c} (namely, @samp{x}), | |
21320 | it is not at the same level; in this case, it is not a term of the | |
21321 | same product as @samp{b} and @samp{c}. However, @kbd{j m} (to select | |
21322 | the whole product @samp{a*b*c} as a term of the sum) followed by | |
21323 | @w{@kbd{j n}} would successfully select the @samp{x}. | |
21324 | ||
21325 | Similarly, @kbd{j p} moves the selection from the @samp{b} in this | |
21326 | sample formula to the @samp{a}. Both commands accept numeric prefix | |
21327 | arguments to move several steps at a time. | |
21328 | ||
21329 | It is interesting to compare Calc's selection commands with the | |
21330 | Emacs Info system's commands for navigating through hierarchically | |
21331 | organized documentation. Calc's @kbd{j n} command is completely | |
21332 | analogous to Info's @kbd{n} command. Likewise, @kbd{j p} maps to | |
21333 | @kbd{p}, @kbd{j 2} maps to @kbd{2}, and Info's @kbd{u} is like @kbd{j m}. | |
21334 | (Note that @kbd{j u} stands for @code{calc-unselect}, not ``up''.) | |
21335 | The Info @kbd{m} command is somewhat similar to Calc's @kbd{j s} and | |
21336 | @kbd{j l}; in each case, you can jump directly to a sub-component | |
21337 | of the hierarchy simply by pointing to it with the cursor. | |
21338 | ||
21339 | @node Displaying Selections, Operating on Selections, Changing Selections, Selecting Subformulas | |
21340 | @subsection Displaying Selections | |
21341 | ||
21342 | @noindent | |
21343 | @kindex j d | |
21344 | @pindex calc-show-selections | |
21345 | The @kbd{j d} (@code{calc-show-selections}) command controls how | |
21346 | selected sub-formulas are displayed. One of the alternatives is | |
21347 | illustrated in the above examples; if we press @kbd{j d} we switch | |
21348 | to the other style in which the selected portion itself is obscured | |
21349 | by @samp{#} signs: | |
21350 | ||
d7b8e6c6 | 21351 | @smallexample |
5d67986c | 21352 | @group |
d7b8e6c6 EZ |
21353 | 3 ... # ___ |
21354 | (a + b) . . . ## # ## + V c | |
21355 | 1* ............... 1* --------------- | |
21356 | . . . . 2 x + 1 | |
d7b8e6c6 | 21357 | @end group |
5d67986c | 21358 | @end smallexample |
d7b8e6c6 EZ |
21359 | |
21360 | @node Operating on Selections, Rearranging with Selections, Displaying Selections, Selecting Subformulas | |
21361 | @subsection Operating on Selections | |
21362 | ||
21363 | @noindent | |
21364 | Once a selection is made, all Calc commands that manipulate items | |
21365 | on the stack will operate on the selected portions of the items | |
21366 | instead. (Note that several stack elements may have selections | |
21367 | at once, though there can be only one selection at a time in any | |
21368 | given stack element.) | |
21369 | ||
21370 | @kindex j e | |
21371 | @pindex calc-enable-selections | |
21372 | The @kbd{j e} (@code{calc-enable-selections}) command disables the | |
21373 | effect that selections have on Calc commands. The current selections | |
21374 | still exist, but Calc commands operate on whole stack elements anyway. | |
21375 | This mode can be identified by the fact that the @samp{*} markers on | |
21376 | the line numbers are gone, even though selections are visible. To | |
21377 | reactivate the selections, press @kbd{j e} again. | |
21378 | ||
21379 | To extract a sub-formula as a new formula, simply select the | |
21380 | sub-formula and press @key{RET}. This normally duplicates the top | |
21381 | stack element; here it duplicates only the selected portion of that | |
21382 | element. | |
21383 | ||
21384 | To replace a sub-formula with something different, you can enter the | |
21385 | new value onto the stack and press @key{TAB}. This normally exchanges | |
21386 | the top two stack elements; here it swaps the value you entered into | |
21387 | the selected portion of the formula, returning the old selected | |
21388 | portion to the top of the stack. | |
21389 | ||
d7b8e6c6 | 21390 | @smallexample |
5d67986c | 21391 | @group |
d7b8e6c6 EZ |
21392 | 3 ... ... ___ |
21393 | (a + b) . . . 17 x y . . . 17 x y + V c | |
21394 | 2* ............... 2* ............. 2: ------------- | |
21395 | . . . . . . . . 2 x + 1 | |
21396 | ||
21397 | 3 3 | |
21398 | 1: 17 x y 1: (a + b) 1: (a + b) | |
d7b8e6c6 | 21399 | @end group |
5d67986c | 21400 | @end smallexample |
d7b8e6c6 EZ |
21401 | |
21402 | In this example we select a sub-formula of our original example, | |
21403 | enter a new formula, @key{TAB} it into place, then deselect to see | |
21404 | the complete, edited formula. | |
21405 | ||
21406 | If you want to swap whole formulas around even though they contain | |
21407 | selections, just use @kbd{j e} before and after. | |
21408 | ||
21409 | @kindex j ' | |
21410 | @pindex calc-enter-selection | |
21411 | The @kbd{j '} (@code{calc-enter-selection}) command is another way | |
21412 | to replace a selected sub-formula. This command does an algebraic | |
21413 | entry just like the regular @kbd{'} key. When you press @key{RET}, | |
21414 | the formula you type replaces the original selection. You can use | |
21415 | the @samp{$} symbol in the formula to refer to the original | |
21416 | selection. If there is no selection in the formula under the cursor, | |
21417 | the cursor is used to make a temporary selection for the purposes of | |
21418 | the command. Thus, to change a term of a formula, all you have to | |
21419 | do is move the Emacs cursor to that term and press @kbd{j '}. | |
21420 | ||
21421 | @kindex j ` | |
21422 | @pindex calc-edit-selection | |
21423 | The @kbd{j `} (@code{calc-edit-selection}) command is a similar | |
21424 | analogue of the @kbd{`} (@code{calc-edit}) command. It edits the | |
21425 | selected sub-formula in a separate buffer. If there is no | |
21426 | selection, it edits the sub-formula indicated by the cursor. | |
21427 | ||
21428 | To delete a sub-formula, press @key{DEL}. This generally replaces | |
21429 | the sub-formula with the constant zero, but in a few suitable contexts | |
21430 | it uses the constant one instead. The @key{DEL} key automatically | |
21431 | deselects and re-simplifies the entire formula afterwards. Thus: | |
21432 | ||
d7b8e6c6 | 21433 | @smallexample |
5d67986c | 21434 | @group |
d7b8e6c6 EZ |
21435 | ### |
21436 | 17 x y + # # 17 x y 17 # y 17 y | |
21437 | 1* ------------- 1: ------- 1* ------- 1: ------- | |
21438 | 2 x + 1 2 x + 1 2 x + 1 2 x + 1 | |
d7b8e6c6 | 21439 | @end group |
5d67986c | 21440 | @end smallexample |
d7b8e6c6 EZ |
21441 | |
21442 | In this example, we first delete the @samp{sqrt(c)} term; Calc | |
21443 | accomplishes this by replacing @samp{sqrt(c)} with zero and | |
21444 | resimplifying. We then delete the @kbd{x} in the numerator; | |
21445 | since this is part of a product, Calc replaces it with @samp{1} | |
21446 | and resimplifies. | |
21447 | ||
21448 | If you select an element of a vector and press @key{DEL}, that | |
21449 | element is deleted from the vector. If you delete one side of | |
21450 | an equation or inequality, only the opposite side remains. | |
21451 | ||
5d67986c | 21452 | @kindex j @key{DEL} |
d7b8e6c6 EZ |
21453 | @pindex calc-del-selection |
21454 | The @kbd{j @key{DEL}} (@code{calc-del-selection}) command is like | |
21455 | @key{DEL} but with the auto-selecting behavior of @kbd{j '} and | |
21456 | @kbd{j `}. It deletes the selected portion of the formula | |
21457 | indicated by the cursor, or, in the absence of a selection, it | |
21458 | deletes the sub-formula indicated by the cursor position. | |
21459 | ||
5d67986c | 21460 | @kindex j @key{RET} |
d7b8e6c6 EZ |
21461 | @pindex calc-grab-selection |
21462 | (There is also an auto-selecting @kbd{j @key{RET}} (@code{calc-copy-selection}) | |
21463 | command.) | |
21464 | ||
21465 | Normal arithmetic operations also apply to sub-formulas. Here we | |
21466 | select the denominator, press @kbd{5 -} to subtract five from the | |
21467 | denominator, press @kbd{n} to negate the denominator, then | |
21468 | press @kbd{Q} to take the square root. | |
21469 | ||
d7b8e6c6 | 21470 | @smallexample |
5d67986c | 21471 | @group |
d7b8e6c6 EZ |
21472 | .. . .. . .. . .. . |
21473 | 1* ....... 1* ....... 1* ....... 1* .......... | |
21474 | 2 x + 1 2 x - 4 4 - 2 x _________ | |
21475 | V 4 - 2 x | |
d7b8e6c6 | 21476 | @end group |
5d67986c | 21477 | @end smallexample |
d7b8e6c6 EZ |
21478 | |
21479 | Certain types of operations on selections are not allowed. For | |
21480 | example, for an arithmetic function like @kbd{-} no more than one of | |
21481 | the arguments may be a selected sub-formula. (As the above example | |
21482 | shows, the result of the subtraction is spliced back into the argument | |
21483 | which had the selection; if there were more than one selection involved, | |
21484 | this would not be well-defined.) If you try to subtract two selections, | |
21485 | the command will abort with an error message. | |
21486 | ||
21487 | Operations on sub-formulas sometimes leave the formula as a whole | |
21488 | in an ``un-natural'' state. Consider negating the @samp{2 x} term | |
21489 | of our sample formula by selecting it and pressing @kbd{n} | |
21490 | (@code{calc-change-sign}).@refill | |
21491 | ||
d7b8e6c6 | 21492 | @smallexample |
5d67986c | 21493 | @group |
d7b8e6c6 EZ |
21494 | .. . .. . |
21495 | 1* .......... 1* ........... | |
21496 | ......... .......... | |
21497 | . . . 2 x . . . -2 x | |
d7b8e6c6 | 21498 | @end group |
5d67986c | 21499 | @end smallexample |
d7b8e6c6 EZ |
21500 | |
21501 | Unselecting the sub-formula reveals that the minus sign, which would | |
21502 | normally have cancelled out with the subtraction automatically, has | |
21503 | not been able to do so because the subtraction was not part of the | |
21504 | selected portion. Pressing @kbd{=} (@code{calc-evaluate}) or doing | |
21505 | any other mathematical operation on the whole formula will cause it | |
21506 | to be simplified. | |
21507 | ||
d7b8e6c6 | 21508 | @smallexample |
5d67986c | 21509 | @group |
d7b8e6c6 EZ |
21510 | 17 y 17 y |
21511 | 1: ----------- 1: ---------- | |
21512 | __________ _________ | |
21513 | V 4 - -2 x V 4 + 2 x | |
d7b8e6c6 | 21514 | @end group |
5d67986c | 21515 | @end smallexample |
d7b8e6c6 EZ |
21516 | |
21517 | @node Rearranging with Selections, , Operating on Selections, Selecting Subformulas | |
21518 | @subsection Rearranging Formulas using Selections | |
21519 | ||
21520 | @noindent | |
21521 | @kindex j R | |
21522 | @pindex calc-commute-right | |
21523 | The @kbd{j R} (@code{calc-commute-right}) command moves the selected | |
21524 | sub-formula to the right in its surrounding formula. Generally the | |
21525 | selection is one term of a sum or product; the sum or product is | |
21526 | rearranged according to the commutative laws of algebra. | |
21527 | ||
5d67986c | 21528 | As with @kbd{j '} and @kbd{j @key{DEL}}, the term under the cursor is used |
d7b8e6c6 EZ |
21529 | if there is no selection in the current formula. All commands described |
21530 | in this section share this property. In this example, we place the | |
21531 | cursor on the @samp{a} and type @kbd{j R}, then repeat. | |
21532 | ||
21533 | @smallexample | |
21534 | 1: a + b - c 1: b + a - c 1: b - c + a | |
21535 | @end smallexample | |
21536 | ||
21537 | @noindent | |
21538 | Note that in the final step above, the @samp{a} is switched with | |
21539 | the @samp{c} but the signs are adjusted accordingly. When moving | |
21540 | terms of sums and products, @kbd{j R} will never change the | |
21541 | mathematical meaning of the formula. | |
21542 | ||
21543 | The selected term may also be an element of a vector or an argument | |
21544 | of a function. The term is exchanged with the one to its right. | |
21545 | In this case, the ``meaning'' of the vector or function may of | |
21546 | course be drastically changed. | |
21547 | ||
21548 | @smallexample | |
21549 | 1: [a, b, c] 1: [b, a, c] 1: [b, c, a] | |
21550 | ||
21551 | 1: f(a, b, c) 1: f(b, a, c) 1: f(b, c, a) | |
21552 | @end smallexample | |
21553 | ||
21554 | @kindex j L | |
21555 | @pindex calc-commute-left | |
21556 | The @kbd{j L} (@code{calc-commute-left}) command is like @kbd{j R} | |
21557 | except that it swaps the selected term with the one to its left. | |
21558 | ||
21559 | With numeric prefix arguments, these commands move the selected | |
21560 | term several steps at a time. It is an error to try to move a | |
21561 | term left or right past the end of its enclosing formula. | |
21562 | With numeric prefix arguments of zero, these commands move the | |
21563 | selected term as far as possible in the given direction. | |
21564 | ||
21565 | @kindex j D | |
21566 | @pindex calc-sel-distribute | |
21567 | The @kbd{j D} (@code{calc-sel-distribute}) command mixes the selected | |
21568 | sum or product into the surrounding formula using the distributive | |
21569 | law. For example, in @samp{a * (b - c)} with the @samp{b - c} | |
21570 | selected, the result is @samp{a b - a c}. This also distributes | |
21571 | products or quotients into surrounding powers, and can also do | |
21572 | transformations like @samp{exp(a + b)} to @samp{exp(a) exp(b)}, | |
21573 | where @samp{a + b} is the selected term, and @samp{ln(a ^ b)} | |
21574 | to @samp{ln(a) b}, where @samp{a ^ b} is the selected term. | |
21575 | ||
21576 | For multiple-term sums or products, @kbd{j D} takes off one term | |
21577 | at a time: @samp{a * (b + c - d)} goes to @samp{a * (c - d) + a b} | |
21578 | with the @samp{c - d} selected so that you can type @kbd{j D} | |
21579 | repeatedly to expand completely. The @kbd{j D} command allows a | |
21580 | numeric prefix argument which specifies the maximum number of | |
21581 | times to expand at once; the default is one time only. | |
21582 | ||
21583 | @vindex DistribRules | |
21584 | The @kbd{j D} command is implemented using rewrite rules. | |
21585 | @xref{Selections with Rewrite Rules}. The rules are stored in | |
21586 | the Calc variable @code{DistribRules}. A convenient way to view | |
21587 | these rules is to use @kbd{s e} (@code{calc-edit-variable}) which | |
0d48e8aa | 21588 | displays and edits the stored value of a variable. Press @kbd{M-# M-#} |
d7b8e6c6 EZ |
21589 | to return from editing mode; be careful not to make any actual changes |
21590 | or else you will affect the behavior of future @kbd{j D} commands! | |
21591 | ||
21592 | To extend @kbd{j D} to handle new cases, just edit @code{DistribRules} | |
21593 | as described above. You can then use the @kbd{s p} command to save | |
21594 | this variable's value permanently for future Calc sessions. | |
21595 | @xref{Operations on Variables}. | |
21596 | ||
21597 | @kindex j M | |
21598 | @pindex calc-sel-merge | |
21599 | @vindex MergeRules | |
21600 | The @kbd{j M} (@code{calc-sel-merge}) command is the complement | |
21601 | of @kbd{j D}; given @samp{a b - a c} with either @samp{a b} or | |
21602 | @samp{a c} selected, the result is @samp{a * (b - c)}. Once | |
21603 | again, @kbd{j M} can also merge calls to functions like @code{exp} | |
21604 | and @code{ln}; examine the variable @code{MergeRules} to see all | |
21605 | the relevant rules. | |
21606 | ||
21607 | @kindex j C | |
21608 | @pindex calc-sel-commute | |
21609 | @vindex CommuteRules | |
21610 | The @kbd{j C} (@code{calc-sel-commute}) command swaps the arguments | |
21611 | of the selected sum, product, or equation. It always behaves as | |
21612 | if @kbd{j b} mode were in effect, i.e., the sum @samp{a + b + c} is | |
21613 | treated as the nested sums @samp{(a + b) + c} by this command. | |
21614 | If you put the cursor on the first @samp{+}, the result is | |
21615 | @samp{(b + a) + c}; if you put the cursor on the second @samp{+}, the | |
21616 | result is @samp{c + (a + b)} (which the default simplifications | |
21617 | will rearrange to @samp{(c + a) + b}). The relevant rules are stored | |
21618 | in the variable @code{CommuteRules}. | |
21619 | ||
21620 | You may need to turn default simplifications off (with the @kbd{m O} | |
21621 | command) in order to get the full benefit of @kbd{j C}. For example, | |
21622 | commuting @samp{a - b} produces @samp{-b + a}, but the default | |
21623 | simplifications will ``simplify'' this right back to @samp{a - b} if | |
21624 | you don't turn them off. The same is true of some of the other | |
21625 | manipulations described in this section. | |
21626 | ||
21627 | @kindex j N | |
21628 | @pindex calc-sel-negate | |
21629 | @vindex NegateRules | |
21630 | The @kbd{j N} (@code{calc-sel-negate}) command replaces the selected | |
21631 | term with the negative of that term, then adjusts the surrounding | |
21632 | formula in order to preserve the meaning. For example, given | |
21633 | @samp{exp(a - b)} where @samp{a - b} is selected, the result is | |
21634 | @samp{1 / exp(b - a)}. By contrast, selecting a term and using the | |
21635 | regular @kbd{n} (@code{calc-change-sign}) command negates the | |
21636 | term without adjusting the surroundings, thus changing the meaning | |
21637 | of the formula as a whole. The rules variable is @code{NegateRules}. | |
21638 | ||
21639 | @kindex j & | |
21640 | @pindex calc-sel-invert | |
21641 | @vindex InvertRules | |
21642 | The @kbd{j &} (@code{calc-sel-invert}) command is similar to @kbd{j N} | |
21643 | except it takes the reciprocal of the selected term. For example, | |
21644 | given @samp{a - ln(b)} with @samp{b} selected, the result is | |
21645 | @samp{a + ln(1/b)}. The rules variable is @code{InvertRules}. | |
21646 | ||
21647 | @kindex j E | |
21648 | @pindex calc-sel-jump-equals | |
21649 | @vindex JumpRules | |
21650 | The @kbd{j E} (@code{calc-sel-jump-equals}) command moves the | |
21651 | selected term from one side of an equation to the other. Given | |
21652 | @samp{a + b = c + d} with @samp{c} selected, the result is | |
21653 | @samp{a + b - c = d}. This command also works if the selected | |
21654 | term is part of a @samp{*}, @samp{/}, or @samp{^} formula. The | |
21655 | relevant rules variable is @code{JumpRules}. | |
21656 | ||
21657 | @kindex j I | |
21658 | @kindex H j I | |
21659 | @pindex calc-sel-isolate | |
21660 | The @kbd{j I} (@code{calc-sel-isolate}) command isolates the | |
21661 | selected term on its side of an equation. It uses the @kbd{a S} | |
21662 | (@code{calc-solve-for}) command to solve the equation, and the | |
21663 | Hyperbolic flag affects it in the same way. @xref{Solving Equations}. | |
21664 | When it applies, @kbd{j I} is often easier to use than @kbd{j E}. | |
21665 | It understands more rules of algebra, and works for inequalities | |
21666 | as well as equations. | |
21667 | ||
21668 | @kindex j * | |
21669 | @kindex j / | |
21670 | @pindex calc-sel-mult-both-sides | |
21671 | @pindex calc-sel-div-both-sides | |
21672 | The @kbd{j *} (@code{calc-sel-mult-both-sides}) command prompts for a | |
21673 | formula using algebraic entry, then multiplies both sides of the | |
21674 | selected quotient or equation by that formula. It simplifies each | |
21675 | side with @kbd{a s} (@code{calc-simplify}) before re-forming the | |
21676 | quotient or equation. You can suppress this simplification by | |
21677 | providing any numeric prefix argument. There is also a @kbd{j /} | |
21678 | (@code{calc-sel-div-both-sides}) which is similar to @kbd{j *} but | |
21679 | dividing instead of multiplying by the factor you enter. | |
21680 | ||
21681 | As a special feature, if the numerator of the quotient is 1, then | |
21682 | the denominator is expanded at the top level using the distributive | |
21683 | law (i.e., using the @kbd{C-u -1 a x} command). Suppose the | |
21684 | formula on the stack is @samp{1 / (sqrt(a) + 1)}, and you wish | |
21685 | to eliminate the square root in the denominator by multiplying both | |
21686 | sides by @samp{sqrt(a) - 1}. Calc's default simplifications would | |
21687 | change the result @samp{(sqrt(a) - 1) / (sqrt(a) - 1) (sqrt(a) + 1)} | |
21688 | right back to the original form by cancellation; Calc expands the | |
21689 | denominator to @samp{sqrt(a) (sqrt(a) - 1) + sqrt(a) - 1} to prevent | |
21690 | this. (You would now want to use an @kbd{a x} command to expand | |
21691 | the rest of the way, whereupon the denominator would cancel out to | |
21692 | the desired form, @samp{a - 1}.) When the numerator is not 1, this | |
21693 | initial expansion is not necessary because Calc's default | |
21694 | simplifications will not notice the potential cancellation. | |
21695 | ||
21696 | If the selection is an inequality, @kbd{j *} and @kbd{j /} will | |
21697 | accept any factor, but will warn unless they can prove the factor | |
21698 | is either positive or negative. (In the latter case the direction | |
21699 | of the inequality will be switched appropriately.) @xref{Declarations}, | |
21700 | for ways to inform Calc that a given variable is positive or | |
21701 | negative. If Calc can't tell for sure what the sign of the factor | |
21702 | will be, it will assume it is positive and display a warning | |
21703 | message. | |
21704 | ||
21705 | For selections that are not quotients, equations, or inequalities, | |
21706 | these commands pull out a multiplicative factor: They divide (or | |
21707 | multiply) by the entered formula, simplify, then multiply (or divide) | |
21708 | back by the formula. | |
21709 | ||
21710 | @kindex j + | |
21711 | @kindex j - | |
21712 | @pindex calc-sel-add-both-sides | |
21713 | @pindex calc-sel-sub-both-sides | |
21714 | The @kbd{j +} (@code{calc-sel-add-both-sides}) and @kbd{j -} | |
21715 | (@code{calc-sel-sub-both-sides}) commands analogously add to or | |
21716 | subtract from both sides of an equation or inequality. For other | |
21717 | types of selections, they extract an additive factor. A numeric | |
21718 | prefix argument suppresses simplification of the intermediate | |
21719 | results. | |
21720 | ||
21721 | @kindex j U | |
21722 | @pindex calc-sel-unpack | |
21723 | The @kbd{j U} (@code{calc-sel-unpack}) command replaces the | |
21724 | selected function call with its argument. For example, given | |
21725 | @samp{a + sin(x^2)} with @samp{sin(x^2)} selected, the result | |
21726 | is @samp{a + x^2}. (The @samp{x^2} will remain selected; if you | |
21727 | wanted to change the @code{sin} to @code{cos}, just press @kbd{C} | |
21728 | now to take the cosine of the selected part.) | |
21729 | ||
21730 | @kindex j v | |
21731 | @pindex calc-sel-evaluate | |
21732 | The @kbd{j v} (@code{calc-sel-evaluate}) command performs the | |
21733 | normal default simplifications on the selected sub-formula. | |
21734 | These are the simplifications that are normally done automatically | |
21735 | on all results, but which may have been partially inhibited by | |
21736 | previous selection-related operations, or turned off altogether | |
21737 | by the @kbd{m O} command. This command is just an auto-selecting | |
21738 | version of the @w{@kbd{a v}} command (@pxref{Algebraic Manipulation}). | |
21739 | ||
21740 | With a numeric prefix argument of 2, @kbd{C-u 2 j v} applies | |
21741 | the @kbd{a s} (@code{calc-simplify}) command to the selected | |
21742 | sub-formula. With a prefix argument of 3 or more, e.g., @kbd{C-u j v} | |
21743 | applies the @kbd{a e} (@code{calc-simplify-extended}) command. | |
21744 | @xref{Simplifying Formulas}. With a negative prefix argument | |
21745 | it simplifies at the top level only, just as with @kbd{a v}. | |
21746 | Here the ``top'' level refers to the top level of the selected | |
21747 | sub-formula. | |
21748 | ||
21749 | @kindex j " | |
21750 | @pindex calc-sel-expand-formula | |
21751 | The @kbd{j "} (@code{calc-sel-expand-formula}) command is to @kbd{a "} | |
21752 | (@pxref{Algebraic Manipulation}) what @kbd{j v} is to @kbd{a v}. | |
21753 | ||
21754 | You can use the @kbd{j r} (@code{calc-rewrite-selection}) command | |
21755 | to define other algebraic operations on sub-formulas. @xref{Rewrite Rules}. | |
21756 | ||
21757 | @node Algebraic Manipulation, Simplifying Formulas, Selecting Subformulas, Algebra | |
21758 | @section Algebraic Manipulation | |
21759 | ||
21760 | @noindent | |
21761 | The commands in this section perform general-purpose algebraic | |
21762 | manipulations. They work on the whole formula at the top of the | |
21763 | stack (unless, of course, you have made a selection in that | |
21764 | formula). | |
21765 | ||
21766 | Many algebra commands prompt for a variable name or formula. If you | |
21767 | answer the prompt with a blank line, the variable or formula is taken | |
21768 | from top-of-stack, and the normal argument for the command is taken | |
21769 | from the second-to-top stack level. | |
21770 | ||
21771 | @kindex a v | |
21772 | @pindex calc-alg-evaluate | |
21773 | The @kbd{a v} (@code{calc-alg-evaluate}) command performs the normal | |
21774 | default simplifications on a formula; for example, @samp{a - -b} is | |
21775 | changed to @samp{a + b}. These simplifications are normally done | |
21776 | automatically on all Calc results, so this command is useful only if | |
21777 | you have turned default simplifications off with an @kbd{m O} | |
21778 | command. @xref{Simplification Modes}. | |
21779 | ||
21780 | It is often more convenient to type @kbd{=}, which is like @kbd{a v} | |
21781 | but which also substitutes stored values for variables in the formula. | |
21782 | Use @kbd{a v} if you want the variables to ignore their stored values. | |
21783 | ||
21784 | If you give a numeric prefix argument of 2 to @kbd{a v}, it simplifies | |
21785 | as if in algebraic simplification mode. This is equivalent to typing | |
21786 | @kbd{a s}; @pxref{Simplifying Formulas}. If you give a numeric prefix | |
21787 | of 3 or more, it uses extended simplification mode (@kbd{a e}). | |
21788 | ||
21789 | If you give a negative prefix argument @i{-1}, @i{-2}, or @i{-3}, | |
21790 | it simplifies in the corresponding mode but only works on the top-level | |
21791 | function call of the formula. For example, @samp{(2 + 3) * (2 + 3)} will | |
21792 | simplify to @samp{(2 + 3)^2}, without simplifying the sub-formulas | |
21793 | @samp{2 + 3}. As another example, typing @kbd{V R +} to sum the vector | |
21794 | @samp{[1, 2, 3, 4]} produces the formula @samp{reduce(add, [1, 2, 3, 4])} | |
21795 | in no-simplify mode. Using @kbd{a v} will evaluate this all the way to | |
21796 | 10; using @kbd{C-u - a v} will evaluate it only to @samp{1 + 2 + 3 + 4}. | |
21797 | (@xref{Reducing and Mapping}.) | |
21798 | ||
21799 | @tindex evalv | |
21800 | @tindex evalvn | |
21801 | The @kbd{=} command corresponds to the @code{evalv} function, and | |
21802 | the related @kbd{N} command, which is like @kbd{=} but temporarily | |
21803 | disables symbolic (@kbd{m s}) mode during the evaluation, corresponds | |
21804 | to the @code{evalvn} function. (These commands interpret their prefix | |
21805 | arguments differently than @kbd{a v}; @kbd{=} treats the prefix as | |
21806 | the number of stack elements to evaluate at once, and @kbd{N} treats | |
21807 | it as a temporary different working precision.) | |
21808 | ||
21809 | The @code{evalvn} function can take an alternate working precision | |
21810 | as an optional second argument. This argument can be either an | |
21811 | integer, to set the precision absolutely, or a vector containing | |
21812 | a single integer, to adjust the precision relative to the current | |
21813 | precision. Note that @code{evalvn} with a larger than current | |
21814 | precision will do the calculation at this higher precision, but the | |
21815 | result will as usual be rounded back down to the current precision | |
21816 | afterward. For example, @samp{evalvn(pi - 3.1415)} at a precision | |
21817 | of 12 will return @samp{9.265359e-5}; @samp{evalvn(pi - 3.1415, 30)} | |
21818 | will return @samp{9.26535897932e-5} (computing a 25-digit result which | |
21819 | is then rounded down to 12); and @samp{evalvn(pi - 3.1415, [-2])} | |
21820 | will return @samp{9.2654e-5}. | |
21821 | ||
21822 | @kindex a " | |
21823 | @pindex calc-expand-formula | |
21824 | The @kbd{a "} (@code{calc-expand-formula}) command expands functions | |
21825 | into their defining formulas wherever possible. For example, | |
21826 | @samp{deg(x^2)} is changed to @samp{180 x^2 / pi}. Most functions, | |
21827 | like @code{sin} and @code{gcd}, are not defined by simple formulas | |
21828 | and so are unaffected by this command. One important class of | |
21829 | functions which @emph{can} be expanded is the user-defined functions | |
21830 | created by the @kbd{Z F} command. @xref{Algebraic Definitions}. | |
21831 | Other functions which @kbd{a "} can expand include the probability | |
21832 | distribution functions, most of the financial functions, and the | |
21833 | hyperbolic and inverse hyperbolic functions. A numeric prefix argument | |
21834 | affects @kbd{a "} in the same way as it does @kbd{a v}: A positive | |
21835 | argument expands all functions in the formula and then simplifies in | |
21836 | various ways; a negative argument expands and simplifies only the | |
21837 | top-level function call. | |
21838 | ||
21839 | @kindex a M | |
21840 | @pindex calc-map-equation | |
21841 | @tindex mapeq | |
21842 | The @kbd{a M} (@code{calc-map-equation}) [@code{mapeq}] command applies | |
21843 | a given function or operator to one or more equations. It is analogous | |
21844 | to @kbd{V M}, which operates on vectors instead of equations. | |
21845 | @pxref{Reducing and Mapping}. For example, @kbd{a M S} changes | |
21846 | @samp{x = y+1} to @samp{sin(x) = sin(y+1)}, and @kbd{a M +} with | |
21847 | @samp{x = y+1} and @cite{6} on the stack produces @samp{x+6 = y+7}. | |
21848 | With two equations on the stack, @kbd{a M +} would add the lefthand | |
21849 | sides together and the righthand sides together to get the two | |
21850 | respective sides of a new equation. | |
21851 | ||
21852 | Mapping also works on inequalities. Mapping two similar inequalities | |
21853 | produces another inequality of the same type. Mapping an inequality | |
21854 | with an equation produces an inequality of the same type. Mapping a | |
21855 | @samp{<=} with a @samp{<} or @samp{!=} (not-equal) produces a @samp{<}. | |
21856 | If inequalities with opposite direction (e.g., @samp{<} and @samp{>}) | |
21857 | are mapped, the direction of the second inequality is reversed to | |
21858 | match the first: Using @kbd{a M +} on @samp{a < b} and @samp{a > 2} | |
21859 | reverses the latter to get @samp{2 < a}, which then allows the | |
21860 | combination @samp{a + 2 < b + a}, which the @kbd{a s} command can | |
21861 | then simplify to get @samp{2 < b}. | |
21862 | ||
21863 | Using @kbd{a M *}, @kbd{a M /}, @kbd{a M n}, or @kbd{a M &} to negate | |
21864 | or invert an inequality will reverse the direction of the inequality. | |
21865 | Other adjustments to inequalities are @emph{not} done automatically; | |
21866 | @kbd{a M S} will change @w{@samp{x < y}} to @samp{sin(x) < sin(y)} even | |
21867 | though this is not true for all values of the variables. | |
21868 | ||
21869 | @kindex H a M | |
21870 | @tindex mapeqp | |
21871 | With the Hyperbolic flag, @kbd{H a M} [@code{mapeqp}] does a plain | |
21872 | mapping operation without reversing the direction of any inequalities. | |
21873 | Thus, @kbd{H a M &} would change @kbd{x > 2} to @kbd{1/x > 0.5}. | |
21874 | (This change is mathematically incorrect, but perhaps you were | |
21875 | fixing an inequality which was already incorrect.) | |
21876 | ||
21877 | @kindex I a M | |
21878 | @tindex mapeqr | |
21879 | With the Inverse flag, @kbd{I a M} [@code{mapeqr}] always reverses | |
21880 | the direction of the inequality. You might use @kbd{I a M C} to | |
21881 | change @samp{x < y} to @samp{cos(x) > cos(y)} if you know you are | |
21882 | working with small positive angles. | |
21883 | ||
21884 | @kindex a b | |
21885 | @pindex calc-substitute | |
21886 | @tindex subst | |
21887 | The @kbd{a b} (@code{calc-substitute}) [@code{subst}] command substitutes | |
21888 | all occurrences | |
21889 | of some variable or sub-expression of an expression with a new | |
21890 | sub-expression. For example, substituting @samp{sin(x)} with @samp{cos(y)} | |
21891 | in @samp{2 sin(x)^2 + x sin(x) + sin(2 x)} produces | |
21892 | @samp{2 cos(y)^2 + x cos(y) + @w{sin(2 x)}}. | |
21893 | Note that this is a purely structural substitution; the lone @samp{x} and | |
21894 | the @samp{sin(2 x)} stayed the same because they did not look like | |
21895 | @samp{sin(x)}. @xref{Rewrite Rules}, for a more general method for | |
21896 | doing substitutions.@refill | |
21897 | ||
21898 | The @kbd{a b} command normally prompts for two formulas, the old | |
21899 | one and the new one. If you enter a blank line for the first | |
21900 | prompt, all three arguments are taken from the stack (new, then old, | |
21901 | then target expression). If you type an old formula but then enter a | |
21902 | blank line for the new one, the new formula is taken from top-of-stack | |
21903 | and the target from second-to-top. If you answer both prompts, the | |
21904 | target is taken from top-of-stack as usual. | |
21905 | ||
21906 | Note that @kbd{a b} has no understanding of commutativity or | |
21907 | associativity. The pattern @samp{x+y} will not match the formula | |
21908 | @samp{y+x}. Also, @samp{y+z} will not match inside the formula @samp{x+y+z} | |
21909 | because the @samp{+} operator is left-associative, so the ``deep | |
21910 | structure'' of that formula is @samp{(x+y) + z}. Use @kbd{d U} | |
21911 | (@code{calc-unformatted-language}) mode to see the true structure of | |
21912 | a formula. The rewrite rule mechanism, discussed later, does not have | |
21913 | these limitations. | |
21914 | ||
21915 | As an algebraic function, @code{subst} takes three arguments: | |
21916 | Target expression, old, new. Note that @code{subst} is always | |
21917 | evaluated immediately, even if its arguments are variables, so if | |
21918 | you wish to put a call to @code{subst} onto the stack you must | |
21919 | turn the default simplifications off first (with @kbd{m O}). | |
21920 | ||
21921 | @node Simplifying Formulas, Polynomials, Algebraic Manipulation, Algebra | |
21922 | @section Simplifying Formulas | |
21923 | ||
21924 | @noindent | |
21925 | @kindex a s | |
21926 | @pindex calc-simplify | |
21927 | @tindex simplify | |
21928 | The @kbd{a s} (@code{calc-simplify}) [@code{simplify}] command applies | |
21929 | various algebraic rules to simplify a formula. This includes rules which | |
21930 | are not part of the default simplifications because they may be too slow | |
21931 | to apply all the time, or may not be desirable all of the time. For | |
21932 | example, non-adjacent terms of sums are combined, as in @samp{a + b + 2 a} | |
21933 | to @samp{b + 3 a}, and some formulas like @samp{sin(arcsin(x))} are | |
21934 | simplified to @samp{x}. | |
21935 | ||
21936 | The sections below describe all the various kinds of algebraic | |
21937 | simplifications Calc provides in full detail. None of Calc's | |
21938 | simplification commands are designed to pull rabbits out of hats; | |
21939 | they simply apply certain specific rules to put formulas into | |
21940 | less redundant or more pleasing forms. Serious algebra in Calc | |
21941 | must be done manually, usually with a combination of selections | |
21942 | and rewrite rules. @xref{Rearranging with Selections}. | |
21943 | @xref{Rewrite Rules}. | |
21944 | ||
21945 | @xref{Simplification Modes}, for commands to control what level of | |
21946 | simplification occurs automatically. Normally only the ``default | |
21947 | simplifications'' occur. | |
21948 | ||
21949 | @menu | |
21950 | * Default Simplifications:: | |
21951 | * Algebraic Simplifications:: | |
21952 | * Unsafe Simplifications:: | |
21953 | * Simplification of Units:: | |
21954 | @end menu | |
21955 | ||
21956 | @node Default Simplifications, Algebraic Simplifications, Simplifying Formulas, Simplifying Formulas | |
21957 | @subsection Default Simplifications | |
21958 | ||
21959 | @noindent | |
21960 | @cindex Default simplifications | |
21961 | This section describes the ``default simplifications,'' those which are | |
21962 | normally applied to all results. For example, if you enter the variable | |
21963 | @cite{x} on the stack twice and push @kbd{+}, Calc's default | |
21964 | simplifications automatically change @cite{x + x} to @cite{2 x}. | |
21965 | ||
21966 | The @kbd{m O} command turns off the default simplifications, so that | |
21967 | @cite{x + x} will remain in this form unless you give an explicit | |
21968 | ``simplify'' command like @kbd{=} or @kbd{a v}. @xref{Algebraic | |
21969 | Manipulation}. The @kbd{m D} command turns the default simplifications | |
21970 | back on. | |
21971 | ||
21972 | The most basic default simplification is the evaluation of functions. | |
21973 | For example, @cite{2 + 3} is evaluated to @cite{5}, and @cite{@t{sqrt}(9)} | |
21974 | is evaluated to @cite{3}. Evaluation does not occur if the arguments | |
21975 | to a function are somehow of the wrong type (@cite{@t{tan}([2,3,4])}, | |
21976 | range (@cite{@t{tan}(90)}), or number (@cite{@t{tan}(3,5)}), or if the | |
21977 | function name is not recognized (@cite{@t{f}(5)}), or if ``symbolic'' | |
21978 | mode (@pxref{Symbolic Mode}) prevents evaluation (@cite{@t{sqrt}(2)}). | |
21979 | ||
21980 | Calc simplifies (evaluates) the arguments to a function before it | |
21981 | simplifies the function itself. Thus @cite{@t{sqrt}(5+4)} is | |
21982 | simplified to @cite{@t{sqrt}(9)} before the @code{sqrt} function | |
21983 | itself is applied. There are very few exceptions to this rule: | |
21984 | @code{quote}, @code{lambda}, and @code{condition} (the @code{::} | |
21985 | operator) do not evaluate their arguments, @code{if} (the @code{? :} | |
21986 | operator) does not evaluate all of its arguments, and @code{evalto} | |
21987 | does not evaluate its lefthand argument. | |
21988 | ||
21989 | Most commands apply the default simplifications to all arguments they | |
21990 | take from the stack, perform a particular operation, then simplify | |
21991 | the result before pushing it back on the stack. In the common special | |
21992 | case of regular arithmetic commands like @kbd{+} and @kbd{Q} [@code{sqrt}], | |
21993 | the arguments are simply popped from the stack and collected into a | |
21994 | suitable function call, which is then simplified (the arguments being | |
21995 | simplified first as part of the process, as described above). | |
21996 | ||
21997 | The default simplifications are too numerous to describe completely | |
21998 | here, but this section will describe the ones that apply to the | |
21999 | major arithmetic operators. This list will be rather technical in | |
22000 | nature, and will probably be interesting to you only if you are | |
22001 | a serious user of Calc's algebra facilities. | |
22002 | ||
22003 | @tex | |
22004 | \bigskip | |
22005 | @end tex | |
22006 | ||
22007 | As well as the simplifications described here, if you have stored | |
22008 | any rewrite rules in the variable @code{EvalRules} then these rules | |
22009 | will also be applied before any built-in default simplifications. | |
22010 | @xref{Automatic Rewrites}, for details. | |
22011 | ||
22012 | @tex | |
22013 | \bigskip | |
22014 | @end tex | |
22015 | ||
22016 | And now, on with the default simplifications: | |
22017 | ||
22018 | Arithmetic operators like @kbd{+} and @kbd{*} always take two | |
22019 | arguments in Calc's internal form. Sums and products of three or | |
22020 | more terms are arranged by the associative law of algebra into | |
22021 | a left-associative form for sums, @cite{((a + b) + c) + d}, and | |
22022 | a right-associative form for products, @cite{a * (b * (c * d))}. | |
22023 | Formulas like @cite{(a + b) + (c + d)} are rearranged to | |
22024 | left-associative form, though this rarely matters since Calc's | |
22025 | algebra commands are designed to hide the inner structure of | |
22026 | sums and products as much as possible. Sums and products in | |
22027 | their proper associative form will be written without parentheses | |
22028 | in the examples below. | |
22029 | ||
22030 | Sums and products are @emph{not} rearranged according to the | |
22031 | commutative law (@cite{a + b} to @cite{b + a}) except in a few | |
22032 | special cases described below. Some algebra programs always | |
22033 | rearrange terms into a canonical order, which enables them to | |
22034 | see that @cite{a b + b a} can be simplified to @cite{2 a b}. | |
22035 | Calc assumes you have put the terms into the order you want | |
22036 | and generally leaves that order alone, with the consequence | |
22037 | that formulas like the above will only be simplified if you | |
22038 | explicitly give the @kbd{a s} command. @xref{Algebraic | |
22039 | Simplifications}. | |
22040 | ||
22041 | Differences @cite{a - b} are treated like sums @cite{a + (-b)} | |
22042 | for purposes of simplification; one of the default simplifications | |
22043 | is to rewrite @cite{a + (-b)} or @cite{(-b) + a}, where @cite{-b} | |
22044 | represents a ``negative-looking'' term, into @cite{a - b} form. | |
22045 | ``Negative-looking'' means negative numbers, negated formulas like | |
22046 | @cite{-x}, and products or quotients in which either term is | |
22047 | negative-looking. | |
22048 | ||
22049 | Other simplifications involving negation are @cite{-(-x)} to @cite{x}; | |
22050 | @cite{-(a b)} or @cite{-(a/b)} where either @cite{a} or @cite{b} is | |
22051 | negative-looking, simplified by negating that term, or else where | |
22052 | @cite{a} or @cite{b} is any number, by negating that number; | |
22053 | @cite{-(a + b)} to @cite{-a - b}, and @cite{-(b - a)} to @cite{a - b}. | |
22054 | (This, and rewriting @cite{(-b) + a} to @cite{a - b}, are the only | |
22055 | cases where the order of terms in a sum is changed by the default | |
22056 | simplifications.) | |
22057 | ||
22058 | The distributive law is used to simplify sums in some cases: | |
22059 | @cite{a x + b x} to @cite{(a + b) x}, where @cite{a} represents | |
22060 | a number or an implicit 1 or @i{-1} (as in @cite{x} or @cite{-x}) | |
22061 | and similarly for @cite{b}. Use the @kbd{a c}, @w{@kbd{a f}}, or | |
22062 | @kbd{j M} commands to merge sums with non-numeric coefficients | |
22063 | using the distributive law. | |
22064 | ||
22065 | The distributive law is only used for sums of two terms, or | |
22066 | for adjacent terms in a larger sum. Thus @cite{a + b + b + c} | |
22067 | is simplified to @cite{a + 2 b + c}, but @cite{a + b + c + b} | |
22068 | is not simplified. The reason is that comparing all terms of a | |
22069 | sum with one another would require time proportional to the | |
22070 | square of the number of terms; Calc relegates potentially slow | |
22071 | operations like this to commands that have to be invoked | |
22072 | explicitly, like @kbd{a s}. | |
22073 | ||
22074 | Finally, @cite{a + 0} and @cite{0 + a} are simplified to @cite{a}. | |
22075 | A consequence of the above rules is that @cite{0 - a} is simplified | |
22076 | to @cite{-a}. | |
22077 | ||
22078 | @tex | |
22079 | \bigskip | |
22080 | @end tex | |
22081 | ||
22082 | The products @cite{1 a} and @cite{a 1} are simplified to @cite{a}; | |
22083 | @cite{(-1) a} and @cite{a (-1)} are simplified to @cite{-a}; | |
22084 | @cite{0 a} and @cite{a 0} are simplified to @cite{0}, except that | |
22085 | in matrix mode where @cite{a} is not provably scalar the result | |
22086 | is the generic zero matrix @samp{idn(0)}, and that if @cite{a} is | |
22087 | infinite the result is @samp{nan}. | |
22088 | ||
22089 | Also, @cite{(-a) b} and @cite{a (-b)} are simplified to @cite{-(a b)}, | |
22090 | where this occurs for negated formulas but not for regular negative | |
22091 | numbers. | |
22092 | ||
22093 | Products are commuted only to move numbers to the front: | |
22094 | @cite{a b 2} is commuted to @cite{2 a b}. | |
22095 | ||
22096 | The product @cite{a (b + c)} is distributed over the sum only if | |
22097 | @cite{a} and at least one of @cite{b} and @cite{c} are numbers: | |
22098 | @cite{2 (x + 3)} goes to @cite{2 x + 6}. The formula | |
22099 | @cite{(-a) (b - c)}, where @cite{-a} is a negative number, is | |
22100 | rewritten to @cite{a (c - b)}. | |
22101 | ||
22102 | The distributive law of products and powers is used for adjacent | |
22103 | terms of the product: @cite{x^a x^b} goes to @c{$x^{a+b}$} | |
22104 | @cite{x^(a+b)} | |
22105 | where @cite{a} is a number, or an implicit 1 (as in @cite{x}), | |
22106 | or the implicit one-half of @cite{@t{sqrt}(x)}, and similarly for | |
22107 | @cite{b}. The result is written using @samp{sqrt} or @samp{1/sqrt} | |
22108 | if the sum of the powers is @cite{1/2} or @cite{-1/2}, respectively. | |
22109 | If the sum of the powers is zero, the product is simplified to | |
22110 | @cite{1} or to @samp{idn(1)} if matrix mode is enabled. | |
22111 | ||
22112 | The product of a negative power times anything but another negative | |
22113 | power is changed to use division: @c{$x^{-2} y$} | |
22114 | @cite{x^(-2) y} goes to @cite{y / x^2} unless matrix mode is | |
22115 | in effect and neither @cite{x} nor @cite{y} are scalar (in which | |
22116 | case it is considered unsafe to rearrange the order of the terms). | |
22117 | ||
22118 | Finally, @cite{a (b/c)} is rewritten to @cite{(a b)/c}, and also | |
22119 | @cite{(a/b) c} is changed to @cite{(a c)/b} unless in matrix mode. | |
22120 | ||
22121 | @tex | |
22122 | \bigskip | |
22123 | @end tex | |
22124 | ||
22125 | Simplifications for quotients are analogous to those for products. | |
22126 | The quotient @cite{0 / x} is simplified to @cite{0}, with the same | |
22127 | exceptions that were noted for @cite{0 x}. Likewise, @cite{x / 1} | |
22128 | and @cite{x / (-1)} are simplified to @cite{x} and @cite{-x}, | |
22129 | respectively. | |
22130 | ||
22131 | The quotient @cite{x / 0} is left unsimplified or changed to an | |
22132 | infinite quantity, as directed by the current infinite mode. | |
22133 | @xref{Infinite Mode}. | |
22134 | ||
22135 | The expression @c{$a / b^{-c}$} | |
22136 | @cite{a / b^(-c)} is changed to @cite{a b^c}, | |
22137 | where @cite{-c} is any negative-looking power. Also, @cite{1 / b^c} | |
22138 | is changed to @c{$b^{-c}$} | |
22139 | @cite{b^(-c)} for any power @cite{c}. | |
22140 | ||
22141 | Also, @cite{(-a) / b} and @cite{a / (-b)} go to @cite{-(a/b)}; | |
22142 | @cite{(a/b) / c} goes to @cite{a / (b c)}; and @cite{a / (b/c)} | |
22143 | goes to @cite{(a c) / b} unless matrix mode prevents this | |
22144 | rearrangement. Similarly, @cite{a / (b:c)} is simplified to | |
22145 | @cite{(c:b) a} for any fraction @cite{b:c}. | |
22146 | ||
22147 | The distributive law is applied to @cite{(a + b) / c} only if | |
22148 | @cite{c} and at least one of @cite{a} and @cite{b} are numbers. | |
22149 | Quotients of powers and square roots are distributed just as | |
22150 | described for multiplication. | |
22151 | ||
22152 | Quotients of products cancel only in the leading terms of the | |
22153 | numerator and denominator. In other words, @cite{a x b / a y b} | |
22154 | is cancelled to @cite{x b / y b} but not to @cite{x / y}. Once | |
22155 | again this is because full cancellation can be slow; use @kbd{a s} | |
22156 | to cancel all terms of the quotient. | |
22157 | ||
22158 | Quotients of negative-looking values are simplified according | |
22159 | to @cite{(-a) / (-b)} to @cite{a / b}, @cite{(-a) / (b - c)} | |
22160 | to @cite{a / (c - b)}, and @cite{(a - b) / (-c)} to @cite{(b - a) / c}. | |
22161 | ||
22162 | @tex | |
22163 | \bigskip | |
22164 | @end tex | |
22165 | ||
22166 | The formula @cite{x^0} is simplified to @cite{1}, or to @samp{idn(1)} | |
22167 | in matrix mode. The formula @cite{0^x} is simplified to @cite{0} | |
22168 | unless @cite{x} is a negative number or complex number, in which | |
22169 | case the result is an infinity or an unsimplified formula according | |
22170 | to the current infinite mode. Note that @cite{0^0} is an | |
22171 | indeterminate form, as evidenced by the fact that the simplifications | |
22172 | for @cite{x^0} and @cite{0^x} conflict when @cite{x=0}. | |
22173 | ||
22174 | Powers of products or quotients @cite{(a b)^c}, @cite{(a/b)^c} | |
22175 | are distributed to @cite{a^c b^c}, @cite{a^c / b^c} only if @cite{c} | |
22176 | is an integer, or if either @cite{a} or @cite{b} are nonnegative | |
22177 | real numbers. Powers of powers @cite{(a^b)^c} are simplified to | |
22178 | @c{$a^{b c}$} | |
22179 | @cite{a^(b c)} only when @cite{c} is an integer and @cite{b c} also | |
22180 | evaluates to an integer. Without these restrictions these simplifications | |
22181 | would not be safe because of problems with principal values. | |
22182 | (In other words, @c{$((-3)^{1/2})^2$} | |
22183 | @cite{((-3)^1:2)^2} is safe to simplify, but | |
22184 | @c{$((-3)^2)^{1/2}$} | |
22185 | @cite{((-3)^2)^1:2} is not.) @xref{Declarations}, for ways to inform | |
22186 | Calc that your variables satisfy these requirements. | |
22187 | ||
22188 | As a special case of this rule, @cite{@t{sqrt}(x)^n} is simplified to | |
22189 | @c{$x^{n/2}$} | |
22190 | @cite{x^(n/2)} only for even integers @cite{n}. | |
22191 | ||
22192 | If @cite{a} is known to be real, @cite{b} is an even integer, and | |
22193 | @cite{c} is a half- or quarter-integer, then @cite{(a^b)^c} is | |
22194 | simplified to @c{$@t{abs}(a^{b c})$} | |
22195 | @cite{@t{abs}(a^(b c))}. | |
22196 | ||
22197 | Also, @cite{(-a)^b} is simplified to @cite{a^b} if @cite{b} is an | |
22198 | even integer, or to @cite{-(a^b)} if @cite{b} is an odd integer, | |
22199 | for any negative-looking expression @cite{-a}. | |
22200 | ||
22201 | Square roots @cite{@t{sqrt}(x)} generally act like one-half powers | |
22202 | @c{$x^{1:2}$} | |
22203 | @cite{x^1:2} for the purposes of the above-listed simplifications. | |
22204 | ||
22205 | Also, note that @c{$1 / x^{1:2}$} | |
22206 | @cite{1 / x^1:2} is changed to @c{$x^{-1:2}$} | |
22207 | @cite{x^(-1:2)}, | |
22208 | but @cite{1 / @t{sqrt}(x)} is left alone. | |
22209 | ||
22210 | @tex | |
22211 | \bigskip | |
22212 | @end tex | |
22213 | ||
22214 | Generic identity matrices (@pxref{Matrix Mode}) are simplified by the | |
22215 | following rules: @cite{@t{idn}(a) + b} to @cite{a + b} if @cite{b} | |
22216 | is provably scalar, or expanded out if @cite{b} is a matrix; | |
22217 | @cite{@t{idn}(a) + @t{idn}(b)} to @cite{@t{idn}(a + b)}; | |
22218 | @cite{-@t{idn}(a)} to @cite{@t{idn}(-a)}; @cite{a @t{idn}(b)} to | |
22219 | @cite{@t{idn}(a b)} if @cite{a} is provably scalar, or to @cite{a b} | |
22220 | if @cite{a} is provably non-scalar; @cite{@t{idn}(a) @t{idn}(b)} | |
22221 | to @cite{@t{idn}(a b)}; analogous simplifications for quotients | |
22222 | involving @code{idn}; and @cite{@t{idn}(a)^n} to @cite{@t{idn}(a^n)} | |
22223 | where @cite{n} is an integer. | |
22224 | ||
22225 | @tex | |
22226 | \bigskip | |
22227 | @end tex | |
22228 | ||
22229 | The @code{floor} function and other integer truncation functions | |
22230 | vanish if the argument is provably integer-valued, so that | |
22231 | @cite{@t{floor}(@t{round}(x))} simplifies to @cite{@t{round}(x)}. | |
22232 | Also, combinations of @code{float}, @code{floor} and its friends, | |
22233 | and @code{ffloor} and its friends, are simplified in appropriate | |
22234 | ways. @xref{Integer Truncation}. | |
22235 | ||
22236 | The expression @cite{@t{abs}(-x)} changes to @cite{@t{abs}(x)}. | |
22237 | The expression @cite{@t{abs}(@t{abs}(x))} changes to @cite{@t{abs}(x)}; | |
22238 | in fact, @cite{@t{abs}(x)} changes to @cite{x} or @cite{-x} if @cite{x} | |
22239 | is provably nonnegative or nonpositive (@pxref{Declarations}). | |
22240 | ||
22241 | While most functions do not recognize the variable @code{i} as an | |
22242 | imaginary number, the @code{arg} function does handle the two cases | |
22243 | @cite{@t{arg}(@t{i})} and @cite{@t{arg}(-@t{i})} just for convenience. | |
22244 | ||
22245 | The expression @cite{@t{conj}(@t{conj}(x))} simplifies to @cite{x}. | |
22246 | Various other expressions involving @code{conj}, @code{re}, and | |
22247 | @code{im} are simplified, especially if some of the arguments are | |
22248 | provably real or involve the constant @code{i}. For example, | |
22249 | @cite{@t{conj}(a + b i)} is changed to @cite{@t{conj}(a) - @t{conj}(b) i}, | |
22250 | or to @cite{a - b i} if @cite{a} and @cite{b} are known to be real. | |
22251 | ||
22252 | Functions like @code{sin} and @code{arctan} generally don't have | |
22253 | any default simplifications beyond simply evaluating the functions | |
22254 | for suitable numeric arguments and infinity. The @kbd{a s} command | |
22255 | described in the next section does provide some simplifications for | |
22256 | these functions, though. | |
22257 | ||
22258 | One important simplification that does occur is that @cite{@t{ln}(@t{e})} | |
22259 | is simplified to 1, and @cite{@t{ln}(@t{e}^x)} is simplified to @cite{x} | |
22260 | for any @cite{x}. This occurs even if you have stored a different | |
22261 | value in the Calc variable @samp{e}; but this would be a bad idea | |
22262 | in any case if you were also using natural logarithms! | |
22263 | ||
5d67986c RS |
22264 | Among the logical functions, @t{(@var{a} <= @var{b})} changes to |
22265 | @t{@var{a} > @var{b}} and so on. Equations and inequalities where both sides | |
d7b8e6c6 EZ |
22266 | are either negative-looking or zero are simplified by negating both sides |
22267 | and reversing the inequality. While it might seem reasonable to simplify | |
22268 | @cite{!!x} to @cite{x}, this would not be valid in general because | |
22269 | @cite{!!2} is 1, not 2. | |
22270 | ||
22271 | Most other Calc functions have few if any default simplifications | |
22272 | defined, aside of course from evaluation when the arguments are | |
22273 | suitable numbers. | |
22274 | ||
22275 | @node Algebraic Simplifications, Unsafe Simplifications, Default Simplifications, Simplifying Formulas | |
22276 | @subsection Algebraic Simplifications | |
22277 | ||
22278 | @noindent | |
22279 | @cindex Algebraic simplifications | |
22280 | The @kbd{a s} command makes simplifications that may be too slow to | |
22281 | do all the time, or that may not be desirable all of the time. | |
22282 | If you find these simplifications are worthwhile, you can type | |
22283 | @kbd{m A} to have Calc apply them automatically. | |
22284 | ||
22285 | This section describes all simplifications that are performed by | |
22286 | the @kbd{a s} command. Note that these occur in addition to the | |
22287 | default simplifications; even if the default simplifications have | |
22288 | been turned off by an @kbd{m O} command, @kbd{a s} will turn them | |
22289 | back on temporarily while it simplifies the formula. | |
22290 | ||
22291 | There is a variable, @code{AlgSimpRules}, in which you can put rewrites | |
22292 | to be applied by @kbd{a s}. Its use is analogous to @code{EvalRules}, | |
22293 | but without the special restrictions. Basically, the simplifier does | |
22294 | @samp{@w{a r} AlgSimpRules} with an infinite repeat count on the whole | |
22295 | expression being simplified, then it traverses the expression applying | |
22296 | the built-in rules described below. If the result is different from | |
22297 | the original expression, the process repeats with the default | |
22298 | simplifications (including @code{EvalRules}), then @code{AlgSimpRules}, | |
22299 | then the built-in simplifications, and so on. | |
22300 | ||
22301 | @tex | |
22302 | \bigskip | |
22303 | @end tex | |
22304 | ||
22305 | Sums are simplified in two ways. Constant terms are commuted to the | |
22306 | end of the sum, so that @cite{a + 2 + b} changes to @cite{a + b + 2}. | |
22307 | The only exception is that a constant will not be commuted away | |
22308 | from the first position of a difference, i.e., @cite{2 - x} is not | |
22309 | commuted to @cite{-x + 2}. | |
22310 | ||
22311 | Also, terms of sums are combined by the distributive law, as in | |
22312 | @cite{x + y + 2 x} to @cite{y + 3 x}. This always occurs for | |
22313 | adjacent terms, but @kbd{a s} compares all pairs of terms including | |
22314 | non-adjacent ones. | |
22315 | ||
22316 | @tex | |
22317 | \bigskip | |
22318 | @end tex | |
22319 | ||
22320 | Products are sorted into a canonical order using the commutative | |
22321 | law. For example, @cite{b c a} is commuted to @cite{a b c}. | |
22322 | This allows easier comparison of products; for example, the default | |
22323 | simplifications will not change @cite{x y + y x} to @cite{2 x y}, | |
22324 | but @kbd{a s} will; it first rewrites the sum to @cite{x y + x y}, | |
22325 | and then the default simplifications are able to recognize a sum | |
22326 | of identical terms. | |
22327 | ||
22328 | The canonical ordering used to sort terms of products has the | |
22329 | property that real-valued numbers, interval forms and infinities | |
22330 | come first, and are sorted into increasing order. The @kbd{V S} | |
22331 | command uses the same ordering when sorting a vector. | |
22332 | ||
22333 | Sorting of terms of products is inhibited when matrix mode is | |
22334 | turned on; in this case, Calc will never exchange the order of | |
22335 | two terms unless it knows at least one of the terms is a scalar. | |
22336 | ||
22337 | Products of powers are distributed by comparing all pairs of | |
22338 | terms, using the same method that the default simplifications | |
22339 | use for adjacent terms of products. | |
22340 | ||
22341 | Even though sums are not sorted, the commutative law is still | |
22342 | taken into account when terms of a product are being compared. | |
22343 | Thus @cite{(x + y) (y + x)} will be simplified to @cite{(x + y)^2}. | |
22344 | A subtle point is that @cite{(x - y) (y - x)} will @emph{not} | |
22345 | be simplified to @cite{-(x - y)^2}; Calc does not notice that | |
22346 | one term can be written as a constant times the other, even if | |
22347 | that constant is @i{-1}. | |
22348 | ||
22349 | A fraction times any expression, @cite{(a:b) x}, is changed to | |
22350 | a quotient involving integers: @cite{a x / b}. This is not | |
22351 | done for floating-point numbers like @cite{0.5}, however. This | |
22352 | is one reason why you may find it convenient to turn Fraction mode | |
22353 | on while doing algebra; @pxref{Fraction Mode}. | |
22354 | ||
22355 | @tex | |
22356 | \bigskip | |
22357 | @end tex | |
22358 | ||
22359 | Quotients are simplified by comparing all terms in the numerator | |
22360 | with all terms in the denominator for possible cancellation using | |
22361 | the distributive law. For example, @cite{a x^2 b / c x^3 d} will | |
22362 | cancel @cite{x^2} from both sides to get @cite{a b / c x d}. | |
22363 | (The terms in the denominator will then be rearranged to @cite{c d x} | |
22364 | as described above.) If there is any common integer or fractional | |
22365 | factor in the numerator and denominator, it is cancelled out; | |
22366 | for example, @cite{(4 x + 6) / 8 x} simplifies to @cite{(2 x + 3) / 4 x}. | |
22367 | ||
22368 | Non-constant common factors are not found even by @kbd{a s}. To | |
22369 | cancel the factor @cite{a} in @cite{(a x + a) / a^2} you could first | |
22370 | use @kbd{j M} on the product @cite{a x} to Merge the numerator to | |
22371 | @cite{a (1+x)}, which can then be simplified successfully. | |
22372 | ||
22373 | @tex | |
22374 | \bigskip | |
22375 | @end tex | |
22376 | ||
22377 | Integer powers of the variable @code{i} are simplified according | |
22378 | to the identity @cite{i^2 = -1}. If you store a new value other | |
22379 | than the complex number @cite{(0,1)} in @code{i}, this simplification | |
22380 | will no longer occur. This is done by @kbd{a s} instead of by default | |
22381 | in case someone (unwisely) uses the name @code{i} for a variable | |
22382 | unrelated to complex numbers; it would be unfortunate if Calc | |
22383 | quietly and automatically changed this formula for reasons the | |
22384 | user might not have been thinking of. | |
22385 | ||
22386 | Square roots of integer or rational arguments are simplified in | |
22387 | several ways. (Note that these will be left unevaluated only in | |
22388 | Symbolic mode.) First, square integer or rational factors are | |
22389 | pulled out so that @cite{@t{sqrt}(8)} is rewritten as | |
22390 | @c{$2\,\t{sqrt}(2)$} | |
22391 | @cite{2 sqrt(2)}. Conceptually speaking this implies factoring | |
22392 | the argument into primes and moving pairs of primes out of the | |
22393 | square root, but for reasons of efficiency Calc only looks for | |
22394 | primes up to 29. | |
22395 | ||
22396 | Square roots in the denominator of a quotient are moved to the | |
22397 | numerator: @cite{1 / @t{sqrt}(3)} changes to @cite{@t{sqrt}(3) / 3}. | |
22398 | The same effect occurs for the square root of a fraction: | |
22399 | @cite{@t{sqrt}(2:3)} changes to @cite{@t{sqrt}(6) / 3}. | |
22400 | ||
22401 | @tex | |
22402 | \bigskip | |
22403 | @end tex | |
22404 | ||
22405 | The @code{%} (modulo) operator is simplified in several ways | |
22406 | when the modulus @cite{M} is a positive real number. First, if | |
22407 | the argument is of the form @cite{x + n} for some real number | |
22408 | @cite{n}, then @cite{n} is itself reduced modulo @cite{M}. For | |
22409 | example, @samp{(x - 23) % 10} is simplified to @samp{(x + 7) % 10}. | |
22410 | ||
22411 | If the argument is multiplied by a constant, and this constant | |
22412 | has a common integer divisor with the modulus, then this factor is | |
22413 | cancelled out. For example, @samp{12 x % 15} is changed to | |
22414 | @samp{3 (4 x % 5)} by factoring out 3. Also, @samp{(12 x + 1) % 15} | |
22415 | is changed to @samp{3 ((4 x + 1:3) % 5)}. While these forms may | |
22416 | not seem ``simpler,'' they allow Calc to discover useful information | |
22417 | about modulo forms in the presence of declarations. | |
22418 | ||
22419 | If the modulus is 1, then Calc can use @code{int} declarations to | |
22420 | evaluate the expression. For example, the idiom @samp{x % 2} is | |
22421 | often used to check whether a number is odd or even. As described | |
22422 | above, @w{@samp{2 n % 2}} and @samp{(2 n + 1) % 2} are simplified to | |
22423 | @samp{2 (n % 1)} and @samp{2 ((n + 1:2) % 1)}, respectively; Calc | |
22424 | can simplify these to 0 and 1 (respectively) if @code{n} has been | |
22425 | declared to be an integer. | |
22426 | ||
22427 | @tex | |
22428 | \bigskip | |
22429 | @end tex | |
22430 | ||
22431 | Trigonometric functions are simplified in several ways. First, | |
22432 | @cite{@t{sin}(@t{arcsin}(x))} is simplified to @cite{x}, and | |
22433 | similarly for @code{cos} and @code{tan}. If the argument to | |
22434 | @code{sin} is negative-looking, it is simplified to @cite{-@t{sin}(x)}, | |
22435 | and similarly for @code{cos} and @code{tan}. Finally, certain | |
22436 | special values of the argument are recognized; | |
22437 | @pxref{Trigonometric and Hyperbolic Functions}. | |
22438 | ||
22439 | Trigonometric functions of inverses of different trigonometric | |
22440 | functions can also be simplified, as in @cite{@t{sin}(@t{arccos}(x))} | |
22441 | to @cite{@t{sqrt}(1 - x^2)}. | |
22442 | ||
22443 | Hyperbolic functions of their inverses and of negative-looking | |
22444 | arguments are also handled, as are exponentials of inverse | |
22445 | hyperbolic functions. | |
22446 | ||
22447 | No simplifications for inverse trigonometric and hyperbolic | |
22448 | functions are known, except for negative arguments of @code{arcsin}, | |
22449 | @code{arctan}, @code{arcsinh}, and @code{arctanh}. Note that | |
22450 | @cite{@t{arcsin}(@t{sin}(x))} can @emph{not} safely change to | |
22451 | @cite{x}, since this only correct within an integer multiple | |
22452 | of @c{$2 \pi$} | |
22453 | @cite{2 pi} radians or 360 degrees. However, | |
22454 | @cite{@t{arcsinh}(@t{sinh}(x))} is simplified to @cite{x} if | |
22455 | @cite{x} is known to be real. | |
22456 | ||
22457 | Several simplifications that apply to logarithms and exponentials | |
22458 | are that @cite{@t{exp}(@t{ln}(x))}, @c{$@t{e}^{\ln(x)}$} | |
22459 | @cite{e^@t{ln}(x)}, and | |
22460 | @c{$10^{{\rm log10}(x)}$} | |
22461 | @cite{10^@t{log10}(x)} all reduce to @cite{x}. | |
22462 | Also, @cite{@t{ln}(@t{exp}(x))}, etc., can reduce to @cite{x} if | |
22463 | @cite{x} is provably real. The form @cite{@t{exp}(x)^y} is simplified | |
22464 | to @cite{@t{exp}(x y)}. If @cite{x} is a suitable multiple of @c{$\pi i$} | |
22465 | @cite{pi i} | |
22466 | (as described above for the trigonometric functions), then @cite{@t{exp}(x)} | |
22467 | or @cite{e^x} will be expanded. Finally, @cite{@t{ln}(x)} is simplified | |
22468 | to a form involving @code{pi} and @code{i} where @cite{x} is provably | |
22469 | negative, positive imaginary, or negative imaginary. | |
22470 | ||
22471 | The error functions @code{erf} and @code{erfc} are simplified when | |
22472 | their arguments are negative-looking or are calls to the @code{conj} | |
22473 | function. | |
22474 | ||
22475 | @tex | |
22476 | \bigskip | |
22477 | @end tex | |
22478 | ||
22479 | Equations and inequalities are simplified by cancelling factors | |
22480 | of products, quotients, or sums on both sides. Inequalities | |
22481 | change sign if a negative multiplicative factor is cancelled. | |
22482 | Non-constant multiplicative factors as in @cite{a b = a c} are | |
22483 | cancelled from equations only if they are provably nonzero (generally | |
22484 | because they were declared so; @pxref{Declarations}). Factors | |
22485 | are cancelled from inequalities only if they are nonzero and their | |
22486 | sign is known. | |
22487 | ||
22488 | Simplification also replaces an equation or inequality with | |
22489 | 1 or 0 (``true'' or ``false'') if it can through the use of | |
22490 | declarations. If @cite{x} is declared to be an integer greater | |
22491 | than 5, then @cite{x < 3}, @cite{x = 3}, and @cite{x = 7.5} are | |
22492 | all simplified to 0, but @cite{x > 3} is simplified to 1. | |
22493 | By a similar analysis, @cite{abs(x) >= 0} is simplified to 1, | |
22494 | as is @cite{x^2 >= 0} if @cite{x} is known to be real. | |
22495 | ||
22496 | @node Unsafe Simplifications, Simplification of Units, Algebraic Simplifications, Simplifying Formulas | |
22497 | @subsection ``Unsafe'' Simplifications | |
22498 | ||
22499 | @noindent | |
22500 | @cindex Unsafe simplifications | |
22501 | @cindex Extended simplification | |
22502 | @kindex a e | |
22503 | @pindex calc-simplify-extended | |
5d67986c RS |
22504 | @ignore |
22505 | @mindex esimpl@idots | |
22506 | @end ignore | |
d7b8e6c6 EZ |
22507 | @tindex esimplify |
22508 | The @kbd{a e} (@code{calc-simplify-extended}) [@code{esimplify}] command | |
22509 | is like @kbd{a s} | |
22510 | except that it applies some additional simplifications which are not | |
22511 | ``safe'' in all cases. Use this only if you know the values in your | |
22512 | formula lie in the restricted ranges for which these simplifications | |
22513 | are valid. The symbolic integrator uses @kbd{a e}; | |
22514 | one effect of this is that the integrator's results must be used with | |
22515 | caution. Where an integral table will often attach conditions like | |
22516 | ``for positive @cite{a} only,'' Calc (like most other symbolic | |
22517 | integration programs) will simply produce an unqualified result.@refill | |
22518 | ||
22519 | Because @kbd{a e}'s simplifications are unsafe, it is sometimes better | |
22520 | to type @kbd{C-u -3 a v}, which does extended simplification only | |
22521 | on the top level of the formula without affecting the sub-formulas. | |
22522 | In fact, @kbd{C-u -3 j v} allows you to target extended simplification | |
22523 | to any specific part of a formula. | |
22524 | ||
22525 | The variable @code{ExtSimpRules} contains rewrites to be applied by | |
22526 | the @kbd{a e} command. These are applied in addition to | |
22527 | @code{EvalRules} and @code{AlgSimpRules}. (The @kbd{a r AlgSimpRules} | |
22528 | step described above is simply followed by an @kbd{a r ExtSimpRules} step.) | |
22529 | ||
22530 | Following is a complete list of ``unsafe'' simplifications performed | |
22531 | by @kbd{a e}. | |
22532 | ||
22533 | @tex | |
22534 | \bigskip | |
22535 | @end tex | |
22536 | ||
22537 | Inverse trigonometric or hyperbolic functions, called with their | |
22538 | corresponding non-inverse functions as arguments, are simplified | |
22539 | by @kbd{a e}. For example, @cite{@t{arcsin}(@t{sin}(x))} changes | |
22540 | to @cite{x}. Also, @cite{@t{arcsin}(@t{cos}(x))} and | |
22541 | @cite{@t{arccos}(@t{sin}(x))} both change to @cite{@t{pi}/2 - x}. | |
22542 | These simplifications are unsafe because they are valid only for | |
22543 | values of @cite{x} in a certain range; outside that range, values | |
22544 | are folded down to the 360-degree range that the inverse trigonometric | |
22545 | functions always produce. | |
22546 | ||
22547 | Powers of powers @cite{(x^a)^b} are simplified to @c{$x^{a b}$} | |
22548 | @cite{x^(a b)} | |
22549 | for all @cite{a} and @cite{b}. These results will be valid only | |
22550 | in a restricted range of @cite{x}; for example, in @c{$(x^2)^{1:2}$} | |
22551 | @cite{(x^2)^1:2} | |
22552 | the powers cancel to get @cite{x}, which is valid for positive values | |
22553 | of @cite{x} but not for negative or complex values. | |
22554 | ||
22555 | Similarly, @cite{@t{sqrt}(x^a)} and @cite{@t{sqrt}(x)^a} are both | |
22556 | simplified (possibly unsafely) to @c{$x^{a/2}$} | |
22557 | @cite{x^(a/2)}. | |
22558 | ||
22559 | Forms like @cite{@t{sqrt}(1 - @t{sin}(x)^2)} are simplified to, e.g., | |
22560 | @cite{@t{cos}(x)}. Calc has identities of this sort for @code{sin}, | |
22561 | @code{cos}, @code{tan}, @code{sinh}, and @code{cosh}. | |
22562 | ||
22563 | Arguments of square roots are partially factored to look for | |
22564 | squared terms that can be extracted. For example, | |
22565 | @cite{@t{sqrt}(a^2 b^3 + a^3 b^2)} simplifies to @cite{a b @t{sqrt}(a+b)}. | |
22566 | ||
22567 | The simplifications of @cite{@t{ln}(@t{exp}(x))}, @cite{@t{ln}(@t{e}^x)}, | |
22568 | and @cite{@t{log10}(10^x)} to @cite{x} are also unsafe because | |
22569 | of problems with principal values (although these simplifications | |
22570 | are safe if @cite{x} is known to be real). | |
22571 | ||
22572 | Common factors are cancelled from products on both sides of an | |
22573 | equation, even if those factors may be zero: @cite{a x / b x} | |
22574 | to @cite{a / b}. Such factors are never cancelled from | |
22575 | inequalities: Even @kbd{a e} is not bold enough to reduce | |
22576 | @cite{a x < b x} to @cite{a < b} (or @cite{a > b}, depending | |
22577 | on whether you believe @cite{x} is positive or negative). | |
22578 | The @kbd{a M /} command can be used to divide a factor out of | |
22579 | both sides of an inequality. | |
22580 | ||
22581 | @node Simplification of Units, , Unsafe Simplifications, Simplifying Formulas | |
22582 | @subsection Simplification of Units | |
22583 | ||
22584 | @noindent | |
22585 | The simplifications described in this section are applied by the | |
22586 | @kbd{u s} (@code{calc-simplify-units}) command. These are in addition | |
22587 | to the regular @kbd{a s} (but not @kbd{a e}) simplifications described | |
22588 | earlier. @xref{Basic Operations on Units}. | |
22589 | ||
22590 | The variable @code{UnitSimpRules} contains rewrites to be applied by | |
22591 | the @kbd{u s} command. These are applied in addition to @code{EvalRules} | |
22592 | and @code{AlgSimpRules}. | |
22593 | ||
22594 | Scalar mode is automatically put into effect when simplifying units. | |
22595 | @xref{Matrix Mode}. | |
22596 | ||
22597 | Sums @cite{a + b} involving units are simplified by extracting the | |
22598 | units of @cite{a} as if by the @kbd{u x} command (call the result | |
22599 | @cite{u_a}), then simplifying the expression @cite{b / u_a} | |
22600 | using @kbd{u b} and @kbd{u s}. If the result has units then the sum | |
22601 | is inconsistent and is left alone. Otherwise, it is rewritten | |
22602 | in terms of the units @cite{u_a}. | |
22603 | ||
22604 | If units auto-ranging mode is enabled, products or quotients in | |
22605 | which the first argument is a number which is out of range for the | |
22606 | leading unit are modified accordingly. | |
22607 | ||
22608 | When cancelling and combining units in products and quotients, | |
22609 | Calc accounts for unit names that differ only in the prefix letter. | |
22610 | For example, @samp{2 km m} is simplified to @samp{2000 m^2}. | |
22611 | However, compatible but different units like @code{ft} and @code{in} | |
22612 | are not combined in this way. | |
22613 | ||
22614 | Quotients @cite{a / b} are simplified in three additional ways. First, | |
22615 | if @cite{b} is a number or a product beginning with a number, Calc | |
22616 | computes the reciprocal of this number and moves it to the numerator. | |
22617 | ||
22618 | Second, for each pair of unit names from the numerator and denominator | |
22619 | of a quotient, if the units are compatible (e.g., they are both | |
22620 | units of area) then they are replaced by the ratio between those | |
22621 | units. For example, in @samp{3 s in N / kg cm} the units | |
22622 | @samp{in / cm} will be replaced by @cite{2.54}. | |
22623 | ||
22624 | Third, if the units in the quotient exactly cancel out, so that | |
22625 | a @kbd{u b} command on the quotient would produce a dimensionless | |
22626 | number for an answer, then the quotient simplifies to that number. | |
22627 | ||
22628 | For powers and square roots, the ``unsafe'' simplifications | |
22629 | @cite{(a b)^c} to @cite{a^c b^c}, @cite{(a/b)^c} to @cite{a^c / b^c}, | |
22630 | and @cite{(a^b)^c} to @c{$a^{b c}$} | |
22631 | @cite{a^(b c)} are done if the powers are | |
22632 | real numbers. (These are safe in the context of units because | |
22633 | all numbers involved can reasonably be assumed to be real.) | |
22634 | ||
22635 | Also, if a unit name is raised to a fractional power, and the | |
22636 | base units in that unit name all occur to powers which are a | |
22637 | multiple of the denominator of the power, then the unit name | |
22638 | is expanded out into its base units, which can then be simplified | |
22639 | according to the previous paragraph. For example, @samp{acre^1.5} | |
22640 | is simplified by noting that @cite{1.5 = 3:2}, that @samp{acre} | |
22641 | is defined in terms of @samp{m^2}, and that the 2 in the power of | |
22642 | @code{m} is a multiple of 2 in @cite{3:2}. Thus, @code{acre^1.5} is | |
22643 | replaced by approximately @c{$(4046 m^2)^{1.5}$} | |
22644 | @cite{(4046 m^2)^1.5}, which is then | |
22645 | changed to @c{$4046^{1.5} \, (m^2)^{1.5}$} | |
22646 | @cite{4046^1.5 (m^2)^1.5}, then to @cite{257440 m^3}. | |
22647 | ||
22648 | The functions @code{float}, @code{frac}, @code{clean}, @code{abs}, | |
22649 | as well as @code{floor} and the other integer truncation functions, | |
22650 | applied to unit names or products or quotients involving units, are | |
22651 | simplified. For example, @samp{round(1.6 in)} is changed to | |
22652 | @samp{round(1.6) round(in)}; the lefthand term evaluates to 2, | |
22653 | and the righthand term simplifies to @code{in}. | |
22654 | ||
22655 | The functions @code{sin}, @code{cos}, and @code{tan} with arguments | |
22656 | that have angular units like @code{rad} or @code{arcmin} are | |
22657 | simplified by converting to base units (radians), then evaluating | |
22658 | with the angular mode temporarily set to radians. | |
22659 | ||
22660 | @node Polynomials, Calculus, Simplifying Formulas, Algebra | |
22661 | @section Polynomials | |
22662 | ||
22663 | A @dfn{polynomial} is a sum of terms which are coefficients times | |
22664 | various powers of a ``base'' variable. For example, @cite{2 x^2 + 3 x - 4} | |
22665 | is a polynomial in @cite{x}. Some formulas can be considered | |
22666 | polynomials in several different variables: @cite{1 + 2 x + 3 y + 4 x y^2} | |
22667 | is a polynomial in both @cite{x} and @cite{y}. Polynomial coefficients | |
22668 | are often numbers, but they may in general be any formulas not | |
22669 | involving the base variable. | |
22670 | ||
22671 | @kindex a f | |
22672 | @pindex calc-factor | |
22673 | @tindex factor | |
22674 | The @kbd{a f} (@code{calc-factor}) [@code{factor}] command factors a | |
22675 | polynomial into a product of terms. For example, the polynomial | |
22676 | @cite{x^3 + 2 x^2 + x} is factored into @samp{x*(x+1)^2}. As another | |
22677 | example, @cite{a c + b d + b c + a d} is factored into the product | |
22678 | @cite{(a + b) (c + d)}. | |
22679 | ||
22680 | Calc currently has three algorithms for factoring. Formulas which are | |
22681 | linear in several variables, such as the second example above, are | |
22682 | merged according to the distributive law. Formulas which are | |
22683 | polynomials in a single variable, with constant integer or fractional | |
22684 | coefficients, are factored into irreducible linear and/or quadratic | |
22685 | terms. The first example above factors into three linear terms | |
22686 | (@cite{x}, @cite{x+1}, and @cite{x+1} again). Finally, formulas | |
22687 | which do not fit the above criteria are handled by the algebraic | |
22688 | rewrite mechanism. | |
22689 | ||
22690 | Calc's polynomial factorization algorithm works by using the general | |
22691 | root-finding command (@w{@kbd{a P}}) to solve for the roots of the | |
22692 | polynomial. It then looks for roots which are rational numbers | |
22693 | or complex-conjugate pairs, and converts these into linear and | |
22694 | quadratic terms, respectively. Because it uses floating-point | |
22695 | arithmetic, it may be unable to find terms that involve large | |
22696 | integers (whose number of digits approaches the current precision). | |
22697 | Also, irreducible factors of degree higher than quadratic are not | |
22698 | found, and polynomials in more than one variable are not treated. | |
22699 | (A more robust factorization algorithm may be included in a future | |
22700 | version of Calc.) | |
22701 | ||
22702 | @vindex FactorRules | |
5d67986c RS |
22703 | @ignore |
22704 | @starindex | |
22705 | @end ignore | |
d7b8e6c6 | 22706 | @tindex thecoefs |
5d67986c RS |
22707 | @ignore |
22708 | @starindex | |
22709 | @end ignore | |
22710 | @ignore | |
22711 | @mindex @idots | |
22712 | @end ignore | |
d7b8e6c6 EZ |
22713 | @tindex thefactors |
22714 | The rewrite-based factorization method uses rules stored in the variable | |
22715 | @code{FactorRules}. @xref{Rewrite Rules}, for a discussion of the | |
22716 | operation of rewrite rules. The default @code{FactorRules} are able | |
22717 | to factor quadratic forms symbolically into two linear terms, | |
22718 | @cite{(a x + b) (c x + d)}. You can edit these rules to include other | |
22719 | cases if you wish. To use the rules, Calc builds the formula | |
22720 | @samp{thecoefs(x, [a, b, c, ...])} where @code{x} is the polynomial | |
22721 | base variable and @code{a}, @code{b}, etc., are polynomial coefficients | |
22722 | (which may be numbers or formulas). The constant term is written first, | |
22723 | i.e., in the @code{a} position. When the rules complete, they should have | |
22724 | changed the formula into the form @samp{thefactors(x, [f1, f2, f3, ...])} | |
22725 | where each @code{fi} should be a factored term, e.g., @samp{x - ai}. | |
22726 | Calc then multiplies these terms together to get the complete | |
22727 | factored form of the polynomial. If the rules do not change the | |
22728 | @code{thecoefs} call to a @code{thefactors} call, @kbd{a f} leaves the | |
22729 | polynomial alone on the assumption that it is unfactorable. (Note that | |
22730 | the function names @code{thecoefs} and @code{thefactors} are used only | |
22731 | as placeholders; there are no actual Calc functions by those names.) | |
22732 | ||
22733 | @kindex H a f | |
22734 | @tindex factors | |
22735 | The @kbd{H a f} [@code{factors}] command also factors a polynomial, | |
22736 | but it returns a list of factors instead of an expression which is the | |
22737 | product of the factors. Each factor is represented by a sub-vector | |
22738 | of the factor, and the power with which it appears. For example, | |
22739 | @cite{x^5 + x^4 - 33 x^3 + 63 x^2} factors to @cite{(x + 7) x^2 (x - 3)^2} | |
22740 | in @kbd{a f}, or to @cite{[ [x, 2], [x+7, 1], [x-3, 2] ]} in @kbd{H a f}. | |
22741 | If there is an overall numeric factor, it always comes first in the list. | |
22742 | The functions @code{factor} and @code{factors} allow a second argument | |
22743 | when written in algebraic form; @samp{factor(x,v)} factors @cite{x} with | |
22744 | respect to the specific variable @cite{v}. The default is to factor with | |
22745 | respect to all the variables that appear in @cite{x}. | |
22746 | ||
22747 | @kindex a c | |
22748 | @pindex calc-collect | |
22749 | @tindex collect | |
22750 | The @kbd{a c} (@code{calc-collect}) [@code{collect}] command rearranges a | |
22751 | formula as a | |
22752 | polynomial in a given variable, ordered in decreasing powers of that | |
22753 | variable. For example, given @cite{1 + 2 x + 3 y + 4 x y^2} on | |
22754 | the stack, @kbd{a c x} would produce @cite{(2 + 4 y^2) x + (1 + 3 y)}, | |
22755 | and @kbd{a c y} would produce @cite{(4 x) y^2 + 3 y + (1 + 2 x)}. | |
22756 | The polynomial will be expanded out using the distributive law as | |
22757 | necessary: Collecting @cite{x} in @cite{(x - 1)^3} produces | |
22758 | @cite{x^3 - 3 x^2 + 3 x - 1}. Terms not involving @cite{x} will | |
22759 | not be expanded. | |
22760 | ||
22761 | The ``variable'' you specify at the prompt can actually be any | |
22762 | expression: @kbd{a c ln(x+1)} will collect together all terms multiplied | |
22763 | by @samp{ln(x+1)} or integer powers thereof. If @samp{x} also appears | |
22764 | in the formula in a context other than @samp{ln(x+1)}, @kbd{a c} will | |
22765 | treat those occurrences as unrelated to @samp{ln(x+1)}, i.e., as constants. | |
22766 | ||
22767 | @kindex a x | |
22768 | @pindex calc-expand | |
22769 | @tindex expand | |
22770 | The @kbd{a x} (@code{calc-expand}) [@code{expand}] command expands an | |
22771 | expression by applying the distributive law everywhere. It applies to | |
22772 | products, quotients, and powers involving sums. By default, it fully | |
22773 | distributes all parts of the expression. With a numeric prefix argument, | |
22774 | the distributive law is applied only the specified number of times, then | |
22775 | the partially expanded expression is left on the stack. | |
22776 | ||
22777 | The @kbd{a x} and @kbd{j D} commands are somewhat redundant. Use | |
22778 | @kbd{a x} if you want to expand all products of sums in your formula. | |
22779 | Use @kbd{j D} if you want to expand a particular specified term of | |
22780 | the formula. There is an exactly analogous correspondence between | |
22781 | @kbd{a f} and @kbd{j M}. (The @kbd{j D} and @kbd{j M} commands | |
22782 | also know many other kinds of expansions, such as | |
22783 | @samp{exp(a + b) = exp(a) exp(b)}, which @kbd{a x} and @kbd{a f} | |
22784 | do not do.) | |
22785 | ||
22786 | Calc's automatic simplifications will sometimes reverse a partial | |
22787 | expansion. For example, the first step in expanding @cite{(x+1)^3} is | |
22788 | to write @cite{(x+1) (x+1)^2}. If @kbd{a x} stops there and tries | |
22789 | to put this formula onto the stack, though, Calc will automatically | |
22790 | simplify it back to @cite{(x+1)^3} form. The solution is to turn | |
22791 | simplification off first (@pxref{Simplification Modes}), or to run | |
22792 | @kbd{a x} without a numeric prefix argument so that it expands all | |
22793 | the way in one step. | |
22794 | ||
22795 | @kindex a a | |
22796 | @pindex calc-apart | |
22797 | @tindex apart | |
22798 | The @kbd{a a} (@code{calc-apart}) [@code{apart}] command expands a | |
22799 | rational function by partial fractions. A rational function is the | |
22800 | quotient of two polynomials; @code{apart} pulls this apart into a | |
22801 | sum of rational functions with simple denominators. In algebraic | |
22802 | notation, the @code{apart} function allows a second argument that | |
22803 | specifies which variable to use as the ``base''; by default, Calc | |
22804 | chooses the base variable automatically. | |
22805 | ||
22806 | @kindex a n | |
22807 | @pindex calc-normalize-rat | |
22808 | @tindex nrat | |
22809 | The @kbd{a n} (@code{calc-normalize-rat}) [@code{nrat}] command | |
22810 | attempts to arrange a formula into a quotient of two polynomials. | |
22811 | For example, given @cite{1 + (a + b/c) / d}, the result would be | |
22812 | @cite{(b + a c + c d) / c d}. The quotient is reduced, so that | |
22813 | @kbd{a n} will simplify @cite{(x^2 + 2x + 1) / (x^2 - 1)} by dividing | |
22814 | out the common factor @cite{x + 1}, yielding @cite{(x + 1) / (x - 1)}. | |
22815 | ||
22816 | @kindex a \ | |
22817 | @pindex calc-poly-div | |
22818 | @tindex pdiv | |
22819 | The @kbd{a \} (@code{calc-poly-div}) [@code{pdiv}] command divides | |
22820 | two polynomials @cite{u} and @cite{v}, yielding a new polynomial | |
22821 | @cite{q}. If several variables occur in the inputs, the inputs are | |
22822 | considered multivariate polynomials. (Calc divides by the variable | |
22823 | with the largest power in @cite{u} first, or, in the case of equal | |
22824 | powers, chooses the variables in alphabetical order.) For example, | |
22825 | dividing @cite{x^2 + 3 x + 2} by @cite{x + 2} yields @cite{x + 1}. | |
22826 | The remainder from the division, if any, is reported at the bottom | |
22827 | of the screen and is also placed in the Trail along with the quotient. | |
22828 | ||
22829 | Using @code{pdiv} in algebraic notation, you can specify the particular | |
5d67986c | 22830 | variable to be used as the base: @code{pdiv(@var{a},@var{b},@var{x})}. |
d7b8e6c6 EZ |
22831 | If @code{pdiv} is given only two arguments (as is always the case with |
22832 | the @kbd{a \} command), then it does a multivariate division as outlined | |
22833 | above. | |
22834 | ||
22835 | @kindex a % | |
22836 | @pindex calc-poly-rem | |
22837 | @tindex prem | |
22838 | The @kbd{a %} (@code{calc-poly-rem}) [@code{prem}] command divides | |
22839 | two polynomials and keeps the remainder @cite{r}. The quotient | |
22840 | @cite{q} is discarded. For any formulas @cite{a} and @cite{b}, the | |
22841 | results of @kbd{a \} and @kbd{a %} satisfy @cite{a = q b + r}. | |
22842 | (This is analogous to plain @kbd{\} and @kbd{%}, which compute the | |
22843 | integer quotient and remainder from dividing two numbers.) | |
22844 | ||
22845 | @kindex a / | |
22846 | @kindex H a / | |
22847 | @pindex calc-poly-div-rem | |
22848 | @tindex pdivrem | |
22849 | @tindex pdivide | |
22850 | The @kbd{a /} (@code{calc-poly-div-rem}) [@code{pdivrem}] command | |
22851 | divides two polynomials and reports both the quotient and the | |
22852 | remainder as a vector @cite{[q, r]}. The @kbd{H a /} [@code{pdivide}] | |
22853 | command divides two polynomials and constructs the formula | |
22854 | @cite{q + r/b} on the stack. (Naturally if the remainder is zero, | |
22855 | this will immediately simplify to @cite{q}.) | |
22856 | ||
22857 | @kindex a g | |
22858 | @pindex calc-poly-gcd | |
22859 | @tindex pgcd | |
22860 | The @kbd{a g} (@code{calc-poly-gcd}) [@code{pgcd}] command computes | |
22861 | the greatest common divisor of two polynomials. (The GCD actually | |
22862 | is unique only to within a constant multiplier; Calc attempts to | |
22863 | choose a GCD which will be unsurprising.) For example, the @kbd{a n} | |
22864 | command uses @kbd{a g} to take the GCD of the numerator and denominator | |
22865 | of a quotient, then divides each by the result using @kbd{a \}. (The | |
22866 | definition of GCD ensures that this division can take place without | |
22867 | leaving a remainder.) | |
22868 | ||
22869 | While the polynomials used in operations like @kbd{a /} and @kbd{a g} | |
22870 | often have integer coefficients, this is not required. Calc can also | |
22871 | deal with polynomials over the rationals or floating-point reals. | |
22872 | Polynomials with modulo-form coefficients are also useful in many | |
22873 | applications; if you enter @samp{(x^2 + 3 x - 1) mod 5}, Calc | |
22874 | automatically transforms this into a polynomial over the field of | |
22875 | integers mod 5: @samp{(1 mod 5) x^2 + (3 mod 5) x + (4 mod 5)}. | |
22876 | ||
22877 | Congratulations and thanks go to Ove Ewerlid | |
22878 | (@code{ewerlid@@mizar.DoCS.UU.SE}), who contributed many of the | |
22879 | polynomial routines used in the above commands. | |
22880 | ||
22881 | @xref{Decomposing Polynomials}, for several useful functions for | |
22882 | extracting the individual coefficients of a polynomial. | |
22883 | ||
22884 | @node Calculus, Solving Equations, Polynomials, Algebra | |
22885 | @section Calculus | |
22886 | ||
22887 | @noindent | |
22888 | The following calculus commands do not automatically simplify their | |
22889 | inputs or outputs using @code{calc-simplify}. You may find it helps | |
22890 | to do this by hand by typing @kbd{a s} or @kbd{a e}. It may also help | |
22891 | to use @kbd{a x} and/or @kbd{a c} to arrange a result in the most | |
22892 | readable way. | |
22893 | ||
22894 | @menu | |
22895 | * Differentiation:: | |
22896 | * Integration:: | |
22897 | * Customizing the Integrator:: | |
22898 | * Numerical Integration:: | |
22899 | * Taylor Series:: | |
22900 | @end menu | |
22901 | ||
22902 | @node Differentiation, Integration, Calculus, Calculus | |
22903 | @subsection Differentiation | |
22904 | ||
22905 | @noindent | |
22906 | @kindex a d | |
22907 | @kindex H a d | |
22908 | @pindex calc-derivative | |
22909 | @tindex deriv | |
22910 | @tindex tderiv | |
22911 | The @kbd{a d} (@code{calc-derivative}) [@code{deriv}] command computes | |
22912 | the derivative of the expression on the top of the stack with respect to | |
22913 | some variable, which it will prompt you to enter. Normally, variables | |
22914 | in the formula other than the specified differentiation variable are | |
22915 | considered constant, i.e., @samp{deriv(y,x)} is reduced to zero. With | |
22916 | the Hyperbolic flag, the @code{tderiv} (total derivative) operation is used | |
22917 | instead, in which derivatives of variables are not reduced to zero | |
22918 | unless those variables are known to be ``constant,'' i.e., independent | |
22919 | of any other variables. (The built-in special variables like @code{pi} | |
22920 | are considered constant, as are variables that have been declared | |
22921 | @code{const}; @pxref{Declarations}.) | |
22922 | ||
22923 | With a numeric prefix argument @var{n}, this command computes the | |
22924 | @var{n}th derivative. | |
22925 | ||
22926 | When working with trigonometric functions, it is best to switch to | |
22927 | radians mode first (with @w{@kbd{m r}}). The derivative of @samp{sin(x)} | |
22928 | in degrees is @samp{(pi/180) cos(x)}, probably not the expected | |
22929 | answer! | |
22930 | ||
22931 | If you use the @code{deriv} function directly in an algebraic formula, | |
22932 | you can write @samp{deriv(f,x,x0)} which represents the derivative | |
22933 | of @cite{f} with respect to @cite{x}, evaluated at the point @c{$x=x_0$} | |
22934 | @cite{x=x0}. | |
22935 | ||
22936 | If the formula being differentiated contains functions which Calc does | |
22937 | not know, the derivatives of those functions are produced by adding | |
22938 | primes (apostrophe characters). For example, @samp{deriv(f(2x), x)} | |
22939 | produces @samp{2 f'(2 x)}, where the function @code{f'} represents the | |
22940 | derivative of @code{f}. | |
22941 | ||
22942 | For functions you have defined with the @kbd{Z F} command, Calc expands | |
22943 | the functions according to their defining formulas unless you have | |
22944 | also defined @code{f'} suitably. For example, suppose we define | |
22945 | @samp{sinc(x) = sin(x)/x} using @kbd{Z F}. If we then differentiate | |
22946 | the formula @samp{sinc(2 x)}, the formula will be expanded to | |
22947 | @samp{sin(2 x) / (2 x)} and differentiated. However, if we also | |
22948 | define @samp{sinc'(x) = dsinc(x)}, say, then Calc will write the | |
22949 | result as @samp{2 dsinc(2 x)}. @xref{Algebraic Definitions}. | |
22950 | ||
22951 | For multi-argument functions @samp{f(x,y,z)}, the derivative with respect | |
22952 | to the first argument is written @samp{f'(x,y,z)}; derivatives with | |
22953 | respect to the other arguments are @samp{f'2(x,y,z)} and @samp{f'3(x,y,z)}. | |
22954 | Various higher-order derivatives can be formed in the obvious way, e.g., | |
22955 | @samp{f'@var{}'(x)} (the second derivative of @code{f}) or | |
22956 | @samp{f'@var{}'2'3(x,y,z)} (@code{f} differentiated with respect to each | |
22957 | argument once).@refill | |
22958 | ||
22959 | @node Integration, Customizing the Integrator, Differentiation, Calculus | |
22960 | @subsection Integration | |
22961 | ||
22962 | @noindent | |
22963 | @kindex a i | |
22964 | @pindex calc-integral | |
22965 | @tindex integ | |
22966 | The @kbd{a i} (@code{calc-integral}) [@code{integ}] command computes the | |
22967 | indefinite integral of the expression on the top of the stack with | |
22968 | respect to a variable. The integrator is not guaranteed to work for | |
22969 | all integrable functions, but it is able to integrate several large | |
22970 | classes of formulas. In particular, any polynomial or rational function | |
22971 | (a polynomial divided by a polynomial) is acceptable. (Rational functions | |
22972 | don't have to be in explicit quotient form, however; @c{$x/(1+x^{-2})$} | |
22973 | @cite{x/(1+x^-2)} | |
22974 | is not strictly a quotient of polynomials, but it is equivalent to | |
22975 | @cite{x^3/(x^2+1)}, which is.) Also, square roots of terms involving | |
22976 | @cite{x} and @cite{x^2} may appear in rational functions being | |
22977 | integrated. Finally, rational functions involving trigonometric or | |
22978 | hyperbolic functions can be integrated. | |
22979 | ||
22980 | @ifinfo | |
22981 | If you use the @code{integ} function directly in an algebraic formula, | |
22982 | you can also write @samp{integ(f,x,v)} which expresses the resulting | |
22983 | indefinite integral in terms of variable @code{v} instead of @code{x}. | |
22984 | With four arguments, @samp{integ(f(x),x,a,b)} represents a definite | |
22985 | integral from @code{a} to @code{b}. | |
22986 | @end ifinfo | |
177c0ea7 | 22987 | @tex |
d7b8e6c6 EZ |
22988 | If you use the @code{integ} function directly in an algebraic formula, |
22989 | you can also write @samp{integ(f,x,v)} which expresses the resulting | |
22990 | indefinite integral in terms of variable @code{v} instead of @code{x}. | |
22991 | With four arguments, @samp{integ(f(x),x,a,b)} represents a definite | |
22992 | integral $\int_a^b f(x) \, dx$. | |
22993 | @end tex | |
22994 | ||
22995 | Please note that the current implementation of Calc's integrator sometimes | |
22996 | produces results that are significantly more complex than they need to | |
22997 | be. For example, the integral Calc finds for @c{$1/(x+\sqrt{x^2+1})$} | |
22998 | @cite{1/(x+sqrt(x^2+1))} | |
22999 | is several times more complicated than the answer Mathematica | |
23000 | returns for the same input, although the two forms are numerically | |
23001 | equivalent. Also, any indefinite integral should be considered to have | |
23002 | an arbitrary constant of integration added to it, although Calc does not | |
23003 | write an explicit constant of integration in its result. For example, | |
23004 | Calc's solution for @c{$1/(1+\tan x)$} | |
23005 | @cite{1/(1+tan(x))} differs from the solution given | |
23006 | in the @emph{CRC Math Tables} by a constant factor of @c{$\pi i / 2$} | |
23007 | @cite{pi i / 2}, | |
23008 | due to a different choice of constant of integration. | |
23009 | ||
23010 | The Calculator remembers all the integrals it has done. If conditions | |
23011 | change in a way that would invalidate the old integrals, say, a switch | |
23012 | from degrees to radians mode, then they will be thrown out. If you | |
23013 | suspect this is not happening when it should, use the | |
23014 | @code{calc-flush-caches} command; @pxref{Caches}. | |
23015 | ||
23016 | @vindex IntegLimit | |
23017 | Calc normally will pursue integration by substitution or integration by | |
23018 | parts up to 3 nested times before abandoning an approach as fruitless. | |
23019 | If the integrator is taking too long, you can lower this limit by storing | |
23020 | a number (like 2) in the variable @code{IntegLimit}. (The @kbd{s I} | |
23021 | command is a convenient way to edit @code{IntegLimit}.) If this variable | |
23022 | has no stored value or does not contain a nonnegative integer, a limit | |
23023 | of 3 is used. The lower this limit is, the greater the chance that Calc | |
23024 | will be unable to integrate a function it could otherwise handle. Raising | |
23025 | this limit allows the Calculator to solve more integrals, though the time | |
23026 | it takes may grow exponentially. You can monitor the integrator's actions | |
23027 | by creating an Emacs buffer called @code{*Trace*}. If such a buffer | |
23028 | exists, the @kbd{a i} command will write a log of its actions there. | |
23029 | ||
23030 | If you want to manipulate integrals in a purely symbolic way, you can | |
23031 | set the integration nesting limit to 0 to prevent all but fast | |
23032 | table-lookup solutions of integrals. You might then wish to define | |
23033 | rewrite rules for integration by parts, various kinds of substitutions, | |
23034 | and so on. @xref{Rewrite Rules}. | |
23035 | ||
23036 | @node Customizing the Integrator, Numerical Integration, Integration, Calculus | |
23037 | @subsection Customizing the Integrator | |
23038 | ||
23039 | @noindent | |
23040 | @vindex IntegRules | |
23041 | Calc has two built-in rewrite rules called @code{IntegRules} and | |
23042 | @code{IntegAfterRules} which you can edit to define new integration | |
23043 | methods. @xref{Rewrite Rules}. At each step of the integration process, | |
23044 | Calc wraps the current integrand in a call to the fictitious function | |
23045 | @samp{integtry(@var{expr},@var{var})}, where @var{expr} is the | |
23046 | integrand and @var{var} is the integration variable. If your rules | |
23047 | rewrite this to be a plain formula (not a call to @code{integtry}), then | |
23048 | Calc will use this formula as the integral of @var{expr}. For example, | |
23049 | the rule @samp{integtry(mysin(x),x) := -mycos(x)} would define a rule to | |
23050 | integrate a function @code{mysin} that acts like the sine function. | |
23051 | Then, putting @samp{4 mysin(2y+1)} on the stack and typing @kbd{a i y} | |
23052 | will produce the integral @samp{-2 mycos(2y+1)}. Note that Calc has | |
23053 | automatically made various transformations on the integral to allow it | |
23054 | to use your rule; integral tables generally give rules for | |
23055 | @samp{mysin(a x + b)}, but you don't need to use this much generality | |
23056 | in your @code{IntegRules}. | |
23057 | ||
23058 | @cindex Exponential integral Ei(x) | |
5d67986c RS |
23059 | @ignore |
23060 | @starindex | |
23061 | @end ignore | |
d7b8e6c6 EZ |
23062 | @tindex Ei |
23063 | As a more serious example, the expression @samp{exp(x)/x} cannot be | |
23064 | integrated in terms of the standard functions, so the ``exponential | |
23065 | integral'' function @c{${\rm Ei}(x)$} | |
23066 | @cite{Ei(x)} was invented to describe it. | |
23067 | We can get Calc to do this integral in terms of a made-up @code{Ei} | |
23068 | function by adding the rule @samp{[integtry(exp(x)/x, x) := Ei(x)]} | |
23069 | to @code{IntegRules}. Now entering @samp{exp(2x)/x} on the stack | |
23070 | and typing @kbd{a i x} yields @samp{Ei(2 x)}. This new rule will | |
23071 | work with Calc's various built-in integration methods (such as | |
23072 | integration by substitution) to solve a variety of other problems | |
23073 | involving @code{Ei}: For example, now Calc will also be able to | |
23074 | integrate @samp{exp(exp(x))} and @samp{ln(ln(x))} (to get @samp{Ei(exp(x))} | |
23075 | and @samp{x ln(ln(x)) - Ei(ln(x))}, respectively). | |
23076 | ||
23077 | Your rule may do further integration by calling @code{integ}. For | |
23078 | example, @samp{integtry(twice(u),x) := twice(integ(u))} allows Calc | |
23079 | to integrate @samp{twice(sin(x))} to get @samp{twice(-cos(x))}. | |
23080 | Note that @code{integ} was called with only one argument. This notation | |
23081 | is allowed only within @code{IntegRules}; it means ``integrate this | |
23082 | with respect to the same integration variable.'' If Calc is unable | |
23083 | to integrate @code{u}, the integration that invoked @code{IntegRules} | |
23084 | also fails. Thus integrating @samp{twice(f(x))} fails, returning the | |
23085 | unevaluated integral @samp{integ(twice(f(x)), x)}. It is still legal | |
23086 | to call @code{integ} with two or more arguments, however; in this case, | |
23087 | if @code{u} is not integrable, @code{twice} itself will still be | |
23088 | integrated: If the above rule is changed to @samp{... := twice(integ(u,x))}, | |
23089 | then integrating @samp{twice(f(x))} will yield @samp{twice(integ(f(x),x))}. | |
23090 | ||
23091 | If a rule instead produces the formula @samp{integsubst(@var{sexpr}, | |
23092 | @var{svar})}, either replacing the top-level @code{integtry} call or | |
23093 | nested anywhere inside the expression, then Calc will apply the | |
23094 | substitution @samp{@var{u} = @var{sexpr}(@var{svar})} to try to | |
23095 | integrate the original @var{expr}. For example, the rule | |
23096 | @samp{sqrt(a) := integsubst(sqrt(x),x)} says that if Calc ever finds | |
23097 | a square root in the integrand, it should attempt the substitution | |
23098 | @samp{u = sqrt(x)}. (This particular rule is unnecessary because | |
23099 | Calc always tries ``obvious'' substitutions where @var{sexpr} actually | |
23100 | appears in the integrand.) The variable @var{svar} may be the same | |
23101 | as the @var{var} that appeared in the call to @code{integtry}, but | |
23102 | it need not be. | |
23103 | ||
23104 | When integrating according to an @code{integsubst}, Calc uses the | |
23105 | equation solver to find the inverse of @var{sexpr} (if the integrand | |
23106 | refers to @var{var} anywhere except in subexpressions that exactly | |
23107 | match @var{sexpr}). It uses the differentiator to find the derivative | |
23108 | of @var{sexpr} and/or its inverse (it has two methods that use one | |
23109 | derivative or the other). You can also specify these items by adding | |
23110 | extra arguments to the @code{integsubst} your rules construct; the | |
23111 | general form is @samp{integsubst(@var{sexpr}, @var{svar}, @var{sinv}, | |
23112 | @var{sprime})}, where @var{sinv} is the inverse of @var{sexpr} (still | |
23113 | written as a function of @var{svar}), and @var{sprime} is the | |
23114 | derivative of @var{sexpr} with respect to @var{svar}. If you don't | |
23115 | specify these things, and Calc is not able to work them out on its | |
23116 | own with the information it knows, then your substitution rule will | |
23117 | work only in very specific, simple cases. | |
23118 | ||
23119 | Calc applies @code{IntegRules} as if by @kbd{C-u 1 a r IntegRules}; | |
23120 | in other words, Calc stops rewriting as soon as any rule in your rule | |
23121 | set succeeds. (If it weren't for this, the @samp{integsubst(sqrt(x),x)} | |
23122 | example above would keep on adding layers of @code{integsubst} calls | |
23123 | forever!) | |
23124 | ||
23125 | @vindex IntegSimpRules | |
23126 | Another set of rules, stored in @code{IntegSimpRules}, are applied | |
23127 | every time the integrator uses @kbd{a s} to simplify an intermediate | |
23128 | result. For example, putting the rule @samp{twice(x) := 2 x} into | |
23129 | @code{IntegSimpRules} would tell Calc to convert the @code{twice} | |
23130 | function into a form it knows whenever integration is attempted. | |
23131 | ||
23132 | One more way to influence the integrator is to define a function with | |
23133 | the @kbd{Z F} command (@pxref{Algebraic Definitions}). Calc's | |
23134 | integrator automatically expands such functions according to their | |
23135 | defining formulas, even if you originally asked for the function to | |
23136 | be left unevaluated for symbolic arguments. (Certain other Calc | |
23137 | systems, such as the differentiator and the equation solver, also | |
23138 | do this.) | |
23139 | ||
23140 | @vindex IntegAfterRules | |
23141 | Sometimes Calc is able to find a solution to your integral, but it | |
23142 | expresses the result in a way that is unnecessarily complicated. If | |
23143 | this happens, you can either use @code{integsubst} as described | |
23144 | above to try to hint at a more direct path to the desired result, or | |
23145 | you can use @code{IntegAfterRules}. This is an extra rule set that | |
23146 | runs after the main integrator returns its result; basically, Calc does | |
23147 | an @kbd{a r IntegAfterRules} on the result before showing it to you. | |
23148 | (It also does an @kbd{a s}, without @code{IntegSimpRules}, after that | |
23149 | to further simplify the result.) For example, Calc's integrator | |
23150 | sometimes produces expressions of the form @samp{ln(1+x) - ln(1-x)}; | |
23151 | the default @code{IntegAfterRules} rewrite this into the more readable | |
23152 | form @samp{2 arctanh(x)}. Note that, unlike @code{IntegRules}, | |
23153 | @code{IntegSimpRules} and @code{IntegAfterRules} are applied any number | |
23154 | of times until no further changes are possible. Rewriting by | |
23155 | @code{IntegAfterRules} occurs only after the main integrator has | |
23156 | finished, not at every step as for @code{IntegRules} and | |
23157 | @code{IntegSimpRules}. | |
23158 | ||
23159 | @node Numerical Integration, Taylor Series, Customizing the Integrator, Calculus | |
23160 | @subsection Numerical Integration | |
23161 | ||
23162 | @noindent | |
23163 | @kindex a I | |
23164 | @pindex calc-num-integral | |
23165 | @tindex ninteg | |
23166 | If you want a purely numerical answer to an integration problem, you can | |
23167 | use the @kbd{a I} (@code{calc-num-integral}) [@code{ninteg}] command. This | |
23168 | command prompts for an integration variable, a lower limit, and an | |
23169 | upper limit. Except for the integration variable, all other variables | |
23170 | that appear in the integrand formula must have stored values. (A stored | |
23171 | value, if any, for the integration variable itself is ignored.) | |
23172 | ||
23173 | Numerical integration works by evaluating your formula at many points in | |
23174 | the specified interval. Calc uses an ``open Romberg'' method; this means | |
23175 | that it does not evaluate the formula actually at the endpoints (so that | |
23176 | it is safe to integrate @samp{sin(x)/x} from zero, for example). Also, | |
23177 | the Romberg method works especially well when the function being | |
23178 | integrated is fairly smooth. If the function is not smooth, Calc will | |
23179 | have to evaluate it at quite a few points before it can accurately | |
23180 | determine the value of the integral. | |
23181 | ||
23182 | Integration is much faster when the current precision is small. It is | |
23183 | best to set the precision to the smallest acceptable number of digits | |
23184 | before you use @kbd{a I}. If Calc appears to be taking too long, press | |
23185 | @kbd{C-g} to halt it and try a lower precision. If Calc still appears | |
23186 | to need hundreds of evaluations, check to make sure your function is | |
23187 | well-behaved in the specified interval. | |
23188 | ||
23189 | It is possible for the lower integration limit to be @samp{-inf} (minus | |
23190 | infinity). Likewise, the upper limit may be plus infinity. Calc | |
23191 | internally transforms the integral into an equivalent one with finite | |
23192 | limits. However, integration to or across singularities is not supported: | |
23193 | The integral of @samp{1/sqrt(x)} from 0 to 1 exists (it can be found | |
23194 | by Calc's symbolic integrator, for example), but @kbd{a I} will fail | |
23195 | because the integrand goes to infinity at one of the endpoints. | |
23196 | ||
23197 | @node Taylor Series, , Numerical Integration, Calculus | |
23198 | @subsection Taylor Series | |
23199 | ||
23200 | @noindent | |
23201 | @kindex a t | |
23202 | @pindex calc-taylor | |
23203 | @tindex taylor | |
23204 | The @kbd{a t} (@code{calc-taylor}) [@code{taylor}] command computes a | |
23205 | power series expansion or Taylor series of a function. You specify the | |
23206 | variable and the desired number of terms. You may give an expression of | |
23207 | the form @samp{@var{var} = @var{a}} or @samp{@var{var} - @var{a}} instead | |
23208 | of just a variable to produce a Taylor expansion about the point @var{a}. | |
23209 | You may specify the number of terms with a numeric prefix argument; | |
23210 | otherwise the command will prompt you for the number of terms. Note that | |
23211 | many series expansions have coefficients of zero for some terms, so you | |
23212 | may appear to get fewer terms than you asked for.@refill | |
23213 | ||
23214 | If the @kbd{a i} command is unable to find a symbolic integral for a | |
23215 | function, you can get an approximation by integrating the function's | |
23216 | Taylor series. | |
23217 | ||
23218 | @node Solving Equations, Numerical Solutions, Calculus, Algebra | |
23219 | @section Solving Equations | |
23220 | ||
23221 | @noindent | |
23222 | @kindex a S | |
23223 | @pindex calc-solve-for | |
23224 | @tindex solve | |
23225 | @cindex Equations, solving | |
23226 | @cindex Solving equations | |
23227 | The @kbd{a S} (@code{calc-solve-for}) [@code{solve}] command rearranges | |
23228 | an equation to solve for a specific variable. An equation is an | |
23229 | expression of the form @cite{L = R}. For example, the command @kbd{a S x} | |
23230 | will rearrange @cite{y = 3x + 6} to the form, @cite{x = y/3 - 2}. If the | |
23231 | input is not an equation, it is treated like an equation of the | |
23232 | form @cite{X = 0}. | |
23233 | ||
23234 | This command also works for inequalities, as in @cite{y < 3x + 6}. | |
23235 | Some inequalities cannot be solved where the analogous equation could | |
23236 | be; for example, solving @c{$a < b \, c$} | |
23237 | @cite{a < b c} for @cite{b} is impossible | |
23238 | without knowing the sign of @cite{c}. In this case, @kbd{a S} will | |
23239 | produce the result @c{$b \mathbin{\hbox{\code{!=}}} a/c$} | |
23240 | @cite{b != a/c} (using the not-equal-to operator) | |
23241 | to signify that the direction of the inequality is now unknown. The | |
23242 | inequality @c{$a \le b \, c$} | |
23243 | @cite{a <= b c} is not even partially solved. | |
23244 | @xref{Declarations}, for a way to tell Calc that the signs of the | |
23245 | variables in a formula are in fact known. | |
23246 | ||
23247 | Two useful commands for working with the result of @kbd{a S} are | |
23248 | @kbd{a .} (@pxref{Logical Operations}), which converts @cite{x = y/3 - 2} | |
23249 | to @cite{y/3 - 2}, and @kbd{s l} (@pxref{Let Command}) which evaluates | |
23250 | another formula with @cite{x} set equal to @cite{y/3 - 2}. | |
23251 | ||
177c0ea7 | 23252 | @menu |
d7b8e6c6 EZ |
23253 | * Multiple Solutions:: |
23254 | * Solving Systems of Equations:: | |
23255 | * Decomposing Polynomials:: | |
23256 | @end menu | |
23257 | ||
23258 | @node Multiple Solutions, Solving Systems of Equations, Solving Equations, Solving Equations | |
23259 | @subsection Multiple Solutions | |
23260 | ||
23261 | @noindent | |
23262 | @kindex H a S | |
23263 | @tindex fsolve | |
23264 | Some equations have more than one solution. The Hyperbolic flag | |
23265 | (@code{H a S}) [@code{fsolve}] tells the solver to report the fully | |
23266 | general family of solutions. It will invent variables @code{n1}, | |
23267 | @code{n2}, @dots{}, which represent independent arbitrary integers, and | |
23268 | @code{s1}, @code{s2}, @dots{}, which represent independent arbitrary | |
23269 | signs (either @i{+1} or @i{-1}). If you don't use the Hyperbolic | |
23270 | flag, Calc will use zero in place of all arbitrary integers, and plus | |
23271 | one in place of all arbitrary signs. Note that variables like @code{n1} | |
23272 | and @code{s1} are not given any special interpretation in Calc except by | |
23273 | the equation solver itself. As usual, you can use the @w{@kbd{s l}} | |
23274 | (@code{calc-let}) command to obtain solutions for various actual values | |
23275 | of these variables. | |
23276 | ||
23277 | For example, @kbd{' x^2 = y @key{RET} H a S x @key{RET}} solves to | |
23278 | get @samp{x = s1 sqrt(y)}, indicating that the two solutions to the | |
23279 | equation are @samp{sqrt(y)} and @samp{-sqrt(y)}. Another way to | |
23280 | think about it is that the square-root operation is really a | |
23281 | two-valued function; since every Calc function must return a | |
23282 | single result, @code{sqrt} chooses to return the positive result. | |
23283 | Then @kbd{H a S} doctors this result using @code{s1} to indicate | |
23284 | the full set of possible values of the mathematical square-root. | |
23285 | ||
23286 | There is a similar phenomenon going the other direction: Suppose | |
23287 | we solve @samp{sqrt(y) = x} for @code{y}. Calc squares both sides | |
23288 | to get @samp{y = x^2}. This is correct, except that it introduces | |
23289 | some dubious solutions. Consider solving @samp{sqrt(y) = -3}: | |
23290 | Calc will report @cite{y = 9} as a valid solution, which is true | |
23291 | in the mathematical sense of square-root, but false (there is no | |
23292 | solution) for the actual Calc positive-valued @code{sqrt}. This | |
23293 | happens for both @kbd{a S} and @kbd{H a S}. | |
23294 | ||
23295 | @cindex @code{GenCount} variable | |
23296 | @vindex GenCount | |
5d67986c RS |
23297 | @ignore |
23298 | @starindex | |
23299 | @end ignore | |
d7b8e6c6 | 23300 | @tindex an |
5d67986c RS |
23301 | @ignore |
23302 | @starindex | |
23303 | @end ignore | |
d7b8e6c6 EZ |
23304 | @tindex as |
23305 | If you store a positive integer in the Calc variable @code{GenCount}, | |
23306 | then Calc will generate formulas of the form @samp{as(@var{n})} for | |
23307 | arbitrary signs, and @samp{an(@var{n})} for arbitrary integers, | |
23308 | where @var{n} represents successive values taken by incrementing | |
23309 | @code{GenCount} by one. While the normal arbitrary sign and | |
23310 | integer symbols start over at @code{s1} and @code{n1} with each | |
23311 | new Calc command, the @code{GenCount} approach will give each | |
23312 | arbitrary value a name that is unique throughout the entire Calc | |
23313 | session. Also, the arbitrary values are function calls instead | |
23314 | of variables, which is advantageous in some cases. For example, | |
23315 | you can make a rewrite rule that recognizes all arbitrary signs | |
23316 | using a pattern like @samp{as(n)}. The @kbd{s l} command only works | |
23317 | on variables, but you can use the @kbd{a b} (@code{calc-substitute}) | |
23318 | command to substitute actual values for function calls like @samp{as(3)}. | |
23319 | ||
23320 | The @kbd{s G} (@code{calc-edit-GenCount}) command is a convenient | |
23321 | way to create or edit this variable. Press @kbd{M-# M-#} to finish. | |
23322 | ||
23323 | If you have not stored a value in @code{GenCount}, or if the value | |
23324 | in that variable is not a positive integer, the regular | |
23325 | @code{s1}/@code{n1} notation is used. | |
23326 | ||
23327 | @kindex I a S | |
23328 | @kindex H I a S | |
23329 | @tindex finv | |
23330 | @tindex ffinv | |
23331 | With the Inverse flag, @kbd{I a S} [@code{finv}] treats the expression | |
23332 | on top of the stack as a function of the specified variable and solves | |
23333 | to find the inverse function, written in terms of the same variable. | |
23334 | For example, @kbd{I a S x} inverts @cite{2x + 6} to @cite{x/2 - 3}. | |
23335 | You can use both Inverse and Hyperbolic [@code{ffinv}] to obtain a | |
23336 | fully general inverse, as described above. | |
23337 | ||
23338 | @kindex a P | |
23339 | @pindex calc-poly-roots | |
23340 | @tindex roots | |
23341 | Some equations, specifically polynomials, have a known, finite number | |
23342 | of solutions. The @kbd{a P} (@code{calc-poly-roots}) [@code{roots}] | |
23343 | command uses @kbd{H a S} to solve an equation in general form, then, for | |
23344 | all arbitrary-sign variables like @code{s1}, and all arbitrary-integer | |
23345 | variables like @code{n1} for which @code{n1} only usefully varies over | |
23346 | a finite range, it expands these variables out to all their possible | |
23347 | values. The results are collected into a vector, which is returned. | |
23348 | For example, @samp{roots(x^4 = 1, x)} returns the four solutions | |
23349 | @samp{[1, -1, (0, 1), (0, -1)]}. Generally an @var{n}th degree | |
23350 | polynomial will always have @var{n} roots on the complex plane. | |
23351 | (If you have given a @code{real} declaration for the solution | |
23352 | variable, then only the real-valued solutions, if any, will be | |
23353 | reported; @pxref{Declarations}.) | |
23354 | ||
23355 | Note that because @kbd{a P} uses @kbd{H a S}, it is able to deliver | |
23356 | symbolic solutions if the polynomial has symbolic coefficients. Also | |
23357 | note that Calc's solver is not able to get exact symbolic solutions | |
23358 | to all polynomials. Polynomials containing powers up to @cite{x^4} | |
23359 | can always be solved exactly; polynomials of higher degree sometimes | |
23360 | can be: @cite{x^6 + x^3 + 1} is converted to @cite{(x^3)^2 + (x^3) + 1}, | |
23361 | which can be solved for @cite{x^3} using the quadratic equation, and then | |
23362 | for @cite{x} by taking cube roots. But in many cases, like | |
23363 | @cite{x^6 + x + 1}, Calc does not know how to rewrite the polynomial | |
23364 | into a form it can solve. The @kbd{a P} command can still deliver a | |
23365 | list of numerical roots, however, provided that symbolic mode (@kbd{m s}) | |
23366 | is not turned on. (If you work with symbolic mode on, recall that the | |
23367 | @kbd{N} (@code{calc-eval-num}) key is a handy way to reevaluate the | |
23368 | formula on the stack with symbolic mode temporarily off.) Naturally, | |
28665d46 | 23369 | @kbd{a P} can only provide numerical roots if the polynomial coefficients |
d7b8e6c6 EZ |
23370 | are all numbers (real or complex). |
23371 | ||
23372 | @node Solving Systems of Equations, Decomposing Polynomials, Multiple Solutions, Solving Equations | |
23373 | @subsection Solving Systems of Equations | |
23374 | ||
23375 | @noindent | |
23376 | @cindex Systems of equations, symbolic | |
23377 | You can also use the commands described above to solve systems of | |
23378 | simultaneous equations. Just create a vector of equations, then | |
23379 | specify a vector of variables for which to solve. (You can omit | |
23380 | the surrounding brackets when entering the vector of variables | |
23381 | at the prompt.) | |
23382 | ||
23383 | For example, putting @samp{[x + y = a, x - y = b]} on the stack | |
23384 | and typing @kbd{a S x,y @key{RET}} produces the vector of solutions | |
23385 | @samp{[x = a - (a-b)/2, y = (a-b)/2]}. The result vector will | |
23386 | have the same length as the variables vector, and the variables | |
23387 | will be listed in the same order there. Note that the solutions | |
23388 | are not always simplified as far as possible; the solution for | |
23389 | @cite{x} here could be improved by an application of the @kbd{a n} | |
23390 | command. | |
23391 | ||
23392 | Calc's algorithm works by trying to eliminate one variable at a | |
23393 | time by solving one of the equations for that variable and then | |
23394 | substituting into the other equations. Calc will try all the | |
23395 | possibilities, but you can speed things up by noting that Calc | |
23396 | first tries to eliminate the first variable with the first | |
23397 | equation, then the second variable with the second equation, | |
23398 | and so on. It also helps to put the simpler (e.g., more linear) | |
23399 | equations toward the front of the list. Calc's algorithm will | |
23400 | solve any system of linear equations, and also many kinds of | |
23401 | nonlinear systems. | |
23402 | ||
5d67986c RS |
23403 | @ignore |
23404 | @starindex | |
23405 | @end ignore | |
d7b8e6c6 EZ |
23406 | @tindex elim |
23407 | Normally there will be as many variables as equations. If you | |
23408 | give fewer variables than equations (an ``over-determined'' system | |
23409 | of equations), Calc will find a partial solution. For example, | |
23410 | typing @kbd{a S y @key{RET}} with the above system of equations | |
23411 | would produce @samp{[y = a - x]}. There are now several ways to | |
23412 | express this solution in terms of the original variables; Calc uses | |
23413 | the first one that it finds. You can control the choice by adding | |
23414 | variable specifiers of the form @samp{elim(@var{v})} to the | |
23415 | variables list. This says that @var{v} should be eliminated from | |
23416 | the equations; the variable will not appear at all in the solution. | |
23417 | For example, typing @kbd{a S y,elim(x)} would yield | |
23418 | @samp{[y = a - (b+a)/2]}. | |
23419 | ||
23420 | If the variables list contains only @code{elim} specifiers, | |
23421 | Calc simply eliminates those variables from the equations | |
23422 | and then returns the resulting set of equations. For example, | |
23423 | @kbd{a S elim(x)} produces @samp{[a - 2 y = b]}. Every variable | |
23424 | eliminated will reduce the number of equations in the system | |
23425 | by one. | |
23426 | ||
23427 | Again, @kbd{a S} gives you one solution to the system of | |
23428 | equations. If there are several solutions, you can use @kbd{H a S} | |
23429 | to get a general family of solutions, or, if there is a finite | |
23430 | number of solutions, you can use @kbd{a P} to get a list. (In | |
23431 | the latter case, the result will take the form of a matrix where | |
23432 | the rows are different solutions and the columns correspond to the | |
23433 | variables you requested.) | |
23434 | ||
23435 | Another way to deal with certain kinds of overdetermined systems of | |
23436 | equations is the @kbd{a F} command, which does least-squares fitting | |
23437 | to satisfy the equations. @xref{Curve Fitting}. | |
23438 | ||
23439 | @node Decomposing Polynomials, , Solving Systems of Equations, Solving Equations | |
23440 | @subsection Decomposing Polynomials | |
23441 | ||
23442 | @noindent | |
5d67986c RS |
23443 | @ignore |
23444 | @starindex | |
23445 | @end ignore | |
d7b8e6c6 EZ |
23446 | @tindex poly |
23447 | The @code{poly} function takes a polynomial and a variable as | |
23448 | arguments, and returns a vector of polynomial coefficients (constant | |
23449 | coefficient first). For example, @samp{poly(x^3 + 2 x, x)} returns | |
23450 | @cite{[0, 2, 0, 1]}. If the input is not a polynomial in @cite{x}, | |
23451 | the call to @code{poly} is left in symbolic form. If the input does | |
23452 | not involve the variable @cite{x}, the input is returned in a list | |
23453 | of length one, representing a polynomial with only a constant | |
23454 | coefficient. The call @samp{poly(x, x)} returns the vector @cite{[0, 1]}. | |
23455 | The last element of the returned vector is guaranteed to be nonzero; | |
23456 | note that @samp{poly(0, x)} returns the empty vector @cite{[]}. | |
23457 | Note also that @cite{x} may actually be any formula; for example, | |
23458 | @samp{poly(sin(x)^2 - sin(x) + 3, sin(x))} returns @cite{[3, -1, 1]}. | |
23459 | ||
23460 | @cindex Coefficients of polynomial | |
23461 | @cindex Degree of polynomial | |
23462 | To get the @cite{x^k} coefficient of polynomial @cite{p}, use | |
23463 | @samp{poly(p, x)_(k+1)}. To get the degree of polynomial @cite{p}, | |
23464 | use @samp{vlen(poly(p, x)) - 1}. For example, @samp{poly((x+1)^4, x)} | |
23465 | returns @samp{[1, 4, 6, 4, 1]}, so @samp{poly((x+1)^4, x)_(2+1)} | |
23466 | gives the @cite{x^2} coefficient of this polynomial, 6. | |
23467 | ||
5d67986c RS |
23468 | @ignore |
23469 | @starindex | |
23470 | @end ignore | |
d7b8e6c6 EZ |
23471 | @tindex gpoly |
23472 | One important feature of the solver is its ability to recognize | |
23473 | formulas which are ``essentially'' polynomials. This ability is | |
23474 | made available to the user through the @code{gpoly} function, which | |
23475 | is used just like @code{poly}: @samp{gpoly(@var{expr}, @var{var})}. | |
23476 | If @var{expr} is a polynomial in some term which includes @var{var}, then | |
23477 | this function will return a vector @samp{[@var{x}, @var{c}, @var{a}]} | |
23478 | where @var{x} is the term that depends on @var{var}, @var{c} is a | |
23479 | vector of polynomial coefficients (like the one returned by @code{poly}), | |
23480 | and @var{a} is a multiplier which is usually 1. Basically, | |
23481 | @samp{@var{expr} = @var{a}*(@var{c}_1 + @var{c}_2 @var{x} + | |
23482 | @var{c}_3 @var{x}^2 + ...)}. The last element of @var{c} is | |
23483 | guaranteed to be non-zero, and @var{c} will not equal @samp{[1]} | |
23484 | (i.e., the trivial decomposition @var{expr} = @var{x} is not | |
23485 | considered a polynomial). One side effect is that @samp{gpoly(x, x)} | |
23486 | and @samp{gpoly(6, x)}, both of which might be expected to recognize | |
23487 | their arguments as polynomials, will not because the decomposition | |
23488 | is considered trivial. | |
23489 | ||
23490 | For example, @samp{gpoly((x-2)^2, x)} returns @samp{[x, [4, -4, 1], 1]}, | |
23491 | since the expanded form of this polynomial is @cite{4 - 4 x + x^2}. | |
23492 | ||
23493 | The term @var{x} may itself be a polynomial in @var{var}. This is | |
23494 | done to reduce the size of the @var{c} vector. For example, | |
23495 | @samp{gpoly(x^4 + x^2 - 1, x)} returns @samp{[x^2, [-1, 1, 1], 1]}, | |
23496 | since a quadratic polynomial in @cite{x^2} is easier to solve than | |
23497 | a quartic polynomial in @cite{x}. | |
23498 | ||
23499 | A few more examples of the kinds of polynomials @code{gpoly} can | |
23500 | discover: | |
23501 | ||
23502 | @smallexample | |
23503 | sin(x) - 1 [sin(x), [-1, 1], 1] | |
23504 | x + 1/x - 1 [x, [1, -1, 1], 1/x] | |
23505 | x + 1/x [x^2, [1, 1], 1/x] | |
23506 | x^3 + 2 x [x^2, [2, 1], x] | |
23507 | x + x^2:3 + sqrt(x) [x^1:6, [1, 1, 0, 1], x^1:2] | |
23508 | x^(2a) + 2 x^a + 5 [x^a, [5, 2, 1], 1] | |
23509 | (exp(-x) + exp(x)) / 2 [e^(2 x), [0.5, 0.5], e^-x] | |
23510 | @end smallexample | |
23511 | ||
23512 | The @code{poly} and @code{gpoly} functions accept a third integer argument | |
23513 | which specifies the largest degree of polynomial that is acceptable. | |
23514 | If this is @cite{n}, then only @var{c} vectors of length @cite{n+1} | |
23515 | or less will be returned. Otherwise, the @code{poly} or @code{gpoly} | |
23516 | call will remain in symbolic form. For example, the equation solver | |
23517 | can handle quartics and smaller polynomials, so it calls | |
23518 | @samp{gpoly(@var{expr}, @var{var}, 4)} to discover whether @var{expr} | |
23519 | can be treated by its linear, quadratic, cubic, or quartic formulas. | |
23520 | ||
5d67986c RS |
23521 | @ignore |
23522 | @starindex | |
23523 | @end ignore | |
d7b8e6c6 EZ |
23524 | @tindex pdeg |
23525 | The @code{pdeg} function computes the degree of a polynomial; | |
23526 | @samp{pdeg(p,x)} is the highest power of @code{x} that appears in | |
23527 | @code{p}. This is the same as @samp{vlen(poly(p,x))-1}, but is | |
23528 | much more efficient. If @code{p} is constant with respect to @code{x}, | |
23529 | then @samp{pdeg(p,x) = 0}. If @code{p} is not a polynomial in @code{x} | |
23530 | (e.g., @samp{pdeg(2 cos(x), x)}, the function remains unevaluated. | |
23531 | It is possible to omit the second argument @code{x}, in which case | |
23532 | @samp{pdeg(p)} returns the highest total degree of any term of the | |
23533 | polynomial, counting all variables that appear in @code{p}. Note | |
23534 | that @code{pdeg(c) = pdeg(c,x) = 0} for any nonzero constant @code{c}; | |
23535 | the degree of the constant zero is considered to be @code{-inf} | |
23536 | (minus infinity). | |
23537 | ||
5d67986c RS |
23538 | @ignore |
23539 | @starindex | |
23540 | @end ignore | |
d7b8e6c6 EZ |
23541 | @tindex plead |
23542 | The @code{plead} function finds the leading term of a polynomial. | |
23543 | Thus @samp{plead(p,x)} is equivalent to @samp{poly(p,x)_vlen(poly(p,x))}, | |
23544 | though again more efficient. In particular, @samp{plead((2x+1)^10, x)} | |
23545 | returns 1024 without expanding out the list of coefficients. The | |
23546 | value of @code{plead(p,x)} will be zero only if @cite{p = 0}. | |
23547 | ||
5d67986c RS |
23548 | @ignore |
23549 | @starindex | |
23550 | @end ignore | |
d7b8e6c6 EZ |
23551 | @tindex pcont |
23552 | The @code{pcont} function finds the @dfn{content} of a polynomial. This | |
23553 | is the greatest common divisor of all the coefficients of the polynomial. | |
23554 | With two arguments, @code{pcont(p,x)} effectively uses @samp{poly(p,x)} | |
23555 | to get a list of coefficients, then uses @code{pgcd} (the polynomial | |
23556 | GCD function) to combine these into an answer. For example, | |
23557 | @samp{pcont(4 x y^2 + 6 x^2 y, x)} is @samp{2 y}. The content is | |
23558 | basically the ``biggest'' polynomial that can be divided into @code{p} | |
23559 | exactly. The sign of the content is the same as the sign of the leading | |
23560 | coefficient. | |
23561 | ||
23562 | With only one argument, @samp{pcont(p)} computes the numerical | |
23563 | content of the polynomial, i.e., the @code{gcd} of the numerical | |
23564 | coefficients of all the terms in the formula. Note that @code{gcd} | |
23565 | is defined on rational numbers as well as integers; it computes | |
23566 | the @code{gcd} of the numerators and the @code{lcm} of the | |
23567 | denominators. Thus @samp{pcont(4:3 x y^2 + 6 x^2 y)} returns 2:3. | |
23568 | Dividing the polynomial by this number will clear all the | |
23569 | denominators, as well as dividing by any common content in the | |
23570 | numerators. The numerical content of a polynomial is negative only | |
23571 | if all the coefficients in the polynomial are negative. | |
23572 | ||
5d67986c RS |
23573 | @ignore |
23574 | @starindex | |
23575 | @end ignore | |
d7b8e6c6 EZ |
23576 | @tindex pprim |
23577 | The @code{pprim} function finds the @dfn{primitive part} of a | |
23578 | polynomial, which is simply the polynomial divided (using @code{pdiv} | |
23579 | if necessary) by its content. If the input polynomial has rational | |
23580 | coefficients, the result will have integer coefficients in simplest | |
23581 | terms. | |
23582 | ||
23583 | @node Numerical Solutions, Curve Fitting, Solving Equations, Algebra | |
23584 | @section Numerical Solutions | |
23585 | ||
23586 | @noindent | |
23587 | Not all equations can be solved symbolically. The commands in this | |
23588 | section use numerical algorithms that can find a solution to a specific | |
23589 | instance of an equation to any desired accuracy. Note that the | |
23590 | numerical commands are slower than their algebraic cousins; it is a | |
23591 | good idea to try @kbd{a S} before resorting to these commands. | |
23592 | ||
23593 | (@xref{Curve Fitting}, for some other, more specialized, operations | |
23594 | on numerical data.) | |
23595 | ||
23596 | @menu | |
23597 | * Root Finding:: | |
23598 | * Minimization:: | |
23599 | * Numerical Systems of Equations:: | |
23600 | @end menu | |
23601 | ||
23602 | @node Root Finding, Minimization, Numerical Solutions, Numerical Solutions | |
23603 | @subsection Root Finding | |
23604 | ||
23605 | @noindent | |
23606 | @kindex a R | |
23607 | @pindex calc-find-root | |
23608 | @tindex root | |
23609 | @cindex Newton's method | |
23610 | @cindex Roots of equations | |
23611 | @cindex Numerical root-finding | |
23612 | The @kbd{a R} (@code{calc-find-root}) [@code{root}] command finds a | |
23613 | numerical solution (or @dfn{root}) of an equation. (This command treats | |
23614 | inequalities the same as equations. If the input is any other kind | |
23615 | of formula, it is interpreted as an equation of the form @cite{X = 0}.) | |
23616 | ||
23617 | The @kbd{a R} command requires an initial guess on the top of the | |
23618 | stack, and a formula in the second-to-top position. It prompts for a | |
23619 | solution variable, which must appear in the formula. All other variables | |
23620 | that appear in the formula must have assigned values, i.e., when | |
23621 | a value is assigned to the solution variable and the formula is | |
23622 | evaluated with @kbd{=}, it should evaluate to a number. Any assigned | |
23623 | value for the solution variable itself is ignored and unaffected by | |
23624 | this command. | |
23625 | ||
23626 | When the command completes, the initial guess is replaced on the stack | |
23627 | by a vector of two numbers: The value of the solution variable that | |
23628 | solves the equation, and the difference between the lefthand and | |
23629 | righthand sides of the equation at that value. Ordinarily, the second | |
23630 | number will be zero or very nearly zero. (Note that Calc uses a | |
23631 | slightly higher precision while finding the root, and thus the second | |
23632 | number may be slightly different from the value you would compute from | |
23633 | the equation yourself.) | |
23634 | ||
23635 | The @kbd{v h} (@code{calc-head}) command is a handy way to extract | |
23636 | the first element of the result vector, discarding the error term. | |
23637 | ||
23638 | The initial guess can be a real number, in which case Calc searches | |
23639 | for a real solution near that number, or a complex number, in which | |
23640 | case Calc searches the whole complex plane near that number for a | |
23641 | solution, or it can be an interval form which restricts the search | |
23642 | to real numbers inside that interval. | |
23643 | ||
23644 | Calc tries to use @kbd{a d} to take the derivative of the equation. | |
23645 | If this succeeds, it uses Newton's method. If the equation is not | |
23646 | differentiable Calc uses a bisection method. (If Newton's method | |
23647 | appears to be going astray, Calc switches over to bisection if it | |
23648 | can, or otherwise gives up. In this case it may help to try again | |
23649 | with a slightly different initial guess.) If the initial guess is a | |
23650 | complex number, the function must be differentiable. | |
23651 | ||
23652 | If the formula (or the difference between the sides of an equation) | |
23653 | is negative at one end of the interval you specify and positive at | |
23654 | the other end, the root finder is guaranteed to find a root. | |
23655 | Otherwise, Calc subdivides the interval into small parts looking for | |
23656 | positive and negative values to bracket the root. When your guess is | |
23657 | an interval, Calc will not look outside that interval for a root. | |
23658 | ||
23659 | @kindex H a R | |
23660 | @tindex wroot | |
23661 | The @kbd{H a R} [@code{wroot}] command is similar to @kbd{a R}, except | |
23662 | that if the initial guess is an interval for which the function has | |
23663 | the same sign at both ends, then rather than subdividing the interval | |
23664 | Calc attempts to widen it to enclose a root. Use this mode if | |
23665 | you are not sure if the function has a root in your interval. | |
23666 | ||
23667 | If the function is not differentiable, and you give a simple number | |
23668 | instead of an interval as your initial guess, Calc uses this widening | |
23669 | process even if you did not type the Hyperbolic flag. (If the function | |
23670 | @emph{is} differentiable, Calc uses Newton's method which does not | |
23671 | require a bounding interval in order to work.) | |
23672 | ||
23673 | If Calc leaves the @code{root} or @code{wroot} function in symbolic | |
23674 | form on the stack, it will normally display an explanation for why | |
23675 | no root was found. If you miss this explanation, press @kbd{w} | |
23676 | (@code{calc-why}) to get it back. | |
23677 | ||
23678 | @node Minimization, Numerical Systems of Equations, Root Finding, Numerical Solutions | |
23679 | @subsection Minimization | |
23680 | ||
23681 | @noindent | |
23682 | @kindex a N | |
23683 | @kindex H a N | |
23684 | @kindex a X | |
23685 | @kindex H a X | |
23686 | @pindex calc-find-minimum | |
23687 | @pindex calc-find-maximum | |
23688 | @tindex minimize | |
23689 | @tindex maximize | |
23690 | @cindex Minimization, numerical | |
23691 | The @kbd{a N} (@code{calc-find-minimum}) [@code{minimize}] command | |
23692 | finds a minimum value for a formula. It is very similar in operation | |
23693 | to @kbd{a R} (@code{calc-find-root}): You give the formula and an initial | |
23694 | guess on the stack, and are prompted for the name of a variable. The guess | |
23695 | may be either a number near the desired minimum, or an interval enclosing | |
23696 | the desired minimum. The function returns a vector containing the | |
269b7745 | 23697 | value of the variable which minimizes the formula's value, along |
d7b8e6c6 EZ |
23698 | with the minimum value itself. |
23699 | ||
23700 | Note that this command looks for a @emph{local} minimum. Many functions | |
23701 | have more than one minimum; some, like @c{$x \sin x$} | |
23702 | @cite{x sin(x)}, have infinitely | |
23703 | many. In fact, there is no easy way to define the ``global'' minimum | |
23704 | of @c{$x \sin x$} | |
23705 | @cite{x sin(x)} but Calc can still locate any particular local minimum | |
23706 | for you. Calc basically goes downhill from the initial guess until it | |
23707 | finds a point at which the function's value is greater both to the left | |
23708 | and to the right. Calc does not use derivatives when minimizing a function. | |
23709 | ||
23710 | If your initial guess is an interval and it looks like the minimum | |
23711 | occurs at one or the other endpoint of the interval, Calc will return | |
23712 | that endpoint only if that endpoint is closed; thus, minimizing @cite{17 x} | |
23713 | over @cite{[2..3]} will return @cite{[2, 38]}, but minimizing over | |
23714 | @cite{(2..3]} would report no minimum found. In general, you should | |
23715 | use closed intervals to find literally the minimum value in that | |
23716 | range of @cite{x}, or open intervals to find the local minimum, if | |
23717 | any, that happens to lie in that range. | |
23718 | ||
23719 | Most functions are smooth and flat near their minimum values. Because | |
23720 | of this flatness, if the current precision is, say, 12 digits, the | |
23721 | variable can only be determined meaningfully to about six digits. Thus | |
23722 | you should set the precision to twice as many digits as you need in your | |
23723 | answer. | |
23724 | ||
5d67986c RS |
23725 | @ignore |
23726 | @mindex wmin@idots | |
23727 | @end ignore | |
d7b8e6c6 | 23728 | @tindex wminimize |
5d67986c RS |
23729 | @ignore |
23730 | @mindex wmax@idots | |
23731 | @end ignore | |
d7b8e6c6 EZ |
23732 | @tindex wmaximize |
23733 | The @kbd{H a N} [@code{wminimize}] command, analogously to @kbd{H a R}, | |
23734 | expands the guess interval to enclose a minimum rather than requiring | |
23735 | that the minimum lie inside the interval you supply. | |
23736 | ||
23737 | The @kbd{a X} (@code{calc-find-maximum}) [@code{maximize}] and | |
23738 | @kbd{H a X} [@code{wmaximize}] commands effectively minimize the | |
23739 | negative of the formula you supply. | |
23740 | ||
23741 | The formula must evaluate to a real number at all points inside the | |
23742 | interval (or near the initial guess if the guess is a number). If | |
23743 | the initial guess is a complex number the variable will be minimized | |
23744 | over the complex numbers; if it is real or an interval it will | |
23745 | be minimized over the reals. | |
23746 | ||
23747 | @node Numerical Systems of Equations, , Minimization, Numerical Solutions | |
23748 | @subsection Systems of Equations | |
23749 | ||
23750 | @noindent | |
23751 | @cindex Systems of equations, numerical | |
23752 | The @kbd{a R} command can also solve systems of equations. In this | |
23753 | case, the equation should instead be a vector of equations, the | |
23754 | guess should instead be a vector of numbers (intervals are not | |
23755 | supported), and the variable should be a vector of variables. You | |
23756 | can omit the brackets while entering the list of variables. Each | |
23757 | equation must be differentiable by each variable for this mode to | |
23758 | work. The result will be a vector of two vectors: The variable | |
23759 | values that solved the system of equations, and the differences | |
23760 | between the sides of the equations with those variable values. | |
23761 | There must be the same number of equations as variables. Since | |
23762 | only plain numbers are allowed as guesses, the Hyperbolic flag has | |
23763 | no effect when solving a system of equations. | |
23764 | ||
23765 | It is also possible to minimize over many variables with @kbd{a N} | |
23766 | (or maximize with @kbd{a X}). Once again the variable name should | |
23767 | be replaced by a vector of variables, and the initial guess should | |
23768 | be an equal-sized vector of initial guesses. But, unlike the case of | |
23769 | multidimensional @kbd{a R}, the formula being minimized should | |
23770 | still be a single formula, @emph{not} a vector. Beware that | |
23771 | multidimensional minimization is currently @emph{very} slow. | |
23772 | ||
23773 | @node Curve Fitting, Summations, Numerical Solutions, Algebra | |
23774 | @section Curve Fitting | |
23775 | ||
23776 | @noindent | |
23777 | The @kbd{a F} command fits a set of data to a @dfn{model formula}, | |
23778 | such as @cite{y = m x + b} where @cite{m} and @cite{b} are parameters | |
23779 | to be determined. For a typical set of measured data there will be | |
23780 | no single @cite{m} and @cite{b} that exactly fit the data; in this | |
23781 | case, Calc chooses values of the parameters that provide the closest | |
23782 | possible fit. | |
23783 | ||
23784 | @menu | |
23785 | * Linear Fits:: | |
23786 | * Polynomial and Multilinear Fits:: | |
23787 | * Error Estimates for Fits:: | |
23788 | * Standard Nonlinear Models:: | |
23789 | * Curve Fitting Details:: | |
23790 | * Interpolation:: | |
23791 | @end menu | |
23792 | ||
23793 | @node Linear Fits, Polynomial and Multilinear Fits, Curve Fitting, Curve Fitting | |
23794 | @subsection Linear Fits | |
23795 | ||
23796 | @noindent | |
23797 | @kindex a F | |
23798 | @pindex calc-curve-fit | |
23799 | @tindex fit | |
23800 | @cindex Linear regression | |
23801 | @cindex Least-squares fits | |
23802 | The @kbd{a F} (@code{calc-curve-fit}) [@code{fit}] command attempts | |
23803 | to fit a set of data (@cite{x} and @cite{y} vectors of numbers) to a | |
23804 | straight line, polynomial, or other function of @cite{x}. For the | |
23805 | moment we will consider only the case of fitting to a line, and we | |
23806 | will ignore the issue of whether or not the model was in fact a good | |
23807 | fit for the data. | |
23808 | ||
23809 | In a standard linear least-squares fit, we have a set of @cite{(x,y)} | |
23810 | data points that we wish to fit to the model @cite{y = m x + b} | |
23811 | by adjusting the parameters @cite{m} and @cite{b} to make the @cite{y} | |
23812 | values calculated from the formula be as close as possible to the actual | |
23813 | @cite{y} values in the data set. (In a polynomial fit, the model is | |
23814 | instead, say, @cite{y = a x^3 + b x^2 + c x + d}. In a multilinear fit, | |
23815 | we have data points of the form @cite{(x_1,x_2,x_3,y)} and our model is | |
23816 | @cite{y = a x_1 + b x_2 + c x_3 + d}. These will be discussed later.) | |
23817 | ||
23818 | In the model formula, variables like @cite{x} and @cite{x_2} are called | |
23819 | the @dfn{independent variables}, and @cite{y} is the @dfn{dependent | |
23820 | variable}. Variables like @cite{m}, @cite{a}, and @cite{b} are called | |
23821 | the @dfn{parameters} of the model. | |
23822 | ||
23823 | The @kbd{a F} command takes the data set to be fitted from the stack. | |
23824 | By default, it expects the data in the form of a matrix. For example, | |
23825 | for a linear or polynomial fit, this would be a @c{$2\times N$} | |
23826 | @asis{2xN} matrix where | |
23827 | the first row is a list of @cite{x} values and the second row has the | |
23828 | corresponding @cite{y} values. For the multilinear fit shown above, | |
23829 | the matrix would have four rows (@cite{x_1}, @cite{x_2}, @cite{x_3}, and | |
23830 | @cite{y}, respectively). | |
23831 | ||
23832 | If you happen to have an @c{$N\times2$} | |
23833 | @asis{Nx2} matrix instead of a @c{$2\times N$} | |
23834 | @asis{2xN} matrix, | |
23835 | just press @kbd{v t} first to transpose the matrix. | |
23836 | ||
23837 | After you type @kbd{a F}, Calc prompts you to select a model. For a | |
23838 | linear fit, press the digit @kbd{1}. | |
23839 | ||
23840 | Calc then prompts for you to name the variables. By default it chooses | |
23841 | high letters like @cite{x} and @cite{y} for independent variables and | |
23842 | low letters like @cite{a} and @cite{b} for parameters. (The dependent | |
23843 | variable doesn't need a name.) The two kinds of variables are separated | |
23844 | by a semicolon. Since you generally care more about the names of the | |
23845 | independent variables than of the parameters, Calc also allows you to | |
23846 | name only those and let the parameters use default names. | |
23847 | ||
23848 | For example, suppose the data matrix | |
23849 | ||
23850 | @ifinfo | |
d7b8e6c6 | 23851 | @example |
5d67986c | 23852 | @group |
d7b8e6c6 EZ |
23853 | [ [ 1, 2, 3, 4, 5 ] |
23854 | [ 5, 7, 9, 11, 13 ] ] | |
d7b8e6c6 | 23855 | @end group |
5d67986c | 23856 | @end example |
d7b8e6c6 EZ |
23857 | @end ifinfo |
23858 | @tex | |
23859 | \turnoffactive | |
23860 | \turnoffactive | |
23861 | \beforedisplay | |
23862 | $$ \pmatrix{ 1 & 2 & 3 & 4 & 5 \cr | |
23863 | 5 & 7 & 9 & 11 & 13 } | |
23864 | $$ | |
23865 | \afterdisplay | |
23866 | @end tex | |
23867 | ||
23868 | @noindent | |
23869 | is on the stack and we wish to do a simple linear fit. Type | |
5d67986c | 23870 | @kbd{a F}, then @kbd{1} for the model, then @key{RET} to use |
d7b8e6c6 EZ |
23871 | the default names. The result will be the formula @cite{3 + 2 x} |
23872 | on the stack. Calc has created the model expression @kbd{a + b x}, | |
23873 | then found the optimal values of @cite{a} and @cite{b} to fit the | |
23874 | data. (In this case, it was able to find an exact fit.) Calc then | |
23875 | substituted those values for @cite{a} and @cite{b} in the model | |
23876 | formula. | |
23877 | ||
23878 | The @kbd{a F} command puts two entries in the trail. One is, as | |
23879 | always, a copy of the result that went to the stack; the other is | |
23880 | a vector of the actual parameter values, written as equations: | |
23881 | @cite{[a = 3, b = 2]}, in case you'd rather read them in a list | |
23882 | than pick them out of the formula. (You can type @kbd{t y} | |
b275eac7 | 23883 | to move this vector to the stack; see @ref{Trail Commands}. |
d7b8e6c6 EZ |
23884 | |
23885 | Specifying a different independent variable name will affect the | |
5d67986c RS |
23886 | resulting formula: @kbd{a F 1 k @key{RET}} produces @kbd{3 + 2 k}. |
23887 | Changing the parameter names (say, @kbd{a F 1 k;b,m @key{RET}}) will affect | |
d7b8e6c6 EZ |
23888 | the equations that go into the trail. |
23889 | ||
23890 | @tex | |
23891 | \bigskip | |
23892 | @end tex | |
23893 | ||
23894 | To see what happens when the fit is not exact, we could change | |
23895 | the number 13 in the data matrix to 14 and try the fit again. | |
23896 | The result is: | |
23897 | ||
23898 | @example | |
23899 | 2.6 + 2.2 x | |
23900 | @end example | |
23901 | ||
5d67986c | 23902 | Evaluating this formula, say with @kbd{v x 5 @key{RET} @key{TAB} V M $ @key{RET}}, shows |
d7b8e6c6 EZ |
23903 | a reasonably close match to the y-values in the data. |
23904 | ||
23905 | @example | |
23906 | [4.8, 7., 9.2, 11.4, 13.6] | |
23907 | @end example | |
23908 | ||
5d67986c | 23909 | Since there is no line which passes through all the @var{n} data points, |
d7b8e6c6 EZ |
23910 | Calc has chosen a line that best approximates the data points using |
23911 | the method of least squares. The idea is to define the @dfn{chi-square} | |
23912 | error measure | |
23913 | ||
23914 | @ifinfo | |
23915 | @example | |
23916 | chi^2 = sum((y_i - (a + b x_i))^2, i, 1, N) | |
23917 | @end example | |
23918 | @end ifinfo | |
23919 | @tex | |
23920 | \turnoffactive | |
23921 | \beforedisplay | |
23922 | $$ \chi^2 = \sum_{i=1}^N (y_i - (a + b x_i))^2 $$ | |
23923 | \afterdisplay | |
23924 | @end tex | |
23925 | ||
23926 | @noindent | |
23927 | which is clearly zero if @cite{a + b x} exactly fits all data points, | |
23928 | and increases as various @cite{a + b x_i} values fail to match the | |
23929 | corresponding @cite{y_i} values. There are several reasons why the | |
23930 | summand is squared, one of them being to ensure that @c{$\chi^2 \ge 0$} | |
23931 | @cite{chi^2 >= 0}. | |
23932 | Least-squares fitting simply chooses the values of @cite{a} and @cite{b} | |
23933 | for which the error @c{$\chi^2$} | |
23934 | @cite{chi^2} is as small as possible. | |
23935 | ||
23936 | Other kinds of models do the same thing but with a different model | |
23937 | formula in place of @cite{a + b x_i}. | |
23938 | ||
23939 | @tex | |
23940 | \bigskip | |
23941 | @end tex | |
23942 | ||
23943 | A numeric prefix argument causes the @kbd{a F} command to take the | |
5d67986c RS |
23944 | data in some other form than one big matrix. A positive argument @var{n} |
23945 | will take @var{N} items from the stack, corresponding to the @var{n} rows | |
23946 | of a data matrix. In the linear case, @var{n} must be 2 since there | |
d7b8e6c6 EZ |
23947 | is always one independent variable and one dependent variable. |
23948 | ||
23949 | A prefix of zero or plain @kbd{C-u} is a compromise; Calc takes two | |
5d67986c | 23950 | items from the stack, an @var{n}-row matrix of @cite{x} values, and a |
d7b8e6c6 EZ |
23951 | vector of @cite{y} values. If there is only one independent variable, |
23952 | the @cite{x} values can be either a one-row matrix or a plain vector, | |
23953 | in which case the @kbd{C-u} prefix is the same as a @w{@kbd{C-u 2}} prefix. | |
23954 | ||
23955 | @node Polynomial and Multilinear Fits, Error Estimates for Fits, Linear Fits, Curve Fitting | |
23956 | @subsection Polynomial and Multilinear Fits | |
23957 | ||
23958 | @noindent | |
23959 | To fit the data to higher-order polynomials, just type one of the | |
23960 | digits @kbd{2} through @kbd{9} when prompted for a model. For example, | |
23961 | we could fit the original data matrix from the previous section | |
23962 | (with 13, not 14) to a parabola instead of a line by typing | |
5d67986c | 23963 | @kbd{a F 2 @key{RET}}. |
d7b8e6c6 EZ |
23964 | |
23965 | @example | |
23966 | 2.00000000001 x - 1.5e-12 x^2 + 2.99999999999 | |
23967 | @end example | |
23968 | ||
23969 | Note that since the constant and linear terms are enough to fit the | |
23970 | data exactly, it's no surprise that Calc chose a tiny contribution | |
23971 | for @cite{x^2}. (The fact that it's not exactly zero is due only | |
23972 | to roundoff error. Since our data are exact integers, we could get | |
23973 | an exact answer by typing @kbd{m f} first to get fraction mode. | |
23974 | Then the @cite{x^2} term would vanish altogether. Usually, though, | |
23975 | the data being fitted will be approximate floats so fraction mode | |
23976 | won't help.) | |
23977 | ||
23978 | Doing the @kbd{a F 2} fit on the data set with 14 instead of 13 | |
23979 | gives a much larger @cite{x^2} contribution, as Calc bends the | |
23980 | line slightly to improve the fit. | |
23981 | ||
23982 | @example | |
23983 | 0.142857142855 x^2 + 1.34285714287 x + 3.59999999998 | |
23984 | @end example | |
23985 | ||
23986 | An important result from the theory of polynomial fitting is that it | |
5d67986c RS |
23987 | is always possible to fit @var{n} data points exactly using a polynomial |
23988 | of degree @i{@var{n}-1}, sometimes called an @dfn{interpolating polynomial}. | |
d7b8e6c6 EZ |
23989 | Using the modified (14) data matrix, a model number of 4 gives |
23990 | a polynomial that exactly matches all five data points: | |
23991 | ||
23992 | @example | |
23993 | 0.04167 x^4 - 0.4167 x^3 + 1.458 x^2 - 0.08333 x + 4. | |
23994 | @end example | |
23995 | ||
23996 | The actual coefficients we get with a precision of 12, like | |
23997 | @cite{0.0416666663588}, clearly suffer from loss of precision. | |
23998 | It is a good idea to increase the working precision to several | |
23999 | digits beyond what you need when you do a fitting operation. | |
24000 | Or, if your data are exact, use fraction mode to get exact | |
24001 | results. | |
24002 | ||
24003 | You can type @kbd{i} instead of a digit at the model prompt to fit | |
24004 | the data exactly to a polynomial. This just counts the number of | |
24005 | columns of the data matrix to choose the degree of the polynomial | |
24006 | automatically. | |
24007 | ||
24008 | Fitting data ``exactly'' to high-degree polynomials is not always | |
24009 | a good idea, though. High-degree polynomials have a tendency to | |
24010 | wiggle uncontrollably in between the fitting data points. Also, | |
24011 | if the exact-fit polynomial is going to be used to interpolate or | |
24012 | extrapolate the data, it is numerically better to use the @kbd{a p} | |
24013 | command described below. @xref{Interpolation}. | |
24014 | ||
24015 | @tex | |
24016 | \bigskip | |
24017 | @end tex | |
24018 | ||
24019 | Another generalization of the linear model is to assume the | |
24020 | @cite{y} values are a sum of linear contributions from several | |
24021 | @cite{x} values. This is a @dfn{multilinear} fit, and it is also | |
24022 | selected by the @kbd{1} digit key. (Calc decides whether the fit | |
24023 | is linear or multilinear by counting the rows in the data matrix.) | |
24024 | ||
24025 | Given the data matrix, | |
24026 | ||
d7b8e6c6 | 24027 | @example |
5d67986c | 24028 | @group |
d7b8e6c6 EZ |
24029 | [ [ 1, 2, 3, 4, 5 ] |
24030 | [ 7, 2, 3, 5, 2 ] | |
24031 | [ 14.5, 15, 18.5, 22.5, 24 ] ] | |
d7b8e6c6 | 24032 | @end group |
5d67986c | 24033 | @end example |
d7b8e6c6 EZ |
24034 | |
24035 | @noindent | |
5d67986c | 24036 | the command @kbd{a F 1 @key{RET}} will call the first row @cite{x} and the |
d7b8e6c6 EZ |
24037 | second row @cite{y}, and will fit the values in the third row to the |
24038 | model @cite{a + b x + c y}. | |
24039 | ||
24040 | @example | |
24041 | 8. + 3. x + 0.5 y | |
24042 | @end example | |
24043 | ||
24044 | Calc can do multilinear fits with any number of independent variables | |
24045 | (i.e., with any number of data rows). | |
24046 | ||
24047 | @tex | |
24048 | \bigskip | |
24049 | @end tex | |
24050 | ||
24051 | Yet another variation is @dfn{homogeneous} linear models, in which | |
24052 | the constant term is known to be zero. In the linear case, this | |
24053 | means the model formula is simply @cite{a x}; in the multilinear | |
24054 | case, the model might be @cite{a x + b y + c z}; and in the polynomial | |
24055 | case, the model could be @cite{a x + b x^2 + c x^3}. You can get | |
24056 | a homogeneous linear or multilinear model by pressing the letter | |
24057 | @kbd{h} followed by a regular model key, like @kbd{1} or @kbd{2}. | |
24058 | ||
24059 | It is certainly possible to have other constrained linear models, | |
24060 | like @cite{2.3 + a x} or @cite{a - 4 x}. While there is no single | |
24061 | key to select models like these, a later section shows how to enter | |
24062 | any desired model by hand. In the first case, for example, you | |
24063 | would enter @kbd{a F ' 2.3 + a x}. | |
24064 | ||
24065 | Another class of models that will work but must be entered by hand | |
24066 | are multinomial fits, e.g., @cite{a + b x + c y + d x^2 + e y^2 + f x y}. | |
24067 | ||
24068 | @node Error Estimates for Fits, Standard Nonlinear Models, Polynomial and Multilinear Fits, Curve Fitting | |
24069 | @subsection Error Estimates for Fits | |
24070 | ||
24071 | @noindent | |
24072 | @kindex H a F | |
24073 | @tindex efit | |
24074 | With the Hyperbolic flag, @kbd{H a F} [@code{efit}] performs the same | |
24075 | fitting operation as @kbd{a F}, but reports the coefficients as error | |
24076 | forms instead of plain numbers. Fitting our two data matrices (first | |
24077 | with 13, then with 14) to a line with @kbd{H a F} gives the results, | |
24078 | ||
24079 | @example | |
24080 | 3. + 2. x | |
24081 | 2.6 +/- 0.382970843103 + 2.2 +/- 0.115470053838 x | |
24082 | @end example | |
24083 | ||
24084 | In the first case the estimated errors are zero because the linear | |
24085 | fit is perfect. In the second case, the errors are nonzero but | |
24086 | moderately small, because the data are still very close to linear. | |
24087 | ||
24088 | It is also possible for the @emph{input} to a fitting operation to | |
24089 | contain error forms. The data values must either all include errors | |
24090 | or all be plain numbers. Error forms can go anywhere but generally | |
24091 | go on the numbers in the last row of the data matrix. If the last | |
24092 | row contains error forms | |
5d67986c RS |
24093 | `@var{y_i}@w{ @t{+/-} }@c{$\sigma_i$} |
24094 | @var{sigma_i}', then the @c{$\chi^2$} | |
d7b8e6c6 EZ |
24095 | @cite{chi^2} |
24096 | statistic is now, | |
24097 | ||
24098 | @ifinfo | |
24099 | @example | |
24100 | chi^2 = sum(((y_i - (a + b x_i)) / sigma_i)^2, i, 1, N) | |
24101 | @end example | |
24102 | @end ifinfo | |
24103 | @tex | |
24104 | \turnoffactive | |
24105 | \beforedisplay | |
24106 | $$ \chi^2 = \sum_{i=1}^N \left(y_i - (a + b x_i) \over \sigma_i\right)^2 $$ | |
24107 | \afterdisplay | |
24108 | @end tex | |
24109 | ||
24110 | @noindent | |
24111 | so that data points with larger error estimates contribute less to | |
24112 | the fitting operation. | |
24113 | ||
24114 | If there are error forms on other rows of the data matrix, all the | |
24115 | errors for a given data point are combined; the square root of the | |
24116 | sum of the squares of the errors forms the @c{$\sigma_i$} | |
24117 | @cite{sigma_i} used for | |
24118 | the data point. | |
24119 | ||
24120 | Both @kbd{a F} and @kbd{H a F} can accept error forms in the input | |
24121 | matrix, although if you are concerned about error analysis you will | |
24122 | probably use @kbd{H a F} so that the output also contains error | |
24123 | estimates. | |
24124 | ||
24125 | If the input contains error forms but all the @c{$\sigma_i$} | |
24126 | @cite{sigma_i} values are | |
24127 | the same, it is easy to see that the resulting fitted model will be | |
24128 | the same as if the input did not have error forms at all (@c{$\chi^2$} | |
24129 | @cite{chi^2} | |
24130 | is simply scaled uniformly by @c{$1 / \sigma^2$} | |
24131 | @cite{1 / sigma^2}, which doesn't affect | |
24132 | where it has a minimum). But there @emph{will} be a difference | |
24133 | in the estimated errors of the coefficients reported by @kbd{H a F}. | |
24134 | ||
28665d46 | 24135 | Consult any text on statistical modeling of data for a discussion |
d7b8e6c6 EZ |
24136 | of where these error estimates come from and how they should be |
24137 | interpreted. | |
24138 | ||
24139 | @tex | |
24140 | \bigskip | |
24141 | @end tex | |
24142 | ||
24143 | @kindex I a F | |
24144 | @tindex xfit | |
24145 | With the Inverse flag, @kbd{I a F} [@code{xfit}] produces even more | |
24146 | information. The result is a vector of six items: | |
24147 | ||
24148 | @enumerate | |
24149 | @item | |
24150 | The model formula with error forms for its coefficients or | |
24151 | parameters. This is the result that @kbd{H a F} would have | |
24152 | produced. | |
24153 | ||
24154 | @item | |
24155 | A vector of ``raw'' parameter values for the model. These are the | |
24156 | polynomial coefficients or other parameters as plain numbers, in the | |
24157 | same order as the parameters appeared in the final prompt of the | |
24158 | @kbd{I a F} command. For polynomials of degree @cite{d}, this vector | |
24159 | will have length @cite{M = d+1} with the constant term first. | |
24160 | ||
24161 | @item | |
24162 | The covariance matrix @cite{C} computed from the fit. This is | |
5d67986c | 24163 | an @var{m}x@var{m} symmetric matrix; the diagonal elements |
d7b8e6c6 EZ |
24164 | @c{$C_{jj}$} |
24165 | @cite{C_j_j} are the variances @c{$\sigma_j^2$} | |
24166 | @cite{sigma_j^2} of the parameters. | |
24167 | The other elements are covariances @c{$\sigma_{ij}^2$} | |
24168 | @cite{sigma_i_j^2} that describe the | |
24169 | correlation between pairs of parameters. (A related set of | |
24170 | numbers, the @dfn{linear correlation coefficients} @c{$r_{ij}$} | |
24171 | @cite{r_i_j}, | |
24172 | are defined as @c{$\sigma_{ij}^2 / \sigma_i \, \sigma_j$} | |
24173 | @cite{sigma_i_j^2 / sigma_i sigma_j}.) | |
24174 | ||
24175 | @item | |
24176 | A vector of @cite{M} ``parameter filter'' functions whose | |
24177 | meanings are described below. If no filters are necessary this | |
24178 | will instead be an empty vector; this is always the case for the | |
24179 | polynomial and multilinear fits described so far. | |
24180 | ||
24181 | @item | |
24182 | The value of @c{$\chi^2$} | |
24183 | @cite{chi^2} for the fit, calculated by the formulas | |
24184 | shown above. This gives a measure of the quality of the fit; | |
24185 | statisticians consider @c{$\chi^2 \approx N - M$} | |
24186 | @cite{chi^2 = N - M} to indicate a moderately good fit | |
24187 | (where again @cite{N} is the number of data points and @cite{M} | |
24188 | is the number of parameters). | |
24189 | ||
24190 | @item | |
24191 | A measure of goodness of fit expressed as a probability @cite{Q}. | |
24192 | This is computed from the @code{utpc} probability distribution | |
24193 | function using @c{$\chi^2$} | |
24194 | @cite{chi^2} with @cite{N - M} degrees of freedom. A | |
24195 | value of 0.5 implies a good fit; some texts recommend that often | |
24196 | @cite{Q = 0.1} or even 0.001 can signify an acceptable fit. In | |
24197 | particular, @c{$\chi^2$} | |
24198 | @cite{chi^2} statistics assume the errors in your inputs | |
24199 | follow a normal (Gaussian) distribution; if they don't, you may | |
24200 | have to accept smaller values of @cite{Q}. | |
24201 | ||
24202 | The @cite{Q} value is computed only if the input included error | |
24203 | estimates. Otherwise, Calc will report the symbol @code{nan} | |
24204 | for @cite{Q}. The reason is that in this case the @c{$\chi^2$} | |
24205 | @cite{chi^2} | |
24206 | value has effectively been used to estimate the original errors | |
24207 | in the input, and thus there is no redundant information left | |
24208 | over to use for a confidence test. | |
24209 | @end enumerate | |
24210 | ||
24211 | @node Standard Nonlinear Models, Curve Fitting Details, Error Estimates for Fits, Curve Fitting | |
24212 | @subsection Standard Nonlinear Models | |
24213 | ||
24214 | @noindent | |
24215 | The @kbd{a F} command also accepts other kinds of models besides | |
24216 | lines and polynomials. Some common models have quick single-key | |
24217 | abbreviations; others must be entered by hand as algebraic formulas. | |
24218 | ||
24219 | Here is a complete list of the standard models recognized by @kbd{a F}: | |
24220 | ||
24221 | @table @kbd | |
24222 | @item 1 | |
24223 | Linear or multilinear. @i{a + b x + c y + d z}. | |
24224 | @item 2-9 | |
24225 | Polynomials. @i{a + b x + c x^2 + d x^3}. | |
24226 | @item e | |
24227 | Exponential. @i{a} @t{exp}@i{(b x)} @t{exp}@i{(c y)}. | |
24228 | @item E | |
24229 | Base-10 exponential. @i{a} @t{10^}@i{(b x)} @t{10^}@i{(c y)}. | |
24230 | @item x | |
24231 | Exponential (alternate notation). @t{exp}@i{(a + b x + c y)}. | |
24232 | @item X | |
24233 | Base-10 exponential (alternate). @t{10^}@i{(a + b x + c y)}. | |
24234 | @item l | |
24235 | Logarithmic. @i{a + b} @t{ln}@i{(x) + c} @t{ln}@i{(y)}. | |
24236 | @item L | |
24237 | Base-10 logarithmic. @i{a + b} @t{log10}@i{(x) + c} @t{log10}@i{(y)}. | |
24238 | @item ^ | |
24239 | General exponential. @i{a b^x c^y}. | |
24240 | @item p | |
24241 | Power law. @i{a x^b y^c}. | |
24242 | @item q | |
24243 | Quadratic. @i{a + b (x-c)^2 + d (x-e)^2}. | |
24244 | @item g | |
24245 | Gaussian. @c{${a \over b \sqrt{2 \pi}} \exp\left( -{1 \over 2} \left( x - c \over b \right)^2 \right)$} | |
24246 | @i{(a / b sqrt(2 pi)) exp(-0.5*((x-c)/b)^2)}. | |
24247 | @end table | |
24248 | ||
24249 | All of these models are used in the usual way; just press the appropriate | |
24250 | letter at the model prompt, and choose variable names if you wish. The | |
24251 | result will be a formula as shown in the above table, with the best-fit | |
24252 | values of the parameters substituted. (You may find it easier to read | |
24253 | the parameter values from the vector that is placed in the trail.) | |
24254 | ||
24255 | All models except Gaussian and polynomials can generalize as shown to any | |
24256 | number of independent variables. Also, all the built-in models have an | |
24257 | additive or multiplicative parameter shown as @cite{a} in the above table | |
24258 | which can be replaced by zero or one, as appropriate, by typing @kbd{h} | |
24259 | before the model key. | |
24260 | ||
24261 | Note that many of these models are essentially equivalent, but express | |
24262 | the parameters slightly differently. For example, @cite{a b^x} and | |
24263 | the other two exponential models are all algebraic rearrangements of | |
24264 | each other. Also, the ``quadratic'' model is just a degree-2 polynomial | |
24265 | with the parameters expressed differently. Use whichever form best | |
24266 | matches the problem. | |
24267 | ||
24268 | The HP-28/48 calculators support four different models for curve | |
24269 | fitting, called @code{LIN}, @code{LOG}, @code{EXP}, and @code{PWR}. | |
24270 | These correspond to Calc models @samp{a + b x}, @samp{a + b ln(x)}, | |
24271 | @samp{a exp(b x)}, and @samp{a x^b}, respectively. In each case, | |
24272 | @cite{a} is what the HP-48 identifies as the ``intercept,'' and | |
24273 | @cite{b} is what it calls the ``slope.'' | |
24274 | ||
24275 | @tex | |
24276 | \bigskip | |
24277 | @end tex | |
24278 | ||
24279 | If the model you want doesn't appear on this list, press @kbd{'} | |
24280 | (the apostrophe key) at the model prompt to enter any algebraic | |
24281 | formula, such as @kbd{m x - b}, as the model. (Not all models | |
24282 | will work, though---see the next section for details.) | |
24283 | ||
24284 | The model can also be an equation like @cite{y = m x + b}. | |
24285 | In this case, Calc thinks of all the rows of the data matrix on | |
24286 | equal terms; this model effectively has two parameters | |
24287 | (@cite{m} and @cite{b}) and two independent variables (@cite{x} | |
24288 | and @cite{y}), with no ``dependent'' variables. Model equations | |
24289 | do not need to take this @cite{y =} form. For example, the | |
24290 | implicit line equation @cite{a x + b y = 1} works fine as a | |
24291 | model. | |
24292 | ||
24293 | When you enter a model, Calc makes an alphabetical list of all | |
24294 | the variables that appear in the model. These are used for the | |
24295 | default parameters, independent variables, and dependent variable | |
24296 | (in that order). If you enter a plain formula (not an equation), | |
24297 | Calc assumes the dependent variable does not appear in the formula | |
24298 | and thus does not need a name. | |
24299 | ||
24300 | For example, if the model formula has the variables @cite{a,mu,sigma,t,x}, | |
24301 | and the data matrix has three rows (meaning two independent variables), | |
24302 | Calc will use @cite{a,mu,sigma} as the default parameters, and the | |
24303 | data rows will be named @cite{t} and @cite{x}, respectively. If you | |
24304 | enter an equation instead of a plain formula, Calc will use @cite{a,mu} | |
24305 | as the parameters, and @cite{sigma,t,x} as the three independent | |
24306 | variables. | |
24307 | ||
24308 | You can, of course, override these choices by entering something | |
24309 | different at the prompt. If you leave some variables out of the list, | |
24310 | those variables must have stored values and those stored values will | |
24311 | be used as constants in the model. (Stored values for the parameters | |
24312 | and independent variables are ignored by the @kbd{a F} command.) | |
24313 | If you list only independent variables, all the remaining variables | |
24314 | in the model formula will become parameters. | |
24315 | ||
24316 | If there are @kbd{$} signs in the model you type, they will stand | |
24317 | for parameters and all other variables (in alphabetical order) | |
24318 | will be independent. Use @kbd{$} for one parameter, @kbd{$$} for | |
24319 | another, and so on. Thus @kbd{$ x + $$} is another way to describe | |
24320 | a linear model. | |
24321 | ||
24322 | If you type a @kbd{$} instead of @kbd{'} at the model prompt itself, | |
24323 | Calc will take the model formula from the stack. (The data must then | |
24324 | appear at the second stack level.) The same conventions are used to | |
24325 | choose which variables in the formula are independent by default and | |
24326 | which are parameters. | |
24327 | ||
24328 | Models taken from the stack can also be expressed as vectors of | |
24329 | two or three elements, @cite{[@var{model}, @var{vars}]} or | |
24330 | @cite{[@var{model}, @var{vars}, @var{params}]}. Each of @var{vars} | |
24331 | and @var{params} may be either a variable or a vector of variables. | |
24332 | (If @var{params} is omitted, all variables in @var{model} except | |
24333 | those listed as @var{vars} are parameters.)@refill | |
24334 | ||
24335 | When you enter a model manually with @kbd{'}, Calc puts a 3-vector | |
24336 | describing the model in the trail so you can get it back if you wish. | |
24337 | ||
24338 | @tex | |
24339 | \bigskip | |
24340 | @end tex | |
24341 | ||
24342 | @vindex Model1 | |
24343 | @vindex Model2 | |
24344 | Finally, you can store a model in one of the Calc variables | |
24345 | @code{Model1} or @code{Model2}, then use this model by typing | |
24346 | @kbd{a F u} or @kbd{a F U} (respectively). The value stored in | |
24347 | the variable can be any of the formats that @kbd{a F $} would | |
24348 | accept for a model on the stack. | |
24349 | ||
24350 | @tex | |
24351 | \bigskip | |
24352 | @end tex | |
24353 | ||
24354 | Calc uses the principal values of inverse functions like @code{ln} | |
24355 | and @code{arcsin} when doing fits. For example, when you enter | |
24356 | the model @samp{y = sin(a t + b)} Calc actually uses the easier | |
24357 | form @samp{arcsin(y) = a t + b}. The @code{arcsin} function always | |
24358 | returns results in the range from @i{-90} to 90 degrees (or the | |
24359 | equivalent range in radians). Suppose you had data that you | |
24360 | believed to represent roughly three oscillations of a sine wave, | |
24361 | so that the argument of the sine might go from zero to @c{$3\times360$} | |
24362 | @i{3*360} degrees. | |
24363 | The above model would appear to be a good way to determine the | |
24364 | true frequency and phase of the sine wave, but in practice it | |
24365 | would fail utterly. The righthand side of the actual model | |
24366 | @samp{arcsin(y) = a t + b} will grow smoothly with @cite{t}, but | |
24367 | the lefthand side will bounce back and forth between @i{-90} and 90. | |
24368 | No values of @cite{a} and @cite{b} can make the two sides match, | |
24369 | even approximately. | |
24370 | ||
24371 | There is no good solution to this problem at present. You could | |
24372 | restrict your data to small enough ranges so that the above problem | |
24373 | doesn't occur (i.e., not straddling any peaks in the sine wave). | |
24374 | Or, in this case, you could use a totally different method such as | |
24375 | Fourier analysis, which is beyond the scope of the @kbd{a F} command. | |
24376 | (Unfortunately, Calc does not currently have any facilities for | |
24377 | taking Fourier and related transforms.) | |
24378 | ||
24379 | @node Curve Fitting Details, Interpolation, Standard Nonlinear Models, Curve Fitting | |
24380 | @subsection Curve Fitting Details | |
24381 | ||
24382 | @noindent | |
24383 | Calc's internal least-squares fitter can only handle multilinear | |
24384 | models. More precisely, it can handle any model of the form | |
24385 | @cite{a f(x,y,z) + b g(x,y,z) + c h(x,y,z)}, where @cite{a,b,c} | |
24386 | are the parameters and @cite{x,y,z} are the independent variables | |
24387 | (of course there can be any number of each, not just three). | |
24388 | ||
24389 | In a simple multilinear or polynomial fit, it is easy to see how | |
24390 | to convert the model into this form. For example, if the model | |
24391 | is @cite{a + b x + c x^2}, then @cite{f(x) = 1}, @cite{g(x) = x}, | |
24392 | and @cite{h(x) = x^2} are suitable functions. | |
24393 | ||
24394 | For other models, Calc uses a variety of algebraic manipulations | |
24395 | to try to put the problem into the form | |
24396 | ||
24397 | @smallexample | |
24398 | Y(x,y,z) = A(a,b,c) F(x,y,z) + B(a,b,c) G(x,y,z) + C(a,b,c) H(x,y,z) | |
24399 | @end smallexample | |
24400 | ||
24401 | @noindent | |
24402 | where @cite{Y,A,B,C,F,G,H} are arbitrary functions. It computes | |
24403 | @cite{Y}, @cite{F}, @cite{G}, and @cite{H} for all the data points, | |
24404 | does a standard linear fit to find the values of @cite{A}, @cite{B}, | |
24405 | and @cite{C}, then uses the equation solver to solve for @cite{a,b,c} | |
24406 | in terms of @cite{A,B,C}. | |
24407 | ||
24408 | A remarkable number of models can be cast into this general form. | |
24409 | We'll look at two examples here to see how it works. The power-law | |
24410 | model @cite{y = a x^b} with two independent variables and two parameters | |
24411 | can be rewritten as follows: | |
24412 | ||
24413 | @example | |
24414 | y = a x^b | |
24415 | y = a exp(b ln(x)) | |
24416 | y = exp(ln(a) + b ln(x)) | |
24417 | ln(y) = ln(a) + b ln(x) | |
24418 | @end example | |
24419 | ||
24420 | @noindent | |
24421 | which matches the desired form with @c{$Y = \ln(y)$} | |
24422 | @cite{Y = ln(y)}, @c{$A = \ln(a)$} | |
24423 | @cite{A = ln(a)}, | |
24424 | @cite{F = 1}, @cite{B = b}, and @c{$G = \ln(x)$} | |
24425 | @cite{G = ln(x)}. Calc thus computes | |
24426 | the logarithms of your @cite{y} and @cite{x} values, does a linear fit | |
24427 | for @cite{A} and @cite{B}, then solves to get @c{$a = \exp(A)$} | |
24428 | @cite{a = exp(A)} and | |
24429 | @cite{b = B}. | |
24430 | ||
24431 | Another interesting example is the ``quadratic'' model, which can | |
24432 | be handled by expanding according to the distributive law. | |
24433 | ||
24434 | @example | |
24435 | y = a + b*(x - c)^2 | |
24436 | y = a + b c^2 - 2 b c x + b x^2 | |
24437 | @end example | |
24438 | ||
24439 | @noindent | |
24440 | which matches with @cite{Y = y}, @cite{A = a + b c^2}, @cite{F = 1}, | |
24441 | @cite{B = -2 b c}, @cite{G = x} (the @i{-2} factor could just as easily | |
24442 | have been put into @cite{G} instead of @cite{B}), @cite{C = b}, and | |
24443 | @cite{H = x^2}. | |
24444 | ||
24445 | The Gaussian model looks quite complicated, but a closer examination | |
24446 | shows that it's actually similar to the quadratic model but with an | |
24447 | exponential that can be brought to the top and moved into @cite{Y}. | |
24448 | ||
24449 | An example of a model that cannot be put into general linear | |
24450 | form is a Gaussian with a constant background added on, i.e., | |
24451 | @cite{d} + the regular Gaussian formula. If you have a model like | |
24452 | this, your best bet is to replace enough of your parameters with | |
24453 | constants to make the model linearizable, then adjust the constants | |
24454 | manually by doing a series of fits. You can compare the fits by | |
24455 | graphing them, by examining the goodness-of-fit measures returned by | |
24456 | @kbd{I a F}, or by some other method suitable to your application. | |
24457 | Note that some models can be linearized in several ways. The | |
5d67986c | 24458 | Gaussian-plus-@var{d} model can be linearized by setting @cite{d} |
d7b8e6c6 EZ |
24459 | (the background) to a constant, or by setting @cite{b} (the standard |
24460 | deviation) and @cite{c} (the mean) to constants. | |
24461 | ||
24462 | To fit a model with constants substituted for some parameters, just | |
24463 | store suitable values in those parameter variables, then omit them | |
24464 | from the list of parameters when you answer the variables prompt. | |
24465 | ||
24466 | @tex | |
24467 | \bigskip | |
24468 | @end tex | |
24469 | ||
24470 | A last desperate step would be to use the general-purpose | |
24471 | @code{minimize} function rather than @code{fit}. After all, both | |
24472 | functions solve the problem of minimizing an expression (the @c{$\chi^2$} | |
24473 | @cite{chi^2} | |
24474 | sum) by adjusting certain parameters in the expression. The @kbd{a F} | |
24475 | command is able to use a vastly more efficient algorithm due to its | |
24476 | special knowledge about linear chi-square sums, but the @kbd{a N} | |
24477 | command can do the same thing by brute force. | |
24478 | ||
24479 | A compromise would be to pick out a few parameters without which the | |
24480 | fit is linearizable, and use @code{minimize} on a call to @code{fit} | |
24481 | which efficiently takes care of the rest of the parameters. The thing | |
24482 | to be minimized would be the value of @c{$\chi^2$} | |
24483 | @cite{chi^2} returned as | |
24484 | the fifth result of the @code{xfit} function: | |
24485 | ||
24486 | @smallexample | |
24487 | minimize(xfit(gaus(a,b,c,d,x), x, [a,b,c], data)_5, d, guess) | |
24488 | @end smallexample | |
24489 | ||
24490 | @noindent | |
24491 | where @code{gaus} represents the Gaussian model with background, | |
24492 | @code{data} represents the data matrix, and @code{guess} represents | |
24493 | the initial guess for @cite{d} that @code{minimize} requires. | |
24494 | This operation will only be, shall we say, extraordinarily slow | |
24495 | rather than astronomically slow (as would be the case if @code{minimize} | |
24496 | were used by itself to solve the problem). | |
24497 | ||
24498 | @tex | |
24499 | \bigskip | |
24500 | @end tex | |
24501 | ||
24502 | The @kbd{I a F} [@code{xfit}] command is somewhat trickier when | |
24503 | nonlinear models are used. The second item in the result is the | |
24504 | vector of ``raw'' parameters @cite{A}, @cite{B}, @cite{C}. The | |
24505 | covariance matrix is written in terms of those raw parameters. | |
24506 | The fifth item is a vector of @dfn{filter} expressions. This | |
24507 | is the empty vector @samp{[]} if the raw parameters were the same | |
24508 | as the requested parameters, i.e., if @cite{A = a}, @cite{B = b}, | |
24509 | and so on (which is always true if the model is already linear | |
24510 | in the parameters as written, e.g., for polynomial fits). If the | |
24511 | parameters had to be rearranged, the fifth item is instead a vector | |
24512 | of one formula per parameter in the original model. The raw | |
24513 | parameters are expressed in these ``filter'' formulas as | |
24514 | @samp{fitdummy(1)} for @cite{A}, @samp{fitdummy(2)} for @cite{B}, | |
24515 | and so on. | |
24516 | ||
24517 | When Calc needs to modify the model to return the result, it replaces | |
24518 | @samp{fitdummy(1)} in all the filters with the first item in the raw | |
24519 | parameters list, and so on for the other raw parameters, then | |
24520 | evaluates the resulting filter formulas to get the actual parameter | |
24521 | values to be substituted into the original model. In the case of | |
24522 | @kbd{H a F} and @kbd{I a F} where the parameters must be error forms, | |
24523 | Calc uses the square roots of the diagonal entries of the covariance | |
24524 | matrix as error values for the raw parameters, then lets Calc's | |
24525 | standard error-form arithmetic take it from there. | |
24526 | ||
24527 | If you use @kbd{I a F} with a nonlinear model, be sure to remember | |
24528 | that the covariance matrix is in terms of the raw parameters, | |
24529 | @emph{not} the actual requested parameters. It's up to you to | |
24530 | figure out how to interpret the covariances in the presence of | |
24531 | nontrivial filter functions. | |
24532 | ||
24533 | Things are also complicated when the input contains error forms. | |
24534 | Suppose there are three independent and dependent variables, @cite{x}, | |
24535 | @cite{y}, and @cite{z}, one or more of which are error forms in the | |
24536 | data. Calc combines all the error values by taking the square root | |
24537 | of the sum of the squares of the errors. It then changes @cite{x} | |
24538 | and @cite{y} to be plain numbers, and makes @cite{z} into an error | |
24539 | form with this combined error. The @cite{Y(x,y,z)} part of the | |
24540 | linearized model is evaluated, and the result should be an error | |
24541 | form. The error part of that result is used for @c{$\sigma_i$} | |
24542 | @cite{sigma_i} for | |
24543 | the data point. If for some reason @cite{Y(x,y,z)} does not return | |
24544 | an error form, the combined error from @cite{z} is used directly | |
24545 | for @c{$\sigma_i$} | |
24546 | @cite{sigma_i}. Finally, @cite{z} is also stripped of its error | |
24547 | for use in computing @cite{F(x,y,z)}, @cite{G(x,y,z)} and so on; | |
24548 | the righthand side of the linearized model is computed in regular | |
24549 | arithmetic with no error forms. | |
24550 | ||
24551 | (While these rules may seem complicated, they are designed to do | |
24552 | the most reasonable thing in the typical case that @cite{Y(x,y,z)} | |
24553 | depends only on the dependent variable @cite{z}, and in fact is | |
24554 | often simply equal to @cite{z}. For common cases like polynomials | |
24555 | and multilinear models, the combined error is simply used as the | |
24556 | @c{$\sigma$} | |
24557 | @cite{sigma} for the data point with no further ado.) | |
24558 | ||
24559 | @tex | |
24560 | \bigskip | |
24561 | @end tex | |
24562 | ||
24563 | @vindex FitRules | |
24564 | It may be the case that the model you wish to use is linearizable, | |
24565 | but Calc's built-in rules are unable to figure it out. Calc uses | |
24566 | its algebraic rewrite mechanism to linearize a model. The rewrite | |
24567 | rules are kept in the variable @code{FitRules}. You can edit this | |
24568 | variable using the @kbd{s e FitRules} command; in fact, there is | |
24569 | a special @kbd{s F} command just for editing @code{FitRules}. | |
24570 | @xref{Operations on Variables}. | |
24571 | ||
24572 | @xref{Rewrite Rules}, for a discussion of rewrite rules. | |
24573 | ||
5d67986c RS |
24574 | @ignore |
24575 | @starindex | |
24576 | @end ignore | |
d7b8e6c6 | 24577 | @tindex fitvar |
5d67986c RS |
24578 | @ignore |
24579 | @starindex | |
24580 | @end ignore | |
24581 | @ignore | |
24582 | @mindex @idots | |
24583 | @end ignore | |
d7b8e6c6 | 24584 | @tindex fitparam |
5d67986c RS |
24585 | @ignore |
24586 | @starindex | |
24587 | @end ignore | |
24588 | @ignore | |
24589 | @mindex @null | |
24590 | @end ignore | |
d7b8e6c6 | 24591 | @tindex fitmodel |
5d67986c RS |
24592 | @ignore |
24593 | @starindex | |
24594 | @end ignore | |
24595 | @ignore | |
24596 | @mindex @null | |
24597 | @end ignore | |
d7b8e6c6 | 24598 | @tindex fitsystem |
5d67986c RS |
24599 | @ignore |
24600 | @starindex | |
24601 | @end ignore | |
24602 | @ignore | |
24603 | @mindex @null | |
24604 | @end ignore | |
d7b8e6c6 EZ |
24605 | @tindex fitdummy |
24606 | Calc uses @code{FitRules} as follows. First, it converts the model | |
24607 | to an equation if necessary and encloses the model equation in a | |
24608 | call to the function @code{fitmodel} (which is not actually a defined | |
24609 | function in Calc; it is only used as a placeholder by the rewrite rules). | |
24610 | Parameter variables are renamed to function calls @samp{fitparam(1)}, | |
24611 | @samp{fitparam(2)}, and so on, and independent variables are renamed | |
24612 | to @samp{fitvar(1)}, @samp{fitvar(2)}, etc. The dependent variable | |
24613 | is the highest-numbered @code{fitvar}. For example, the power law | |
24614 | model @cite{a x^b} is converted to @cite{y = a x^b}, then to | |
24615 | ||
d7b8e6c6 | 24616 | @smallexample |
5d67986c | 24617 | @group |
d7b8e6c6 | 24618 | fitmodel(fitvar(2) = fitparam(1) fitvar(1)^fitparam(2)) |
d7b8e6c6 | 24619 | @end group |
5d67986c | 24620 | @end smallexample |
d7b8e6c6 EZ |
24621 | |
24622 | Calc then applies the rewrites as if by @samp{C-u 0 a r FitRules}. | |
24623 | (The zero prefix means that rewriting should continue until no further | |
24624 | changes are possible.) | |
24625 | ||
24626 | When rewriting is complete, the @code{fitmodel} call should have | |
24627 | been replaced by a @code{fitsystem} call that looks like this: | |
24628 | ||
24629 | @example | |
24630 | fitsystem(@var{Y}, @var{FGH}, @var{abc}) | |
24631 | @end example | |
24632 | ||
24633 | @noindent | |
24634 | where @var{Y} is a formula that describes the function @cite{Y(x,y,z)}, | |
24635 | @var{FGH} is the vector of formulas @cite{[F(x,y,z), G(x,y,z), H(x,y,z)]}, | |
24636 | and @var{abc} is the vector of parameter filters which refer to the | |
24637 | raw parameters as @samp{fitdummy(1)} for @cite{A}, @samp{fitdummy(2)} | |
24638 | for @cite{B}, etc. While the number of raw parameters (the length of | |
24639 | the @var{FGH} vector) is usually the same as the number of original | |
24640 | parameters (the length of the @var{abc} vector), this is not required. | |
24641 | ||
24642 | The power law model eventually boils down to | |
24643 | ||
d7b8e6c6 | 24644 | @smallexample |
5d67986c | 24645 | @group |
d7b8e6c6 EZ |
24646 | fitsystem(ln(fitvar(2)), |
24647 | [1, ln(fitvar(1))], | |
24648 | [exp(fitdummy(1)), fitdummy(2)]) | |
d7b8e6c6 | 24649 | @end group |
5d67986c | 24650 | @end smallexample |
d7b8e6c6 EZ |
24651 | |
24652 | The actual implementation of @code{FitRules} is complicated; it | |
24653 | proceeds in four phases. First, common rearrangements are done | |
24654 | to try to bring linear terms together and to isolate functions like | |
24655 | @code{exp} and @code{ln} either all the way ``out'' (so that they | |
24656 | can be put into @var{Y}) or all the way ``in'' (so that they can | |
24657 | be put into @var{abc} or @var{FGH}). In particular, all | |
24658 | non-constant powers are converted to logs-and-exponentials form, | |
24659 | and the distributive law is used to expand products of sums. | |
24660 | Quotients are rewritten to use the @samp{fitinv} function, where | |
24661 | @samp{fitinv(x)} represents @cite{1/x} while the @code{FitRules} | |
24662 | are operating. (The use of @code{fitinv} makes recognition of | |
24663 | linear-looking forms easier.) If you modify @code{FitRules}, you | |
24664 | will probably only need to modify the rules for this phase. | |
24665 | ||
24666 | Phase two, whose rules can actually also apply during phases one | |
24667 | and three, first rewrites @code{fitmodel} to a two-argument | |
24668 | form @samp{fitmodel(@var{Y}, @var{model})}, where @var{Y} is | |
24669 | initially zero and @var{model} has been changed from @cite{a=b} | |
24670 | to @cite{a-b} form. It then tries to peel off invertible functions | |
24671 | from the outside of @var{model} and put them into @var{Y} instead, | |
24672 | calling the equation solver to invert the functions. Finally, when | |
24673 | this is no longer possible, the @code{fitmodel} is changed to a | |
24674 | four-argument @code{fitsystem}, where the fourth argument is | |
24675 | @var{model} and the @var{FGH} and @var{abc} vectors are initially | |
24676 | empty. (The last vector is really @var{ABC}, corresponding to | |
24677 | raw parameters, for now.) | |
24678 | ||
24679 | Phase three converts a sum of items in the @var{model} to a sum | |
24680 | of @samp{fitpart(@var{a}, @var{b}, @var{c})} terms which represent | |
24681 | terms @samp{@var{a}*@var{b}*@var{c}} of the sum, where @var{a} | |
24682 | is all factors that do not involve any variables, @var{b} is all | |
24683 | factors that involve only parameters, and @var{c} is the factors | |
24684 | that involve only independent variables. (If this decomposition | |
24685 | is not possible, the rule set will not complete and Calc will | |
24686 | complain that the model is too complex.) Then @code{fitpart}s | |
24687 | with equal @var{b} or @var{c} components are merged back together | |
24688 | using the distributive law in order to minimize the number of | |
24689 | raw parameters needed. | |
24690 | ||
24691 | Phase four moves the @code{fitpart} terms into the @var{FGH} and | |
24692 | @var{ABC} vectors. Also, some of the algebraic expansions that | |
24693 | were done in phase 1 are undone now to make the formulas more | |
24694 | computationally efficient. Finally, it calls the solver one more | |
24695 | time to convert the @var{ABC} vector to an @var{abc} vector, and | |
24696 | removes the fourth @var{model} argument (which by now will be zero) | |
24697 | to obtain the three-argument @code{fitsystem} that the linear | |
24698 | least-squares solver wants to see. | |
24699 | ||
5d67986c RS |
24700 | @ignore |
24701 | @starindex | |
24702 | @end ignore | |
24703 | @ignore | |
24704 | @mindex hasfit@idots | |
24705 | @end ignore | |
d7b8e6c6 | 24706 | @tindex hasfitparams |
5d67986c RS |
24707 | @ignore |
24708 | @starindex | |
24709 | @end ignore | |
24710 | @ignore | |
24711 | @mindex @null | |
24712 | @end ignore | |
d7b8e6c6 EZ |
24713 | @tindex hasfitvars |
24714 | Two functions which are useful in connection with @code{FitRules} | |
24715 | are @samp{hasfitparams(x)} and @samp{hasfitvars(x)}, which check | |
24716 | whether @cite{x} refers to any parameters or independent variables, | |
24717 | respectively. Specifically, these functions return ``true'' if the | |
24718 | argument contains any @code{fitparam} (or @code{fitvar}) function | |
24719 | calls, and ``false'' otherwise. (Recall that ``true'' means a | |
24720 | nonzero number, and ``false'' means zero. The actual nonzero number | |
24721 | returned is the largest @var{n} from all the @samp{fitparam(@var{n})}s | |
24722 | or @samp{fitvar(@var{n})}s, respectively, that appear in the formula.) | |
24723 | ||
24724 | @tex | |
24725 | \bigskip | |
24726 | @end tex | |
24727 | ||
24728 | The @code{fit} function in algebraic notation normally takes four | |
24729 | arguments, @samp{fit(@var{model}, @var{vars}, @var{params}, @var{data})}, | |
24730 | where @var{model} is the model formula as it would be typed after | |
24731 | @kbd{a F '}, @var{vars} is the independent variable or a vector of | |
24732 | independent variables, @var{params} likewise gives the parameter(s), | |
24733 | and @var{data} is the data matrix. Note that the length of @var{vars} | |
24734 | must be equal to the number of rows in @var{data} if @var{model} is | |
24735 | an equation, or one less than the number of rows if @var{model} is | |
24736 | a plain formula. (Actually, a name for the dependent variable is | |
24737 | allowed but will be ignored in the plain-formula case.) | |
24738 | ||
24739 | If @var{params} is omitted, the parameters are all variables in | |
24740 | @var{model} except those that appear in @var{vars}. If @var{vars} | |
24741 | is also omitted, Calc sorts all the variables that appear in | |
24742 | @var{model} alphabetically and uses the higher ones for @var{vars} | |
24743 | and the lower ones for @var{params}. | |
24744 | ||
24745 | Alternatively, @samp{fit(@var{modelvec}, @var{data})} is allowed | |
24746 | where @var{modelvec} is a 2- or 3-vector describing the model | |
24747 | and variables, as discussed previously. | |
24748 | ||
24749 | If Calc is unable to do the fit, the @code{fit} function is left | |
24750 | in symbolic form, ordinarily with an explanatory message. The | |
24751 | message will be ``Model expression is too complex'' if the | |
24752 | linearizer was unable to put the model into the required form. | |
24753 | ||
24754 | The @code{efit} (corresponding to @kbd{H a F}) and @code{xfit} | |
24755 | (for @kbd{I a F}) functions are completely analogous. | |
24756 | ||
24757 | @node Interpolation, , Curve Fitting Details, Curve Fitting | |
24758 | @subsection Polynomial Interpolation | |
24759 | ||
24760 | @kindex a p | |
24761 | @pindex calc-poly-interp | |
24762 | @tindex polint | |
24763 | The @kbd{a p} (@code{calc-poly-interp}) [@code{polint}] command does | |
24764 | a polynomial interpolation at a particular @cite{x} value. It takes | |
24765 | two arguments from the stack: A data matrix of the sort used by | |
24766 | @kbd{a F}, and a single number which represents the desired @cite{x} | |
24767 | value. Calc effectively does an exact polynomial fit as if by @kbd{a F i}, | |
24768 | then substitutes the @cite{x} value into the result in order to get an | |
24769 | approximate @cite{y} value based on the fit. (Calc does not actually | |
24770 | use @kbd{a F i}, however; it uses a direct method which is both more | |
24771 | efficient and more numerically stable.) | |
24772 | ||
24773 | The result of @kbd{a p} is actually a vector of two values: The @cite{y} | |
24774 | value approximation, and an error measure @cite{dy} that reflects Calc's | |
24775 | estimation of the probable error of the approximation at that value of | |
24776 | @cite{x}. If the input @cite{x} is equal to any of the @cite{x} values | |
24777 | in the data matrix, the output @cite{y} will be the corresponding @cite{y} | |
24778 | value from the matrix, and the output @cite{dy} will be exactly zero. | |
24779 | ||
24780 | A prefix argument of 2 causes @kbd{a p} to take separate x- and | |
24781 | y-vectors from the stack instead of one data matrix. | |
24782 | ||
24783 | If @cite{x} is a vector of numbers, @kbd{a p} will return a matrix of | |
24784 | interpolated results for each of those @cite{x} values. (The matrix will | |
24785 | have two columns, the @cite{y} values and the @cite{dy} values.) | |
24786 | If @cite{x} is a formula instead of a number, the @code{polint} function | |
24787 | remains in symbolic form; use the @kbd{a "} command to expand it out to | |
24788 | a formula that describes the fit in symbolic terms. | |
24789 | ||
24790 | In all cases, the @kbd{a p} command leaves the data vectors or matrix | |
24791 | on the stack. Only the @cite{x} value is replaced by the result. | |
24792 | ||
24793 | @kindex H a p | |
24794 | @tindex ratint | |
24795 | The @kbd{H a p} [@code{ratint}] command does a rational function | |
24796 | interpolation. It is used exactly like @kbd{a p}, except that it | |
24797 | uses as its model the quotient of two polynomials. If there are | |
24798 | @cite{N} data points, the numerator and denominator polynomials will | |
24799 | each have degree @cite{N/2} (if @cite{N} is odd, the denominator will | |
24800 | have degree one higher than the numerator). | |
24801 | ||
24802 | Rational approximations have the advantage that they can accurately | |
24803 | describe functions that have poles (points at which the function's value | |
24804 | goes to infinity, so that the denominator polynomial of the approximation | |
24805 | goes to zero). If @cite{x} corresponds to a pole of the fitted rational | |
24806 | function, then the result will be a division by zero. If Infinite mode | |
24807 | is enabled, the result will be @samp{[uinf, uinf]}. | |
24808 | ||
24809 | There is no way to get the actual coefficients of the rational function | |
24810 | used by @kbd{H a p}. (The algorithm never generates these coefficients | |
24811 | explicitly, and quotients of polynomials are beyond @w{@kbd{a F}}'s | |
24812 | capabilities to fit.) | |
24813 | ||
24814 | @node Summations, Logical Operations, Curve Fitting, Algebra | |
24815 | @section Summations | |
24816 | ||
24817 | @noindent | |
24818 | @cindex Summation of a series | |
24819 | @kindex a + | |
24820 | @pindex calc-summation | |
24821 | @tindex sum | |
24822 | The @kbd{a +} (@code{calc-summation}) [@code{sum}] command computes | |
24823 | the sum of a formula over a certain range of index values. The formula | |
24824 | is taken from the top of the stack; the command prompts for the | |
24825 | name of the summation index variable, the lower limit of the | |
24826 | sum (any formula), and the upper limit of the sum. If you | |
24827 | enter a blank line at any of these prompts, that prompt and | |
24828 | any later ones are answered by reading additional elements from | |
5d67986c | 24829 | the stack. Thus, @kbd{' k^2 @key{RET} ' k @key{RET} 1 @key{RET} 5 @key{RET} a + @key{RET}} |
d7b8e6c6 EZ |
24830 | produces the result 55. |
24831 | @tex | |
24832 | \turnoffactive | |
24833 | $$ \sum_{k=1}^5 k^2 = 55 $$ | |
24834 | @end tex | |
24835 | ||
24836 | The choice of index variable is arbitrary, but it's best not to | |
24837 | use a variable with a stored value. In particular, while | |
24838 | @code{i} is often a favorite index variable, it should be avoided | |
24839 | in Calc because @code{i} has the imaginary constant @cite{(0, 1)} | |
24840 | as a value. If you pressed @kbd{=} on a sum over @code{i}, it would | |
24841 | be changed to a nonsensical sum over the ``variable'' @cite{(0, 1)}! | |
24842 | If you really want to use @code{i} as an index variable, use | |
5d67986c | 24843 | @w{@kbd{s u i @key{RET}}} first to ``unstore'' this variable. |
d7b8e6c6 EZ |
24844 | (@xref{Storing Variables}.) |
24845 | ||
24846 | A numeric prefix argument steps the index by that amount rather | |
5d67986c | 24847 | than by one. Thus @kbd{' a_k @key{RET} C-u -2 a + k @key{RET} 10 @key{RET} 0 @key{RET}} |
d7b8e6c6 EZ |
24848 | yields @samp{a_10 + a_8 + a_6 + a_4 + a_2 + a_0}. A prefix |
24849 | argument of plain @kbd{C-u} causes @kbd{a +} to prompt for the | |
24850 | step value, in which case you can enter any formula or enter | |
24851 | a blank line to take the step value from the stack. With the | |
24852 | @kbd{C-u} prefix, @kbd{a +} can take up to five arguments from | |
24853 | the stack: The formula, the variable, the lower limit, the | |
24854 | upper limit, and (at the top of the stack), the step value. | |
24855 | ||
24856 | Calc knows how to do certain sums in closed form. For example, | |
24857 | @samp{sum(6 k^2, k, 1, n) = @w{2 n^3} + 3 n^2 + n}. In particular, | |
24858 | this is possible if the formula being summed is polynomial or | |
24859 | exponential in the index variable. Sums of logarithms are | |
24860 | transformed into logarithms of products. Sums of trigonometric | |
24861 | and hyperbolic functions are transformed to sums of exponentials | |
24862 | and then done in closed form. Also, of course, sums in which the | |
24863 | lower and upper limits are both numbers can always be evaluated | |
24864 | just by grinding them out, although Calc will use closed forms | |
24865 | whenever it can for the sake of efficiency. | |
24866 | ||
24867 | The notation for sums in algebraic formulas is | |
24868 | @samp{sum(@var{expr}, @var{var}, @var{low}, @var{high}, @var{step})}. | |
24869 | If @var{step} is omitted, it defaults to one. If @var{high} is | |
24870 | omitted, @var{low} is actually the upper limit and the lower limit | |
24871 | is one. If @var{low} is also omitted, the limits are @samp{-inf} | |
24872 | and @samp{inf}, respectively. | |
24873 | ||
24874 | Infinite sums can sometimes be evaluated: @samp{sum(.5^k, k, 1, inf)} | |
24875 | returns @cite{1}. This is done by evaluating the sum in closed | |
24876 | form (to @samp{1. - 0.5^n} in this case), then evaluating this | |
24877 | formula with @code{n} set to @code{inf}. Calc's usual rules | |
24878 | for ``infinite'' arithmetic can find the answer from there. If | |
24879 | infinite arithmetic yields a @samp{nan}, or if the sum cannot be | |
24880 | solved in closed form, Calc leaves the @code{sum} function in | |
24881 | symbolic form. @xref{Infinities}. | |
24882 | ||
24883 | As a special feature, if the limits are infinite (or omitted, as | |
24884 | described above) but the formula includes vectors subscripted by | |
24885 | expressions that involve the iteration variable, Calc narrows | |
24886 | the limits to include only the range of integers which result in | |
24887 | legal subscripts for the vector. For example, the sum | |
24888 | @samp{sum(k [a,b,c,d,e,f,g]_(2k),k)} evaluates to @samp{b + 2 d + 3 f}. | |
24889 | ||
24890 | The limits of a sum do not need to be integers. For example, | |
24891 | @samp{sum(a_k, k, 0, 2 n, n)} produces @samp{a_0 + a_n + a_(2 n)}. | |
24892 | Calc computes the number of iterations using the formula | |
24893 | @samp{1 + (@var{high} - @var{low}) / @var{step}}, which must, | |
24894 | after simplification as if by @kbd{a s}, evaluate to an integer. | |
24895 | ||
24896 | If the number of iterations according to the above formula does | |
24897 | not come out to an integer, the sum is illegal and will be left | |
24898 | in symbolic form. However, closed forms are still supplied, and | |
24899 | you are on your honor not to misuse the resulting formulas by | |
24900 | substituting mismatched bounds into them. For example, | |
24901 | @samp{sum(k, k, 1, 10, 2)} is invalid, but Calc will go ahead and | |
24902 | evaluate the closed form solution for the limits 1 and 10 to get | |
24903 | the rather dubious answer, 29.25. | |
24904 | ||
24905 | If the lower limit is greater than the upper limit (assuming a | |
24906 | positive step size), the result is generally zero. However, | |
24907 | Calc only guarantees a zero result when the upper limit is | |
24908 | exactly one step less than the lower limit, i.e., if the number | |
24909 | of iterations is @i{-1}. Thus @samp{sum(f(k), k, n, n-1)} is zero | |
24910 | but the sum from @samp{n} to @samp{n-2} may report a nonzero value | |
24911 | if Calc used a closed form solution. | |
24912 | ||
24913 | Calc's logical predicates like @cite{a < b} return 1 for ``true'' | |
24914 | and 0 for ``false.'' @xref{Logical Operations}. This can be | |
24915 | used to advantage for building conditional sums. For example, | |
24916 | @samp{sum(prime(k)*k^2, k, 1, 20)} is the sum of the squares of all | |
24917 | prime numbers from 1 to 20; the @code{prime} predicate returns 1 if | |
24918 | its argument is prime and 0 otherwise. You can read this expression | |
24919 | as ``the sum of @cite{k^2}, where @cite{k} is prime.'' Indeed, | |
24920 | @samp{sum(prime(k)*k^2, k)} would represent the sum of @emph{all} primes | |
24921 | squared, since the limits default to plus and minus infinity, but | |
24922 | there are no such sums that Calc's built-in rules can do in | |
24923 | closed form. | |
24924 | ||
24925 | As another example, @samp{sum((k != k_0) * f(k), k, 1, n)} is the | |
24926 | sum of @cite{f(k)} for all @cite{k} from 1 to @cite{n}, excluding | |
24927 | one value @cite{k_0}. Slightly more tricky is the summand | |
24928 | @samp{(k != k_0) / (k - k_0)}, which is an attempt to describe | |
24929 | the sum of all @cite{1/(k-k_0)} except at @cite{k = k_0}, where | |
24930 | this would be a division by zero. But at @cite{k = k_0}, this | |
24931 | formula works out to the indeterminate form @cite{0 / 0}, which | |
24932 | Calc will not assume is zero. Better would be to use | |
24933 | @samp{(k != k_0) ? 1/(k-k_0) : 0}; the @samp{? :} operator does | |
24934 | an ``if-then-else'' test: This expression says, ``if @c{$k \ne k_0$} | |
24935 | @cite{k != k_0}, | |
24936 | then @cite{1/(k-k_0)}, else zero.'' Now the formula @cite{1/(k-k_0)} | |
24937 | will not even be evaluated by Calc when @cite{k = k_0}. | |
24938 | ||
24939 | @cindex Alternating sums | |
24940 | @kindex a - | |
24941 | @pindex calc-alt-summation | |
24942 | @tindex asum | |
24943 | The @kbd{a -} (@code{calc-alt-summation}) [@code{asum}] command | |
24944 | computes an alternating sum. Successive terms of the sequence | |
24945 | are given alternating signs, with the first term (corresponding | |
24946 | to the lower index value) being positive. Alternating sums | |
24947 | are converted to normal sums with an extra term of the form | |
24948 | @samp{(-1)^(k-@var{low})}. This formula is adjusted appropriately | |
24949 | if the step value is other than one. For example, the Taylor | |
24950 | series for the sine function is @samp{asum(x^k / k!, k, 1, inf, 2)}. | |
24951 | (Calc cannot evaluate this infinite series, but it can approximate | |
24952 | it if you replace @code{inf} with any particular odd number.) | |
24953 | Calc converts this series to a regular sum with a step of one, | |
24954 | namely @samp{sum((-1)^k x^(2k+1) / (2k+1)!, k, 0, inf)}. | |
24955 | ||
24956 | @cindex Product of a sequence | |
24957 | @kindex a * | |
24958 | @pindex calc-product | |
24959 | @tindex prod | |
24960 | The @kbd{a *} (@code{calc-product}) [@code{prod}] command is | |
24961 | the analogous way to take a product of many terms. Calc also knows | |
24962 | some closed forms for products, such as @samp{prod(k, k, 1, n) = n!}. | |
24963 | Conditional products can be written @samp{prod(k^prime(k), k, 1, n)} | |
24964 | or @samp{prod(prime(k) ? k : 1, k, 1, n)}. | |
24965 | ||
24966 | @kindex a T | |
24967 | @pindex calc-tabulate | |
24968 | @tindex table | |
24969 | The @kbd{a T} (@code{calc-tabulate}) [@code{table}] command | |
24970 | evaluates a formula at a series of iterated index values, just | |
24971 | like @code{sum} and @code{prod}, but its result is simply a | |
24972 | vector of the results. For example, @samp{table(a_i, i, 1, 7, 2)} | |
24973 | produces @samp{[a_1, a_3, a_5, a_7]}. | |
24974 | ||
24975 | @node Logical Operations, Rewrite Rules, Summations, Algebra | |
24976 | @section Logical Operations | |
24977 | ||
24978 | @noindent | |
24979 | The following commands and algebraic functions return true/false values, | |
24980 | where 1 represents ``true'' and 0 represents ``false.'' In cases where | |
24981 | a truth value is required (such as for the condition part of a rewrite | |
24982 | rule, or as the condition for a @w{@kbd{Z [ Z ]}} control structure), any | |
24983 | nonzero value is accepted to mean ``true.'' (Specifically, anything | |
24984 | for which @code{dnonzero} returns 1 is ``true,'' and anything for | |
24985 | which @code{dnonzero} returns 0 or cannot decide is assumed ``false.'' | |
24986 | Note that this means that @w{@kbd{Z [ Z ]}} will execute the ``then'' | |
24987 | portion if its condition is provably true, but it will execute the | |
24988 | ``else'' portion for any condition like @cite{a = b} that is not | |
24989 | provably true, even if it might be true. Algebraic functions that | |
24990 | have conditions as arguments, like @code{? :} and @code{&&}, remain | |
24991 | unevaluated if the condition is neither provably true nor provably | |
24992 | false. @xref{Declarations}.) | |
24993 | ||
24994 | @kindex a = | |
24995 | @pindex calc-equal-to | |
24996 | @tindex eq | |
24997 | @tindex = | |
24998 | @tindex == | |
24999 | The @kbd{a =} (@code{calc-equal-to}) command, or @samp{eq(a,b)} function | |
25000 | (which can also be written @samp{a = b} or @samp{a == b} in an algebraic | |
25001 | formula) is true if @cite{a} and @cite{b} are equal, either because they | |
25002 | are identical expressions, or because they are numbers which are | |
25003 | numerically equal. (Thus the integer 1 is considered equal to the float | |
25004 | 1.0.) If the equality of @cite{a} and @cite{b} cannot be determined, | |
25005 | the comparison is left in symbolic form. Note that as a command, this | |
25006 | operation pops two values from the stack and pushes back either a 1 or | |
25007 | a 0, or a formula @samp{a = b} if the values' equality cannot be determined. | |
25008 | ||
25009 | Many Calc commands use @samp{=} formulas to represent @dfn{equations}. | |
25010 | For example, the @kbd{a S} (@code{calc-solve-for}) command rearranges | |
25011 | an equation to solve for a given variable. The @kbd{a M} | |
25012 | (@code{calc-map-equation}) command can be used to apply any | |
25013 | function to both sides of an equation; for example, @kbd{2 a M *} | |
25014 | multiplies both sides of the equation by two. Note that just | |
25015 | @kbd{2 *} would not do the same thing; it would produce the formula | |
25016 | @samp{2 (a = b)} which represents 2 if the equality is true or | |
25017 | zero if not. | |
25018 | ||
25019 | The @code{eq} function with more than two arguments (e.g., @kbd{C-u 3 a =} | |
25020 | or @samp{a = b = c}) tests if all of its arguments are equal. In | |
25021 | algebraic notation, the @samp{=} operator is unusual in that it is | |
25022 | neither left- nor right-associative: @samp{a = b = c} is not the | |
25023 | same as @samp{(a = b) = c} or @samp{a = (b = c)} (which each compare | |
25024 | one variable with the 1 or 0 that results from comparing two other | |
25025 | variables). | |
25026 | ||
25027 | @kindex a # | |
25028 | @pindex calc-not-equal-to | |
25029 | @tindex neq | |
25030 | @tindex != | |
25031 | The @kbd{a #} (@code{calc-not-equal-to}) command, or @samp{neq(a,b)} or | |
25032 | @samp{a != b} function, is true if @cite{a} and @cite{b} are not equal. | |
25033 | This also works with more than two arguments; @samp{a != b != c != d} | |
25034 | tests that all four of @cite{a}, @cite{b}, @cite{c}, and @cite{d} are | |
25035 | distinct numbers. | |
25036 | ||
25037 | @kindex a < | |
25038 | @tindex lt | |
5d67986c RS |
25039 | @ignore |
25040 | @mindex @idots | |
25041 | @end ignore | |
d7b8e6c6 | 25042 | @kindex a > |
5d67986c RS |
25043 | @ignore |
25044 | @mindex @null | |
25045 | @end ignore | |
d7b8e6c6 | 25046 | @kindex a [ |
5d67986c RS |
25047 | @ignore |
25048 | @mindex @null | |
25049 | @end ignore | |
d7b8e6c6 EZ |
25050 | @kindex a ] |
25051 | @pindex calc-less-than | |
25052 | @pindex calc-greater-than | |
25053 | @pindex calc-less-equal | |
25054 | @pindex calc-greater-equal | |
5d67986c RS |
25055 | @ignore |
25056 | @mindex @null | |
25057 | @end ignore | |
d7b8e6c6 | 25058 | @tindex gt |
5d67986c RS |
25059 | @ignore |
25060 | @mindex @null | |
25061 | @end ignore | |
d7b8e6c6 | 25062 | @tindex leq |
5d67986c RS |
25063 | @ignore |
25064 | @mindex @null | |
25065 | @end ignore | |
d7b8e6c6 | 25066 | @tindex geq |
5d67986c RS |
25067 | @ignore |
25068 | @mindex @null | |
25069 | @end ignore | |
d7b8e6c6 | 25070 | @tindex < |
5d67986c RS |
25071 | @ignore |
25072 | @mindex @null | |
25073 | @end ignore | |
d7b8e6c6 | 25074 | @tindex > |
5d67986c RS |
25075 | @ignore |
25076 | @mindex @null | |
25077 | @end ignore | |
d7b8e6c6 | 25078 | @tindex <= |
5d67986c RS |
25079 | @ignore |
25080 | @mindex @null | |
25081 | @end ignore | |
d7b8e6c6 EZ |
25082 | @tindex >= |
25083 | The @kbd{a <} (@code{calc-less-than}) [@samp{lt(a,b)} or @samp{a < b}] | |
25084 | operation is true if @cite{a} is less than @cite{b}. Similar functions | |
25085 | are @kbd{a >} (@code{calc-greater-than}) [@samp{gt(a,b)} or @samp{a > b}], | |
25086 | @kbd{a [} (@code{calc-less-equal}) [@samp{leq(a,b)} or @samp{a <= b}], and | |
25087 | @kbd{a ]} (@code{calc-greater-equal}) [@samp{geq(a,b)} or @samp{a >= b}]. | |
25088 | ||
25089 | While the inequality functions like @code{lt} do not accept more | |
25090 | than two arguments, the syntax @w{@samp{a <= b < c}} is translated to an | |
25091 | equivalent expression involving intervals: @samp{b in [a .. c)}. | |
25092 | (See the description of @code{in} below.) All four combinations | |
25093 | of @samp{<} and @samp{<=} are allowed, or any of the four combinations | |
25094 | of @samp{>} and @samp{>=}. Four-argument constructions like | |
25095 | @samp{a < b < c < d}, and mixtures like @w{@samp{a < b = c}} that | |
25096 | involve both equalities and inequalities, are not allowed. | |
25097 | ||
25098 | @kindex a . | |
25099 | @pindex calc-remove-equal | |
25100 | @tindex rmeq | |
25101 | The @kbd{a .} (@code{calc-remove-equal}) [@code{rmeq}] command extracts | |
25102 | the righthand side of the equation or inequality on the top of the | |
25103 | stack. It also works elementwise on vectors. For example, if | |
25104 | @samp{[x = 2.34, y = z / 2]} is on the stack, then @kbd{a .} produces | |
25105 | @samp{[2.34, z / 2]}. As a special case, if the righthand side is a | |
25106 | variable and the lefthand side is a number (as in @samp{2.34 = x}), then | |
25107 | Calc keeps the lefthand side instead. Finally, this command works with | |
25108 | assignments @samp{x := 2.34} as well as equations, always taking the | |
25109 | the righthand side, and for @samp{=>} (evaluates-to) operators, always | |
25110 | taking the lefthand side. | |
25111 | ||
25112 | @kindex a & | |
25113 | @pindex calc-logical-and | |
25114 | @tindex land | |
25115 | @tindex && | |
25116 | The @kbd{a &} (@code{calc-logical-and}) [@samp{land(a,b)} or @samp{a && b}] | |
25117 | function is true if both of its arguments are true, i.e., are | |
25118 | non-zero numbers. In this case, the result will be either @cite{a} or | |
25119 | @cite{b}, chosen arbitrarily. If either argument is zero, the result is | |
25120 | zero. Otherwise, the formula is left in symbolic form. | |
25121 | ||
25122 | @kindex a | | |
25123 | @pindex calc-logical-or | |
25124 | @tindex lor | |
25125 | @tindex || | |
25126 | The @kbd{a |} (@code{calc-logical-or}) [@samp{lor(a,b)} or @samp{a || b}] | |
25127 | function is true if either or both of its arguments are true (nonzero). | |
25128 | The result is whichever argument was nonzero, choosing arbitrarily if both | |
25129 | are nonzero. If both @cite{a} and @cite{b} are zero, the result is | |
25130 | zero. | |
25131 | ||
25132 | @kindex a ! | |
25133 | @pindex calc-logical-not | |
25134 | @tindex lnot | |
25135 | @tindex ! | |
25136 | The @kbd{a !} (@code{calc-logical-not}) [@samp{lnot(a)} or @samp{!@: a}] | |
25137 | function is true if @cite{a} is false (zero), or false if @cite{a} is | |
25138 | true (nonzero). It is left in symbolic form if @cite{a} is not a | |
25139 | number. | |
25140 | ||
25141 | @kindex a : | |
25142 | @pindex calc-logical-if | |
25143 | @tindex if | |
5d67986c RS |
25144 | @ignore |
25145 | @mindex ? : | |
25146 | @end ignore | |
d7b8e6c6 | 25147 | @tindex ? |
5d67986c RS |
25148 | @ignore |
25149 | @mindex @null | |
25150 | @end ignore | |
d7b8e6c6 EZ |
25151 | @tindex : |
25152 | @cindex Arguments, not evaluated | |
25153 | The @kbd{a :} (@code{calc-logical-if}) [@samp{if(a,b,c)} or @samp{a ? b :@: c}] | |
25154 | function is equal to either @cite{b} or @cite{c} if @cite{a} is a nonzero | |
25155 | number or zero, respectively. If @cite{a} is not a number, the test is | |
25156 | left in symbolic form and neither @cite{b} nor @cite{c} is evaluated in | |
25157 | any way. In algebraic formulas, this is one of the few Calc functions | |
25158 | whose arguments are not automatically evaluated when the function itself | |
25159 | is evaluated. The others are @code{lambda}, @code{quote}, and | |
25160 | @code{condition}. | |
25161 | ||
25162 | One minor surprise to watch out for is that the formula @samp{a?3:4} | |
25163 | will not work because the @samp{3:4} is parsed as a fraction instead of | |
25164 | as three separate symbols. Type something like @samp{a ? 3 : 4} or | |
25165 | @samp{a?(3):4} instead. | |
25166 | ||
25167 | As a special case, if @cite{a} evaluates to a vector, then both @cite{b} | |
25168 | and @cite{c} are evaluated; the result is a vector of the same length | |
25169 | as @cite{a} whose elements are chosen from corresponding elements of | |
25170 | @cite{b} and @cite{c} according to whether each element of @cite{a} | |
25171 | is zero or nonzero. Each of @cite{b} and @cite{c} must be either a | |
25172 | vector of the same length as @cite{a}, or a non-vector which is matched | |
25173 | with all elements of @cite{a}. | |
25174 | ||
25175 | @kindex a @{ | |
25176 | @pindex calc-in-set | |
25177 | @tindex in | |
25178 | The @kbd{a @{} (@code{calc-in-set}) [@samp{in(a,b)}] function is true if | |
25179 | the number @cite{a} is in the set of numbers represented by @cite{b}. | |
25180 | If @cite{b} is an interval form, @cite{a} must be one of the values | |
25181 | encompassed by the interval. If @cite{b} is a vector, @cite{a} must be | |
25182 | equal to one of the elements of the vector. (If any vector elements are | |
25183 | intervals, @cite{a} must be in any of the intervals.) If @cite{b} is a | |
25184 | plain number, @cite{a} must be numerically equal to @cite{b}. | |
25185 | @xref{Set Operations}, for a group of commands that manipulate sets | |
25186 | of this sort. | |
25187 | ||
5d67986c RS |
25188 | @ignore |
25189 | @starindex | |
25190 | @end ignore | |
d7b8e6c6 EZ |
25191 | @tindex typeof |
25192 | The @samp{typeof(a)} function produces an integer or variable which | |
25193 | characterizes @cite{a}. If @cite{a} is a number, vector, or variable, | |
25194 | the result will be one of the following numbers: | |
25195 | ||
25196 | @example | |
25197 | 1 Integer | |
25198 | 2 Fraction | |
25199 | 3 Floating-point number | |
25200 | 4 HMS form | |
25201 | 5 Rectangular complex number | |
25202 | 6 Polar complex number | |
25203 | 7 Error form | |
25204 | 8 Interval form | |
25205 | 9 Modulo form | |
25206 | 10 Date-only form | |
25207 | 11 Date/time form | |
25208 | 12 Infinity (inf, uinf, or nan) | |
25209 | 100 Variable | |
25210 | 101 Vector (but not a matrix) | |
25211 | 102 Matrix | |
25212 | @end example | |
25213 | ||
25214 | Otherwise, @cite{a} is a formula, and the result is a variable which | |
25215 | represents the name of the top-level function call. | |
25216 | ||
5d67986c RS |
25217 | @ignore |
25218 | @starindex | |
25219 | @end ignore | |
d7b8e6c6 | 25220 | @tindex integer |
5d67986c RS |
25221 | @ignore |
25222 | @starindex | |
25223 | @end ignore | |
d7b8e6c6 | 25224 | @tindex real |
5d67986c RS |
25225 | @ignore |
25226 | @starindex | |
25227 | @end ignore | |
d7b8e6c6 EZ |
25228 | @tindex constant |
25229 | The @samp{integer(a)} function returns true if @cite{a} is an integer. | |
25230 | The @samp{real(a)} function | |
25231 | is true if @cite{a} is a real number, either integer, fraction, or | |
25232 | float. The @samp{constant(a)} function returns true if @cite{a} is | |
25233 | any of the objects for which @code{typeof} would produce an integer | |
25234 | code result except for variables, and provided that the components of | |
25235 | an object like a vector or error form are themselves constant. | |
25236 | Note that infinities do not satisfy any of these tests, nor do | |
25237 | special constants like @code{pi} and @code{e}.@refill | |
25238 | ||
25239 | @xref{Declarations}, for a set of similar functions that recognize | |
25240 | formulas as well as actual numbers. For example, @samp{dint(floor(x))} | |
25241 | is true because @samp{floor(x)} is provably integer-valued, but | |
25242 | @samp{integer(floor(x))} does not because @samp{floor(x)} is not | |
25243 | literally an integer constant. | |
25244 | ||
5d67986c RS |
25245 | @ignore |
25246 | @starindex | |
25247 | @end ignore | |
d7b8e6c6 EZ |
25248 | @tindex refers |
25249 | The @samp{refers(a,b)} function is true if the variable (or sub-expression) | |
25250 | @cite{b} appears in @cite{a}, or false otherwise. Unlike the other | |
25251 | tests described here, this function returns a definite ``no'' answer | |
25252 | even if its arguments are still in symbolic form. The only case where | |
25253 | @code{refers} will be left unevaluated is if @cite{a} is a plain | |
25254 | variable (different from @cite{b}). | |
25255 | ||
5d67986c RS |
25256 | @ignore |
25257 | @starindex | |
25258 | @end ignore | |
d7b8e6c6 EZ |
25259 | @tindex negative |
25260 | The @samp{negative(a)} function returns true if @cite{a} ``looks'' negative, | |
25261 | because it is a negative number, because it is of the form @cite{-x}, | |
25262 | or because it is a product or quotient with a term that looks negative. | |
25263 | This is most useful in rewrite rules. Beware that @samp{negative(a)} | |
25264 | evaluates to 1 or 0 for @emph{any} argument @cite{a}, so it can only | |
25265 | be stored in a formula if the default simplifications are turned off | |
25266 | first with @kbd{m O} (or if it appears in an unevaluated context such | |
25267 | as a rewrite rule condition). | |
25268 | ||
5d67986c RS |
25269 | @ignore |
25270 | @starindex | |
25271 | @end ignore | |
d7b8e6c6 EZ |
25272 | @tindex variable |
25273 | The @samp{variable(a)} function is true if @cite{a} is a variable, | |
25274 | or false if not. If @cite{a} is a function call, this test is left | |
25275 | in symbolic form. Built-in variables like @code{pi} and @code{inf} | |
25276 | are considered variables like any others by this test. | |
25277 | ||
5d67986c RS |
25278 | @ignore |
25279 | @starindex | |
25280 | @end ignore | |
d7b8e6c6 EZ |
25281 | @tindex nonvar |
25282 | The @samp{nonvar(a)} function is true if @cite{a} is a non-variable. | |
25283 | If its argument is a variable it is left unsimplified; it never | |
25284 | actually returns zero. However, since Calc's condition-testing | |
25285 | commands consider ``false'' anything not provably true, this is | |
25286 | often good enough. | |
25287 | ||
5d67986c RS |
25288 | @ignore |
25289 | @starindex | |
25290 | @end ignore | |
d7b8e6c6 | 25291 | @tindex lin |
5d67986c RS |
25292 | @ignore |
25293 | @starindex | |
25294 | @end ignore | |
d7b8e6c6 | 25295 | @tindex linnt |
5d67986c RS |
25296 | @ignore |
25297 | @starindex | |
25298 | @end ignore | |
d7b8e6c6 | 25299 | @tindex islin |
5d67986c RS |
25300 | @ignore |
25301 | @starindex | |
25302 | @end ignore | |
d7b8e6c6 EZ |
25303 | @tindex islinnt |
25304 | @cindex Linearity testing | |
25305 | The functions @code{lin}, @code{linnt}, @code{islin}, and @code{islinnt} | |
25306 | check if an expression is ``linear,'' i.e., can be written in the form | |
25307 | @cite{a + b x} for some constants @cite{a} and @cite{b}, and some | |
25308 | variable or subformula @cite{x}. The function @samp{islin(f,x)} checks | |
25309 | if formula @cite{f} is linear in @cite{x}, returning 1 if so. For | |
25310 | example, @samp{islin(x,x)}, @samp{islin(-x,x)}, @samp{islin(3,x)}, and | |
25311 | @samp{islin(x y / 3 - 2, x)} all return 1. The @samp{lin(f,x)} function | |
25312 | is similar, except that instead of returning 1 it returns the vector | |
25313 | @cite{[a, b, x]}. For the above examples, this vector would be | |
25314 | @cite{[0, 1, x]}, @cite{[0, -1, x]}, @cite{[3, 0, x]}, and | |
25315 | @cite{[-2, y/3, x]}, respectively. Both @code{lin} and @code{islin} | |
25316 | generally remain unevaluated for expressions which are not linear, | |
25317 | e.g., @samp{lin(2 x^2, x)} and @samp{lin(sin(x), x)}. The second | |
25318 | argument can also be a formula; @samp{islin(2 + 3 sin(x), sin(x))} | |
25319 | returns true. | |
25320 | ||
25321 | The @code{linnt} and @code{islinnt} functions perform a similar check, | |
25322 | but require a ``non-trivial'' linear form, which means that the | |
25323 | @cite{b} coefficient must be non-zero. For example, @samp{lin(2,x)} | |
25324 | returns @cite{[2, 0, x]} and @samp{lin(y,x)} returns @cite{[y, 0, x]}, | |
25325 | but @samp{linnt(2,x)} and @samp{linnt(y,x)} are left unevaluated | |
25326 | (in other words, these formulas are considered to be only ``trivially'' | |
25327 | linear in @cite{x}). | |
25328 | ||
25329 | All four linearity-testing functions allow you to omit the second | |
25330 | argument, in which case the input may be linear in any non-constant | |
25331 | formula. Here, the @cite{a=0}, @cite{b=1} case is also considered | |
25332 | trivial, and only constant values for @cite{a} and @cite{b} are | |
25333 | recognized. Thus, @samp{lin(2 x y)} returns @cite{[0, 2, x y]}, | |
25334 | @samp{lin(2 - x y)} returns @cite{[2, -1, x y]}, and @samp{lin(x y)} | |
25335 | returns @cite{[0, 1, x y]}. The @code{linnt} function would allow the | |
25336 | first two cases but not the third. Also, neither @code{lin} nor | |
25337 | @code{linnt} accept plain constants as linear in the one-argument | |
25338 | case: @samp{islin(2,x)} is true, but @samp{islin(2)} is false. | |
25339 | ||
5d67986c RS |
25340 | @ignore |
25341 | @starindex | |
25342 | @end ignore | |
d7b8e6c6 EZ |
25343 | @tindex istrue |
25344 | The @samp{istrue(a)} function returns 1 if @cite{a} is a nonzero | |
25345 | number or provably nonzero formula, or 0 if @cite{a} is anything else. | |
25346 | Calls to @code{istrue} can only be manipulated if @kbd{m O} mode is | |
25347 | used to make sure they are not evaluated prematurely. (Note that | |
25348 | declarations are used when deciding whether a formula is true; | |
25349 | @code{istrue} returns 1 when @code{dnonzero} would return 1, and | |
25350 | it returns 0 when @code{dnonzero} would return 0 or leave itself | |
25351 | in symbolic form.) | |
25352 | ||
25353 | @node Rewrite Rules, , Logical Operations, Algebra | |
25354 | @section Rewrite Rules | |
25355 | ||
25356 | @noindent | |
25357 | @cindex Rewrite rules | |
25358 | @cindex Transformations | |
25359 | @cindex Pattern matching | |
25360 | @kindex a r | |
25361 | @pindex calc-rewrite | |
25362 | @tindex rewrite | |
25363 | The @kbd{a r} (@code{calc-rewrite}) [@code{rewrite}] command makes | |
25364 | substitutions in a formula according to a specified pattern or patterns | |
25365 | known as @dfn{rewrite rules}. Whereas @kbd{a b} (@code{calc-substitute}) | |
25366 | matches literally, so that substituting @samp{sin(x)} with @samp{cos(x)} | |
25367 | matches only the @code{sin} function applied to the variable @code{x}, | |
25368 | rewrite rules match general kinds of formulas; rewriting using the rule | |
25369 | @samp{sin(x) := cos(x)} matches @code{sin} of any argument and replaces | |
25370 | it with @code{cos} of that same argument. The only significance of the | |
25371 | name @code{x} is that the same name is used on both sides of the rule. | |
25372 | ||
25373 | Rewrite rules rearrange formulas already in Calc's memory. | |
25374 | @xref{Syntax Tables}, to read about @dfn{syntax rules}, which are | |
25375 | similar to algebraic rewrite rules but operate when new algebraic | |
25376 | entries are being parsed, converting strings of characters into | |
25377 | Calc formulas. | |
25378 | ||
25379 | @menu | |
25380 | * Entering Rewrite Rules:: | |
25381 | * Basic Rewrite Rules:: | |
25382 | * Conditional Rewrite Rules:: | |
25383 | * Algebraic Properties of Rewrite Rules:: | |
25384 | * Other Features of Rewrite Rules:: | |
25385 | * Composing Patterns in Rewrite Rules:: | |
25386 | * Nested Formulas with Rewrite Rules:: | |
25387 | * Multi-Phase Rewrite Rules:: | |
25388 | * Selections with Rewrite Rules:: | |
25389 | * Matching Commands:: | |
25390 | * Automatic Rewrites:: | |
25391 | * Debugging Rewrites:: | |
25392 | * Examples of Rewrite Rules:: | |
25393 | @end menu | |
25394 | ||
25395 | @node Entering Rewrite Rules, Basic Rewrite Rules, Rewrite Rules, Rewrite Rules | |
25396 | @subsection Entering Rewrite Rules | |
25397 | ||
25398 | @noindent | |
25399 | Rewrite rules normally use the ``assignment'' operator | |
25400 | @samp{@var{old} := @var{new}}. | |
25401 | This operator is equivalent to the function call @samp{assign(old, new)}. | |
25402 | The @code{assign} function is undefined by itself in Calc, so an | |
25403 | assignment formula such as a rewrite rule will be left alone by ordinary | |
25404 | Calc commands. But certain commands, like the rewrite system, interpret | |
25405 | assignments in special ways.@refill | |
25406 | ||
25407 | For example, the rule @samp{sin(x)^2 := 1-cos(x)^2} says to replace | |
25408 | every occurrence of the sine of something, squared, with one minus the | |
25409 | square of the cosine of that same thing. All by itself as a formula | |
25410 | on the stack it does nothing, but when given to the @kbd{a r} command | |
25411 | it turns that command into a sine-squared-to-cosine-squared converter. | |
25412 | ||
25413 | To specify a set of rules to be applied all at once, make a vector of | |
25414 | rules. | |
25415 | ||
25416 | When @kbd{a r} prompts you to enter the rewrite rules, you can answer | |
25417 | in several ways: | |
25418 | ||
25419 | @enumerate | |
25420 | @item | |
5d67986c | 25421 | With a rule: @kbd{f(x) := g(x) @key{RET}}. |
d7b8e6c6 | 25422 | @item |
5d67986c | 25423 | With a vector of rules: @kbd{[f1(x) := g1(x), f2(x) := g2(x)] @key{RET}}. |
d7b8e6c6 EZ |
25424 | (You can omit the enclosing square brackets if you wish.) |
25425 | @item | |
25426 | With the name of a variable that contains the rule or rules vector: | |
5d67986c | 25427 | @kbd{myrules @key{RET}}. |
d7b8e6c6 EZ |
25428 | @item |
25429 | With any formula except a rule, a vector, or a variable name; this | |
25430 | will be interpreted as the @var{old} half of a rewrite rule, | |
25431 | and you will be prompted a second time for the @var{new} half: | |
25432 | @kbd{f(x) @key{RET} g(x) @key{RET}}. | |
25433 | @item | |
25434 | With a blank line, in which case the rule, rules vector, or variable | |
25435 | will be taken from the top of the stack (and the formula to be | |
25436 | rewritten will come from the second-to-top position). | |
25437 | @end enumerate | |
25438 | ||
25439 | If you enter the rules directly (as opposed to using rules stored | |
25440 | in a variable), those rules will be put into the Trail so that you | |
25441 | can retrieve them later. @xref{Trail Commands}. | |
25442 | ||
25443 | It is most convenient to store rules you use often in a variable and | |
25444 | invoke them by giving the variable name. The @kbd{s e} | |
25445 | (@code{calc-edit-variable}) command is an easy way to create or edit a | |
25446 | rule set stored in a variable. You may also wish to use @kbd{s p} | |
25447 | (@code{calc-permanent-variable}) to save your rules permanently; | |
25448 | @pxref{Operations on Variables}.@refill | |
25449 | ||
25450 | Rewrite rules are compiled into a special internal form for faster | |
25451 | matching. If you enter a rule set directly it must be recompiled | |
25452 | every time. If you store the rules in a variable and refer to them | |
25453 | through that variable, they will be compiled once and saved away | |
25454 | along with the variable for later reference. This is another good | |
25455 | reason to store your rules in a variable. | |
25456 | ||
25457 | Calc also accepts an obsolete notation for rules, as vectors | |
25458 | @samp{[@var{old}, @var{new}]}. But because it is easily confused with a | |
25459 | vector of two rules, the use of this notation is no longer recommended. | |
25460 | ||
25461 | @node Basic Rewrite Rules, Conditional Rewrite Rules, Entering Rewrite Rules, Rewrite Rules | |
25462 | @subsection Basic Rewrite Rules | |
25463 | ||
25464 | @noindent | |
25465 | To match a particular formula @cite{x} with a particular rewrite rule | |
25466 | @samp{@var{old} := @var{new}}, Calc compares the structure of @cite{x} with | |
25467 | the structure of @var{old}. Variables that appear in @var{old} are | |
25468 | treated as @dfn{meta-variables}; the corresponding positions in @cite{x} | |
25469 | may contain any sub-formulas. For example, the pattern @samp{f(x,y)} | |
25470 | would match the expression @samp{f(12, a+1)} with the meta-variable | |
25471 | @samp{x} corresponding to 12 and with @samp{y} corresponding to | |
25472 | @samp{a+1}. However, this pattern would not match @samp{f(12)} or | |
25473 | @samp{g(12, a+1)}, since there is no assignment of the meta-variables | |
25474 | that will make the pattern match these expressions. Notice that if | |
25475 | the pattern is a single meta-variable, it will match any expression. | |
25476 | ||
25477 | If a given meta-variable appears more than once in @var{old}, the | |
25478 | corresponding sub-formulas of @cite{x} must be identical. Thus | |
25479 | the pattern @samp{f(x,x)} would match @samp{f(12, 12)} and | |
25480 | @samp{f(a+1, a+1)} but not @samp{f(12, a+1)} or @samp{f(a+b, b+a)}. | |
25481 | (@xref{Conditional Rewrite Rules}, for a way to match the latter.) | |
25482 | ||
25483 | Things other than variables must match exactly between the pattern | |
25484 | and the target formula. To match a particular variable exactly, use | |
25485 | the pseudo-function @samp{quote(v)} in the pattern. For example, the | |
25486 | pattern @samp{x+quote(y)} matches @samp{x+y}, @samp{2+y}, or | |
25487 | @samp{sin(a)+y}. | |
25488 | ||
25489 | The special variable names @samp{e}, @samp{pi}, @samp{i}, @samp{phi}, | |
25490 | @samp{gamma}, @samp{inf}, @samp{uinf}, and @samp{nan} always match | |
25491 | literally. Thus the pattern @samp{sin(d + e + f)} acts exactly like | |
25492 | @samp{sin(d + quote(e) + f)}. | |
25493 | ||
25494 | If the @var{old} pattern is found to match a given formula, that | |
25495 | formula is replaced by @var{new}, where any occurrences in @var{new} | |
25496 | of meta-variables from the pattern are replaced with the sub-formulas | |
25497 | that they matched. Thus, applying the rule @samp{f(x,y) := g(y+x,x)} | |
25498 | to @samp{f(12, a+1)} would produce @samp{g(a+13, 12)}. | |
25499 | ||
25500 | The normal @kbd{a r} command applies rewrite rules over and over | |
25501 | throughout the target formula until no further changes are possible | |
25502 | (up to a limit of 100 times). Use @kbd{C-u 1 a r} to make only one | |
25503 | change at a time. | |
25504 | ||
25505 | @node Conditional Rewrite Rules, Algebraic Properties of Rewrite Rules, Basic Rewrite Rules, Rewrite Rules | |
25506 | @subsection Conditional Rewrite Rules | |
25507 | ||
25508 | @noindent | |
25509 | A rewrite rule can also be @dfn{conditional}, written in the form | |
25510 | @samp{@var{old} := @var{new} :: @var{cond}}. (There is also the obsolete | |
25511 | form @samp{[@var{old}, @var{new}, @var{cond}]}.) If a @var{cond} part | |
25512 | is present in the | |
25513 | rule, this is an additional condition that must be satisfied before | |
25514 | the rule is accepted. Once @var{old} has been successfully matched | |
25515 | to the target expression, @var{cond} is evaluated (with all the | |
25516 | meta-variables substituted for the values they matched) and simplified | |
25517 | with @kbd{a s} (@code{calc-simplify}). If the result is a nonzero | |
25518 | number or any other object known to be nonzero (@pxref{Declarations}), | |
25519 | the rule is accepted. If the result is zero or if it is a symbolic | |
25520 | formula that is not known to be nonzero, the rule is rejected. | |
25521 | @xref{Logical Operations}, for a number of functions that return | |
25522 | 1 or 0 according to the results of various tests.@refill | |
25523 | ||
25524 | For example, the formula @samp{n > 0} simplifies to 1 or 0 if @cite{n} | |
25525 | is replaced by a positive or nonpositive number, respectively (or if | |
25526 | @cite{n} has been declared to be positive or nonpositive). Thus, | |
25527 | the rule @samp{f(x,y) := g(y+x,x) :: x+y > 0} would apply to | |
25528 | @samp{f(0, 4)} but not to @samp{f(-3, 2)} or @samp{f(12, a+1)} | |
25529 | (assuming no outstanding declarations for @cite{a}). In the case of | |
25530 | @samp{f(-3, 2)}, the condition can be shown not to be satisfied; in | |
25531 | the case of @samp{f(12, a+1)}, the condition merely cannot be shown | |
25532 | to be satisfied, but that is enough to reject the rule. | |
25533 | ||
25534 | While Calc will use declarations to reason about variables in the | |
25535 | formula being rewritten, declarations do not apply to meta-variables. | |
25536 | For example, the rule @samp{f(a) := g(a+1)} will match for any values | |
25537 | of @samp{a}, such as complex numbers, vectors, or formulas, even if | |
25538 | @samp{a} has been declared to be real or scalar. If you want the | |
25539 | meta-variable @samp{a} to match only literal real numbers, use | |
25540 | @samp{f(a) := g(a+1) :: real(a)}. If you want @samp{a} to match only | |
25541 | reals and formulas which are provably real, use @samp{dreal(a)} as | |
25542 | the condition. | |
25543 | ||
25544 | The @samp{::} operator is a shorthand for the @code{condition} | |
25545 | function; @samp{@var{old} := @var{new} :: @var{cond}} is equivalent to | |
25546 | the formula @samp{condition(assign(@var{old}, @var{new}), @var{cond})}. | |
25547 | ||
25548 | If you have several conditions, you can use @samp{... :: c1 :: c2 :: c3} | |
25549 | or @samp{... :: c1 && c2 && c3}. The two are entirely equivalent. | |
25550 | ||
25551 | It is also possible to embed conditions inside the pattern: | |
25552 | @samp{f(x :: x>0, y) := g(y+x, x)}. This is purely a notational | |
25553 | convenience, though; where a condition appears in a rule has no | |
25554 | effect on when it is tested. The rewrite-rule compiler automatically | |
25555 | decides when it is best to test each condition while a rule is being | |
25556 | matched. | |
25557 | ||
25558 | Certain conditions are handled as special cases by the rewrite rule | |
25559 | system and are tested very efficiently: Where @cite{x} is any | |
25560 | meta-variable, these conditions are @samp{integer(x)}, @samp{real(x)}, | |
25561 | @samp{constant(x)}, @samp{negative(x)}, @samp{x >= y} where @cite{y} | |
25562 | is either a constant or another meta-variable and @samp{>=} may be | |
25563 | replaced by any of the six relational operators, and @samp{x % a = b} | |
25564 | where @cite{a} and @cite{b} are constants. Other conditions, like | |
25565 | @samp{x >= y+1} or @samp{dreal(x)}, will be less efficient to check | |
25566 | since Calc must bring the whole evaluator and simplifier into play. | |
25567 | ||
25568 | An interesting property of @samp{::} is that neither of its arguments | |
25569 | will be touched by Calc's default simplifications. This is important | |
25570 | because conditions often are expressions that cannot safely be | |
25571 | evaluated early. For example, the @code{typeof} function never | |
25572 | remains in symbolic form; entering @samp{typeof(a)} will put the | |
25573 | number 100 (the type code for variables like @samp{a}) on the stack. | |
25574 | But putting the condition @samp{... :: typeof(a) = 6} on the stack | |
25575 | is safe since @samp{::} prevents the @code{typeof} from being | |
25576 | evaluated until the condition is actually used by the rewrite system. | |
25577 | ||
25578 | Since @samp{::} protects its lefthand side, too, you can use a dummy | |
25579 | condition to protect a rule that must itself not evaluate early. | |
25580 | For example, it's not safe to put @samp{a(f,x) := apply(f, [x])} on | |
25581 | the stack because it will immediately evaluate to @samp{a(f,x) := f(x)}, | |
25582 | where the meta-variable-ness of @code{f} on the righthand side has been | |
25583 | lost. But @samp{a(f,x) := apply(f, [x]) :: 1} is safe, and of course | |
25584 | the condition @samp{1} is always true (nonzero) so it has no effect on | |
25585 | the functioning of the rule. (The rewrite compiler will ensure that | |
25586 | it doesn't even impact the speed of matching the rule.) | |
25587 | ||
25588 | @node Algebraic Properties of Rewrite Rules, Other Features of Rewrite Rules, Conditional Rewrite Rules, Rewrite Rules | |
25589 | @subsection Algebraic Properties of Rewrite Rules | |
25590 | ||
25591 | @noindent | |
25592 | The rewrite mechanism understands the algebraic properties of functions | |
25593 | like @samp{+} and @samp{*}. In particular, pattern matching takes | |
25594 | the associativity and commutativity of the following functions into | |
25595 | account: | |
25596 | ||
25597 | @smallexample | |
25598 | + - * = != && || and or xor vint vunion vxor gcd lcm max min beta | |
25599 | @end smallexample | |
25600 | ||
25601 | For example, the rewrite rule: | |
25602 | ||
25603 | @example | |
25604 | a x + b x := (a + b) x | |
25605 | @end example | |
25606 | ||
25607 | @noindent | |
25608 | will match formulas of the form, | |
25609 | ||
25610 | @example | |
25611 | a x + b x, x a + x b, a x + x b, x a + b x | |
25612 | @end example | |
25613 | ||
25614 | Rewrites also understand the relationship between the @samp{+} and @samp{-} | |
25615 | operators. The above rewrite rule will also match the formulas, | |
25616 | ||
25617 | @example | |
25618 | a x - b x, x a - x b, a x - x b, x a - b x | |
25619 | @end example | |
25620 | ||
25621 | @noindent | |
25622 | by matching @samp{b} in the pattern to @samp{-b} from the formula. | |
25623 | ||
25624 | Applied to a sum of many terms like @samp{r + a x + s + b x + t}, this | |
25625 | pattern will check all pairs of terms for possible matches. The rewrite | |
25626 | will take whichever suitable pair it discovers first. | |
25627 | ||
25628 | In general, a pattern using an associative operator like @samp{a + b} | |
5d67986c | 25629 | will try @var{2 n} different ways to match a sum of @var{n} terms |
d7b8e6c6 EZ |
25630 | like @samp{x + y + z - w}. First, @samp{a} is matched against each |
25631 | of @samp{x}, @samp{y}, @samp{z}, and @samp{-w} in turn, with @samp{b} | |
25632 | being matched to the remainders @samp{y + z - w}, @samp{x + z - w}, etc. | |
25633 | If none of these succeed, then @samp{b} is matched against each of the | |
25634 | four terms with @samp{a} matching the remainder. Half-and-half matches, | |
25635 | like @samp{(x + y) + (z - w)}, are not tried. | |
25636 | ||
25637 | Note that @samp{*} is not commutative when applied to matrices, but | |
25638 | rewrite rules pretend that it is. If you type @kbd{m v} to enable | |
25639 | matrix mode (@pxref{Matrix Mode}), rewrite rules will match @samp{*} | |
25640 | literally, ignoring its usual commutativity property. (In the | |
25641 | current implementation, the associativity also vanishes---it is as | |
25642 | if the pattern had been enclosed in a @code{plain} marker; see below.) | |
25643 | If you are applying rewrites to formulas with matrices, it's best to | |
25644 | enable matrix mode first to prevent algebraically incorrect rewrites | |
25645 | from occurring. | |
25646 | ||
25647 | The pattern @samp{-x} will actually match any expression. For example, | |
25648 | the rule | |
25649 | ||
25650 | @example | |
25651 | f(-x) := -f(x) | |
25652 | @end example | |
25653 | ||
25654 | @noindent | |
25655 | will rewrite @samp{f(a)} to @samp{-f(-a)}. To avoid this, either use | |
25656 | a @code{plain} marker as described below, or add a @samp{negative(x)} | |
25657 | condition. The @code{negative} function is true if its argument | |
25658 | ``looks'' negative, for example, because it is a negative number or | |
25659 | because it is a formula like @samp{-x}. The new rule using this | |
25660 | condition is: | |
25661 | ||
25662 | @example | |
25663 | f(x) := -f(-x) :: negative(x) @r{or, equivalently,} | |
25664 | f(-x) := -f(x) :: negative(-x) | |
25665 | @end example | |
25666 | ||
25667 | In the same way, the pattern @samp{x - y} will match the sum @samp{a + b} | |
25668 | by matching @samp{y} to @samp{-b}. | |
25669 | ||
25670 | The pattern @samp{a b} will also match the formula @samp{x/y} if | |
25671 | @samp{y} is a number. Thus the rule @samp{a x + @w{b x} := (a+b) x} | |
25672 | will also convert @samp{a x + x / 2} to @samp{(a + 0.5) x} (or | |
25673 | @samp{(a + 1:2) x}, depending on the current fraction mode). | |
25674 | ||
25675 | Calc will @emph{not} take other liberties with @samp{*}, @samp{/}, and | |
25676 | @samp{^}. For example, the pattern @samp{f(a b)} will not match | |
25677 | @samp{f(x^2)}, and @samp{f(a + b)} will not match @samp{f(2 x)}, even | |
25678 | though conceivably these patterns could match with @samp{a = b = x}. | |
25679 | Nor will @samp{f(a b)} match @samp{f(x / y)} if @samp{y} is not a | |
25680 | constant, even though it could be considered to match with @samp{a = x} | |
25681 | and @samp{b = 1/y}. The reasons are partly for efficiency, and partly | |
25682 | because while few mathematical operations are substantively different | |
25683 | for addition and subtraction, often it is preferable to treat the cases | |
25684 | of multiplication, division, and integer powers separately. | |
25685 | ||
25686 | Even more subtle is the rule set | |
25687 | ||
25688 | @example | |
25689 | [ f(a) + f(b) := f(a + b), -f(a) := f(-a) ] | |
25690 | @end example | |
25691 | ||
25692 | @noindent | |
25693 | attempting to match @samp{f(x) - f(y)}. You might think that Calc | |
25694 | will view this subtraction as @samp{f(x) + (-f(y))} and then apply | |
25695 | the above two rules in turn, but actually this will not work because | |
25696 | Calc only does this when considering rules for @samp{+} (like the | |
25697 | first rule in this set). So it will see first that @samp{f(x) + (-f(y))} | |
25698 | does not match @samp{f(a) + f(b)} for any assignments of the | |
25699 | meta-variables, and then it will see that @samp{f(x) - f(y)} does | |
25700 | not match @samp{-f(a)} for any assignment of @samp{a}. Because Calc | |
25701 | tries only one rule at a time, it will not be able to rewrite | |
25702 | @samp{f(x) - f(y)} with this rule set. An explicit @samp{f(a) - f(b)} | |
25703 | rule will have to be added. | |
25704 | ||
25705 | Another thing patterns will @emph{not} do is break up complex numbers. | |
25706 | The pattern @samp{myconj(a + @w{b i)} := a - b i} will work for formulas | |
25707 | involving the special constant @samp{i} (such as @samp{3 - 4 i}), but | |
25708 | it will not match actual complex numbers like @samp{(3, -4)}. A version | |
25709 | of the above rule for complex numbers would be | |
25710 | ||
25711 | @example | |
25712 | myconj(a) := re(a) - im(a) (0,1) :: im(a) != 0 | |
25713 | @end example | |
25714 | ||
25715 | @noindent | |
25716 | (Because the @code{re} and @code{im} functions understand the properties | |
25717 | of the special constant @samp{i}, this rule will also work for | |
25718 | @samp{3 - 4 i}. In fact, this particular rule would probably be better | |
25719 | without the @samp{im(a) != 0} condition, since if @samp{im(a) = 0} the | |
25720 | righthand side of the rule will still give the correct answer for the | |
25721 | conjugate of a real number.) | |
25722 | ||
25723 | It is also possible to specify optional arguments in patterns. The rule | |
25724 | ||
25725 | @example | |
25726 | opt(a) x + opt(b) (x^opt(c) + opt(d)) := f(a, b, c, d) | |
25727 | @end example | |
25728 | ||
25729 | @noindent | |
25730 | will match the formula | |
25731 | ||
25732 | @example | |
25733 | 5 (x^2 - 4) + 3 x | |
25734 | @end example | |
25735 | ||
25736 | @noindent | |
25737 | in a fairly straightforward manner, but it will also match reduced | |
25738 | formulas like | |
25739 | ||
25740 | @example | |
25741 | x + x^2, 2(x + 1) - x, x + x | |
25742 | @end example | |
25743 | ||
25744 | @noindent | |
25745 | producing, respectively, | |
25746 | ||
25747 | @example | |
25748 | f(1, 1, 2, 0), f(-1, 2, 1, 1), f(1, 1, 1, 0) | |
25749 | @end example | |
25750 | ||
25751 | (The latter two formulas can be entered only if default simplifications | |
25752 | have been turned off with @kbd{m O}.) | |
25753 | ||
25754 | The default value for a term of a sum is zero. The default value | |
25755 | for a part of a product, for a power, or for the denominator of a | |
25756 | quotient, is one. Also, @samp{-x} matches the pattern @samp{opt(a) b} | |
25757 | with @samp{a = -1}. | |
25758 | ||
25759 | In particular, the distributive-law rule can be refined to | |
25760 | ||
25761 | @example | |
25762 | opt(a) x + opt(b) x := (a + b) x | |
25763 | @end example | |
25764 | ||
25765 | @noindent | |
25766 | so that it will convert, e.g., @samp{a x - x}, to @samp{(a - 1) x}. | |
25767 | ||
25768 | The pattern @samp{opt(a) + opt(b) x} matches almost any formulas which | |
25769 | are linear in @samp{x}. You can also use the @code{lin} and @code{islin} | |
25770 | functions with rewrite conditions to test for this; @pxref{Logical | |
25771 | Operations}. These functions are not as convenient to use in rewrite | |
25772 | rules, but they recognize more kinds of formulas as linear: | |
25773 | @samp{x/z} is considered linear with @cite{b = 1/z} by @code{lin}, | |
25774 | but it will not match the above pattern because that pattern calls | |
25775 | for a multiplication, not a division. | |
25776 | ||
25777 | As another example, the obvious rule to replace @samp{sin(x)^2 + cos(x)^2} | |
25778 | by 1, | |
25779 | ||
25780 | @example | |
25781 | sin(x)^2 + cos(x)^2 := 1 | |
25782 | @end example | |
25783 | ||
25784 | @noindent | |
25785 | misses many cases because the sine and cosine may both be multiplied by | |
25786 | an equal factor. Here's a more successful rule: | |
25787 | ||
25788 | @example | |
25789 | opt(a) sin(x)^2 + opt(a) cos(x)^2 := a | |
25790 | @end example | |
25791 | ||
25792 | Note that this rule will @emph{not} match @samp{sin(x)^2 + 6 cos(x)^2} | |
25793 | because one @cite{a} would have ``matched'' 1 while the other matched 6. | |
25794 | ||
25795 | Calc automatically converts a rule like | |
25796 | ||
25797 | @example | |
25798 | f(x-1, x) := g(x) | |
25799 | @end example | |
25800 | ||
25801 | @noindent | |
25802 | into the form | |
25803 | ||
25804 | @example | |
25805 | f(temp, x) := g(x) :: temp = x-1 | |
25806 | @end example | |
25807 | ||
25808 | @noindent | |
25809 | (where @code{temp} stands for a new, invented meta-variable that | |
25810 | doesn't actually have a name). This modified rule will successfully | |
25811 | match @samp{f(6, 7)}, binding @samp{temp} and @samp{x} to 6 and 7, | |
25812 | respectively, then verifying that they differ by one even though | |
25813 | @samp{6} does not superficially look like @samp{x-1}. | |
25814 | ||
25815 | However, Calc does not solve equations to interpret a rule. The | |
25816 | following rule, | |
25817 | ||
25818 | @example | |
25819 | f(x-1, x+1) := g(x) | |
25820 | @end example | |
25821 | ||
25822 | @noindent | |
25823 | will not work. That is, it will match @samp{f(a - 1 + b, a + 1 + b)} | |
25824 | but not @samp{f(6, 8)}. Calc always interprets at least one occurrence | |
25825 | of a variable by literal matching. If the variable appears ``isolated'' | |
25826 | then Calc is smart enough to use it for literal matching. But in this | |
25827 | last example, Calc is forced to rewrite the rule to @samp{f(x-1, temp) | |
25828 | := g(x) :: temp = x+1} where the @samp{x-1} term must correspond to an | |
25829 | actual ``something-minus-one'' in the target formula. | |
25830 | ||
25831 | A successful way to write this would be @samp{f(x, x+2) := g(x+1)}. | |
25832 | You could make this resemble the original form more closely by using | |
25833 | @code{let} notation, which is described in the next section: | |
25834 | ||
25835 | @example | |
25836 | f(xm1, x+1) := g(x) :: let(x := xm1+1) | |
25837 | @end example | |
25838 | ||
25839 | Calc does this rewriting or ``conditionalizing'' for any sub-pattern | |
25840 | which involves only the functions in the following list, operating | |
25841 | only on constants and meta-variables which have already been matched | |
25842 | elsewhere in the pattern. When matching a function call, Calc is | |
25843 | careful to match arguments which are plain variables before arguments | |
25844 | which are calls to any of the functions below, so that a pattern like | |
25845 | @samp{f(x-1, x)} can be conditionalized even though the isolated | |
25846 | @samp{x} comes after the @samp{x-1}. | |
25847 | ||
25848 | @smallexample | |
25849 | + - * / \ % ^ abs sign round rounde roundu trunc floor ceil | |
25850 | max min re im conj arg | |
25851 | @end smallexample | |
25852 | ||
25853 | You can suppress all of the special treatments described in this | |
25854 | section by surrounding a function call with a @code{plain} marker. | |
25855 | This marker causes the function call which is its argument to be | |
25856 | matched literally, without regard to commutativity, associativity, | |
25857 | negation, or conditionalization. When you use @code{plain}, the | |
25858 | ``deep structure'' of the formula being matched can show through. | |
25859 | For example, | |
25860 | ||
25861 | @example | |
25862 | plain(a - a b) := f(a, b) | |
25863 | @end example | |
25864 | ||
25865 | @noindent | |
25866 | will match only literal subtractions. However, the @code{plain} | |
25867 | marker does not affect its arguments' arguments. In this case, | |
25868 | commutativity and associativity is still considered while matching | |
25869 | the @w{@samp{a b}} sub-pattern, so the whole pattern will match | |
25870 | @samp{x - y x} as well as @samp{x - x y}. We could go still | |
25871 | further and use | |
25872 | ||
25873 | @example | |
25874 | plain(a - plain(a b)) := f(a, b) | |
25875 | @end example | |
25876 | ||
25877 | @noindent | |
25878 | which would do a completely strict match for the pattern. | |
25879 | ||
25880 | By contrast, the @code{quote} marker means that not only the | |
25881 | function name but also the arguments must be literally the same. | |
25882 | The above pattern will match @samp{x - x y} but | |
25883 | ||
25884 | @example | |
25885 | quote(a - a b) := f(a, b) | |
25886 | @end example | |
25887 | ||
25888 | @noindent | |
25889 | will match only the single formula @samp{a - a b}. Also, | |
25890 | ||
25891 | @example | |
25892 | quote(a - quote(a b)) := f(a, b) | |
25893 | @end example | |
25894 | ||
25895 | @noindent | |
25896 | will match only @samp{a - quote(a b)}---probably not the desired | |
25897 | effect! | |
25898 | ||
25899 | A certain amount of algebra is also done when substituting the | |
25900 | meta-variables on the righthand side of a rule. For example, | |
25901 | in the rule | |
25902 | ||
25903 | @example | |
25904 | a + f(b) := f(a + b) | |
25905 | @end example | |
25906 | ||
25907 | @noindent | |
25908 | matching @samp{f(x) - y} would produce @samp{f((-y) + x)} if | |
25909 | taken literally, but the rewrite mechanism will simplify the | |
25910 | righthand side to @samp{f(x - y)} automatically. (Of course, | |
25911 | the default simplifications would do this anyway, so this | |
25912 | special simplification is only noticeable if you have turned the | |
25913 | default simplifications off.) This rewriting is done only when | |
25914 | a meta-variable expands to a ``negative-looking'' expression. | |
25915 | If this simplification is not desirable, you can use a @code{plain} | |
25916 | marker on the righthand side: | |
25917 | ||
25918 | @example | |
25919 | a + f(b) := f(plain(a + b)) | |
25920 | @end example | |
25921 | ||
25922 | @noindent | |
25923 | In this example, we are still allowing the pattern-matcher to | |
25924 | use all the algebra it can muster, but the righthand side will | |
25925 | always simplify to a literal addition like @samp{f((-y) + x)}. | |
25926 | ||
25927 | @node Other Features of Rewrite Rules, Composing Patterns in Rewrite Rules, Algebraic Properties of Rewrite Rules, Rewrite Rules | |
25928 | @subsection Other Features of Rewrite Rules | |
25929 | ||
25930 | @noindent | |
25931 | Certain ``function names'' serve as markers in rewrite rules. | |
25932 | Here is a complete list of these markers. First are listed the | |
25933 | markers that work inside a pattern; then come the markers that | |
25934 | work in the righthand side of a rule. | |
25935 | ||
5d67986c RS |
25936 | @ignore |
25937 | @starindex | |
25938 | @end ignore | |
d7b8e6c6 EZ |
25939 | @tindex import |
25940 | One kind of marker, @samp{import(x)}, takes the place of a whole | |
25941 | rule. Here @cite{x} is the name of a variable containing another | |
25942 | rule set; those rules are ``spliced into'' the rule set that | |
25943 | imports them. For example, if @samp{[f(a+b) := f(a) + f(b), | |
25944 | f(a b) := a f(b) :: real(a)]} is stored in variable @samp{linearF}, | |
25945 | then the rule set @samp{[f(0) := 0, import(linearF)]} will apply | |
25946 | all three rules. It is possible to modify the imported rules | |
25947 | slightly: @samp{import(x, v1, x1, v2, x2, @dots{})} imports | |
25948 | the rule set @cite{x} with all occurrences of @c{$v_1$} | |
25949 | @cite{v1}, as either | |
25950 | a variable name or a function name, replaced with @c{$x_1$} | |
25951 | @cite{x1} and | |
25952 | so on. (If @c{$v_1$} | |
25953 | @cite{v1} is used as a function name, then @c{$x_1$} | |
25954 | @cite{x1} | |
25955 | must be either a function name itself or a @w{@samp{< >}} nameless | |
25956 | function; @pxref{Specifying Operators}.) For example, @samp{[g(0) := 0, | |
25957 | import(linearF, f, g)]} applies the linearity rules to the function | |
25958 | @samp{g} instead of @samp{f}. Imports can be nested, but the | |
25959 | import-with-renaming feature may fail to rename sub-imports properly. | |
25960 | ||
25961 | The special functions allowed in patterns are: | |
25962 | ||
25963 | @table @samp | |
25964 | @item quote(x) | |
5d67986c RS |
25965 | @ignore |
25966 | @starindex | |
25967 | @end ignore | |
d7b8e6c6 EZ |
25968 | @tindex quote |
25969 | This pattern matches exactly @cite{x}; variable names in @cite{x} are | |
25970 | not interpreted as meta-variables. The only flexibility is that | |
25971 | numbers are compared for numeric equality, so that the pattern | |
25972 | @samp{f(quote(12))} will match both @samp{f(12)} and @samp{f(12.0)}. | |
25973 | (Numbers are always treated this way by the rewrite mechanism: | |
25974 | The rule @samp{f(x,x) := g(x)} will match @samp{f(12, 12.0)}. | |
25975 | The rewrite may produce either @samp{g(12)} or @samp{g(12.0)} | |
25976 | as a result in this case.) | |
25977 | ||
25978 | @item plain(x) | |
5d67986c RS |
25979 | @ignore |
25980 | @starindex | |
25981 | @end ignore | |
d7b8e6c6 EZ |
25982 | @tindex plain |
25983 | Here @cite{x} must be a function call @samp{f(x1,x2,@dots{})}. This | |
25984 | pattern matches a call to function @cite{f} with the specified | |
25985 | argument patterns. No special knowledge of the properties of the | |
25986 | function @cite{f} is used in this case; @samp{+} is not commutative or | |
25987 | associative. Unlike @code{quote}, the arguments @samp{x1,x2,@dots{}} | |
25988 | are treated as patterns. If you wish them to be treated ``plainly'' | |
25989 | as well, you must enclose them with more @code{plain} markers: | |
25990 | @samp{plain(plain(@w{-a}) + plain(b c))}. | |
25991 | ||
25992 | @item opt(x,def) | |
5d67986c RS |
25993 | @ignore |
25994 | @starindex | |
25995 | @end ignore | |
d7b8e6c6 EZ |
25996 | @tindex opt |
25997 | Here @cite{x} must be a variable name. This must appear as an | |
25998 | argument to a function or an element of a vector; it specifies that | |
25999 | the argument or element is optional. | |
26000 | As an argument to @samp{+}, @samp{-}, @samp{*}, @samp{&&}, or @samp{||}, | |
26001 | or as the second argument to @samp{/} or @samp{^}, the value @var{def} | |
26002 | may be omitted. The pattern @samp{x + opt(y)} matches a sum by | |
26003 | binding one summand to @cite{x} and the other to @cite{y}, and it | |
26004 | matches anything else by binding the whole expression to @cite{x} and | |
26005 | zero to @cite{y}. The other operators above work similarly.@refill | |
26006 | ||
28665d46 | 26007 | For general miscellaneous functions, the default value @code{def} |
d7b8e6c6 EZ |
26008 | must be specified. Optional arguments are dropped starting with |
26009 | the rightmost one during matching. For example, the pattern | |
26010 | @samp{f(opt(a,0), b, opt(c,b))} will match @samp{f(b)}, @samp{f(a,b)}, | |
26011 | or @samp{f(a,b,c)}. Default values of zero and @cite{b} are | |
26012 | supplied in this example for the omitted arguments. Note that | |
26013 | the literal variable @cite{b} will be the default in the latter | |
26014 | case, @emph{not} the value that matched the meta-variable @cite{b}. | |
26015 | In other words, the default @var{def} is effectively quoted. | |
26016 | ||
26017 | @item condition(x,c) | |
5d67986c RS |
26018 | @ignore |
26019 | @starindex | |
26020 | @end ignore | |
d7b8e6c6 EZ |
26021 | @tindex condition |
26022 | @tindex :: | |
26023 | This matches the pattern @cite{x}, with the attached condition | |
26024 | @cite{c}. It is the same as @samp{x :: c}. | |
26025 | ||
26026 | @item pand(x,y) | |
5d67986c RS |
26027 | @ignore |
26028 | @starindex | |
26029 | @end ignore | |
d7b8e6c6 EZ |
26030 | @tindex pand |
26031 | @tindex &&& | |
26032 | This matches anything that matches both pattern @cite{x} and | |
26033 | pattern @cite{y}. It is the same as @samp{x &&& y}. | |
26034 | @pxref{Composing Patterns in Rewrite Rules}. | |
26035 | ||
26036 | @item por(x,y) | |
5d67986c RS |
26037 | @ignore |
26038 | @starindex | |
26039 | @end ignore | |
d7b8e6c6 EZ |
26040 | @tindex por |
26041 | @tindex ||| | |
26042 | This matches anything that matches either pattern @cite{x} or | |
26043 | pattern @cite{y}. It is the same as @w{@samp{x ||| y}}. | |
26044 | ||
26045 | @item pnot(x) | |
5d67986c RS |
26046 | @ignore |
26047 | @starindex | |
26048 | @end ignore | |
d7b8e6c6 EZ |
26049 | @tindex pnot |
26050 | @tindex !!! | |
26051 | This matches anything that does not match pattern @cite{x}. | |
26052 | It is the same as @samp{!!! x}. | |
26053 | ||
26054 | @item cons(h,t) | |
5d67986c RS |
26055 | @ignore |
26056 | @mindex cons | |
26057 | @end ignore | |
d7b8e6c6 EZ |
26058 | @tindex cons (rewrites) |
26059 | This matches any vector of one or more elements. The first | |
26060 | element is matched to @cite{h}; a vector of the remaining | |
26061 | elements is matched to @cite{t}. Note that vectors of fixed | |
26062 | length can also be matched as actual vectors: The rule | |
26063 | @samp{cons(a,cons(b,[])) := cons(a+b,[])} is equivalent | |
26064 | to the rule @samp{[a,b] := [a+b]}. | |
26065 | ||
26066 | @item rcons(t,h) | |
5d67986c RS |
26067 | @ignore |
26068 | @mindex rcons | |
26069 | @end ignore | |
d7b8e6c6 EZ |
26070 | @tindex rcons (rewrites) |
26071 | This is like @code{cons}, except that the @emph{last} element | |
26072 | is matched to @cite{h}, with the remaining elements matched | |
26073 | to @cite{t}. | |
26074 | ||
26075 | @item apply(f,args) | |
5d67986c RS |
26076 | @ignore |
26077 | @mindex apply | |
26078 | @end ignore | |
d7b8e6c6 EZ |
26079 | @tindex apply (rewrites) |
26080 | This matches any function call. The name of the function, in | |
26081 | the form of a variable, is matched to @cite{f}. The arguments | |
26082 | of the function, as a vector of zero or more objects, are | |
26083 | matched to @samp{args}. Constants, variables, and vectors | |
26084 | do @emph{not} match an @code{apply} pattern. For example, | |
26085 | @samp{apply(f,x)} matches any function call, @samp{apply(quote(f),x)} | |
26086 | matches any call to the function @samp{f}, @samp{apply(f,[a,b])} | |
26087 | matches any function call with exactly two arguments, and | |
26088 | @samp{apply(quote(f), cons(a,cons(b,x)))} matches any call | |
26089 | to the function @samp{f} with two or more arguments. Another | |
26090 | way to implement the latter, if the rest of the rule does not | |
26091 | need to refer to the first two arguments of @samp{f} by name, | |
26092 | would be @samp{apply(quote(f), x :: vlen(x) >= 2)}. | |
26093 | Here's a more interesting sample use of @code{apply}: | |
26094 | ||
26095 | @example | |
26096 | apply(f,[x+n]) := n + apply(f,[x]) | |
26097 | :: in(f, [floor,ceil,round,trunc]) :: integer(n) | |
26098 | @end example | |
26099 | ||
26100 | Note, however, that this will be slower to match than a rule | |
26101 | set with four separate rules. The reason is that Calc sorts | |
26102 | the rules of a rule set according to top-level function name; | |
26103 | if the top-level function is @code{apply}, Calc must try the | |
26104 | rule for every single formula and sub-formula. If the top-level | |
26105 | function in the pattern is, say, @code{floor}, then Calc invokes | |
26106 | the rule only for sub-formulas which are calls to @code{floor}. | |
26107 | ||
26108 | Formulas normally written with operators like @code{+} are still | |
26109 | considered function calls: @code{apply(f,x)} matches @samp{a+b} | |
26110 | with @samp{f = add}, @samp{x = [a,b]}. | |
26111 | ||
26112 | You must use @code{apply} for meta-variables with function names | |
26113 | on both sides of a rewrite rule: @samp{apply(f, [x]) := f(x+1)} | |
26114 | is @emph{not} correct, because it rewrites @samp{spam(6)} into | |
26115 | @samp{f(7)}. The righthand side should be @samp{apply(f, [x+1])}. | |
26116 | Also note that you will have to use no-simplify (@kbd{m O}) | |
26117 | mode when entering this rule so that the @code{apply} isn't | |
26118 | evaluated immediately to get the new rule @samp{f(x) := f(x+1)}. | |
26119 | Or, use @kbd{s e} to enter the rule without going through the stack, | |
26120 | or enter the rule as @samp{apply(f, [x]) := apply(f, [x+1]) @w{:: 1}}. | |
26121 | @xref{Conditional Rewrite Rules}. | |
26122 | ||
26123 | @item select(x) | |
5d67986c RS |
26124 | @ignore |
26125 | @starindex | |
26126 | @end ignore | |
d7b8e6c6 EZ |
26127 | @tindex select |
26128 | This is used for applying rules to formulas with selections; | |
26129 | @pxref{Selections with Rewrite Rules}. | |
26130 | @end table | |
26131 | ||
26132 | Special functions for the righthand sides of rules are: | |
26133 | ||
26134 | @table @samp | |
26135 | @item quote(x) | |
26136 | The notation @samp{quote(x)} is changed to @samp{x} when the | |
26137 | righthand side is used. As far as the rewrite rule is concerned, | |
26138 | @code{quote} is invisible. However, @code{quote} has the special | |
26139 | property in Calc that its argument is not evaluated. Thus, | |
26140 | while it will not work to put the rule @samp{t(a) := typeof(a)} | |
26141 | on the stack because @samp{typeof(a)} is evaluated immediately | |
26142 | to produce @samp{t(a) := 100}, you can use @code{quote} to | |
26143 | protect the righthand side: @samp{t(a) := quote(typeof(a))}. | |
26144 | (@xref{Conditional Rewrite Rules}, for another trick for | |
26145 | protecting rules from evaluation.) | |
26146 | ||
26147 | @item plain(x) | |
26148 | Special properties of and simplifications for the function call | |
26149 | @cite{x} are not used. One interesting case where @code{plain} | |
26150 | is useful is the rule, @samp{q(x) := quote(x)}, trying to expand a | |
26151 | shorthand notation for the @code{quote} function. This rule will | |
26152 | not work as shown; instead of replacing @samp{q(foo)} with | |
26153 | @samp{quote(foo)}, it will replace it with @samp{foo}! The correct | |
26154 | rule would be @samp{q(x) := plain(quote(x))}. | |
26155 | ||
26156 | @item cons(h,t) | |
26157 | Where @cite{t} is a vector, this is converted into an expanded | |
26158 | vector during rewrite processing. Note that @code{cons} is a regular | |
26159 | Calc function which normally does this anyway; the only way @code{cons} | |
26160 | is treated specially by rewrites is that @code{cons} on the righthand | |
26161 | side of a rule will be evaluated even if default simplifications | |
26162 | have been turned off. | |
26163 | ||
26164 | @item rcons(t,h) | |
26165 | Analogous to @code{cons} except putting @cite{h} at the @emph{end} of | |
26166 | the vector @cite{t}. | |
26167 | ||
26168 | @item apply(f,args) | |
26169 | Where @cite{f} is a variable and @var{args} is a vector, this | |
26170 | is converted to a function call. Once again, note that @code{apply} | |
26171 | is also a regular Calc function. | |
26172 | ||
26173 | @item eval(x) | |
5d67986c RS |
26174 | @ignore |
26175 | @starindex | |
26176 | @end ignore | |
d7b8e6c6 EZ |
26177 | @tindex eval |
26178 | The formula @cite{x} is handled in the usual way, then the | |
26179 | default simplifications are applied to it even if they have | |
26180 | been turned off normally. This allows you to treat any function | |
26181 | similarly to the way @code{cons} and @code{apply} are always | |
26182 | treated. However, there is a slight difference: @samp{cons(2+3, [])} | |
26183 | with default simplifications off will be converted to @samp{[2+3]}, | |
26184 | whereas @samp{eval(cons(2+3, []))} will be converted to @samp{[5]}. | |
26185 | ||
26186 | @item evalsimp(x) | |
5d67986c RS |
26187 | @ignore |
26188 | @starindex | |
26189 | @end ignore | |
d7b8e6c6 EZ |
26190 | @tindex evalsimp |
26191 | The formula @cite{x} has meta-variables substituted in the usual | |
26192 | way, then algebraically simplified as if by the @kbd{a s} command. | |
26193 | ||
26194 | @item evalextsimp(x) | |
5d67986c RS |
26195 | @ignore |
26196 | @starindex | |
26197 | @end ignore | |
d7b8e6c6 EZ |
26198 | @tindex evalextsimp |
26199 | The formula @cite{x} has meta-variables substituted in the normal | |
26200 | way, then ``extendedly'' simplified as if by the @kbd{a e} command. | |
26201 | ||
26202 | @item select(x) | |
26203 | @xref{Selections with Rewrite Rules}. | |
26204 | @end table | |
26205 | ||
26206 | There are also some special functions you can use in conditions. | |
26207 | ||
26208 | @table @samp | |
26209 | @item let(v := x) | |
5d67986c RS |
26210 | @ignore |
26211 | @starindex | |
26212 | @end ignore | |
d7b8e6c6 EZ |
26213 | @tindex let |
26214 | The expression @cite{x} is evaluated with meta-variables substituted. | |
26215 | The @kbd{a s} command's simplifications are @emph{not} applied by | |
26216 | default, but @cite{x} can include calls to @code{evalsimp} or | |
26217 | @code{evalextsimp} as described above to invoke higher levels | |
26218 | of simplification. The | |
26219 | result of @cite{x} is then bound to the meta-variable @cite{v}. As | |
26220 | usual, if this meta-variable has already been matched to something | |
26221 | else the two values must be equal; if the meta-variable is new then | |
26222 | it is bound to the result of the expression. This variable can then | |
26223 | appear in later conditions, and on the righthand side of the rule. | |
26224 | In fact, @cite{v} may be any pattern in which case the result of | |
26225 | evaluating @cite{x} is matched to that pattern, binding any | |
26226 | meta-variables that appear in that pattern. Note that @code{let} | |
26227 | can only appear by itself as a condition, or as one term of an | |
26228 | @samp{&&} which is a whole condition: It cannot be inside | |
26229 | an @samp{||} term or otherwise buried.@refill | |
26230 | ||
26231 | The alternate, equivalent form @samp{let(v, x)} is also recognized. | |
26232 | Note that the use of @samp{:=} by @code{let}, while still being | |
26233 | assignment-like in character, is unrelated to the use of @samp{:=} | |
26234 | in the main part of a rewrite rule. | |
26235 | ||
26236 | As an example, @samp{f(a) := g(ia) :: let(ia := 1/a) :: constant(ia)} | |
26237 | replaces @samp{f(a)} with @samp{g} of the inverse of @samp{a}, if | |
26238 | that inverse exists and is constant. For example, if @samp{a} is a | |
26239 | singular matrix the operation @samp{1/a} is left unsimplified and | |
26240 | @samp{constant(ia)} fails, but if @samp{a} is an invertible matrix | |
26241 | then the rule succeeds. Without @code{let} there would be no way | |
26242 | to express this rule that didn't have to invert the matrix twice. | |
26243 | Note that, because the meta-variable @samp{ia} is otherwise unbound | |
26244 | in this rule, the @code{let} condition itself always ``succeeds'' | |
26245 | because no matter what @samp{1/a} evaluates to, it can successfully | |
26246 | be bound to @code{ia}.@refill | |
26247 | ||
26248 | Here's another example, for integrating cosines of linear | |
26249 | terms: @samp{myint(cos(y),x) := sin(y)/b :: let([a,b,x] := lin(y,x))}. | |
26250 | The @code{lin} function returns a 3-vector if its argument is linear, | |
26251 | or leaves itself unevaluated if not. But an unevaluated @code{lin} | |
26252 | call will not match the 3-vector on the lefthand side of the @code{let}, | |
26253 | so this @code{let} both verifies that @code{y} is linear, and binds | |
26254 | the coefficients @code{a} and @code{b} for use elsewhere in the rule. | |
26255 | (It would have been possible to use @samp{sin(a x + b)/b} for the | |
26256 | righthand side instead, but using @samp{sin(y)/b} avoids gratuitous | |
26257 | rearrangement of the argument of the sine.)@refill | |
26258 | ||
5d67986c RS |
26259 | @ignore |
26260 | @starindex | |
26261 | @end ignore | |
d7b8e6c6 EZ |
26262 | @tindex ierf |
26263 | Similarly, here is a rule that implements an inverse-@code{erf} | |
26264 | function. It uses @code{root} to search for a solution. If | |
26265 | @code{root} succeeds, it will return a vector of two numbers | |
26266 | where the first number is the desired solution. If no solution | |
26267 | is found, @code{root} remains in symbolic form. So we use | |
26268 | @code{let} to check that the result was indeed a vector. | |
26269 | ||
26270 | @example | |
26271 | ierf(x) := y :: let([y,z] := root(erf(a) = x, a, .5)) | |
26272 | @end example | |
26273 | ||
26274 | @item matches(v,p) | |
26275 | The meta-variable @var{v}, which must already have been matched | |
26276 | to something elsewhere in the rule, is compared against pattern | |
26277 | @var{p}. Since @code{matches} is a standard Calc function, it | |
26278 | can appear anywhere in a condition. But if it appears alone or | |
26279 | as a term of a top-level @samp{&&}, then you get the special | |
26280 | extra feature that meta-variables which are bound to things | |
26281 | inside @var{p} can be used elsewhere in the surrounding rewrite | |
26282 | rule. | |
26283 | ||
26284 | The only real difference between @samp{let(p := v)} and | |
26285 | @samp{matches(v, p)} is that the former evaluates @samp{v} using | |
26286 | the default simplifications, while the latter does not. | |
26287 | ||
26288 | @item remember | |
26289 | @vindex remember | |
26290 | This is actually a variable, not a function. If @code{remember} | |
26291 | appears as a condition in a rule, then when that rule succeeds | |
26292 | the original expression and rewritten expression are added to the | |
26293 | front of the rule set that contained the rule. If the rule set | |
26294 | was not stored in a variable, @code{remember} is ignored. The | |
26295 | lefthand side is enclosed in @code{quote} in the added rule if it | |
26296 | contains any variables. | |
26297 | ||
26298 | For example, the rule @samp{f(n) := n f(n-1) :: remember} applied | |
26299 | to @samp{f(7)} will add the rule @samp{f(7) := 7 f(6)} to the front | |
26300 | of the rule set. The rule set @code{EvalRules} works slightly | |
26301 | differently: There, the evaluation of @samp{f(6)} will complete before | |
26302 | the result is added to the rule set, in this case as @samp{f(7) := 5040}. | |
26303 | Thus @code{remember} is most useful inside @code{EvalRules}. | |
26304 | ||
26305 | It is up to you to ensure that the optimization performed by | |
26306 | @code{remember} is safe. For example, the rule @samp{foo(n) := n | |
26307 | :: evalv(eatfoo) > 0 :: remember} is a bad idea (@code{evalv} is | |
26308 | the function equivalent of the @kbd{=} command); if the variable | |
26309 | @code{eatfoo} ever contains 1, rules like @samp{foo(7) := 7} will | |
26310 | be added to the rule set and will continue to operate even if | |
26311 | @code{eatfoo} is later changed to 0. | |
26312 | ||
26313 | @item remember(c) | |
5d67986c RS |
26314 | @ignore |
26315 | @starindex | |
26316 | @end ignore | |
d7b8e6c6 EZ |
26317 | @tindex remember |
26318 | Remember the match as described above, but only if condition @cite{c} | |
26319 | is true. For example, @samp{remember(n % 4 = 0)} in the above factorial | |
26320 | rule remembers only every fourth result. Note that @samp{remember(1)} | |
26321 | is equivalent to @samp{remember}, and @samp{remember(0)} has no effect. | |
26322 | @end table | |
26323 | ||
26324 | @node Composing Patterns in Rewrite Rules, Nested Formulas with Rewrite Rules, Other Features of Rewrite Rules, Rewrite Rules | |
26325 | @subsection Composing Patterns in Rewrite Rules | |
26326 | ||
26327 | @noindent | |
26328 | There are three operators, @samp{&&&}, @samp{|||}, and @samp{!!!}, | |
26329 | that combine rewrite patterns to make larger patterns. The | |
26330 | combinations are ``and,'' ``or,'' and ``not,'' respectively, and | |
26331 | these operators are the pattern equivalents of @samp{&&}, @samp{||} | |
26332 | and @samp{!} (which operate on zero-or-nonzero logical values). | |
26333 | ||
26334 | Note that @samp{&&&}, @samp{|||}, and @samp{!!!} are left in symbolic | |
26335 | form by all regular Calc features; they have special meaning only in | |
26336 | the context of rewrite rule patterns. | |
26337 | ||
26338 | The pattern @samp{@var{p1} &&& @var{p2}} matches anything that | |
26339 | matches both @var{p1} and @var{p2}. One especially useful case is | |
26340 | when one of @var{p1} or @var{p2} is a meta-variable. For example, | |
26341 | here is a rule that operates on error forms: | |
26342 | ||
26343 | @example | |
26344 | f(x &&& a +/- b, x) := g(x) | |
26345 | @end example | |
26346 | ||
26347 | This does the same thing, but is arguably simpler than, the rule | |
26348 | ||
26349 | @example | |
26350 | f(a +/- b, a +/- b) := g(a +/- b) | |
26351 | @end example | |
26352 | ||
5d67986c RS |
26353 | @ignore |
26354 | @starindex | |
26355 | @end ignore | |
d7b8e6c6 EZ |
26356 | @tindex ends |
26357 | Here's another interesting example: | |
26358 | ||
26359 | @example | |
26360 | ends(cons(a, x) &&& rcons(y, b)) := [a, b] | |
26361 | @end example | |
26362 | ||
26363 | @noindent | |
26364 | which effectively clips out the middle of a vector leaving just | |
26365 | the first and last elements. This rule will change a one-element | |
26366 | vector @samp{[a]} to @samp{[a, a]}. The similar rule | |
26367 | ||
26368 | @example | |
26369 | ends(cons(a, rcons(y, b))) := [a, b] | |
26370 | @end example | |
26371 | ||
26372 | @noindent | |
26373 | would do the same thing except that it would fail to match a | |
26374 | one-element vector. | |
26375 | ||
26376 | @tex | |
26377 | \bigskip | |
26378 | @end tex | |
26379 | ||
26380 | The pattern @samp{@var{p1} ||| @var{p2}} matches anything that | |
26381 | matches either @var{p1} or @var{p2}. Calc first tries matching | |
26382 | against @var{p1}; if that fails, it goes on to try @var{p2}. | |
26383 | ||
5d67986c RS |
26384 | @ignore |
26385 | @starindex | |
26386 | @end ignore | |
d7b8e6c6 EZ |
26387 | @tindex curve |
26388 | A simple example of @samp{|||} is | |
26389 | ||
26390 | @example | |
26391 | curve(inf ||| -inf) := 0 | |
26392 | @end example | |
26393 | ||
26394 | @noindent | |
26395 | which converts both @samp{curve(inf)} and @samp{curve(-inf)} to zero. | |
26396 | ||
26397 | Here is a larger example: | |
26398 | ||
26399 | @example | |
26400 | log(a, b) ||| (ln(a) :: let(b := e)) := mylog(a, b) | |
26401 | @end example | |
26402 | ||
26403 | This matches both generalized and natural logarithms in a single rule. | |
26404 | Note that the @samp{::} term must be enclosed in parentheses because | |
26405 | that operator has lower precedence than @samp{|||} or @samp{:=}. | |
26406 | ||
26407 | (In practice this rule would probably include a third alternative, | |
26408 | omitted here for brevity, to take care of @code{log10}.) | |
26409 | ||
26410 | While Calc generally treats interior conditions exactly the same as | |
26411 | conditions on the outside of a rule, it does guarantee that if all the | |
26412 | variables in the condition are special names like @code{e}, or already | |
26413 | bound in the pattern to which the condition is attached (say, if | |
26414 | @samp{a} had appeared in this condition), then Calc will process this | |
26415 | condition right after matching the pattern to the left of the @samp{::}. | |
26416 | Thus, we know that @samp{b} will be bound to @samp{e} only if the | |
26417 | @code{ln} branch of the @samp{|||} was taken. | |
26418 | ||
26419 | Note that this rule was careful to bind the same set of meta-variables | |
26420 | on both sides of the @samp{|||}. Calc does not check this, but if | |
26421 | you bind a certain meta-variable only in one branch and then use that | |
26422 | meta-variable elsewhere in the rule, results are unpredictable: | |
26423 | ||
26424 | @example | |
26425 | f(a,b) ||| g(b) := h(a,b) | |
26426 | @end example | |
26427 | ||
26428 | Here if the pattern matches @samp{g(17)}, Calc makes no promises about | |
26429 | the value that will be substituted for @samp{a} on the righthand side. | |
26430 | ||
26431 | @tex | |
26432 | \bigskip | |
26433 | @end tex | |
26434 | ||
26435 | The pattern @samp{!!! @var{pat}} matches anything that does not | |
26436 | match @var{pat}. Any meta-variables that are bound while matching | |
26437 | @var{pat} remain unbound outside of @var{pat}. | |
26438 | ||
26439 | For example, | |
26440 | ||
26441 | @example | |
26442 | f(x &&& !!! a +/- b, !!![]) := g(x) | |
26443 | @end example | |
26444 | ||
26445 | @noindent | |
26446 | converts @code{f} whose first argument is anything @emph{except} an | |
26447 | error form, and whose second argument is not the empty vector, into | |
26448 | a similar call to @code{g} (but without the second argument). | |
26449 | ||
26450 | If we know that the second argument will be a vector (empty or not), | |
26451 | then an equivalent rule would be: | |
26452 | ||
26453 | @example | |
26454 | f(x, y) := g(x) :: typeof(x) != 7 :: vlen(y) > 0 | |
26455 | @end example | |
26456 | ||
26457 | @noindent | |
26458 | where of course 7 is the @code{typeof} code for error forms. | |
26459 | Another final condition, that works for any kind of @samp{y}, | |
26460 | would be @samp{!istrue(y == [])}. (The @code{istrue} function | |
26461 | returns an explicit 0 if its argument was left in symbolic form; | |
26462 | plain @samp{!(y == [])} or @samp{y != []} would not work to replace | |
26463 | @samp{!!![]} since these would be left unsimplified, and thus cause | |
26464 | the rule to fail, if @samp{y} was something like a variable name.) | |
26465 | ||
26466 | It is possible for a @samp{!!!} to refer to meta-variables bound | |
26467 | elsewhere in the pattern. For example, | |
26468 | ||
26469 | @example | |
26470 | f(a, !!!a) := g(a) | |
26471 | @end example | |
26472 | ||
26473 | @noindent | |
26474 | matches any call to @code{f} with different arguments, changing | |
26475 | this to @code{g} with only the first argument. | |
26476 | ||
26477 | If a function call is to be matched and one of the argument patterns | |
26478 | contains a @samp{!!!} somewhere inside it, that argument will be | |
26479 | matched last. Thus | |
26480 | ||
26481 | @example | |
26482 | f(!!!a, a) := g(a) | |
26483 | @end example | |
26484 | ||
26485 | @noindent | |
26486 | will be careful to bind @samp{a} to the second argument of @code{f} | |
26487 | before testing the first argument. If Calc had tried to match the | |
26488 | first argument of @code{f} first, the results would have been | |
28665d46 | 26489 | disastrous: since @code{a} was unbound so far, the pattern @samp{a} |
d7b8e6c6 EZ |
26490 | would have matched anything at all, and the pattern @samp{!!!a} |
26491 | therefore would @emph{not} have matched anything at all! | |
26492 | ||
26493 | @node Nested Formulas with Rewrite Rules, Multi-Phase Rewrite Rules, Composing Patterns in Rewrite Rules, Rewrite Rules | |
26494 | @subsection Nested Formulas with Rewrite Rules | |
26495 | ||
26496 | @noindent | |
26497 | When @kbd{a r} (@code{calc-rewrite}) is used, it takes an expression from | |
26498 | the top of the stack and attempts to match any of the specified rules | |
26499 | to any part of the expression, starting with the whole expression | |
26500 | and then, if that fails, trying deeper and deeper sub-expressions. | |
26501 | For each part of the expression, the rules are tried in the order | |
26502 | they appear in the rules vector. The first rule to match the first | |
26503 | sub-expression wins; it replaces the matched sub-expression according | |
26504 | to the @var{new} part of the rule. | |
26505 | ||
26506 | Often, the rule set will match and change the formula several times. | |
26507 | The top-level formula is first matched and substituted repeatedly until | |
26508 | it no longer matches the pattern; then, sub-formulas are tried, and | |
26509 | so on. Once every part of the formula has gotten its chance, the | |
26510 | rewrite mechanism starts over again with the top-level formula | |
26511 | (in case a substitution of one of its arguments has caused it again | |
26512 | to match). This continues until no further matches can be made | |
26513 | anywhere in the formula. | |
26514 | ||
26515 | It is possible for a rule set to get into an infinite loop. The | |
26516 | most obvious case, replacing a formula with itself, is not a problem | |
26517 | because a rule is not considered to ``succeed'' unless the righthand | |
26518 | side actually comes out to something different than the original | |
26519 | formula or sub-formula that was matched. But if you accidentally | |
26520 | had both @samp{ln(a b) := ln(a) + ln(b)} and the reverse | |
26521 | @samp{ln(a) + ln(b) := ln(a b)} in your rule set, Calc would | |
26522 | run forever switching a formula back and forth between the two | |
26523 | forms. | |
26524 | ||
26525 | To avoid disaster, Calc normally stops after 100 changes have been | |
26526 | made to the formula. This will be enough for most multiple rewrites, | |
26527 | but it will keep an endless loop of rewrites from locking up the | |
26528 | computer forever. (On most systems, you can also type @kbd{C-g} to | |
26529 | halt any Emacs command prematurely.) | |
26530 | ||
26531 | To change this limit, give a positive numeric prefix argument. | |
26532 | In particular, @kbd{M-1 a r} applies only one rewrite at a time, | |
26533 | useful when you are first testing your rule (or just if repeated | |
26534 | rewriting is not what is called for by your application). | |
26535 | ||
5d67986c RS |
26536 | @ignore |
26537 | @starindex | |
26538 | @end ignore | |
26539 | @ignore | |
26540 | @mindex iter@idots | |
26541 | @end ignore | |
d7b8e6c6 EZ |
26542 | @tindex iterations |
26543 | You can also put a ``function call'' @samp{iterations(@var{n})} | |
26544 | in place of a rule anywhere in your rules vector (but usually at | |
26545 | the top). Then, @var{n} will be used instead of 100 as the default | |
26546 | number of iterations for this rule set. You can use | |
26547 | @samp{iterations(inf)} if you want no iteration limit by default. | |
26548 | A prefix argument will override the @code{iterations} limit in the | |
26549 | rule set. | |
26550 | ||
26551 | @example | |
26552 | [ iterations(1), | |
26553 | f(x) := f(x+1) ] | |
26554 | @end example | |
26555 | ||
26556 | More precisely, the limit controls the number of ``iterations,'' | |
26557 | where each iteration is a successful matching of a rule pattern whose | |
26558 | righthand side, after substituting meta-variables and applying the | |
26559 | default simplifications, is different from the original sub-formula | |
26560 | that was matched. | |
26561 | ||
26562 | A prefix argument of zero sets the limit to infinity. Use with caution! | |
26563 | ||
26564 | Given a negative numeric prefix argument, @kbd{a r} will match and | |
26565 | substitute the top-level expression up to that many times, but | |
26566 | will not attempt to match the rules to any sub-expressions. | |
26567 | ||
26568 | In a formula, @code{rewrite(@var{expr}, @var{rules}, @var{n})} | |
26569 | does a rewriting operation. Here @var{expr} is the expression | |
26570 | being rewritten, @var{rules} is the rule, vector of rules, or | |
26571 | variable containing the rules, and @var{n} is the optional | |
26572 | iteration limit, which may be a positive integer, a negative | |
26573 | integer, or @samp{inf} or @samp{-inf}. If @var{n} is omitted | |
26574 | the @code{iterations} value from the rule set is used; if both | |
26575 | are omitted, 100 is used. | |
26576 | ||
26577 | @node Multi-Phase Rewrite Rules, Selections with Rewrite Rules, Nested Formulas with Rewrite Rules, Rewrite Rules | |
26578 | @subsection Multi-Phase Rewrite Rules | |
26579 | ||
26580 | @noindent | |
26581 | It is possible to separate a rewrite rule set into several @dfn{phases}. | |
26582 | During each phase, certain rules will be enabled while certain others | |
26583 | will be disabled. A @dfn{phase schedule} controls the order in which | |
26584 | phases occur during the rewriting process. | |
26585 | ||
5d67986c RS |
26586 | @ignore |
26587 | @starindex | |
26588 | @end ignore | |
d7b8e6c6 EZ |
26589 | @tindex phase |
26590 | @vindex all | |
26591 | If a call to the marker function @code{phase} appears in the rules | |
26592 | vector in place of a rule, all rules following that point will be | |
26593 | members of the phase(s) identified in the arguments to @code{phase}. | |
26594 | Phases are given integer numbers. The markers @samp{phase()} and | |
26595 | @samp{phase(all)} both mean the following rules belong to all phases; | |
26596 | this is the default at the start of the rule set. | |
26597 | ||
26598 | If you do not explicitly schedule the phases, Calc sorts all phase | |
26599 | numbers that appear in the rule set and executes the phases in | |
26600 | ascending order. For example, the rule set | |
26601 | ||
d7b8e6c6 | 26602 | @example |
5d67986c | 26603 | @group |
d7b8e6c6 EZ |
26604 | [ f0(x) := g0(x), |
26605 | phase(1), | |
26606 | f1(x) := g1(x), | |
26607 | phase(2), | |
26608 | f2(x) := g2(x), | |
26609 | phase(3), | |
26610 | f3(x) := g3(x), | |
26611 | phase(1,2), | |
26612 | f4(x) := g4(x) ] | |
d7b8e6c6 | 26613 | @end group |
5d67986c | 26614 | @end example |
d7b8e6c6 EZ |
26615 | |
26616 | @noindent | |
26617 | has three phases, 1 through 3. Phase 1 consists of the @code{f0}, | |
26618 | @code{f1}, and @code{f4} rules (in that order). Phase 2 consists of | |
26619 | @code{f0}, @code{f2}, and @code{f4}. Phase 3 consists of @code{f0} | |
26620 | and @code{f3}. | |
26621 | ||
26622 | When Calc rewrites a formula using this rule set, it first rewrites | |
26623 | the formula using only the phase 1 rules until no further changes are | |
26624 | possible. Then it switches to the phase 2 rule set and continues | |
26625 | until no further changes occur, then finally rewrites with phase 3. | |
26626 | When no more phase 3 rules apply, rewriting finishes. (This is | |
26627 | assuming @kbd{a r} with a large enough prefix argument to allow the | |
26628 | rewriting to run to completion; the sequence just described stops | |
26629 | early if the number of iterations specified in the prefix argument, | |
26630 | 100 by default, is reached.) | |
26631 | ||
26632 | During each phase, Calc descends through the nested levels of the | |
26633 | formula as described previously. (@xref{Nested Formulas with Rewrite | |
26634 | Rules}.) Rewriting starts at the top of the formula, then works its | |
26635 | way down to the parts, then goes back to the top and works down again. | |
26636 | The phase 2 rules do not begin until no phase 1 rules apply anywhere | |
26637 | in the formula. | |
26638 | ||
5d67986c RS |
26639 | @ignore |
26640 | @starindex | |
26641 | @end ignore | |
d7b8e6c6 EZ |
26642 | @tindex schedule |
26643 | A @code{schedule} marker appearing in the rule set (anywhere, but | |
26644 | conventionally at the top) changes the default schedule of phases. | |
26645 | In the simplest case, @code{schedule} has a sequence of phase numbers | |
26646 | for arguments; each phase number is invoked in turn until the | |
26647 | arguments to @code{schedule} are exhausted. Thus adding | |
26648 | @samp{schedule(3,2,1)} at the top of the above rule set would | |
26649 | reverse the order of the phases; @samp{schedule(1,2,3)} would have | |
26650 | no effect since this is the default schedule; and @samp{schedule(1,2,1,3)} | |
26651 | would give phase 1 a second chance after phase 2 has completed, before | |
26652 | moving on to phase 3. | |
26653 | ||
26654 | Any argument to @code{schedule} can instead be a vector of phase | |
26655 | numbers (or even of sub-vectors). Then the sub-sequence of phases | |
26656 | described by the vector are tried repeatedly until no change occurs | |
26657 | in any phase in the sequence. For example, @samp{schedule([1, 2], 3)} | |
26658 | tries phase 1, then phase 2, then, if either phase made any changes | |
26659 | to the formula, repeats these two phases until they can make no | |
26660 | further progress. Finally, it goes on to phase 3 for finishing | |
26661 | touches. | |
26662 | ||
26663 | Also, items in @code{schedule} can be variable names as well as | |
26664 | numbers. A variable name is interpreted as the name of a function | |
26665 | to call on the whole formula. For example, @samp{schedule(1, simplify)} | |
26666 | says to apply the phase-1 rules (presumably, all of them), then to | |
26667 | call @code{simplify} which is the function name equivalent of @kbd{a s}. | |
26668 | Likewise, @samp{schedule([1, simplify])} says to alternate between | |
26669 | phase 1 and @kbd{a s} until no further changes occur. | |
26670 | ||
26671 | Phases can be used purely to improve efficiency; if it is known that | |
26672 | a certain group of rules will apply only at the beginning of rewriting, | |
26673 | and a certain other group will apply only at the end, then rewriting | |
26674 | will be faster if these groups are identified as separate phases. | |
26675 | Once the phase 1 rules are done, Calc can put them aside and no longer | |
26676 | spend any time on them while it works on phase 2. | |
26677 | ||
26678 | There are also some problems that can only be solved with several | |
26679 | rewrite phases. For a real-world example of a multi-phase rule set, | |
26680 | examine the set @code{FitRules}, which is used by the curve-fitting | |
26681 | command to convert a model expression to linear form. | |
26682 | @xref{Curve Fitting Details}. This set is divided into four phases. | |
26683 | The first phase rewrites certain kinds of expressions to be more | |
26684 | easily linearizable, but less computationally efficient. After the | |
26685 | linear components have been picked out, the final phase includes the | |
26686 | opposite rewrites to put each component back into an efficient form. | |
26687 | If both sets of rules were included in one big phase, Calc could get | |
26688 | into an infinite loop going back and forth between the two forms. | |
26689 | ||
26690 | Elsewhere in @code{FitRules}, the components are first isolated, | |
26691 | then recombined where possible to reduce the complexity of the linear | |
26692 | fit, then finally packaged one component at a time into vectors. | |
26693 | If the packaging rules were allowed to begin before the recombining | |
26694 | rules were finished, some components might be put away into vectors | |
26695 | before they had a chance to recombine. By putting these rules in | |
26696 | two separate phases, this problem is neatly avoided. | |
26697 | ||
26698 | @node Selections with Rewrite Rules, Matching Commands, Multi-Phase Rewrite Rules, Rewrite Rules | |
26699 | @subsection Selections with Rewrite Rules | |
26700 | ||
26701 | @noindent | |
26702 | If a sub-formula of the current formula is selected (as by @kbd{j s}; | |
26703 | @pxref{Selecting Subformulas}), the @kbd{a r} (@code{calc-rewrite}) | |
26704 | command applies only to that sub-formula. Together with a negative | |
26705 | prefix argument, you can use this fact to apply a rewrite to one | |
26706 | specific part of a formula without affecting any other parts. | |
26707 | ||
26708 | @kindex j r | |
26709 | @pindex calc-rewrite-selection | |
26710 | The @kbd{j r} (@code{calc-rewrite-selection}) command allows more | |
26711 | sophisticated operations on selections. This command prompts for | |
26712 | the rules in the same way as @kbd{a r}, but it then applies those | |
26713 | rules to the whole formula in question even though a sub-formula | |
26714 | of it has been selected. However, the selected sub-formula will | |
26715 | first have been surrounded by a @samp{select( )} function call. | |
26716 | (Calc's evaluator does not understand the function name @code{select}; | |
26717 | this is only a tag used by the @kbd{j r} command.) | |
26718 | ||
26719 | For example, suppose the formula on the stack is @samp{2 (a + b)^2} | |
26720 | and the sub-formula @samp{a + b} is selected. This formula will | |
26721 | be rewritten to @samp{2 select(a + b)^2} and then the rewrite | |
26722 | rules will be applied in the usual way. The rewrite rules can | |
26723 | include references to @code{select} to tell where in the pattern | |
26724 | the selected sub-formula should appear. | |
26725 | ||
26726 | If there is still exactly one @samp{select( )} function call in | |
26727 | the formula after rewriting is done, it indicates which part of | |
26728 | the formula should be selected afterwards. Otherwise, the | |
26729 | formula will be unselected. | |
26730 | ||
26731 | You can make @kbd{j r} act much like @kbd{a r} by enclosing both parts | |
26732 | of the rewrite rule with @samp{select()}. However, @kbd{j r} | |
26733 | allows you to use the current selection in more flexible ways. | |
26734 | Suppose you wished to make a rule which removed the exponent from | |
26735 | the selected term; the rule @samp{select(a)^x := select(a)} would | |
26736 | work. In the above example, it would rewrite @samp{2 select(a + b)^2} | |
26737 | to @samp{2 select(a + b)}. This would then be returned to the | |
26738 | stack as @samp{2 (a + b)} with the @samp{a + b} selected. | |
26739 | ||
26740 | The @kbd{j r} command uses one iteration by default, unlike | |
26741 | @kbd{a r} which defaults to 100 iterations. A numeric prefix | |
26742 | argument affects @kbd{j r} in the same way as @kbd{a r}. | |
26743 | @xref{Nested Formulas with Rewrite Rules}. | |
26744 | ||
26745 | As with other selection commands, @kbd{j r} operates on the stack | |
26746 | entry that contains the cursor. (If the cursor is on the top-of-stack | |
26747 | @samp{.} marker, it works as if the cursor were on the formula | |
26748 | at stack level 1.) | |
26749 | ||
26750 | If you don't specify a set of rules, the rules are taken from the | |
26751 | top of the stack, just as with @kbd{a r}. In this case, the | |
26752 | cursor must indicate stack entry 2 or above as the formula to be | |
26753 | rewritten (otherwise the same formula would be used as both the | |
26754 | target and the rewrite rules). | |
26755 | ||
26756 | If the indicated formula has no selection, the cursor position within | |
26757 | the formula temporarily selects a sub-formula for the purposes of this | |
26758 | command. If the cursor is not on any sub-formula (e.g., it is in | |
26759 | the line-number area to the left of the formula), the @samp{select( )} | |
26760 | markers are ignored by the rewrite mechanism and the rules are allowed | |
26761 | to apply anywhere in the formula. | |
26762 | ||
26763 | As a special feature, the normal @kbd{a r} command also ignores | |
26764 | @samp{select( )} calls in rewrite rules. For example, if you used the | |
26765 | above rule @samp{select(a)^x := select(a)} with @kbd{a r}, it would apply | |
26766 | the rule as if it were @samp{a^x := a}. Thus, you can write general | |
26767 | purpose rules with @samp{select( )} hints inside them so that they | |
26768 | will ``do the right thing'' in both @kbd{a r} and @kbd{j r}, | |
26769 | both with and without selections. | |
26770 | ||
26771 | @node Matching Commands, Automatic Rewrites, Selections with Rewrite Rules, Rewrite Rules | |
26772 | @subsection Matching Commands | |
26773 | ||
26774 | @noindent | |
26775 | @kindex a m | |
26776 | @pindex calc-match | |
26777 | @tindex match | |
26778 | The @kbd{a m} (@code{calc-match}) [@code{match}] function takes a | |
26779 | vector of formulas and a rewrite-rule-style pattern, and produces | |
26780 | a vector of all formulas which match the pattern. The command | |
26781 | prompts you to enter the pattern; as for @kbd{a r}, you can enter | |
26782 | a single pattern (i.e., a formula with meta-variables), or a | |
26783 | vector of patterns, or a variable which contains patterns, or | |
26784 | you can give a blank response in which case the patterns are taken | |
26785 | from the top of the stack. The pattern set will be compiled once | |
26786 | and saved if it is stored in a variable. If there are several | |
26787 | patterns in the set, vector elements are kept if they match any | |
26788 | of the patterns. | |
26789 | ||
26790 | For example, @samp{match(a+b, [x, x+y, x-y, 7, x+y+z])} | |
26791 | will return @samp{[x+y, x-y, x+y+z]}. | |
26792 | ||
26793 | The @code{import} mechanism is not available for pattern sets. | |
26794 | ||
26795 | The @kbd{a m} command can also be used to extract all vector elements | |
26796 | which satisfy any condition: The pattern @samp{x :: x>0} will select | |
26797 | all the positive vector elements. | |
26798 | ||
26799 | @kindex I a m | |
26800 | @tindex matchnot | |
26801 | With the Inverse flag [@code{matchnot}], this command extracts all | |
26802 | vector elements which do @emph{not} match the given pattern. | |
26803 | ||
5d67986c RS |
26804 | @ignore |
26805 | @starindex | |
26806 | @end ignore | |
d7b8e6c6 EZ |
26807 | @tindex matches |
26808 | There is also a function @samp{matches(@var{x}, @var{p})} which | |
26809 | evaluates to 1 if expression @var{x} matches pattern @var{p}, or | |
26810 | to 0 otherwise. This is sometimes useful for including into the | |
26811 | conditional clauses of other rewrite rules. | |
26812 | ||
5d67986c RS |
26813 | @ignore |
26814 | @starindex | |
26815 | @end ignore | |
d7b8e6c6 EZ |
26816 | @tindex vmatches |
26817 | The function @code{vmatches} is just like @code{matches}, except | |
26818 | that if the match succeeds it returns a vector of assignments to | |
26819 | the meta-variables instead of the number 1. For example, | |
26820 | @samp{vmatches(f(1,2), f(a,b))} returns @samp{[a := 1, b := 2]}. | |
26821 | If the match fails, the function returns the number 0. | |
26822 | ||
26823 | @node Automatic Rewrites, Debugging Rewrites, Matching Commands, Rewrite Rules | |
26824 | @subsection Automatic Rewrites | |
26825 | ||
26826 | @noindent | |
26827 | @cindex @code{EvalRules} variable | |
26828 | @vindex EvalRules | |
26829 | It is possible to get Calc to apply a set of rewrite rules on all | |
26830 | results, effectively adding to the built-in set of default | |
26831 | simplifications. To do this, simply store your rule set in the | |
26832 | variable @code{EvalRules}. There is a convenient @kbd{s E} command | |
26833 | for editing @code{EvalRules}; @pxref{Operations on Variables}. | |
26834 | ||
26835 | For example, suppose you want @samp{sin(a + b)} to be expanded out | |
26836 | to @samp{sin(b) cos(a) + cos(b) sin(a)} wherever it appears, and | |
26837 | similarly for @samp{cos(a + b)}. The corresponding rewrite rule | |
26838 | set would be, | |
26839 | ||
d7b8e6c6 | 26840 | @smallexample |
5d67986c | 26841 | @group |
d7b8e6c6 EZ |
26842 | [ sin(a + b) := cos(a) sin(b) + sin(a) cos(b), |
26843 | cos(a + b) := cos(a) cos(b) - sin(a) sin(b) ] | |
d7b8e6c6 | 26844 | @end group |
5d67986c | 26845 | @end smallexample |
d7b8e6c6 EZ |
26846 | |
26847 | To apply these manually, you could put them in a variable called | |
26848 | @code{trigexp} and then use @kbd{a r trigexp} every time you wanted | |
26849 | to expand trig functions. But if instead you store them in the | |
26850 | variable @code{EvalRules}, they will automatically be applied to all | |
26851 | sines and cosines of sums. Then, with @samp{2 x} and @samp{45} on | |
26852 | the stack, typing @kbd{+ S} will (assuming degrees mode) result in | |
26853 | @samp{0.7071 sin(2 x) + 0.7071 cos(2 x)} automatically. | |
26854 | ||
26855 | As each level of a formula is evaluated, the rules from | |
26856 | @code{EvalRules} are applied before the default simplifications. | |
26857 | Rewriting continues until no further @code{EvalRules} apply. | |
26858 | Note that this is different from the usual order of application of | |
26859 | rewrite rules: @code{EvalRules} works from the bottom up, simplifying | |
26860 | the arguments to a function before the function itself, while @kbd{a r} | |
26861 | applies rules from the top down. | |
26862 | ||
26863 | Because the @code{EvalRules} are tried first, you can use them to | |
26864 | override the normal behavior of any built-in Calc function. | |
26865 | ||
26866 | It is important not to write a rule that will get into an infinite | |
26867 | loop. For example, the rule set @samp{[f(0) := 1, f(n) := n f(n-1)]} | |
26868 | appears to be a good definition of a factorial function, but it is | |
26869 | unsafe. Imagine what happens if @samp{f(2.5)} is simplified. Calc | |
26870 | will continue to subtract 1 from this argument forever without reaching | |
26871 | zero. A safer second rule would be @samp{f(n) := n f(n-1) :: n>0}. | |
26872 | Another dangerous rule is @samp{g(x, y) := g(y, x)}. Rewriting | |
26873 | @samp{g(2, 4)}, this would bounce back and forth between that and | |
26874 | @samp{g(4, 2)} forever. If an infinite loop in @code{EvalRules} | |
26875 | occurs, Emacs will eventually stop with a ``Computation got stuck | |
26876 | or ran too long'' message. | |
26877 | ||
26878 | Another subtle difference between @code{EvalRules} and regular rewrites | |
26879 | concerns rules that rewrite a formula into an identical formula. For | |
26880 | example, @samp{f(n) := f(floor(n))} ``fails to match'' when @cite{n} is | |
26881 | already an integer. But in @code{EvalRules} this case is detected only | |
26882 | if the righthand side literally becomes the original formula before any | |
26883 | further simplification. This means that @samp{f(n) := f(floor(n))} will | |
26884 | get into an infinite loop if it occurs in @code{EvalRules}. Calc will | |
26885 | replace @samp{f(6)} with @samp{f(floor(6))}, which is different from | |
26886 | @samp{f(6)}, so it will consider the rule to have matched and will | |
26887 | continue simplifying that formula; first the argument is simplified | |
26888 | to get @samp{f(6)}, then the rule matches again to get @samp{f(floor(6))} | |
26889 | again, ad infinitum. A much safer rule would check its argument first, | |
26890 | say, with @samp{f(n) := f(floor(n)) :: !dint(n)}. | |
26891 | ||
26892 | (What really happens is that the rewrite mechanism substitutes the | |
26893 | meta-variables in the righthand side of a rule, compares to see if the | |
26894 | result is the same as the original formula and fails if so, then uses | |
26895 | the default simplifications to simplify the result and compares again | |
26896 | (and again fails if the formula has simplified back to its original | |
26897 | form). The only special wrinkle for the @code{EvalRules} is that the | |
26898 | same rules will come back into play when the default simplifications | |
26899 | are used. What Calc wants to do is build @samp{f(floor(6))}, see that | |
26900 | this is different from the original formula, simplify to @samp{f(6)}, | |
26901 | see that this is the same as the original formula, and thus halt the | |
26902 | rewriting. But while simplifying, @samp{f(6)} will again trigger | |
26903 | the same @code{EvalRules} rule and Calc will get into a loop inside | |
26904 | the rewrite mechanism itself.) | |
26905 | ||
26906 | The @code{phase}, @code{schedule}, and @code{iterations} markers do | |
26907 | not work in @code{EvalRules}. If the rule set is divided into phases, | |
26908 | only the phase 1 rules are applied, and the schedule is ignored. | |
26909 | The rules are always repeated as many times as possible. | |
26910 | ||
26911 | The @code{EvalRules} are applied to all function calls in a formula, | |
26912 | but not to numbers (and other number-like objects like error forms), | |
26913 | nor to vectors or individual variable names. (Though they will apply | |
26914 | to @emph{components} of vectors and error forms when appropriate.) You | |
26915 | might try to make a variable @code{phihat} which automatically expands | |
26916 | to its definition without the need to press @kbd{=} by writing the | |
26917 | rule @samp{quote(phihat) := (1-sqrt(5))/2}, but unfortunately this rule | |
26918 | will not work as part of @code{EvalRules}. | |
26919 | ||
26920 | Finally, another limitation is that Calc sometimes calls its built-in | |
26921 | functions directly rather than going through the default simplifications. | |
26922 | When it does this, @code{EvalRules} will not be able to override those | |
26923 | functions. For example, when you take the absolute value of the complex | |
26924 | number @cite{(2, 3)}, Calc computes @samp{sqrt(2*2 + 3*3)} by calling | |
26925 | the multiplication, addition, and square root functions directly rather | |
26926 | than applying the default simplifications to this formula. So an | |
26927 | @code{EvalRules} rule that (perversely) rewrites @samp{sqrt(13) := 6} | |
26928 | would not apply. (However, if you put Calc into symbolic mode so that | |
26929 | @samp{sqrt(13)} will be left in symbolic form by the built-in square | |
26930 | root function, your rule will be able to apply. But if the complex | |
26931 | number were @cite{(3,4)}, so that @samp{sqrt(25)} must be calculated, | |
26932 | then symbolic mode will not help because @samp{sqrt(25)} can be | |
26933 | evaluated exactly to 5.) | |
26934 | ||
26935 | One subtle restriction that normally only manifests itself with | |
26936 | @code{EvalRules} is that while a given rewrite rule is in the process | |
26937 | of being checked, that same rule cannot be recursively applied. Calc | |
26938 | effectively removes the rule from its rule set while checking the rule, | |
26939 | then puts it back once the match succeeds or fails. (The technical | |
26940 | reason for this is that compiled pattern programs are not reentrant.) | |
26941 | For example, consider the rule @samp{foo(x) := x :: foo(x/2) > 0} | |
26942 | attempting to match @samp{foo(8)}. This rule will be inactive while | |
26943 | the condition @samp{foo(4) > 0} is checked, even though it might be | |
26944 | an integral part of evaluating that condition. Note that this is not | |
26945 | a problem for the more usual recursive type of rule, such as | |
26946 | @samp{foo(x) := foo(x/2)}, because there the rule has succeeded and | |
26947 | been reactivated by the time the righthand side is evaluated. | |
26948 | ||
26949 | If @code{EvalRules} has no stored value (its default state), or if | |
26950 | anything but a vector is stored in it, then it is ignored. | |
26951 | ||
26952 | Even though Calc's rewrite mechanism is designed to compare rewrite | |
26953 | rules to formulas as quickly as possible, storing rules in | |
26954 | @code{EvalRules} may make Calc run substantially slower. This is | |
26955 | particularly true of rules where the top-level call is a commonly used | |
26956 | function, or is not fixed. The rule @samp{f(n) := n f(n-1) :: n>0} will | |
26957 | only activate the rewrite mechanism for calls to the function @code{f}, | |
26958 | but @samp{lg(n) + lg(m) := lg(n m)} will check every @samp{+} operator. | |
5d67986c RS |
26959 | |
26960 | @smallexample | |
26961 | apply(f, [a*b]) := apply(f, [a]) + apply(f, [b]) :: in(f, [ln, log10]) | |
26962 | @end smallexample | |
26963 | ||
26964 | @noindent | |
26965 | may seem more ``efficient'' than two separate rules for @code{ln} and | |
26966 | @code{log10}, but actually it is vastly less efficient because rules | |
26967 | with @code{apply} as the top-level pattern must be tested against | |
26968 | @emph{every} function call that is simplified. | |
d7b8e6c6 EZ |
26969 | |
26970 | @cindex @code{AlgSimpRules} variable | |
26971 | @vindex AlgSimpRules | |
26972 | Suppose you want @samp{sin(a + b)} to be expanded out not all the time, | |
26973 | but only when @kbd{a s} is used to simplify the formula. The variable | |
26974 | @code{AlgSimpRules} holds rules for this purpose. The @kbd{a s} command | |
26975 | will apply @code{EvalRules} and @code{AlgSimpRules} to the formula, as | |
26976 | well as all of its built-in simplifications. | |
26977 | ||
26978 | Most of the special limitations for @code{EvalRules} don't apply to | |
26979 | @code{AlgSimpRules}. Calc simply does an @kbd{a r AlgSimpRules} | |
26980 | command with an infinite repeat count as the first step of @kbd{a s}. | |
26981 | It then applies its own built-in simplifications throughout the | |
26982 | formula, and then repeats these two steps (along with applying the | |
26983 | default simplifications) until no further changes are possible. | |
26984 | ||
26985 | @cindex @code{ExtSimpRules} variable | |
26986 | @cindex @code{UnitSimpRules} variable | |
26987 | @vindex ExtSimpRules | |
26988 | @vindex UnitSimpRules | |
26989 | There are also @code{ExtSimpRules} and @code{UnitSimpRules} variables | |
26990 | that are used by @kbd{a e} and @kbd{u s}, respectively; these commands | |
26991 | also apply @code{EvalRules} and @code{AlgSimpRules}. The variable | |
26992 | @code{IntegSimpRules} contains simplification rules that are used | |
26993 | only during integration by @kbd{a i}. | |
26994 | ||
26995 | @node Debugging Rewrites, Examples of Rewrite Rules, Automatic Rewrites, Rewrite Rules | |
26996 | @subsection Debugging Rewrites | |
26997 | ||
26998 | @noindent | |
26999 | If a buffer named @samp{*Trace*} exists, the rewrite mechanism will | |
27000 | record some useful information there as it operates. The original | |
27001 | formula is written there, as is the result of each successful rewrite, | |
27002 | and the final result of the rewriting. All phase changes are also | |
27003 | noted. | |
27004 | ||
27005 | Calc always appends to @samp{*Trace*}. You must empty this buffer | |
27006 | yourself periodically if it is in danger of growing unwieldy. | |
27007 | ||
27008 | Note that the rewriting mechanism is substantially slower when the | |
27009 | @samp{*Trace*} buffer exists, even if the buffer is not visible on | |
27010 | the screen. Once you are done, you will probably want to kill this | |
27011 | buffer (with @kbd{C-x k *Trace* @key{RET}}). If you leave it in | |
27012 | existence and forget about it, all your future rewrite commands will | |
27013 | be needlessly slow. | |
27014 | ||
27015 | @node Examples of Rewrite Rules, , Debugging Rewrites, Rewrite Rules | |
27016 | @subsection Examples of Rewrite Rules | |
27017 | ||
27018 | @noindent | |
27019 | Returning to the example of substituting the pattern | |
27020 | @samp{sin(x)^2 + cos(x)^2} with 1, we saw that the rule | |
27021 | @samp{opt(a) sin(x)^2 + opt(a) cos(x)^2 := a} does a good job of | |
27022 | finding suitable cases. Another solution would be to use the rule | |
27023 | @samp{cos(x)^2 := 1 - sin(x)^2}, followed by algebraic simplification | |
27024 | if necessary. This rule will be the most effective way to do the job, | |
27025 | but at the expense of making some changes that you might not desire.@refill | |
27026 | ||
27027 | Another algebraic rewrite rule is @samp{exp(x+y) := exp(x) exp(y)}. | |
27028 | To make this work with the @w{@kbd{j r}} command so that it can be | |
27029 | easily targeted to a particular exponential in a large formula, | |
27030 | you might wish to write the rule as @samp{select(exp(x+y)) := | |
27031 | select(exp(x) exp(y))}. The @samp{select} markers will be | |
27032 | ignored by the regular @kbd{a r} command | |
27033 | (@pxref{Selections with Rewrite Rules}).@refill | |
27034 | ||
27035 | A surprisingly useful rewrite rule is @samp{a/(b-c) := a*(b+c)/(b^2-c^2)}. | |
27036 | This will simplify the formula whenever @cite{b} and/or @cite{c} can | |
27037 | be made simpler by squaring. For example, applying this rule to | |
27038 | @samp{2 / (sqrt(2) + 3)} yields @samp{6:7 - 2:7 sqrt(2)} (assuming | |
27039 | Symbolic Mode has been enabled to keep the square root from being | |
28665d46 | 27040 | evaluated to a floating-point approximation). This rule is also |
d7b8e6c6 EZ |
27041 | useful when working with symbolic complex numbers, e.g., |
27042 | @samp{(a + b i) / (c + d i)}. | |
27043 | ||
27044 | As another example, we could define our own ``triangular numbers'' function | |
27045 | with the rules @samp{[tri(0) := 0, tri(n) := n + tri(n-1) :: n>0]}. Enter | |
27046 | this vector and store it in a variable: @kbd{@w{s t} trirules}. Now, given | |
27047 | a suitable formula like @samp{tri(5)} on the stack, type @samp{a r trirules} | |
27048 | to apply these rules repeatedly. After six applications, @kbd{a r} will | |
27049 | stop with 15 on the stack. Once these rules are debugged, it would probably | |
27050 | be most useful to add them to @code{EvalRules} so that Calc will evaluate | |
27051 | the new @code{tri} function automatically. We could then use @kbd{Z K} on | |
5d67986c | 27052 | the keyboard macro @kbd{' tri($) @key{RET}} to make a command that applies |
d7b8e6c6 EZ |
27053 | @code{tri} to the value on the top of the stack. @xref{Programming}. |
27054 | ||
27055 | @cindex Quaternions | |
27056 | The following rule set, contributed by @c{Fran\c cois} | |
27057 | @asis{Francois} Pinard, implements | |
27058 | @dfn{quaternions}, a generalization of the concept of complex numbers. | |
27059 | Quaternions have four components, and are here represented by function | |
27060 | calls @samp{quat(@var{w}, [@var{x}, @var{y}, @var{z}])} with ``real | |
27061 | part'' @var{w} and the three ``imaginary'' parts collected into a | |
27062 | vector. Various arithmetical operations on quaternions are supported. | |
27063 | To use these rules, either add them to @code{EvalRules}, or create a | |
27064 | command based on @kbd{a r} for simplifying quaternion formulas. | |
27065 | A convenient way to enter quaternions would be a command defined by | |
27066 | a keyboard macro containing: @kbd{' quat($$$$, [$$$, $$, $]) @key{RET}}. | |
27067 | ||
27068 | @smallexample | |
27069 | [ quat(w, x, y, z) := quat(w, [x, y, z]), | |
27070 | quat(w, [0, 0, 0]) := w, | |
27071 | abs(quat(w, v)) := hypot(w, v), | |
27072 | -quat(w, v) := quat(-w, -v), | |
27073 | r + quat(w, v) := quat(r + w, v) :: real(r), | |
27074 | r - quat(w, v) := quat(r - w, -v) :: real(r), | |
27075 | quat(w1, v1) + quat(w2, v2) := quat(w1 + w2, v1 + v2), | |
27076 | r * quat(w, v) := quat(r * w, r * v) :: real(r), | |
27077 | plain(quat(w1, v1) * quat(w2, v2)) | |
27078 | := quat(w1 * w2 - v1 * v2, w1 * v2 + w2 * v1 + cross(v1, v2)), | |
27079 | quat(w1, v1) / r := quat(w1 / r, v1 / r) :: real(r), | |
27080 | z / quat(w, v) := z * quatinv(quat(w, v)), | |
27081 | quatinv(quat(w, v)) := quat(w, -v) / (w^2 + v^2), | |
27082 | quatsqr(quat(w, v)) := quat(w^2 - v^2, 2 * w * v), | |
27083 | quat(w, v)^k := quatsqr(quat(w, v)^(k / 2)) | |
27084 | :: integer(k) :: k > 0 :: k % 2 = 0, | |
27085 | quat(w, v)^k := quatsqr(quat(w, v)^((k - 1) / 2)) * quat(w, v) | |
27086 | :: integer(k) :: k > 2, | |
27087 | quat(w, v)^-k := quatinv(quat(w, v)^k) :: integer(k) :: k > 0 ] | |
27088 | @end smallexample | |
27089 | ||
27090 | Quaternions, like matrices, have non-commutative multiplication. | |
27091 | In other words, @cite{q1 * q2 = q2 * q1} is not necessarily true if | |
27092 | @cite{q1} and @cite{q2} are @code{quat} forms. The @samp{quat*quat} | |
27093 | rule above uses @code{plain} to prevent Calc from rearranging the | |
27094 | product. It may also be wise to add the line @samp{[quat(), matrix]} | |
27095 | to the @code{Decls} matrix, to ensure that Calc's other algebraic | |
27096 | operations will not rearrange a quaternion product. @xref{Declarations}. | |
27097 | ||
27098 | These rules also accept a four-argument @code{quat} form, converting | |
27099 | it to the preferred form in the first rule. If you would rather see | |
27100 | results in the four-argument form, just append the two items | |
27101 | @samp{phase(2), quat(w, [x, y, z]) := quat(w, x, y, z)} to the end | |
27102 | of the rule set. (But remember that multi-phase rule sets don't work | |
27103 | in @code{EvalRules}.) | |
27104 | ||
27105 | @node Units, Store and Recall, Algebra, Top | |
27106 | @chapter Operating on Units | |
27107 | ||
27108 | @noindent | |
27109 | One special interpretation of algebraic formulas is as numbers with units. | |
27110 | For example, the formula @samp{5 m / s^2} can be read ``five meters | |
27111 | per second squared.'' The commands in this chapter help you | |
27112 | manipulate units expressions in this form. Units-related commands | |
27113 | begin with the @kbd{u} prefix key. | |
27114 | ||
27115 | @menu | |
27116 | * Basic Operations on Units:: | |
27117 | * The Units Table:: | |
27118 | * Predefined Units:: | |
27119 | * User-Defined Units:: | |
27120 | @end menu | |
27121 | ||
27122 | @node Basic Operations on Units, The Units Table, Units, Units | |
27123 | @section Basic Operations on Units | |
27124 | ||
27125 | @noindent | |
27126 | A @dfn{units expression} is a formula which is basically a number | |
27127 | multiplied and/or divided by one or more @dfn{unit names}, which may | |
27128 | optionally be raised to integer powers. Actually, the value part need not | |
27129 | be a number; any product or quotient involving unit names is a units | |
27130 | expression. Many of the units commands will also accept any formula, | |
27131 | where the command applies to all units expressions which appear in the | |
27132 | formula. | |
27133 | ||
27134 | A unit name is a variable whose name appears in the @dfn{unit table}, | |
27135 | or a variable whose name is a prefix character like @samp{k} (for ``kilo'') | |
27136 | or @samp{u} (for ``micro'') followed by a name in the unit table. | |
27137 | A substantial table of built-in units is provided with Calc; | |
27138 | @pxref{Predefined Units}. You can also define your own unit names; | |
27139 | @pxref{User-Defined Units}.@refill | |
27140 | ||
27141 | Note that if the value part of a units expression is exactly @samp{1}, | |
27142 | it will be removed by the Calculator's automatic algebra routines: The | |
27143 | formula @samp{1 mm} is ``simplified'' to @samp{mm}. This is only a | |
27144 | display anomaly, however; @samp{mm} will work just fine as a | |
27145 | representation of one millimeter.@refill | |
27146 | ||
27147 | You may find that Algebraic Mode (@pxref{Algebraic Entry}) makes working | |
27148 | with units expressions easier. Otherwise, you will have to remember | |
27149 | to hit the apostrophe key every time you wish to enter units. | |
27150 | ||
27151 | @kindex u s | |
27152 | @pindex calc-simplify-units | |
5d67986c RS |
27153 | @ignore |
27154 | @mindex usimpl@idots | |
27155 | @end ignore | |
d7b8e6c6 EZ |
27156 | @tindex usimplify |
27157 | The @kbd{u s} (@code{calc-simplify-units}) [@code{usimplify}] command | |
27158 | simplifies a units | |
27159 | expression. It uses @kbd{a s} (@code{calc-simplify}) to simplify the | |
27160 | expression first as a regular algebraic formula; it then looks for | |
27161 | features that can be further simplified by converting one object's units | |
27162 | to be compatible with another's. For example, @samp{5 m + 23 mm} will | |
27163 | simplify to @samp{5.023 m}. When different but compatible units are | |
27164 | added, the righthand term's units are converted to match those of the | |
27165 | lefthand term. @xref{Simplification Modes}, for a way to have this done | |
27166 | automatically at all times.@refill | |
27167 | ||
27168 | Units simplification also handles quotients of two units with the same | |
27169 | dimensionality, as in @w{@samp{2 in s/L cm}} to @samp{5.08 s/L}; fractional | |
27170 | powers of unit expressions, as in @samp{sqrt(9 mm^2)} to @samp{3 mm} and | |
27171 | @samp{sqrt(9 acre)} to a quantity in meters; and @code{floor}, | |
27172 | @code{ceil}, @code{round}, @code{rounde}, @code{roundu}, @code{trunc}, | |
27173 | @code{float}, @code{frac}, @code{abs}, and @code{clean} | |
27174 | applied to units expressions, in which case | |
27175 | the operation in question is applied only to the numeric part of the | |
27176 | expression. Finally, trigonometric functions of quantities with units | |
27177 | of angle are evaluated, regardless of the current angular mode.@refill | |
27178 | ||
27179 | @kindex u c | |
27180 | @pindex calc-convert-units | |
27181 | The @kbd{u c} (@code{calc-convert-units}) command converts a units | |
27182 | expression to new, compatible units. For example, given the units | |
27183 | expression @samp{55 mph}, typing @kbd{u c m/s @key{RET}} produces | |
27184 | @samp{24.5872 m/s}. If the units you request are inconsistent with | |
27185 | the original units, the number will be converted into your units | |
27186 | times whatever ``remainder'' units are left over. For example, | |
27187 | converting @samp{55 mph} into acres produces @samp{6.08e-3 acre / m s}. | |
27188 | (Recall that multiplication binds more strongly than division in Calc | |
27189 | formulas, so the units here are acres per meter-second.) Remainder | |
27190 | units are expressed in terms of ``fundamental'' units like @samp{m} and | |
27191 | @samp{s}, regardless of the input units. | |
27192 | ||
27193 | One special exception is that if you specify a single unit name, and | |
27194 | a compatible unit appears somewhere in the units expression, then | |
27195 | that compatible unit will be converted to the new unit and the | |
27196 | remaining units in the expression will be left alone. For example, | |
27197 | given the input @samp{980 cm/s^2}, the command @kbd{u c ms} will | |
27198 | change the @samp{s} to @samp{ms} to get @samp{9.8e-4 cm/ms^2}. | |
27199 | The ``remainder unit'' @samp{cm} is left alone rather than being | |
27200 | changed to the base unit @samp{m}. | |
27201 | ||
27202 | You can use explicit unit conversion instead of the @kbd{u s} command | |
27203 | to gain more control over the units of the result of an expression. | |
27204 | For example, given @samp{5 m + 23 mm}, you can type @kbd{u c m} or | |
27205 | @kbd{u c mm} to express the result in either meters or millimeters. | |
27206 | (For that matter, you could type @kbd{u c fath} to express the result | |
27207 | in fathoms, if you preferred!) | |
27208 | ||
27209 | In place of a specific set of units, you can also enter one of the | |
27210 | units system names @code{si}, @code{mks} (equivalent), or @code{cgs}. | |
27211 | For example, @kbd{u c si @key{RET}} converts the expression into | |
27212 | International System of Units (SI) base units. Also, @kbd{u c base} | |
27213 | converts to Calc's base units, which are the same as @code{si} units | |
27214 | except that @code{base} uses @samp{g} as the fundamental unit of mass | |
27215 | whereas @code{si} uses @samp{kg}. | |
27216 | ||
27217 | @cindex Composite units | |
27218 | The @kbd{u c} command also accepts @dfn{composite units}, which | |
27219 | are expressed as the sum of several compatible unit names. For | |
27220 | example, converting @samp{30.5 in} to units @samp{mi+ft+in} (miles, | |
27221 | feet, and inches) produces @samp{2 ft + 6.5 in}. Calc first | |
27222 | sorts the unit names into order of decreasing relative size. | |
27223 | It then accounts for as much of the input quantity as it can | |
27224 | using an integer number times the largest unit, then moves on | |
27225 | to the next smaller unit, and so on. Only the smallest unit | |
27226 | may have a non-integer amount attached in the result. A few | |
27227 | standard unit names exist for common combinations, such as | |
27228 | @code{mfi} for @samp{mi+ft+in}, and @code{tpo} for @samp{ton+lb+oz}. | |
27229 | Composite units are expanded as if by @kbd{a x}, so that | |
27230 | @samp{(ft+in)/hr} is first converted to @samp{ft/hr+in/hr}. | |
27231 | ||
27232 | If the value on the stack does not contain any units, @kbd{u c} will | |
27233 | prompt first for the old units which this value should be considered | |
27234 | to have, then for the new units. Assuming the old and new units you | |
27235 | give are consistent with each other, the result also will not contain | |
5d67986c | 27236 | any units. For example, @kbd{@w{u c} cm @key{RET} in @key{RET}} converts the number |
d7b8e6c6 EZ |
27237 | 2 on the stack to 5.08. |
27238 | ||
27239 | @kindex u b | |
27240 | @pindex calc-base-units | |
27241 | The @kbd{u b} (@code{calc-base-units}) command is shorthand for | |
27242 | @kbd{u c base}; it converts the units expression on the top of the | |
27243 | stack into @code{base} units. If @kbd{u s} does not simplify a | |
27244 | units expression as far as you would like, try @kbd{u b}. | |
27245 | ||
27246 | The @kbd{u c} and @kbd{u b} commands treat temperature units (like | |
27247 | @samp{degC} and @samp{K}) as relative temperatures. For example, | |
27248 | @kbd{u c} converts @samp{10 degC} to @samp{18 degF}: A change of 10 | |
27249 | degrees Celsius corresponds to a change of 18 degrees Fahrenheit. | |
27250 | ||
27251 | @kindex u t | |
27252 | @pindex calc-convert-temperature | |
27253 | @cindex Temperature conversion | |
27254 | The @kbd{u t} (@code{calc-convert-temperature}) command converts | |
27255 | absolute temperatures. The value on the stack must be a simple units | |
27256 | expression with units of temperature only. This command would convert | |
27257 | @samp{10 degC} to @samp{50 degF}, the equivalent temperature on the | |
27258 | Fahrenheit scale.@refill | |
27259 | ||
27260 | @kindex u r | |
27261 | @pindex calc-remove-units | |
27262 | @kindex u x | |
27263 | @pindex calc-extract-units | |
27264 | The @kbd{u r} (@code{calc-remove-units}) command removes units from the | |
27265 | formula at the top of the stack. The @kbd{u x} | |
27266 | (@code{calc-extract-units}) command extracts only the units portion of a | |
27267 | formula. These commands essentially replace every term of the formula | |
27268 | that does or doesn't (respectively) look like a unit name by the | |
27269 | constant 1, then resimplify the formula.@refill | |
27270 | ||
27271 | @kindex u a | |
27272 | @pindex calc-autorange-units | |
27273 | The @kbd{u a} (@code{calc-autorange-units}) command turns on and off a | |
27274 | mode in which unit prefixes like @code{k} (``kilo'') are automatically | |
27275 | applied to keep the numeric part of a units expression in a reasonable | |
27276 | range. This mode affects @kbd{u s} and all units conversion commands | |
27277 | except @kbd{u b}. For example, with autoranging on, @samp{12345 Hz} | |
27278 | will be simplified to @samp{12.345 kHz}. Autoranging is useful for | |
27279 | some kinds of units (like @code{Hz} and @code{m}), but is probably | |
27280 | undesirable for non-metric units like @code{ft} and @code{tbsp}. | |
27281 | (Composite units are more appropriate for those; see above.) | |
27282 | ||
27283 | Autoranging always applies the prefix to the leftmost unit name. | |
27284 | Calc chooses the largest prefix that causes the number to be greater | |
27285 | than or equal to 1.0. Thus an increasing sequence of adjusted times | |
27286 | would be @samp{1 ms, 10 ms, 100 ms, 1 s, 10 s, 100 s, 1 ks}. | |
27287 | Generally the rule of thumb is that the number will be adjusted | |
27288 | to be in the interval @samp{[1 .. 1000)}, although there are several | |
27289 | exceptions to this rule. First, if the unit has a power then this | |
27290 | is not possible; @samp{0.1 s^2} simplifies to @samp{100000 ms^2}. | |
27291 | Second, the ``centi-'' prefix is allowed to form @code{cm} (centimeters), | |
27292 | but will not apply to other units. The ``deci-,'' ``deka-,'' and | |
27293 | ``hecto-'' prefixes are never used. Thus the allowable interval is | |
27294 | @samp{[1 .. 10)} for millimeters and @samp{[1 .. 100)} for centimeters. | |
27295 | Finally, a prefix will not be added to a unit if the resulting name | |
27296 | is also the actual name of another unit; @samp{1e-15 t} would normally | |
27297 | be considered a ``femto-ton,'' but it is written as @samp{1000 at} | |
27298 | (1000 atto-tons) instead because @code{ft} would be confused with feet. | |
27299 | ||
27300 | @node The Units Table, Predefined Units, Basic Operations on Units, Units | |
27301 | @section The Units Table | |
27302 | ||
27303 | @noindent | |
27304 | @kindex u v | |
27305 | @pindex calc-enter-units-table | |
27306 | The @kbd{u v} (@code{calc-enter-units-table}) command displays the units table | |
27307 | in another buffer called @code{*Units Table*}. Each entry in this table | |
27308 | gives the unit name as it would appear in an expression, the definition | |
27309 | of the unit in terms of simpler units, and a full name or description of | |
27310 | the unit. Fundamental units are defined as themselves; these are the | |
27311 | units produced by the @kbd{u b} command. The fundamental units are | |
27312 | meters, seconds, grams, kelvins, amperes, candelas, moles, radians, | |
27313 | and steradians. | |
27314 | ||
27315 | The Units Table buffer also displays the Unit Prefix Table. Note that | |
27316 | two prefixes, ``kilo'' and ``hecto,'' accept either upper- or lower-case | |
27317 | prefix letters. @samp{Meg} is also accepted as a synonym for the @samp{M} | |
27318 | prefix. Whenever a unit name can be interpreted as either a built-in name | |
27319 | or a prefix followed by another built-in name, the former interpretation | |
27320 | wins. For example, @samp{2 pt} means two pints, not two pico-tons. | |
27321 | ||
27322 | The Units Table buffer, once created, is not rebuilt unless you define | |
27323 | new units. To force the buffer to be rebuilt, give any numeric prefix | |
27324 | argument to @kbd{u v}. | |
27325 | ||
27326 | @kindex u V | |
27327 | @pindex calc-view-units-table | |
27328 | The @kbd{u V} (@code{calc-view-units-table}) command is like @kbd{u v} except | |
27329 | that the cursor is not moved into the Units Table buffer. You can | |
27330 | type @kbd{u V} again to remove the Units Table from the display. To | |
27331 | return from the Units Table buffer after a @kbd{u v}, type @kbd{M-# c} | |
27332 | again or use the regular Emacs @w{@kbd{C-x o}} (@code{other-window}) | |
27333 | command. You can also kill the buffer with @kbd{C-x k} if you wish; | |
27334 | the actual units table is safely stored inside the Calculator. | |
27335 | ||
27336 | @kindex u g | |
27337 | @pindex calc-get-unit-definition | |
27338 | The @kbd{u g} (@code{calc-get-unit-definition}) command retrieves a unit's | |
27339 | defining expression and pushes it onto the Calculator stack. For example, | |
27340 | @kbd{u g in} will produce the expression @samp{2.54 cm}. This is the | |
27341 | same definition for the unit that would appear in the Units Table buffer. | |
27342 | Note that this command works only for actual unit names; @kbd{u g km} | |
27343 | will report that no such unit exists, for example, because @code{km} is | |
27344 | really the unit @code{m} with a @code{k} (``kilo'') prefix. To see a | |
27345 | definition of a unit in terms of base units, it is easier to push the | |
27346 | unit name on the stack and then reduce it to base units with @kbd{u b}. | |
27347 | ||
27348 | @kindex u e | |
27349 | @pindex calc-explain-units | |
27350 | The @kbd{u e} (@code{calc-explain-units}) command displays an English | |
27351 | description of the units of the expression on the stack. For example, | |
27352 | for the expression @samp{62 km^2 g / s^2 mol K}, the description is | |
27353 | ``Square-Kilometer Gram per (Second-squared Mole Degree-Kelvin).'' This | |
27354 | command uses the English descriptions that appear in the righthand | |
27355 | column of the Units Table. | |
27356 | ||
27357 | @node Predefined Units, User-Defined Units, The Units Table, Units | |
27358 | @section Predefined Units | |
27359 | ||
27360 | @noindent | |
27361 | Since the exact definitions of many kinds of units have evolved over the | |
27362 | years, and since certain countries sometimes have local differences in | |
27363 | their definitions, it is a good idea to examine Calc's definition of a | |
27364 | unit before depending on its exact value. For example, there are three | |
27365 | different units for gallons, corresponding to the US (@code{gal}), | |
27366 | Canadian (@code{galC}), and British (@code{galUK}) definitions. Also, | |
27367 | note that @code{oz} is a standard ounce of mass, @code{ozt} is a Troy | |
27368 | ounce, and @code{ozfl} is a fluid ounce. | |
27369 | ||
27370 | The temperature units corresponding to degrees Kelvin and Centigrade | |
27371 | (Celsius) are the same in this table, since most units commands treat | |
27372 | temperatures as being relative. The @code{calc-convert-temperature} | |
27373 | command has special rules for handling the different absolute magnitudes | |
27374 | of the various temperature scales. | |
27375 | ||
27376 | The unit of volume ``liters'' can be referred to by either the lower-case | |
27377 | @code{l} or the upper-case @code{L}. | |
27378 | ||
27379 | The unit @code{A} stands for Amperes; the name @code{Ang} is used | |
27380 | @tex | |
27381 | for \AA ngstroms. | |
27382 | @end tex | |
27383 | @ifinfo | |
27384 | for Angstroms. | |
27385 | @end ifinfo | |
27386 | ||
27387 | The unit @code{pt} stands for pints; the name @code{point} stands for | |
27388 | a typographical point, defined by @samp{72 point = 1 in}. There is | |
27389 | also @code{tpt}, which stands for a printer's point as defined by the | |
27390 | @TeX{} typesetting system: @samp{72.27 tpt = 1 in}. | |
27391 | ||
27392 | The unit @code{e} stands for the elementary (electron) unit of charge; | |
27393 | because algebra command could mistake this for the special constant | |
27394 | @cite{e}, Calc provides the alternate unit name @code{ech} which is | |
27395 | preferable to @code{e}. | |
27396 | ||
27397 | The name @code{g} stands for one gram of mass; there is also @code{gf}, | |
27398 | one gram of force. (Likewise for @kbd{lb}, pounds, and @kbd{lbf}.) | |
27399 | Meanwhile, one ``@cite{g}'' of acceleration is denoted @code{ga}. | |
27400 | ||
27401 | The unit @code{ton} is a U.S. ton of @samp{2000 lb}, and @code{t} is | |
27402 | a metric ton of @samp{1000 kg}. | |
27403 | ||
27404 | The names @code{s} (or @code{sec}) and @code{min} refer to units of | |
27405 | time; @code{arcsec} and @code{arcmin} are units of angle. | |
27406 | ||
27407 | Some ``units'' are really physical constants; for example, @code{c} | |
27408 | represents the speed of light, and @code{h} represents Planck's | |
27409 | constant. You can use these just like other units: converting | |
27410 | @samp{.5 c} to @samp{m/s} expresses one-half the speed of light in | |
27411 | meters per second. You can also use this merely as a handy reference; | |
27412 | the @kbd{u g} command gets the definition of one of these constants | |
27413 | in its normal terms, and @kbd{u b} expresses the definition in base | |
27414 | units. | |
27415 | ||
27416 | Two units, @code{pi} and @code{fsc} (the fine structure constant, | |
27417 | approximately @i{1/137}) are dimensionless. The units simplification | |
27418 | commands simply treat these names as equivalent to their corresponding | |
27419 | values. However you can, for example, use @kbd{u c} to convert a pure | |
27420 | number into multiples of the fine structure constant, or @kbd{u b} to | |
27421 | convert this back into a pure number. (When @kbd{u c} prompts for the | |
27422 | ``old units,'' just enter a blank line to signify that the value | |
27423 | really is unitless.) | |
27424 | ||
27425 | @c Describe angular units, luminosity vs. steradians problem. | |
27426 | ||
27427 | @node User-Defined Units, , Predefined Units, Units | |
27428 | @section User-Defined Units | |
27429 | ||
27430 | @noindent | |
27431 | Calc provides ways to get quick access to your selected ``favorite'' | |
27432 | units, as well as ways to define your own new units. | |
27433 | ||
27434 | @kindex u 0-9 | |
27435 | @pindex calc-quick-units | |
27436 | @vindex Units | |
27437 | @cindex @code{Units} variable | |
27438 | @cindex Quick units | |
27439 | To select your favorite units, store a vector of unit names or | |
27440 | expressions in the Calc variable @code{Units}. The @kbd{u 1} | |
27441 | through @kbd{u 9} commands (@code{calc-quick-units}) provide access | |
27442 | to these units. If the value on the top of the stack is a plain | |
27443 | number (with no units attached), then @kbd{u 1} gives it the | |
27444 | specified units. (Basically, it multiplies the number by the | |
27445 | first item in the @code{Units} vector.) If the number on the | |
27446 | stack @emph{does} have units, then @kbd{u 1} converts that number | |
27447 | to the new units. For example, suppose the vector @samp{[in, ft]} | |
27448 | is stored in @code{Units}. Then @kbd{30 u 1} will create the | |
27449 | expression @samp{30 in}, and @kbd{u 2} will convert that expression | |
27450 | to @samp{2.5 ft}. | |
27451 | ||
27452 | The @kbd{u 0} command accesses the tenth element of @code{Units}. | |
27453 | Only ten quick units may be defined at a time. If the @code{Units} | |
27454 | variable has no stored value (the default), or if its value is not | |
27455 | a vector, then the quick-units commands will not function. The | |
27456 | @kbd{s U} command is a convenient way to edit the @code{Units} | |
27457 | variable; @pxref{Operations on Variables}. | |
27458 | ||
27459 | @kindex u d | |
27460 | @pindex calc-define-unit | |
27461 | @cindex User-defined units | |
27462 | The @kbd{u d} (@code{calc-define-unit}) command records the units | |
27463 | expression on the top of the stack as the definition for a new, | |
27464 | user-defined unit. For example, putting @samp{16.5 ft} on the stack and | |
27465 | typing @kbd{u d rod} defines the new unit @samp{rod} to be equivalent to | |
27466 | 16.5 feet. The unit conversion and simplification commands will now | |
27467 | treat @code{rod} just like any other unit of length. You will also be | |
27468 | prompted for an optional English description of the unit, which will | |
27469 | appear in the Units Table. | |
27470 | ||
27471 | @kindex u u | |
27472 | @pindex calc-undefine-unit | |
27473 | The @kbd{u u} (@code{calc-undefine-unit}) command removes a user-defined | |
27474 | unit. It is not possible to remove one of the predefined units, | |
27475 | however. | |
27476 | ||
27477 | If you define a unit with an existing unit name, your new definition | |
27478 | will replace the original definition of that unit. If the unit was a | |
27479 | predefined unit, the old definition will not be replaced, only | |
27480 | ``shadowed.'' The built-in definition will reappear if you later use | |
27481 | @kbd{u u} to remove the shadowing definition. | |
27482 | ||
27483 | To create a new fundamental unit, use either 1 or the unit name itself | |
27484 | as the defining expression. Otherwise the expression can involve any | |
27485 | other units that you like (except for composite units like @samp{mfi}). | |
27486 | You can create a new composite unit with a sum of other units as the | |
27487 | defining expression. The next unit operation like @kbd{u c} or @kbd{u v} | |
27488 | will rebuild the internal unit table incorporating your modifications. | |
27489 | Note that erroneous definitions (such as two units defined in terms of | |
27490 | each other) will not be detected until the unit table is next rebuilt; | |
27491 | @kbd{u v} is a convenient way to force this to happen. | |
27492 | ||
27493 | Temperature units are treated specially inside the Calculator; it is not | |
27494 | possible to create user-defined temperature units. | |
27495 | ||
27496 | @kindex u p | |
27497 | @pindex calc-permanent-units | |
27498 | @cindex @file{.emacs} file, user-defined units | |
27499 | The @kbd{u p} (@code{calc-permanent-units}) command stores the user-defined | |
27500 | units in your @file{.emacs} file, so that the units will still be | |
27501 | available in subsequent Emacs sessions. If there was already a set of | |
27502 | user-defined units in your @file{.emacs} file, it is replaced by the | |
27503 | new set. (@xref{General Mode Commands}, for a way to tell Calc to use | |
27504 | a different file instead of @file{.emacs}.) | |
27505 | ||
27506 | @node Store and Recall, Graphics, Units, Top | |
27507 | @chapter Storing and Recalling | |
27508 | ||
27509 | @noindent | |
27510 | Calculator variables are really just Lisp variables that contain numbers | |
27511 | or formulas in a form that Calc can understand. The commands in this | |
27512 | section allow you to manipulate variables conveniently. Commands related | |
27513 | to variables use the @kbd{s} prefix key. | |
27514 | ||
27515 | @menu | |
27516 | * Storing Variables:: | |
27517 | * Recalling Variables:: | |
27518 | * Operations on Variables:: | |
27519 | * Let Command:: | |
27520 | * Evaluates-To Operator:: | |
27521 | @end menu | |
27522 | ||
27523 | @node Storing Variables, Recalling Variables, Store and Recall, Store and Recall | |
27524 | @section Storing Variables | |
27525 | ||
27526 | @noindent | |
27527 | @kindex s s | |
27528 | @pindex calc-store | |
27529 | @cindex Storing variables | |
27530 | @cindex Quick variables | |
27531 | @vindex q0 | |
27532 | @vindex q9 | |
27533 | The @kbd{s s} (@code{calc-store}) command stores the value at the top of | |
27534 | the stack into a specified variable. It prompts you to enter the | |
27535 | name of the variable. If you press a single digit, the value is stored | |
27536 | immediately in one of the ``quick'' variables @code{var-q0} through | |
27537 | @code{var-q9}. Or you can enter any variable name. The prefix @samp{var-} | |
27538 | is supplied for you; when a name appears in a formula (as in @samp{a+q2}) | |
27539 | the prefix @samp{var-} is also supplied there, so normally you can simply | |
27540 | forget about @samp{var-} everywhere. Its only purpose is to enable you to | |
27541 | use Calc variables without fear of accidentally clobbering some variable in | |
27542 | another Emacs package. If you really want to store in an arbitrary Lisp | |
27543 | variable, just backspace over the @samp{var-}. | |
27544 | ||
27545 | @kindex s t | |
27546 | @pindex calc-store-into | |
27547 | The @kbd{s s} command leaves the stored value on the stack. There is | |
27548 | also an @kbd{s t} (@code{calc-store-into}) command, which removes a | |
27549 | value from the stack and stores it in a variable. | |
27550 | ||
27551 | If the top of stack value is an equation @samp{a = 7} or assignment | |
27552 | @samp{a := 7} with a variable on the lefthand side, then Calc will | |
27553 | assign that variable with that value by default, i.e., if you type | |
27554 | @kbd{s s @key{RET}} or @kbd{s t @key{RET}}. In this example, the | |
27555 | value 7 would be stored in the variable @samp{a}. (If you do type | |
27556 | a variable name at the prompt, the top-of-stack value is stored in | |
27557 | its entirety, even if it is an equation: @samp{s s b @key{RET}} | |
27558 | with @samp{a := 7} on the stack stores @samp{a := 7} in @code{b}.) | |
27559 | ||
27560 | In fact, the top of stack value can be a vector of equations or | |
27561 | assignments with different variables on their lefthand sides; the | |
27562 | default will be to store all the variables with their corresponding | |
27563 | righthand sides simultaneously. | |
27564 | ||
27565 | It is also possible to type an equation or assignment directly at | |
27566 | the prompt for the @kbd{s s} or @kbd{s t} command: @kbd{s s foo = 7}. | |
27567 | In this case the expression to the right of the @kbd{=} or @kbd{:=} | |
27568 | symbol is evaluated as if by the @kbd{=} command, and that value is | |
27569 | stored in the variable. No value is taken from the stack; @kbd{s s} | |
27570 | and @kbd{s t} are equivalent when used in this way. | |
27571 | ||
27572 | @kindex s 0-9 | |
27573 | @kindex t 0-9 | |
27574 | The prefix keys @kbd{s} and @kbd{t} may be followed immediately by a | |
27575 | digit; @kbd{s 9} is equivalent to @kbd{s s 9}, and @kbd{t 9} is | |
27576 | equivalent to @kbd{s t 9}. (The @kbd{t} prefix is otherwise used | |
27577 | for trail and time/date commands.) | |
27578 | ||
27579 | @kindex s + | |
27580 | @kindex s - | |
5d67986c RS |
27581 | @ignore |
27582 | @mindex @idots | |
27583 | @end ignore | |
d7b8e6c6 | 27584 | @kindex s * |
5d67986c RS |
27585 | @ignore |
27586 | @mindex @null | |
27587 | @end ignore | |
d7b8e6c6 | 27588 | @kindex s / |
5d67986c RS |
27589 | @ignore |
27590 | @mindex @null | |
27591 | @end ignore | |
d7b8e6c6 | 27592 | @kindex s ^ |
5d67986c RS |
27593 | @ignore |
27594 | @mindex @null | |
27595 | @end ignore | |
d7b8e6c6 | 27596 | @kindex s | |
5d67986c RS |
27597 | @ignore |
27598 | @mindex @null | |
27599 | @end ignore | |
d7b8e6c6 | 27600 | @kindex s n |
5d67986c RS |
27601 | @ignore |
27602 | @mindex @null | |
27603 | @end ignore | |
d7b8e6c6 | 27604 | @kindex s & |
5d67986c RS |
27605 | @ignore |
27606 | @mindex @null | |
27607 | @end ignore | |
d7b8e6c6 | 27608 | @kindex s [ |
5d67986c RS |
27609 | @ignore |
27610 | @mindex @null | |
27611 | @end ignore | |
d7b8e6c6 EZ |
27612 | @kindex s ] |
27613 | @pindex calc-store-plus | |
27614 | @pindex calc-store-minus | |
27615 | @pindex calc-store-times | |
27616 | @pindex calc-store-div | |
27617 | @pindex calc-store-power | |
27618 | @pindex calc-store-concat | |
27619 | @pindex calc-store-neg | |
27620 | @pindex calc-store-inv | |
27621 | @pindex calc-store-decr | |
27622 | @pindex calc-store-incr | |
27623 | There are also several ``arithmetic store'' commands. For example, | |
27624 | @kbd{s +} removes a value from the stack and adds it to the specified | |
27625 | variable. The other arithmetic stores are @kbd{s -}, @kbd{s *}, @kbd{s /}, | |
27626 | @kbd{s ^}, and @w{@kbd{s |}} (vector concatenation), plus @kbd{s n} and | |
27627 | @kbd{s &} which negate or invert the value in a variable, and @w{@kbd{s [}} | |
27628 | and @kbd{s ]} which decrease or increase a variable by one. | |
27629 | ||
27630 | All the arithmetic stores accept the Inverse prefix to reverse the | |
27631 | order of the operands. If @cite{v} represents the contents of the | |
27632 | variable, and @cite{a} is the value drawn from the stack, then regular | |
27633 | @w{@kbd{s -}} assigns @c{$v \coloneq v - a$} | |
27634 | @cite{v := v - a}, but @kbd{I s -} assigns | |
27635 | @c{$v \coloneq a - v$} | |
27636 | @cite{v := a - v}. While @kbd{I s *} might seem pointless, it is | |
27637 | useful if matrix multiplication is involved. Actually, all the | |
27638 | arithmetic stores use formulas designed to behave usefully both | |
27639 | forwards and backwards: | |
27640 | ||
d7b8e6c6 | 27641 | @example |
5d67986c | 27642 | @group |
d7b8e6c6 EZ |
27643 | s + v := v + a v := a + v |
27644 | s - v := v - a v := a - v | |
27645 | s * v := v * a v := a * v | |
27646 | s / v := v / a v := a / v | |
27647 | s ^ v := v ^ a v := a ^ v | |
27648 | s | v := v | a v := a | v | |
27649 | s n v := v / (-1) v := (-1) / v | |
27650 | s & v := v ^ (-1) v := (-1) ^ v | |
27651 | s [ v := v - 1 v := 1 - v | |
27652 | s ] v := v - (-1) v := (-1) - v | |
d7b8e6c6 | 27653 | @end group |
5d67986c | 27654 | @end example |
d7b8e6c6 EZ |
27655 | |
27656 | In the last four cases, a numeric prefix argument will be used in | |
27657 | place of the number one. (For example, @kbd{M-2 s ]} increases | |
27658 | a variable by 2, and @kbd{M-2 I s ]} replaces a variable by | |
27659 | minus-two minus the variable. | |
27660 | ||
27661 | The first six arithmetic stores can also be typed @kbd{s t +}, @kbd{s t -}, | |
27662 | etc. The commands @kbd{s s +}, @kbd{s s -}, and so on are analogous | |
27663 | arithmetic stores that don't remove the value @cite{a} from the stack. | |
27664 | ||
27665 | All arithmetic stores report the new value of the variable in the | |
27666 | Trail for your information. They signal an error if the variable | |
27667 | previously had no stored value. If default simplifications have been | |
27668 | turned off, the arithmetic stores temporarily turn them on for numeric | |
27669 | arguments only (i.e., they temporarily do an @kbd{m N} command). | |
27670 | @xref{Simplification Modes}. Large vectors put in the trail by | |
27671 | these commands always use abbreviated (@kbd{t .}) mode. | |
27672 | ||
27673 | @kindex s m | |
27674 | @pindex calc-store-map | |
27675 | The @kbd{s m} command is a general way to adjust a variable's value | |
27676 | using any Calc function. It is a ``mapping'' command analogous to | |
27677 | @kbd{V M}, @kbd{V R}, etc. @xref{Reducing and Mapping}, to see | |
27678 | how to specify a function for a mapping command. Basically, | |
27679 | all you do is type the Calc command key that would invoke that | |
27680 | function normally. For example, @kbd{s m n} applies the @kbd{n} | |
27681 | key to negate the contents of the variable, so @kbd{s m n} is | |
27682 | equivalent to @kbd{s n}. Also, @kbd{s m Q} takes the square root | |
27683 | of the value stored in a variable, @kbd{s m v v} uses @kbd{v v} to | |
27684 | reverse the vector stored in the variable, and @kbd{s m H I S} | |
27685 | takes the hyperbolic arcsine of the variable contents. | |
27686 | ||
27687 | If the mapping function takes two or more arguments, the additional | |
27688 | arguments are taken from the stack; the old value of the variable | |
27689 | is provided as the first argument. Thus @kbd{s m -} with @cite{a} | |
27690 | on the stack computes @cite{v - a}, just like @kbd{s -}. With the | |
27691 | Inverse prefix, the variable's original value becomes the @emph{last} | |
27692 | argument instead of the first. Thus @kbd{I s m -} is also | |
27693 | equivalent to @kbd{I s -}. | |
27694 | ||
27695 | @kindex s x | |
27696 | @pindex calc-store-exchange | |
27697 | The @kbd{s x} (@code{calc-store-exchange}) command exchanges the value | |
27698 | of a variable with the value on the top of the stack. Naturally, the | |
27699 | variable must already have a stored value for this to work. | |
27700 | ||
27701 | You can type an equation or assignment at the @kbd{s x} prompt. The | |
27702 | command @kbd{s x a=6} takes no values from the stack; instead, it | |
27703 | pushes the old value of @samp{a} on the stack and stores @samp{a = 6}. | |
27704 | ||
27705 | @kindex s u | |
27706 | @pindex calc-unstore | |
27707 | @cindex Void variables | |
27708 | @cindex Un-storing variables | |
27709 | Until you store something in them, variables are ``void,'' that is, they | |
27710 | contain no value at all. If they appear in an algebraic formula they | |
27711 | will be left alone even if you press @kbd{=} (@code{calc-evaluate}). | |
27712 | The @kbd{s u} (@code{calc-unstore}) command returns a variable to the | |
27713 | void state.@refill | |
27714 | ||
27715 | The only variables with predefined values are the ``special constants'' | |
27716 | @code{pi}, @code{e}, @code{i}, @code{phi}, and @code{gamma}. You are free | |
27717 | to unstore these variables or to store new values into them if you like, | |
27718 | although some of the algebraic-manipulation functions may assume these | |
27719 | variables represent their standard values. Calc displays a warning if | |
27720 | you change the value of one of these variables, or of one of the other | |
27721 | special variables @code{inf}, @code{uinf}, and @code{nan} (which are | |
27722 | normally void). | |
27723 | ||
27724 | Note that @code{var-pi} doesn't actually have 3.14159265359 stored | |
27725 | in it, but rather a special magic value that evaluates to @c{$\pi$} | |
27726 | @cite{pi} | |
27727 | at the current precision. Likewise @code{var-e}, @code{var-i}, and | |
27728 | @code{var-phi} evaluate according to the current precision or polar mode. | |
27729 | If you recall a value from @code{pi} and store it back, this magic | |
27730 | property will be lost. | |
27731 | ||
27732 | @kindex s c | |
27733 | @pindex calc-copy-variable | |
27734 | The @kbd{s c} (@code{calc-copy-variable}) command copies the stored | |
27735 | value of one variable to another. It differs from a simple @kbd{s r} | |
27736 | followed by an @kbd{s t} in two important ways. First, the value never | |
27737 | goes on the stack and thus is never rounded, evaluated, or simplified | |
27738 | in any way; it is not even rounded down to the current precision. | |
27739 | Second, the ``magic'' contents of a variable like @code{var-e} can | |
27740 | be copied into another variable with this command, perhaps because | |
27741 | you need to unstore @code{var-e} right now but you wish to put it | |
27742 | back when you're done. The @kbd{s c} command is the only way to | |
27743 | manipulate these magic values intact. | |
27744 | ||
27745 | @node Recalling Variables, Operations on Variables, Storing Variables, Store and Recall | |
27746 | @section Recalling Variables | |
27747 | ||
27748 | @noindent | |
27749 | @kindex s r | |
27750 | @pindex calc-recall | |
27751 | @cindex Recalling variables | |
27752 | The most straightforward way to extract the stored value from a variable | |
27753 | is to use the @kbd{s r} (@code{calc-recall}) command. This command prompts | |
27754 | for a variable name (similarly to @code{calc-store}), looks up the value | |
27755 | of the specified variable, and pushes that value onto the stack. It is | |
27756 | an error to try to recall a void variable. | |
27757 | ||
27758 | It is also possible to recall the value from a variable by evaluating a | |
27759 | formula containing that variable. For example, @kbd{' a @key{RET} =} is | |
27760 | the same as @kbd{s r a @key{RET}} except that if the variable is void, the | |
27761 | former will simply leave the formula @samp{a} on the stack whereas the | |
27762 | latter will produce an error message. | |
27763 | ||
27764 | @kindex r 0-9 | |
27765 | The @kbd{r} prefix may be followed by a digit, so that @kbd{r 9} is | |
27766 | equivalent to @kbd{s r 9}. (The @kbd{r} prefix is otherwise unused | |
27767 | in the current version of Calc.) | |
27768 | ||
27769 | @node Operations on Variables, Let Command, Recalling Variables, Store and Recall | |
27770 | @section Other Operations on Variables | |
27771 | ||
27772 | @noindent | |
27773 | @kindex s e | |
27774 | @pindex calc-edit-variable | |
27775 | The @kbd{s e} (@code{calc-edit-variable}) command edits the stored | |
27776 | value of a variable without ever putting that value on the stack | |
27777 | or simplifying or evaluating the value. It prompts for the name of | |
27778 | the variable to edit. If the variable has no stored value, the | |
27779 | editing buffer will start out empty. If the editing buffer is | |
0d48e8aa | 27780 | empty when you press @kbd{M-# M-#} to finish, the variable will |
d7b8e6c6 EZ |
27781 | be made void. @xref{Editing Stack Entries}, for a general |
27782 | description of editing. | |
27783 | ||
27784 | The @kbd{s e} command is especially useful for creating and editing | |
27785 | rewrite rules which are stored in variables. Sometimes these rules | |
27786 | contain formulas which must not be evaluated until the rules are | |
27787 | actually used. (For example, they may refer to @samp{deriv(x,y)}, | |
27788 | where @code{x} will someday become some expression involving @code{y}; | |
27789 | if you let Calc evaluate the rule while you are defining it, Calc will | |
27790 | replace @samp{deriv(x,y)} with 0 because the formula @code{x} does | |
27791 | not itself refer to @code{y}.) By contrast, recalling the variable, | |
27792 | editing with @kbd{`}, and storing will evaluate the variable's value | |
27793 | as a side effect of putting the value on the stack. | |
27794 | ||
27795 | @kindex s A | |
27796 | @kindex s D | |
5d67986c RS |
27797 | @ignore |
27798 | @mindex @idots | |
27799 | @end ignore | |
d7b8e6c6 | 27800 | @kindex s E |
5d67986c RS |
27801 | @ignore |
27802 | @mindex @null | |
27803 | @end ignore | |
d7b8e6c6 | 27804 | @kindex s F |
5d67986c RS |
27805 | @ignore |
27806 | @mindex @null | |
27807 | @end ignore | |
d7b8e6c6 | 27808 | @kindex s G |
5d67986c RS |
27809 | @ignore |
27810 | @mindex @null | |
27811 | @end ignore | |
d7b8e6c6 | 27812 | @kindex s H |
5d67986c RS |
27813 | @ignore |
27814 | @mindex @null | |
27815 | @end ignore | |
d7b8e6c6 | 27816 | @kindex s I |
5d67986c RS |
27817 | @ignore |
27818 | @mindex @null | |
27819 | @end ignore | |
d7b8e6c6 | 27820 | @kindex s L |
5d67986c RS |
27821 | @ignore |
27822 | @mindex @null | |
27823 | @end ignore | |
d7b8e6c6 | 27824 | @kindex s P |
5d67986c RS |
27825 | @ignore |
27826 | @mindex @null | |
27827 | @end ignore | |
d7b8e6c6 | 27828 | @kindex s R |
5d67986c RS |
27829 | @ignore |
27830 | @mindex @null | |
27831 | @end ignore | |
d7b8e6c6 | 27832 | @kindex s T |
5d67986c RS |
27833 | @ignore |
27834 | @mindex @null | |
27835 | @end ignore | |
d7b8e6c6 | 27836 | @kindex s U |
5d67986c RS |
27837 | @ignore |
27838 | @mindex @null | |
27839 | @end ignore | |
d7b8e6c6 EZ |
27840 | @kindex s X |
27841 | @pindex calc-store-AlgSimpRules | |
27842 | @pindex calc-store-Decls | |
27843 | @pindex calc-store-EvalRules | |
27844 | @pindex calc-store-FitRules | |
27845 | @pindex calc-store-GenCount | |
27846 | @pindex calc-store-Holidays | |
27847 | @pindex calc-store-IntegLimit | |
27848 | @pindex calc-store-LineStyles | |
27849 | @pindex calc-store-PointStyles | |
27850 | @pindex calc-store-PlotRejects | |
27851 | @pindex calc-store-TimeZone | |
27852 | @pindex calc-store-Units | |
27853 | @pindex calc-store-ExtSimpRules | |
27854 | There are several special-purpose variable-editing commands that | |
27855 | use the @kbd{s} prefix followed by a shifted letter: | |
27856 | ||
27857 | @table @kbd | |
27858 | @item s A | |
27859 | Edit @code{AlgSimpRules}. @xref{Algebraic Simplifications}. | |
27860 | @item s D | |
27861 | Edit @code{Decls}. @xref{Declarations}. | |
27862 | @item s E | |
27863 | Edit @code{EvalRules}. @xref{Default Simplifications}. | |
27864 | @item s F | |
27865 | Edit @code{FitRules}. @xref{Curve Fitting}. | |
27866 | @item s G | |
27867 | Edit @code{GenCount}. @xref{Solving Equations}. | |
27868 | @item s H | |
27869 | Edit @code{Holidays}. @xref{Business Days}. | |
27870 | @item s I | |
27871 | Edit @code{IntegLimit}. @xref{Calculus}. | |
27872 | @item s L | |
27873 | Edit @code{LineStyles}. @xref{Graphics}. | |
27874 | @item s P | |
27875 | Edit @code{PointStyles}. @xref{Graphics}. | |
27876 | @item s R | |
27877 | Edit @code{PlotRejects}. @xref{Graphics}. | |
27878 | @item s T | |
27879 | Edit @code{TimeZone}. @xref{Time Zones}. | |
27880 | @item s U | |
27881 | Edit @code{Units}. @xref{User-Defined Units}. | |
27882 | @item s X | |
27883 | Edit @code{ExtSimpRules}. @xref{Unsafe Simplifications}. | |
27884 | @end table | |
27885 | ||
27886 | These commands are just versions of @kbd{s e} that use fixed variable | |
27887 | names rather than prompting for the variable name. | |
27888 | ||
27889 | @kindex s p | |
27890 | @pindex calc-permanent-variable | |
27891 | @cindex Storing variables | |
27892 | @cindex Permanent variables | |
28665d46 | 27893 | @cindex @file{.emacs} file, variables |
d7b8e6c6 EZ |
27894 | The @kbd{s p} (@code{calc-permanent-variable}) command saves a |
27895 | variable's value permanently in your @file{.emacs} file, so that its | |
27896 | value will still be available in future Emacs sessions. You can | |
27897 | re-execute @w{@kbd{s p}} later on to update the saved value, but the | |
27898 | only way to remove a saved variable is to edit your @file{.emacs} file | |
27899 | by hand. (@xref{General Mode Commands}, for a way to tell Calc to | |
27900 | use a different file instead of @file{.emacs}.) | |
27901 | ||
27902 | If you do not specify the name of a variable to save (i.e., | |
27903 | @kbd{s p @key{RET}}), all @samp{var-} variables with defined values | |
27904 | are saved except for the special constants @code{pi}, @code{e}, | |
27905 | @code{i}, @code{phi}, and @code{gamma}; the variables @code{TimeZone} | |
27906 | and @code{PlotRejects}; | |
27907 | @code{FitRules}, @code{DistribRules}, and other built-in rewrite | |
27908 | rules; and @code{PlotData@var{n}} variables generated | |
27909 | by the graphics commands. (You can still save these variables by | |
27910 | explicitly naming them in an @kbd{s p} command.)@refill | |
27911 | ||
27912 | @kindex s i | |
27913 | @pindex calc-insert-variables | |
27914 | The @kbd{s i} (@code{calc-insert-variables}) command writes | |
27915 | the values of all @samp{var-} variables into a specified buffer. | |
27916 | The variables are written in the form of Lisp @code{setq} commands | |
27917 | which store the values in string form. You can place these commands | |
27918 | in your @file{.emacs} buffer if you wish, though in this case it | |
27919 | would be easier to use @kbd{s p @key{RET}}. (Note that @kbd{s i} | |
27920 | omits the same set of variables as @w{@kbd{s p @key{RET}}}; the difference | |
27921 | is that @kbd{s i} will store the variables in any buffer, and it also | |
27922 | stores in a more human-readable format.) | |
27923 | ||
27924 | @node Let Command, Evaluates-To Operator, Operations on Variables, Store and Recall | |
27925 | @section The Let Command | |
27926 | ||
27927 | @noindent | |
27928 | @kindex s l | |
27929 | @pindex calc-let | |
27930 | @cindex Variables, temporary assignment | |
27931 | @cindex Temporary assignment to variables | |
27932 | If you have an expression like @samp{a+b^2} on the stack and you wish to | |
27933 | compute its value where @cite{b=3}, you can simply store 3 in @cite{b} and | |
27934 | then press @kbd{=} to reevaluate the formula. This has the side-effect | |
27935 | of leaving the stored value of 3 in @cite{b} for future operations. | |
27936 | ||
27937 | The @kbd{s l} (@code{calc-let}) command evaluates a formula under a | |
27938 | @emph{temporary} assignment of a variable. It stores the value on the | |
27939 | top of the stack into the specified variable, then evaluates the | |
27940 | second-to-top stack entry, then restores the original value (or lack of one) | |
27941 | in the variable. Thus after @kbd{'@w{ }a+b^2 @key{RET} 3 s l b @key{RET}}, | |
27942 | the stack will contain the formula @samp{a + 9}. The subsequent command | |
27943 | @kbd{@w{5 s l a} @key{RET}} will replace this formula with the number 14. | |
27944 | The variables @samp{a} and @samp{b} are not permanently affected in any way | |
27945 | by these commands. | |
27946 | ||
27947 | The value on the top of the stack may be an equation or assignment, or | |
27948 | a vector of equations or assignments, in which case the default will be | |
27949 | analogous to the case of @kbd{s t @key{RET}}. @xref{Storing Variables}. | |
27950 | ||
27951 | Also, you can answer the variable-name prompt with an equation or | |
5d67986c RS |
27952 | assignment: @kbd{s l b=3 @key{RET}} is the same as storing 3 on the stack |
27953 | and typing @kbd{s l b @key{RET}}. | |
d7b8e6c6 EZ |
27954 | |
27955 | The @kbd{a b} (@code{calc-substitute}) command is another way to substitute | |
27956 | a variable with a value in a formula. It does an actual substitution | |
27957 | rather than temporarily assigning the variable and evaluating. For | |
27958 | example, letting @cite{n=2} in @samp{f(n pi)} with @kbd{a b} will | |
27959 | produce @samp{f(2 pi)}, whereas @kbd{s l} would give @samp{f(6.28)} | |
27960 | since the evaluation step will also evaluate @code{pi}. | |
27961 | ||
27962 | @node Evaluates-To Operator, , Let Command, Store and Recall | |
27963 | @section The Evaluates-To Operator | |
27964 | ||
27965 | @noindent | |
27966 | @tindex evalto | |
27967 | @tindex => | |
27968 | @cindex Evaluates-to operator | |
27969 | @cindex @samp{=>} operator | |
27970 | The special algebraic symbol @samp{=>} is known as the @dfn{evaluates-to | |
27971 | operator}. (It will show up as an @code{evalto} function call in | |
27972 | other language modes like Pascal and @TeX{}.) This is a binary | |
27973 | operator, that is, it has a lefthand and a righthand argument, | |
27974 | although it can be entered with the righthand argument omitted. | |
27975 | ||
27976 | A formula like @samp{@var{a} => @var{b}} is evaluated by Calc as | |
27977 | follows: First, @var{a} is not simplified or modified in any | |
27978 | way. The previous value of argument @var{b} is thrown away; the | |
27979 | formula @var{a} is then copied and evaluated as if by the @kbd{=} | |
27980 | command according to all current modes and stored variable values, | |
27981 | and the result is installed as the new value of @var{b}. | |
27982 | ||
27983 | For example, suppose you enter the algebraic formula @samp{2 + 3 => 17}. | |
27984 | The number 17 is ignored, and the lefthand argument is left in its | |
27985 | unevaluated form; the result is the formula @samp{2 + 3 => 5}. | |
27986 | ||
27987 | @kindex s = | |
27988 | @pindex calc-evalto | |
27989 | You can enter an @samp{=>} formula either directly using algebraic | |
27990 | entry (in which case the righthand side may be omitted since it is | |
27991 | going to be replaced right away anyhow), or by using the @kbd{s =} | |
27992 | (@code{calc-evalto}) command, which takes @var{a} from the stack | |
27993 | and replaces it with @samp{@var{a} => @var{b}}. | |
27994 | ||
27995 | Calc keeps track of all @samp{=>} operators on the stack, and | |
27996 | recomputes them whenever anything changes that might affect their | |
27997 | values, i.e., a mode setting or variable value. This occurs only | |
27998 | if the @samp{=>} operator is at the top level of the formula, or | |
27999 | if it is part of a top-level vector. In other words, pushing | |
28000 | @samp{2 + (a => 17)} will change the 17 to the actual value of | |
28001 | @samp{a} when you enter the formula, but the result will not be | |
28002 | dynamically updated when @samp{a} is changed later because the | |
28003 | @samp{=>} operator is buried inside a sum. However, a vector | |
28004 | of @samp{=>} operators will be recomputed, since it is convenient | |
28005 | to push a vector like @samp{[a =>, b =>, c =>]} on the stack to | |
28006 | make a concise display of all the variables in your problem. | |
28007 | (Another way to do this would be to use @samp{[a, b, c] =>}, | |
28008 | which provides a slightly different format of display. You | |
28009 | can use whichever you find easiest to read.) | |
28010 | ||
28011 | @kindex m C | |
28012 | @pindex calc-auto-recompute | |
28013 | The @kbd{m C} (@code{calc-auto-recompute}) command allows you to | |
28014 | turn this automatic recomputation on or off. If you turn | |
28015 | recomputation off, you must explicitly recompute an @samp{=>} | |
28016 | operator on the stack in one of the usual ways, such as by | |
28017 | pressing @kbd{=}. Turning recomputation off temporarily can save | |
28018 | a lot of time if you will be changing several modes or variables | |
28019 | before you look at the @samp{=>} entries again. | |
28020 | ||
28021 | Most commands are not especially useful with @samp{=>} operators | |
28022 | as arguments. For example, given @samp{x + 2 => 17}, it won't | |
28023 | work to type @kbd{1 +} to get @samp{x + 3 => 18}. If you want | |
28024 | to operate on the lefthand side of the @samp{=>} operator on | |
28025 | the top of the stack, type @kbd{j 1} (that's the digit ``one'') | |
28026 | to select the lefthand side, execute your commands, then type | |
28027 | @kbd{j u} to unselect. | |
28028 | ||
28029 | All current modes apply when an @samp{=>} operator is computed, | |
28030 | including the current simplification mode. Recall that the | |
28031 | formula @samp{x + y + x} is not handled by Calc's default | |
28032 | simplifications, but the @kbd{a s} command will reduce it to | |
28033 | the simpler form @samp{y + 2 x}. You can also type @kbd{m A} | |
28034 | to enable an algebraic-simplification mode in which the | |
28035 | equivalent of @kbd{a s} is used on all of Calc's results. | |
28036 | If you enter @samp{x + y + x =>} normally, the result will | |
28037 | be @samp{x + y + x => x + y + x}. If you change to | |
28038 | algebraic-simplification mode, the result will be | |
28039 | @samp{x + y + x => y + 2 x}. However, just pressing @kbd{a s} | |
28040 | once will have no effect on @samp{x + y + x => x + y + x}, | |
28041 | because the righthand side depends only on the lefthand side | |
28042 | and the current mode settings, and the lefthand side is not | |
28043 | affected by commands like @kbd{a s}. | |
28044 | ||
28045 | The ``let'' command (@kbd{s l}) has an interesting interaction | |
28046 | with the @samp{=>} operator. The @kbd{s l} command evaluates the | |
28047 | second-to-top stack entry with the top stack entry supplying | |
28048 | a temporary value for a given variable. As you might expect, | |
28049 | if that stack entry is an @samp{=>} operator its righthand | |
28050 | side will temporarily show this value for the variable. In | |
28051 | fact, all @samp{=>}s on the stack will be updated if they refer | |
28052 | to that variable. But this change is temporary in the sense | |
28053 | that the next command that causes Calc to look at those stack | |
28054 | entries will make them revert to the old variable value. | |
28055 | ||
d7b8e6c6 | 28056 | @smallexample |
5d67986c | 28057 | @group |
d7b8e6c6 EZ |
28058 | 2: a => a 2: a => 17 2: a => a |
28059 | 1: a + 1 => a + 1 1: a + 1 => 18 1: a + 1 => a + 1 | |
28060 | . . . | |
28061 | ||
5d67986c | 28062 | 17 s l a @key{RET} p 8 @key{RET} |
d7b8e6c6 | 28063 | @end group |
5d67986c | 28064 | @end smallexample |
d7b8e6c6 EZ |
28065 | |
28066 | Here the @kbd{p 8} command changes the current precision, | |
28067 | thus causing the @samp{=>} forms to be recomputed after the | |
5d67986c | 28068 | influence of the ``let'' is gone. The @kbd{d @key{SPC}} command |
d7b8e6c6 EZ |
28069 | (@code{calc-refresh}) is a handy way to force the @samp{=>} |
28070 | operators on the stack to be recomputed without any other | |
28071 | side effects. | |
28072 | ||
28073 | @kindex s : | |
28074 | @pindex calc-assign | |
28075 | @tindex assign | |
28076 | @tindex := | |
28077 | Embedded Mode also uses @samp{=>} operators. In embedded mode, | |
28078 | the lefthand side of an @samp{=>} operator can refer to variables | |
28079 | assigned elsewhere in the file by @samp{:=} operators. The | |
28080 | assignment operator @samp{a := 17} does not actually do anything | |
28081 | by itself. But Embedded Mode recognizes it and marks it as a sort | |
28082 | of file-local definition of the variable. You can enter @samp{:=} | |
28083 | operators in algebraic mode, or by using the @kbd{s :} | |
28084 | (@code{calc-assign}) [@code{assign}] command which takes a variable | |
28085 | and value from the stack and replaces them with an assignment. | |
28086 | ||
28087 | @xref{TeX Language Mode}, for the way @samp{=>} appears in | |
28088 | @TeX{} language output. The @dfn{eqn} mode gives similar | |
28089 | treatment to @samp{=>}. | |
28090 | ||
28091 | @node Graphics, Kill and Yank, Store and Recall, Top | |
28092 | @chapter Graphics | |
28093 | ||
28094 | @noindent | |
28095 | The commands for graphing data begin with the @kbd{g} prefix key. Calc | |
28096 | uses GNUPLOT 2.0 or 3.0 to do graphics. These commands will only work | |
28097 | if GNUPLOT is available on your system. (While GNUPLOT sounds like | |
28098 | a relative of GNU Emacs, it is actually completely unrelated. | |
28099 | However, it is free software and can be obtained from the Free | |
28100 | Software Foundation's machine @samp{prep.ai.mit.edu}.) | |
28101 | ||
28102 | @vindex calc-gnuplot-name | |
28103 | If you have GNUPLOT installed on your system but Calc is unable to | |
28104 | find it, you may need to set the @code{calc-gnuplot-name} variable | |
28105 | in your @file{.emacs} file. You may also need to set some Lisp | |
28106 | variables to show Calc how to run GNUPLOT on your system; these | |
28107 | are described under @kbd{g D} and @kbd{g O} below. If you are | |
28108 | using the X window system, Calc will configure GNUPLOT for you | |
28109 | automatically. If you have GNUPLOT 3.0 and you are not using X, | |
28110 | Calc will configure GNUPLOT to display graphs using simple character | |
28111 | graphics that will work on any terminal. | |
28112 | ||
28113 | @menu | |
28114 | * Basic Graphics:: | |
28115 | * Three Dimensional Graphics:: | |
28116 | * Managing Curves:: | |
28117 | * Graphics Options:: | |
28118 | * Devices:: | |
28119 | @end menu | |
28120 | ||
28121 | @node Basic Graphics, Three Dimensional Graphics, Graphics, Graphics | |
28122 | @section Basic Graphics | |
28123 | ||
28124 | @noindent | |
28125 | @kindex g f | |
28126 | @pindex calc-graph-fast | |
28127 | The easiest graphics command is @kbd{g f} (@code{calc-graph-fast}). | |
28128 | This command takes two vectors of equal length from the stack. | |
28129 | The vector at the top of the stack represents the ``y'' values of | |
28130 | the various data points. The vector in the second-to-top position | |
28131 | represents the corresponding ``x'' values. This command runs | |
28132 | GNUPLOT (if it has not already been started by previous graphing | |
28133 | commands) and displays the set of data points. The points will | |
28134 | be connected by lines, and there will also be some kind of symbol | |
28135 | to indicate the points themselves. | |
28136 | ||
28137 | The ``x'' entry may instead be an interval form, in which case suitable | |
28138 | ``x'' values are interpolated between the minimum and maximum values of | |
28139 | the interval (whether the interval is open or closed is ignored). | |
28140 | ||
28141 | The ``x'' entry may also be a number, in which case Calc uses the | |
28142 | sequence of ``x'' values @cite{x}, @cite{x+1}, @cite{x+2}, etc. | |
28143 | (Generally the number 0 or 1 would be used for @cite{x} in this case.) | |
28144 | ||
28145 | The ``y'' entry may be any formula instead of a vector. Calc effectively | |
28146 | uses @kbd{N} (@code{calc-eval-num}) to evaluate variables in the formula; | |
28147 | the result of this must be a formula in a single (unassigned) variable. | |
28148 | The formula is plotted with this variable taking on the various ``x'' | |
28149 | values. Graphs of formulas by default use lines without symbols at the | |
28150 | computed data points. Note that if neither ``x'' nor ``y'' is a vector, | |
28151 | Calc guesses at a reasonable number of data points to use. See the | |
28152 | @kbd{g N} command below. (The ``x'' values must be either a vector | |
28153 | or an interval if ``y'' is a formula.) | |
28154 | ||
5d67986c RS |
28155 | @ignore |
28156 | @starindex | |
28157 | @end ignore | |
d7b8e6c6 EZ |
28158 | @tindex xy |
28159 | If ``y'' is (or evaluates to) a formula of the form | |
28160 | @samp{xy(@var{x}, @var{y})} then the result is a | |
28161 | parametric plot. The two arguments of the fictitious @code{xy} function | |
28162 | are used as the ``x'' and ``y'' coordinates of the curve, respectively. | |
28163 | In this case the ``x'' vector or interval you specified is not directly | |
28164 | visible in the graph. For example, if ``x'' is the interval @samp{[0..360]} | |
28165 | and ``y'' is the formula @samp{xy(sin(t), cos(t))}, the resulting graph | |
28166 | will be a circle.@refill | |
28167 | ||
28168 | Also, ``x'' and ``y'' may each be variable names, in which case Calc | |
28169 | looks for suitable vectors, intervals, or formulas stored in those | |
28170 | variables. | |
28171 | ||
28172 | The ``x'' and ``y'' values for the data points (as pulled from the vectors, | |
28173 | calculated from the formulas, or interpolated from the intervals) should | |
28174 | be real numbers (integers, fractions, or floats). If either the ``x'' | |
28175 | value or the ``y'' value of a given data point is not a real number, that | |
28176 | data point will be omitted from the graph. The points on either side | |
28177 | of the invalid point will @emph{not} be connected by a line. | |
28178 | ||
28179 | See the documentation for @kbd{g a} below for a description of the way | |
28180 | numeric prefix arguments affect @kbd{g f}. | |
28181 | ||
28182 | @cindex @code{PlotRejects} variable | |
28183 | @vindex PlotRejects | |
28184 | If you store an empty vector in the variable @code{PlotRejects} | |
28185 | (i.e., @kbd{[ ] s t PlotRejects}), Calc will append information to | |
28186 | this vector for every data point which was rejected because its | |
28187 | ``x'' or ``y'' values were not real numbers. The result will be | |
28188 | a matrix where each row holds the curve number, data point number, | |
28189 | ``x'' value, and ``y'' value for a rejected data point. | |
28190 | @xref{Evaluates-To Operator}, for a handy way to keep tabs on the | |
28191 | current value of @code{PlotRejects}. @xref{Operations on Variables}, | |
28192 | for the @kbd{s R} command which is another easy way to examine | |
28193 | @code{PlotRejects}. | |
28194 | ||
28195 | @kindex g c | |
28196 | @pindex calc-graph-clear | |
28197 | To clear the graphics display, type @kbd{g c} (@code{calc-graph-clear}). | |
28198 | If the GNUPLOT output device is an X window, the window will go away. | |
28199 | Effects on other kinds of output devices will vary. You don't need | |
28200 | to use @kbd{g c} if you don't want to---if you give another @kbd{g f} | |
28201 | or @kbd{g p} command later on, it will reuse the existing graphics | |
28202 | window if there is one. | |
28203 | ||
28204 | @node Three Dimensional Graphics, Managing Curves, Basic Graphics, Graphics | |
28205 | @section Three-Dimensional Graphics | |
28206 | ||
28207 | @kindex g F | |
28208 | @pindex calc-graph-fast-3d | |
28209 | The @kbd{g F} (@code{calc-graph-fast-3d}) command makes a three-dimensional | |
28210 | graph. It works only if you have GNUPLOT 3.0 or later; with GNUPLOT 2.0, | |
28211 | you will see a GNUPLOT error message if you try this command. | |
28212 | ||
28213 | The @kbd{g F} command takes three values from the stack, called ``x'', | |
28214 | ``y'', and ``z'', respectively. As was the case for 2D graphs, there | |
28215 | are several options for these values. | |
28216 | ||
28217 | In the first case, ``x'' and ``y'' are each vectors (not necessarily of | |
28218 | the same length); either or both may instead be interval forms. The | |
28219 | ``z'' value must be a matrix with the same number of rows as elements | |
28220 | in ``x'', and the same number of columns as elements in ``y''. The | |
28221 | result is a surface plot where @c{$z_{ij}$} | |
28222 | @cite{z_ij} is the height of the point | |
28223 | at coordinate @cite{(x_i, y_j)} on the surface. The 3D graph will | |
28224 | be displayed from a certain default viewpoint; you can change this | |
28225 | viewpoint by adding a @samp{set view} to the @samp{*Gnuplot Commands*} | |
28226 | buffer as described later. See the GNUPLOT 3.0 documentation for a | |
28227 | description of the @samp{set view} command. | |
28228 | ||
28229 | Each point in the matrix will be displayed as a dot in the graph, | |
28230 | and these points will be connected by a grid of lines (@dfn{isolines}). | |
28231 | ||
28232 | In the second case, ``x'', ``y'', and ``z'' are all vectors of equal | |
28233 | length. The resulting graph displays a 3D line instead of a surface, | |
28234 | where the coordinates of points along the line are successive triplets | |
28235 | of values from the input vectors. | |
28236 | ||
28237 | In the third case, ``x'' and ``y'' are vectors or interval forms, and | |
28238 | ``z'' is any formula involving two variables (not counting variables | |
28239 | with assigned values). These variables are sorted into alphabetical | |
28240 | order; the first takes on values from ``x'' and the second takes on | |
28241 | values from ``y'' to form a matrix of results that are graphed as a | |
28242 | 3D surface. | |
28243 | ||
5d67986c RS |
28244 | @ignore |
28245 | @starindex | |
28246 | @end ignore | |
d7b8e6c6 EZ |
28247 | @tindex xyz |
28248 | If the ``z'' formula evaluates to a call to the fictitious function | |
28249 | @samp{xyz(@var{x}, @var{y}, @var{z})}, then the result is a | |
28250 | ``parametric surface.'' In this case, the axes of the graph are | |
28251 | taken from the @var{x} and @var{y} values in these calls, and the | |
28252 | ``x'' and ``y'' values from the input vectors or intervals are used only | |
28253 | to specify the range of inputs to the formula. For example, plotting | |
28254 | @samp{[0..360], [0..180], xyz(sin(x)*sin(y), cos(x)*sin(y), cos(y))} | |
28255 | will draw a sphere. (Since the default resolution for 3D plots is | |
28256 | 5 steps in each of ``x'' and ``y'', this will draw a very crude | |
28257 | sphere. You could use the @kbd{g N} command, described below, to | |
28258 | increase this resolution, or specify the ``x'' and ``y'' values as | |
28259 | vectors with more than 5 elements. | |
28260 | ||
28261 | It is also possible to have a function in a regular @kbd{g f} plot | |
28262 | evaluate to an @code{xyz} call. Since @kbd{g f} plots a line, not | |
28263 | a surface, the result will be a 3D parametric line. For example, | |
28264 | @samp{[[0..720], xyz(sin(x), cos(x), x)]} will plot two turns of a | |
28265 | helix (a three-dimensional spiral). | |
28266 | ||
28267 | As for @kbd{g f}, each of ``x'', ``y'', and ``z'' may instead be | |
28268 | variables containing the relevant data. | |
28269 | ||
28270 | @node Managing Curves, Graphics Options, Three Dimensional Graphics, Graphics | |
28271 | @section Managing Curves | |
28272 | ||
28273 | @noindent | |
28274 | The @kbd{g f} command is really shorthand for the following commands: | |
28275 | @kbd{C-u g d g a g p}. Likewise, @w{@kbd{g F}} is shorthand for | |
28276 | @kbd{C-u g d g A g p}. You can gain more control over your graph | |
28277 | by using these commands directly. | |
28278 | ||
28279 | @kindex g a | |
28280 | @pindex calc-graph-add | |
28281 | The @kbd{g a} (@code{calc-graph-add}) command adds the ``curve'' | |
28282 | represented by the two values on the top of the stack to the current | |
28283 | graph. You can have any number of curves in the same graph. When | |
28284 | you give the @kbd{g p} command, all the curves will be drawn superimposed | |
28285 | on the same axes. | |
28286 | ||
28287 | The @kbd{g a} command (and many others that affect the current graph) | |
28288 | will cause a special buffer, @samp{*Gnuplot Commands*}, to be displayed | |
28289 | in another window. This buffer is a template of the commands that will | |
28290 | be sent to GNUPLOT when it is time to draw the graph. The first | |
28291 | @kbd{g a} command adds a @code{plot} command to this buffer. Succeeding | |
28292 | @kbd{g a} commands add extra curves onto that @code{plot} command. | |
28293 | Other graph-related commands put other GNUPLOT commands into this | |
28294 | buffer. In normal usage you never need to work with this buffer | |
28295 | directly, but you can if you wish. The only constraint is that there | |
28296 | must be only one @code{plot} command, and it must be the last command | |
28297 | in the buffer. If you want to save and later restore a complete graph | |
28298 | configuration, you can use regular Emacs commands to save and restore | |
28299 | the contents of the @samp{*Gnuplot Commands*} buffer. | |
28300 | ||
28301 | @vindex PlotData1 | |
28302 | @vindex PlotData2 | |
28303 | If the values on the stack are not variable names, @kbd{g a} will invent | |
28304 | variable names for them (of the form @samp{PlotData@var{n}}) and store | |
28305 | the values in those variables. The ``x'' and ``y'' variables are what | |
28306 | go into the @code{plot} command in the template. If you add a curve | |
28307 | that uses a certain variable and then later change that variable, you | |
28308 | can replot the graph without having to delete and re-add the curve. | |
28309 | That's because the variable name, not the vector, interval or formula | |
28310 | itself, is what was added by @kbd{g a}. | |
28311 | ||
28312 | A numeric prefix argument on @kbd{g a} or @kbd{g f} changes the way | |
28313 | stack entries are interpreted as curves. With a positive prefix | |
28314 | argument @cite{n}, the top @cite{n} stack entries are ``y'' values | |
28315 | for @cite{n} different curves which share a common ``x'' value in | |
28316 | the @cite{n+1}st stack entry. (Thus @kbd{g a} with no prefix | |
28317 | argument is equivalent to @kbd{C-u 1 g a}.) | |
28318 | ||
28319 | A prefix of zero or plain @kbd{C-u} means to take two stack entries, | |
28320 | ``x'' and ``y'' as usual, but to interpret ``y'' as a vector of | |
28321 | ``y'' values for several curves that share a common ``x''. | |
28322 | ||
28323 | A negative prefix argument tells Calc to read @cite{n} vectors from | |
28324 | the stack; each vector @cite{[x, y]} describes an independent curve. | |
28325 | This is the only form of @kbd{g a} that creates several curves at once | |
28326 | that don't have common ``x'' values. (Of course, the range of ``x'' | |
28327 | values covered by all the curves ought to be roughly the same if | |
28328 | they are to look nice on the same graph.) | |
28329 | ||
28330 | For example, to plot @c{$\sin n x$} | |
28331 | @cite{sin(n x)} for integers @cite{n} | |
28332 | from 1 to 5, you could use @kbd{v x} to create a vector of integers | |
28333 | (@cite{n}), then @kbd{V M '} or @kbd{V M $} to map @samp{sin(n x)} | |
28334 | across this vector. The resulting vector of formulas is suitable | |
28335 | for use as the ``y'' argument to a @kbd{C-u g a} or @kbd{C-u g f} | |
28336 | command. | |
28337 | ||
28338 | @kindex g A | |
28339 | @pindex calc-graph-add-3d | |
28340 | The @kbd{g A} (@code{calc-graph-add-3d}) command adds a 3D curve | |
28341 | to the graph. It is not legal to intermix 2D and 3D curves in a | |
28342 | single graph. This command takes three arguments, ``x'', ``y'', | |
28343 | and ``z'', from the stack. With a positive prefix @cite{n}, it | |
28344 | takes @cite{n+2} arguments (common ``x'' and ``y'', plus @cite{n} | |
28345 | separate ``z''s). With a zero prefix, it takes three stack entries | |
28346 | but the ``z'' entry is a vector of curve values. With a negative | |
28347 | prefix @cite{-n}, it takes @cite{n} vectors of the form @cite{[x, y, z]}. | |
28348 | The @kbd{g A} command works by adding a @code{splot} (surface-plot) | |
28349 | command to the @samp{*Gnuplot Commands*} buffer. | |
28350 | ||
28351 | (Although @kbd{g a} adds a 2D @code{plot} command to the | |
28352 | @samp{*Gnuplot Commands*} buffer, Calc changes this to @code{splot} | |
28353 | before sending it to GNUPLOT if it notices that the data points are | |
28354 | evaluating to @code{xyz} calls. It will not work to mix 2D and 3D | |
28355 | @kbd{g a} curves in a single graph, although Calc does not currently | |
28356 | check for this.) | |
28357 | ||
28358 | @kindex g d | |
28359 | @pindex calc-graph-delete | |
28360 | The @kbd{g d} (@code{calc-graph-delete}) command deletes the most | |
28361 | recently added curve from the graph. It has no effect if there are | |
28362 | no curves in the graph. With a numeric prefix argument of any kind, | |
28363 | it deletes all of the curves from the graph. | |
28364 | ||
28365 | @kindex g H | |
28366 | @pindex calc-graph-hide | |
28367 | The @kbd{g H} (@code{calc-graph-hide}) command ``hides'' or ``unhides'' | |
28368 | the most recently added curve. A hidden curve will not appear in | |
28369 | the actual plot, but information about it such as its name and line and | |
28370 | point styles will be retained. | |
28371 | ||
28372 | @kindex g j | |
28373 | @pindex calc-graph-juggle | |
28374 | The @kbd{g j} (@code{calc-graph-juggle}) command moves the curve | |
28375 | at the end of the list (the ``most recently added curve'') to the | |
28376 | front of the list. The next-most-recent curve is thus exposed for | |
28377 | @w{@kbd{g d}} or similar commands to use. With @kbd{g j} you can work | |
28378 | with any curve in the graph even though curve-related commands only | |
28379 | affect the last curve in the list. | |
28380 | ||
28381 | @kindex g p | |
28382 | @pindex calc-graph-plot | |
28383 | The @kbd{g p} (@code{calc-graph-plot}) command uses GNUPLOT to draw | |
28384 | the graph described in the @samp{*Gnuplot Commands*} buffer. Any | |
28385 | GNUPLOT parameters which are not defined by commands in this buffer | |
28386 | are reset to their default values. The variables named in the @code{plot} | |
28387 | command are written to a temporary data file and the variable names | |
28388 | are then replaced by the file name in the template. The resulting | |
28389 | plotting commands are fed to the GNUPLOT program. See the documentation | |
28390 | for the GNUPLOT program for more specific information. All temporary | |
28391 | files are removed when Emacs or GNUPLOT exits. | |
28392 | ||
28393 | If you give a formula for ``y'', Calc will remember all the values that | |
28394 | it calculates for the formula so that later plots can reuse these values. | |
28395 | Calc throws out these saved values when you change any circumstances | |
28396 | that may affect the data, such as switching from Degrees to Radians | |
28397 | mode, or changing the value of a parameter in the formula. You can | |
28398 | force Calc to recompute the data from scratch by giving a negative | |
28399 | numeric prefix argument to @kbd{g p}. | |
28400 | ||
28401 | Calc uses a fairly rough step size when graphing formulas over intervals. | |
28402 | This is to ensure quick response. You can ``refine'' a plot by giving | |
28403 | a positive numeric prefix argument to @kbd{g p}. Calc goes through | |
28404 | the data points it has computed and saved from previous plots of the | |
28405 | function, and computes and inserts a new data point midway between | |
28406 | each of the existing points. You can refine a plot any number of times, | |
28407 | but beware that the amount of calculation involved doubles each time. | |
28408 | ||
28409 | Calc does not remember computed values for 3D graphs. This means the | |
28410 | numerix prefix argument, if any, to @kbd{g p} is effectively ignored if | |
28411 | the current graph is three-dimensional. | |
28412 | ||
28413 | @kindex g P | |
28414 | @pindex calc-graph-print | |
28415 | The @kbd{g P} (@code{calc-graph-print}) command is like @kbd{g p}, | |
28416 | except that it sends the output to a printer instead of to the | |
28417 | screen. More precisely, @kbd{g p} looks for @samp{set terminal} | |
28418 | or @samp{set output} commands in the @samp{*Gnuplot Commands*} buffer; | |
28419 | lacking these it uses the default settings. However, @kbd{g P} | |
28420 | ignores @samp{set terminal} and @samp{set output} commands and | |
28421 | uses a different set of default values. All of these values are | |
28422 | controlled by the @kbd{g D} and @kbd{g O} commands discussed below. | |
28423 | Provided everything is set up properly, @kbd{g p} will plot to | |
28424 | the screen unless you have specified otherwise and @kbd{g P} will | |
28425 | always plot to the printer. | |
28426 | ||
28427 | @node Graphics Options, Devices, Managing Curves, Graphics | |
28428 | @section Graphics Options | |
28429 | ||
28430 | @noindent | |
28431 | @kindex g g | |
28432 | @pindex calc-graph-grid | |
28433 | The @kbd{g g} (@code{calc-graph-grid}) command turns the ``grid'' | |
28434 | on and off. It is off by default; tick marks appear only at the | |
28435 | edges of the graph. With the grid turned on, dotted lines appear | |
28436 | across the graph at each tick mark. Note that this command only | |
28437 | changes the setting in @samp{*Gnuplot Commands*}; to see the effects | |
28438 | of the change you must give another @kbd{g p} command. | |
28439 | ||
28440 | @kindex g b | |
28441 | @pindex calc-graph-border | |
28442 | The @kbd{g b} (@code{calc-graph-border}) command turns the border | |
28443 | (the box that surrounds the graph) on and off. It is on by default. | |
28444 | This command will only work with GNUPLOT 3.0 and later versions. | |
28445 | ||
28446 | @kindex g k | |
28447 | @pindex calc-graph-key | |
28448 | The @kbd{g k} (@code{calc-graph-key}) command turns the ``key'' | |
28449 | on and off. The key is a chart in the corner of the graph that | |
28450 | shows the correspondence between curves and line styles. It is | |
28451 | off by default, and is only really useful if you have several | |
28452 | curves on the same graph. | |
28453 | ||
28454 | @kindex g N | |
28455 | @pindex calc-graph-num-points | |
28456 | The @kbd{g N} (@code{calc-graph-num-points}) command allows you | |
28457 | to select the number of data points in the graph. This only affects | |
28458 | curves where neither ``x'' nor ``y'' is specified as a vector. | |
28459 | Enter a blank line to revert to the default value (initially 15). | |
28460 | With no prefix argument, this command affects only the current graph. | |
28461 | With a positive prefix argument this command changes or, if you enter | |
28462 | a blank line, displays the default number of points used for all | |
28463 | graphs created by @kbd{g a} that don't specify the resolution explicitly. | |
28464 | With a negative prefix argument, this command changes or displays | |
28465 | the default value (initially 5) used for 3D graphs created by @kbd{g A}. | |
28466 | Note that a 3D setting of 5 means that a total of @cite{5^2 = 25} points | |
28467 | will be computed for the surface. | |
28468 | ||
28469 | Data values in the graph of a function are normally computed to a | |
28470 | precision of five digits, regardless of the current precision at the | |
28471 | time. This is usually more than adequate, but there are cases where | |
28472 | it will not be. For example, plotting @cite{1 + x} with @cite{x} in the | |
28473 | interval @samp{[0 ..@: 1e-6]} will round all the data points down | |
28474 | to 1.0! Putting the command @samp{set precision @var{n}} in the | |
28475 | @samp{*Gnuplot Commands*} buffer will cause the data to be computed | |
28476 | at precision @var{n} instead of 5. Since this is such a rare case, | |
28477 | there is no keystroke-based command to set the precision. | |
28478 | ||
28479 | @kindex g h | |
28480 | @pindex calc-graph-header | |
28481 | The @kbd{g h} (@code{calc-graph-header}) command sets the title | |
28482 | for the graph. This will show up centered above the graph. | |
28483 | The default title is blank (no title). | |
28484 | ||
28485 | @kindex g n | |
28486 | @pindex calc-graph-name | |
28487 | The @kbd{g n} (@code{calc-graph-name}) command sets the title of an | |
28488 | individual curve. Like the other curve-manipulating commands, it | |
28489 | affects the most recently added curve, i.e., the last curve on the | |
28490 | list in the @samp{*Gnuplot Commands*} buffer. To set the title of | |
28491 | the other curves you must first juggle them to the end of the list | |
28492 | with @kbd{g j}, or edit the @samp{*Gnuplot Commands*} buffer by hand. | |
28493 | Curve titles appear in the key; if the key is turned off they are | |
28494 | not used. | |
28495 | ||
28496 | @kindex g t | |
28497 | @kindex g T | |
28498 | @pindex calc-graph-title-x | |
28499 | @pindex calc-graph-title-y | |
28500 | The @kbd{g t} (@code{calc-graph-title-x}) and @kbd{g T} | |
28501 | (@code{calc-graph-title-y}) commands set the titles on the ``x'' | |
28502 | and ``y'' axes, respectively. These titles appear next to the | |
28503 | tick marks on the left and bottom edges of the graph, respectively. | |
28504 | Calc does not have commands to control the tick marks themselves, | |
28505 | but you can edit them into the @samp{*Gnuplot Commands*} buffer if | |
28506 | you wish. See the GNUPLOT documentation for details. | |
28507 | ||
28508 | @kindex g r | |
28509 | @kindex g R | |
28510 | @pindex calc-graph-range-x | |
28511 | @pindex calc-graph-range-y | |
28512 | The @kbd{g r} (@code{calc-graph-range-x}) and @kbd{g R} | |
28513 | (@code{calc-graph-range-y}) commands set the range of values on the | |
28514 | ``x'' and ``y'' axes, respectively. You are prompted to enter a | |
28515 | suitable range. This should be either a pair of numbers of the | |
28516 | form, @samp{@var{min}:@var{max}}, or a blank line to revert to the | |
28517 | default behavior of setting the range based on the range of values | |
28518 | in the data, or @samp{$} to take the range from the top of the stack. | |
28519 | Ranges on the stack can be represented as either interval forms or | |
28520 | vectors: @samp{[@var{min} ..@: @var{max}]} or @samp{[@var{min}, @var{max}]}. | |
28521 | ||
28522 | @kindex g l | |
28523 | @kindex g L | |
28524 | @pindex calc-graph-log-x | |
28525 | @pindex calc-graph-log-y | |
28526 | The @kbd{g l} (@code{calc-graph-log-x}) and @kbd{g L} (@code{calc-graph-log-y}) | |
28527 | commands allow you to set either or both of the axes of the graph to | |
28528 | be logarithmic instead of linear. | |
28529 | ||
28530 | @kindex g C-l | |
28531 | @kindex g C-r | |
28532 | @kindex g C-t | |
28533 | @pindex calc-graph-log-z | |
28534 | @pindex calc-graph-range-z | |
28535 | @pindex calc-graph-title-z | |
28536 | For 3D plots, @kbd{g C-t}, @kbd{g C-r}, and @kbd{g C-l} (those are | |
28537 | letters with the Control key held down) are the corresponding commands | |
28538 | for the ``z'' axis. | |
28539 | ||
28540 | @kindex g z | |
28541 | @kindex g Z | |
28542 | @pindex calc-graph-zero-x | |
28543 | @pindex calc-graph-zero-y | |
28544 | The @kbd{g z} (@code{calc-graph-zero-x}) and @kbd{g Z} | |
28545 | (@code{calc-graph-zero-y}) commands control whether a dotted line is | |
28546 | drawn to indicate the ``x'' and/or ``y'' zero axes. (These are the same | |
28547 | dotted lines that would be drawn there anyway if you used @kbd{g g} to | |
28548 | turn the ``grid'' feature on.) Zero-axis lines are on by default, and | |
28549 | may be turned off only in GNUPLOT 3.0 and later versions. They are | |
28550 | not available for 3D plots. | |
28551 | ||
28552 | @kindex g s | |
28553 | @pindex calc-graph-line-style | |
28554 | The @kbd{g s} (@code{calc-graph-line-style}) command turns the connecting | |
28555 | lines on or off for the most recently added curve, and optionally selects | |
28556 | the style of lines to be used for that curve. Plain @kbd{g s} simply | |
28557 | toggles the lines on and off. With a numeric prefix argument, @kbd{g s} | |
28558 | turns lines on and sets a particular line style. Line style numbers | |
28559 | start at one and their meanings vary depending on the output device. | |
28560 | GNUPLOT guarantees that there will be at least six different line styles | |
28561 | available for any device. | |
28562 | ||
28563 | @kindex g S | |
28564 | @pindex calc-graph-point-style | |
28565 | The @kbd{g S} (@code{calc-graph-point-style}) command similarly turns | |
28566 | the symbols at the data points on or off, or sets the point style. | |
28567 | If you turn both lines and points off, the data points will show as | |
28568 | tiny dots. | |
28569 | ||
28570 | @cindex @code{LineStyles} variable | |
28571 | @cindex @code{PointStyles} variable | |
28572 | @vindex LineStyles | |
28573 | @vindex PointStyles | |
28574 | Another way to specify curve styles is with the @code{LineStyles} and | |
28575 | @code{PointStyles} variables. These variables initially have no stored | |
28576 | values, but if you store a vector of integers in one of these variables, | |
28577 | the @kbd{g a} and @kbd{g f} commands will use those style numbers | |
28578 | instead of the defaults for new curves that are added to the graph. | |
28579 | An entry should be a positive integer for a specific style, or 0 to let | |
28580 | the style be chosen automatically, or @i{-1} to turn off lines or points | |
28581 | altogether. If there are more curves than elements in the vector, the | |
28582 | last few curves will continue to have the default styles. Of course, | |
28583 | you can later use @kbd{g s} and @kbd{g S} to change any of these styles. | |
28584 | ||
5d67986c | 28585 | For example, @kbd{'[2 -1 3] @key{RET} s t LineStyles} causes the first curve |
d7b8e6c6 EZ |
28586 | to have lines in style number 2, the second curve to have no connecting |
28587 | lines, and the third curve to have lines in style 3. Point styles will | |
28588 | still be assigned automatically, but you could store another vector in | |
28589 | @code{PointStyles} to define them, too. | |
28590 | ||
28591 | @node Devices, , Graphics Options, Graphics | |
28592 | @section Graphical Devices | |
28593 | ||
28594 | @noindent | |
28595 | @kindex g D | |
28596 | @pindex calc-graph-device | |
28597 | The @kbd{g D} (@code{calc-graph-device}) command sets the device name | |
28598 | (or ``terminal name'' in GNUPLOT lingo) to be used by @kbd{g p} commands | |
28599 | on this graph. It does not affect the permanent default device name. | |
28600 | If you enter a blank name, the device name reverts to the default. | |
28601 | Enter @samp{?} to see a list of supported devices. | |
28602 | ||
28603 | With a positive numeric prefix argument, @kbd{g D} instead sets | |
28604 | the default device name, used by all plots in the future which do | |
28605 | not override it with a plain @kbd{g D} command. If you enter a | |
28606 | blank line this command shows you the current default. The special | |
28607 | name @code{default} signifies that Calc should choose @code{x11} if | |
28608 | the X window system is in use (as indicated by the presence of a | |
28609 | @code{DISPLAY} environment variable), or otherwise @code{dumb} under | |
28610 | GNUPLOT 3.0 and later, or @code{postscript} under GNUPLOT 2.0. | |
28611 | This is the initial default value. | |
28612 | ||
28613 | The @code{dumb} device is an interface to ``dumb terminals,'' i.e., | |
28614 | terminals with no special graphics facilities. It writes a crude | |
28615 | picture of the graph composed of characters like @code{-} and @code{|} | |
28616 | to a buffer called @samp{*Gnuplot Trail*}, which Calc then displays. | |
28617 | The graph is made the same size as the Emacs screen, which on most | |
28618 | dumb terminals will be @c{$80\times24$} | |
28619 | @asis{80x24} characters. The graph is displayed in | |
28620 | an Emacs ``recursive edit''; type @kbd{q} or @kbd{M-# M-#} to exit | |
28621 | the recursive edit and return to Calc. Note that the @code{dumb} | |
28622 | device is present only in GNUPLOT 3.0 and later versions. | |
28623 | ||
28624 | The word @code{dumb} may be followed by two numbers separated by | |
28625 | spaces. These are the desired width and height of the graph in | |
28626 | characters. Also, the device name @code{big} is like @code{dumb} | |
28627 | but creates a graph four times the width and height of the Emacs | |
28628 | screen. You will then have to scroll around to view the entire | |
28629 | graph. In the @samp{*Gnuplot Trail*} buffer, @key{SPC}, @key{DEL}, | |
28630 | @kbd{<}, and @kbd{>} are defined to scroll by one screenful in each | |
28631 | of the four directions. | |
28632 | ||
28633 | With a negative numeric prefix argument, @kbd{g D} sets or displays | |
28634 | the device name used by @kbd{g P} (@code{calc-graph-print}). This | |
28635 | is initially @code{postscript}. If you don't have a PostScript | |
28636 | printer, you may decide once again to use @code{dumb} to create a | |
28637 | plot on any text-only printer. | |
28638 | ||
28639 | @kindex g O | |
28640 | @pindex calc-graph-output | |
28641 | The @kbd{g O} (@code{calc-graph-output}) command sets the name of | |
28642 | the output file used by GNUPLOT. For some devices, notably @code{x11}, | |
28643 | there is no output file and this information is not used. Many other | |
28644 | ``devices'' are really file formats like @code{postscript}; in these | |
28645 | cases the output in the desired format goes into the file you name | |
5d67986c | 28646 | with @kbd{g O}. Type @kbd{g O stdout @key{RET}} to set GNUPLOT to write |
d7b8e6c6 EZ |
28647 | to its standard output stream, i.e., to @samp{*Gnuplot Trail*}. |
28648 | This is the default setting. | |
28649 | ||
28650 | Another special output name is @code{tty}, which means that GNUPLOT | |
28651 | is going to write graphics commands directly to its standard output, | |
28652 | which you wish Emacs to pass through to your terminal. Tektronix | |
28653 | graphics terminals, among other devices, operate this way. Calc does | |
28654 | this by telling GNUPLOT to write to a temporary file, then running a | |
28655 | sub-shell executing the command @samp{cat tempfile >/dev/tty}. On | |
28656 | typical Unix systems, this will copy the temporary file directly to | |
28657 | the terminal, bypassing Emacs entirely. You will have to type @kbd{C-l} | |
28658 | to Emacs afterwards to refresh the screen. | |
28659 | ||
28660 | Once again, @kbd{g O} with a positive or negative prefix argument | |
28661 | sets the default or printer output file names, respectively. In each | |
28662 | case you can specify @code{auto}, which causes Calc to invent a temporary | |
28663 | file name for each @kbd{g p} (or @kbd{g P}) command. This temporary file | |
28664 | will be deleted once it has been displayed or printed. If the output file | |
28665 | name is not @code{auto}, the file is not automatically deleted. | |
28666 | ||
28667 | The default and printer devices and output files can be saved | |
28668 | permanently by the @kbd{m m} (@code{calc-save-modes}) command. The | |
28669 | default number of data points (see @kbd{g N}) and the X geometry | |
28670 | (see @kbd{g X}) are also saved. Other graph information is @emph{not} | |
28671 | saved; you can save a graph's configuration simply by saving the contents | |
28672 | of the @samp{*Gnuplot Commands*} buffer. | |
28673 | ||
28674 | @vindex calc-gnuplot-plot-command | |
28675 | @vindex calc-gnuplot-default-device | |
28676 | @vindex calc-gnuplot-default-output | |
28677 | @vindex calc-gnuplot-print-command | |
28678 | @vindex calc-gnuplot-print-device | |
28679 | @vindex calc-gnuplot-print-output | |
28680 | If you are installing Calc you may wish to configure the default and | |
28681 | printer devices and output files for the whole system. The relevant | |
28682 | Lisp variables are @code{calc-gnuplot-default-device} and @code{-output}, | |
28683 | and @code{calc-gnuplot-print-device} and @code{-output}. The output | |
28684 | file names must be either strings as described above, or Lisp | |
28685 | expressions which are evaluated on the fly to get the output file names. | |
28686 | ||
28687 | Other important Lisp variables are @code{calc-gnuplot-plot-command} and | |
28688 | @code{calc-gnuplot-print-command}, which give the system commands to | |
28689 | display or print the output of GNUPLOT, respectively. These may be | |
28690 | @code{nil} if no command is necessary, or strings which can include | |
28691 | @samp{%s} to signify the name of the file to be displayed or printed. | |
28692 | Or, these variables may contain Lisp expressions which are evaluated | |
28693 | to display or print the output. | |
28694 | ||
28695 | @kindex g x | |
28696 | @pindex calc-graph-display | |
28697 | The @kbd{g x} (@code{calc-graph-display}) command lets you specify | |
28698 | on which X window system display your graphs should be drawn. Enter | |
28699 | a blank line to see the current display name. This command has no | |
28700 | effect unless the current device is @code{x11}. | |
28701 | ||
28702 | @kindex g X | |
28703 | @pindex calc-graph-geometry | |
28704 | The @kbd{g X} (@code{calc-graph-geometry}) command is a similar | |
28705 | command for specifying the position and size of the X window. | |
28706 | The normal value is @code{default}, which generally means your | |
28707 | window manager will let you place the window interactively. | |
28708 | Entering @samp{800x500+0+0} would create an 800-by-500 pixel | |
28709 | window in the upper-left corner of the screen. | |
28710 | ||
28711 | The buffer called @samp{*Gnuplot Trail*} holds a transcript of the | |
28712 | session with GNUPLOT. This shows the commands Calc has ``typed'' to | |
28713 | GNUPLOT and the responses it has received. Calc tries to notice when an | |
28714 | error message has appeared here and display the buffer for you when | |
28715 | this happens. You can check this buffer yourself if you suspect | |
28716 | something has gone wrong. | |
28717 | ||
28718 | @kindex g C | |
28719 | @pindex calc-graph-command | |
28720 | The @kbd{g C} (@code{calc-graph-command}) command prompts you to | |
28721 | enter any line of text, then simply sends that line to the current | |
28722 | GNUPLOT process. The @samp{*Gnuplot Trail*} buffer looks deceptively | |
28723 | like a Shell buffer but you can't type commands in it yourself. | |
28724 | Instead, you must use @kbd{g C} for this purpose. | |
28725 | ||
28726 | @kindex g v | |
28727 | @kindex g V | |
28728 | @pindex calc-graph-view-commands | |
28729 | @pindex calc-graph-view-trail | |
28730 | The @kbd{g v} (@code{calc-graph-view-commands}) and @kbd{g V} | |
28731 | (@code{calc-graph-view-trail}) commands display the @samp{*Gnuplot Commands*} | |
28732 | and @samp{*Gnuplot Trail*} buffers, respectively, in another window. | |
28733 | This happens automatically when Calc thinks there is something you | |
28734 | will want to see in either of these buffers. If you type @kbd{g v} | |
28735 | or @kbd{g V} when the relevant buffer is already displayed, the | |
28736 | buffer is hidden again. | |
28737 | ||
28738 | One reason to use @kbd{g v} is to add your own commands to the | |
28739 | @samp{*Gnuplot Commands*} buffer. Press @kbd{g v}, then use | |
28740 | @kbd{C-x o} to switch into that window. For example, GNUPLOT has | |
28741 | @samp{set label} and @samp{set arrow} commands that allow you to | |
28742 | annotate your plots. Since Calc doesn't understand these commands, | |
28743 | you have to add them to the @samp{*Gnuplot Commands*} buffer | |
28744 | yourself, then use @w{@kbd{g p}} to replot using these new commands. Note | |
28745 | that your commands must appear @emph{before} the @code{plot} command. | |
28746 | To get help on any GNUPLOT feature, type, e.g., @kbd{g C help set label}. | |
5d67986c | 28747 | You may have to type @kbd{g C @key{RET}} a few times to clear the |
d7b8e6c6 EZ |
28748 | ``press return for more'' or ``subtopic of @dots{}'' requests. |
28749 | Note that Calc always sends commands (like @samp{set nolabel}) to | |
28750 | reset all plotting parameters to the defaults before each plot, so | |
28751 | to delete a label all you need to do is delete the @samp{set label} | |
28752 | line you added (or comment it out with @samp{#}) and then replot | |
28753 | with @kbd{g p}. | |
28754 | ||
28755 | @kindex g q | |
28756 | @pindex calc-graph-quit | |
28757 | You can use @kbd{g q} (@code{calc-graph-quit}) to kill the GNUPLOT | |
28758 | process that is running. The next graphing command you give will | |
28759 | start a fresh GNUPLOT process. The word @samp{Graph} appears in | |
28760 | the Calc window's mode line whenever a GNUPLOT process is currently | |
28761 | running. The GNUPLOT process is automatically killed when you | |
28762 | exit Emacs if you haven't killed it manually by then. | |
28763 | ||
28764 | @kindex g K | |
28765 | @pindex calc-graph-kill | |
28766 | The @kbd{g K} (@code{calc-graph-kill}) command is like @kbd{g q} | |
28767 | except that it also views the @samp{*Gnuplot Trail*} buffer so that | |
28768 | you can see the process being killed. This is better if you are | |
28769 | killing GNUPLOT because you think it has gotten stuck. | |
28770 | ||
28771 | @node Kill and Yank, Keypad Mode, Graphics, Top | |
28772 | @chapter Kill and Yank Functions | |
28773 | ||
28774 | @noindent | |
28775 | The commands in this chapter move information between the Calculator and | |
28776 | other Emacs editing buffers. | |
28777 | ||
28778 | In many cases Embedded Mode is an easier and more natural way to | |
28779 | work with Calc from a regular editing buffer. @xref{Embedded Mode}. | |
28780 | ||
28781 | @menu | |
28782 | * Killing From Stack:: | |
28783 | * Yanking Into Stack:: | |
28784 | * Grabbing From Buffers:: | |
28785 | * Yanking Into Buffers:: | |
28786 | * X Cut and Paste:: | |
28787 | @end menu | |
28788 | ||
28789 | @node Killing From Stack, Yanking Into Stack, Kill and Yank, Kill and Yank | |
28790 | @section Killing from the Stack | |
28791 | ||
28792 | @noindent | |
28793 | @kindex C-k | |
28794 | @pindex calc-kill | |
28795 | @kindex M-k | |
28796 | @pindex calc-copy-as-kill | |
28797 | @kindex C-w | |
28798 | @pindex calc-kill-region | |
28799 | @kindex M-w | |
28800 | @pindex calc-copy-region-as-kill | |
28801 | @cindex Kill ring | |
28802 | @dfn{Kill} commands are Emacs commands that insert text into the | |
28803 | ``kill ring,'' from which it can later be ``yanked'' by a @kbd{C-y} | |
28804 | command. Three common kill commands in normal Emacs are @kbd{C-k}, which | |
28805 | kills one line, @kbd{C-w}, which kills the region between mark and point, | |
28806 | and @kbd{M-w}, which puts the region into the kill ring without actually | |
28807 | deleting it. All of these commands work in the Calculator, too. Also, | |
28808 | @kbd{M-k} has been provided to complete the set; it puts the current line | |
28809 | into the kill ring without deleting anything. | |
28810 | ||
28811 | The kill commands are unusual in that they pay attention to the location | |
28812 | of the cursor in the Calculator buffer. If the cursor is on or below the | |
28813 | bottom line, the kill commands operate on the top of the stack. Otherwise, | |
28814 | they operate on whatever stack element the cursor is on. Calc's kill | |
28815 | commands always operate on whole stack entries. (They act the same as their | |
28816 | standard Emacs cousins except they ``round up'' the specified region to | |
28817 | encompass full lines.) The text is copied into the kill ring exactly as | |
28818 | it appears on the screen, including line numbers if they are enabled. | |
28819 | ||
28820 | A numeric prefix argument to @kbd{C-k} or @kbd{M-k} affects the number | |
28821 | of lines killed. A positive argument kills the current line and @cite{n-1} | |
28822 | lines below it. A negative argument kills the @cite{-n} lines above the | |
28823 | current line. Again this mirrors the behavior of the standard Emacs | |
28824 | @kbd{C-k} command. Although a whole line is always deleted, @kbd{C-k} | |
28825 | with no argument copies only the number itself into the kill ring, whereas | |
28826 | @kbd{C-k} with a prefix argument of 1 copies the number with its trailing | |
28827 | newline. | |
28828 | ||
28829 | @node Yanking Into Stack, Grabbing From Buffers, Killing From Stack, Kill and Yank | |
28830 | @section Yanking into the Stack | |
28831 | ||
28832 | @noindent | |
28833 | @kindex C-y | |
28834 | @pindex calc-yank | |
28835 | The @kbd{C-y} command yanks the most recently killed text back into the | |
28836 | Calculator. It pushes this value onto the top of the stack regardless of | |
28837 | the cursor position. In general it re-parses the killed text as a number | |
28838 | or formula (or a list of these separated by commas or newlines). However if | |
28839 | the thing being yanked is something that was just killed from the Calculator | |
28840 | itself, its full internal structure is yanked. For example, if you have | |
28841 | set the floating-point display mode to show only four significant digits, | |
28842 | then killing and re-yanking 3.14159 (which displays as 3.142) will yank the | |
28843 | full 3.14159, even though yanking it into any other buffer would yank the | |
28844 | number in its displayed form, 3.142. (Since the default display modes | |
28845 | show all objects to their full precision, this feature normally makes no | |
28846 | difference.) | |
28847 | ||
28848 | @node Grabbing From Buffers, Yanking Into Buffers, Yanking Into Stack, Kill and Yank | |
28849 | @section Grabbing from Other Buffers | |
28850 | ||
28851 | @noindent | |
28852 | @kindex M-# g | |
28853 | @pindex calc-grab-region | |
28854 | The @kbd{M-# g} (@code{calc-grab-region}) command takes the text between | |
28855 | point and mark in the current buffer and attempts to parse it as a | |
28856 | vector of values. Basically, it wraps the text in vector brackets | |
28857 | @samp{[ ]} unless the text already is enclosed in vector brackets, | |
28858 | then reads the text as if it were an algebraic entry. The contents | |
28859 | of the vector may be numbers, formulas, or any other Calc objects. | |
28860 | If the @kbd{M-# g} command works successfully, it does an automatic | |
28861 | @kbd{M-# c} to enter the Calculator buffer. | |
28862 | ||
28863 | A numeric prefix argument grabs the specified number of lines around | |
28864 | point, ignoring the mark. A positive prefix grabs from point to the | |
28865 | @cite{n}th following newline (so that @kbd{M-1 M-# g} grabs from point | |
28866 | to the end of the current line); a negative prefix grabs from point | |
28867 | back to the @cite{n+1}st preceding newline. In these cases the text | |
28868 | that is grabbed is exactly the same as the text that @kbd{C-k} would | |
28869 | delete given that prefix argument. | |
28870 | ||
28871 | A prefix of zero grabs the current line; point may be anywhere on the | |
28872 | line. | |
28873 | ||
28874 | A plain @kbd{C-u} prefix interprets the region between point and mark | |
28875 | as a single number or formula rather than a vector. For example, | |
28876 | @kbd{M-# g} on the text @samp{2 a b} produces the vector of three | |
28877 | values @samp{[2, a, b]}, but @kbd{C-u M-# g} on the same region | |
28878 | reads a formula which is a product of three things: @samp{2 a b}. | |
28879 | (The text @samp{a + b}, on the other hand, will be grabbed as a | |
28880 | vector of one element by plain @kbd{M-# g} because the interpretation | |
28881 | @samp{[a, +, b]} would be a syntax error.) | |
28882 | ||
28883 | If a different language has been specified (@pxref{Language Modes}), | |
28884 | the grabbed text will be interpreted according to that language. | |
28885 | ||
28886 | @kindex M-# r | |
28887 | @pindex calc-grab-rectangle | |
28888 | The @kbd{M-# r} (@code{calc-grab-rectangle}) command takes the text between | |
28889 | point and mark and attempts to parse it as a matrix. If point and mark | |
28890 | are both in the leftmost column, the lines in between are parsed in their | |
28891 | entirety. Otherwise, point and mark define the corners of a rectangle | |
28892 | whose contents are parsed. | |
28893 | ||
28894 | Each line of the grabbed area becomes a row of the matrix. The result | |
28895 | will actually be a vector of vectors, which Calc will treat as a matrix | |
28896 | only if every row contains the same number of values. | |
28897 | ||
28898 | If a line contains a portion surrounded by square brackets (or curly | |
28899 | braces), that portion is interpreted as a vector which becomes a row | |
28900 | of the matrix. Any text surrounding the bracketed portion on the line | |
28901 | is ignored. | |
28902 | ||
28903 | Otherwise, the entire line is interpreted as a row vector as if it | |
28904 | were surrounded by square brackets. Leading line numbers (in the | |
28905 | format used in the Calc stack buffer) are ignored. If you wish to | |
28906 | force this interpretation (even if the line contains bracketed | |
28907 | portions), give a negative numeric prefix argument to the | |
28908 | @kbd{M-# r} command. | |
28909 | ||
28910 | If you give a numeric prefix argument of zero or plain @kbd{C-u}, each | |
28911 | line is instead interpreted as a single formula which is converted into | |
28912 | a one-element vector. Thus the result of @kbd{C-u M-# r} will be a | |
28913 | one-column matrix. For example, suppose one line of the data is the | |
28914 | expression @samp{2 a}. A plain @w{@kbd{M-# r}} will interpret this as | |
28915 | @samp{[2 a]}, which in turn is read as a two-element vector that forms | |
28916 | one row of the matrix. But a @kbd{C-u M-# r} will interpret this row | |
28917 | as @samp{[2*a]}. | |
28918 | ||
28919 | If you give a positive numeric prefix argument @var{n}, then each line | |
28920 | will be split up into columns of width @var{n}; each column is parsed | |
28921 | separately as a matrix element. If a line contained | |
28922 | @w{@samp{2 +/- 3 4 +/- 5}}, then grabbing with a prefix argument of 8 | |
28923 | would correctly split the line into two error forms.@refill | |
28924 | ||
28925 | @xref{Matrix Functions}, to see how to pull the matrix apart into its | |
28926 | constituent rows and columns. (If it is a @c{$1\times1$} | |
28927 | @asis{1x1} matrix, just hit @kbd{v u} | |
28928 | (@code{calc-unpack}) twice.) | |
28929 | ||
28930 | @kindex M-# : | |
28931 | @kindex M-# _ | |
28932 | @pindex calc-grab-sum-across | |
28933 | @pindex calc-grab-sum-down | |
28934 | @cindex Summing rows and columns of data | |
28935 | The @kbd{M-# :} (@code{calc-grab-sum-down}) command is a handy way to | |
28936 | grab a rectangle of data and sum its columns. It is equivalent to | |
28937 | typing @kbd{M-# r}, followed by @kbd{V R : +} (the vector reduction | |
28938 | command that sums the columns of a matrix; @pxref{Reducing}). The | |
28939 | result of the command will be a vector of numbers, one for each column | |
28940 | in the input data. The @kbd{M-# _} (@code{calc-grab-sum-across}) command | |
28941 | similarly grabs a rectangle and sums its rows by executing @w{@kbd{V R _ +}}. | |
28942 | ||
28943 | As well as being more convenient, @kbd{M-# :} and @kbd{M-# _} are also | |
28944 | much faster because they don't actually place the grabbed vector on | |
28945 | the stack. In a @kbd{M-# r V R : +} sequence, formatting the vector | |
28946 | for display on the stack takes a large fraction of the total time | |
28947 | (unless you have planned ahead and used @kbd{v .} and @kbd{t .} modes). | |
28948 | ||
28949 | For example, suppose we have a column of numbers in a file which we | |
28950 | wish to sum. Go to one corner of the column and press @kbd{C-@@} to | |
28951 | set the mark; go to the other corner and type @kbd{M-# :}. Since there | |
28952 | is only one column, the result will be a vector of one number, the sum. | |
28953 | (You can type @kbd{v u} to unpack this vector into a plain number if | |
28954 | you want to do further arithmetic with it.) | |
28955 | ||
28956 | To compute the product of the column of numbers, we would have to do | |
28957 | it ``by hand'' since there's no special grab-and-multiply command. | |
28958 | Use @kbd{M-# r} to grab the column of numbers into the calculator in | |
28959 | the form of a column matrix. The statistics command @kbd{u *} is a | |
28960 | handy way to find the product of a vector or matrix of numbers. | |
28961 | @xref{Statistical Operations}. Another approach would be to use | |
28962 | an explicit column reduction command, @kbd{V R : *}. | |
28963 | ||
28964 | @node Yanking Into Buffers, X Cut and Paste, Grabbing From Buffers, Kill and Yank | |
28965 | @section Yanking into Other Buffers | |
28966 | ||
28967 | @noindent | |
28968 | @kindex y | |
28969 | @pindex calc-copy-to-buffer | |
28970 | The plain @kbd{y} (@code{calc-copy-to-buffer}) command inserts the number | |
28971 | at the top of the stack into the most recently used normal editing buffer. | |
28972 | (More specifically, this is the most recently used buffer which is displayed | |
28973 | in a window and whose name does not begin with @samp{*}. If there is no | |
28974 | such buffer, this is the most recently used buffer except for Calculator | |
28975 | and Calc Trail buffers.) The number is inserted exactly as it appears and | |
28976 | without a newline. (If line-numbering is enabled, the line number is | |
28977 | normally not included.) The number is @emph{not} removed from the stack. | |
28978 | ||
28979 | With a prefix argument, @kbd{y} inserts several numbers, one per line. | |
28980 | A positive argument inserts the specified number of values from the top | |
28981 | of the stack. A negative argument inserts the @cite{n}th value from the | |
28982 | top of the stack. An argument of zero inserts the entire stack. Note | |
28983 | that @kbd{y} with an argument of 1 is slightly different from @kbd{y} | |
28984 | with no argument; the former always copies full lines, whereas the | |
28985 | latter strips off the trailing newline. | |
28986 | ||
28987 | With a lone @kbd{C-u} as a prefix argument, @kbd{y} @emph{replaces} the | |
28988 | region in the other buffer with the yanked text, then quits the | |
28989 | Calculator, leaving you in that buffer. A typical use would be to use | |
28990 | @kbd{M-# g} to read a region of data into the Calculator, operate on the | |
28991 | data to produce a new matrix, then type @kbd{C-u y} to replace the | |
28992 | original data with the new data. One might wish to alter the matrix | |
28993 | display style (@pxref{Vector and Matrix Formats}) or change the current | |
28994 | display language (@pxref{Language Modes}) before doing this. Also, note | |
28995 | that this command replaces a linear region of text (as grabbed by | |
28996 | @kbd{M-# g}), not a rectangle (as grabbed by @kbd{M-# r}).@refill | |
28997 | ||
28998 | If the editing buffer is in overwrite (as opposed to insert) mode, | |
28999 | and the @kbd{C-u} prefix was not used, then the yanked number will | |
29000 | overwrite the characters following point rather than being inserted | |
29001 | before those characters. The usual conventions of overwrite mode | |
29002 | are observed; for example, characters will be inserted at the end of | |
29003 | a line rather than overflowing onto the next line. Yanking a multi-line | |
29004 | object such as a matrix in overwrite mode overwrites the next @var{n} | |
29005 | lines in the buffer, lengthening or shortening each line as necessary. | |
29006 | Finally, if the thing being yanked is a simple integer or floating-point | |
29007 | number (like @samp{-1.2345e-3}) and the characters following point also | |
29008 | make up such a number, then Calc will replace that number with the new | |
29009 | number, lengthening or shortening as necessary. The concept of | |
29010 | ``overwrite mode'' has thus been generalized from overwriting characters | |
29011 | to overwriting one complete number with another. | |
29012 | ||
29013 | @kindex M-# y | |
29014 | The @kbd{M-# y} key sequence is equivalent to @kbd{y} except that | |
29015 | it can be typed anywhere, not just in Calc. This provides an easy | |
29016 | way to guarantee that Calc knows which editing buffer you want to use! | |
29017 | ||
29018 | @node X Cut and Paste, , Yanking Into Buffers, Kill and Yank | |
29019 | @section X Cut and Paste | |
29020 | ||
29021 | @noindent | |
29022 | If you are using Emacs with the X window system, there is an easier | |
29023 | way to move small amounts of data into and out of the calculator: | |
29024 | Use the mouse-oriented cut and paste facilities of X. | |
29025 | ||
29026 | The default bindings for a three-button mouse cause the left button | |
29027 | to move the Emacs cursor to the given place, the right button to | |
29028 | select the text between the cursor and the clicked location, and | |
29029 | the middle button to yank the selection into the buffer at the | |
29030 | clicked location. So, if you have a Calc window and an editing | |
29031 | window on your Emacs screen, you can use left-click/right-click | |
29032 | to select a number, vector, or formula from one window, then | |
29033 | middle-click to paste that value into the other window. When you | |
29034 | paste text into the Calc window, Calc interprets it as an algebraic | |
29035 | entry. It doesn't matter where you click in the Calc window; the | |
29036 | new value is always pushed onto the top of the stack. | |
29037 | ||
29038 | The @code{xterm} program that is typically used for general-purpose | |
29039 | shell windows in X interprets the mouse buttons in the same way. | |
29040 | So you can use the mouse to move data between Calc and any other | |
29041 | Unix program. One nice feature of @code{xterm} is that a double | |
29042 | left-click selects one word, and a triple left-click selects a | |
29043 | whole line. So you can usually transfer a single number into Calc | |
29044 | just by double-clicking on it in the shell, then middle-clicking | |
29045 | in the Calc window. | |
29046 | ||
29047 | @node Keypad Mode, Embedded Mode, Kill and Yank, Introduction | |
29048 | @chapter ``Keypad'' Mode | |
29049 | ||
29050 | @noindent | |
29051 | @kindex M-# k | |
29052 | @pindex calc-keypad | |
29053 | The @kbd{M-# k} (@code{calc-keypad}) command starts the Calculator | |
29054 | and displays a picture of a calculator-style keypad. If you are using | |
29055 | the X window system, you can click on any of the ``keys'' in the | |
29056 | keypad using the left mouse button to operate the calculator. | |
29057 | The original window remains the selected window; in keypad mode | |
29058 | you can type in your file while simultaneously performing | |
29059 | calculations with the mouse. | |
29060 | ||
29061 | @pindex full-calc-keypad | |
29062 | If you have used @kbd{M-# b} first, @kbd{M-# k} instead invokes | |
29063 | the @code{full-calc-keypad} command, which takes over the whole | |
29064 | Emacs screen and displays the keypad, the Calc stack, and the Calc | |
29065 | trail all at once. This mode would normally be used when running | |
29066 | Calc standalone (@pxref{Standalone Operation}). | |
29067 | ||
29068 | If you aren't using the X window system, you must switch into | |
29069 | the @samp{*Calc Keypad*} window, place the cursor on the desired | |
29070 | ``key,'' and type @key{SPC} or @key{RET}. If you think this | |
29071 | is easier than using Calc normally, go right ahead. | |
29072 | ||
29073 | Calc commands are more or less the same in keypad mode. Certain | |
29074 | keypad keys differ slightly from the corresponding normal Calc | |
29075 | keystrokes; all such deviations are described below. | |
29076 | ||
29077 | Keypad Mode includes many more commands than will fit on the keypad | |
29078 | at once. Click the right mouse button [@code{calc-keypad-menu}] | |
29079 | to switch to the next menu. The bottom five rows of the keypad | |
29080 | stay the same; the top three rows change to a new set of commands. | |
29081 | To return to earlier menus, click the middle mouse button | |
29082 | [@code{calc-keypad-menu-back}] or simply advance through the menus | |
29083 | until you wrap around. Typing @key{TAB} inside the keypad window | |
29084 | is equivalent to clicking the right mouse button there. | |
29085 | ||
29086 | You can always click the @key{EXEC} button and type any normal | |
29087 | Calc key sequence. This is equivalent to switching into the | |
29088 | Calc buffer, typing the keys, then switching back to your | |
29089 | original buffer. | |
29090 | ||
29091 | @menu | |
29092 | * Keypad Main Menu:: | |
29093 | * Keypad Functions Menu:: | |
29094 | * Keypad Binary Menu:: | |
29095 | * Keypad Vectors Menu:: | |
29096 | * Keypad Modes Menu:: | |
29097 | @end menu | |
29098 | ||
29099 | @node Keypad Main Menu, Keypad Functions Menu, Keypad Mode, Keypad Mode | |
29100 | @section Main Menu | |
29101 | ||
d7b8e6c6 | 29102 | @smallexample |
5d67986c | 29103 | @group |
d7b8e6c6 EZ |
29104 | |----+-----Calc 2.00-----+----1 |
29105 | |FLR |CEIL|RND |TRNC|CLN2|FLT | | |
29106 | |----+----+----+----+----+----| | |
29107 | | LN |EXP | |ABS |IDIV|MOD | | |
29108 | |----+----+----+----+----+----| | |
29109 | |SIN |COS |TAN |SQRT|y^x |1/x | | |
29110 | |----+----+----+----+----+----| | |
29111 | | ENTER |+/- |EEX |UNDO| <- | | |
29112 | |-----+---+-+--+--+-+---++----| | |
29113 | | INV | 7 | 8 | 9 | / | | |
29114 | |-----+-----+-----+-----+-----| | |
29115 | | HYP | 4 | 5 | 6 | * | | |
29116 | |-----+-----+-----+-----+-----| | |
29117 | |EXEC | 1 | 2 | 3 | - | | |
29118 | |-----+-----+-----+-----+-----| | |
29119 | | OFF | 0 | . | PI | + | | |
29120 | |-----+-----+-----+-----+-----+ | |
d7b8e6c6 | 29121 | @end group |
5d67986c | 29122 | @end smallexample |
d7b8e6c6 EZ |
29123 | |
29124 | @noindent | |
29125 | This is the menu that appears the first time you start Keypad Mode. | |
29126 | It will show up in a vertical window on the right side of your screen. | |
29127 | Above this menu is the traditional Calc stack display. On a 24-line | |
29128 | screen you will be able to see the top three stack entries. | |
29129 | ||
29130 | The ten digit keys, decimal point, and @key{EEX} key are used for | |
29131 | entering numbers in the obvious way. @key{EEX} begins entry of an | |
29132 | exponent in scientific notation. Just as with regular Calc, the | |
29133 | number is pushed onto the stack as soon as you press @key{ENTER} | |
29134 | or any other function key. | |
29135 | ||
29136 | The @key{+/-} key corresponds to normal Calc's @kbd{n} key. During | |
29137 | numeric entry it changes the sign of the number or of the exponent. | |
29138 | At other times it changes the sign of the number on the top of the | |
29139 | stack. | |
29140 | ||
29141 | The @key{INV} and @key{HYP} keys modify other keys. As well as | |
29142 | having the effects described elsewhere in this manual, Keypad Mode | |
29143 | defines several other ``inverse'' operations. These are described | |
29144 | below and in the following sections. | |
29145 | ||
29146 | The @key{ENTER} key finishes the current numeric entry, or otherwise | |
29147 | duplicates the top entry on the stack. | |
29148 | ||
29149 | The @key{UNDO} key undoes the most recent Calc operation. | |
29150 | @kbd{INV UNDO} is the ``redo'' command, and @kbd{HYP UNDO} is | |
5d67986c | 29151 | ``last arguments'' (@kbd{M-@key{RET}}). |
d7b8e6c6 EZ |
29152 | |
29153 | The @key{<-} key acts as a ``backspace'' during numeric entry. | |
29154 | At other times it removes the top stack entry. @kbd{INV <-} | |
29155 | clears the entire stack. @kbd{HYP <-} takes an integer from | |
29156 | the stack, then removes that many additional stack elements. | |
29157 | ||
29158 | The @key{EXEC} key prompts you to enter any keystroke sequence | |
29159 | that would normally work in Calc mode. This can include a | |
29160 | numeric prefix if you wish. It is also possible simply to | |
29161 | switch into the Calc window and type commands in it; there is | |
29162 | nothing ``magic'' about this window when Keypad Mode is active. | |
29163 | ||
29164 | The other keys in this display perform their obvious calculator | |
29165 | functions. @key{CLN2} rounds the top-of-stack by temporarily | |
29166 | reducing the precision by 2 digits. @key{FLT} converts an | |
29167 | integer or fraction on the top of the stack to floating-point. | |
29168 | ||
29169 | The @key{INV} and @key{HYP} keys combined with several of these keys | |
29170 | give you access to some common functions even if the appropriate menu | |
29171 | is not displayed. Obviously you don't need to learn these keys | |
29172 | unless you find yourself wasting time switching among the menus. | |
29173 | ||
29174 | @table @kbd | |
29175 | @item INV +/- | |
29176 | is the same as @key{1/x}. | |
29177 | @item INV + | |
29178 | is the same as @key{SQRT}. | |
29179 | @item INV - | |
29180 | is the same as @key{CONJ}. | |
29181 | @item INV * | |
29182 | is the same as @key{y^x}. | |
29183 | @item INV / | |
29184 | is the same as @key{INV y^x} (the @cite{x}th root of @cite{y}). | |
29185 | @item HYP/INV 1 | |
29186 | are the same as @key{SIN} / @kbd{INV SIN}. | |
29187 | @item HYP/INV 2 | |
29188 | are the same as @key{COS} / @kbd{INV COS}. | |
29189 | @item HYP/INV 3 | |
29190 | are the same as @key{TAN} / @kbd{INV TAN}. | |
29191 | @item INV/HYP 4 | |
29192 | are the same as @key{LN} / @kbd{HYP LN}. | |
29193 | @item INV/HYP 5 | |
29194 | are the same as @key{EXP} / @kbd{HYP EXP}. | |
29195 | @item INV 6 | |
29196 | is the same as @key{ABS}. | |
29197 | @item INV 7 | |
29198 | is the same as @key{RND} (@code{calc-round}). | |
29199 | @item INV 8 | |
29200 | is the same as @key{CLN2}. | |
29201 | @item INV 9 | |
29202 | is the same as @key{FLT} (@code{calc-float}). | |
29203 | @item INV 0 | |
29204 | is the same as @key{IMAG}. | |
29205 | @item INV . | |
29206 | is the same as @key{PREC}. | |
29207 | @item INV ENTER | |
29208 | is the same as @key{SWAP}. | |
29209 | @item HYP ENTER | |
29210 | is the same as @key{RLL3}. | |
29211 | @item INV HYP ENTER | |
29212 | is the same as @key{OVER}. | |
29213 | @item HYP +/- | |
29214 | packs the top two stack entries as an error form. | |
29215 | @item HYP EEX | |
29216 | packs the top two stack entries as a modulo form. | |
29217 | @item INV EEX | |
29218 | creates an interval form; this removes an integer which is one | |
29219 | of 0 @samp{[]}, 1 @samp{[)}, 2 @samp{(]} or 3 @samp{()}, followed | |
29220 | by the two limits of the interval. | |
29221 | @end table | |
29222 | ||
29223 | The @kbd{OFF} key turns Calc off; typing @kbd{M-# k} or @kbd{M-# M-#} | |
29224 | again has the same effect. This is analogous to typing @kbd{q} or | |
29225 | hitting @kbd{M-# c} again in the normal calculator. If Calc is | |
29226 | running standalone (the @code{full-calc-keypad} command appeared in the | |
29227 | command line that started Emacs), then @kbd{OFF} is replaced with | |
29228 | @kbd{EXIT}; clicking on this actually exits Emacs itself. | |
29229 | ||
29230 | @node Keypad Functions Menu, Keypad Binary Menu, Keypad Main Menu, Keypad Mode | |
29231 | @section Functions Menu | |
29232 | ||
d7b8e6c6 | 29233 | @smallexample |
5d67986c | 29234 | @group |
d7b8e6c6 EZ |
29235 | |----+----+----+----+----+----2 |
29236 | |IGAM|BETA|IBET|ERF |BESJ|BESY| | |
29237 | |----+----+----+----+----+----| | |
29238 | |IMAG|CONJ| RE |ATN2|RAND|RAGN| | |
29239 | |----+----+----+----+----+----| | |
29240 | |GCD |FACT|DFCT|BNOM|PERM|NXTP| | |
29241 | |----+----+----+----+----+----| | |
d7b8e6c6 | 29242 | @end group |
5d67986c | 29243 | @end smallexample |
d7b8e6c6 EZ |
29244 | |
29245 | @noindent | |
29246 | This menu provides various operations from the @kbd{f} and @kbd{k} | |
29247 | prefix keys. | |
29248 | ||
29249 | @key{IMAG} multiplies the number on the stack by the imaginary | |
29250 | number @cite{i = (0, 1)}. | |
29251 | ||
29252 | @key{RE} extracts the real part a complex number. @kbd{INV RE} | |
29253 | extracts the imaginary part. | |
29254 | ||
29255 | @key{RAND} takes a number from the top of the stack and computes | |
29256 | a random number greater than or equal to zero but less than that | |
29257 | number. (@xref{Random Numbers}.) @key{RAGN} is the ``random | |
29258 | again'' command; it computes another random number using the | |
29259 | same limit as last time. | |
29260 | ||
29261 | @key{INV GCD} computes the LCM (least common multiple) function. | |
29262 | ||
29263 | @key{INV FACT} is the gamma function. @c{$\Gamma(x) = (x-1)!$} | |
29264 | @cite{gamma(x) = (x-1)!}. | |
29265 | ||
29266 | @key{PERM} is the number-of-permutations function, which is on the | |
29267 | @kbd{H k c} key in normal Calc. | |
29268 | ||
29269 | @key{NXTP} finds the next prime after a number. @kbd{INV NXTP} | |
29270 | finds the previous prime. | |
29271 | ||
29272 | @node Keypad Binary Menu, Keypad Vectors Menu, Keypad Functions Menu, Keypad Mode | |
29273 | @section Binary Menu | |
29274 | ||
d7b8e6c6 | 29275 | @smallexample |
5d67986c | 29276 | @group |
d7b8e6c6 EZ |
29277 | |----+----+----+----+----+----3 |
29278 | |AND | OR |XOR |NOT |LSH |RSH | | |
29279 | |----+----+----+----+----+----| | |
29280 | |DEC |HEX |OCT |BIN |WSIZ|ARSH| | |
29281 | |----+----+----+----+----+----| | |
29282 | | A | B | C | D | E | F | | |
29283 | |----+----+----+----+----+----| | |
d7b8e6c6 | 29284 | @end group |
5d67986c | 29285 | @end smallexample |
d7b8e6c6 EZ |
29286 | |
29287 | @noindent | |
29288 | The keys in this menu perform operations on binary integers. | |
29289 | Note that both logical and arithmetic right-shifts are provided. | |
29290 | @key{INV LSH} rotates one bit to the left. | |
29291 | ||
29292 | The ``difference'' function (normally on @kbd{b d}) is on @key{INV AND}. | |
29293 | The ``clip'' function (normally on @w{@kbd{b c}}) is on @key{INV NOT}. | |
29294 | ||
29295 | The @key{DEC}, @key{HEX}, @key{OCT}, and @key{BIN} keys select the | |
29296 | current radix for display and entry of numbers: Decimal, hexadecimal, | |
29297 | octal, or binary. The six letter keys @key{A} through @key{F} are used | |
29298 | for entering hexadecimal numbers. | |
29299 | ||
29300 | The @key{WSIZ} key displays the current word size for binary operations | |
29301 | and allows you to enter a new word size. You can respond to the prompt | |
29302 | using either the keyboard or the digits and @key{ENTER} from the keypad. | |
29303 | The initial word size is 32 bits. | |
29304 | ||
29305 | @node Keypad Vectors Menu, Keypad Modes Menu, Keypad Binary Menu, Keypad Mode | |
29306 | @section Vectors Menu | |
29307 | ||
d7b8e6c6 | 29308 | @smallexample |
5d67986c | 29309 | @group |
d7b8e6c6 EZ |
29310 | |----+----+----+----+----+----4 |
29311 | |SUM |PROD|MAX |MAP*|MAP^|MAP$| | |
29312 | |----+----+----+----+----+----| | |
29313 | |MINV|MDET|MTRN|IDNT|CRSS|"x" | | |
29314 | |----+----+----+----+----+----| | |
29315 | |PACK|UNPK|INDX|BLD |LEN |... | | |
29316 | |----+----+----+----+----+----| | |
d7b8e6c6 | 29317 | @end group |
5d67986c | 29318 | @end smallexample |
d7b8e6c6 EZ |
29319 | |
29320 | @noindent | |
29321 | The keys in this menu operate on vectors and matrices. | |
29322 | ||
29323 | @key{PACK} removes an integer @var{n} from the top of the stack; | |
29324 | the next @var{n} stack elements are removed and packed into a vector, | |
29325 | which is replaced onto the stack. Thus the sequence | |
29326 | @kbd{1 ENTER 3 ENTER 5 ENTER 3 PACK} enters the vector | |
29327 | @samp{[1, 3, 5]} onto the stack. To enter a matrix, build each row | |
29328 | on the stack as a vector, then use a final @key{PACK} to collect the | |
29329 | rows into a matrix. | |
29330 | ||
29331 | @key{UNPK} unpacks the vector on the stack, pushing each of its | |
29332 | components separately. | |
29333 | ||
29334 | @key{INDX} removes an integer @var{n}, then builds a vector of | |
29335 | integers from 1 to @var{n}. @kbd{INV INDX} takes three numbers | |
29336 | from the stack: The vector size @var{n}, the starting number, | |
29337 | and the increment. @kbd{BLD} takes an integer @var{n} and any | |
29338 | value @var{x} and builds a vector of @var{n} copies of @var{x}. | |
29339 | ||
29340 | @key{IDNT} removes an integer @var{n}, then builds an @var{n}-by-@var{n} | |
29341 | identity matrix. | |
29342 | ||
29343 | @key{LEN} replaces a vector by its length, an integer. | |
29344 | ||
29345 | @key{...} turns on or off ``abbreviated'' display mode for large vectors. | |
29346 | ||
29347 | @key{MINV}, @key{MDET}, @key{MTRN}, and @key{CROSS} are the matrix | |
29348 | inverse, determinant, and transpose, and vector cross product. | |
29349 | ||
29350 | @key{SUM} replaces a vector by the sum of its elements. It is | |
29351 | equivalent to @kbd{u +} in normal Calc (@pxref{Statistical Operations}). | |
29352 | @key{PROD} computes the product of the elements of a vector, and | |
29353 | @key{MAX} computes the maximum of all the elements of a vector. | |
29354 | ||
29355 | @key{INV SUM} computes the alternating sum of the first element | |
29356 | minus the second, plus the third, minus the fourth, and so on. | |
29357 | @key{INV MAX} computes the minimum of the vector elements. | |
29358 | ||
29359 | @key{HYP SUM} computes the mean of the vector elements. | |
29360 | @key{HYP PROD} computes the sample standard deviation. | |
29361 | @key{HYP MAX} computes the median. | |
29362 | ||
29363 | @key{MAP*} multiplies two vectors elementwise. It is equivalent | |
29364 | to the @kbd{V M *} command. @key{MAP^} computes powers elementwise. | |
29365 | The arguments must be vectors of equal length, or one must be a vector | |
29366 | and the other must be a plain number. For example, @kbd{2 MAP^} squares | |
29367 | all the elements of a vector. | |
29368 | ||
29369 | @key{MAP$} maps the formula on the top of the stack across the | |
29370 | vector in the second-to-top position. If the formula contains | |
29371 | several variables, Calc takes that many vectors starting at the | |
29372 | second-to-top position and matches them to the variables in | |
29373 | alphabetical order. The result is a vector of the same size as | |
29374 | the input vectors, whose elements are the formula evaluated with | |
29375 | the variables set to the various sets of numbers in those vectors. | |
29376 | For example, you could simulate @key{MAP^} using @key{MAP$} with | |
29377 | the formula @samp{x^y}. | |
29378 | ||
29379 | The @kbd{"x"} key pushes the variable name @cite{x} onto the | |
29380 | stack. To build the formula @cite{x^2 + 6}, you would use the | |
29381 | key sequence @kbd{"x" 2 y^x 6 +}. This formula would then be | |
29382 | suitable for use with the @key{MAP$} key described above. | |
29383 | With @key{INV}, @key{HYP}, or @key{INV} and @key{HYP}, the | |
29384 | @kbd{"x"} key pushes the variable names @cite{y}, @cite{z}, and | |
29385 | @cite{t}, respectively. | |
29386 | ||
29387 | @node Keypad Modes Menu, , Keypad Vectors Menu, Keypad Mode | |
29388 | @section Modes Menu | |
29389 | ||
d7b8e6c6 | 29390 | @smallexample |
5d67986c | 29391 | @group |
d7b8e6c6 EZ |
29392 | |----+----+----+----+----+----5 |
29393 | |FLT |FIX |SCI |ENG |GRP | | | |
29394 | |----+----+----+----+----+----| | |
29395 | |RAD |DEG |FRAC|POLR|SYMB|PREC| | |
29396 | |----+----+----+----+----+----| | |
29397 | |SWAP|RLL3|RLL4|OVER|STO |RCL | | |
29398 | |----+----+----+----+----+----| | |
d7b8e6c6 | 29399 | @end group |
5d67986c | 29400 | @end smallexample |
d7b8e6c6 EZ |
29401 | |
29402 | @noindent | |
29403 | The keys in this menu manipulate modes, variables, and the stack. | |
29404 | ||
29405 | The @key{FLT}, @key{FIX}, @key{SCI}, and @key{ENG} keys select | |
29406 | floating-point, fixed-point, scientific, or engineering notation. | |
29407 | @key{FIX} displays two digits after the decimal by default; the | |
29408 | others display full precision. With the @key{INV} prefix, these | |
29409 | keys pop a number-of-digits argument from the stack. | |
29410 | ||
29411 | The @key{GRP} key turns grouping of digits with commas on or off. | |
29412 | @kbd{INV GRP} enables grouping to the right of the decimal point as | |
29413 | well as to the left. | |
29414 | ||
29415 | The @key{RAD} and @key{DEG} keys switch between radians and degrees | |
29416 | for trigonometric functions. | |
29417 | ||
29418 | The @key{FRAC} key turns Fraction mode on or off. This affects | |
29419 | whether commands like @kbd{/} with integer arguments produce | |
29420 | fractional or floating-point results. | |
29421 | ||
29422 | The @key{POLR} key turns Polar mode on or off, determining whether | |
29423 | polar or rectangular complex numbers are used by default. | |
29424 | ||
29425 | The @key{SYMB} key turns Symbolic mode on or off, in which | |
29426 | operations that would produce inexact floating-point results | |
29427 | are left unevaluated as algebraic formulas. | |
29428 | ||
29429 | The @key{PREC} key selects the current precision. Answer with | |
29430 | the keyboard or with the keypad digit and @key{ENTER} keys. | |
29431 | ||
29432 | The @key{SWAP} key exchanges the top two stack elements. | |
29433 | The @key{RLL3} key rotates the top three stack elements upwards. | |
29434 | The @key{RLL4} key rotates the top four stack elements upwards. | |
29435 | The @key{OVER} key duplicates the second-to-top stack element. | |
29436 | ||
29437 | The @key{STO} and @key{RCL} keys are analogous to @kbd{s t} and | |
29438 | @kbd{s r} in regular Calc. @xref{Store and Recall}. Click the | |
29439 | @key{STO} or @key{RCL} key, then one of the ten digits. (Named | |
29440 | variables are not available in Keypad Mode.) You can also use, | |
29441 | for example, @kbd{STO + 3} to add to register 3. | |
29442 | ||
29443 | @node Embedded Mode, Programming, Keypad Mode, Top | |
29444 | @chapter Embedded Mode | |
29445 | ||
29446 | @noindent | |
29447 | Embedded Mode in Calc provides an alternative to copying numbers | |
29448 | and formulas back and forth between editing buffers and the Calc | |
29449 | stack. In Embedded Mode, your editing buffer becomes temporarily | |
29450 | linked to the stack and this copying is taken care of automatically. | |
29451 | ||
29452 | @menu | |
29453 | * Basic Embedded Mode:: | |
29454 | * More About Embedded Mode:: | |
29455 | * Assignments in Embedded Mode:: | |
29456 | * Mode Settings in Embedded Mode:: | |
29457 | * Customizing Embedded Mode:: | |
29458 | @end menu | |
29459 | ||
29460 | @node Basic Embedded Mode, More About Embedded Mode, Embedded Mode, Embedded Mode | |
29461 | @section Basic Embedded Mode | |
29462 | ||
29463 | @noindent | |
29464 | @kindex M-# e | |
29465 | @pindex calc-embedded | |
29466 | To enter Embedded mode, position the Emacs point (cursor) on a | |
29467 | formula in any buffer and press @kbd{M-# e} (@code{calc-embedded}). | |
29468 | Note that @kbd{M-# e} is not to be used in the Calc stack buffer | |
29469 | like most Calc commands, but rather in regular editing buffers that | |
29470 | are visiting your own files. | |
29471 | ||
29472 | Calc normally scans backward and forward in the buffer for the | |
29473 | nearest opening and closing @dfn{formula delimiters}. The simplest | |
29474 | delimiters are blank lines. Other delimiters that Embedded Mode | |
29475 | understands are: | |
29476 | ||
29477 | @enumerate | |
29478 | @item | |
29479 | The @TeX{} and La@TeX{} math delimiters @samp{$ $}, @samp{$$ $$}, | |
29480 | @samp{\[ \]}, and @samp{\( \)}; | |
29481 | @item | |
29482 | Lines beginning with @samp{\begin} and @samp{\end}; | |
29483 | @item | |
29484 | Lines beginning with @samp{@@} (Texinfo delimiters). | |
29485 | @item | |
29486 | Lines beginning with @samp{.EQ} and @samp{.EN} (@dfn{eqn} delimiters); | |
29487 | @item | |
29488 | Lines containing a single @samp{%} or @samp{.\"} symbol and nothing else. | |
29489 | @end enumerate | |
29490 | ||
29491 | @xref{Customizing Embedded Mode}, to see how to make Calc recognize | |
29492 | your own favorite delimiters. Delimiters like @samp{$ $} can appear | |
29493 | on their own separate lines or in-line with the formula. | |
29494 | ||
29495 | If you give a positive or negative numeric prefix argument, Calc | |
29496 | instead uses the current point as one end of the formula, and moves | |
29497 | forward or backward (respectively) by that many lines to find the | |
29498 | other end. Explicit delimiters are not necessary in this case. | |
29499 | ||
29500 | With a prefix argument of zero, Calc uses the current region | |
29501 | (delimited by point and mark) instead of formula delimiters. | |
29502 | ||
29503 | @kindex M-# w | |
29504 | @pindex calc-embedded-word | |
29505 | With a prefix argument of @kbd{C-u} only, Calc scans for the first | |
29506 | non-numeric character (i.e., the first character that is not a | |
29507 | digit, sign, decimal point, or upper- or lower-case @samp{e}) | |
29508 | forward and backward to delimit the formula. @kbd{M-# w} | |
29509 | (@code{calc-embedded-word}) is equivalent to @kbd{C-u M-# e}. | |
29510 | ||
29511 | When you enable Embedded mode for a formula, Calc reads the text | |
29512 | between the delimiters and tries to interpret it as a Calc formula. | |
29513 | It's best if the current Calc language mode is correct for the | |
29514 | formula, but Calc can generally identify @TeX{} formulas and | |
29515 | Big-style formulas even if the language mode is wrong. If Calc | |
29516 | can't make sense of the formula, it beeps and refuses to enter | |
29517 | Embedded mode. But if the current language is wrong, Calc can | |
29518 | sometimes parse the formula successfully (but incorrectly); | |
29519 | for example, the C expression @samp{atan(a[1])} can be parsed | |
29520 | in Normal language mode, but the @code{atan} won't correspond to | |
29521 | the built-in @code{arctan} function, and the @samp{a[1]} will be | |
29522 | interpreted as @samp{a} times the vector @samp{[1]}! | |
29523 | ||
29524 | If you press @kbd{M-# e} or @kbd{M-# w} to activate an embedded | |
29525 | formula which is blank, say with the cursor on the space between | |
29526 | the two delimiters @samp{$ $}, Calc will immediately prompt for | |
29527 | an algebraic entry. | |
29528 | ||
29529 | Only one formula in one buffer can be enabled at a time. If you | |
29530 | move to another area of the current buffer and give Calc commands, | |
29531 | Calc turns Embedded mode off for the old formula and then tries | |
29532 | to restart Embedded mode at the new position. Other buffers are | |
29533 | not affected by Embedded mode. | |
29534 | ||
29535 | When Embedded mode begins, Calc pushes the current formula onto | |
29536 | the stack. No Calc stack window is created; however, Calc copies | |
29537 | the top-of-stack position into the original buffer at all times. | |
29538 | You can create a Calc window by hand with @kbd{M-# o} if you | |
29539 | find you need to see the entire stack. | |
29540 | ||
29541 | For example, typing @kbd{M-# e} while somewhere in the formula | |
29542 | @samp{n>2} in the following line enables Embedded mode on that | |
29543 | inequality: | |
29544 | ||
29545 | @example | |
29546 | We define $F_n = F_(n-1)+F_(n-2)$ for all $n>2$. | |
29547 | @end example | |
29548 | ||
29549 | @noindent | |
29550 | The formula @cite{n>2} will be pushed onto the Calc stack, and | |
29551 | the top of stack will be copied back into the editing buffer. | |
29552 | This means that spaces will appear around the @samp{>} symbol | |
29553 | to match Calc's usual display style: | |
29554 | ||
29555 | @example | |
29556 | We define $F_n = F_(n-1)+F_(n-2)$ for all $n > 2$. | |
29557 | @end example | |
29558 | ||
29559 | @noindent | |
29560 | No spaces have appeared around the @samp{+} sign because it's | |
29561 | in a different formula, one which we have not yet touched with | |
29562 | Embedded mode. | |
29563 | ||
29564 | Now that Embedded mode is enabled, keys you type in this buffer | |
29565 | are interpreted as Calc commands. At this point we might use | |
29566 | the ``commute'' command @kbd{j C} to reverse the inequality. | |
29567 | This is a selection-based command for which we first need to | |
29568 | move the cursor onto the operator (@samp{>} in this case) that | |
29569 | needs to be commuted. | |
29570 | ||
29571 | @example | |
29572 | We define $F_n = F_(n-1)+F_(n-2)$ for all $2 < n$. | |
29573 | @end example | |
29574 | ||
29575 | The @kbd{M-# o} command is a useful way to open a Calc window | |
29576 | without actually selecting that window. Giving this command | |
29577 | verifies that @samp{2 < n} is also on the Calc stack. Typing | |
5d67986c | 29578 | @kbd{17 @key{RET}} would produce: |
d7b8e6c6 EZ |
29579 | |
29580 | @example | |
29581 | We define $F_n = F_(n-1)+F_(n-2)$ for all $17$. | |
29582 | @end example | |
29583 | ||
29584 | @noindent | |
29585 | with @samp{2 < n} and @samp{17} on the stack; typing @key{TAB} | |
29586 | at this point will exchange the two stack values and restore | |
29587 | @samp{2 < n} to the embedded formula. Even though you can't | |
29588 | normally see the stack in Embedded mode, it is still there and | |
29589 | it still operates in the same way. But, as with old-fashioned | |
29590 | RPN calculators, you can only see the value at the top of the | |
29591 | stack at any given time (unless you use @kbd{M-# o}). | |
29592 | ||
29593 | Typing @kbd{M-# e} again turns Embedded mode off. The Calc | |
29594 | window reveals that the formula @w{@samp{2 < n}} is automatically | |
29595 | removed from the stack, but the @samp{17} is not. Entering | |
29596 | Embedded mode always pushes one thing onto the stack, and | |
29597 | leaving Embedded mode always removes one thing. Anything else | |
29598 | that happens on the stack is entirely your business as far as | |
29599 | Embedded mode is concerned. | |
29600 | ||
29601 | If you press @kbd{M-# e} in the wrong place by accident, it is | |
29602 | possible that Calc will be able to parse the nearby text as a | |
29603 | formula and will mangle that text in an attempt to redisplay it | |
29604 | ``properly'' in the current language mode. If this happens, | |
29605 | press @kbd{M-# e} again to exit Embedded mode, then give the | |
29606 | regular Emacs ``undo'' command (@kbd{C-_} or @kbd{C-x u}) to put | |
29607 | the text back the way it was before Calc edited it. Note that Calc's | |
29608 | own Undo command (typed before you turn Embedded mode back off) | |
29609 | will not do you any good, because as far as Calc is concerned | |
29610 | you haven't done anything with this formula yet. | |
29611 | ||
29612 | @node More About Embedded Mode, Assignments in Embedded Mode, Basic Embedded Mode, Embedded Mode | |
29613 | @section More About Embedded Mode | |
29614 | ||
29615 | @noindent | |
29616 | When Embedded mode ``activates'' a formula, i.e., when it examines | |
29617 | the formula for the first time since the buffer was created or | |
29618 | loaded, Calc tries to sense the language in which the formula was | |
29619 | written. If the formula contains any @TeX{}-like @samp{\} sequences, | |
29620 | it is parsed (i.e., read) in @TeX{} mode. If the formula appears to | |
29621 | be written in multi-line Big mode, it is parsed in Big mode. Otherwise, | |
29622 | it is parsed according to the current language mode. | |
29623 | ||
29624 | Note that Calc does not change the current language mode according | |
29625 | to what it finds. Even though it can read a @TeX{} formula when | |
29626 | not in @TeX{} mode, it will immediately rewrite this formula using | |
29627 | whatever language mode is in effect. You must then type @kbd{d T} | |
29628 | to switch Calc permanently into @TeX{} mode if that is what you | |
29629 | desire. | |
29630 | ||
29631 | @tex | |
29632 | \bigskip | |
29633 | @end tex | |
29634 | ||
29635 | @kindex d p | |
29636 | @pindex calc-show-plain | |
29637 | Calc's parser is unable to read certain kinds of formulas. For | |
29638 | example, with @kbd{v ]} (@code{calc-matrix-brackets}) you can | |
29639 | specify matrix display styles which the parser is unable to | |
29640 | recognize as matrices. The @kbd{d p} (@code{calc-show-plain}) | |
29641 | command turns on a mode in which a ``plain'' version of a | |
29642 | formula is placed in front of the fully-formatted version. | |
29643 | When Calc reads a formula that has such a plain version in | |
29644 | front, it reads the plain version and ignores the formatted | |
29645 | version. | |
29646 | ||
29647 | Plain formulas are preceded and followed by @samp{%%%} signs | |
29648 | by default. This notation has the advantage that the @samp{%} | |
29649 | character begins a comment in @TeX{}, so if your formula is | |
29650 | embedded in a @TeX{} document its plain version will be | |
29651 | invisible in the final printed copy. @xref{Customizing | |
29652 | Embedded Mode}, to see how to change the ``plain'' formula | |
29653 | delimiters, say to something that @dfn{eqn} or some other | |
29654 | formatter will treat as a comment. | |
29655 | ||
29656 | There are several notations which Calc's parser for ``big'' | |
29657 | formatted formulas can't yet recognize. In particular, it can't | |
29658 | read the large symbols for @code{sum}, @code{prod}, and @code{integ}, | |
29659 | and it can't handle @samp{=>} with the righthand argument omitted. | |
29660 | Also, Calc won't recognize special formats you have defined with | |
29661 | the @kbd{Z C} command (@pxref{User-Defined Compositions}). In | |
29662 | these cases it is important to use ``plain'' mode to make sure | |
29663 | Calc will be able to read your formula later. | |
29664 | ||
29665 | Another example where ``plain'' mode is important is if you have | |
29666 | specified a float mode with few digits of precision. Normally | |
29667 | any digits that are computed but not displayed will simply be | |
29668 | lost when you save and re-load your embedded buffer, but ``plain'' | |
29669 | mode allows you to make sure that the complete number is present | |
29670 | in the file as well as the rounded-down number. | |
29671 | ||
29672 | @tex | |
29673 | \bigskip | |
29674 | @end tex | |
29675 | ||
29676 | Embedded buffers remember active formulas for as long as they | |
29677 | exist in Emacs memory. Suppose you have an embedded formula | |
29678 | which is @c{$\pi$} | |
29679 | @cite{pi} to the normal 12 decimal places, and then | |
29680 | type @w{@kbd{C-u 5 d n}} to display only five decimal places. | |
29681 | If you then type @kbd{d n}, all 12 places reappear because the | |
29682 | full number is still there on the Calc stack. More surprisingly, | |
29683 | even if you exit Embedded mode and later re-enter it for that | |
29684 | formula, typing @kbd{d n} will restore all 12 places because | |
29685 | each buffer remembers all its active formulas. However, if you | |
29686 | save the buffer in a file and reload it in a new Emacs session, | |
29687 | all non-displayed digits will have been lost unless you used | |
29688 | ``plain'' mode. | |
29689 | ||
29690 | @tex | |
29691 | \bigskip | |
29692 | @end tex | |
29693 | ||
29694 | In some applications of Embedded mode, you will want to have a | |
29695 | sequence of copies of a formula that show its evolution as you | |
29696 | work on it. For example, you might want to have a sequence | |
29697 | like this in your file (elaborating here on the example from | |
29698 | the ``Getting Started'' chapter): | |
29699 | ||
29700 | @smallexample | |
29701 | The derivative of | |
29702 | ||
29703 | ln(ln(x)) | |
29704 | ||
29705 | is | |
29706 | ||
29707 | @r{(the derivative of }ln(ln(x))@r{)} | |
29708 | ||
29709 | whose value at x = 2 is | |
29710 | ||
29711 | @r{(the value)} | |
29712 | ||
29713 | and at x = 3 is | |
29714 | ||
29715 | @r{(the value)} | |
29716 | @end smallexample | |
29717 | ||
29718 | @kindex M-# d | |
29719 | @pindex calc-embedded-duplicate | |
29720 | The @kbd{M-# d} (@code{calc-embedded-duplicate}) command is a | |
29721 | handy way to make sequences like this. If you type @kbd{M-# d}, | |
29722 | the formula under the cursor (which may or may not have Embedded | |
29723 | mode enabled for it at the time) is copied immediately below and | |
29724 | Embedded mode is then enabled for that copy. | |
29725 | ||
29726 | For this example, you would start with just | |
29727 | ||
29728 | @smallexample | |
29729 | The derivative of | |
29730 | ||
29731 | ln(ln(x)) | |
29732 | @end smallexample | |
29733 | ||
29734 | @noindent | |
29735 | and press @kbd{M-# d} with the cursor on this formula. The result | |
29736 | is | |
29737 | ||
29738 | @smallexample | |
29739 | The derivative of | |
29740 | ||
29741 | ln(ln(x)) | |
29742 | ||
29743 | ||
29744 | ln(ln(x)) | |
29745 | @end smallexample | |
29746 | ||
29747 | @noindent | |
29748 | with the second copy of the formula enabled in Embedded mode. | |
5d67986c | 29749 | You can now press @kbd{a d x @key{RET}} to take the derivative, and |
d7b8e6c6 | 29750 | @kbd{M-# d M-# d} to make two more copies of the derivative. |
5d67986c | 29751 | To complete the computations, type @kbd{3 s l x @key{RET}} to evaluate |
d7b8e6c6 | 29752 | the last formula, then move up to the second-to-last formula |
5d67986c | 29753 | and type @kbd{2 s l x @key{RET}}. |
d7b8e6c6 EZ |
29754 | |
29755 | Finally, you would want to press @kbd{M-# e} to exit Embedded | |
29756 | mode, then go up and insert the necessary text in between the | |
29757 | various formulas and numbers. | |
29758 | ||
29759 | @tex | |
29760 | \bigskip | |
29761 | @end tex | |
29762 | ||
29763 | @kindex M-# f | |
29764 | @kindex M-# ' | |
29765 | @pindex calc-embedded-new-formula | |
29766 | The @kbd{M-# f} (@code{calc-embedded-new-formula}) command | |
29767 | creates a new embedded formula at the current point. It inserts | |
29768 | some default delimiters, which are usually just blank lines, | |
29769 | and then does an algebraic entry to get the formula (which is | |
29770 | then enabled for Embedded mode). This is just shorthand for | |
29771 | typing the delimiters yourself, positioning the cursor between | |
29772 | the new delimiters, and pressing @kbd{M-# e}. The key sequence | |
29773 | @kbd{M-# '} is equivalent to @kbd{M-# f}. | |
29774 | ||
29775 | @kindex M-# n | |
29776 | @kindex M-# p | |
29777 | @pindex calc-embedded-next | |
29778 | @pindex calc-embedded-previous | |
29779 | The @kbd{M-# n} (@code{calc-embedded-next}) and @kbd{M-# p} | |
29780 | (@code{calc-embedded-previous}) commands move the cursor to the | |
29781 | next or previous active embedded formula in the buffer. They | |
29782 | can take positive or negative prefix arguments to move by several | |
29783 | formulas. Note that these commands do not actually examine the | |
29784 | text of the buffer looking for formulas; they only see formulas | |
29785 | which have previously been activated in Embedded mode. In fact, | |
29786 | @kbd{M-# n} and @kbd{M-# p} are a useful way to tell which | |
29787 | embedded formulas are currently active. Also, note that these | |
29788 | commands do not enable Embedded mode on the next or previous | |
29789 | formula, they just move the cursor. (By the way, @kbd{M-# n} is | |
29790 | not as awkward to type as it may seem, because @kbd{M-#} ignores | |
29791 | Shift and Meta on the second keystroke: @kbd{M-# M-N} can be typed | |
29792 | by holding down Shift and Meta and alternately typing two keys.) | |
29793 | ||
29794 | @kindex M-# ` | |
29795 | @pindex calc-embedded-edit | |
29796 | The @kbd{M-# `} (@code{calc-embedded-edit}) command edits the | |
29797 | embedded formula at the current point as if by @kbd{`} (@code{calc-edit}). | |
29798 | Embedded mode does not have to be enabled for this to work. Press | |
29799 | @kbd{M-# M-#} to finish the edit, or @kbd{M-# x} to cancel. | |
29800 | ||
29801 | @node Assignments in Embedded Mode, Mode Settings in Embedded Mode, More About Embedded Mode, Embedded Mode | |
29802 | @section Assignments in Embedded Mode | |
29803 | ||
29804 | @noindent | |
29805 | The @samp{:=} (assignment) and @samp{=>} (``evaluates-to'') operators | |
29806 | are especially useful in Embedded mode. They allow you to make | |
29807 | a definition in one formula, then refer to that definition in | |
29808 | other formulas embedded in the same buffer. | |
29809 | ||
29810 | An embedded formula which is an assignment to a variable, as in | |
29811 | ||
29812 | @example | |
29813 | foo := 5 | |
29814 | @end example | |
29815 | ||
29816 | @noindent | |
29817 | records @cite{5} as the stored value of @code{foo} for the | |
29818 | purposes of Embedded mode operations in the current buffer. It | |
29819 | does @emph{not} actually store @cite{5} as the ``global'' value | |
29820 | of @code{foo}, however. Regular Calc operations, and Embedded | |
29821 | formulas in other buffers, will not see this assignment. | |
29822 | ||
29823 | One way to use this assigned value is simply to create an | |
29824 | Embedded formula elsewhere that refers to @code{foo}, and to press | |
29825 | @kbd{=} in that formula. However, this permanently replaces the | |
29826 | @code{foo} in the formula with its current value. More interesting | |
29827 | is to use @samp{=>} elsewhere: | |
29828 | ||
29829 | @example | |
29830 | foo + 7 => 12 | |
29831 | @end example | |
29832 | ||
29833 | @xref{Evaluates-To Operator}, for a general discussion of @samp{=>}. | |
29834 | ||
29835 | If you move back and change the assignment to @code{foo}, any | |
29836 | @samp{=>} formulas which refer to it are automatically updated. | |
29837 | ||
29838 | @example | |
29839 | foo := 17 | |
29840 | ||
29841 | foo + 7 => 24 | |
29842 | @end example | |
29843 | ||
29844 | The obvious question then is, @emph{how} can one easily change the | |
29845 | assignment to @code{foo}? If you simply select the formula in | |
29846 | Embedded mode and type 17, the assignment itself will be replaced | |
29847 | by the 17. The effect on the other formula will be that the | |
29848 | variable @code{foo} becomes unassigned: | |
29849 | ||
29850 | @example | |
29851 | 17 | |
29852 | ||
29853 | foo + 7 => foo + 7 | |
29854 | @end example | |
29855 | ||
29856 | The right thing to do is first to use a selection command (@kbd{j 2} | |
29857 | will do the trick) to select the righthand side of the assignment. | |
5d67986c | 29858 | Then, @kbd{17 @key{TAB} @key{DEL}} will swap the 17 into place (@pxref{Selecting |
d7b8e6c6 EZ |
29859 | Subformulas}, to see how this works). |
29860 | ||
29861 | @kindex M-# j | |
29862 | @pindex calc-embedded-select | |
29863 | The @kbd{M-# j} (@code{calc-embedded-select}) command provides an | |
28665d46 | 29864 | easy way to operate on assignments. It is just like @kbd{M-# e}, |
d7b8e6c6 EZ |
29865 | except that if the enabled formula is an assignment, it uses |
29866 | @kbd{j 2} to select the righthand side. If the enabled formula | |
29867 | is an evaluates-to, it uses @kbd{j 1} to select the lefthand side. | |
29868 | A formula can also be a combination of both: | |
29869 | ||
29870 | @example | |
29871 | bar := foo + 3 => 20 | |
29872 | @end example | |
29873 | ||
29874 | @noindent | |
29875 | in which case @kbd{M-# j} will select the middle part (@samp{foo + 3}). | |
29876 | ||
29877 | The formula is automatically deselected when you leave Embedded | |
29878 | mode. | |
29879 | ||
29880 | @kindex M-# u | |
29881 | @kindex M-# = | |
29882 | @pindex calc-embedded-update | |
29883 | Another way to change the assignment to @code{foo} would simply be | |
29884 | to edit the number using regular Emacs editing rather than Embedded | |
29885 | mode. Then, we have to find a way to get Embedded mode to notice | |
29886 | the change. The @kbd{M-# u} or @kbd{M-# =} | |
29887 | (@code{calc-embedded-update-formula}) command is a convenient way | |
29888 | to do this.@refill | |
29889 | ||
29890 | @example | |
29891 | foo := 6 | |
29892 | ||
29893 | foo + 7 => 13 | |
29894 | @end example | |
29895 | ||
29896 | Pressing @kbd{M-# u} is much like pressing @kbd{M-# e = M-# e}, that | |
29897 | is, temporarily enabling Embedded mode for the formula under the | |
29898 | cursor and then evaluating it with @kbd{=}. But @kbd{M-# u} does | |
29899 | not actually use @kbd{M-# e}, and in fact another formula somewhere | |
29900 | else can be enabled in Embedded mode while you use @kbd{M-# u} and | |
29901 | that formula will not be disturbed. | |
29902 | ||
29903 | With a numeric prefix argument, @kbd{M-# u} updates all active | |
29904 | @samp{=>} formulas in the buffer. Formulas which have not yet | |
29905 | been activated in Embedded mode, and formulas which do not have | |
29906 | @samp{=>} as their top-level operator, are not affected by this. | |
29907 | (This is useful only if you have used @kbd{m C}; see below.) | |
29908 | ||
29909 | With a plain @kbd{C-u} prefix, @kbd{C-u M-# u} updates only in the | |
29910 | region between mark and point rather than in the whole buffer. | |
29911 | ||
29912 | @kbd{M-# u} is also a handy way to activate a formula, such as an | |
29913 | @samp{=>} formula that has freshly been typed in or loaded from a | |
29914 | file. | |
29915 | ||
29916 | @kindex M-# a | |
29917 | @pindex calc-embedded-activate | |
29918 | The @kbd{M-# a} (@code{calc-embedded-activate}) command scans | |
29919 | through the current buffer and activates all embedded formulas | |
29920 | that contain @samp{:=} or @samp{=>} symbols. This does not mean | |
29921 | that Embedded mode is actually turned on, but only that the | |
29922 | formulas' positions are registered with Embedded mode so that | |
29923 | the @samp{=>} values can be properly updated as assignments are | |
29924 | changed. | |
29925 | ||
29926 | It is a good idea to type @kbd{M-# a} right after loading a file | |
29927 | that uses embedded @samp{=>} operators. Emacs includes a nifty | |
29928 | ``buffer-local variables'' feature that you can use to do this | |
29929 | automatically. The idea is to place near the end of your file | |
29930 | a few lines that look like this: | |
29931 | ||
29932 | @example | |
29933 | --- Local Variables: --- | |
29934 | --- eval:(calc-embedded-activate) --- | |
29935 | --- End: --- | |
29936 | @end example | |
29937 | ||
29938 | @noindent | |
29939 | where the leading and trailing @samp{---} can be replaced by | |
29940 | any suitable strings (which must be the same on all three lines) | |
29941 | or omitted altogether; in a @TeX{} file, @samp{%} would be a good | |
29942 | leading string and no trailing string would be necessary. In a | |
29943 | C program, @samp{/*} and @samp{*/} would be good leading and | |
29944 | trailing strings. | |
29945 | ||
29946 | When Emacs loads a file into memory, it checks for a Local Variables | |
29947 | section like this one at the end of the file. If it finds this | |
29948 | section, it does the specified things (in this case, running | |
29949 | @kbd{M-# a} automatically) before editing of the file begins. | |
29950 | The Local Variables section must be within 3000 characters of the | |
29951 | end of the file for Emacs to find it, and it must be in the last | |
29952 | page of the file if the file has any page separators. | |
29953 | @xref{File Variables, , Local Variables in Files, emacs, the | |
29954 | Emacs manual}. | |
29955 | ||
29956 | Note that @kbd{M-# a} does not update the formulas it finds. | |
29957 | To do this, type, say, @kbd{M-1 M-# u} after @w{@kbd{M-# a}}. | |
29958 | Generally this should not be a problem, though, because the | |
29959 | formulas will have been up-to-date already when the file was | |
29960 | saved. | |
29961 | ||
29962 | Normally, @kbd{M-# a} activates all the formulas it finds, but | |
29963 | any previous active formulas remain active as well. With a | |
29964 | positive numeric prefix argument, @kbd{M-# a} first deactivates | |
29965 | all current active formulas, then actives the ones it finds in | |
29966 | its scan of the buffer. With a negative prefix argument, | |
29967 | @kbd{M-# a} simply deactivates all formulas. | |
29968 | ||
29969 | Embedded mode has two symbols, @samp{Active} and @samp{~Active}, | |
29970 | which it puts next to the major mode name in a buffer's mode line. | |
29971 | It puts @samp{Active} if it has reason to believe that all | |
29972 | formulas in the buffer are active, because you have typed @kbd{M-# a} | |
29973 | and Calc has not since had to deactivate any formulas (which can | |
29974 | happen if Calc goes to update an @samp{=>} formula somewhere because | |
29975 | a variable changed, and finds that the formula is no longer there | |
29976 | due to some kind of editing outside of Embedded mode). Calc puts | |
29977 | @samp{~Active} in the mode line if some, but probably not all, | |
29978 | formulas in the buffer are active. This happens if you activate | |
29979 | a few formulas one at a time but never use @kbd{M-# a}, or if you | |
29980 | used @kbd{M-# a} but then Calc had to deactivate a formula | |
29981 | because it lost track of it. If neither of these symbols appears | |
29982 | in the mode line, no embedded formulas are active in the buffer | |
29983 | (e.g., before Embedded mode has been used, or after a @kbd{M-- M-# a}). | |
29984 | ||
29985 | Embedded formulas can refer to assignments both before and after them | |
29986 | in the buffer. If there are several assignments to a variable, the | |
29987 | nearest preceding assignment is used if there is one, otherwise the | |
29988 | following assignment is used. | |
29989 | ||
29990 | @example | |
29991 | x => 1 | |
29992 | ||
29993 | x := 1 | |
29994 | ||
29995 | x => 1 | |
29996 | ||
29997 | x := 2 | |
29998 | ||
29999 | x => 2 | |
30000 | @end example | |
30001 | ||
30002 | As well as simple variables, you can also assign to subscript | |
30003 | expressions of the form @samp{@var{var}_@var{number}} (as in | |
30004 | @code{x_0}), or @samp{@var{var}_@var{var}} (as in @code{x_max}). | |
30005 | Assignments to other kinds of objects can be represented by Calc, | |
30006 | but the automatic linkage between assignments and references works | |
30007 | only for plain variables and these two kinds of subscript expressions. | |
30008 | ||
30009 | If there are no assignments to a given variable, the global | |
30010 | stored value for the variable is used (@pxref{Storing Variables}), | |
30011 | or, if no value is stored, the variable is left in symbolic form. | |
30012 | Note that global stored values will be lost when the file is saved | |
30013 | and loaded in a later Emacs session, unless you have used the | |
30014 | @kbd{s p} (@code{calc-permanent-variable}) command to save them; | |
30015 | @pxref{Operations on Variables}. | |
30016 | ||
30017 | The @kbd{m C} (@code{calc-auto-recompute}) command turns automatic | |
30018 | recomputation of @samp{=>} forms on and off. If you turn automatic | |
30019 | recomputation off, you will have to use @kbd{M-# u} to update these | |
30020 | formulas manually after an assignment has been changed. If you | |
30021 | plan to change several assignments at once, it may be more efficient | |
30022 | to type @kbd{m C}, change all the assignments, then use @kbd{M-1 M-# u} | |
30023 | to update the entire buffer afterwards. The @kbd{m C} command also | |
30024 | controls @samp{=>} formulas on the stack; @pxref{Evaluates-To | |
30025 | Operator}. When you turn automatic recomputation back on, the | |
30026 | stack will be updated but the Embedded buffer will not; you must | |
30027 | use @kbd{M-# u} to update the buffer by hand. | |
30028 | ||
30029 | @node Mode Settings in Embedded Mode, Customizing Embedded Mode, Assignments in Embedded Mode, Embedded Mode | |
30030 | @section Mode Settings in Embedded Mode | |
30031 | ||
30032 | @noindent | |
30033 | Embedded Mode has a rather complicated mechanism for handling mode | |
30034 | settings in Embedded formulas. It is possible to put annotations | |
30035 | in the file that specify mode settings either global to the entire | |
30036 | file or local to a particular formula or formulas. In the latter | |
30037 | case, different modes can be specified for use when a formula | |
30038 | is the enabled Embedded Mode formula. | |
30039 | ||
30040 | When you give any mode-setting command, like @kbd{m f} (for fraction | |
30041 | mode) or @kbd{d s} (for scientific notation), Embedded Mode adds | |
30042 | a line like the following one to the file just before the opening | |
30043 | delimiter of the formula. | |
30044 | ||
30045 | @example | |
30046 | % [calc-mode: fractions: t] | |
30047 | % [calc-mode: float-format: (sci 0)] | |
30048 | @end example | |
30049 | ||
30050 | When Calc interprets an embedded formula, it scans the text before | |
30051 | the formula for mode-setting annotations like these and sets the | |
30052 | Calc buffer to match these modes. Modes not explicitly described | |
30053 | in the file are not changed. Calc scans all the way to the top of | |
30054 | the file, or up to a line of the form | |
30055 | ||
30056 | @example | |
30057 | % [calc-defaults] | |
30058 | @end example | |
30059 | ||
30060 | @noindent | |
30061 | which you can insert at strategic places in the file if this backward | |
30062 | scan is getting too slow, or just to provide a barrier between one | |
30063 | ``zone'' of mode settings and another. | |
30064 | ||
30065 | If the file contains several annotations for the same mode, the | |
30066 | closest one before the formula is used. Annotations after the | |
30067 | formula are never used (except for global annotations, described | |
30068 | below). | |
30069 | ||
30070 | The scan does not look for the leading @samp{% }, only for the | |
30071 | square brackets and the text they enclose. You can edit the mode | |
30072 | annotations to a style that works better in context if you wish. | |
30073 | @xref{Customizing Embedded Mode}, to see how to change the style | |
30074 | that Calc uses when it generates the annotations. You can write | |
30075 | mode annotations into the file yourself if you know the syntax; | |
30076 | the easiest way to find the syntax for a given mode is to let | |
30077 | Calc write the annotation for it once and see what it does. | |
30078 | ||
30079 | If you give a mode-changing command for a mode that already has | |
30080 | a suitable annotation just above the current formula, Calc will | |
30081 | modify that annotation rather than generating a new, conflicting | |
30082 | one. | |
30083 | ||
30084 | Mode annotations have three parts, separated by colons. (Spaces | |
30085 | after the colons are optional.) The first identifies the kind | |
30086 | of mode setting, the second is a name for the mode itself, and | |
30087 | the third is the value in the form of a Lisp symbol, number, | |
30088 | or list. Annotations with unrecognizable text in the first or | |
30089 | second parts are ignored. The third part is not checked to make | |
30090 | sure the value is of a legal type or range; if you write an | |
30091 | annotation by hand, be sure to give a proper value or results | |
30092 | will be unpredictable. Mode-setting annotations are case-sensitive. | |
30093 | ||
30094 | While Embedded Mode is enabled, the word @code{Local} appears in | |
30095 | the mode line. This is to show that mode setting commands generate | |
30096 | annotations that are ``local'' to the current formula or set of | |
30097 | formulas. The @kbd{m R} (@code{calc-mode-record-mode}) command | |
30098 | causes Calc to generate different kinds of annotations. Pressing | |
30099 | @kbd{m R} repeatedly cycles through the possible modes. | |
30100 | ||
30101 | @code{LocEdit} and @code{LocPerm} modes generate annotations | |
30102 | that look like this, respectively: | |
30103 | ||
30104 | @example | |
30105 | % [calc-edit-mode: float-format: (sci 0)] | |
30106 | % [calc-perm-mode: float-format: (sci 5)] | |
30107 | @end example | |
30108 | ||
30109 | The first kind of annotation will be used only while a formula | |
30110 | is enabled in Embedded Mode. The second kind will be used only | |
30111 | when the formula is @emph{not} enabled. (Whether the formula | |
30112 | is ``active'' or not, i.e., whether Calc has seen this formula | |
30113 | yet, is not relevant here.) | |
30114 | ||
30115 | @code{Global} mode generates an annotation like this at the end | |
30116 | of the file: | |
30117 | ||
30118 | @example | |
30119 | % [calc-global-mode: fractions t] | |
30120 | @end example | |
30121 | ||
30122 | Global mode annotations affect all formulas throughout the file, | |
30123 | and may appear anywhere in the file. This allows you to tuck your | |
30124 | mode annotations somewhere out of the way, say, on a new page of | |
30125 | the file, as long as those mode settings are suitable for all | |
30126 | formulas in the file. | |
30127 | ||
30128 | Enabling a formula with @kbd{M-# e} causes a fresh scan for local | |
30129 | mode annotations; you will have to use this after adding annotations | |
30130 | above a formula by hand to get the formula to notice them. Updating | |
30131 | a formula with @kbd{M-# u} will also re-scan the local modes, but | |
30132 | global modes are only re-scanned by @kbd{M-# a}. | |
30133 | ||
30134 | Another way that modes can get out of date is if you add a local | |
30135 | mode annotation to a formula that has another formula after it. | |
30136 | In this example, we have used the @kbd{d s} command while the | |
30137 | first of the two embedded formulas is active. But the second | |
30138 | formula has not changed its style to match, even though by the | |
30139 | rules of reading annotations the @samp{(sci 0)} applies to it, too. | |
30140 | ||
30141 | @example | |
30142 | % [calc-mode: float-format: (sci 0)] | |
30143 | 1.23e2 | |
30144 | ||
30145 | 456. | |
30146 | @end example | |
30147 | ||
30148 | We would have to go down to the other formula and press @kbd{M-# u} | |
30149 | on it in order to get it to notice the new annotation. | |
30150 | ||
30151 | Two more mode-recording modes selectable by @kbd{m R} are @code{Save} | |
30152 | (which works even outside of Embedded Mode), in which mode settings | |
30153 | are recorded permanently in your Emacs startup file @file{~/.emacs} | |
30154 | rather than by annotating the current document, and no-recording | |
30155 | mode (where there is no symbol like @code{Save} or @code{Local} in | |
30156 | the mode line), in which mode-changing commands do not leave any | |
30157 | annotations at all. | |
30158 | ||
30159 | When Embedded Mode is not enabled, mode-recording modes except | |
30160 | for @code{Save} have no effect. | |
30161 | ||
30162 | @node Customizing Embedded Mode, , Mode Settings in Embedded Mode, Embedded Mode | |
30163 | @section Customizing Embedded Mode | |
30164 | ||
30165 | @noindent | |
30166 | You can modify Embedded Mode's behavior by setting various Lisp | |
30167 | variables described here. Use @kbd{M-x set-variable} or | |
30168 | @kbd{M-x edit-options} to adjust a variable on the fly, or | |
30169 | put a suitable @code{setq} statement in your @file{~/.emacs} | |
30170 | file to set a variable permanently. (Another possibility would | |
30171 | be to use a file-local variable annotation at the end of the | |
30172 | file; @pxref{File Variables, , Local Variables in Files, emacs, the | |
30173 | Emacs manual}.) | |
30174 | ||
30175 | While none of these variables will be buffer-local by default, you | |
30176 | can make any of them local to any embedded-mode buffer. (Their | |
30177 | values in the @samp{*Calculator*} buffer are never used.) | |
30178 | ||
30179 | @vindex calc-embedded-open-formula | |
30180 | The @code{calc-embedded-open-formula} variable holds a regular | |
30181 | expression for the opening delimiter of a formula. @xref{Regexp Search, | |
30182 | , Regular Expression Search, emacs, the Emacs manual}, to see | |
30183 | how regular expressions work. Basically, a regular expression is a | |
30184 | pattern that Calc can search for. A regular expression that considers | |
30185 | blank lines, @samp{$}, and @samp{$$} to be opening delimiters is | |
30186 | @code{"\\`\\|^\n\\|\\$\\$?"}. Just in case the meaning of this | |
30187 | regular expression is not completely plain, let's go through it | |
30188 | in detail. | |
30189 | ||
30190 | The surrounding @samp{" "} marks quote the text between them as a | |
30191 | Lisp string. If you left them off, @code{set-variable} or | |
30192 | @code{edit-options} would try to read the regular expression as a | |
30193 | Lisp program. | |
30194 | ||
30195 | The most obvious property of this regular expression is that it | |
30196 | contains indecently many backslashes. There are actually two levels | |
30197 | of backslash usage going on here. First, when Lisp reads a quoted | |
30198 | string, all pairs of characters beginning with a backslash are | |
30199 | interpreted as special characters. Here, @code{\n} changes to a | |
30200 | new-line character, and @code{\\} changes to a single backslash. | |
30201 | So the actual regular expression seen by Calc is | |
30202 | @samp{\`\|^ @r{(newline)} \|\$\$?}. | |
30203 | ||
30204 | Regular expressions also consider pairs beginning with backslash | |
30205 | to have special meanings. Sometimes the backslash is used to quote | |
30206 | a character that otherwise would have a special meaning in a regular | |
30207 | expression, like @samp{$}, which normally means ``end-of-line,'' | |
30208 | or @samp{?}, which means that the preceding item is optional. So | |
30209 | @samp{\$\$?} matches either one or two dollar signs. | |
30210 | ||
30211 | The other codes in this regular expression are @samp{^}, which matches | |
30212 | ``beginning-of-line,'' @samp{\|}, which means ``or,'' and @samp{\`}, | |
30213 | which matches ``beginning-of-buffer.'' So the whole pattern means | |
30214 | that a formula begins at the beginning of the buffer, or on a newline | |
30215 | that occurs at the beginning of a line (i.e., a blank line), or at | |
30216 | one or two dollar signs. | |
30217 | ||
30218 | The default value of @code{calc-embedded-open-formula} looks just | |
30219 | like this example, with several more alternatives added on to | |
30220 | recognize various other common kinds of delimiters. | |
30221 | ||
30222 | By the way, the reason to use @samp{^\n} rather than @samp{^$} | |
30223 | or @samp{\n\n}, which also would appear to match blank lines, | |
30224 | is that the former expression actually ``consumes'' only one | |
30225 | newline character as @emph{part of} the delimiter, whereas the | |
30226 | latter expressions consume zero or two newlines, respectively. | |
30227 | The former choice gives the most natural behavior when Calc | |
30228 | must operate on a whole formula including its delimiters. | |
30229 | ||
30230 | See the Emacs manual for complete details on regular expressions. | |
30231 | But just for your convenience, here is a list of all characters | |
30232 | which must be quoted with backslash (like @samp{\$}) to avoid | |
30233 | some special interpretation: @samp{. * + ? [ ] ^ $ \}. (Note | |
30234 | the backslash in this list; for example, to match @samp{\[} you | |
30235 | must use @code{"\\\\\\["}. An exercise for the reader is to | |
30236 | account for each of these six backslashes!) | |
30237 | ||
30238 | @vindex calc-embedded-close-formula | |
30239 | The @code{calc-embedded-close-formula} variable holds a regular | |
30240 | expression for the closing delimiter of a formula. A closing | |
30241 | regular expression to match the above example would be | |
30242 | @code{"\\'\\|\n$\\|\\$\\$?"}. This is almost the same as the | |
30243 | other one, except it now uses @samp{\'} (``end-of-buffer'') and | |
30244 | @samp{\n$} (newline occurring at end of line, yet another way | |
30245 | of describing a blank line that is more appropriate for this | |
30246 | case). | |
30247 | ||
30248 | @vindex calc-embedded-open-word | |
30249 | @vindex calc-embedded-close-word | |
30250 | The @code{calc-embedded-open-word} and @code{calc-embedded-close-word} | |
30251 | variables are similar expressions used when you type @kbd{M-# w} | |
30252 | instead of @kbd{M-# e} to enable Embedded mode. | |
30253 | ||
30254 | @vindex calc-embedded-open-plain | |
30255 | The @code{calc-embedded-open-plain} variable is a string which | |
30256 | begins a ``plain'' formula written in front of the formatted | |
30257 | formula when @kbd{d p} mode is turned on. Note that this is an | |
30258 | actual string, not a regular expression, because Calc must be able | |
30259 | to write this string into a buffer as well as to recognize it. | |
30260 | The default string is @code{"%%% "} (note the trailing space). | |
30261 | ||
30262 | @vindex calc-embedded-close-plain | |
30263 | The @code{calc-embedded-close-plain} variable is a string which | |
30264 | ends a ``plain'' formula. The default is @code{" %%%\n"}. Without | |
30265 | the trailing newline here, the first line of a ``big'' mode formula | |
30266 | that followed might be shifted over with respect to the other lines. | |
30267 | ||
30268 | @vindex calc-embedded-open-new-formula | |
30269 | The @code{calc-embedded-open-new-formula} variable is a string | |
30270 | which is inserted at the front of a new formula when you type | |
30271 | @kbd{M-# f}. Its default value is @code{"\n\n"}. If this | |
30272 | string begins with a newline character and the @kbd{M-# f} is | |
30273 | typed at the beginning of a line, @kbd{M-# f} will skip this | |
30274 | first newline to avoid introducing unnecessary blank lines in | |
30275 | the file. | |
30276 | ||
30277 | @vindex calc-embedded-close-new-formula | |
30278 | The @code{calc-embedded-close-new-formula} variable is the corresponding | |
30279 | string which is inserted at the end of a new formula. Its default | |
30280 | value is also @code{"\n\n"}. The final newline is omitted by | |
30281 | @w{@kbd{M-# f}} if typed at the end of a line. (It follows that if | |
30282 | @kbd{M-# f} is typed on a blank line, both a leading opening | |
30283 | newline and a trailing closing newline are omitted.) | |
30284 | ||
30285 | @vindex calc-embedded-announce-formula | |
30286 | The @code{calc-embedded-announce-formula} variable is a regular | |
30287 | expression which is sure to be followed by an embedded formula. | |
30288 | The @kbd{M-# a} command searches for this pattern as well as for | |
30289 | @samp{=>} and @samp{:=} operators. Note that @kbd{M-# a} will | |
30290 | not activate just anything surrounded by formula delimiters; after | |
30291 | all, blank lines are considered formula delimiters by default! | |
30292 | But if your language includes a delimiter which can only occur | |
30293 | actually in front of a formula, you can take advantage of it here. | |
30294 | The default pattern is @code{"%Embed\n\\(% .*\n\\)*"}, which | |
30295 | checks for @samp{%Embed} followed by any number of lines beginning | |
30296 | with @samp{%} and a space. This last is important to make Calc | |
30297 | consider mode annotations part of the pattern, so that the formula's | |
30298 | opening delimiter really is sure to follow the pattern. | |
30299 | ||
30300 | @vindex calc-embedded-open-mode | |
30301 | The @code{calc-embedded-open-mode} variable is a string (not a | |
30302 | regular expression) which should precede a mode annotation. | |
30303 | Calc never scans for this string; Calc always looks for the | |
30304 | annotation itself. But this is the string that is inserted before | |
30305 | the opening bracket when Calc adds an annotation on its own. | |
30306 | The default is @code{"% "}. | |
30307 | ||
30308 | @vindex calc-embedded-close-mode | |
30309 | The @code{calc-embedded-close-mode} variable is a string which | |
30310 | follows a mode annotation written by Calc. Its default value | |
30311 | is simply a newline, @code{"\n"}. If you change this, it is a | |
30312 | good idea still to end with a newline so that mode annotations | |
30313 | will appear on lines by themselves. | |
30314 | ||
30315 | @node Programming, Installation, Embedded Mode, Top | |
30316 | @chapter Programming | |
30317 | ||
30318 | @noindent | |
30319 | There are several ways to ``program'' the Emacs Calculator, depending | |
30320 | on the nature of the problem you need to solve. | |
30321 | ||
30322 | @enumerate | |
30323 | @item | |
30324 | @dfn{Keyboard macros} allow you to record a sequence of keystrokes | |
30325 | and play them back at a later time. This is just the standard Emacs | |
30326 | keyboard macro mechanism, dressed up with a few more features such | |
30327 | as loops and conditionals. | |
30328 | ||
30329 | @item | |
30330 | @dfn{Algebraic definitions} allow you to use any formula to define a | |
30331 | new function. This function can then be used in algebraic formulas or | |
30332 | as an interactive command. | |
30333 | ||
30334 | @item | |
30335 | @dfn{Rewrite rules} are discussed in the section on algebra commands. | |
30336 | @xref{Rewrite Rules}. If you put your rewrite rules in the variable | |
30337 | @code{EvalRules}, they will be applied automatically to all Calc | |
30338 | results in just the same way as an internal ``rule'' is applied to | |
30339 | evaluate @samp{sqrt(9)} to 3 and so on. @xref{Automatic Rewrites}. | |
30340 | ||
30341 | @item | |
30342 | @dfn{Lisp} is the programming language that Calc (and most of Emacs) | |
30343 | is written in. If the above techniques aren't powerful enough, you | |
30344 | can write Lisp functions to do anything that built-in Calc commands | |
30345 | can do. Lisp code is also somewhat faster than keyboard macros or | |
30346 | rewrite rules. | |
30347 | @end enumerate | |
30348 | ||
30349 | @kindex z | |
30350 | Programming features are available through the @kbd{z} and @kbd{Z} | |
30351 | prefix keys. New commands that you define are two-key sequences | |
30352 | beginning with @kbd{z}. Commands for managing these definitions | |
30353 | use the shift-@kbd{Z} prefix. (The @kbd{Z T} (@code{calc-timing}) | |
30354 | command is described elsewhere; @pxref{Troubleshooting Commands}. | |
30355 | The @kbd{Z C} (@code{calc-user-define-composition}) command is also | |
30356 | described elsewhere; @pxref{User-Defined Compositions}.) | |
30357 | ||
30358 | @menu | |
30359 | * Creating User Keys:: | |
30360 | * Keyboard Macros:: | |
30361 | * Invocation Macros:: | |
30362 | * Algebraic Definitions:: | |
30363 | * Lisp Definitions:: | |
30364 | @end menu | |
30365 | ||
30366 | @node Creating User Keys, Keyboard Macros, Programming, Programming | |
30367 | @section Creating User Keys | |
30368 | ||
30369 | @noindent | |
30370 | @kindex Z D | |
30371 | @pindex calc-user-define | |
30372 | Any Calculator command may be bound to a key using the @kbd{Z D} | |
30373 | (@code{calc-user-define}) command. Actually, it is bound to a two-key | |
30374 | sequence beginning with the lower-case @kbd{z} prefix. | |
30375 | ||
30376 | The @kbd{Z D} command first prompts for the key to define. For example, | |
30377 | press @kbd{Z D a} to define the new key sequence @kbd{z a}. You are then | |
30378 | prompted for the name of the Calculator command that this key should | |
30379 | run. For example, the @code{calc-sincos} command is not normally | |
30380 | available on a key. Typing @kbd{Z D s sincos @key{RET}} programs the | |
30381 | @kbd{z s} key sequence to run @code{calc-sincos}. This definition will remain | |
30382 | in effect for the rest of this Emacs session, or until you redefine | |
30383 | @kbd{z s} to be something else. | |
30384 | ||
30385 | You can actually bind any Emacs command to a @kbd{z} key sequence by | |
30386 | backspacing over the @samp{calc-} when you are prompted for the command name. | |
30387 | ||
30388 | As with any other prefix key, you can type @kbd{z ?} to see a list of | |
30389 | all the two-key sequences you have defined that start with @kbd{z}. | |
30390 | Initially, no @kbd{z} sequences (except @kbd{z ?} itself) are defined. | |
30391 | ||
30392 | User keys are typically letters, but may in fact be any key. | |
30393 | (@key{META}-keys are not permitted, nor are a terminal's special | |
30394 | function keys which generate multi-character sequences when pressed.) | |
30395 | You can define different commands on the shifted and unshifted versions | |
30396 | of a letter if you wish. | |
30397 | ||
30398 | @kindex Z U | |
30399 | @pindex calc-user-undefine | |
30400 | The @kbd{Z U} (@code{calc-user-undefine}) command unbinds a user key. | |
30401 | For example, the key sequence @kbd{Z U s} will undefine the @code{sincos} | |
30402 | key we defined above. | |
30403 | ||
30404 | @kindex Z P | |
30405 | @pindex calc-user-define-permanent | |
30406 | @cindex Storing user definitions | |
30407 | @cindex Permanent user definitions | |
30408 | @cindex @file{.emacs} file, user-defined commands | |
30409 | The @kbd{Z P} (@code{calc-user-define-permanent}) command makes a key | |
30410 | binding permanent so that it will remain in effect even in future Emacs | |
30411 | sessions. (It does this by adding a suitable bit of Lisp code into | |
30412 | your @file{.emacs} file.) For example, @kbd{Z P s} would register | |
30413 | our @code{sincos} command permanently. If you later wish to unregister | |
30414 | this command you must edit your @file{.emacs} file by hand. | |
30415 | (@xref{General Mode Commands}, for a way to tell Calc to use a | |
30416 | different file instead of @file{.emacs}.) | |
30417 | ||
30418 | The @kbd{Z P} command also saves the user definition, if any, for the | |
30419 | command bound to the key. After @kbd{Z F} and @kbd{Z C}, a given user | |
30420 | key could invoke a command, which in turn calls an algebraic function, | |
30421 | which might have one or more special display formats. A single @kbd{Z P} | |
30422 | command will save all of these definitions. | |
30423 | ||
30424 | To save a command or function without its key binding (or if there is | |
30425 | no key binding for the command or function), type @kbd{'} (the apostrophe) | |
30426 | when prompted for a key. Then, type the function name, or backspace | |
30427 | to change the @samp{calcFunc-} prefix to @samp{calc-} and enter a | |
30428 | command name. (If the command you give implies a function, the function | |
30429 | will be saved, and if the function has any display formats, those will | |
30430 | be saved, but not the other way around: Saving a function will not save | |
30431 | any commands or key bindings associated with the function.) | |
30432 | ||
30433 | @kindex Z E | |
30434 | @pindex calc-user-define-edit | |
30435 | @cindex Editing user definitions | |
30436 | The @kbd{Z E} (@code{calc-user-define-edit}) command edits the definition | |
30437 | of a user key. This works for keys that have been defined by either | |
30438 | keyboard macros or formulas; further details are contained in the relevant | |
30439 | following sections. | |
30440 | ||
30441 | @node Keyboard Macros, Invocation Macros, Creating User Keys, Programming | |
30442 | @section Programming with Keyboard Macros | |
30443 | ||
30444 | @noindent | |
30445 | @kindex X | |
30446 | @cindex Programming with keyboard macros | |
30447 | @cindex Keyboard macros | |
30448 | The easiest way to ``program'' the Emacs Calculator is to use standard | |
30449 | keyboard macros. Press @w{@kbd{C-x (}} to begin recording a macro. From | |
30450 | this point on, keystrokes you type will be saved away as well as | |
30451 | performing their usual functions. Press @kbd{C-x )} to end recording. | |
30452 | Press shift-@kbd{X} (or the standard Emacs key sequence @kbd{C-x e}) to | |
30453 | execute your keyboard macro by replaying the recorded keystrokes. | |
30454 | @xref{Keyboard Macros, , , emacs, the Emacs Manual}, for further | |
30455 | information.@refill | |
30456 | ||
30457 | When you use @kbd{X} to invoke a keyboard macro, the entire macro is | |
30458 | treated as a single command by the undo and trail features. The stack | |
30459 | display buffer is not updated during macro execution, but is instead | |
30460 | fixed up once the macro completes. Thus, commands defined with keyboard | |
30461 | macros are convenient and efficient. The @kbd{C-x e} command, on the | |
30462 | other hand, invokes the keyboard macro with no special treatment: Each | |
30463 | command in the macro will record its own undo information and trail entry, | |
30464 | and update the stack buffer accordingly. If your macro uses features | |
30465 | outside of Calc's control to operate on the contents of the Calc stack | |
30466 | buffer, or if it includes Undo, Redo, or last-arguments commands, you | |
30467 | must use @kbd{C-x e} to make sure the buffer and undo list are up-to-date | |
30468 | at all times. You could also consider using @kbd{K} (@code{calc-keep-args}) | |
30469 | instead of @kbd{M-@key{RET}} (@code{calc-last-args}). | |
30470 | ||
30471 | Calc extends the standard Emacs keyboard macros in several ways. | |
30472 | Keyboard macros can be used to create user-defined commands. Keyboard | |
30473 | macros can include conditional and iteration structures, somewhat | |
30474 | analogous to those provided by a traditional programmable calculator. | |
30475 | ||
30476 | @menu | |
30477 | * Naming Keyboard Macros:: | |
30478 | * Conditionals in Macros:: | |
30479 | * Loops in Macros:: | |
30480 | * Local Values in Macros:: | |
30481 | * Queries in Macros:: | |
30482 | @end menu | |
30483 | ||
30484 | @node Naming Keyboard Macros, Conditionals in Macros, Keyboard Macros, Keyboard Macros | |
30485 | @subsection Naming Keyboard Macros | |
30486 | ||
30487 | @noindent | |
30488 | @kindex Z K | |
30489 | @pindex calc-user-define-kbd-macro | |
30490 | Once you have defined a keyboard macro, you can bind it to a @kbd{z} | |
30491 | key sequence with the @kbd{Z K} (@code{calc-user-define-kbd-macro}) command. | |
30492 | This command prompts first for a key, then for a command name. For | |
30493 | example, if you type @kbd{C-x ( n @key{TAB} n @key{TAB} C-x )} you will | |
30494 | define a keyboard macro which negates the top two numbers on the stack | |
30495 | (@key{TAB} swaps the top two stack elements). Now you can type | |
30496 | @kbd{Z K n @key{RET}} to define this keyboard macro onto the @kbd{z n} key | |
30497 | sequence. The default command name (if you answer the second prompt with | |
30498 | just the @key{RET} key as in this example) will be something like | |
30499 | @samp{calc-User-n}. The keyboard macro will now be available as both | |
30500 | @kbd{z n} and @kbd{M-x calc-User-n}. You can backspace and enter a more | |
30501 | descriptive command name if you wish.@refill | |
30502 | ||
30503 | Macros defined by @kbd{Z K} act like single commands; they are executed | |
30504 | in the same way as by the @kbd{X} key. If you wish to define the macro | |
30505 | as a standard no-frills Emacs macro (to be executed as if by @kbd{C-x e}), | |
30506 | give a negative prefix argument to @kbd{Z K}. | |
30507 | ||
30508 | Once you have bound your keyboard macro to a key, you can use | |
30509 | @kbd{Z P} to register it permanently with Emacs. @xref{Creating User Keys}. | |
30510 | ||
30511 | @cindex Keyboard macros, editing | |
30512 | The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has | |
30513 | been defined by a keyboard macro tries to use the @code{edit-kbd-macro} | |
30514 | command to edit the macro. This command may be found in the | |
30515 | @file{macedit} package, a copy of which comes with Calc. It decomposes | |
30516 | the macro definition into full Emacs command names, like @code{calc-pop} | |
30517 | and @code{calc-add}. Type @kbd{M-# M-#} to finish editing and update | |
30518 | the definition stored on the key, or, to cancel the edit, type | |
30519 | @kbd{M-# x}.@refill | |
30520 | ||
30521 | If you give a negative numeric prefix argument to @kbd{Z E}, the keyboard | |
30522 | macro is edited in spelled-out keystroke form. For example, the editing | |
5d67986c | 30523 | buffer might contain the nine characters @w{@samp{1 @key{RET} 2 +}}. When you press |
d7b8e6c6 EZ |
30524 | @kbd{M-# M-#}, the @code{read-kbd-macro} feature of the @file{macedit} |
30525 | package is used to reinterpret these key names. The | |
30526 | notations @code{RET}, @code{LFD}, @code{TAB}, @code{SPC}, @code{DEL}, and | |
30527 | @code{NUL} must be written in all uppercase, as must the prefixes @code{C-} | |
30528 | and @code{M-}. Spaces and line breaks are ignored. Other characters are | |
30529 | copied verbatim into the keyboard macro. Basically, the notation is the | |
30530 | same as is used in all of this manual's examples, except that the manual | |
5d67986c RS |
30531 | takes some liberties with spaces: When we say @kbd{' [1 2 3] @key{RET}}, we take |
30532 | it for granted that it is clear we really mean @kbd{' [1 @key{SPC} 2 @key{SPC} 3] @key{RET}}, | |
d7b8e6c6 EZ |
30533 | which is what @code{read-kbd-macro} wants to see.@refill |
30534 | ||
30535 | If @file{macedit} is not available, @kbd{Z E} edits the keyboard macro | |
30536 | in ``raw'' form; the editing buffer simply contains characters like | |
30537 | @samp{1^M2+} (here @samp{^M} represents the carriage-return character). | |
30538 | Editing in this mode, you will have to use @kbd{C-q} to enter new | |
30539 | control characters into the buffer.@refill | |
30540 | ||
30541 | @kindex M-# m | |
30542 | @pindex read-kbd-macro | |
30543 | The @kbd{M-# m} (@code{read-kbd-macro}) command reads an Emacs ``region'' | |
30544 | of spelled-out keystrokes and defines it as the current keyboard macro. | |
30545 | It is a convenient way to define a keyboard macro that has been stored | |
30546 | in a file, or to define a macro without executing it at the same time. | |
30547 | The @kbd{M-# m} command works only if @file{macedit} is present. | |
30548 | ||
30549 | @node Conditionals in Macros, Loops in Macros, Naming Keyboard Macros, Keyboard Macros | |
30550 | @subsection Conditionals in Keyboard Macros | |
30551 | ||
30552 | @noindent | |
30553 | @kindex Z [ | |
30554 | @kindex Z ] | |
30555 | @pindex calc-kbd-if | |
30556 | @pindex calc-kbd-else | |
30557 | @pindex calc-kbd-else-if | |
30558 | @pindex calc-kbd-end-if | |
30559 | @cindex Conditional structures | |
30560 | The @kbd{Z [} (@code{calc-kbd-if}) and @kbd{Z ]} (@code{calc-kbd-end-if}) | |
30561 | commands allow you to put simple tests in a keyboard macro. When Calc | |
30562 | sees the @kbd{Z [}, it pops an object from the stack and, if the object is | |
30563 | a non-zero value, continues executing keystrokes. But if the object is | |
30564 | zero, or if it is not provably nonzero, Calc skips ahead to the matching | |
30565 | @kbd{Z ]} keystroke. @xref{Logical Operations}, for a set of commands for | |
30566 | performing tests which conveniently produce 1 for true and 0 for false. | |
30567 | ||
30568 | For example, @kbd{@key{RET} 0 a < Z [ n Z ]} implements an absolute-value | |
30569 | function in the form of a keyboard macro. This macro duplicates the | |
30570 | number on the top of the stack, pushes zero and compares using @kbd{a <} | |
30571 | (@code{calc-less-than}), then, if the number was less than zero, | |
30572 | executes @kbd{n} (@code{calc-change-sign}). Otherwise, the change-sign | |
30573 | command is skipped. | |
30574 | ||
30575 | To program this macro, type @kbd{C-x (}, type the above sequence of | |
30576 | keystrokes, then type @kbd{C-x )}. Note that the keystrokes will be | |
30577 | executed while you are making the definition as well as when you later | |
30578 | re-execute the macro by typing @kbd{X}. Thus you should make sure a | |
30579 | suitable number is on the stack before defining the macro so that you | |
30580 | don't get a stack-underflow error during the definition process. | |
30581 | ||
30582 | Conditionals can be nested arbitrarily. However, there should be exactly | |
30583 | one @kbd{Z ]} for each @kbd{Z [} in a keyboard macro. | |
30584 | ||
30585 | @kindex Z : | |
30586 | The @kbd{Z :} (@code{calc-kbd-else}) command allows you to choose between | |
30587 | two keystroke sequences. The general format is @kbd{@var{cond} Z [ | |
30588 | @var{then-part} Z : @var{else-part} Z ]}. If @var{cond} is true | |
30589 | (i.e., if the top of stack contains a non-zero number after @var{cond} | |
30590 | has been executed), the @var{then-part} will be executed and the | |
30591 | @var{else-part} will be skipped. Otherwise, the @var{then-part} will | |
30592 | be skipped and the @var{else-part} will be executed. | |
30593 | ||
30594 | @kindex Z | | |
30595 | The @kbd{Z |} (@code{calc-kbd-else-if}) command allows you to choose | |
30596 | between any number of alternatives. For example, | |
30597 | @kbd{@var{cond1} Z [ @var{part1} Z : @var{cond2} Z | @var{part2} Z : | |
30598 | @var{part3} Z ]} will execute @var{part1} if @var{cond1} is true, | |
30599 | otherwise it will execute @var{part2} if @var{cond2} is true, otherwise | |
30600 | it will execute @var{part3}. | |
30601 | ||
30602 | More precisely, @kbd{Z [} pops a number and conditionally skips to the | |
30603 | next matching @kbd{Z :} or @kbd{Z ]} key. @w{@kbd{Z ]}} has no effect when | |
30604 | actually executed. @kbd{Z :} skips to the next matching @kbd{Z ]}. | |
30605 | @kbd{Z |} pops a number and conditionally skips to the next matching | |
30606 | @kbd{Z :} or @kbd{Z ]}; thus, @kbd{Z [} and @kbd{Z |} are functionally | |
30607 | equivalent except that @kbd{Z [} participates in nesting but @kbd{Z |} | |
30608 | does not. | |
30609 | ||
30610 | Calc's conditional and looping constructs work by scanning the | |
30611 | keyboard macro for occurrences of character sequences like @samp{Z:} | |
30612 | and @samp{Z]}. One side-effect of this is that if you use these | |
30613 | constructs you must be careful that these character pairs do not | |
30614 | occur by accident in other parts of the macros. Since Calc rarely | |
30615 | uses shift-@kbd{Z} for any purpose except as a prefix character, this | |
30616 | is not likely to be a problem. Another side-effect is that it will | |
30617 | not work to define your own custom key bindings for these commands. | |
30618 | Only the standard shift-@kbd{Z} bindings will work correctly. | |
30619 | ||
30620 | @kindex Z C-g | |
30621 | If Calc gets stuck while skipping characters during the definition of a | |
30622 | macro, type @kbd{Z C-g} to cancel the definition. (Typing plain @kbd{C-g} | |
30623 | actually adds a @kbd{C-g} keystroke to the macro.) | |
30624 | ||
30625 | @node Loops in Macros, Local Values in Macros, Conditionals in Macros, Keyboard Macros | |
30626 | @subsection Loops in Keyboard Macros | |
30627 | ||
30628 | @noindent | |
30629 | @kindex Z < | |
30630 | @kindex Z > | |
30631 | @pindex calc-kbd-repeat | |
30632 | @pindex calc-kbd-end-repeat | |
30633 | @cindex Looping structures | |
30634 | @cindex Iterative structures | |
30635 | The @kbd{Z <} (@code{calc-kbd-repeat}) and @kbd{Z >} | |
30636 | (@code{calc-kbd-end-repeat}) commands pop a number from the stack, | |
30637 | which must be an integer, then repeat the keystrokes between the brackets | |
30638 | the specified number of times. If the integer is zero or negative, the | |
30639 | body is skipped altogether. For example, @kbd{1 @key{TAB} Z < 2 * Z >} | |
30640 | computes two to a nonnegative integer power. First, we push 1 on the | |
30641 | stack and then swap the integer argument back to the top. The @kbd{Z <} | |
30642 | pops that argument leaving the 1 back on top of the stack. Then, we | |
30643 | repeat a multiply-by-two step however many times.@refill | |
30644 | ||
30645 | Once again, the keyboard macro is executed as it is being entered. | |
30646 | In this case it is especially important to set up reasonable initial | |
30647 | conditions before making the definition: Suppose the integer 1000 just | |
30648 | happened to be sitting on the stack before we typed the above definition! | |
30649 | Another approach is to enter a harmless dummy definition for the macro, | |
30650 | then go back and edit in the real one with a @kbd{Z E} command. Yet | |
30651 | another approach is to type the macro as written-out keystroke names | |
30652 | in a buffer, then use @kbd{M-# m} (@code{read-kbd-macro}) to read the | |
30653 | macro. | |
30654 | ||
30655 | @kindex Z / | |
30656 | @pindex calc-break | |
30657 | The @kbd{Z /} (@code{calc-kbd-break}) command allows you to break out | |
30658 | of a keyboard macro loop prematurely. It pops an object from the stack; | |
30659 | if that object is true (a non-zero number), control jumps out of the | |
30660 | innermost enclosing @kbd{Z <} @dots{} @kbd{Z >} loop and continues | |
30661 | after the @kbd{Z >}. If the object is false, the @kbd{Z /} has no | |
30662 | effect. Thus @kbd{@var{cond} Z /} is similar to @samp{if (@var{cond}) break;} | |
30663 | in the C language.@refill | |
30664 | ||
30665 | @kindex Z ( | |
30666 | @kindex Z ) | |
30667 | @pindex calc-kbd-for | |
30668 | @pindex calc-kbd-end-for | |
30669 | The @kbd{Z (} (@code{calc-kbd-for}) and @kbd{Z )} (@code{calc-kbd-end-for}) | |
30670 | commands are similar to @kbd{Z <} and @kbd{Z >}, except that they make the | |
30671 | value of the counter available inside the loop. The general layout is | |
30672 | @kbd{@var{init} @var{final} Z ( @var{body} @var{step} Z )}. The @kbd{Z (} | |
30673 | command pops initial and final values from the stack. It then creates | |
30674 | a temporary internal counter and initializes it with the value @var{init}. | |
30675 | The @kbd{Z (} command then repeatedly pushes the counter value onto the | |
30676 | stack and executes @var{body} and @var{step}, adding @var{step} to the | |
30677 | counter each time until the loop finishes.@refill | |
30678 | ||
30679 | @cindex Summations (by keyboard macros) | |
30680 | By default, the loop finishes when the counter becomes greater than (or | |
30681 | less than) @var{final}, assuming @var{initial} is less than (greater | |
30682 | than) @var{final}. If @var{initial} is equal to @var{final}, the body | |
30683 | executes exactly once. The body of the loop always executes at least | |
30684 | once. For example, @kbd{0 1 10 Z ( 2 ^ + 1 Z )} computes the sum of the | |
30685 | squares of the integers from 1 to 10, in steps of 1. | |
30686 | ||
30687 | If you give a numeric prefix argument of 1 to @kbd{Z (}, the loop is | |
30688 | forced to use upward-counting conventions. In this case, if @var{initial} | |
30689 | is greater than @var{final} the body will not be executed at all. | |
30690 | Note that @var{step} may still be negative in this loop; the prefix | |
30691 | argument merely constrains the loop-finished test. Likewise, a prefix | |
30692 | argument of @i{-1} forces downward-counting conventions. | |
30693 | ||
30694 | @kindex Z @{ | |
30695 | @kindex Z @} | |
30696 | @pindex calc-kbd-loop | |
30697 | @pindex calc-kbd-end-loop | |
30698 | The @kbd{Z @{} (@code{calc-kbd-loop}) and @kbd{Z @}} | |
30699 | (@code{calc-kbd-end-loop}) commands are similar to @kbd{Z <} and | |
30700 | @kbd{Z >}, except that they do not pop a count from the stack---they | |
30701 | effectively create an infinite loop. Every @kbd{Z @{} @dots{} @kbd{Z @}} | |
30702 | loop ought to include at least one @kbd{Z /} to make sure the loop | |
30703 | doesn't run forever. (If any error message occurs which causes Emacs | |
30704 | to beep, the keyboard macro will also be halted; this is a standard | |
30705 | feature of Emacs. You can also generally press @kbd{C-g} to halt a | |
30706 | running keyboard macro, although not all versions of Unix support | |
30707 | this feature.) | |
30708 | ||
30709 | The conditional and looping constructs are not actually tied to | |
30710 | keyboard macros, but they are most often used in that context. | |
30711 | For example, the keystrokes @kbd{10 Z < 23 @key{RET} Z >} push | |
30712 | ten copies of 23 onto the stack. This can be typed ``live'' just | |
30713 | as easily as in a macro definition. | |
30714 | ||
30715 | @xref{Conditionals in Macros}, for some additional notes about | |
30716 | conditional and looping commands. | |
30717 | ||
30718 | @node Local Values in Macros, Queries in Macros, Loops in Macros, Keyboard Macros | |
30719 | @subsection Local Values in Macros | |
30720 | ||
30721 | @noindent | |
30722 | @cindex Local variables | |
30723 | @cindex Restoring saved modes | |
30724 | Keyboard macros sometimes want to operate under known conditions | |
30725 | without affecting surrounding conditions. For example, a keyboard | |
30726 | macro may wish to turn on Fraction Mode, or set a particular | |
30727 | precision, independent of the user's normal setting for those | |
30728 | modes. | |
30729 | ||
30730 | @kindex Z ` | |
30731 | @kindex Z ' | |
30732 | @pindex calc-kbd-push | |
30733 | @pindex calc-kbd-pop | |
30734 | Macros also sometimes need to use local variables. Assignments to | |
30735 | local variables inside the macro should not affect any variables | |
30736 | outside the macro. The @kbd{Z `} (@code{calc-kbd-push}) and @kbd{Z '} | |
30737 | (@code{calc-kbd-pop}) commands give you both of these capabilities. | |
30738 | ||
30739 | When you type @kbd{Z `} (with a backquote or accent grave character), | |
30740 | the values of various mode settings are saved away. The ten ``quick'' | |
30741 | variables @code{q0} through @code{q9} are also saved. When | |
30742 | you type @w{@kbd{Z '}} (with an apostrophe), these values are restored. | |
30743 | Pairs of @kbd{Z `} and @kbd{Z '} commands may be nested. | |
30744 | ||
30745 | If a keyboard macro halts due to an error in between a @kbd{Z `} and | |
30746 | a @kbd{Z '}, the saved values will be restored correctly even though | |
30747 | the macro never reaches the @kbd{Z '} command. Thus you can use | |
30748 | @kbd{Z `} and @kbd{Z '} without having to worry about what happens | |
30749 | in exceptional conditions. | |
30750 | ||
30751 | If you type @kbd{Z `} ``live'' (not in a keyboard macro), Calc puts | |
30752 | you into a ``recursive edit.'' You can tell you are in a recursive | |
30753 | edit because there will be extra square brackets in the mode line, | |
30754 | as in @samp{[(Calculator)]}. These brackets will go away when you | |
30755 | type the matching @kbd{Z '} command. The modes and quick variables | |
30756 | will be saved and restored in just the same way as if actual keyboard | |
30757 | macros were involved. | |
30758 | ||
30759 | The modes saved by @kbd{Z `} and @kbd{Z '} are the current precision | |
30760 | and binary word size, the angular mode (Deg, Rad, or HMS), the | |
30761 | simplification mode, Algebraic mode, Symbolic mode, Infinite mode, | |
30762 | Matrix or Scalar mode, Fraction mode, and the current complex mode | |
30763 | (Polar or Rectangular). The ten ``quick'' variables' values (or lack | |
30764 | thereof) are also saved. | |
30765 | ||
30766 | Most mode-setting commands act as toggles, but with a numeric prefix | |
30767 | they force the mode either on (positive prefix) or off (negative | |
30768 | or zero prefix). Since you don't know what the environment might | |
30769 | be when you invoke your macro, it's best to use prefix arguments | |
30770 | for all mode-setting commands inside the macro. | |
30771 | ||
30772 | In fact, @kbd{C-u Z `} is like @kbd{Z `} except that it sets the modes | |
30773 | listed above to their default values. As usual, the matching @kbd{Z '} | |
30774 | will restore the modes to their settings from before the @kbd{C-u Z `}. | |
30775 | Also, @w{@kbd{Z `}} with a negative prefix argument resets algebraic mode | |
30776 | to its default (off) but leaves the other modes the same as they were | |
30777 | outside the construct. | |
30778 | ||
30779 | The contents of the stack and trail, values of non-quick variables, and | |
30780 | other settings such as the language mode and the various display modes, | |
30781 | are @emph{not} affected by @kbd{Z `} and @kbd{Z '}. | |
30782 | ||
30783 | @node Queries in Macros, , Local Values in Macros, Keyboard Macros | |
30784 | @subsection Queries in Keyboard Macros | |
30785 | ||
30786 | @noindent | |
30787 | @kindex Z = | |
30788 | @pindex calc-kbd-report | |
30789 | The @kbd{Z =} (@code{calc-kbd-report}) command displays an informative | |
30790 | message including the value on the top of the stack. You are prompted | |
30791 | to enter a string. That string, along with the top-of-stack value, | |
30792 | is displayed unless @kbd{m w} (@code{calc-working}) has been used | |
30793 | to turn such messages off. | |
30794 | ||
30795 | @kindex Z # | |
30796 | @pindex calc-kbd-query | |
30797 | The @kbd{Z #} (@code{calc-kbd-query}) command displays a prompt message | |
30798 | (which you enter during macro definition), then does an algebraic entry | |
30799 | which takes its input from the keyboard, even during macro execution. | |
30800 | This command allows your keyboard macros to accept numbers or formulas | |
30801 | as interactive input. All the normal conventions of algebraic input, | |
30802 | including the use of @kbd{$} characters, are supported. | |
30803 | ||
6b61353c | 30804 | @xref{Keyboard Macro Query, , , emacs, the Emacs Manual}, for a description of |
d7b8e6c6 EZ |
30805 | @kbd{C-x q} (@code{kbd-macro-query}), the standard Emacs way to accept |
30806 | keyboard input during a keyboard macro. In particular, you can use | |
30807 | @kbd{C-x q} to enter a recursive edit, which allows the user to perform | |
30808 | any Calculator operations interactively before pressing @kbd{C-M-c} to | |
30809 | return control to the keyboard macro. | |
30810 | ||
30811 | @node Invocation Macros, Algebraic Definitions, Keyboard Macros, Programming | |
30812 | @section Invocation Macros | |
30813 | ||
30814 | @kindex M-# z | |
30815 | @kindex Z I | |
30816 | @pindex calc-user-invocation | |
30817 | @pindex calc-user-define-invocation | |
30818 | Calc provides one special keyboard macro, called up by @kbd{M-# z} | |
30819 | (@code{calc-user-invocation}), that is intended to allow you to define | |
30820 | your own special way of starting Calc. To define this ``invocation | |
30821 | macro,'' create the macro in the usual way with @kbd{C-x (} and | |
30822 | @kbd{C-x )}, then type @kbd{Z I} (@code{calc-user-define-invocation}). | |
30823 | There is only one invocation macro, so you don't need to type any | |
30824 | additional letters after @kbd{Z I}. From now on, you can type | |
30825 | @kbd{M-# z} at any time to execute your invocation macro. | |
30826 | ||
30827 | For example, suppose you find yourself often grabbing rectangles of | |
30828 | numbers into Calc and multiplying their columns. You can do this | |
30829 | by typing @kbd{M-# r} to grab, and @kbd{V R : *} to multiply columns. | |
30830 | To make this into an invocation macro, just type @kbd{C-x ( M-# r | |
30831 | V R : * C-x )}, then @kbd{Z I}. Then, to multiply a rectangle of data, | |
30832 | just mark the data in its buffer in the usual way and type @kbd{M-# z}. | |
30833 | ||
30834 | Invocation macros are treated like regular Emacs keyboard macros; | |
30835 | all the special features described above for @kbd{Z K}-style macros | |
30836 | do not apply. @kbd{M-# z} is just like @kbd{C-x e}, except that it | |
30837 | uses the macro that was last stored by @kbd{Z I}. (In fact, the | |
30838 | macro does not even have to have anything to do with Calc!) | |
30839 | ||
30840 | The @kbd{m m} command saves the last invocation macro defined by | |
30841 | @kbd{Z I} along with all the other Calc mode settings. | |
30842 | @xref{General Mode Commands}. | |
30843 | ||
30844 | @node Algebraic Definitions, Lisp Definitions, Invocation Macros, Programming | |
30845 | @section Programming with Formulas | |
30846 | ||
30847 | @noindent | |
30848 | @kindex Z F | |
30849 | @pindex calc-user-define-formula | |
30850 | @cindex Programming with algebraic formulas | |
30851 | Another way to create a new Calculator command uses algebraic formulas. | |
30852 | The @kbd{Z F} (@code{calc-user-define-formula}) command stores the | |
30853 | formula at the top of the stack as the definition for a key. This | |
30854 | command prompts for five things: The key, the command name, the function | |
30855 | name, the argument list, and the behavior of the command when given | |
30856 | non-numeric arguments. | |
30857 | ||
30858 | For example, suppose we type @kbd{' a+2b @key{RET}} to push the formula | |
30859 | @samp{a + 2*b} onto the stack. We now type @kbd{Z F m} to define this | |
30860 | formula on the @kbd{z m} key sequence. The next prompt is for a command | |
30861 | name, beginning with @samp{calc-}, which should be the long (@kbd{M-x}) form | |
30862 | for the new command. If you simply press @key{RET}, a default name like | |
30863 | @code{calc-User-m} will be constructed. In our example, suppose we enter | |
30864 | @kbd{spam @key{RET}} to define the new command as @code{calc-spam}. | |
30865 | ||
30866 | If you want to give the formula a long-style name only, you can press | |
30867 | @key{SPC} or @key{RET} when asked which single key to use. For example | |
30868 | @kbd{Z F @key{RET} spam @key{RET}} defines the new command as | |
30869 | @kbd{M-x calc-spam}, with no keyboard equivalent. | |
30870 | ||
30871 | The third prompt is for a function name. The default is to use the same | |
30872 | name as the command name but with @samp{calcFunc-} in place of | |
30873 | @samp{calc-}. This is the name you will use if you want to enter your | |
30874 | new function in an algebraic formula. Suppose we enter @kbd{yow @key{RET}}. | |
30875 | Then the new function can be invoked by pushing two numbers on the | |
30876 | stack and typing @kbd{z m} or @kbd{x spam}, or by entering the algebraic | |
30877 | formula @samp{yow(x,y)}.@refill | |
30878 | ||
30879 | The fourth prompt is for the function's argument list. This is used to | |
30880 | associate values on the stack with the variables that appear in the formula. | |
30881 | The default is a list of all variables which appear in the formula, sorted | |
30882 | into alphabetical order. In our case, the default would be @samp{(a b)}. | |
30883 | This means that, when the user types @kbd{z m}, the Calculator will remove | |
30884 | two numbers from the stack, substitute these numbers for @samp{a} and | |
30885 | @samp{b} (respectively) in the formula, then simplify the formula and | |
30886 | push the result on the stack. In other words, @kbd{10 @key{RET} 100 z m} | |
30887 | would replace the 10 and 100 on the stack with the number 210, which is | |
30888 | @cite{a + 2 b} with @cite{a=10} and @cite{b=100}. Likewise, the formula | |
30889 | @samp{yow(10, 100)} will be evaluated by substituting @cite{a=10} and | |
30890 | @cite{b=100} in the definition. | |
30891 | ||
30892 | You can rearrange the order of the names before pressing @key{RET} to | |
30893 | control which stack positions go to which variables in the formula. If | |
30894 | you remove a variable from the argument list, that variable will be left | |
30895 | in symbolic form by the command. Thus using an argument list of @samp{(b)} | |
30896 | for our function would cause @kbd{10 z m} to replace the 10 on the stack | |
30897 | with the formula @samp{a + 20}. If we had used an argument list of | |
30898 | @samp{(b a)}, the result with inputs 10 and 100 would have been 120. | |
30899 | ||
30900 | You can also put a nameless function on the stack instead of just a | |
30901 | formula, as in @samp{<a, b : a + 2 b>}. @xref{Specifying Operators}. | |
30902 | In this example, the command will be defined by the formula @samp{a + 2 b} | |
30903 | using the argument list @samp{(a b)}. | |
30904 | ||
30905 | The final prompt is a y-or-n question concerning what to do if symbolic | |
30906 | arguments are given to your function. If you answer @kbd{y}, then | |
30907 | executing @kbd{z m} (using the original argument list @samp{(a b)}) with | |
30908 | arguments @cite{10} and @cite{x} will leave the function in symbolic | |
30909 | form, i.e., @samp{yow(10,x)}. On the other hand, if you answer @kbd{n}, | |
30910 | then the formula will always be expanded, even for non-constant | |
30911 | arguments: @samp{10 + 2 x}. If you never plan to feed algebraic | |
30912 | formulas to your new function, it doesn't matter how you answer this | |
30913 | question.@refill | |
30914 | ||
30915 | If you answered @kbd{y} to this question you can still cause a function | |
30916 | call to be expanded by typing @kbd{a "} (@code{calc-expand-formula}). | |
30917 | Also, Calc will expand the function if necessary when you take a | |
30918 | derivative or integral or solve an equation involving the function. | |
30919 | ||
30920 | @kindex Z G | |
30921 | @pindex calc-get-user-defn | |
30922 | Once you have defined a formula on a key, you can retrieve this formula | |
30923 | with the @kbd{Z G} (@code{calc-user-define-get-defn}) command. Press a | |
30924 | key, and this command pushes the formula that was used to define that | |
30925 | key onto the stack. Actually, it pushes a nameless function that | |
30926 | specifies both the argument list and the defining formula. You will get | |
30927 | an error message if the key is undefined, or if the key was not defined | |
30928 | by a @kbd{Z F} command.@refill | |
30929 | ||
30930 | The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has | |
30931 | been defined by a formula uses a variant of the @code{calc-edit} command | |
30932 | to edit the defining formula. Press @kbd{M-# M-#} to finish editing and | |
30933 | store the new formula back in the definition, or @kbd{M-# x} to | |
30934 | cancel the edit. (The argument list and other properties of the | |
30935 | definition are unchanged; to adjust the argument list, you can use | |
30936 | @kbd{Z G} to grab the function onto the stack, edit with @kbd{`}, and | |
30937 | then re-execute the @kbd{Z F} command.) | |
30938 | ||
30939 | As usual, the @kbd{Z P} command records your definition permanently. | |
30940 | In this case it will permanently record all three of the relevant | |
30941 | definitions: the key, the command, and the function. | |
30942 | ||
30943 | You may find it useful to turn off the default simplifications with | |
30944 | @kbd{m O} (@code{calc-no-simplify-mode}) when entering a formula to be | |
30945 | used as a function definition. For example, the formula @samp{deriv(a^2,v)} | |
30946 | which might be used to define a new function @samp{dsqr(a,v)} will be | |
30947 | ``simplified'' to 0 immediately upon entry since @code{deriv} considers | |
30948 | @cite{a} to be constant with respect to @cite{v}. Turning off | |
30949 | default simplifications cures this problem: The definition will be stored | |
30950 | in symbolic form without ever activating the @code{deriv} function. Press | |
30951 | @kbd{m D} to turn the default simplifications back on afterwards. | |
30952 | ||
30953 | @node Lisp Definitions, , Algebraic Definitions, Programming | |
30954 | @section Programming with Lisp | |
30955 | ||
30956 | @noindent | |
30957 | The Calculator can be programmed quite extensively in Lisp. All you | |
30958 | do is write a normal Lisp function definition, but with @code{defmath} | |
30959 | in place of @code{defun}. This has the same form as @code{defun}, but it | |
30960 | automagically replaces calls to standard Lisp functions like @code{+} and | |
30961 | @code{zerop} with calls to the corresponding functions in Calc's own library. | |
30962 | Thus you can write natural-looking Lisp code which operates on all of the | |
30963 | standard Calculator data types. You can then use @kbd{Z D} if you wish to | |
30964 | bind your new command to a @kbd{z}-prefix key sequence. The @kbd{Z E} command | |
30965 | will not edit a Lisp-based definition. | |
30966 | ||
30967 | Emacs Lisp is described in the GNU Emacs Lisp Reference Manual. This section | |
30968 | assumes a familiarity with Lisp programming concepts; if you do not know | |
30969 | Lisp, you may find keyboard macros or rewrite rules to be an easier way | |
30970 | to program the Calculator. | |
30971 | ||
30972 | This section first discusses ways to write commands, functions, or | |
30973 | small programs to be executed inside of Calc. Then it discusses how | |
30974 | your own separate programs are able to call Calc from the outside. | |
30975 | Finally, there is a list of internal Calc functions and data structures | |
30976 | for the true Lisp enthusiast. | |
30977 | ||
30978 | @menu | |
30979 | * Defining Functions:: | |
30980 | * Defining Simple Commands:: | |
30981 | * Defining Stack Commands:: | |
30982 | * Argument Qualifiers:: | |
30983 | * Example Definitions:: | |
30984 | ||
30985 | * Calling Calc from Your Programs:: | |
30986 | * Internals:: | |
30987 | @end menu | |
30988 | ||
30989 | @node Defining Functions, Defining Simple Commands, Lisp Definitions, Lisp Definitions | |
30990 | @subsection Defining New Functions | |
30991 | ||
30992 | @noindent | |
30993 | @findex defmath | |
30994 | The @code{defmath} function (actually a Lisp macro) is like @code{defun} | |
30995 | except that code in the body of the definition can make use of the full | |
30996 | range of Calculator data types. The prefix @samp{calcFunc-} is added | |
30997 | to the specified name to get the actual Lisp function name. As a simple | |
30998 | example, | |
30999 | ||
31000 | @example | |
31001 | (defmath myfact (n) | |
31002 | (if (> n 0) | |
31003 | (* n (myfact (1- n))) | |
31004 | 1)) | |
31005 | @end example | |
31006 | ||
31007 | @noindent | |
31008 | This actually expands to the code, | |
31009 | ||
31010 | @example | |
31011 | (defun calcFunc-myfact (n) | |
31012 | (if (math-posp n) | |
31013 | (math-mul n (calcFunc-myfact (math-add n -1))) | |
31014 | 1)) | |
31015 | @end example | |
31016 | ||
31017 | @noindent | |
31018 | This function can be used in algebraic expressions, e.g., @samp{myfact(5)}. | |
31019 | ||
31020 | The @samp{myfact} function as it is defined above has the bug that an | |
31021 | expression @samp{myfact(a+b)} will be simplified to 1 because the | |
31022 | formula @samp{a+b} is not considered to be @code{posp}. A robust | |
31023 | factorial function would be written along the following lines: | |
31024 | ||
31025 | @smallexample | |
31026 | (defmath myfact (n) | |
31027 | (if (> n 0) | |
31028 | (* n (myfact (1- n))) | |
31029 | (if (= n 0) | |
31030 | 1 | |
31031 | nil))) ; this could be simplified as: (and (= n 0) 1) | |
31032 | @end smallexample | |
31033 | ||
31034 | If a function returns @code{nil}, it is left unsimplified by the Calculator | |
31035 | (except that its arguments will be simplified). Thus, @samp{myfact(a+1+2)} | |
31036 | will be simplified to @samp{myfact(a+3)} but no further. Beware that every | |
31037 | time the Calculator reexamines this formula it will attempt to resimplify | |
31038 | it, so your function ought to detect the returning-@code{nil} case as | |
31039 | efficiently as possible. | |
31040 | ||
31041 | The following standard Lisp functions are treated by @code{defmath}: | |
31042 | @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^} or | |
31043 | @code{expt}, @code{=}, @code{<}, @code{>}, @code{<=}, @code{>=}, | |
31044 | @code{/=}, @code{1+}, @code{1-}, @code{logand}, @code{logior}, @code{logxor}, | |
31045 | @code{logandc2}, @code{lognot}. Also, @code{~=} is an abbreviation for | |
31046 | @code{math-nearly-equal}, which is useful in implementing Taylor series.@refill | |
31047 | ||
31048 | For other functions @var{func}, if a function by the name | |
31049 | @samp{calcFunc-@var{func}} exists it is used, otherwise if a function by the | |
31050 | name @samp{math-@var{func}} exists it is used, otherwise if @var{func} itself | |
31051 | is defined as a function it is used, otherwise @samp{calcFunc-@var{func}} is | |
31052 | used on the assumption that this is a to-be-defined math function. Also, if | |
31053 | the function name is quoted as in @samp{('integerp a)} the function name is | |
31054 | always used exactly as written (but not quoted).@refill | |
31055 | ||
31056 | Variable names have @samp{var-} prepended to them unless they appear in | |
31057 | the function's argument list or in an enclosing @code{let}, @code{let*}, | |
31058 | @code{for}, or @code{foreach} form, | |
31059 | or their names already contain a @samp{-} character. Thus a reference to | |
31060 | @samp{foo} is the same as a reference to @samp{var-foo}.@refill | |
31061 | ||
31062 | A few other Lisp extensions are available in @code{defmath} definitions: | |
31063 | ||
31064 | @itemize @bullet | |
31065 | @item | |
31066 | The @code{elt} function accepts any number of index variables. | |
31067 | Note that Calc vectors are stored as Lisp lists whose first | |
31068 | element is the symbol @code{vec}; thus, @samp{(elt v 2)} yields | |
31069 | the second element of vector @code{v}, and @samp{(elt m i j)} | |
31070 | yields one element of a Calc matrix. | |
31071 | ||
31072 | @item | |
31073 | The @code{setq} function has been extended to act like the Common | |
31074 | Lisp @code{setf} function. (The name @code{setf} is recognized as | |
31075 | a synonym of @code{setq}.) Specifically, the first argument of | |
31076 | @code{setq} can be an @code{nth}, @code{elt}, @code{car}, or @code{cdr} form, | |
31077 | in which case the effect is to store into the specified | |
31078 | element of a list. Thus, @samp{(setq (elt m i j) x)} stores @cite{x} | |
31079 | into one element of a matrix. | |
31080 | ||
31081 | @item | |
31082 | A @code{for} looping construct is available. For example, | |
31083 | @samp{(for ((i 0 10)) body)} executes @code{body} once for each | |
31084 | binding of @cite{i} from zero to 10. This is like a @code{let} | |
31085 | form in that @cite{i} is temporarily bound to the loop count | |
31086 | without disturbing its value outside the @code{for} construct. | |
31087 | Nested loops, as in @samp{(for ((i 0 10) (j 0 (1- i) 2)) body)}, | |
31088 | are also available. For each value of @cite{i} from zero to 10, | |
31089 | @cite{j} counts from 0 to @cite{i-1} in steps of two. Note that | |
31090 | @code{for} has the same general outline as @code{let*}, except | |
31091 | that each element of the header is a list of three or four | |
31092 | things, not just two. | |
31093 | ||
31094 | @item | |
31095 | The @code{foreach} construct loops over elements of a list. | |
31096 | For example, @samp{(foreach ((x (cdr v))) body)} executes | |
31097 | @code{body} with @cite{x} bound to each element of Calc vector | |
31098 | @cite{v} in turn. The purpose of @code{cdr} here is to skip over | |
31099 | the initial @code{vec} symbol in the vector. | |
31100 | ||
31101 | @item | |
31102 | The @code{break} function breaks out of the innermost enclosing | |
31103 | @code{while}, @code{for}, or @code{foreach} loop. If given a | |
31104 | value, as in @samp{(break x)}, this value is returned by the | |
31105 | loop. (Lisp loops otherwise always return @code{nil}.) | |
31106 | ||
31107 | @item | |
31108 | The @code{return} function prematurely returns from the enclosing | |
31109 | function. For example, @samp{(return (+ x y))} returns @cite{x+y} | |
31110 | as the value of a function. You can use @code{return} anywhere | |
31111 | inside the body of the function. | |
31112 | @end itemize | |
31113 | ||
31114 | Non-integer numbers (and extremely large integers) cannot be included | |
31115 | directly into a @code{defmath} definition. This is because the Lisp | |
31116 | reader will fail to parse them long before @code{defmath} ever gets control. | |
31117 | Instead, use the notation, @samp{:"3.1415"}. In fact, any algebraic | |
31118 | formula can go between the quotes. For example, | |
31119 | ||
31120 | @smallexample | |
31121 | (defmath sqexp (x) ; sqexp(x) == sqrt(exp(x)) == exp(x*0.5) | |
31122 | (and (numberp x) | |
31123 | (exp :"x * 0.5"))) | |
31124 | @end smallexample | |
31125 | ||
31126 | expands to | |
31127 | ||
31128 | @smallexample | |
31129 | (defun calcFunc-sqexp (x) | |
31130 | (and (math-numberp x) | |
31131 | (calcFunc-exp (math-mul x '(float 5 -1))))) | |
31132 | @end smallexample | |
31133 | ||
31134 | Note the use of @code{numberp} as a guard to ensure that the argument is | |
31135 | a number first, returning @code{nil} if not. The exponential function | |
31136 | could itself have been included in the expression, if we had preferred: | |
31137 | @samp{:"exp(x * 0.5)"}. As another example, the multiplication-and-recursion | |
31138 | step of @code{myfact} could have been written | |
31139 | ||
31140 | @example | |
31141 | :"n * myfact(n-1)" | |
31142 | @end example | |
31143 | ||
31144 | If a file named @file{.emacs} exists in your home directory, Emacs reads | |
31145 | and executes the Lisp forms in this file as it starts up. While it may | |
31146 | seem like a good idea to put your favorite @code{defmath} commands here, | |
31147 | this has the unfortunate side-effect that parts of the Calculator must be | |
31148 | loaded in to process the @code{defmath} commands whether or not you will | |
31149 | actually use the Calculator! A better effect can be had by writing | |
31150 | ||
31151 | @example | |
31152 | (put 'calc-define 'thing '(progn | |
31153 | (defmath ... ) | |
31154 | (defmath ... ) | |
31155 | )) | |
31156 | @end example | |
31157 | ||
31158 | @noindent | |
31159 | @vindex calc-define | |
31160 | The @code{put} function adds a @dfn{property} to a symbol. Each Lisp | |
31161 | symbol has a list of properties associated with it. Here we add a | |
31162 | property with a name of @code{thing} and a @samp{(progn ...)} form as | |
31163 | its value. When Calc starts up, and at the start of every Calc command, | |
31164 | the property list for the symbol @code{calc-define} is checked and the | |
31165 | values of any properties found are evaluated as Lisp forms. The | |
31166 | properties are removed as they are evaluated. The property names | |
31167 | (like @code{thing}) are not used; you should choose something like the | |
31168 | name of your project so as not to conflict with other properties. | |
31169 | ||
31170 | The net effect is that you can put the above code in your @file{.emacs} | |
31171 | file and it will not be executed until Calc is loaded. Or, you can put | |
31172 | that same code in another file which you load by hand either before or | |
31173 | after Calc itself is loaded. | |
31174 | ||
31175 | The properties of @code{calc-define} are evaluated in the same order | |
31176 | that they were added. They can assume that the Calc modules @file{calc.el}, | |
31177 | @file{calc-ext.el}, and @file{calc-macs.el} have been fully loaded, and | |
31178 | that the @samp{*Calculator*} buffer will be the current buffer. | |
31179 | ||
31180 | If your @code{calc-define} property only defines algebraic functions, | |
31181 | you can be sure that it will have been evaluated before Calc tries to | |
31182 | call your function, even if the file defining the property is loaded | |
31183 | after Calc is loaded. But if the property defines commands or key | |
31184 | sequences, it may not be evaluated soon enough. (Suppose it defines the | |
31185 | new command @code{tweak-calc}; the user can load your file, then type | |
31186 | @kbd{M-x tweak-calc} before Calc has had chance to do anything.) To | |
31187 | protect against this situation, you can put | |
31188 | ||
31189 | @example | |
31190 | (run-hooks 'calc-check-defines) | |
31191 | @end example | |
31192 | ||
31193 | @findex calc-check-defines | |
31194 | @noindent | |
31195 | at the end of your file. The @code{calc-check-defines} function is what | |
31196 | looks for and evaluates properties on @code{calc-define}; @code{run-hooks} | |
31197 | has the advantage that it is quietly ignored if @code{calc-check-defines} | |
31198 | is not yet defined because Calc has not yet been loaded. | |
31199 | ||
31200 | Examples of things that ought to be enclosed in a @code{calc-define} | |
31201 | property are @code{defmath} calls, @code{define-key} calls that modify | |
31202 | the Calc key map, and any calls that redefine things defined inside Calc. | |
31203 | Ordinary @code{defun}s need not be enclosed with @code{calc-define}. | |
31204 | ||
31205 | @node Defining Simple Commands, Defining Stack Commands, Defining Functions, Lisp Definitions | |
31206 | @subsection Defining New Simple Commands | |
31207 | ||
31208 | @noindent | |
31209 | @findex interactive | |
31210 | If a @code{defmath} form contains an @code{interactive} clause, it defines | |
31211 | a Calculator command. Actually such a @code{defmath} results in @emph{two} | |
31212 | function definitions: One, a @samp{calcFunc-} function as was just described, | |
31213 | with the @code{interactive} clause removed. Two, a @samp{calc-} function | |
31214 | with a suitable @code{interactive} clause and some sort of wrapper to make | |
31215 | the command work in the Calc environment. | |
31216 | ||
31217 | In the simple case, the @code{interactive} clause has the same form as | |
31218 | for normal Emacs Lisp commands: | |
31219 | ||
31220 | @smallexample | |
31221 | (defmath increase-precision (delta) | |
31222 | "Increase precision by DELTA." ; This is the "documentation string" | |
31223 | (interactive "p") ; Register this as a M-x-able command | |
31224 | (setq calc-internal-prec (+ calc-internal-prec delta))) | |
31225 | @end smallexample | |
31226 | ||
31227 | This expands to the pair of definitions, | |
31228 | ||
31229 | @smallexample | |
31230 | (defun calc-increase-precision (delta) | |
31231 | "Increase precision by DELTA." | |
31232 | (interactive "p") | |
31233 | (calc-wrapper | |
31234 | (setq calc-internal-prec (math-add calc-internal-prec delta)))) | |
31235 | ||
31236 | (defun calcFunc-increase-precision (delta) | |
31237 | "Increase precision by DELTA." | |
31238 | (setq calc-internal-prec (math-add calc-internal-prec delta))) | |
31239 | @end smallexample | |
31240 | ||
31241 | @noindent | |
31242 | where in this case the latter function would never really be used! Note | |
31243 | that since the Calculator stores small integers as plain Lisp integers, | |
31244 | the @code{math-add} function will work just as well as the native | |
31245 | @code{+} even when the intent is to operate on native Lisp integers. | |
31246 | ||
31247 | @findex calc-wrapper | |
31248 | The @samp{calc-wrapper} call invokes a macro which surrounds the body of | |
31249 | the function with code that looks roughly like this: | |
31250 | ||
31251 | @smallexample | |
31252 | (let ((calc-command-flags nil)) | |
31253 | (unwind-protect | |
31254 | (save-excursion | |
31255 | (calc-select-buffer) | |
31256 | @emph{body of function} | |
31257 | @emph{renumber stack} | |
31258 | @emph{clear} Working @emph{message}) | |
31259 | @emph{realign cursor and window} | |
31260 | @emph{clear Inverse, Hyperbolic, and Keep Args flags} | |
31261 | @emph{update Emacs mode line})) | |
31262 | @end smallexample | |
31263 | ||
31264 | @findex calc-select-buffer | |
31265 | The @code{calc-select-buffer} function selects the @samp{*Calculator*} | |
31266 | buffer if necessary, say, because the command was invoked from inside | |
31267 | the @samp{*Calc Trail*} window. | |
31268 | ||
31269 | @findex calc-set-command-flag | |
5d67986c RS |
31270 | You can call, for example, @code{(calc-set-command-flag 'no-align)} to |
31271 | set the above-mentioned command flags. Calc routines recognize the | |
31272 | following command flags: | |
d7b8e6c6 EZ |
31273 | |
31274 | @table @code | |
31275 | @item renum-stack | |
31276 | Stack line numbers @samp{1:}, @samp{2:}, and so on must be renumbered | |
31277 | after this command completes. This is set by routines like | |
31278 | @code{calc-push}. | |
31279 | ||
31280 | @item clear-message | |
31281 | Calc should call @samp{(message "")} if this command completes normally | |
31282 | (to clear a ``Working@dots{}'' message out of the echo area). | |
31283 | ||
31284 | @item no-align | |
31285 | Do not move the cursor back to the @samp{.} top-of-stack marker. | |
31286 | ||
31287 | @item position-point | |
31288 | Use the variables @code{calc-position-point-line} and | |
31289 | @code{calc-position-point-column} to position the cursor after | |
31290 | this command finishes. | |
31291 | ||
31292 | @item keep-flags | |
31293 | Do not clear @code{calc-inverse-flag}, @code{calc-hyperbolic-flag}, | |
31294 | and @code{calc-keep-args-flag} at the end of this command. | |
31295 | ||
31296 | @item do-edit | |
31297 | Switch to buffer @samp{*Calc Edit*} after this command. | |
31298 | ||
31299 | @item hold-trail | |
31300 | Do not move trail pointer to end of trail when something is recorded | |
31301 | there. | |
31302 | @end table | |
31303 | ||
31304 | @kindex Y | |
31305 | @kindex Y ? | |
31306 | @vindex calc-Y-help-msgs | |
31307 | Calc reserves a special prefix key, shift-@kbd{Y}, for user-written | |
31308 | extensions to Calc. There are no built-in commands that work with | |
31309 | this prefix key; you must call @code{define-key} from Lisp (probably | |
31310 | from inside a @code{calc-define} property) to add to it. Initially only | |
31311 | @kbd{Y ?} is defined; it takes help messages from a list of strings | |
31312 | (initially @code{nil}) in the variable @code{calc-Y-help-msgs}. All | |
31313 | other undefined keys except for @kbd{Y} are reserved for use by | |
31314 | future versions of Calc. | |
31315 | ||
31316 | If you are writing a Calc enhancement which you expect to give to | |
31317 | others, it is best to minimize the number of @kbd{Y}-key sequences | |
31318 | you use. In fact, if you have more than one key sequence you should | |
31319 | consider defining three-key sequences with a @kbd{Y}, then a key that | |
31320 | stands for your package, then a third key for the particular command | |
31321 | within your package. | |
31322 | ||
31323 | Users may wish to install several Calc enhancements, and it is possible | |
31324 | that several enhancements will choose to use the same key. In the | |
31325 | example below, a variable @code{inc-prec-base-key} has been defined | |
31326 | to contain the key that identifies the @code{inc-prec} package. Its | |
31327 | value is initially @code{"P"}, but a user can change this variable | |
31328 | if necessary without having to modify the file. | |
31329 | ||
31330 | Here is a complete file, @file{inc-prec.el}, which makes a @kbd{Y P I} | |
31331 | command that increases the precision, and a @kbd{Y P D} command that | |
31332 | decreases the precision. | |
31333 | ||
31334 | @smallexample | |
31335 | ;;; Increase and decrease Calc precision. Dave Gillespie, 5/31/91. | |
31336 | ;;; (Include copyright or copyleft stuff here.) | |
31337 | ||
31338 | (defvar inc-prec-base-key "P" | |
31339 | "Base key for inc-prec.el commands.") | |
31340 | ||
31341 | (put 'calc-define 'inc-prec '(progn | |
31342 | ||
31343 | (define-key calc-mode-map (format "Y%sI" inc-prec-base-key) | |
31344 | 'increase-precision) | |
31345 | (define-key calc-mode-map (format "Y%sD" inc-prec-base-key) | |
31346 | 'decrease-precision) | |
31347 | ||
31348 | (setq calc-Y-help-msgs | |
31349 | (cons (format "%s + Inc-prec, Dec-prec" inc-prec-base-key) | |
31350 | calc-Y-help-msgs)) | |
31351 | ||
31352 | (defmath increase-precision (delta) | |
31353 | "Increase precision by DELTA." | |
31354 | (interactive "p") | |
31355 | (setq calc-internal-prec (+ calc-internal-prec delta))) | |
31356 | ||
31357 | (defmath decrease-precision (delta) | |
31358 | "Decrease precision by DELTA." | |
31359 | (interactive "p") | |
31360 | (setq calc-internal-prec (- calc-internal-prec delta))) | |
31361 | ||
31362 | )) ; end of calc-define property | |
31363 | ||
31364 | (run-hooks 'calc-check-defines) | |
31365 | @end smallexample | |
31366 | ||
31367 | @node Defining Stack Commands, Argument Qualifiers, Defining Simple Commands, Lisp Definitions | |
31368 | @subsection Defining New Stack-Based Commands | |
31369 | ||
31370 | @noindent | |
31371 | To define a new computational command which takes and/or leaves arguments | |
31372 | on the stack, a special form of @code{interactive} clause is used. | |
31373 | ||
31374 | @example | |
31375 | (interactive @var{num} @var{tag}) | |
31376 | @end example | |
31377 | ||
31378 | @noindent | |
31379 | where @var{num} is an integer, and @var{tag} is a string. The effect is | |
31380 | to pop @var{num} values off the stack, resimplify them by calling | |
31381 | @code{calc-normalize}, and hand them to your function according to the | |
31382 | function's argument list. Your function may include @code{&optional} and | |
31383 | @code{&rest} parameters, so long as calling the function with @var{num} | |
31384 | parameters is legal. | |
31385 | ||
31386 | Your function must return either a number or a formula in a form | |
31387 | acceptable to Calc, or a list of such numbers or formulas. These value(s) | |
31388 | are pushed onto the stack when the function completes. They are also | |
31389 | recorded in the Calc Trail buffer on a line beginning with @var{tag}, | |
31390 | a string of (normally) four characters or less. If you omit @var{tag} | |
31391 | or use @code{nil} as a tag, the result is not recorded in the trail. | |
31392 | ||
31393 | As an example, the definition | |
31394 | ||
31395 | @smallexample | |
31396 | (defmath myfact (n) | |
31397 | "Compute the factorial of the integer at the top of the stack." | |
31398 | (interactive 1 "fact") | |
31399 | (if (> n 0) | |
31400 | (* n (myfact (1- n))) | |
31401 | (and (= n 0) 1))) | |
31402 | @end smallexample | |
31403 | ||
31404 | @noindent | |
31405 | is a version of the factorial function shown previously which can be used | |
31406 | as a command as well as an algebraic function. It expands to | |
31407 | ||
31408 | @smallexample | |
31409 | (defun calc-myfact () | |
31410 | "Compute the factorial of the integer at the top of the stack." | |
31411 | (interactive) | |
31412 | (calc-slow-wrapper | |
31413 | (calc-enter-result 1 "fact" | |
31414 | (cons 'calcFunc-myfact (calc-top-list-n 1))))) | |
31415 | ||
31416 | (defun calcFunc-myfact (n) | |
31417 | "Compute the factorial of the integer at the top of the stack." | |
31418 | (if (math-posp n) | |
31419 | (math-mul n (calcFunc-myfact (math-add n -1))) | |
31420 | (and (math-zerop n) 1))) | |
31421 | @end smallexample | |
31422 | ||
31423 | @findex calc-slow-wrapper | |
31424 | The @code{calc-slow-wrapper} function is a version of @code{calc-wrapper} | |
31425 | that automatically puts up a @samp{Working...} message before the | |
31426 | computation begins. (This message can be turned off by the user | |
31427 | with an @kbd{m w} (@code{calc-working}) command.) | |
31428 | ||
31429 | @findex calc-top-list-n | |
31430 | The @code{calc-top-list-n} function returns a list of the specified number | |
31431 | of values from the top of the stack. It resimplifies each value by | |
31432 | calling @code{calc-normalize}. If its argument is zero it returns an | |
31433 | empty list. It does not actually remove these values from the stack. | |
31434 | ||
31435 | @findex calc-enter-result | |
31436 | The @code{calc-enter-result} function takes an integer @var{num} and string | |
31437 | @var{tag} as described above, plus a third argument which is either a | |
31438 | Calculator data object or a list of such objects. These objects are | |
31439 | resimplified and pushed onto the stack after popping the specified number | |
31440 | of values from the stack. If @var{tag} is non-@code{nil}, the values | |
31441 | being pushed are also recorded in the trail. | |
31442 | ||
31443 | Note that if @code{calcFunc-myfact} returns @code{nil} this represents | |
31444 | ``leave the function in symbolic form.'' To return an actual empty list, | |
31445 | in the sense that @code{calc-enter-result} will push zero elements back | |
31446 | onto the stack, you should return the special value @samp{'(nil)}, a list | |
31447 | containing the single symbol @code{nil}. | |
31448 | ||
31449 | The @code{interactive} declaration can actually contain a limited | |
31450 | Emacs-style code string as well which comes just before @var{num} and | |
31451 | @var{tag}. Currently the only Emacs code supported is @samp{"p"}, as in | |
31452 | ||
31453 | @example | |
31454 | (defmath foo (a b &optional c) | |
31455 | (interactive "p" 2 "foo") | |
31456 | @var{body}) | |
31457 | @end example | |
31458 | ||
31459 | In this example, the command @code{calc-foo} will evaluate the expression | |
31460 | @samp{foo(a,b)} if executed with no argument, or @samp{foo(a,b,n)} if | |
31461 | executed with a numeric prefix argument of @cite{n}. | |
31462 | ||
31463 | The other code string allowed is @samp{"m"} (unrelated to the usual @samp{"m"} | |
31464 | code as used with @code{defun}). It uses the numeric prefix argument as the | |
31465 | number of objects to remove from the stack and pass to the function. | |
31466 | In this case, the integer @var{num} serves as a default number of | |
31467 | arguments to be used when no prefix is supplied. | |
31468 | ||
31469 | @node Argument Qualifiers, Example Definitions, Defining Stack Commands, Lisp Definitions | |
31470 | @subsection Argument Qualifiers | |
31471 | ||
31472 | @noindent | |
31473 | Anywhere a parameter name can appear in the parameter list you can also use | |
31474 | an @dfn{argument qualifier}. Thus the general form of a definition is: | |
31475 | ||
31476 | @example | |
31477 | (defmath @var{name} (@var{param} @var{param...} | |
31478 | &optional @var{param} @var{param...} | |
31479 | &rest @var{param}) | |
31480 | @var{body}) | |
31481 | @end example | |
31482 | ||
31483 | @noindent | |
31484 | where each @var{param} is either a symbol or a list of the form | |
31485 | ||
31486 | @example | |
31487 | (@var{qual} @var{param}) | |
31488 | @end example | |
31489 | ||
31490 | The following qualifiers are recognized: | |
31491 | ||
31492 | @table @samp | |
31493 | @item complete | |
31494 | @findex complete | |
31495 | The argument must not be an incomplete vector, interval, or complex number. | |
31496 | (This is rarely needed since the Calculator itself will never call your | |
31497 | function with an incomplete argument. But there is nothing stopping your | |
31498 | own Lisp code from calling your function with an incomplete argument.)@refill | |
31499 | ||
31500 | @item integer | |
31501 | @findex integer | |
31502 | The argument must be an integer. If it is an integer-valued float | |
31503 | it will be accepted but converted to integer form. Non-integers and | |
31504 | formulas are rejected. | |
31505 | ||
31506 | @item natnum | |
31507 | @findex natnum | |
31508 | Like @samp{integer}, but the argument must be non-negative. | |
31509 | ||
31510 | @item fixnum | |
31511 | @findex fixnum | |
31512 | Like @samp{integer}, but the argument must fit into a native Lisp integer, | |
31513 | which on most systems means less than 2^23 in absolute value. The | |
31514 | argument is converted into Lisp-integer form if necessary. | |
31515 | ||
31516 | @item float | |
31517 | @findex float | |
31518 | The argument is converted to floating-point format if it is a number or | |
31519 | vector. If it is a formula it is left alone. (The argument is never | |
31520 | actually rejected by this qualifier.) | |
31521 | ||
31522 | @item @var{pred} | |
31523 | The argument must satisfy predicate @var{pred}, which is one of the | |
31524 | standard Calculator predicates. @xref{Predicates}. | |
31525 | ||
31526 | @item not-@var{pred} | |
31527 | The argument must @emph{not} satisfy predicate @var{pred}. | |
31528 | @end table | |
31529 | ||
31530 | For example, | |
31531 | ||
31532 | @example | |
31533 | (defmath foo (a (constp (not-matrixp b)) &optional (float c) | |
31534 | &rest (integer d)) | |
31535 | @var{body}) | |
31536 | @end example | |
31537 | ||
31538 | @noindent | |
31539 | expands to | |
31540 | ||
31541 | @example | |
31542 | (defun calcFunc-foo (a b &optional c &rest d) | |
31543 | (and (math-matrixp b) | |
31544 | (math-reject-arg b 'not-matrixp)) | |
31545 | (or (math-constp b) | |
31546 | (math-reject-arg b 'constp)) | |
31547 | (and c (setq c (math-check-float c))) | |
31548 | (setq d (mapcar 'math-check-integer d)) | |
31549 | @var{body}) | |
31550 | @end example | |
31551 | ||
31552 | @noindent | |
31553 | which performs the necessary checks and conversions before executing the | |
31554 | body of the function. | |
31555 | ||
31556 | @node Example Definitions, Calling Calc from Your Programs, Argument Qualifiers, Lisp Definitions | |
31557 | @subsection Example Definitions | |
31558 | ||
31559 | @noindent | |
31560 | This section includes some Lisp programming examples on a larger scale. | |
31561 | These programs make use of some of the Calculator's internal functions; | |
31562 | @pxref{Internals}. | |
31563 | ||
31564 | @menu | |
31565 | * Bit Counting Example:: | |
31566 | * Sine Example:: | |
31567 | @end menu | |
31568 | ||
31569 | @node Bit Counting Example, Sine Example, Example Definitions, Example Definitions | |
31570 | @subsubsection Bit-Counting | |
31571 | ||
31572 | @noindent | |
5d67986c RS |
31573 | @ignore |
31574 | @starindex | |
31575 | @end ignore | |
d7b8e6c6 EZ |
31576 | @tindex bcount |
31577 | Calc does not include a built-in function for counting the number of | |
31578 | ``one'' bits in a binary integer. It's easy to invent one using @kbd{b u} | |
31579 | to convert the integer to a set, and @kbd{V #} to count the elements of | |
31580 | that set; let's write a function that counts the bits without having to | |
31581 | create an intermediate set. | |
31582 | ||
31583 | @smallexample | |
31584 | (defmath bcount ((natnum n)) | |
31585 | (interactive 1 "bcnt") | |
31586 | (let ((count 0)) | |
31587 | (while (> n 0) | |
31588 | (if (oddp n) | |
31589 | (setq count (1+ count))) | |
31590 | (setq n (lsh n -1))) | |
31591 | count)) | |
31592 | @end smallexample | |
31593 | ||
31594 | @noindent | |
31595 | When this is expanded by @code{defmath}, it will become the following | |
31596 | Emacs Lisp function: | |
31597 | ||
31598 | @smallexample | |
31599 | (defun calcFunc-bcount (n) | |
31600 | (setq n (math-check-natnum n)) | |
31601 | (let ((count 0)) | |
31602 | (while (math-posp n) | |
31603 | (if (math-oddp n) | |
31604 | (setq count (math-add count 1))) | |
31605 | (setq n (calcFunc-lsh n -1))) | |
31606 | count)) | |
31607 | @end smallexample | |
31608 | ||
31609 | If the input numbers are large, this function involves a fair amount | |
31610 | of arithmetic. A binary right shift is essentially a division by two; | |
31611 | recall that Calc stores integers in decimal form so bit shifts must | |
31612 | involve actual division. | |
31613 | ||
31614 | To gain a bit more efficiency, we could divide the integer into | |
5d67986c | 31615 | @var{n}-bit chunks, each of which can be handled quickly because |
d7b8e6c6 EZ |
31616 | they fit into Lisp integers. It turns out that Calc's arithmetic |
31617 | routines are especially fast when dividing by an integer less than | |
5d67986c | 31618 | 1000, so we can set @var{n = 9} bits and use repeated division by 512: |
d7b8e6c6 EZ |
31619 | |
31620 | @smallexample | |
31621 | (defmath bcount ((natnum n)) | |
31622 | (interactive 1 "bcnt") | |
31623 | (let ((count 0)) | |
31624 | (while (not (fixnump n)) | |
31625 | (let ((qr (idivmod n 512))) | |
31626 | (setq count (+ count (bcount-fixnum (cdr qr))) | |
31627 | n (car qr)))) | |
31628 | (+ count (bcount-fixnum n)))) | |
31629 | ||
31630 | (defun bcount-fixnum (n) | |
31631 | (let ((count 0)) | |
31632 | (while (> n 0) | |
31633 | (setq count (+ count (logand n 1)) | |
31634 | n (lsh n -1))) | |
31635 | count)) | |
31636 | @end smallexample | |
31637 | ||
31638 | @noindent | |
31639 | Note that the second function uses @code{defun}, not @code{defmath}. | |
31640 | Because this function deals only with native Lisp integers (``fixnums''), | |
31641 | it can use the actual Emacs @code{+} and related functions rather | |
31642 | than the slower but more general Calc equivalents which @code{defmath} | |
31643 | uses. | |
31644 | ||
31645 | The @code{idivmod} function does an integer division, returning both | |
31646 | the quotient and the remainder at once. Again, note that while it | |
31647 | might seem that @samp{(logand n 511)} and @samp{(lsh n -9)} are | |
31648 | more efficient ways to split off the bottom nine bits of @code{n}, | |
31649 | actually they are less efficient because each operation is really | |
31650 | a division by 512 in disguise; @code{idivmod} allows us to do the | |
31651 | same thing with a single division by 512. | |
31652 | ||
31653 | @node Sine Example, , Bit Counting Example, Example Definitions | |
31654 | @subsubsection The Sine Function | |
31655 | ||
31656 | @noindent | |
5d67986c RS |
31657 | @ignore |
31658 | @starindex | |
31659 | @end ignore | |
d7b8e6c6 EZ |
31660 | @tindex mysin |
31661 | A somewhat limited sine function could be defined as follows, using the | |
31662 | well-known Taylor series expansion for @c{$\sin x$} | |
31663 | @samp{sin(x)}: | |
31664 | ||
31665 | @smallexample | |
31666 | (defmath mysin ((float (anglep x))) | |
31667 | (interactive 1 "mysn") | |
31668 | (setq x (to-radians x)) ; Convert from current angular mode. | |
31669 | (let ((sum x) ; Initial term of Taylor expansion of sin. | |
31670 | newsum | |
31671 | (nfact 1) ; "nfact" equals "n" factorial at all times. | |
31672 | (xnegsqr :"-(x^2)")) ; "xnegsqr" equals -x^2. | |
31673 | (for ((n 3 100 2)) ; Upper limit of 100 is a good precaution. | |
31674 | (working "mysin" sum) ; Display "Working" message, if enabled. | |
31675 | (setq nfact (* nfact (1- n) n) | |
31676 | x (* x xnegsqr) | |
31677 | newsum (+ sum (/ x nfact))) | |
31678 | (if (~= newsum sum) ; If newsum is "nearly equal to" sum, | |
31679 | (break)) ; then we are done. | |
31680 | (setq sum newsum)) | |
31681 | sum)) | |
31682 | @end smallexample | |
31683 | ||
31684 | The actual @code{sin} function in Calc works by first reducing the problem | |
31685 | to a sine or cosine of a nonnegative number less than @c{$\pi \over 4$} | |
31686 | @cite{pi/4}. This | |
31687 | ensures that the Taylor series will converge quickly. Also, the calculation | |
31688 | is carried out with two extra digits of precision to guard against cumulative | |
31689 | round-off in @samp{sum}. Finally, complex arguments are allowed and handled | |
31690 | by a separate algorithm. | |
31691 | ||
31692 | @smallexample | |
31693 | (defmath mysin ((float (scalarp x))) | |
31694 | (interactive 1 "mysn") | |
31695 | (setq x (to-radians x)) ; Convert from current angular mode. | |
31696 | (with-extra-prec 2 ; Evaluate with extra precision. | |
31697 | (cond ((complexp x) | |
31698 | (mysin-complex x)) | |
31699 | ((< x 0) | |
31700 | (- (mysin-raw (- x))) ; Always call mysin-raw with x >= 0. | |
31701 | (t (mysin-raw x)))))) | |
31702 | ||
31703 | (defmath mysin-raw (x) | |
31704 | (cond ((>= x 7) | |
31705 | (mysin-raw (% x (two-pi)))) ; Now x < 7. | |
31706 | ((> x (pi-over-2)) | |
31707 | (- (mysin-raw (- x (pi))))) ; Now -pi/2 <= x <= pi/2. | |
31708 | ((> x (pi-over-4)) | |
31709 | (mycos-raw (- x (pi-over-2)))) ; Now -pi/2 <= x <= pi/4. | |
31710 | ((< x (- (pi-over-4))) | |
31711 | (- (mycos-raw (+ x (pi-over-2))))) ; Now -pi/4 <= x <= pi/4, | |
31712 | (t (mysin-series x)))) ; so the series will be efficient. | |
31713 | @end smallexample | |
31714 | ||
31715 | @noindent | |
31716 | where @code{mysin-complex} is an appropriate function to handle complex | |
31717 | numbers, @code{mysin-series} is the routine to compute the sine Taylor | |
31718 | series as before, and @code{mycos-raw} is a function analogous to | |
31719 | @code{mysin-raw} for cosines. | |
31720 | ||
31721 | The strategy is to ensure that @cite{x} is nonnegative before calling | |
31722 | @code{mysin-raw}. This function then recursively reduces its argument | |
31723 | to a suitable range, namely, plus-or-minus @c{$\pi \over 4$} | |
31724 | @cite{pi/4}. Note that each | |
31725 | test, and particularly the first comparison against 7, is designed so | |
28665d46 | 31726 | that small roundoff errors cannot produce an infinite loop. (Suppose |
d7b8e6c6 EZ |
31727 | we compared with @samp{(two-pi)} instead; if due to roundoff problems |
31728 | the modulo operator ever returned @samp{(two-pi)} exactly, an infinite | |
31729 | recursion could result!) We use modulo only for arguments that will | |
31730 | clearly get reduced, knowing that the next rule will catch any reductions | |
31731 | that this rule misses. | |
31732 | ||
31733 | If a program is being written for general use, it is important to code | |
31734 | it carefully as shown in this second example. For quick-and-dirty programs, | |
31735 | when you know that your own use of the sine function will never encounter | |
31736 | a large argument, a simpler program like the first one shown is fine. | |
31737 | ||
31738 | @node Calling Calc from Your Programs, Internals, Example Definitions, Lisp Definitions | |
31739 | @subsection Calling Calc from Your Lisp Programs | |
31740 | ||
31741 | @noindent | |
31742 | A later section (@pxref{Internals}) gives a full description of | |
31743 | Calc's internal Lisp functions. It's not hard to call Calc from | |
31744 | inside your programs, but the number of these functions can be daunting. | |
31745 | So Calc provides one special ``programmer-friendly'' function called | |
31746 | @code{calc-eval} that can be made to do just about everything you | |
31747 | need. It's not as fast as the low-level Calc functions, but it's | |
31748 | much simpler to use! | |
31749 | ||
31750 | It may seem that @code{calc-eval} itself has a daunting number of | |
31751 | options, but they all stem from one simple operation. | |
31752 | ||
31753 | In its simplest manifestation, @samp{(calc-eval "1+2")} parses the | |
31754 | string @code{"1+2"} as if it were a Calc algebraic entry and returns | |
31755 | the result formatted as a string: @code{"3"}. | |
31756 | ||
31757 | Since @code{calc-eval} is on the list of recommended @code{autoload} | |
31758 | functions, you don't need to make any special preparations to load | |
31759 | Calc before calling @code{calc-eval} the first time. Calc will be | |
31760 | loaded and initialized for you. | |
31761 | ||
31762 | All the Calc modes that are currently in effect will be used when | |
31763 | evaluating the expression and formatting the result. | |
31764 | ||
31765 | @ifinfo | |
31766 | @example | |
31767 | ||
31768 | @end example | |
31769 | @end ifinfo | |
31770 | @subsubsection Additional Arguments to @code{calc-eval} | |
31771 | ||
31772 | @noindent | |
31773 | If the input string parses to a list of expressions, Calc returns | |
31774 | the results separated by @code{", "}. You can specify a different | |
31775 | separator by giving a second string argument to @code{calc-eval}: | |
31776 | @samp{(calc-eval "1+2,3+4" ";")} returns @code{"3;7"}. | |
31777 | ||
31778 | The ``separator'' can also be any of several Lisp symbols which | |
31779 | request other behaviors from @code{calc-eval}. These are discussed | |
31780 | one by one below. | |
31781 | ||
31782 | You can give additional arguments to be substituted for | |
31783 | @samp{$}, @samp{$$}, and so on in the main expression. For | |
31784 | example, @samp{(calc-eval "$/$$" nil "7" "1+1")} evaluates the | |
31785 | expression @code{"7/(1+1)"} to yield the result @code{"3.5"} | |
31786 | (assuming Fraction mode is not in effect). Note the @code{nil} | |
31787 | used as a placeholder for the item-separator argument. | |
31788 | ||
31789 | @ifinfo | |
31790 | @example | |
31791 | ||
31792 | @end example | |
31793 | @end ifinfo | |
31794 | @subsubsection Error Handling | |
31795 | ||
31796 | @noindent | |
31797 | If @code{calc-eval} encounters an error, it returns a list containing | |
31798 | the character position of the error, plus a suitable message as a | |
31799 | string. Note that @samp{1 / 0} is @emph{not} an error by Calc's | |
31800 | standards; it simply returns the string @code{"1 / 0"} which is the | |
31801 | division left in symbolic form. But @samp{(calc-eval "1/")} will | |
31802 | return the list @samp{(2 "Expected a number")}. | |
31803 | ||
31804 | If you bind the variable @code{calc-eval-error} to @code{t} | |
31805 | using a @code{let} form surrounding the call to @code{calc-eval}, | |
31806 | errors instead call the Emacs @code{error} function which aborts | |
31807 | to the Emacs command loop with a beep and an error message. | |
31808 | ||
31809 | If you bind this variable to the symbol @code{string}, error messages | |
31810 | are returned as strings instead of lists. The character position is | |
31811 | ignored. | |
31812 | ||
31813 | As a courtesy to other Lisp code which may be using Calc, be sure | |
31814 | to bind @code{calc-eval-error} using @code{let} rather than changing | |
31815 | it permanently with @code{setq}. | |
31816 | ||
31817 | @ifinfo | |
31818 | @example | |
31819 | ||
31820 | @end example | |
31821 | @end ifinfo | |
31822 | @subsubsection Numbers Only | |
31823 | ||
31824 | @noindent | |
31825 | Sometimes it is preferable to treat @samp{1 / 0} as an error | |
31826 | rather than returning a symbolic result. If you pass the symbol | |
31827 | @code{num} as the second argument to @code{calc-eval}, results | |
31828 | that are not constants are treated as errors. The error message | |
31829 | reported is the first @code{calc-why} message if there is one, | |
31830 | or otherwise ``Number expected.'' | |
31831 | ||
31832 | A result is ``constant'' if it is a number, vector, or other | |
31833 | object that does not include variables or function calls. If it | |
31834 | is a vector, the components must themselves be constants. | |
31835 | ||
31836 | @ifinfo | |
31837 | @example | |
31838 | ||
31839 | @end example | |
31840 | @end ifinfo | |
31841 | @subsubsection Default Modes | |
31842 | ||
31843 | @noindent | |
31844 | If the first argument to @code{calc-eval} is a list whose first | |
31845 | element is a formula string, then @code{calc-eval} sets all the | |
31846 | various Calc modes to their default values while the formula is | |
31847 | evaluated and formatted. For example, the precision is set to 12 | |
31848 | digits, digit grouping is turned off, and the normal language | |
31849 | mode is used. | |
31850 | ||
31851 | This same principle applies to the other options discussed below. | |
31852 | If the first argument would normally be @var{x}, then it can also | |
31853 | be the list @samp{(@var{x})} to use the default mode settings. | |
31854 | ||
31855 | If there are other elements in the list, they are taken as | |
31856 | variable-name/value pairs which override the default mode | |
31857 | settings. Look at the documentation at the front of the | |
31858 | @file{calc.el} file to find the names of the Lisp variables for | |
31859 | the various modes. The mode settings are restored to their | |
31860 | original values when @code{calc-eval} is done. | |
31861 | ||
31862 | For example, @samp{(calc-eval '("$+$$" calc-internal-prec 8) 'num a b)} | |
31863 | computes the sum of two numbers, requiring a numeric result, and | |
31864 | using default mode settings except that the precision is 8 instead | |
31865 | of the default of 12. | |
31866 | ||
31867 | It's usually best to use this form of @code{calc-eval} unless your | |
31868 | program actually considers the interaction with Calc's mode settings | |
31869 | to be a feature. This will avoid all sorts of potential ``gotchas''; | |
31870 | consider what happens with @samp{(calc-eval "sqrt(2)" 'num)} | |
31871 | when the user has left Calc in symbolic mode or no-simplify mode. | |
31872 | ||
31873 | As another example, @samp{(equal (calc-eval '("$<$$") nil a b) "1")} | |
31874 | checks if the number in string @cite{a} is less than the one in | |
31875 | string @cite{b}. Without using a list, the integer 1 might | |
31876 | come out in a variety of formats which would be hard to test for | |
31877 | conveniently: @code{"1"}, @code{"8#1"}, @code{"00001"}. (But | |
31878 | see ``Predicates'' mode, below.) | |
31879 | ||
31880 | @ifinfo | |
31881 | @example | |
31882 | ||
31883 | @end example | |
31884 | @end ifinfo | |
31885 | @subsubsection Raw Numbers | |
31886 | ||
31887 | @noindent | |
31888 | Normally all input and output for @code{calc-eval} is done with strings. | |
31889 | You can do arithmetic with, say, @samp{(calc-eval "$+$$" nil a b)} | |
31890 | in place of @samp{(+ a b)}, but this is very inefficient since the | |
31891 | numbers must be converted to and from string format as they are passed | |
31892 | from one @code{calc-eval} to the next. | |
31893 | ||
31894 | If the separator is the symbol @code{raw}, the result will be returned | |
31895 | as a raw Calc data structure rather than a string. You can read about | |
31896 | how these objects look in the following sections, but usually you can | |
31897 | treat them as ``black box'' objects with no important internal | |
31898 | structure. | |
31899 | ||
31900 | There is also a @code{rawnum} symbol, which is a combination of | |
28665d46 | 31901 | @code{raw} (returning a raw Calc object) and @code{num} (signaling |
d7b8e6c6 EZ |
31902 | an error if that object is not a constant). |
31903 | ||
31904 | You can pass a raw Calc object to @code{calc-eval} in place of a | |
31905 | string, either as the formula itself or as one of the @samp{$} | |
31906 | arguments. Thus @samp{(calc-eval "$+$$" 'raw a b)} is an | |
31907 | addition function that operates on raw Calc objects. Of course | |
31908 | in this case it would be easier to call the low-level @code{math-add} | |
31909 | function in Calc, if you can remember its name. | |
31910 | ||
31911 | In particular, note that a plain Lisp integer is acceptable to Calc | |
31912 | as a raw object. (All Lisp integers are accepted on input, but | |
31913 | integers of more than six decimal digits are converted to ``big-integer'' | |
31914 | form for output. @xref{Data Type Formats}.) | |
31915 | ||
31916 | When it comes time to display the object, just use @samp{(calc-eval a)} | |
31917 | to format it as a string. | |
31918 | ||
31919 | It is an error if the input expression evaluates to a list of | |
31920 | values. The separator symbol @code{list} is like @code{raw} | |
31921 | except that it returns a list of one or more raw Calc objects. | |
31922 | ||
31923 | Note that a Lisp string is not a valid Calc object, nor is a list | |
31924 | containing a string. Thus you can still safely distinguish all the | |
31925 | various kinds of error returns discussed above. | |
31926 | ||
31927 | @ifinfo | |
31928 | @example | |
31929 | ||
31930 | @end example | |
31931 | @end ifinfo | |
31932 | @subsubsection Predicates | |
31933 | ||
31934 | @noindent | |
31935 | If the separator symbol is @code{pred}, the result of the formula is | |
31936 | treated as a true/false value; @code{calc-eval} returns @code{t} or | |
31937 | @code{nil}, respectively. A value is considered ``true'' if it is a | |
31938 | non-zero number, or false if it is zero or if it is not a number. | |
31939 | ||
31940 | For example, @samp{(calc-eval "$<$$" 'pred a b)} tests whether | |
31941 | one value is less than another. | |
31942 | ||
31943 | As usual, it is also possible for @code{calc-eval} to return one of | |
31944 | the error indicators described above. Lisp will interpret such an | |
31945 | indicator as ``true'' if you don't check for it explicitly. If you | |
31946 | wish to have an error register as ``false'', use something like | |
31947 | @samp{(eq (calc-eval ...) t)}. | |
31948 | ||
31949 | @ifinfo | |
31950 | @example | |
31951 | ||
31952 | @end example | |
31953 | @end ifinfo | |
31954 | @subsubsection Variable Values | |
31955 | ||
31956 | @noindent | |
31957 | Variables in the formula passed to @code{calc-eval} are not normally | |
31958 | replaced by their values. If you wish this, you can use the | |
31959 | @code{evalv} function (@pxref{Algebraic Manipulation}). For example, | |
31960 | if 4 is stored in Calc variable @code{a} (i.e., in Lisp variable | |
31961 | @code{var-a}), then @samp{(calc-eval "a+pi")} will return the | |
31962 | formula @code{"a + pi"}, but @samp{(calc-eval "evalv(a+pi)")} | |
31963 | will return @code{"7.14159265359"}. | |
31964 | ||
31965 | To store in a Calc variable, just use @code{setq} to store in the | |
31966 | corresponding Lisp variable. (This is obtained by prepending | |
31967 | @samp{var-} to the Calc variable name.) Calc routines will | |
31968 | understand either string or raw form values stored in variables, | |
31969 | although raw data objects are much more efficient. For example, | |
31970 | to increment the Calc variable @code{a}: | |
31971 | ||
31972 | @example | |
31973 | (setq var-a (calc-eval "evalv(a+1)" 'raw)) | |
31974 | @end example | |
31975 | ||
31976 | @ifinfo | |
31977 | @example | |
31978 | ||
31979 | @end example | |
31980 | @end ifinfo | |
31981 | @subsubsection Stack Access | |
31982 | ||
31983 | @noindent | |
31984 | If the separator symbol is @code{push}, the formula argument is | |
31985 | evaluated (with possible @samp{$} expansions, as usual). The | |
31986 | result is pushed onto the Calc stack. The return value is @code{nil} | |
31987 | (unless there is an error from evaluating the formula, in which | |
31988 | case the return value depends on @code{calc-eval-error} in the | |
31989 | usual way). | |
31990 | ||
31991 | If the separator symbol is @code{pop}, the first argument to | |
31992 | @code{calc-eval} must be an integer instead of a string. That | |
31993 | many values are popped from the stack and thrown away. A negative | |
31994 | argument deletes the entry at that stack level. The return value | |
31995 | is the number of elements remaining in the stack after popping; | |
31996 | @samp{(calc-eval 0 'pop)} is a good way to measure the size of | |
31997 | the stack. | |
31998 | ||
31999 | If the separator symbol is @code{top}, the first argument to | |
32000 | @code{calc-eval} must again be an integer. The value at that | |
32001 | stack level is formatted as a string and returned. Thus | |
32002 | @samp{(calc-eval 1 'top)} returns the top-of-stack value. If the | |
32003 | integer is out of range, @code{nil} is returned. | |
32004 | ||
32005 | The separator symbol @code{rawtop} is just like @code{top} except | |
32006 | that the stack entry is returned as a raw Calc object instead of | |
32007 | as a string. | |
32008 | ||
32009 | In all of these cases the first argument can be made a list in | |
32010 | order to force the default mode settings, as described above. | |
32011 | Thus @samp{(calc-eval '(2 calc-number-radix 16) 'top)} returns the | |
32012 | second-to-top stack entry, formatted as a string using the default | |
32013 | instead of current display modes, except that the radix is | |
32014 | hexadecimal instead of decimal. | |
32015 | ||
32016 | It is, of course, polite to put the Calc stack back the way you | |
32017 | found it when you are done, unless the user of your program is | |
32018 | actually expecting it to affect the stack. | |
32019 | ||
32020 | Note that you do not actually have to switch into the @samp{*Calculator*} | |
32021 | buffer in order to use @code{calc-eval}; it temporarily switches into | |
32022 | the stack buffer if necessary. | |
32023 | ||
32024 | @ifinfo | |
32025 | @example | |
32026 | ||
32027 | @end example | |
32028 | @end ifinfo | |
32029 | @subsubsection Keyboard Macros | |
32030 | ||
32031 | @noindent | |
32032 | If the separator symbol is @code{macro}, the first argument must be a | |
32033 | string of characters which Calc can execute as a sequence of keystrokes. | |
32034 | This switches into the Calc buffer for the duration of the macro. | |
32035 | For example, @samp{(calc-eval "vx5\rVR+" 'macro)} pushes the | |
32036 | vector @samp{[1,2,3,4,5]} on the stack and then replaces it | |
32037 | with the sum of those numbers. Note that @samp{\r} is the Lisp | |
32038 | notation for the carriage-return, @key{RET}, character. | |
32039 | ||
32040 | If your keyboard macro wishes to pop the stack, @samp{\C-d} is | |
32041 | safer than @samp{\177} (the @key{DEL} character) because some | |
32042 | installations may have switched the meanings of @key{DEL} and | |
32043 | @kbd{C-h}. Calc always interprets @kbd{C-d} as a synonym for | |
32044 | ``pop-stack'' regardless of key mapping. | |
32045 | ||
32046 | If you provide a third argument to @code{calc-eval}, evaluation | |
32047 | of the keyboard macro will leave a record in the Trail using | |
32048 | that argument as a tag string. Normally the Trail is unaffected. | |
32049 | ||
32050 | The return value in this case is always @code{nil}. | |
32051 | ||
32052 | @ifinfo | |
32053 | @example | |
32054 | ||
32055 | @end example | |
32056 | @end ifinfo | |
32057 | @subsubsection Lisp Evaluation | |
32058 | ||
32059 | @noindent | |
32060 | Finally, if the separator symbol is @code{eval}, then the Lisp | |
32061 | @code{eval} function is called on the first argument, which must | |
32062 | be a Lisp expression rather than a Calc formula. Remember to | |
32063 | quote the expression so that it is not evaluated until inside | |
32064 | @code{calc-eval}. | |
32065 | ||
32066 | The difference from plain @code{eval} is that @code{calc-eval} | |
32067 | switches to the Calc buffer before evaluating the expression. | |
32068 | For example, @samp{(calc-eval '(setq calc-internal-prec 17) 'eval)} | |
32069 | will correctly affect the buffer-local Calc precision variable. | |
32070 | ||
32071 | An alternative would be @samp{(calc-eval '(calc-precision 17) 'eval)}. | |
32072 | This is evaluating a call to the function that is normally invoked | |
32073 | by the @kbd{p} key, giving it 17 as its ``numeric prefix argument.'' | |
32074 | Note that this function will leave a message in the echo area as | |
32075 | a side effect. Also, all Calc functions switch to the Calc buffer | |
32076 | automatically if not invoked from there, so the above call is | |
32077 | also equivalent to @samp{(calc-precision 17)} by itself. | |
32078 | In all cases, Calc uses @code{save-excursion} to switch back to | |
32079 | your original buffer when it is done. | |
32080 | ||
32081 | As usual the first argument can be a list that begins with a Lisp | |
32082 | expression to use default instead of current mode settings. | |
32083 | ||
32084 | The result of @code{calc-eval} in this usage is just the result | |
32085 | returned by the evaluated Lisp expression. | |
32086 | ||
32087 | @ifinfo | |
32088 | @example | |
32089 | ||
32090 | @end example | |
32091 | @end ifinfo | |
32092 | @subsubsection Example | |
32093 | ||
32094 | @noindent | |
32095 | @findex convert-temp | |
32096 | Here is a sample Emacs command that uses @code{calc-eval}. Suppose | |
32097 | you have a document with lots of references to temperatures on the | |
32098 | Fahrenheit scale, say ``98.6 F'', and you wish to convert these | |
32099 | references to Centigrade. The following command does this conversion. | |
32100 | Place the Emacs cursor right after the letter ``F'' and invoke the | |
32101 | command to change ``98.6 F'' to ``37 C''. Or, if the temperature is | |
32102 | already in Centigrade form, the command changes it back to Fahrenheit. | |
32103 | ||
32104 | @example | |
32105 | (defun convert-temp () | |
32106 | (interactive) | |
32107 | (save-excursion | |
32108 | (re-search-backward "[^-.0-9]\\([-.0-9]+\\) *\\([FC]\\)") | |
32109 | (let* ((top1 (match-beginning 1)) | |
32110 | (bot1 (match-end 1)) | |
32111 | (number (buffer-substring top1 bot1)) | |
32112 | (top2 (match-beginning 2)) | |
32113 | (bot2 (match-end 2)) | |
32114 | (type (buffer-substring top2 bot2))) | |
32115 | (if (equal type "F") | |
32116 | (setq type "C" | |
32117 | number (calc-eval "($ - 32)*5/9" nil number)) | |
32118 | (setq type "F" | |
32119 | number (calc-eval "$*9/5 + 32" nil number))) | |
32120 | (goto-char top2) | |
32121 | (delete-region top2 bot2) | |
32122 | (insert-before-markers type) | |
32123 | (goto-char top1) | |
32124 | (delete-region top1 bot1) | |
32125 | (if (string-match "\\.$" number) ; change "37." to "37" | |
32126 | (setq number (substring number 0 -1))) | |
32127 | (insert number)))) | |
32128 | @end example | |
32129 | ||
32130 | Note the use of @code{insert-before-markers} when changing between | |
32131 | ``F'' and ``C'', so that the character winds up before the cursor | |
32132 | instead of after it. | |
32133 | ||
32134 | @node Internals, , Calling Calc from Your Programs, Lisp Definitions | |
32135 | @subsection Calculator Internals | |
32136 | ||
32137 | @noindent | |
32138 | This section describes the Lisp functions defined by the Calculator that | |
32139 | may be of use to user-written Calculator programs (as described in the | |
32140 | rest of this chapter). These functions are shown by their names as they | |
32141 | conventionally appear in @code{defmath}. Their full Lisp names are | |
32142 | generally gotten by prepending @samp{calcFunc-} or @samp{math-} to their | |
32143 | apparent names. (Names that begin with @samp{calc-} are already in | |
32144 | their full Lisp form.) You can use the actual full names instead if you | |
32145 | prefer them, or if you are calling these functions from regular Lisp. | |
32146 | ||
32147 | The functions described here are scattered throughout the various | |
32148 | Calc component files. Note that @file{calc.el} includes @code{autoload}s | |
32149 | for only a few component files; when Calc wants to call an advanced | |
32150 | function it calls @samp{(calc-extensions)} first; this function | |
32151 | autoloads @file{calc-ext.el}, which in turn autoloads all the functions | |
32152 | in the remaining component files. | |
32153 | ||
32154 | Because @code{defmath} itself uses the extensions, user-written code | |
32155 | generally always executes with the extensions already loaded, so | |
32156 | normally you can use any Calc function and be confident that it will | |
32157 | be autoloaded for you when necessary. If you are doing something | |
32158 | special, check carefully to make sure each function you are using is | |
32159 | from @file{calc.el} or its components, and call @samp{(calc-extensions)} | |
32160 | before using any function based in @file{calc-ext.el} if you can't | |
32161 | prove this file will already be loaded. | |
32162 | ||
32163 | @menu | |
32164 | * Data Type Formats:: | |
32165 | * Interactive Lisp Functions:: | |
32166 | * Stack Lisp Functions:: | |
32167 | * Predicates:: | |
32168 | * Computational Lisp Functions:: | |
32169 | * Vector Lisp Functions:: | |
32170 | * Symbolic Lisp Functions:: | |
32171 | * Formatting Lisp Functions:: | |
32172 | * Hooks:: | |
32173 | @end menu | |
32174 | ||
32175 | @node Data Type Formats, Interactive Lisp Functions, Internals, Internals | |
32176 | @subsubsection Data Type Formats | |
32177 | ||
32178 | @noindent | |
32179 | Integers are stored in either of two ways, depending on their magnitude. | |
32180 | Integers less than one million in absolute value are stored as standard | |
32181 | Lisp integers. This is the only storage format for Calc data objects | |
32182 | which is not a Lisp list. | |
32183 | ||
32184 | Large integers are stored as lists of the form @samp{(bigpos @var{d0} | |
32185 | @var{d1} @var{d2} @dots{})} for positive integers 1000000 or more, or | |
32186 | @samp{(bigneg @var{d0} @var{d1} @var{d2} @dots{})} for negative integers | |
32187 | @i{-1000000} or less. Each @var{d} is a base-1000 ``digit,'' a Lisp integer | |
32188 | from 0 to 999. The least significant digit is @var{d0}; the last digit, | |
32189 | @var{dn}, which is always nonzero, is the most significant digit. For | |
32190 | example, the integer @i{-12345678} is stored as @samp{(bigneg 678 345 12)}. | |
32191 | ||
32192 | The distinction between small and large integers is entirely hidden from | |
32193 | the user. In @code{defmath} definitions, the Lisp predicate @code{integerp} | |
32194 | returns true for either kind of integer, and in general both big and small | |
32195 | integers are accepted anywhere the word ``integer'' is used in this manual. | |
32196 | If the distinction must be made, native Lisp integers are called @dfn{fixnums} | |
32197 | and large integers are called @dfn{bignums}. | |
32198 | ||
32199 | Fractions are stored as a list of the form, @samp{(frac @var{n} @var{d})} | |
32200 | where @var{n} is an integer (big or small) numerator, @var{d} is an | |
32201 | integer denominator greater than one, and @var{n} and @var{d} are relatively | |
32202 | prime. Note that fractions where @var{d} is one are automatically converted | |
32203 | to plain integers by all math routines; fractions where @var{d} is negative | |
32204 | are normalized by negating the numerator and denominator. | |
32205 | ||
32206 | Floating-point numbers are stored in the form, @samp{(float @var{mant} | |
32207 | @var{exp})}, where @var{mant} (the ``mantissa'') is an integer less than | |
32208 | @samp{10^@var{p}} in absolute value (@var{p} represents the current | |
32209 | precision), and @var{exp} (the ``exponent'') is a fixnum. The value of | |
32210 | the float is @samp{@var{mant} * 10^@var{exp}}. For example, the number | |
32211 | @i{-3.14} is stored as @samp{(float -314 -2) = -314*10^-2}. Other constraints | |
32212 | are that the number 0.0 is always stored as @samp{(float 0 0)}, and, | |
32213 | except for the 0.0 case, the rightmost base-10 digit of @var{mant} is | |
32214 | always nonzero. (If the rightmost digit is zero, the number is | |
32215 | rearranged by dividing @var{mant} by ten and incrementing @var{exp}.)@refill | |
32216 | ||
32217 | Rectangular complex numbers are stored in the form @samp{(cplx @var{re} | |
32218 | @var{im})}, where @var{re} and @var{im} are each real numbers, either | |
32219 | integers, fractions, or floats. The value is @samp{@var{re} + @var{im}i}. | |
32220 | The @var{im} part is nonzero; complex numbers with zero imaginary | |
32221 | components are converted to real numbers automatically.@refill | |
32222 | ||
32223 | Polar complex numbers are stored in the form @samp{(polar @var{r} | |
32224 | @var{theta})}, where @var{r} is a positive real value and @var{theta} | |
32225 | is a real value or HMS form representing an angle. This angle is | |
32226 | usually normalized to lie in the interval @samp{(-180 ..@: 180)} degrees, | |
32227 | or @samp{(-pi ..@: pi)} radians, according to the current angular mode. | |
32228 | If the angle is 0 the value is converted to a real number automatically. | |
32229 | (If the angle is 180 degrees, the value is usually also converted to a | |
32230 | negative real number.)@refill | |
32231 | ||
32232 | Hours-minutes-seconds forms are stored as @samp{(hms @var{h} @var{m} | |
32233 | @var{s})}, where @var{h} is an integer or an integer-valued float (i.e., | |
32234 | a float with @samp{@var{exp} >= 0}), @var{m} is an integer or integer-valued | |
32235 | float in the range @w{@samp{[0 ..@: 60)}}, and @var{s} is any real number | |
32236 | in the range @samp{[0 ..@: 60)}.@refill | |
32237 | ||
32238 | Date forms are stored as @samp{(date @var{n})}, where @var{n} is | |
32239 | a real number that counts days since midnight on the morning of | |
32240 | January 1, 1 AD. If @var{n} is an integer, this is a pure date | |
32241 | form. If @var{n} is a fraction or float, this is a date/time form. | |
32242 | ||
32243 | Modulo forms are stored as @samp{(mod @var{n} @var{m})}, where @var{m} is a | |
32244 | positive real number or HMS form, and @var{n} is a real number or HMS | |
32245 | form in the range @samp{[0 ..@: @var{m})}. | |
32246 | ||
32247 | Error forms are stored as @samp{(sdev @var{x} @var{sigma})}, where @var{x} | |
32248 | is the mean value and @var{sigma} is the standard deviation. Each | |
32249 | component is either a number, an HMS form, or a symbolic object | |
32250 | (a variable or function call). If @var{sigma} is zero, the value is | |
32251 | converted to a plain real number. If @var{sigma} is negative or | |
32252 | complex, it is automatically normalized to be a positive real. | |
32253 | ||
32254 | Interval forms are stored as @samp{(intv @var{mask} @var{lo} @var{hi})}, | |
32255 | where @var{mask} is one of the integers 0, 1, 2, or 3, and @var{lo} and | |
32256 | @var{hi} are real numbers, HMS forms, or symbolic objects. The @var{mask} | |
32257 | is a binary integer where 1 represents the fact that the interval is | |
32258 | closed on the high end, and 2 represents the fact that it is closed on | |
32259 | the low end. (Thus 3 represents a fully closed interval.) The interval | |
32260 | @w{@samp{(intv 3 @var{x} @var{x})}} is converted to the plain number @var{x}; | |
32261 | intervals @samp{(intv @var{mask} @var{x} @var{x})} for any other @var{mask} | |
32262 | represent empty intervals. If @var{hi} is less than @var{lo}, the interval | |
32263 | is converted to a standard empty interval by replacing @var{hi} with @var{lo}. | |
32264 | ||
32265 | Vectors are stored as @samp{(vec @var{v1} @var{v2} @dots{})}, where @var{v1} | |
32266 | is the first element of the vector, @var{v2} is the second, and so on. | |
32267 | An empty vector is stored as @samp{(vec)}. A matrix is simply a vector | |
32268 | where all @var{v}'s are themselves vectors of equal lengths. Note that | |
32269 | Calc vectors are unrelated to the Emacs Lisp ``vector'' type, which is | |
32270 | generally unused by Calc data structures. | |
32271 | ||
32272 | Variables are stored as @samp{(var @var{name} @var{sym})}, where | |
32273 | @var{name} is a Lisp symbol whose print name is used as the visible name | |
32274 | of the variable, and @var{sym} is a Lisp symbol in which the variable's | |
32275 | value is actually stored. Thus, @samp{(var pi var-pi)} represents the | |
32276 | special constant @samp{pi}. Almost always, the form is @samp{(var | |
32277 | @var{v} var-@var{v})}. If the variable name was entered with @code{#} | |
32278 | signs (which are converted to hyphens internally), the form is | |
32279 | @samp{(var @var{u} @var{v})}, where @var{u} is a symbol whose name | |
32280 | contains @code{#} characters, and @var{v} is a symbol that contains | |
32281 | @code{-} characters instead. The value of a variable is the Calc | |
32282 | object stored in its @var{sym} symbol's value cell. If the symbol's | |
32283 | value cell is void or if it contains @code{nil}, the variable has no | |
32284 | value. Special constants have the form @samp{(special-const | |
32285 | @var{value})} stored in their value cell, where @var{value} is a formula | |
32286 | which is evaluated when the constant's value is requested. Variables | |
32287 | which represent units are not stored in any special way; they are units | |
32288 | only because their names appear in the units table. If the value | |
32289 | cell contains a string, it is parsed to get the variable's value when | |
32290 | the variable is used.@refill | |
32291 | ||
32292 | A Lisp list with any other symbol as the first element is a function call. | |
32293 | The symbols @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^}, | |
32294 | and @code{|} represent special binary operators; these lists are always | |
32295 | of the form @samp{(@var{op} @var{lhs} @var{rhs})} where @var{lhs} is the | |
32296 | sub-formula on the lefthand side and @var{rhs} is the sub-formula on the | |
32297 | right. The symbol @code{neg} represents unary negation; this list is always | |
32298 | of the form @samp{(neg @var{arg})}. Any other symbol @var{func} represents a | |
32299 | function that would be displayed in function-call notation; the symbol | |
32300 | @var{func} is in general always of the form @samp{calcFunc-@var{name}}. | |
32301 | The function cell of the symbol @var{func} should contain a Lisp function | |
32302 | for evaluating a call to @var{func}. This function is passed the remaining | |
32303 | elements of the list (themselves already evaluated) as arguments; such | |
32304 | functions should return @code{nil} or call @code{reject-arg} to signify | |
32305 | that they should be left in symbolic form, or they should return a Calc | |
32306 | object which represents their value, or a list of such objects if they | |
32307 | wish to return multiple values. (The latter case is allowed only for | |
32308 | functions which are the outer-level call in an expression whose value is | |
32309 | about to be pushed on the stack; this feature is considered obsolete | |
32310 | and is not used by any built-in Calc functions.)@refill | |
32311 | ||
32312 | @node Interactive Lisp Functions, Stack Lisp Functions, Data Type Formats, Internals | |
32313 | @subsubsection Interactive Functions | |
32314 | ||
32315 | @noindent | |
32316 | The functions described here are used in implementing interactive Calc | |
32317 | commands. Note that this list is not exhaustive! If there is an | |
32318 | existing command that behaves similarly to the one you want to define, | |
32319 | you may find helpful tricks by checking the source code for that command. | |
32320 | ||
32321 | @defun calc-set-command-flag flag | |
32322 | Set the command flag @var{flag}. This is generally a Lisp symbol, but | |
32323 | may in fact be anything. The effect is to add @var{flag} to the list | |
32324 | stored in the variable @code{calc-command-flags}, unless it is already | |
32325 | there. @xref{Defining Simple Commands}. | |
32326 | @end defun | |
32327 | ||
32328 | @defun calc-clear-command-flag flag | |
32329 | If @var{flag} appears among the list of currently-set command flags, | |
32330 | remove it from that list. | |
32331 | @end defun | |
32332 | ||
32333 | @defun calc-record-undo rec | |
32334 | Add the ``undo record'' @var{rec} to the list of steps to take if the | |
32335 | current operation should need to be undone. Stack push and pop functions | |
32336 | automatically call @code{calc-record-undo}, so the kinds of undo records | |
32337 | you might need to create take the form @samp{(set @var{sym} @var{value})}, | |
32338 | which says that the Lisp variable @var{sym} was changed and had previously | |
32339 | contained @var{value}; @samp{(store @var{var} @var{value})} which says that | |
32340 | the Calc variable @var{var} (a string which is the name of the symbol that | |
32341 | contains the variable's value) was stored and its previous value was | |
32342 | @var{value} (either a Calc data object, or @code{nil} if the variable was | |
32343 | previously void); or @samp{(eval @var{undo} @var{redo} @var{args} @dots{})}, | |
32344 | which means that to undo requires calling the function @samp{(@var{undo} | |
32345 | @var{args} @dots{})} and, if the undo is later redone, calling | |
32346 | @samp{(@var{redo} @var{args} @dots{})}.@refill | |
32347 | @end defun | |
32348 | ||
32349 | @defun calc-record-why msg args | |
32350 | Record the error or warning message @var{msg}, which is normally a string. | |
32351 | This message will be replayed if the user types @kbd{w} (@code{calc-why}); | |
32352 | if the message string begins with a @samp{*}, it is considered important | |
32353 | enough to display even if the user doesn't type @kbd{w}. If one or more | |
32354 | @var{args} are present, the displayed message will be of the form, | |
32355 | @samp{@var{msg}: @var{arg1}, @var{arg2}, @dots{}}, where the arguments are | |
32356 | formatted on the assumption that they are either strings or Calc objects of | |
32357 | some sort. If @var{msg} is a symbol, it is the name of a Calc predicate | |
32358 | (such as @code{integerp} or @code{numvecp}) which the arguments did not | |
32359 | satisfy; it is expanded to a suitable string such as ``Expected an | |
32360 | integer.'' The @code{reject-arg} function calls @code{calc-record-why} | |
32361 | automatically; @pxref{Predicates}.@refill | |
32362 | @end defun | |
32363 | ||
32364 | @defun calc-is-inverse | |
32365 | This predicate returns true if the current command is inverse, | |
32366 | i.e., if the Inverse (@kbd{I} key) flag was set. | |
32367 | @end defun | |
32368 | ||
32369 | @defun calc-is-hyperbolic | |
32370 | This predicate is the analogous function for the @kbd{H} key. | |
32371 | @end defun | |
32372 | ||
32373 | @node Stack Lisp Functions, Predicates, Interactive Lisp Functions, Internals | |
32374 | @subsubsection Stack-Oriented Functions | |
32375 | ||
32376 | @noindent | |
32377 | The functions described here perform various operations on the Calc | |
32378 | stack and trail. They are to be used in interactive Calc commands. | |
32379 | ||
32380 | @defun calc-push-list vals n | |
32381 | Push the Calc objects in list @var{vals} onto the stack at stack level | |
32382 | @var{n}. If @var{n} is omitted it defaults to 1, so that the elements | |
32383 | are pushed at the top of the stack. If @var{n} is greater than 1, the | |
32384 | elements will be inserted into the stack so that the last element will | |
32385 | end up at level @var{n}, the next-to-last at level @var{n}+1, etc. | |
32386 | The elements of @var{vals} are assumed to be valid Calc objects, and | |
32387 | are not evaluated, rounded, or renormalized in any way. If @var{vals} | |
32388 | is an empty list, nothing happens.@refill | |
32389 | ||
32390 | The stack elements are pushed without any sub-formula selections. | |
32391 | You can give an optional third argument to this function, which must | |
32392 | be a list the same size as @var{vals} of selections. Each selection | |
32393 | must be @code{eq} to some sub-formula of the corresponding formula | |
32394 | in @var{vals}, or @code{nil} if that formula should have no selection. | |
32395 | @end defun | |
32396 | ||
32397 | @defun calc-top-list n m | |
32398 | Return a list of the @var{n} objects starting at level @var{m} of the | |
32399 | stack. If @var{m} is omitted it defaults to 1, so that the elements are | |
32400 | taken from the top of the stack. If @var{n} is omitted, it also | |
32401 | defaults to 1, so that the top stack element (in the form of a | |
32402 | one-element list) is returned. If @var{m} is greater than 1, the | |
32403 | @var{m}th stack element will be at the end of the list, the @var{m}+1st | |
32404 | element will be next-to-last, etc. If @var{n} or @var{m} are out of | |
32405 | range, the command is aborted with a suitable error message. If @var{n} | |
32406 | is zero, the function returns an empty list. The stack elements are not | |
32407 | evaluated, rounded, or renormalized.@refill | |
32408 | ||
32409 | If any stack elements contain selections, and selections have not | |
32410 | been disabled by the @kbd{j e} (@code{calc-enable-selections}) command, | |
32411 | this function returns the selected portions rather than the entire | |
32412 | stack elements. It can be given a third ``selection-mode'' argument | |
32413 | which selects other behaviors. If it is the symbol @code{t}, then | |
32414 | a selection in any of the requested stack elements produces an | |
32415 | ``illegal operation on selections'' error. If it is the symbol @code{full}, | |
32416 | the whole stack entry is always returned regardless of selections. | |
32417 | If it is the symbol @code{sel}, the selected portion is always returned, | |
32418 | or @code{nil} if there is no selection. (This mode ignores the @kbd{j e} | |
32419 | command.) If the symbol is @code{entry}, the complete stack entry in | |
32420 | list form is returned; the first element of this list will be the whole | |
32421 | formula, and the third element will be the selection (or @code{nil}). | |
32422 | @end defun | |
32423 | ||
32424 | @defun calc-pop-stack n m | |
32425 | Remove the specified elements from the stack. The parameters @var{n} | |
32426 | and @var{m} are defined the same as for @code{calc-top-list}. The return | |
32427 | value of @code{calc-pop-stack} is uninteresting. | |
32428 | ||
32429 | If there are any selected sub-formulas among the popped elements, and | |
32430 | @kbd{j e} has not been used to disable selections, this produces an | |
32431 | error without changing the stack. If you supply an optional third | |
32432 | argument of @code{t}, the stack elements are popped even if they | |
32433 | contain selections. | |
32434 | @end defun | |
32435 | ||
32436 | @defun calc-record-list vals tag | |
32437 | This function records one or more results in the trail. The @var{vals} | |
32438 | are a list of strings or Calc objects. The @var{tag} is the four-character | |
32439 | tag string to identify the values. If @var{tag} is omitted, a blank tag | |
32440 | will be used. | |
32441 | @end defun | |
32442 | ||
32443 | @defun calc-normalize n | |
32444 | This function takes a Calc object and ``normalizes'' it. At the very | |
32445 | least this involves re-rounding floating-point values according to the | |
32446 | current precision and other similar jobs. Also, unless the user has | |
32447 | selected no-simplify mode (@pxref{Simplification Modes}), this involves | |
32448 | actually evaluating a formula object by executing the function calls | |
32449 | it contains, and possibly also doing algebraic simplification, etc. | |
32450 | @end defun | |
32451 | ||
32452 | @defun calc-top-list-n n m | |
32453 | This function is identical to @code{calc-top-list}, except that it calls | |
32454 | @code{calc-normalize} on the values that it takes from the stack. They | |
32455 | are also passed through @code{check-complete}, so that incomplete | |
32456 | objects will be rejected with an error message. All computational | |
32457 | commands should use this in preference to @code{calc-top-list}; the only | |
32458 | standard Calc commands that operate on the stack without normalizing | |
32459 | are stack management commands like @code{calc-enter} and @code{calc-roll-up}. | |
32460 | This function accepts the same optional selection-mode argument as | |
32461 | @code{calc-top-list}. | |
32462 | @end defun | |
32463 | ||
32464 | @defun calc-top-n m | |
32465 | This function is a convenient form of @code{calc-top-list-n} in which only | |
32466 | a single element of the stack is taken and returned, rather than a list | |
32467 | of elements. This also accepts an optional selection-mode argument. | |
32468 | @end defun | |
32469 | ||
32470 | @defun calc-enter-result n tag vals | |
32471 | This function is a convenient interface to most of the above functions. | |
32472 | The @var{vals} argument should be either a single Calc object, or a list | |
32473 | of Calc objects; the object or objects are normalized, and the top @var{n} | |
32474 | stack entries are replaced by the normalized objects. If @var{tag} is | |
32475 | non-@code{nil}, the normalized objects are also recorded in the trail. | |
32476 | A typical stack-based computational command would take the form, | |
32477 | ||
32478 | @smallexample | |
32479 | (calc-enter-result @var{n} @var{tag} (cons 'calcFunc-@var{func} | |
32480 | (calc-top-list-n @var{n}))) | |
32481 | @end smallexample | |
32482 | ||
32483 | If any of the @var{n} stack elements replaced contain sub-formula | |
32484 | selections, and selections have not been disabled by @kbd{j e}, | |
32485 | this function takes one of two courses of action. If @var{n} is | |
32486 | equal to the number of elements in @var{vals}, then each element of | |
32487 | @var{vals} is spliced into the corresponding selection; this is what | |
32488 | happens when you use the @key{TAB} key, or when you use a unary | |
32489 | arithmetic operation like @code{sqrt}. If @var{vals} has only one | |
32490 | element but @var{n} is greater than one, there must be only one | |
32491 | selection among the top @var{n} stack elements; the element from | |
32492 | @var{vals} is spliced into that selection. This is what happens when | |
32493 | you use a binary arithmetic operation like @kbd{+}. Any other | |
32494 | combination of @var{n} and @var{vals} is an error when selections | |
32495 | are present. | |
32496 | @end defun | |
32497 | ||
32498 | @defun calc-unary-op tag func arg | |
32499 | This function implements a unary operator that allows a numeric prefix | |
32500 | argument to apply the operator over many stack entries. If the prefix | |
32501 | argument @var{arg} is @code{nil}, this uses @code{calc-enter-result} | |
32502 | as outlined above. Otherwise, it maps the function over several stack | |
32503 | elements; @pxref{Prefix Arguments}. For example,@refill | |
32504 | ||
32505 | @smallexample | |
32506 | (defun calc-zeta (arg) | |
32507 | (interactive "P") | |
32508 | (calc-unary-op "zeta" 'calcFunc-zeta arg)) | |
32509 | @end smallexample | |
32510 | @end defun | |
32511 | ||
32512 | @defun calc-binary-op tag func arg ident unary | |
32513 | This function implements a binary operator, analogously to | |
32514 | @code{calc-unary-op}. The optional @var{ident} and @var{unary} | |
32515 | arguments specify the behavior when the prefix argument is zero or | |
32516 | one, respectively. If the prefix is zero, the value @var{ident} | |
32517 | is pushed onto the stack, if specified, otherwise an error message | |
32518 | is displayed. If the prefix is one, the unary function @var{unary} | |
32519 | is applied to the top stack element, or, if @var{unary} is not | |
32520 | specified, nothing happens. When the argument is two or more, | |
32521 | the binary function @var{func} is reduced across the top @var{arg} | |
32522 | stack elements; when the argument is negative, the function is | |
32523 | mapped between the next-to-top @i{-@var{arg}} stack elements and the | |
32524 | top element.@refill | |
32525 | @end defun | |
32526 | ||
32527 | @defun calc-stack-size | |
32528 | Return the number of elements on the stack as an integer. This count | |
32529 | does not include elements that have been temporarily hidden by stack | |
32530 | truncation; @pxref{Truncating the Stack}. | |
32531 | @end defun | |
32532 | ||
32533 | @defun calc-cursor-stack-index n | |
32534 | Move the point to the @var{n}th stack entry. If @var{n} is zero, this | |
32535 | will be the @samp{.} line. If @var{n} is from 1 to the current stack size, | |
32536 | this will be the beginning of the first line of that stack entry's display. | |
32537 | If line numbers are enabled, this will move to the first character of the | |
32538 | line number, not the stack entry itself.@refill | |
32539 | @end defun | |
32540 | ||
32541 | @defun calc-substack-height n | |
32542 | Return the number of lines between the beginning of the @var{n}th stack | |
32543 | entry and the bottom of the buffer. If @var{n} is zero, this | |
32544 | will be one (assuming no stack truncation). If all stack entries are | |
32545 | one line long (i.e., no matrices are displayed), the return value will | |
32546 | be equal @var{n}+1 as long as @var{n} is in range. (Note that in Big | |
32547 | mode, the return value includes the blank lines that separate stack | |
32548 | entries.)@refill | |
32549 | @end defun | |
32550 | ||
32551 | @defun calc-refresh | |
32552 | Erase the @code{*Calculator*} buffer and reformat its contents from memory. | |
32553 | This must be called after changing any parameter, such as the current | |
32554 | display radix, which might change the appearance of existing stack | |
32555 | entries. (During a keyboard macro invoked by the @kbd{X} key, refreshing | |
32556 | is suppressed, but a flag is set so that the entire stack will be refreshed | |
32557 | rather than just the top few elements when the macro finishes.)@refill | |
32558 | @end defun | |
32559 | ||
32560 | @node Predicates, Computational Lisp Functions, Stack Lisp Functions, Internals | |
32561 | @subsubsection Predicates | |
32562 | ||
32563 | @noindent | |
32564 | The functions described here are predicates, that is, they return a | |
32565 | true/false value where @code{nil} means false and anything else means | |
32566 | true. These predicates are expanded by @code{defmath}, for example, | |
32567 | from @code{zerop} to @code{math-zerop}. In many cases they correspond | |
32568 | to native Lisp functions by the same name, but are extended to cover | |
32569 | the full range of Calc data types. | |
32570 | ||
32571 | @defun zerop x | |
32572 | Returns true if @var{x} is numerically zero, in any of the Calc data | |
32573 | types. (Note that for some types, such as error forms and intervals, | |
32574 | it never makes sense to return true.) In @code{defmath}, the expression | |
32575 | @samp{(= x 0)} will automatically be converted to @samp{(math-zerop x)}, | |
32576 | and @samp{(/= x 0)} will be converted to @samp{(not (math-zerop x))}. | |
32577 | @end defun | |
32578 | ||
32579 | @defun negp x | |
32580 | Returns true if @var{x} is negative. This accepts negative real numbers | |
32581 | of various types, negative HMS and date forms, and intervals in which | |
32582 | all included values are negative. In @code{defmath}, the expression | |
32583 | @samp{(< x 0)} will automatically be converted to @samp{(math-negp x)}, | |
32584 | and @samp{(>= x 0)} will be converted to @samp{(not (math-negp x))}. | |
32585 | @end defun | |
32586 | ||
32587 | @defun posp x | |
32588 | Returns true if @var{x} is positive (and non-zero). For complex | |
32589 | numbers, none of these three predicates will return true. | |
32590 | @end defun | |
32591 | ||
32592 | @defun looks-negp x | |
32593 | Returns true if @var{x} is ``negative-looking.'' This returns true if | |
32594 | @var{x} is a negative number, or a formula with a leading minus sign | |
32595 | such as @samp{-a/b}. In other words, this is an object which can be | |
32596 | made simpler by calling @code{(- @var{x})}. | |
32597 | @end defun | |
32598 | ||
32599 | @defun integerp x | |
32600 | Returns true if @var{x} is an integer of any size. | |
32601 | @end defun | |
32602 | ||
32603 | @defun fixnump x | |
32604 | Returns true if @var{x} is a native Lisp integer. | |
32605 | @end defun | |
32606 | ||
32607 | @defun natnump x | |
32608 | Returns true if @var{x} is a nonnegative integer of any size. | |
32609 | @end defun | |
32610 | ||
32611 | @defun fixnatnump x | |
32612 | Returns true if @var{x} is a nonnegative Lisp integer. | |
32613 | @end defun | |
32614 | ||
32615 | @defun num-integerp x | |
32616 | Returns true if @var{x} is numerically an integer, i.e., either a | |
32617 | true integer or a float with no significant digits to the right of | |
32618 | the decimal point. | |
32619 | @end defun | |
32620 | ||
32621 | @defun messy-integerp x | |
32622 | Returns true if @var{x} is numerically, but not literally, an integer. | |
32623 | A value is @code{num-integerp} if it is @code{integerp} or | |
32624 | @code{messy-integerp} (but it is never both at once). | |
32625 | @end defun | |
32626 | ||
32627 | @defun num-natnump x | |
32628 | Returns true if @var{x} is numerically a nonnegative integer. | |
32629 | @end defun | |
32630 | ||
32631 | @defun evenp x | |
32632 | Returns true if @var{x} is an even integer. | |
32633 | @end defun | |
32634 | ||
32635 | @defun looks-evenp x | |
32636 | Returns true if @var{x} is an even integer, or a formula with a leading | |
32637 | multiplicative coefficient which is an even integer. | |
32638 | @end defun | |
32639 | ||
32640 | @defun oddp x | |
32641 | Returns true if @var{x} is an odd integer. | |
32642 | @end defun | |
32643 | ||
32644 | @defun ratp x | |
32645 | Returns true if @var{x} is a rational number, i.e., an integer or a | |
32646 | fraction. | |
32647 | @end defun | |
32648 | ||
32649 | @defun realp x | |
32650 | Returns true if @var{x} is a real number, i.e., an integer, fraction, | |
32651 | or floating-point number. | |
32652 | @end defun | |
32653 | ||
32654 | @defun anglep x | |
32655 | Returns true if @var{x} is a real number or HMS form. | |
32656 | @end defun | |
32657 | ||
32658 | @defun floatp x | |
32659 | Returns true if @var{x} is a float, or a complex number, error form, | |
32660 | interval, date form, or modulo form in which at least one component | |
32661 | is a float. | |
32662 | @end defun | |
32663 | ||
32664 | @defun complexp x | |
32665 | Returns true if @var{x} is a rectangular or polar complex number | |
32666 | (but not a real number). | |
32667 | @end defun | |
32668 | ||
32669 | @defun rect-complexp x | |
32670 | Returns true if @var{x} is a rectangular complex number. | |
32671 | @end defun | |
32672 | ||
32673 | @defun polar-complexp x | |
32674 | Returns true if @var{x} is a polar complex number. | |
32675 | @end defun | |
32676 | ||
32677 | @defun numberp x | |
32678 | Returns true if @var{x} is a real number or a complex number. | |
32679 | @end defun | |
32680 | ||
32681 | @defun scalarp x | |
32682 | Returns true if @var{x} is a real or complex number or an HMS form. | |
32683 | @end defun | |
32684 | ||
32685 | @defun vectorp x | |
32686 | Returns true if @var{x} is a vector (this simply checks if its argument | |
32687 | is a list whose first element is the symbol @code{vec}). | |
32688 | @end defun | |
32689 | ||
32690 | @defun numvecp x | |
32691 | Returns true if @var{x} is a number or vector. | |
32692 | @end defun | |
32693 | ||
32694 | @defun matrixp x | |
32695 | Returns true if @var{x} is a matrix, i.e., a vector of one or more vectors, | |
32696 | all of the same size. | |
32697 | @end defun | |
32698 | ||
32699 | @defun square-matrixp x | |
32700 | Returns true if @var{x} is a square matrix. | |
32701 | @end defun | |
32702 | ||
32703 | @defun objectp x | |
32704 | Returns true if @var{x} is any numeric Calc object, including real and | |
32705 | complex numbers, HMS forms, date forms, error forms, intervals, and | |
32706 | modulo forms. (Note that error forms and intervals may include formulas | |
32707 | as their components; see @code{constp} below.) | |
32708 | @end defun | |
32709 | ||
32710 | @defun objvecp x | |
32711 | Returns true if @var{x} is an object or a vector. This also accepts | |
32712 | incomplete objects, but it rejects variables and formulas (except as | |
32713 | mentioned above for @code{objectp}). | |
32714 | @end defun | |
32715 | ||
32716 | @defun primp x | |
32717 | Returns true if @var{x} is a ``primitive'' or ``atomic'' Calc object, | |
32718 | i.e., one whose components cannot be regarded as sub-formulas. This | |
32719 | includes variables, and all @code{objectp} types except error forms | |
32720 | and intervals. | |
32721 | @end defun | |
32722 | ||
32723 | @defun constp x | |
32724 | Returns true if @var{x} is constant, i.e., a real or complex number, | |
32725 | HMS form, date form, or error form, interval, or vector all of whose | |
32726 | components are @code{constp}. | |
32727 | @end defun | |
32728 | ||
32729 | @defun lessp x y | |
32730 | Returns true if @var{x} is numerically less than @var{y}. Returns false | |
32731 | if @var{x} is greater than or equal to @var{y}, or if the order is | |
32732 | undefined or cannot be determined. Generally speaking, this works | |
32733 | by checking whether @samp{@var{x} - @var{y}} is @code{negp}. In | |
32734 | @code{defmath}, the expression @samp{(< x y)} will automatically be | |
32735 | converted to @samp{(lessp x y)}; expressions involving @code{>}, @code{<=}, | |
32736 | and @code{>=} are similarly converted in terms of @code{lessp}.@refill | |
32737 | @end defun | |
32738 | ||
32739 | @defun beforep x y | |
32740 | Returns true if @var{x} comes before @var{y} in a canonical ordering | |
32741 | of Calc objects. If @var{x} and @var{y} are both real numbers, this | |
32742 | will be the same as @code{lessp}. But whereas @code{lessp} considers | |
32743 | other types of objects to be unordered, @code{beforep} puts any two | |
32744 | objects into a definite, consistent order. The @code{beforep} | |
32745 | function is used by the @kbd{V S} vector-sorting command, and also | |
32746 | by @kbd{a s} to put the terms of a product into canonical order: | |
32747 | This allows @samp{x y + y x} to be simplified easily to @samp{2 x y}. | |
32748 | @end defun | |
32749 | ||
32750 | @defun equal x y | |
32751 | This is the standard Lisp @code{equal} predicate; it returns true if | |
32752 | @var{x} and @var{y} are structurally identical. This is the usual way | |
32753 | to compare numbers for equality, but note that @code{equal} will treat | |
32754 | 0 and 0.0 as different. | |
32755 | @end defun | |
32756 | ||
32757 | @defun math-equal x y | |
32758 | Returns true if @var{x} and @var{y} are numerically equal, either because | |
32759 | they are @code{equal}, or because their difference is @code{zerop}. In | |
32760 | @code{defmath}, the expression @samp{(= x y)} will automatically be | |
32761 | converted to @samp{(math-equal x y)}. | |
32762 | @end defun | |
32763 | ||
32764 | @defun equal-int x n | |
32765 | Returns true if @var{x} and @var{n} are numerically equal, where @var{n} | |
32766 | is a fixnum which is not a multiple of 10. This will automatically be | |
32767 | used by @code{defmath} in place of the more general @code{math-equal} | |
32768 | whenever possible.@refill | |
32769 | @end defun | |
32770 | ||
32771 | @defun nearly-equal x y | |
32772 | Returns true if @var{x} and @var{y}, as floating-point numbers, are | |
32773 | equal except possibly in the last decimal place. For example, | |
32774 | 314.159 and 314.166 are considered nearly equal if the current | |
32775 | precision is 6 (since they differ by 7 units), but not if the current | |
32776 | precision is 7 (since they differ by 70 units). Most functions which | |
32777 | use series expansions use @code{with-extra-prec} to evaluate the | |
32778 | series with 2 extra digits of precision, then use @code{nearly-equal} | |
32779 | to decide when the series has converged; this guards against cumulative | |
32780 | error in the series evaluation without doing extra work which would be | |
32781 | lost when the result is rounded back down to the current precision. | |
32782 | In @code{defmath}, this can be written @samp{(~= @var{x} @var{y})}. | |
32783 | The @var{x} and @var{y} can be numbers of any kind, including complex. | |
32784 | @end defun | |
32785 | ||
32786 | @defun nearly-zerop x y | |
32787 | Returns true if @var{x} is nearly zero, compared to @var{y}. This | |
32788 | checks whether @var{x} plus @var{y} would by be @code{nearly-equal} | |
32789 | to @var{y} itself, to within the current precision, in other words, | |
32790 | if adding @var{x} to @var{y} would have a negligible effect on @var{y} | |
32791 | due to roundoff error. @var{X} may be a real or complex number, but | |
32792 | @var{y} must be real. | |
32793 | @end defun | |
32794 | ||
32795 | @defun is-true x | |
32796 | Return true if the formula @var{x} represents a true value in | |
32797 | Calc, not Lisp, terms. It tests if @var{x} is a non-zero number | |
32798 | or a provably non-zero formula. | |
32799 | @end defun | |
32800 | ||
32801 | @defun reject-arg val pred | |
32802 | Abort the current function evaluation due to unacceptable argument values. | |
32803 | This calls @samp{(calc-record-why @var{pred} @var{val})}, then signals a | |
32804 | Lisp error which @code{normalize} will trap. The net effect is that the | |
32805 | function call which led here will be left in symbolic form.@refill | |
32806 | @end defun | |
32807 | ||
32808 | @defun inexact-value | |
32809 | If Symbolic Mode is enabled, this will signal an error that causes | |
32810 | @code{normalize} to leave the formula in symbolic form, with the message | |
32811 | ``Inexact result.'' (This function has no effect when not in Symbolic Mode.) | |
32812 | Note that if your function calls @samp{(sin 5)} in Symbolic Mode, the | |
32813 | @code{sin} function will call @code{inexact-value}, which will cause your | |
32814 | function to be left unsimplified. You may instead wish to call | |
32815 | @samp{(normalize (list 'calcFunc-sin 5))}, which in Symbolic Mode will | |
32816 | return the formula @samp{sin(5)} to your function.@refill | |
32817 | @end defun | |
32818 | ||
32819 | @defun overflow | |
32820 | This signals an error that will be reported as a floating-point overflow. | |
32821 | @end defun | |
32822 | ||
32823 | @defun underflow | |
32824 | This signals a floating-point underflow. | |
32825 | @end defun | |
32826 | ||
32827 | @node Computational Lisp Functions, Vector Lisp Functions, Predicates, Internals | |
32828 | @subsubsection Computational Functions | |
32829 | ||
32830 | @noindent | |
32831 | The functions described here do the actual computational work of the | |
32832 | Calculator. In addition to these, note that any function described in | |
32833 | the main body of this manual may be called from Lisp; for example, if | |
32834 | the documentation refers to the @code{calc-sqrt} [@code{sqrt}] command, | |
32835 | this means @code{calc-sqrt} is an interactive stack-based square-root | |
32836 | command and @code{sqrt} (which @code{defmath} expands to @code{calcFunc-sqrt}) | |
32837 | is the actual Lisp function for taking square roots.@refill | |
32838 | ||
32839 | The functions @code{math-add}, @code{math-sub}, @code{math-mul}, | |
32840 | @code{math-div}, @code{math-mod}, and @code{math-neg} are not included | |
32841 | in this list, since @code{defmath} allows you to write native Lisp | |
32842 | @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, and unary @code{-}, | |
32843 | respectively, instead.@refill | |
32844 | ||
32845 | @defun normalize val | |
32846 | (Full form: @code{math-normalize}.) | |
32847 | Reduce the value @var{val} to standard form. For example, if @var{val} | |
32848 | is a fixnum, it will be converted to a bignum if it is too large, and | |
32849 | if @var{val} is a bignum it will be normalized by clipping off trailing | |
32850 | (i.e., most-significant) zero digits and converting to a fixnum if it is | |
32851 | small. All the various data types are similarly converted to their standard | |
32852 | forms. Variables are left alone, but function calls are actually evaluated | |
32853 | in formulas. For example, normalizing @samp{(+ 2 (calcFunc-abs -4))} will | |
32854 | return 6.@refill | |
32855 | ||
32856 | If a function call fails, because the function is void or has the wrong | |
32857 | number of parameters, or because it returns @code{nil} or calls | |
32858 | @code{reject-arg} or @code{inexact-result}, @code{normalize} returns | |
32859 | the formula still in symbolic form.@refill | |
32860 | ||
32861 | If the current Simplification Mode is ``none'' or ``numeric arguments | |
32862 | only,'' @code{normalize} will act appropriately. However, the more | |
32863 | powerful simplification modes (like algebraic simplification) are | |
32864 | not handled by @code{normalize}. They are handled by @code{calc-normalize}, | |
32865 | which calls @code{normalize} and possibly some other routines, such | |
32866 | as @code{simplify} or @code{simplify-units}. Programs generally will | |
32867 | never call @code{calc-normalize} except when popping or pushing values | |
32868 | on the stack.@refill | |
32869 | @end defun | |
32870 | ||
32871 | @defun evaluate-expr expr | |
32872 | Replace all variables in @var{expr} that have values with their values, | |
32873 | then use @code{normalize} to simplify the result. This is what happens | |
32874 | when you press the @kbd{=} key interactively.@refill | |
32875 | @end defun | |
32876 | ||
32877 | @defmac with-extra-prec n body | |
32878 | Evaluate the Lisp forms in @var{body} with precision increased by @var{n} | |
32879 | digits. This is a macro which expands to | |
32880 | ||
32881 | @smallexample | |
32882 | (math-normalize | |
32883 | (let ((calc-internal-prec (+ calc-internal-prec @var{n}))) | |
32884 | @var{body})) | |
32885 | @end smallexample | |
32886 | ||
32887 | The surrounding call to @code{math-normalize} causes a floating-point | |
32888 | result to be rounded down to the original precision afterwards. This | |
32889 | is important because some arithmetic operations assume a number's | |
32890 | mantissa contains no more digits than the current precision allows. | |
32891 | @end defmac | |
32892 | ||
32893 | @defun make-frac n d | |
32894 | Build a fraction @samp{@var{n}:@var{d}}. This is equivalent to calling | |
32895 | @samp{(normalize (list 'frac @var{n} @var{d}))}, but more efficient. | |
32896 | @end defun | |
32897 | ||
32898 | @defun make-float mant exp | |
32899 | Build a floating-point value out of @var{mant} and @var{exp}, both | |
32900 | of which are arbitrary integers. This function will return a | |
32901 | properly normalized float value, or signal an overflow or underflow | |
32902 | if @var{exp} is out of range. | |
32903 | @end defun | |
32904 | ||
32905 | @defun make-sdev x sigma | |
32906 | Build an error form out of @var{x} and the absolute value of @var{sigma}. | |
32907 | If @var{sigma} is zero, the result is the number @var{x} directly. | |
32908 | If @var{sigma} is negative or complex, its absolute value is used. | |
32909 | If @var{x} or @var{sigma} is not a valid type of object for use in | |
32910 | error forms, this calls @code{reject-arg}. | |
32911 | @end defun | |
32912 | ||
32913 | @defun make-intv mask lo hi | |
32914 | Build an interval form out of @var{mask} (which is assumed to be an | |
32915 | integer from 0 to 3), and the limits @var{lo} and @var{hi}. If | |
32916 | @var{lo} is greater than @var{hi}, an empty interval form is returned. | |
32917 | This calls @code{reject-arg} if @var{lo} or @var{hi} is unsuitable. | |
32918 | @end defun | |
32919 | ||
32920 | @defun sort-intv mask lo hi | |
32921 | Build an interval form, similar to @code{make-intv}, except that if | |
32922 | @var{lo} is less than @var{hi} they are simply exchanged, and the | |
32923 | bits of @var{mask} are swapped accordingly. | |
32924 | @end defun | |
32925 | ||
32926 | @defun make-mod n m | |
32927 | Build a modulo form out of @var{n} and the modulus @var{m}. Since modulo | |
32928 | forms do not allow formulas as their components, if @var{n} or @var{m} | |
32929 | is not a real number or HMS form the result will be a formula which | |
32930 | is a call to @code{makemod}, the algebraic version of this function. | |
32931 | @end defun | |
32932 | ||
32933 | @defun float x | |
32934 | Convert @var{x} to floating-point form. Integers and fractions are | |
32935 | converted to numerically equivalent floats; components of complex | |
32936 | numbers, vectors, HMS forms, date forms, error forms, intervals, and | |
32937 | modulo forms are recursively floated. If the argument is a variable | |
32938 | or formula, this calls @code{reject-arg}. | |
32939 | @end defun | |
32940 | ||
32941 | @defun compare x y | |
32942 | Compare the numbers @var{x} and @var{y}, and return @i{-1} if | |
32943 | @samp{(lessp @var{x} @var{y})}, 1 if @samp{(lessp @var{y} @var{x})}, | |
32944 | 0 if @samp{(math-equal @var{x} @var{y})}, or 2 if the order is | |
32945 | undefined or cannot be determined.@refill | |
32946 | @end defun | |
32947 | ||
32948 | @defun numdigs n | |
32949 | Return the number of digits of integer @var{n}, effectively | |
32950 | @samp{ceil(log10(@var{n}))}, but much more efficient. Zero is | |
32951 | considered to have zero digits. | |
32952 | @end defun | |
32953 | ||
32954 | @defun scale-int x n | |
32955 | Shift integer @var{x} left @var{n} decimal digits, or right @i{-@var{n}} | |
32956 | digits with truncation toward zero. | |
32957 | @end defun | |
32958 | ||
32959 | @defun scale-rounding x n | |
32960 | Like @code{scale-int}, except that a right shift rounds to the nearest | |
32961 | integer rather than truncating. | |
32962 | @end defun | |
32963 | ||
32964 | @defun fixnum n | |
32965 | Return the integer @var{n} as a fixnum, i.e., a native Lisp integer. | |
32966 | If @var{n} is outside the permissible range for Lisp integers (usually | |
32967 | 24 binary bits) the result is undefined. | |
32968 | @end defun | |
32969 | ||
32970 | @defun sqr x | |
32971 | Compute the square of @var{x}; short for @samp{(* @var{x} @var{x})}. | |
32972 | @end defun | |
32973 | ||
32974 | @defun quotient x y | |
32975 | Divide integer @var{x} by integer @var{y}; return an integer quotient | |
32976 | and discard the remainder. If @var{x} or @var{y} is negative, the | |
32977 | direction of rounding is undefined. | |
32978 | @end defun | |
32979 | ||
32980 | @defun idiv x y | |
32981 | Perform an integer division; if @var{x} and @var{y} are both nonnegative | |
32982 | integers, this uses the @code{quotient} function, otherwise it computes | |
32983 | @samp{floor(@var{x}/@var{y})}. Thus the result is well-defined but | |
32984 | slower than for @code{quotient}. | |
32985 | @end defun | |
32986 | ||
32987 | @defun imod x y | |
32988 | Divide integer @var{x} by integer @var{y}; return the integer remainder | |
32989 | and discard the quotient. Like @code{quotient}, this works only for | |
32990 | integer arguments and is not well-defined for negative arguments. | |
32991 | For a more well-defined result, use @samp{(% @var{x} @var{y})}. | |
32992 | @end defun | |
32993 | ||
32994 | @defun idivmod x y | |
32995 | Divide integer @var{x} by integer @var{y}; return a cons cell whose | |
32996 | @code{car} is @samp{(quotient @var{x} @var{y})} and whose @code{cdr} | |
32997 | is @samp{(imod @var{x} @var{y})}.@refill | |
32998 | @end defun | |
32999 | ||
33000 | @defun pow x y | |
33001 | Compute @var{x} to the power @var{y}. In @code{defmath} code, this can | |
33002 | also be written @samp{(^ @var{x} @var{y})} or | |
33003 | @w{@samp{(expt @var{x} @var{y})}}.@refill | |
33004 | @end defun | |
33005 | ||
33006 | @defun abs-approx x | |
33007 | Compute a fast approximation to the absolute value of @var{x}. For | |
33008 | example, for a rectangular complex number the result is the sum of | |
33009 | the absolute values of the components. | |
33010 | @end defun | |
33011 | ||
33012 | @findex two-pi | |
33013 | @findex pi-over-2 | |
33014 | @findex pi-over-4 | |
33015 | @findex pi-over-180 | |
33016 | @findex sqrt-two-pi | |
33017 | @findex sqrt-e | |
33018 | @findex e | |
33019 | @findex ln-2 | |
33020 | @findex ln-10 | |
33021 | @defun pi | |
33022 | The function @samp{(pi)} computes @samp{pi} to the current precision. | |
33023 | Other related constant-generating functions are @code{two-pi}, | |
33024 | @code{pi-over-2}, @code{pi-over-4}, @code{pi-over-180}, @code{sqrt-two-pi}, | |
33025 | @code{e}, @code{sqrt-e}, @code{ln-2}, and @code{ln-10}. Each function | |
33026 | returns a floating-point value in the current precision, and each uses | |
33027 | caching so that all calls after the first are essentially free.@refill | |
33028 | @end defun | |
33029 | ||
33030 | @defmac math-defcache @var{func} @var{initial} @var{form} | |
33031 | This macro, usually used as a top-level call like @code{defun} or | |
33032 | @code{defvar}, defines a new cached constant analogous to @code{pi}, etc. | |
33033 | It defines a function @code{func} which returns the requested value; | |
33034 | if @var{initial} is non-@code{nil} it must be a @samp{(float @dots{})} | |
33035 | form which serves as an initial value for the cache. If @var{func} | |
33036 | is called when the cache is empty or does not have enough digits to | |
33037 | satisfy the current precision, the Lisp expression @var{form} is evaluated | |
33038 | with the current precision increased by four, and the result minus its | |
33039 | two least significant digits is stored in the cache. For example, | |
33040 | calling @samp{(pi)} with a precision of 30 computes @samp{pi} to 34 | |
33041 | digits, rounds it down to 32 digits for future use, then rounds it | |
33042 | again to 30 digits for use in the present request.@refill | |
33043 | @end defmac | |
33044 | ||
33045 | @findex half-circle | |
33046 | @findex quarter-circle | |
33047 | @defun full-circle symb | |
33048 | If the current angular mode is Degrees or HMS, this function returns the | |
33049 | integer 360. In Radians mode, this function returns either the | |
33050 | corresponding value in radians to the current precision, or the formula | |
33051 | @samp{2*pi}, depending on the Symbolic Mode. There are also similar | |
33052 | function @code{half-circle} and @code{quarter-circle}. | |
33053 | @end defun | |
33054 | ||
33055 | @defun power-of-2 n | |
33056 | Compute two to the integer power @var{n}, as a (potentially very large) | |
33057 | integer. Powers of two are cached, so only the first call for a | |
33058 | particular @var{n} is expensive. | |
33059 | @end defun | |
33060 | ||
33061 | @defun integer-log2 n | |
33062 | Compute the base-2 logarithm of @var{n}, which must be an integer which | |
33063 | is a power of two. If @var{n} is not a power of two, this function will | |
33064 | return @code{nil}. | |
33065 | @end defun | |
33066 | ||
33067 | @defun div-mod a b m | |
33068 | Divide @var{a} by @var{b}, modulo @var{m}. This returns @code{nil} if | |
33069 | there is no solution, or if any of the arguments are not integers.@refill | |
33070 | @end defun | |
33071 | ||
33072 | @defun pow-mod a b m | |
33073 | Compute @var{a} to the power @var{b}, modulo @var{m}. If @var{a}, | |
33074 | @var{b}, and @var{m} are integers, this uses an especially efficient | |
33075 | algorithm. Otherwise, it simply computes @samp{(% (^ a b) m)}. | |
33076 | @end defun | |
33077 | ||
33078 | @defun isqrt n | |
33079 | Compute the integer square root of @var{n}. This is the square root | |
33080 | of @var{n} rounded down toward zero, i.e., @samp{floor(sqrt(@var{n}))}. | |
33081 | If @var{n} is itself an integer, the computation is especially efficient. | |
33082 | @end defun | |
33083 | ||
33084 | @defun to-hms a ang | |
33085 | Convert the argument @var{a} into an HMS form. If @var{ang} is specified, | |
33086 | it is the angular mode in which to interpret @var{a}, either @code{deg} | |
33087 | or @code{rad}. Otherwise, the current angular mode is used. If @var{a} | |
33088 | is already an HMS form it is returned as-is. | |
33089 | @end defun | |
33090 | ||
33091 | @defun from-hms a ang | |
33092 | Convert the HMS form @var{a} into a real number. If @var{ang} is specified, | |
33093 | it is the angular mode in which to express the result, otherwise the | |
33094 | current angular mode is used. If @var{a} is already a real number, it | |
33095 | is returned as-is. | |
33096 | @end defun | |
33097 | ||
33098 | @defun to-radians a | |
33099 | Convert the number or HMS form @var{a} to radians from the current | |
33100 | angular mode. | |
33101 | @end defun | |
33102 | ||
33103 | @defun from-radians a | |
33104 | Convert the number @var{a} from radians to the current angular mode. | |
33105 | If @var{a} is a formula, this returns the formula @samp{deg(@var{a})}. | |
33106 | @end defun | |
33107 | ||
33108 | @defun to-radians-2 a | |
33109 | Like @code{to-radians}, except that in Symbolic Mode a degrees to | |
33110 | radians conversion yields a formula like @samp{@var{a}*pi/180}. | |
33111 | @end defun | |
33112 | ||
33113 | @defun from-radians-2 a | |
33114 | Like @code{from-radians}, except that in Symbolic Mode a radians to | |
33115 | degrees conversion yields a formula like @samp{@var{a}*180/pi}. | |
33116 | @end defun | |
33117 | ||
33118 | @defun random-digit | |
33119 | Produce a random base-1000 digit in the range 0 to 999. | |
33120 | @end defun | |
33121 | ||
33122 | @defun random-digits n | |
33123 | Produce a random @var{n}-digit integer; this will be an integer | |
33124 | in the interval @samp{[0, 10^@var{n})}. | |
33125 | @end defun | |
33126 | ||
33127 | @defun random-float | |
33128 | Produce a random float in the interval @samp{[0, 1)}. | |
33129 | @end defun | |
33130 | ||
33131 | @defun prime-test n iters | |
33132 | Determine whether the integer @var{n} is prime. Return a list which has | |
33133 | one of these forms: @samp{(nil @var{f})} means the number is non-prime | |
33134 | because it was found to be divisible by @var{f}; @samp{(nil)} means it | |
33135 | was found to be non-prime by table look-up (so no factors are known); | |
33136 | @samp{(nil unknown)} means it is definitely non-prime but no factors | |
33137 | are known because @var{n} was large enough that Fermat's probabilistic | |
33138 | test had to be used; @samp{(t)} means the number is definitely prime; | |
33139 | and @samp{(maybe @var{i} @var{p})} means that Fermat's test, after @var{i} | |
33140 | iterations, is @var{p} percent sure that the number is prime. The | |
33141 | @var{iters} parameter is the number of Fermat iterations to use, in the | |
33142 | case that this is necessary. If @code{prime-test} returns ``maybe,'' | |
33143 | you can call it again with the same @var{n} to get a greater certainty; | |
33144 | @code{prime-test} remembers where it left off.@refill | |
33145 | @end defun | |
33146 | ||
33147 | @defun to-simple-fraction f | |
33148 | If @var{f} is a floating-point number which can be represented exactly | |
33149 | as a small rational number. return that number, else return @var{f}. | |
33150 | For example, 0.75 would be converted to 3:4. This function is very | |
33151 | fast. | |
33152 | @end defun | |
33153 | ||
33154 | @defun to-fraction f tol | |
33155 | Find a rational approximation to floating-point number @var{f} to within | |
33156 | a specified tolerance @var{tol}; this corresponds to the algebraic | |
33157 | function @code{frac}, and can be rather slow. | |
33158 | @end defun | |
33159 | ||
33160 | @defun quarter-integer n | |
33161 | If @var{n} is an integer or integer-valued float, this function | |
33162 | returns zero. If @var{n} is a half-integer (i.e., an integer plus | |
33163 | @i{1:2} or 0.5), it returns 2. If @var{n} is a quarter-integer, | |
33164 | it returns 1 or 3. If @var{n} is anything else, this function | |
33165 | returns @code{nil}. | |
33166 | @end defun | |
33167 | ||
33168 | @node Vector Lisp Functions, Symbolic Lisp Functions, Computational Lisp Functions, Internals | |
33169 | @subsubsection Vector Functions | |
33170 | ||
33171 | @noindent | |
33172 | The functions described here perform various operations on vectors and | |
33173 | matrices. | |
33174 | ||
33175 | @defun math-concat x y | |
33176 | Do a vector concatenation; this operation is written @samp{@var{x} | @var{y}} | |
33177 | in a symbolic formula. @xref{Building Vectors}. | |
33178 | @end defun | |
33179 | ||
33180 | @defun vec-length v | |
33181 | Return the length of vector @var{v}. If @var{v} is not a vector, the | |
33182 | result is zero. If @var{v} is a matrix, this returns the number of | |
33183 | rows in the matrix. | |
33184 | @end defun | |
33185 | ||
33186 | @defun mat-dimens m | |
33187 | Determine the dimensions of vector or matrix @var{m}. If @var{m} is not | |
33188 | a vector, the result is an empty list. If @var{m} is a plain vector | |
33189 | but not a matrix, the result is a one-element list containing the length | |
33190 | of the vector. If @var{m} is a matrix with @var{r} rows and @var{c} columns, | |
33191 | the result is the list @samp{(@var{r} @var{c})}. Higher-order tensors | |
33192 | produce lists of more than two dimensions. Note that the object | |
33193 | @samp{[[1, 2, 3], [4, 5]]} is a vector of vectors not all the same size, | |
33194 | and is treated by this and other Calc routines as a plain vector of two | |
33195 | elements.@refill | |
33196 | @end defun | |
33197 | ||
33198 | @defun dimension-error | |
33199 | Abort the current function with a message of ``Dimension error.'' | |
33200 | The Calculator will leave the function being evaluated in symbolic | |
33201 | form; this is really just a special case of @code{reject-arg}. | |
33202 | @end defun | |
33203 | ||
33204 | @defun build-vector args | |
5d67986c | 33205 | Return a Calc vector with @var{args} as elements. |
d7b8e6c6 EZ |
33206 | For example, @samp{(build-vector 1 2 3)} returns the Calc vector |
33207 | @samp{[1, 2, 3]}, stored internally as the list @samp{(vec 1 2 3)}. | |
33208 | @end defun | |
33209 | ||
33210 | @defun make-vec obj dims | |
33211 | Return a Calc vector or matrix all of whose elements are equal to | |
33212 | @var{obj}. For example, @samp{(make-vec 27 3 4)} returns a 3x4 matrix | |
33213 | filled with 27's. | |
33214 | @end defun | |
33215 | ||
33216 | @defun row-matrix v | |
33217 | If @var{v} is a plain vector, convert it into a row matrix, i.e., | |
33218 | a matrix whose single row is @var{v}. If @var{v} is already a matrix, | |
33219 | leave it alone. | |
33220 | @end defun | |
33221 | ||
33222 | @defun col-matrix v | |
33223 | If @var{v} is a plain vector, convert it into a column matrix, i.e., a | |
33224 | matrix with each element of @var{v} as a separate row. If @var{v} is | |
33225 | already a matrix, leave it alone. | |
33226 | @end defun | |
33227 | ||
33228 | @defun map-vec f v | |
33229 | Map the Lisp function @var{f} over the Calc vector @var{v}. For example, | |
33230 | @samp{(map-vec 'math-floor v)} returns a vector of the floored components | |
33231 | of vector @var{v}. | |
33232 | @end defun | |
33233 | ||
33234 | @defun map-vec-2 f a b | |
33235 | Map the Lisp function @var{f} over the two vectors @var{a} and @var{b}. | |
33236 | If @var{a} and @var{b} are vectors of equal length, the result is a | |
33237 | vector of the results of calling @samp{(@var{f} @var{ai} @var{bi})} | |
33238 | for each pair of elements @var{ai} and @var{bi}. If either @var{a} or | |
33239 | @var{b} is a scalar, it is matched with each value of the other vector. | |
33240 | For example, @samp{(map-vec-2 'math-add v 1)} returns the vector @var{v} | |
33241 | with each element increased by one. Note that using @samp{'+} would not | |
33242 | work here, since @code{defmath} does not expand function names everywhere, | |
33243 | just where they are in the function position of a Lisp expression.@refill | |
33244 | @end defun | |
33245 | ||
33246 | @defun reduce-vec f v | |
33247 | Reduce the function @var{f} over the vector @var{v}. For example, if | |
33248 | @var{v} is @samp{[10, 20, 30, 40]}, this calls @samp{(f (f (f 10 20) 30) 40)}. | |
33249 | If @var{v} is a matrix, this reduces over the rows of @var{v}. | |
33250 | @end defun | |
33251 | ||
33252 | @defun reduce-cols f m | |
33253 | Reduce the function @var{f} over the columns of matrix @var{m}. For | |
33254 | example, if @var{m} is @samp{[[1, 2], [3, 4], [5, 6]]}, the result | |
33255 | is a vector of the two elements @samp{(f (f 1 3) 5)} and @samp{(f (f 2 4) 6)}. | |
33256 | @end defun | |
33257 | ||
33258 | @defun mat-row m n | |
33259 | Return the @var{n}th row of matrix @var{m}. This is equivalent to | |
33260 | @samp{(elt m n)}. For a slower but safer version, use @code{mrow}. | |
33261 | (@xref{Extracting Elements}.) | |
33262 | @end defun | |
33263 | ||
33264 | @defun mat-col m n | |
33265 | Return the @var{n}th column of matrix @var{m}, in the form of a vector. | |
33266 | The arguments are not checked for correctness. | |
33267 | @end defun | |
33268 | ||
33269 | @defun mat-less-row m n | |
33270 | Return a copy of matrix @var{m} with its @var{n}th row deleted. The | |
33271 | number @var{n} must be in range from 1 to the number of rows in @var{m}. | |
33272 | @end defun | |
33273 | ||
33274 | @defun mat-less-col m n | |
33275 | Return a copy of matrix @var{m} with its @var{n}th column deleted. | |
33276 | @end defun | |
33277 | ||
33278 | @defun transpose m | |
33279 | Return the transpose of matrix @var{m}. | |
33280 | @end defun | |
33281 | ||
33282 | @defun flatten-vector v | |
33283 | Flatten nested vector @var{v} into a vector of scalars. For example, | |
33284 | if @var{v} is @samp{[[1, 2, 3], [4, 5]]} the result is @samp{[1, 2, 3, 4, 5]}. | |
33285 | @end defun | |
33286 | ||
33287 | @defun copy-matrix m | |
33288 | If @var{m} is a matrix, return a copy of @var{m}. This maps | |
33289 | @code{copy-sequence} over the rows of @var{m}; in Lisp terms, each | |
33290 | element of the result matrix will be @code{eq} to the corresponding | |
33291 | element of @var{m}, but none of the @code{cons} cells that make up | |
33292 | the structure of the matrix will be @code{eq}. If @var{m} is a plain | |
33293 | vector, this is the same as @code{copy-sequence}.@refill | |
33294 | @end defun | |
33295 | ||
33296 | @defun swap-rows m r1 r2 | |
33297 | Exchange rows @var{r1} and @var{r2} of matrix @var{m} in-place. In | |
33298 | other words, unlike most of the other functions described here, this | |
33299 | function changes @var{m} itself rather than building up a new result | |
33300 | matrix. The return value is @var{m}, i.e., @samp{(eq (swap-rows m 1 2) m)} | |
33301 | is true, with the side effect of exchanging the first two rows of | |
33302 | @var{m}.@refill | |
33303 | @end defun | |
33304 | ||
33305 | @node Symbolic Lisp Functions, Formatting Lisp Functions, Vector Lisp Functions, Internals | |
33306 | @subsubsection Symbolic Functions | |
33307 | ||
33308 | @noindent | |
33309 | The functions described here operate on symbolic formulas in the | |
33310 | Calculator. | |
33311 | ||
33312 | @defun calc-prepare-selection num | |
33313 | Prepare a stack entry for selection operations. If @var{num} is | |
33314 | omitted, the stack entry containing the cursor is used; otherwise, | |
33315 | it is the number of the stack entry to use. This function stores | |
33316 | useful information about the current stack entry into a set of | |
33317 | variables. @code{calc-selection-cache-num} contains the number of | |
33318 | the stack entry involved (equal to @var{num} if you specified it); | |
33319 | @code{calc-selection-cache-entry} contains the stack entry as a | |
33320 | list (such as @code{calc-top-list} would return with @code{entry} | |
33321 | as the selection mode); and @code{calc-selection-cache-comp} contains | |
33322 | a special ``tagged'' composition (@pxref{Formatting Lisp Functions}) | |
33323 | which allows Calc to relate cursor positions in the buffer with | |
33324 | their corresponding sub-formulas. | |
33325 | ||
33326 | A slight complication arises in the selection mechanism because | |
33327 | formulas may contain small integers. For example, in the vector | |
33328 | @samp{[1, 2, 1]} the first and last elements are @code{eq} to each | |
33329 | other; selections are recorded as the actual Lisp object that | |
33330 | appears somewhere in the tree of the whole formula, but storing | |
33331 | @code{1} would falsely select both @code{1}'s in the vector. So | |
33332 | @code{calc-prepare-selection} also checks the stack entry and | |
33333 | replaces any plain integers with ``complex number'' lists of the form | |
33334 | @samp{(cplx @var{n} 0)}. This list will be displayed the same as a | |
33335 | plain @var{n} and the change will be completely invisible to the | |
33336 | user, but it will guarantee that no two sub-formulas of the stack | |
33337 | entry will be @code{eq} to each other. Next time the stack entry | |
33338 | is involved in a computation, @code{calc-normalize} will replace | |
33339 | these lists with plain numbers again, again invisibly to the user. | |
33340 | @end defun | |
33341 | ||
33342 | @defun calc-encase-atoms x | |
33343 | This modifies the formula @var{x} to ensure that each part of the | |
33344 | formula is a unique atom, using the @samp{(cplx @var{n} 0)} trick | |
33345 | described above. This function may use @code{setcar} to modify | |
33346 | the formula in-place. | |
33347 | @end defun | |
33348 | ||
33349 | @defun calc-find-selected-part | |
33350 | Find the smallest sub-formula of the current formula that contains | |
33351 | the cursor. This assumes @code{calc-prepare-selection} has been | |
33352 | called already. If the cursor is not actually on any part of the | |
33353 | formula, this returns @code{nil}. | |
33354 | @end defun | |
33355 | ||
33356 | @defun calc-change-current-selection selection | |
33357 | Change the currently prepared stack element's selection to | |
33358 | @var{selection}, which should be @code{eq} to some sub-formula | |
33359 | of the stack element, or @code{nil} to unselect the formula. | |
33360 | The stack element's appearance in the Calc buffer is adjusted | |
33361 | to reflect the new selection. | |
33362 | @end defun | |
33363 | ||
33364 | @defun calc-find-nth-part expr n | |
33365 | Return the @var{n}th sub-formula of @var{expr}. This function is used | |
33366 | by the selection commands, and (unless @kbd{j b} has been used) treats | |
33367 | sums and products as flat many-element formulas. Thus if @var{expr} | |
33368 | is @samp{((a + b) - c) + d}, calling @code{calc-find-nth-part} with | |
33369 | @var{n} equal to four will return @samp{d}. | |
33370 | @end defun | |
33371 | ||
33372 | @defun calc-find-parent-formula expr part | |
33373 | Return the sub-formula of @var{expr} which immediately contains | |
33374 | @var{part}. If @var{expr} is @samp{a*b + (c+1)*d} and @var{part} | |
33375 | is @code{eq} to the @samp{c+1} term of @var{expr}, then this function | |
33376 | will return @samp{(c+1)*d}. If @var{part} turns out not to be a | |
33377 | sub-formula of @var{expr}, the function returns @code{nil}. If | |
33378 | @var{part} is @code{eq} to @var{expr}, the function returns @code{t}. | |
33379 | This function does not take associativity into account. | |
33380 | @end defun | |
33381 | ||
33382 | @defun calc-find-assoc-parent-formula expr part | |
33383 | This is the same as @code{calc-find-parent-formula}, except that | |
33384 | (unless @kbd{j b} has been used) it continues widening the selection | |
33385 | to contain a complete level of the formula. Given @samp{a} from | |
33386 | @samp{((a + b) - c) + d}, @code{calc-find-parent-formula} will | |
33387 | return @samp{a + b} but @code{calc-find-assoc-parent-formula} will | |
33388 | return the whole expression. | |
33389 | @end defun | |
33390 | ||
33391 | @defun calc-grow-assoc-formula expr part | |
33392 | This expands sub-formula @var{part} of @var{expr} to encompass a | |
33393 | complete level of the formula. If @var{part} and its immediate | |
33394 | parent are not compatible associative operators, or if @kbd{j b} | |
33395 | has been used, this simply returns @var{part}. | |
33396 | @end defun | |
33397 | ||
33398 | @defun calc-find-sub-formula expr part | |
33399 | This finds the immediate sub-formula of @var{expr} which contains | |
33400 | @var{part}. It returns an index @var{n} such that | |
33401 | @samp{(calc-find-nth-part @var{expr} @var{n})} would return @var{part}. | |
33402 | If @var{part} is not a sub-formula of @var{expr}, it returns @code{nil}. | |
33403 | If @var{part} is @code{eq} to @var{expr}, it returns @code{t}. This | |
33404 | function does not take associativity into account. | |
33405 | @end defun | |
33406 | ||
33407 | @defun calc-replace-sub-formula expr old new | |
33408 | This function returns a copy of formula @var{expr}, with the | |
33409 | sub-formula that is @code{eq} to @var{old} replaced by @var{new}. | |
33410 | @end defun | |
33411 | ||
33412 | @defun simplify expr | |
33413 | Simplify the expression @var{expr} by applying various algebraic rules. | |
33414 | This is what the @w{@kbd{a s}} (@code{calc-simplify}) command uses. This | |
33415 | always returns a copy of the expression; the structure @var{expr} points | |
33416 | to remains unchanged in memory. | |
33417 | ||
33418 | More precisely, here is what @code{simplify} does: The expression is | |
33419 | first normalized and evaluated by calling @code{normalize}. If any | |
33420 | @code{AlgSimpRules} have been defined, they are then applied. Then | |
33421 | the expression is traversed in a depth-first, bottom-up fashion; at | |
33422 | each level, any simplifications that can be made are made until no | |
33423 | further changes are possible. Once the entire formula has been | |
33424 | traversed in this way, it is compared with the original formula (from | |
33425 | before the call to @code{normalize}) and, if it has changed, | |
33426 | the entire procedure is repeated (starting with @code{normalize}) | |
33427 | until no further changes occur. Usually only two iterations are | |
33428 | needed:@: one to simplify the formula, and another to verify that no | |
33429 | further simplifications were possible. | |
33430 | @end defun | |
33431 | ||
33432 | @defun simplify-extended expr | |
33433 | Simplify the expression @var{expr}, with additional rules enabled that | |
33434 | help do a more thorough job, while not being entirely ``safe'' in all | |
33435 | circumstances. (For example, this mode will simplify @samp{sqrt(x^2)} | |
33436 | to @samp{x}, which is only valid when @var{x} is positive.) This is | |
33437 | implemented by temporarily binding the variable @code{math-living-dangerously} | |
33438 | to @code{t} (using a @code{let} form) and calling @code{simplify}. | |
33439 | Dangerous simplification rules are written to check this variable | |
33440 | before taking any action.@refill | |
33441 | @end defun | |
33442 | ||
33443 | @defun simplify-units expr | |
33444 | Simplify the expression @var{expr}, treating variable names as units | |
33445 | whenever possible. This works by binding the variable | |
33446 | @code{math-simplifying-units} to @code{t} while calling @code{simplify}. | |
33447 | @end defun | |
33448 | ||
33449 | @defmac math-defsimplify funcs body | |
33450 | Register a new simplification rule; this is normally called as a top-level | |
33451 | form, like @code{defun} or @code{defmath}. If @var{funcs} is a symbol | |
33452 | (like @code{+} or @code{calcFunc-sqrt}), this simplification rule is | |
33453 | applied to the formulas which are calls to the specified function. Or, | |
33454 | @var{funcs} can be a list of such symbols; the rule applies to all | |
33455 | functions on the list. The @var{body} is written like the body of a | |
33456 | function with a single argument called @code{expr}. The body will be | |
33457 | executed with @code{expr} bound to a formula which is a call to one of | |
33458 | the functions @var{funcs}. If the function body returns @code{nil}, or | |
33459 | if it returns a result @code{equal} to the original @code{expr}, it is | |
33460 | ignored and Calc goes on to try the next simplification rule that applies. | |
33461 | If the function body returns something different, that new formula is | |
33462 | substituted for @var{expr} in the original formula.@refill | |
33463 | ||
33464 | At each point in the formula, rules are tried in the order of the | |
33465 | original calls to @code{math-defsimplify}; the search stops after the | |
33466 | first rule that makes a change. Thus later rules for that same | |
33467 | function will not have a chance to trigger until the next iteration | |
33468 | of the main @code{simplify} loop. | |
33469 | ||
33470 | Note that, since @code{defmath} is not being used here, @var{body} must | |
33471 | be written in true Lisp code without the conveniences that @code{defmath} | |
33472 | provides. If you prefer, you can have @var{body} simply call another | |
33473 | function (defined with @code{defmath}) which does the real work. | |
33474 | ||
33475 | The arguments of a function call will already have been simplified | |
33476 | before any rules for the call itself are invoked. Since a new argument | |
33477 | list is consed up when this happens, this means that the rule's body is | |
33478 | allowed to rearrange the function's arguments destructively if that is | |
33479 | convenient. Here is a typical example of a simplification rule: | |
33480 | ||
33481 | @smallexample | |
33482 | (math-defsimplify calcFunc-arcsinh | |
33483 | (or (and (math-looks-negp (nth 1 expr)) | |
33484 | (math-neg (list 'calcFunc-arcsinh | |
33485 | (math-neg (nth 1 expr))))) | |
33486 | (and (eq (car-safe (nth 1 expr)) 'calcFunc-sinh) | |
33487 | (or math-living-dangerously | |
33488 | (math-known-realp (nth 1 (nth 1 expr)))) | |
33489 | (nth 1 (nth 1 expr))))) | |
33490 | @end smallexample | |
33491 | ||
33492 | This is really a pair of rules written with one @code{math-defsimplify} | |
33493 | for convenience; the first replaces @samp{arcsinh(-x)} with | |
33494 | @samp{-arcsinh(x)}, and the second, which is safe only for real @samp{x}, | |
33495 | replaces @samp{arcsinh(sinh(x))} with @samp{x}.@refill | |
33496 | @end defmac | |
33497 | ||
33498 | @defun common-constant-factor expr | |
33499 | Check @var{expr} to see if it is a sum of terms all multiplied by the | |
33500 | same rational value. If so, return this value. If not, return @code{nil}. | |
33501 | For example, if called on @samp{6x + 9y + 12z}, it would return 3, since | |
33502 | 3 is a common factor of all the terms. | |
33503 | @end defun | |
33504 | ||
33505 | @defun cancel-common-factor expr factor | |
33506 | Assuming @var{expr} is a sum with @var{factor} as a common factor, | |
33507 | divide each term of the sum by @var{factor}. This is done by | |
33508 | destructively modifying parts of @var{expr}, on the assumption that | |
33509 | it is being used by a simplification rule (where such things are | |
33510 | allowed; see above). For example, consider this built-in rule for | |
33511 | square roots: | |
33512 | ||
33513 | @smallexample | |
33514 | (math-defsimplify calcFunc-sqrt | |
33515 | (let ((fac (math-common-constant-factor (nth 1 expr)))) | |
33516 | (and fac (not (eq fac 1)) | |
33517 | (math-mul (math-normalize (list 'calcFunc-sqrt fac)) | |
33518 | (math-normalize | |
33519 | (list 'calcFunc-sqrt | |
33520 | (math-cancel-common-factor | |
33521 | (nth 1 expr) fac))))))) | |
33522 | @end smallexample | |
33523 | @end defun | |
33524 | ||
33525 | @defun frac-gcd a b | |
33526 | Compute a ``rational GCD'' of @var{a} and @var{b}, which must both be | |
33527 | rational numbers. This is the fraction composed of the GCD of the | |
33528 | numerators of @var{a} and @var{b}, over the GCD of the denominators. | |
33529 | It is used by @code{common-constant-factor}. Note that the standard | |
33530 | @code{gcd} function uses the LCM to combine the denominators.@refill | |
33531 | @end defun | |
33532 | ||
33533 | @defun map-tree func expr many | |
33534 | Try applying Lisp function @var{func} to various sub-expressions of | |
33535 | @var{expr}. Initially, call @var{func} with @var{expr} itself as an | |
33536 | argument. If this returns an expression which is not @code{equal} to | |
33537 | @var{expr}, apply @var{func} again until eventually it does return | |
33538 | @var{expr} with no changes. Then, if @var{expr} is a function call, | |
33539 | recursively apply @var{func} to each of the arguments. This keeps going | |
33540 | until no changes occur anywhere in the expression; this final expression | |
33541 | is returned by @code{map-tree}. Note that, unlike simplification rules, | |
33542 | @var{func} functions may @emph{not} make destructive changes to | |
33543 | @var{expr}. If a third argument @var{many} is provided, it is an | |
33544 | integer which says how many times @var{func} may be applied; the | |
33545 | default, as described above, is infinitely many times.@refill | |
33546 | @end defun | |
33547 | ||
33548 | @defun compile-rewrites rules | |
33549 | Compile the rewrite rule set specified by @var{rules}, which should | |
33550 | be a formula that is either a vector or a variable name. If the latter, | |
33551 | the compiled rules are saved so that later @code{compile-rules} calls | |
33552 | for that same variable can return immediately. If there are problems | |
33553 | with the rules, this function calls @code{error} with a suitable | |
33554 | message. | |
33555 | @end defun | |
33556 | ||
33557 | @defun apply-rewrites expr crules heads | |
33558 | Apply the compiled rewrite rule set @var{crules} to the expression | |
33559 | @var{expr}. This will make only one rewrite and only checks at the | |
33560 | top level of the expression. The result @code{nil} if no rules | |
33561 | matched, or if the only rules that matched did not actually change | |
33562 | the expression. The @var{heads} argument is optional; if is given, | |
33563 | it should be a list of all function names that (may) appear in | |
33564 | @var{expr}. The rewrite compiler tags each rule with the | |
33565 | rarest-looking function name in the rule; if you specify @var{heads}, | |
33566 | @code{apply-rewrites} can use this information to narrow its search | |
33567 | down to just a few rules in the rule set. | |
33568 | @end defun | |
33569 | ||
33570 | @defun rewrite-heads expr | |
33571 | Compute a @var{heads} list for @var{expr} suitable for use with | |
33572 | @code{apply-rewrites}, as discussed above. | |
33573 | @end defun | |
33574 | ||
33575 | @defun rewrite expr rules many | |
33576 | This is an all-in-one rewrite function. It compiles the rule set | |
33577 | specified by @var{rules}, then uses @code{map-tree} to apply the | |
33578 | rules throughout @var{expr} up to @var{many} (default infinity) | |
33579 | times. | |
33580 | @end defun | |
33581 | ||
33582 | @defun match-patterns pat vec not-flag | |
33583 | Given a Calc vector @var{vec} and an uncompiled pattern set or | |
33584 | pattern set variable @var{pat}, this function returns a new vector | |
33585 | of all elements of @var{vec} which do (or don't, if @var{not-flag} is | |
33586 | non-@code{nil}) match any of the patterns in @var{pat}. | |
33587 | @end defun | |
33588 | ||
33589 | @defun deriv expr var value symb | |
33590 | Compute the derivative of @var{expr} with respect to variable @var{var} | |
33591 | (which may actually be any sub-expression). If @var{value} is specified, | |
33592 | the derivative is evaluated at the value of @var{var}; otherwise, the | |
33593 | derivative is left in terms of @var{var}. If the expression contains | |
33594 | functions for which no derivative formula is known, new derivative | |
33595 | functions are invented by adding primes to the names; @pxref{Calculus}. | |
33596 | However, if @var{symb} is non-@code{nil}, the presence of undifferentiable | |
33597 | functions in @var{expr} instead cancels the whole differentiation, and | |
33598 | @code{deriv} returns @code{nil} instead. | |
33599 | ||
33600 | Derivatives of an @var{n}-argument function can be defined by | |
33601 | adding a @code{math-derivative-@var{n}} property to the property list | |
33602 | of the symbol for the function's derivative, which will be the | |
33603 | function name followed by an apostrophe. The value of the property | |
33604 | should be a Lisp function; it is called with the same arguments as the | |
33605 | original function call that is being differentiated. It should return | |
33606 | a formula for the derivative. For example, the derivative of @code{ln} | |
33607 | is defined by | |
33608 | ||
33609 | @smallexample | |
33610 | (put 'calcFunc-ln\' 'math-derivative-1 | |
33611 | (function (lambda (u) (math-div 1 u)))) | |
33612 | @end smallexample | |
33613 | ||
33614 | The two-argument @code{log} function has two derivatives, | |
33615 | @smallexample | |
33616 | (put 'calcFunc-log\' 'math-derivative-2 ; d(log(x,b)) / dx | |
33617 | (function (lambda (x b) ... ))) | |
33618 | (put 'calcFunc-log\'2 'math-derivative-2 ; d(log(x,b)) / db | |
33619 | (function (lambda (x b) ... ))) | |
33620 | @end smallexample | |
33621 | @end defun | |
33622 | ||
33623 | @defun tderiv expr var value symb | |
33624 | Compute the total derivative of @var{expr}. This is the same as | |
33625 | @code{deriv}, except that variables other than @var{var} are not | |
33626 | assumed to be constant with respect to @var{var}. | |
33627 | @end defun | |
33628 | ||
33629 | @defun integ expr var low high | |
33630 | Compute the integral of @var{expr} with respect to @var{var}. | |
33631 | @xref{Calculus}, for further details. | |
33632 | @end defun | |
33633 | ||
33634 | @defmac math-defintegral funcs body | |
33635 | Define a rule for integrating a function or functions of one argument; | |
33636 | this macro is very similar in format to @code{math-defsimplify}. | |
33637 | The main difference is that here @var{body} is the body of a function | |
33638 | with a single argument @code{u} which is bound to the argument to the | |
33639 | function being integrated, not the function call itself. Also, the | |
33640 | variable of integration is available as @code{math-integ-var}. If | |
33641 | evaluation of the integral requires doing further integrals, the body | |
33642 | should call @samp{(math-integral @var{x})} to find the integral of | |
33643 | @var{x} with respect to @code{math-integ-var}; this function returns | |
33644 | @code{nil} if the integral could not be done. Some examples: | |
33645 | ||
33646 | @smallexample | |
33647 | (math-defintegral calcFunc-conj | |
33648 | (let ((int (math-integral u))) | |
33649 | (and int | |
33650 | (list 'calcFunc-conj int)))) | |
33651 | ||
33652 | (math-defintegral calcFunc-cos | |
33653 | (and (equal u math-integ-var) | |
33654 | (math-from-radians-2 (list 'calcFunc-sin u)))) | |
33655 | @end smallexample | |
33656 | ||
33657 | In the @code{cos} example, we define only the integral of @samp{cos(x) dx}, | |
33658 | relying on the general integration-by-substitution facility to handle | |
33659 | cosines of more complicated arguments. An integration rule should return | |
33660 | @code{nil} if it can't do the integral; if several rules are defined for | |
33661 | the same function, they are tried in order until one returns a non-@code{nil} | |
33662 | result.@refill | |
33663 | @end defmac | |
33664 | ||
33665 | @defmac math-defintegral-2 funcs body | |
33666 | Define a rule for integrating a function or functions of two arguments. | |
33667 | This is exactly analogous to @code{math-defintegral}, except that @var{body} | |
33668 | is written as the body of a function with two arguments, @var{u} and | |
33669 | @var{v}.@refill | |
33670 | @end defmac | |
33671 | ||
33672 | @defun solve-for lhs rhs var full | |
33673 | Attempt to solve the equation @samp{@var{lhs} = @var{rhs}} by isolating | |
33674 | the variable @var{var} on the lefthand side; return the resulting righthand | |
33675 | side, or @code{nil} if the equation cannot be solved. The variable | |
33676 | @var{var} must appear at least once in @var{lhs} or @var{rhs}. Note that | |
33677 | the return value is a formula which does not contain @var{var}; this is | |
33678 | different from the user-level @code{solve} and @code{finv} functions, | |
33679 | which return a rearranged equation or a functional inverse, respectively. | |
33680 | If @var{full} is non-@code{nil}, a full solution including dummy signs | |
33681 | and dummy integers will be produced. User-defined inverses are provided | |
33682 | as properties in a manner similar to derivatives:@refill | |
33683 | ||
33684 | @smallexample | |
33685 | (put 'calcFunc-ln 'math-inverse | |
33686 | (function (lambda (x) (list 'calcFunc-exp x)))) | |
33687 | @end smallexample | |
33688 | ||
33689 | This function can call @samp{(math-solve-get-sign @var{x})} to create | |
33690 | a new arbitrary sign variable, returning @var{x} times that sign, and | |
33691 | @samp{(math-solve-get-int @var{x})} to create a new arbitrary integer | |
33692 | variable multiplied by @var{x}. These functions simply return @var{x} | |
33693 | if the caller requested a non-``full'' solution. | |
33694 | @end defun | |
33695 | ||
33696 | @defun solve-eqn expr var full | |
33697 | This version of @code{solve-for} takes an expression which will | |
33698 | typically be an equation or inequality. (If it is not, it will be | |
33699 | interpreted as the equation @samp{@var{expr} = 0}.) It returns an | |
33700 | equation or inequality, or @code{nil} if no solution could be found. | |
33701 | @end defun | |
33702 | ||
33703 | @defun solve-system exprs vars full | |
33704 | This function solves a system of equations. Generally, @var{exprs} | |
33705 | and @var{vars} will be vectors of equal length. | |
33706 | @xref{Solving Systems of Equations}, for other options. | |
33707 | @end defun | |
33708 | ||
33709 | @defun expr-contains expr var | |
33710 | Returns a non-@code{nil} value if @var{var} occurs as a subexpression | |
33711 | of @var{expr}. | |
33712 | ||
33713 | This function might seem at first to be identical to | |
33714 | @code{calc-find-sub-formula}. The key difference is that | |
33715 | @code{expr-contains} uses @code{equal} to test for matches, whereas | |
33716 | @code{calc-find-sub-formula} uses @code{eq}. In the formula | |
33717 | @samp{f(a, a)}, the two @samp{a}s will be @code{equal} but not | |
33718 | @code{eq} to each other.@refill | |
33719 | @end defun | |
33720 | ||
33721 | @defun expr-contains-count expr var | |
33722 | Returns the number of occurrences of @var{var} as a subexpression | |
33723 | of @var{expr}, or @code{nil} if there are no occurrences.@refill | |
33724 | @end defun | |
33725 | ||
33726 | @defun expr-depends expr var | |
33727 | Returns true if @var{expr} refers to any variable the occurs in @var{var}. | |
33728 | In other words, it checks if @var{expr} and @var{var} have any variables | |
33729 | in common. | |
33730 | @end defun | |
33731 | ||
33732 | @defun expr-contains-vars expr | |
33733 | Return true if @var{expr} contains any variables, or @code{nil} if @var{expr} | |
33734 | contains only constants and functions with constant arguments. | |
33735 | @end defun | |
33736 | ||
33737 | @defun expr-subst expr old new | |
33738 | Returns a copy of @var{expr}, with all occurrences of @var{old} replaced | |
33739 | by @var{new}. This treats @code{lambda} forms specially with respect | |
33740 | to the dummy argument variables, so that the effect is always to return | |
33741 | @var{expr} evaluated at @var{old} = @var{new}.@refill | |
33742 | @end defun | |
33743 | ||
33744 | @defun multi-subst expr old new | |
33745 | This is like @code{expr-subst}, except that @var{old} and @var{new} | |
33746 | are lists of expressions to be substituted simultaneously. If one | |
33747 | list is shorter than the other, trailing elements of the longer list | |
33748 | are ignored. | |
33749 | @end defun | |
33750 | ||
33751 | @defun expr-weight expr | |
33752 | Returns the ``weight'' of @var{expr}, basically a count of the total | |
33753 | number of objects and function calls that appear in @var{expr}. For | |
33754 | ``primitive'' objects, this will be one. | |
33755 | @end defun | |
33756 | ||
33757 | @defun expr-height expr | |
33758 | Returns the ``height'' of @var{expr}, which is the deepest level to | |
33759 | which function calls are nested. (Note that @samp{@var{a} + @var{b}} | |
33760 | counts as a function call.) For primitive objects, this returns zero.@refill | |
33761 | @end defun | |
33762 | ||
33763 | @defun polynomial-p expr var | |
33764 | Check if @var{expr} is a polynomial in variable (or sub-expression) | |
33765 | @var{var}. If so, return the degree of the polynomial, that is, the | |
33766 | highest power of @var{var} that appears in @var{expr}. For example, | |
33767 | for @samp{(x^2 + 3)^3 + 4} this would return 6. This function returns | |
33768 | @code{nil} unless @var{expr}, when expanded out by @kbd{a x} | |
33769 | (@code{calc-expand}), would consist of a sum of terms in which @var{var} | |
33770 | appears only raised to nonnegative integer powers. Note that if | |
33771 | @var{var} does not occur in @var{expr}, then @var{expr} is considered | |
33772 | a polynomial of degree 0.@refill | |
33773 | @end defun | |
33774 | ||
33775 | @defun is-polynomial expr var degree loose | |
33776 | Check if @var{expr} is a polynomial in variable or sub-expression | |
33777 | @var{var}, and, if so, return a list representation of the polynomial | |
33778 | where the elements of the list are coefficients of successive powers of | |
33779 | @var{var}: @samp{@var{a} + @var{b} x + @var{c} x^3} would produce the | |
33780 | list @samp{(@var{a} @var{b} 0 @var{c})}, and @samp{(x + 1)^2} would | |
33781 | produce the list @samp{(1 2 1)}. The highest element of the list will | |
33782 | be non-zero, with the special exception that if @var{expr} is the | |
33783 | constant zero, the returned value will be @samp{(0)}. Return @code{nil} | |
33784 | if @var{expr} is not a polynomial in @var{var}. If @var{degree} is | |
33785 | specified, this will not consider polynomials of degree higher than that | |
33786 | value. This is a good precaution because otherwise an input of | |
33787 | @samp{(x+1)^1000} will cause a huge coefficient list to be built. If | |
33788 | @var{loose} is non-@code{nil}, then a looser definition of a polynomial | |
33789 | is used in which coefficients are no longer required not to depend on | |
33790 | @var{var}, but are only required not to take the form of polynomials | |
33791 | themselves. For example, @samp{sin(x) x^2 + cos(x)} is a loose | |
33792 | polynomial with coefficients @samp{((calcFunc-cos x) 0 (calcFunc-sin | |
33793 | x))}. The result will never be @code{nil} in loose mode, since any | |
33794 | expression can be interpreted as a ``constant'' loose polynomial.@refill | |
33795 | @end defun | |
33796 | ||
33797 | @defun polynomial-base expr pred | |
33798 | Check if @var{expr} is a polynomial in any variable that occurs in it; | |
33799 | if so, return that variable. (If @var{expr} is a multivariate polynomial, | |
33800 | this chooses one variable arbitrarily.) If @var{pred} is specified, it should | |
33801 | be a Lisp function which is called as @samp{(@var{pred} @var{subexpr})}, | |
33802 | and which should return true if @code{mpb-top-expr} (a global name for | |
33803 | the original @var{expr}) is a suitable polynomial in @var{subexpr}. | |
33804 | The default predicate uses @samp{(polynomial-p mpb-top-expr @var{subexpr})}; | |
33805 | you can use @var{pred} to specify additional conditions. Or, you could | |
33806 | have @var{pred} build up a list of every suitable @var{subexpr} that | |
33807 | is found.@refill | |
33808 | @end defun | |
33809 | ||
33810 | @defun poly-simplify poly | |
33811 | Simplify polynomial coefficient list @var{poly} by (destructively) | |
33812 | clipping off trailing zeros. | |
33813 | @end defun | |
33814 | ||
33815 | @defun poly-mix a ac b bc | |
33816 | Mix two polynomial lists @var{a} and @var{b} (in the form returned by | |
33817 | @code{is-polynomial}) in a linear combination with coefficient expressions | |
33818 | @var{ac} and @var{bc}. The result is a (not necessarily simplified) | |
33819 | polynomial list representing @samp{@var{ac} @var{a} + @var{bc} @var{b}}.@refill | |
33820 | @end defun | |
33821 | ||
33822 | @defun poly-mul a b | |
33823 | Multiply two polynomial coefficient lists @var{a} and @var{b}. The | |
33824 | result will be in simplified form if the inputs were simplified. | |
33825 | @end defun | |
33826 | ||
33827 | @defun build-polynomial-expr poly var | |
33828 | Construct a Calc formula which represents the polynomial coefficient | |
33829 | list @var{poly} applied to variable @var{var}. The @kbd{a c} | |
33830 | (@code{calc-collect}) command uses @code{is-polynomial} to turn an | |
33831 | expression into a coefficient list, then @code{build-polynomial-expr} | |
33832 | to turn the list back into an expression in regular form.@refill | |
33833 | @end defun | |
33834 | ||
33835 | @defun check-unit-name var | |
33836 | Check if @var{var} is a variable which can be interpreted as a unit | |
33837 | name. If so, return the units table entry for that unit. This | |
33838 | will be a list whose first element is the unit name (not counting | |
33839 | prefix characters) as a symbol and whose second element is the | |
33840 | Calc expression which defines the unit. (Refer to the Calc sources | |
33841 | for details on the remaining elements of this list.) If @var{var} | |
33842 | is not a variable or is not a unit name, return @code{nil}. | |
33843 | @end defun | |
33844 | ||
33845 | @defun units-in-expr-p expr sub-exprs | |
33846 | Return true if @var{expr} contains any variables which can be | |
33847 | interpreted as units. If @var{sub-exprs} is @code{t}, the entire | |
33848 | expression is searched. If @var{sub-exprs} is @code{nil}, this | |
33849 | checks whether @var{expr} is directly a units expression.@refill | |
33850 | @end defun | |
33851 | ||
33852 | @defun single-units-in-expr-p expr | |
33853 | Check whether @var{expr} contains exactly one units variable. If so, | |
33854 | return the units table entry for the variable. If @var{expr} does | |
33855 | not contain any units, return @code{nil}. If @var{expr} contains | |
33856 | two or more units, return the symbol @code{wrong}. | |
33857 | @end defun | |
33858 | ||
33859 | @defun to-standard-units expr which | |
33860 | Convert units expression @var{expr} to base units. If @var{which} | |
33861 | is @code{nil}, use Calc's native base units. Otherwise, @var{which} | |
33862 | can specify a units system, which is a list of two-element lists, | |
33863 | where the first element is a Calc base symbol name and the second | |
33864 | is an expression to substitute for it.@refill | |
33865 | @end defun | |
33866 | ||
33867 | @defun remove-units expr | |
33868 | Return a copy of @var{expr} with all units variables replaced by ones. | |
33869 | This expression is generally normalized before use. | |
33870 | @end defun | |
33871 | ||
33872 | @defun extract-units expr | |
33873 | Return a copy of @var{expr} with everything but units variables replaced | |
33874 | by ones. | |
33875 | @end defun | |
33876 | ||
33877 | @node Formatting Lisp Functions, Hooks, Symbolic Lisp Functions, Internals | |
33878 | @subsubsection I/O and Formatting Functions | |
33879 | ||
33880 | @noindent | |
33881 | The functions described here are responsible for parsing and formatting | |
33882 | Calc numbers and formulas. | |
33883 | ||
33884 | @defun calc-eval str sep arg1 arg2 @dots{} | |
33885 | This is the simplest interface to the Calculator from another Lisp program. | |
33886 | @xref{Calling Calc from Your Programs}. | |
33887 | @end defun | |
33888 | ||
33889 | @defun read-number str | |
33890 | If string @var{str} contains a valid Calc number, either integer, | |
33891 | fraction, float, or HMS form, this function parses and returns that | |
33892 | number. Otherwise, it returns @code{nil}. | |
33893 | @end defun | |
33894 | ||
33895 | @defun read-expr str | |
33896 | Read an algebraic expression from string @var{str}. If @var{str} does | |
33897 | not have the form of a valid expression, return a list of the form | |
33898 | @samp{(error @var{pos} @var{msg})} where @var{pos} is an integer index | |
33899 | into @var{str} of the general location of the error, and @var{msg} is | |
33900 | a string describing the problem.@refill | |
33901 | @end defun | |
33902 | ||
33903 | @defun read-exprs str | |
33904 | Read a list of expressions separated by commas, and return it as a | |
33905 | Lisp list. If an error occurs in any expressions, an error list as | |
33906 | shown above is returned instead. | |
33907 | @end defun | |
33908 | ||
33909 | @defun calc-do-alg-entry initial prompt no-norm | |
33910 | Read an algebraic formula or formulas using the minibuffer. All | |
33911 | conventions of regular algebraic entry are observed. The return value | |
33912 | is a list of Calc formulas; there will be more than one if the user | |
33913 | entered a list of values separated by commas. The result is @code{nil} | |
33914 | if the user presses Return with a blank line. If @var{initial} is | |
33915 | given, it is a string which the minibuffer will initially contain. | |
33916 | If @var{prompt} is given, it is the prompt string to use; the default | |
33917 | is ``Algebraic:''. If @var{no-norm} is @code{t}, the formulas will | |
33918 | be returned exactly as parsed; otherwise, they will be passed through | |
33919 | @code{calc-normalize} first.@refill | |
33920 | ||
33921 | To support the use of @kbd{$} characters in the algebraic entry, use | |
33922 | @code{let} to bind @code{calc-dollar-values} to a list of the values | |
33923 | to be substituted for @kbd{$}, @kbd{$$}, and so on, and bind | |
33924 | @code{calc-dollar-used} to 0. Upon return, @code{calc-dollar-used} | |
33925 | will have been changed to the highest number of consecutive @kbd{$}s | |
33926 | that actually appeared in the input.@refill | |
33927 | @end defun | |
33928 | ||
33929 | @defun format-number a | |
33930 | Convert the real or complex number or HMS form @var{a} to string form. | |
33931 | @end defun | |
33932 | ||
33933 | @defun format-flat-expr a prec | |
33934 | Convert the arbitrary Calc number or formula @var{a} to string form, | |
33935 | in the style used by the trail buffer and the @code{calc-edit} command. | |
33936 | This is a simple format designed | |
33937 | mostly to guarantee the string is of a form that can be re-parsed by | |
33938 | @code{read-expr}. Most formatting modes, such as digit grouping, | |
33939 | complex number format, and point character, are ignored to ensure the | |
33940 | result will be re-readable. The @var{prec} parameter is normally 0; if | |
33941 | you pass a large integer like 1000 instead, the expression will be | |
33942 | surrounded by parentheses unless it is a plain number or variable name.@refill | |
33943 | @end defun | |
33944 | ||
33945 | @defun format-nice-expr a width | |
33946 | This is like @code{format-flat-expr} (with @var{prec} equal to 0), | |
33947 | except that newlines will be inserted to keep lines down to the | |
33948 | specified @var{width}, and vectors that look like matrices or rewrite | |
33949 | rules are written in a pseudo-matrix format. The @code{calc-edit} | |
33950 | command uses this when only one stack entry is being edited. | |
33951 | @end defun | |
33952 | ||
33953 | @defun format-value a width | |
33954 | Convert the Calc number or formula @var{a} to string form, using the | |
269b7745 | 33955 | format seen in the stack buffer. Beware the string returned may |
d7b8e6c6 EZ |
33956 | not be re-readable by @code{read-expr}, for example, because of digit |
33957 | grouping. Multi-line objects like matrices produce strings that | |
33958 | contain newline characters to separate the lines. The @var{w} | |
33959 | parameter, if given, is the target window size for which to format | |
33960 | the expressions. If @var{w} is omitted, the width of the Calculator | |
33961 | window is used.@refill | |
33962 | @end defun | |
33963 | ||
33964 | @defun compose-expr a prec | |
33965 | Format the Calc number or formula @var{a} according to the current | |
33966 | language mode, returning a ``composition.'' To learn about the | |
33967 | structure of compositions, see the comments in the Calc source code. | |
33968 | You can specify the format of a given type of function call by putting | |
33969 | a @code{math-compose-@var{lang}} property on the function's symbol, | |
33970 | whose value is a Lisp function that takes @var{a} and @var{prec} as | |
33971 | arguments and returns a composition. Here @var{lang} is a language | |
33972 | mode name, one of @code{normal}, @code{big}, @code{c}, @code{pascal}, | |
33973 | @code{fortran}, @code{tex}, @code{eqn}, @code{math}, or @code{maple}. | |
33974 | In Big mode, Calc actually tries @code{math-compose-big} first, then | |
33975 | tries @code{math-compose-normal}. If this property does not exist, | |
33976 | or if the function returns @code{nil}, the function is written in the | |
33977 | normal function-call notation for that language. | |
33978 | @end defun | |
33979 | ||
33980 | @defun composition-to-string c w | |
33981 | Convert a composition structure returned by @code{compose-expr} into | |
33982 | a string. Multi-line compositions convert to strings containing | |
33983 | newline characters. The target window size is given by @var{w}. | |
33984 | The @code{format-value} function basically calls @code{compose-expr} | |
33985 | followed by @code{composition-to-string}. | |
33986 | @end defun | |
33987 | ||
33988 | @defun comp-width c | |
33989 | Compute the width in characters of composition @var{c}. | |
33990 | @end defun | |
33991 | ||
33992 | @defun comp-height c | |
33993 | Compute the height in lines of composition @var{c}. | |
33994 | @end defun | |
33995 | ||
33996 | @defun comp-ascent c | |
33997 | Compute the portion of the height of composition @var{c} which is on or | |
33998 | above the baseline. For a one-line composition, this will be one. | |
33999 | @end defun | |
34000 | ||
34001 | @defun comp-descent c | |
34002 | Compute the portion of the height of composition @var{c} which is below | |
34003 | the baseline. For a one-line composition, this will be zero. | |
34004 | @end defun | |
34005 | ||
34006 | @defun comp-first-char c | |
34007 | If composition @var{c} is a ``flat'' composition, return the first | |
34008 | (leftmost) character of the composition as an integer. Otherwise, | |
34009 | return @code{nil}.@refill | |
34010 | @end defun | |
34011 | ||
34012 | @defun comp-last-char c | |
34013 | If composition @var{c} is a ``flat'' composition, return the last | |
34014 | (rightmost) character, otherwise return @code{nil}. | |
34015 | @end defun | |
34016 | ||
34017 | @comment @node Lisp Variables, Hooks, Formatting Lisp Functions, Internals | |
34018 | @comment @subsubsection Lisp Variables | |
177c0ea7 | 34019 | @comment |
d7b8e6c6 EZ |
34020 | @comment @noindent |
34021 | @comment (This section is currently unfinished.) | |
34022 | ||
34023 | @node Hooks, , Formatting Lisp Functions, Internals | |
34024 | @subsubsection Hooks | |
34025 | ||
34026 | @noindent | |
34027 | Hooks are variables which contain Lisp functions (or lists of functions) | |
34028 | which are called at various times. Calc defines a number of hooks | |
34029 | that help you to customize it in various ways. Calc uses the Lisp | |
34030 | function @code{run-hooks} to invoke the hooks shown below. Several | |
34031 | other customization-related variables are also described here. | |
34032 | ||
34033 | @defvar calc-load-hook | |
34034 | This hook is called at the end of @file{calc.el}, after the file has | |
34035 | been loaded, before any functions in it have been called, but after | |
34036 | @code{calc-mode-map} and similar variables have been set up. | |
34037 | @end defvar | |
34038 | ||
34039 | @defvar calc-ext-load-hook | |
34040 | This hook is called at the end of @file{calc-ext.el}. | |
34041 | @end defvar | |
34042 | ||
34043 | @defvar calc-start-hook | |
34044 | This hook is called as the last step in a @kbd{M-x calc} command. | |
34045 | At this point, the Calc buffer has been created and initialized if | |
34046 | necessary, the Calc window and trail window have been created, | |
34047 | and the ``Welcome to Calc'' message has been displayed. | |
34048 | @end defvar | |
34049 | ||
34050 | @defvar calc-mode-hook | |
34051 | This hook is called when the Calc buffer is being created. Usually | |
34052 | this will only happen once per Emacs session. The hook is called | |
34053 | after Emacs has switched to the new buffer, the mode-settings file | |
34054 | has been read if necessary, and all other buffer-local variables | |
34055 | have been set up. After this hook returns, Calc will perform a | |
34056 | @code{calc-refresh} operation, set up the mode line display, then | |
34057 | evaluate any deferred @code{calc-define} properties that have not | |
34058 | been evaluated yet. | |
34059 | @end defvar | |
34060 | ||
34061 | @defvar calc-trail-mode-hook | |
34062 | This hook is called when the Calc Trail buffer is being created. | |
34063 | It is called as the very last step of setting up the Trail buffer. | |
34064 | Like @code{calc-mode-hook}, this will normally happen only once | |
34065 | per Emacs session. | |
34066 | @end defvar | |
34067 | ||
34068 | @defvar calc-end-hook | |
34069 | This hook is called by @code{calc-quit}, generally because the user | |
34070 | presses @kbd{q} or @kbd{M-# c} while in Calc. The Calc buffer will | |
34071 | be the current buffer. The hook is called as the very first | |
34072 | step, before the Calc window is destroyed. | |
34073 | @end defvar | |
34074 | ||
34075 | @defvar calc-window-hook | |
34076 | If this hook exists, it is called to create the Calc window. | |
34077 | Upon return, this new Calc window should be the current window. | |
34078 | (The Calc buffer will already be the current buffer when the | |
34079 | hook is called.) If the hook is not defined, Calc will | |
34080 | generally use @code{split-window}, @code{set-window-buffer}, | |
34081 | and @code{select-window} to create the Calc window. | |
34082 | @end defvar | |
34083 | ||
34084 | @defvar calc-trail-window-hook | |
34085 | If this hook exists, it is called to create the Calc Trail window. | |
34086 | The variable @code{calc-trail-buffer} will contain the buffer | |
34087 | which the window should use. Unlike @code{calc-window-hook}, | |
34088 | this hook must @emph{not} switch into the new window. | |
34089 | @end defvar | |
34090 | ||
34091 | @defvar calc-edit-mode-hook | |
34092 | This hook is called by @code{calc-edit} (and the other ``edit'' | |
34093 | commands) when the temporary editing buffer is being created. | |
34094 | The buffer will have been selected and set up to be in | |
34095 | @code{calc-edit-mode}, but will not yet have been filled with | |
34096 | text. (In fact it may still have leftover text from a previous | |
34097 | @code{calc-edit} command.) | |
34098 | @end defvar | |
34099 | ||
34100 | @defvar calc-mode-save-hook | |
34101 | This hook is called by the @code{calc-save-modes} command, | |
34102 | after Calc's own mode features have been inserted into the | |
34103 | @file{.emacs} buffer and just before the ``End of mode settings'' | |
34104 | message is inserted. | |
34105 | @end defvar | |
34106 | ||
34107 | @defvar calc-reset-hook | |
34108 | This hook is called after @kbd{M-# 0} (@code{calc-reset}) has | |
34109 | reset all modes. The Calc buffer will be the current buffer. | |
34110 | @end defvar | |
34111 | ||
34112 | @defvar calc-other-modes | |
34113 | This variable contains a list of strings. The strings are | |
34114 | concatenated at the end of the modes portion of the Calc | |
34115 | mode line (after standard modes such as ``Deg'', ``Inv'' and | |
34116 | ``Hyp''). Each string should be a short, single word followed | |
34117 | by a space. The variable is @code{nil} by default. | |
34118 | @end defvar | |
34119 | ||
34120 | @defvar calc-mode-map | |
34121 | This is the keymap that is used by Calc mode. The best time | |
34122 | to adjust it is probably in a @code{calc-mode-hook}. If the | |
34123 | Calc extensions package (@file{calc-ext.el}) has not yet been | |
34124 | loaded, many of these keys will be bound to @code{calc-missing-key}, | |
34125 | which is a command that loads the extensions package and | |
34126 | ``retypes'' the key. If your @code{calc-mode-hook} rebinds | |
34127 | one of these keys, it will probably be overridden when the | |
34128 | extensions are loaded. | |
34129 | @end defvar | |
34130 | ||
34131 | @defvar calc-digit-map | |
34132 | This is the keymap that is used during numeric entry. Numeric | |
34133 | entry uses the minibuffer, but this map binds every non-numeric | |
34134 | key to @code{calcDigit-nondigit} which generally calls | |
34135 | @code{exit-minibuffer} and ``retypes'' the key. | |
34136 | @end defvar | |
34137 | ||
34138 | @defvar calc-alg-ent-map | |
34139 | This is the keymap that is used during algebraic entry. This is | |
34140 | mostly a copy of @code{minibuffer-local-map}. | |
34141 | @end defvar | |
34142 | ||
34143 | @defvar calc-store-var-map | |
34144 | This is the keymap that is used during entry of variable names for | |
34145 | commands like @code{calc-store} and @code{calc-recall}. This is | |
34146 | mostly a copy of @code{minibuffer-local-completion-map}. | |
34147 | @end defvar | |
34148 | ||
34149 | @defvar calc-edit-mode-map | |
34150 | This is the (sparse) keymap used by @code{calc-edit} and other | |
34151 | temporary editing commands. It binds @key{RET}, @key{LFD}, | |
34152 | and @kbd{C-c C-c} to @code{calc-edit-finish}. | |
34153 | @end defvar | |
34154 | ||
34155 | @defvar calc-mode-var-list | |
34156 | This is a list of variables which are saved by @code{calc-save-modes}. | |
34157 | Each entry is a list of two items, the variable (as a Lisp symbol) | |
34158 | and its default value. When modes are being saved, each variable | |
34159 | is compared with its default value (using @code{equal}) and any | |
34160 | non-default variables are written out. | |
34161 | @end defvar | |
34162 | ||
34163 | @defvar calc-local-var-list | |
34164 | This is a list of variables which should be buffer-local to the | |
34165 | Calc buffer. Each entry is a variable name (as a Lisp symbol). | |
34166 | These variables also have their default values manipulated by | |
34167 | the @code{calc} and @code{calc-quit} commands; @pxref{Multiple Calculators}. | |
34168 | Since @code{calc-mode-hook} is called after this list has been | |
34169 | used the first time, your hook should add a variable to the | |
34170 | list and also call @code{make-local-variable} itself. | |
34171 | @end defvar | |
34172 | ||
34173 | @node Installation, Reporting Bugs, Programming, Top | |
34174 | @appendix Installation | |
34175 | ||
34176 | @noindent | |
ed7899e8 CW |
34177 | As of Calc 2.02g, Calc is integrated with GNU Emacs, and thus requires |
34178 | no separate installation of its Lisp files and this manual. | |
d7b8e6c6 | 34179 | |
d7b8e6c6 EZ |
34180 | @appendixsec The GNUPLOT Program |
34181 | ||
34182 | @noindent | |
34183 | Calc's graphing commands use the GNUPLOT program. If you have GNUPLOT | |
34184 | but you must type some command other than @file{gnuplot} to get it, | |
34185 | you should add a command to set the Lisp variable @code{calc-gnuplot-name} | |
34186 | to the appropriate file name. You may also need to change the variables | |
34187 | @code{calc-gnuplot-plot-command} and @code{calc-gnuplot-print-command} in | |
34188 | order to get correct displays and hardcopies, respectively, of your | |
34189 | plots.@refill | |
34190 | ||
34191 | @ifinfo | |
34192 | @example | |
34193 | ||
d7b8e6c6 EZ |
34194 | @end example |
34195 | @end ifinfo | |
34196 | @appendixsec Printed Documentation | |
34197 | ||
34198 | @noindent | |
34199 | Because the Calc manual is so large, you should only make a printed | |
34200 | copy if you really need it. To print the manual, you will need the | |
34201 | @TeX{} typesetting program (this is a free program by Donald Knuth | |
34202 | at Stanford University) as well as the @file{texindex} program and | |
34203 | @file{texinfo.tex} file, both of which can be obtained from the FSF | |
ed7899e8 | 34204 | as part of the @code{texinfo} package.@refill |
d7b8e6c6 | 34205 | |
0d48e8aa | 34206 | To print the Calc manual in one huge 470 page tome, you will need the |
ed7899e8 | 34207 | source code to this manual, @file{calc.texi}, available as part of the |
0d48e8aa EZ |
34208 | Emacs source. Once you have this file, type @kbd{texi2dvi calc.texi}. |
34209 | Alternatively, change to the @file{man} subdirectory of the Emacs | |
34210 | source distribution, and type @kbd{make calc.dvi}. (Don't worry if you | |
34211 | get some ``overfull box'' warnings while @TeX{} runs.) | |
d7b8e6c6 EZ |
34212 | |
34213 | The result will be a device-independent output file called | |
34214 | @file{calc.dvi}, which you must print in whatever way is right | |
34215 | for your system. On many systems, the command is | |
34216 | ||
34217 | @example | |
34218 | lpr -d calc.dvi | |
34219 | @end example | |
34220 | ||
0d48e8aa EZ |
34221 | @noindent |
34222 | or | |
34223 | ||
34224 | @example | |
34225 | dvips calc.dvi | |
34226 | @end example | |
34227 | ||
34228 | @c the bumpoddpages macro was deleted | |
34229 | @ignore | |
d7b8e6c6 EZ |
34230 | @cindex Marginal notes, adjusting |
34231 | Marginal notes for each function and key sequence normally alternate | |
34232 | between the left and right sides of the page, which is correct if the | |
34233 | manual is going to be bound as double-sided pages. Near the top of | |
ed7899e8 | 34234 | the file @file{calc.texi} you will find alternate definitions of |
d7b8e6c6 EZ |
34235 | the @code{\bumpoddpages} macro that put the marginal notes always on |
34236 | the same side, best if you plan to be binding single-sided pages. | |
0d48e8aa | 34237 | @end ignore |
d7b8e6c6 | 34238 | |
d7b8e6c6 EZ |
34239 | @appendixsec Settings File |
34240 | ||
34241 | @noindent | |
34242 | @vindex calc-settings-file | |
34243 | Another variable you might want to set is @code{calc-settings-file}, | |
34244 | which holds the file name in which commands like @kbd{m m} and @kbd{Z P} | |
34245 | store ``permanent'' definitions. The default value for this variable | |
34246 | is @code{"~/.emacs"}. If @code{calc-settings-file} does not contain | |
34247 | @code{".emacs"} as a substring, and if the variable | |
34248 | @code{calc-loaded-settings-file} is @code{nil}, then Calc will | |
34249 | automatically load your settings file (if it exists) the first time | |
34250 | Calc is invoked.@refill | |
34251 | ||
34252 | @ifinfo | |
34253 | @example | |
34254 | ||
34255 | @end example | |
34256 | @end ifinfo | |
34257 | @appendixsec Testing the Installation | |
34258 | ||
34259 | @noindent | |
34260 | To test your installation of Calc, start a new Emacs and type @kbd{M-# c} | |
34261 | to make sure the autoloads and key bindings work. Type @kbd{M-# i} | |
34262 | to make sure Calc can find its Info documentation. Press @kbd{q} to | |
34263 | exit the Info system and @kbd{M-# c} to re-enter the Calculator. | |
34264 | Type @kbd{20 S} to compute the sine of 20 degrees; this will test the | |
34265 | autoloading of the extensions modules. The result should be | |
34266 | 0.342020143326. Finally, press @kbd{M-# c} again to make sure the | |
34267 | Calculator can exit. | |
34268 | ||
34269 | You may also wish to test the GNUPLOT interface; to plot a sine wave, | |
5d67986c | 34270 | type @kbd{' [0 ..@: 360], sin(x) @key{RET} g f}. Type @kbd{g q} when you |
d7b8e6c6 EZ |
34271 | are done viewing the plot. |
34272 | ||
34273 | Calc is now ready to use. If you wish to go through the Calc Tutorial, | |
34274 | press @kbd{M-# t} to begin. | |
34275 | @example | |
34276 | ||
34277 | @end example | |
d7b8e6c6 EZ |
34278 | @node Reporting Bugs, Summary, Installation, Top |
34279 | @appendix Reporting Bugs | |
34280 | ||
34281 | @noindent | |
ed7899e8 | 34282 | If you find a bug in Calc, send e-mail to Colin Walters, |
d7b8e6c6 EZ |
34283 | |
34284 | @example | |
ed7899e8 CW |
34285 | walters@@debian.org @r{or} |
34286 | walters@@verbum.org | |
d7b8e6c6 EZ |
34287 | @end example |
34288 | ||
34289 | @noindent | |
ed7899e8 CW |
34290 | (In the following text, ``I'' refers to the original Calc author, Dave |
34291 | Gillespie). | |
34292 | ||
d7b8e6c6 EZ |
34293 | While I cannot guarantee that I will have time to work on your bug, |
34294 | I do try to fix bugs quickly whenever I can. | |
34295 | ||
ed7899e8 | 34296 | The latest version of Calc is available from Savannah, in the Emacs |
450c6476 | 34297 | CVS tree. See @uref{http://savannah.gnu.org/projects/emacs}. |
d7b8e6c6 | 34298 | |
ed7899e8 | 34299 | There is an automatic command @kbd{M-x report-calc-bug} which helps |
d7b8e6c6 EZ |
34300 | you to report bugs. This command prompts you for a brief subject |
34301 | line, then leaves you in a mail editing buffer. Type @kbd{C-c C-c} to | |
34302 | send your mail. Make sure your subject line indicates that you are | |
ed7899e8 CW |
34303 | reporting a Calc bug; this command sends mail to the maintainer's |
34304 | regular mailbox. | |
d7b8e6c6 EZ |
34305 | |
34306 | If you have suggestions for additional features for Calc, I would | |
34307 | love to hear them. Some have dared to suggest that Calc is already | |
34308 | top-heavy with features; I really don't see what they're talking | |
34309 | about, so, if you have ideas, send them right in. (I may even have | |
34310 | time to implement them!) | |
34311 | ||
34312 | At the front of the source file, @file{calc.el}, is a list of ideas for | |
34313 | future work which I have not had time to do. If any enthusiastic souls | |
34314 | wish to take it upon themselves to work on these, I would be delighted. | |
34315 | Please let me know if you plan to contribute to Calc so I can coordinate | |
34316 | your efforts with mine and those of others. I will do my best to help | |
34317 | you in whatever way I can. | |
34318 | ||
34319 | @c [summary] | |
34320 | @node Summary, Key Index, Reporting Bugs, Top | |
34321 | @appendix Calc Summary | |
34322 | ||
34323 | @noindent | |
34324 | This section includes a complete list of Calc 2.02 keystroke commands. | |
34325 | Each line lists the stack entries used by the command (top-of-stack | |
34326 | last), the keystrokes themselves, the prompts asked by the command, | |
34327 | and the result of the command (also with top-of-stack last). | |
34328 | The result is expressed using the equivalent algebraic function. | |
34329 | Commands which put no results on the stack show the full @kbd{M-x} | |
34330 | command name in that position. Numbers preceding the result or | |
34331 | command name refer to notes at the end. | |
34332 | ||
34333 | Algebraic functions and @kbd{M-x} commands that don't have corresponding | |
34334 | keystrokes are not listed in this summary. | |
34335 | @xref{Command Index}. @xref{Function Index}. | |
34336 | ||
34337 | @iftex | |
34338 | @begingroup | |
34339 | @tex | |
34340 | \vskip-2\baselineskip \null | |
34341 | \gdef\sumrow#1{\sumrowx#1\relax}% | |
34342 | \gdef\sumrowx#1\:#2\:#3\:#4\:#5\:#6\relax{% | |
34343 | \leavevmode% | |
5d67986c RS |
34344 | {\smallfonts |
34345 | \hbox to5em{\sl\hss#1}% | |
34346 | \hbox to5em{\tt#2\hss}% | |
34347 | \hbox to4em{\sl#3\hss}% | |
34348 | \hbox to5em{\rm\hss#4}% | |
d7b8e6c6 | 34349 | \thinspace% |
5d67986c RS |
34350 | {\tt#5}% |
34351 | {\sl#6}% | |
34352 | }}% | |
34353 | \gdef\sumlpar{{\rm(}}% | |
34354 | \gdef\sumrpar{{\rm)}}% | |
34355 | \gdef\sumcomma{{\rm,\thinspace}}% | |
34356 | \gdef\sumexcl{{\rm!}}% | |
d7b8e6c6 EZ |
34357 | \gdef\sumbreak{\vskip-2.5\baselineskip\goodbreak}% |
34358 | \gdef\minus#1{{\tt-}}% | |
34359 | @end tex | |
34360 | @let@:=@sumsep | |
34361 | @let@r=@sumrow | |
34362 | @catcode`@(=@active @let(=@sumlpar | |
34363 | @catcode`@)=@active @let)=@sumrpar | |
34364 | @catcode`@,=@active @let,=@sumcomma | |
34365 | @catcode`@!=@active @let!=@sumexcl | |
34366 | @end iftex | |
34367 | @format | |
34368 | @iftex | |
34369 | @advance@baselineskip-2.5pt | |
d7b8e6c6 EZ |
34370 | @let@c@sumbreak |
34371 | @end iftex | |
34372 | @r{ @: M-# a @: @: 33 @:calc-embedded-activate@:} | |
34373 | @r{ @: M-# b @: @: @:calc-big-or-small@:} | |
34374 | @r{ @: M-# c @: @: @:calc@:} | |
34375 | @r{ @: M-# d @: @: @:calc-embedded-duplicate@:} | |
34376 | @r{ @: M-# e @: @: 34 @:calc-embedded@:} | |
34377 | @r{ @: M-# f @:formula @: @:calc-embedded-new-formula@:} | |
34378 | @r{ @: M-# g @: @: 35 @:calc-grab-region@:} | |
34379 | @r{ @: M-# i @: @: @:calc-info@:} | |
34380 | @r{ @: M-# j @: @: @:calc-embedded-select@:} | |
34381 | @r{ @: M-# k @: @: @:calc-keypad@:} | |
34382 | @r{ @: M-# l @: @: @:calc-load-everything@:} | |
34383 | @r{ @: M-# m @: @: @:read-kbd-macro@:} | |
34384 | @r{ @: M-# n @: @: 4 @:calc-embedded-next@:} | |
34385 | @r{ @: M-# o @: @: @:calc-other-window@:} | |
34386 | @r{ @: M-# p @: @: 4 @:calc-embedded-previous@:} | |
34387 | @r{ @: M-# q @:formula @: @:quick-calc@:} | |
34388 | @r{ @: M-# r @: @: 36 @:calc-grab-rectangle@:} | |
34389 | @r{ @: M-# s @: @: @:calc-info-summary@:} | |
34390 | @r{ @: M-# t @: @: @:calc-tutorial@:} | |
34391 | @r{ @: M-# u @: @: @:calc-embedded-update@:} | |
34392 | @r{ @: M-# w @: @: @:calc-embedded-word@:} | |
34393 | @r{ @: M-# x @: @: @:calc-quit@:} | |
34394 | @r{ @: M-# y @: @:1,28,49 @:calc-copy-to-buffer@:} | |
34395 | @r{ @: M-# z @: @: @:calc-user-invocation@:} | |
34396 | @r{ @: M-# : @: @: 36 @:calc-grab-sum-down@:} | |
34397 | @r{ @: M-# _ @: @: 36 @:calc-grab-sum-across@:} | |
34398 | @r{ @: M-# ` @:editing @: 30 @:calc-embedded-edit@:} | |
34399 | @r{ @: M-# 0 @:(zero) @: @:calc-reset@:} | |
177c0ea7 JB |
34400 | |
34401 | @c | |
d7b8e6c6 EZ |
34402 | @r{ @: 0-9 @:number @: @:@:number} |
34403 | @r{ @: . @:number @: @:@:0.number} | |
34404 | @r{ @: _ @:number @: @:-@:number} | |
34405 | @r{ @: e @:number @: @:@:1e number} | |
34406 | @r{ @: # @:number @: @:@:current-radix@t{#}number} | |
34407 | @r{ @: P @:(in number) @: @:+/-@:} | |
34408 | @r{ @: M @:(in number) @: @:mod@:} | |
34409 | @r{ @: @@ ' " @: (in number)@: @:@:HMS form} | |
34410 | @r{ @: h m s @: (in number)@: @:@:HMS form} | |
34411 | ||
177c0ea7 | 34412 | @c |
d7b8e6c6 EZ |
34413 | @r{ @: ' @:formula @: 37,46 @:@:formula} |
34414 | @r{ @: $ @:formula @: 37,46 @:$@:formula} | |
34415 | @r{ @: " @:string @: 37,46 @:@:string} | |
177c0ea7 JB |
34416 | |
34417 | @c | |
d7b8e6c6 EZ |
34418 | @r{ a b@: + @: @: 2 @:add@:(a,b) a+b} |
34419 | @r{ a b@: - @: @: 2 @:sub@:(a,b) a@minus{}b} | |
34420 | @r{ a b@: * @: @: 2 @:mul@:(a,b) a b, a*b} | |
34421 | @r{ a b@: / @: @: 2 @:div@:(a,b) a/b} | |
34422 | @r{ a b@: ^ @: @: 2 @:pow@:(a,b) a^b} | |
34423 | @r{ a b@: I ^ @: @: 2 @:nroot@:(a,b) a^(1/b)} | |
34424 | @r{ a b@: % @: @: 2 @:mod@:(a,b) a%b} | |
34425 | @r{ a b@: \ @: @: 2 @:idiv@:(a,b) a\b} | |
34426 | @r{ a b@: : @: @: 2 @:fdiv@:(a,b)} | |
34427 | @r{ a b@: | @: @: 2 @:vconcat@:(a,b) a|b} | |
34428 | @r{ a b@: I | @: @: @:vconcat@:(b,a) b|a} | |
34429 | @r{ a b@: H | @: @: 2 @:append@:(a,b)} | |
34430 | @r{ a b@: I H | @: @: @:append@:(b,a)} | |
34431 | @r{ a@: & @: @: 1 @:inv@:(a) 1/a} | |
34432 | @r{ a@: ! @: @: 1 @:fact@:(a) a!} | |
34433 | @r{ a@: = @: @: 1 @:evalv@:(a)} | |
34434 | @r{ a@: M-% @: @: @:percent@:(a) a%} | |
177c0ea7 JB |
34435 | |
34436 | @c | |
5d67986c RS |
34437 | @r{ ... a@: @key{RET} @: @: 1 @:@:... a a} |
34438 | @r{ ... a@: @key{SPC} @: @: 1 @:@:... a a} | |
34439 | @r{... a b@: @key{TAB} @: @: 3 @:@:... b a} | |
34440 | @r{. a b c@: M-@key{TAB} @: @: 3 @:@:... b c a} | |
34441 | @r{... a b@: @key{LFD} @: @: 1 @:@:... a b a} | |
34442 | @r{ ... a@: @key{DEL} @: @: 1 @:@:...} | |
34443 | @r{... a b@: M-@key{DEL} @: @: 1 @:@:... b} | |
34444 | @r{ @: M-@key{RET} @: @: 4 @:calc-last-args@:} | |
d7b8e6c6 | 34445 | @r{ a@: ` @:editing @: 1,30 @:calc-edit@:} |
177c0ea7 JB |
34446 | |
34447 | @c | |
d7b8e6c6 EZ |
34448 | @r{ ... a@: C-d @: @: 1 @:@:...} |
34449 | @r{ @: C-k @: @: 27 @:calc-kill@:} | |
34450 | @r{ @: C-w @: @: 27 @:calc-kill-region@:} | |
34451 | @r{ @: C-y @: @: @:calc-yank@:} | |
34452 | @r{ @: C-_ @: @: 4 @:calc-undo@:} | |
34453 | @r{ @: M-k @: @: 27 @:calc-copy-as-kill@:} | |
34454 | @r{ @: M-w @: @: 27 @:calc-copy-region-as-kill@:} | |
177c0ea7 JB |
34455 | |
34456 | @c | |
d7b8e6c6 EZ |
34457 | @r{ @: [ @: @: @:@:[...} |
34458 | @r{[.. a b@: ] @: @: @:@:[a,b]} | |
34459 | @r{ @: ( @: @: @:@:(...} | |
34460 | @r{(.. a b@: ) @: @: @:@:(a,b)} | |
34461 | @r{ @: , @: @: @:@:vector or rect complex} | |
34462 | @r{ @: ; @: @: @:@:matrix or polar complex} | |
34463 | @r{ @: .. @: @: @:@:interval} | |
34464 | ||
177c0ea7 | 34465 | @c |
d7b8e6c6 EZ |
34466 | @r{ @: ~ @: @: @:calc-num-prefix@:} |
34467 | @r{ @: < @: @: 4 @:calc-scroll-left@:} | |
34468 | @r{ @: > @: @: 4 @:calc-scroll-right@:} | |
34469 | @r{ @: @{ @: @: 4 @:calc-scroll-down@:} | |
34470 | @r{ @: @} @: @: 4 @:calc-scroll-up@:} | |
34471 | @r{ @: ? @: @: @:calc-help@:} | |
177c0ea7 JB |
34472 | |
34473 | @c | |
d7b8e6c6 EZ |
34474 | @r{ a@: n @: @: 1 @:neg@:(a) @minus{}a} |
34475 | @r{ @: o @: @: 4 @:calc-realign@:} | |
34476 | @r{ @: p @:precision @: 31 @:calc-precision@:} | |
34477 | @r{ @: q @: @: @:calc-quit@:} | |
34478 | @r{ @: w @: @: @:calc-why@:} | |
34479 | @r{ @: x @:command @: @:M-x calc-@:command} | |
34480 | @r{ a@: y @: @:1,28,49 @:calc-copy-to-buffer@:} | |
177c0ea7 JB |
34481 | |
34482 | @c | |
d7b8e6c6 EZ |
34483 | @r{ a@: A @: @: 1 @:abs@:(a)} |
34484 | @r{ a b@: B @: @: 2 @:log@:(a,b)} | |
34485 | @r{ a b@: I B @: @: 2 @:alog@:(a,b) b^a} | |
34486 | @r{ a@: C @: @: 1 @:cos@:(a)} | |
34487 | @r{ a@: I C @: @: 1 @:arccos@:(a)} | |
34488 | @r{ a@: H C @: @: 1 @:cosh@:(a)} | |
34489 | @r{ a@: I H C @: @: 1 @:arccosh@:(a)} | |
34490 | @r{ @: D @: @: 4 @:calc-redo@:} | |
34491 | @r{ a@: E @: @: 1 @:exp@:(a)} | |
34492 | @r{ a@: H E @: @: 1 @:exp10@:(a) 10.^a} | |
34493 | @r{ a@: F @: @: 1,11 @:floor@:(a,d)} | |
34494 | @r{ a@: I F @: @: 1,11 @:ceil@:(a,d)} | |
34495 | @r{ a@: H F @: @: 1,11 @:ffloor@:(a,d)} | |
34496 | @r{ a@: I H F @: @: 1,11 @:fceil@:(a,d)} | |
34497 | @r{ a@: G @: @: 1 @:arg@:(a)} | |
34498 | @r{ @: H @:command @: 32 @:@:Hyperbolic} | |
34499 | @r{ @: I @:command @: 32 @:@:Inverse} | |
34500 | @r{ a@: J @: @: 1 @:conj@:(a)} | |
34501 | @r{ @: K @:command @: 32 @:@:Keep-args} | |
34502 | @r{ a@: L @: @: 1 @:ln@:(a)} | |
34503 | @r{ a@: H L @: @: 1 @:log10@:(a)} | |
34504 | @r{ @: M @: @: @:calc-more-recursion-depth@:} | |
34505 | @r{ @: I M @: @: @:calc-less-recursion-depth@:} | |
34506 | @r{ a@: N @: @: 5 @:evalvn@:(a)} | |
34507 | @r{ @: P @: @: @:@:pi} | |
34508 | @r{ @: I P @: @: @:@:gamma} | |
34509 | @r{ @: H P @: @: @:@:e} | |
34510 | @r{ @: I H P @: @: @:@:phi} | |
34511 | @r{ a@: Q @: @: 1 @:sqrt@:(a)} | |
34512 | @r{ a@: I Q @: @: 1 @:sqr@:(a) a^2} | |
34513 | @r{ a@: R @: @: 1,11 @:round@:(a,d)} | |
34514 | @r{ a@: I R @: @: 1,11 @:trunc@:(a,d)} | |
34515 | @r{ a@: H R @: @: 1,11 @:fround@:(a,d)} | |
34516 | @r{ a@: I H R @: @: 1,11 @:ftrunc@:(a,d)} | |
34517 | @r{ a@: S @: @: 1 @:sin@:(a)} | |
34518 | @r{ a@: I S @: @: 1 @:arcsin@:(a)} | |
34519 | @r{ a@: H S @: @: 1 @:sinh@:(a)} | |
34520 | @r{ a@: I H S @: @: 1 @:arcsinh@:(a)} | |
34521 | @r{ a@: T @: @: 1 @:tan@:(a)} | |
34522 | @r{ a@: I T @: @: 1 @:arctan@:(a)} | |
34523 | @r{ a@: H T @: @: 1 @:tanh@:(a)} | |
34524 | @r{ a@: I H T @: @: 1 @:arctanh@:(a)} | |
34525 | @r{ @: U @: @: 4 @:calc-undo@:} | |
34526 | @r{ @: X @: @: 4 @:calc-call-last-kbd-macro@:} | |
177c0ea7 JB |
34527 | |
34528 | @c | |
d7b8e6c6 EZ |
34529 | @r{ a b@: a = @: @: 2 @:eq@:(a,b) a=b} |
34530 | @r{ a b@: a # @: @: 2 @:neq@:(a,b) a!=b} | |
34531 | @r{ a b@: a < @: @: 2 @:lt@:(a,b) a<b} | |
34532 | @r{ a b@: a > @: @: 2 @:gt@:(a,b) a>b} | |
34533 | @r{ a b@: a [ @: @: 2 @:leq@:(a,b) a<=b} | |
34534 | @r{ a b@: a ] @: @: 2 @:geq@:(a,b) a>=b} | |
34535 | @r{ a b@: a @{ @: @: 2 @:in@:(a,b)} | |
34536 | @r{ a b@: a & @: @: 2,45 @:land@:(a,b) a&&b} | |
34537 | @r{ a b@: a | @: @: 2,45 @:lor@:(a,b) a||b} | |
34538 | @r{ a@: a ! @: @: 1,45 @:lnot@:(a) !a} | |
34539 | @r{ a b c@: a : @: @: 45 @:if@:(a,b,c) a?b:c} | |
34540 | @r{ a@: a . @: @: 1 @:rmeq@:(a)} | |
34541 | @r{ a@: a " @: @: 7,8 @:calc-expand-formula@:} | |
177c0ea7 JB |
34542 | |
34543 | @c | |
d7b8e6c6 EZ |
34544 | @r{ a@: a + @:i, l, h @: 6,38 @:sum@:(a,i,l,h)} |
34545 | @r{ a@: a - @:i, l, h @: 6,38 @:asum@:(a,i,l,h)} | |
34546 | @r{ a@: a * @:i, l, h @: 6,38 @:prod@:(a,i,l,h)} | |
34547 | @r{ a b@: a _ @: @: 2 @:subscr@:(a,b) a_b} | |
177c0ea7 JB |
34548 | |
34549 | @c | |
d7b8e6c6 EZ |
34550 | @r{ a b@: a \ @: @: 2 @:pdiv@:(a,b)} |
34551 | @r{ a b@: a % @: @: 2 @:prem@:(a,b)} | |
34552 | @r{ a b@: a / @: @: 2 @:pdivrem@:(a,b) [q,r]} | |
34553 | @r{ a b@: H a / @: @: 2 @:pdivide@:(a,b) q+r/b} | |
177c0ea7 JB |
34554 | |
34555 | @c | |
d7b8e6c6 EZ |
34556 | @r{ a@: a a @: @: 1 @:apart@:(a)} |
34557 | @r{ a@: a b @:old, new @: 38 @:subst@:(a,old,new)} | |
34558 | @r{ a@: a c @:v @: 38 @:collect@:(a,v)} | |
34559 | @r{ a@: a d @:v @: 4,38 @:deriv@:(a,v)} | |
34560 | @r{ a@: H a d @:v @: 4,38 @:tderiv@:(a,v)} | |
34561 | @r{ a@: a e @: @: @:esimplify@:(a)} | |
34562 | @r{ a@: a f @: @: 1 @:factor@:(a)} | |
34563 | @r{ a@: H a f @: @: 1 @:factors@:(a)} | |
34564 | @r{ a b@: a g @: @: 2 @:pgcd@:(a,b)} | |
34565 | @r{ a@: a i @:v @: 38 @:integ@:(a,v)} | |
34566 | @r{ a@: a m @:pats @: 38 @:match@:(a,pats)} | |
34567 | @r{ a@: I a m @:pats @: 38 @:matchnot@:(a,pats)} | |
34568 | @r{ data x@: a p @: @: 28 @:polint@:(data,x)} | |
34569 | @r{ data x@: H a p @: @: 28 @:ratint@:(data,x)} | |
34570 | @r{ a@: a n @: @: 1 @:nrat@:(a)} | |
34571 | @r{ a@: a r @:rules @:4,8,38 @:rewrite@:(a,rules,n)} | |
34572 | @r{ a@: a s @: @: @:simplify@:(a)} | |
34573 | @r{ a@: a t @:v, n @: 31,39 @:taylor@:(a,v,n)} | |
34574 | @r{ a@: a v @: @: 7,8 @:calc-alg-evaluate@:} | |
34575 | @r{ a@: a x @: @: 4,8 @:expand@:(a)} | |
177c0ea7 JB |
34576 | |
34577 | @c | |
d7b8e6c6 EZ |
34578 | @r{ data@: a F @:model, vars @: 48 @:fit@:(m,iv,pv,data)} |
34579 | @r{ data@: I a F @:model, vars @: 48 @:xfit@:(m,iv,pv,data)} | |
34580 | @r{ data@: H a F @:model, vars @: 48 @:efit@:(m,iv,pv,data)} | |
34581 | @r{ a@: a I @:v, l, h @: 38 @:ninteg@:(a,v,l,h)} | |
34582 | @r{ a b@: a M @:op @: 22 @:mapeq@:(op,a,b)} | |
34583 | @r{ a b@: I a M @:op @: 22 @:mapeqr@:(op,a,b)} | |
34584 | @r{ a b@: H a M @:op @: 22 @:mapeqp@:(op,a,b)} | |
34585 | @r{ a g@: a N @:v @: 38 @:minimize@:(a,v,g)} | |
34586 | @r{ a g@: H a N @:v @: 38 @:wminimize@:(a,v,g)} | |
34587 | @r{ a@: a P @:v @: 38 @:roots@:(a,v)} | |
34588 | @r{ a g@: a R @:v @: 38 @:root@:(a,v,g)} | |
34589 | @r{ a g@: H a R @:v @: 38 @:wroot@:(a,v,g)} | |
34590 | @r{ a@: a S @:v @: 38 @:solve@:(a,v)} | |
34591 | @r{ a@: I a S @:v @: 38 @:finv@:(a,v)} | |
34592 | @r{ a@: H a S @:v @: 38 @:fsolve@:(a,v)} | |
34593 | @r{ a@: I H a S @:v @: 38 @:ffinv@:(a,v)} | |
34594 | @r{ a@: a T @:i, l, h @: 6,38 @:table@:(a,i,l,h)} | |
34595 | @r{ a g@: a X @:v @: 38 @:maximize@:(a,v,g)} | |
34596 | @r{ a g@: H a X @:v @: 38 @:wmaximize@:(a,v,g)} | |
177c0ea7 JB |
34597 | |
34598 | @c | |
d7b8e6c6 EZ |
34599 | @r{ a b@: b a @: @: 9 @:and@:(a,b,w)} |
34600 | @r{ a@: b c @: @: 9 @:clip@:(a,w)} | |
34601 | @r{ a b@: b d @: @: 9 @:diff@:(a,b,w)} | |
34602 | @r{ a@: b l @: @: 10 @:lsh@:(a,n,w)} | |
34603 | @r{ a n@: H b l @: @: 9 @:lsh@:(a,n,w)} | |
34604 | @r{ a@: b n @: @: 9 @:not@:(a,w)} | |
34605 | @r{ a b@: b o @: @: 9 @:or@:(a,b,w)} | |
34606 | @r{ v@: b p @: @: 1 @:vpack@:(v)} | |
34607 | @r{ a@: b r @: @: 10 @:rsh@:(a,n,w)} | |
34608 | @r{ a n@: H b r @: @: 9 @:rsh@:(a,n,w)} | |
34609 | @r{ a@: b t @: @: 10 @:rot@:(a,n,w)} | |
34610 | @r{ a n@: H b t @: @: 9 @:rot@:(a,n,w)} | |
34611 | @r{ a@: b u @: @: 1 @:vunpack@:(a)} | |
34612 | @r{ @: b w @:w @: 9,50 @:calc-word-size@:} | |
34613 | @r{ a b@: b x @: @: 9 @:xor@:(a,b,w)} | |
177c0ea7 JB |
34614 | |
34615 | @c | |
d7b8e6c6 EZ |
34616 | @r{c s l p@: b D @: @: @:ddb@:(c,s,l,p)} |
34617 | @r{ r n p@: b F @: @: @:fv@:(r,n,p)} | |
34618 | @r{ r n p@: I b F @: @: @:fvb@:(r,n,p)} | |
34619 | @r{ r n p@: H b F @: @: @:fvl@:(r,n,p)} | |
34620 | @r{ v@: b I @: @: 19 @:irr@:(v)} | |
34621 | @r{ v@: I b I @: @: 19 @:irrb@:(v)} | |
34622 | @r{ a@: b L @: @: 10 @:ash@:(a,n,w)} | |
34623 | @r{ a n@: H b L @: @: 9 @:ash@:(a,n,w)} | |
34624 | @r{ r n a@: b M @: @: @:pmt@:(r,n,a)} | |
34625 | @r{ r n a@: I b M @: @: @:pmtb@:(r,n,a)} | |
34626 | @r{ r n a@: H b M @: @: @:pmtl@:(r,n,a)} | |
34627 | @r{ r v@: b N @: @: 19 @:npv@:(r,v)} | |
34628 | @r{ r v@: I b N @: @: 19 @:npvb@:(r,v)} | |
34629 | @r{ r n p@: b P @: @: @:pv@:(r,n,p)} | |
34630 | @r{ r n p@: I b P @: @: @:pvb@:(r,n,p)} | |
34631 | @r{ r n p@: H b P @: @: @:pvl@:(r,n,p)} | |
34632 | @r{ a@: b R @: @: 10 @:rash@:(a,n,w)} | |
34633 | @r{ a n@: H b R @: @: 9 @:rash@:(a,n,w)} | |
34634 | @r{ c s l@: b S @: @: @:sln@:(c,s,l)} | |
34635 | @r{ n p a@: b T @: @: @:rate@:(n,p,a)} | |
34636 | @r{ n p a@: I b T @: @: @:rateb@:(n,p,a)} | |
34637 | @r{ n p a@: H b T @: @: @:ratel@:(n,p,a)} | |
34638 | @r{c s l p@: b Y @: @: @:syd@:(c,s,l,p)} | |
34639 | ||
34640 | @r{ r p a@: b # @: @: @:nper@:(r,p,a)} | |
34641 | @r{ r p a@: I b # @: @: @:nperb@:(r,p,a)} | |
34642 | @r{ r p a@: H b # @: @: @:nperl@:(r,p,a)} | |
34643 | @r{ a b@: b % @: @: @:relch@:(a,b)} | |
177c0ea7 JB |
34644 | |
34645 | @c | |
d7b8e6c6 EZ |
34646 | @r{ a@: c c @: @: 5 @:pclean@:(a,p)} |
34647 | @r{ a@: c 0-9 @: @: @:pclean@:(a,p)} | |
34648 | @r{ a@: H c c @: @: 5 @:clean@:(a,p)} | |
34649 | @r{ a@: H c 0-9 @: @: @:clean@:(a,p)} | |
34650 | @r{ a@: c d @: @: 1 @:deg@:(a)} | |
34651 | @r{ a@: c f @: @: 1 @:pfloat@:(a)} | |
34652 | @r{ a@: H c f @: @: 1 @:float@:(a)} | |
34653 | @r{ a@: c h @: @: 1 @:hms@:(a)} | |
34654 | @r{ a@: c p @: @: @:polar@:(a)} | |
34655 | @r{ a@: I c p @: @: @:rect@:(a)} | |
34656 | @r{ a@: c r @: @: 1 @:rad@:(a)} | |
177c0ea7 JB |
34657 | |
34658 | @c | |
d7b8e6c6 EZ |
34659 | @r{ a@: c F @: @: 5 @:pfrac@:(a,p)} |
34660 | @r{ a@: H c F @: @: 5 @:frac@:(a,p)} | |
177c0ea7 JB |
34661 | |
34662 | @c | |
d7b8e6c6 | 34663 | @r{ a@: c % @: @: @:percent@:(a*100)} |
177c0ea7 JB |
34664 | |
34665 | @c | |
d7b8e6c6 EZ |
34666 | @r{ @: d . @:char @: 50 @:calc-point-char@:} |
34667 | @r{ @: d , @:char @: 50 @:calc-group-char@:} | |
34668 | @r{ @: d < @: @: 13,50 @:calc-left-justify@:} | |
34669 | @r{ @: d = @: @: 13,50 @:calc-center-justify@:} | |
34670 | @r{ @: d > @: @: 13,50 @:calc-right-justify@:} | |
34671 | @r{ @: d @{ @:label @: 50 @:calc-left-label@:} | |
34672 | @r{ @: d @} @:label @: 50 @:calc-right-label@:} | |
34673 | @r{ @: d [ @: @: 4 @:calc-truncate-up@:} | |
34674 | @r{ @: d ] @: @: 4 @:calc-truncate-down@:} | |
34675 | @r{ @: d " @: @: 12,50 @:calc-display-strings@:} | |
5d67986c RS |
34676 | @r{ @: d @key{SPC} @: @: @:calc-refresh@:} |
34677 | @r{ @: d @key{RET} @: @: 1 @:calc-refresh-top@:} | |
177c0ea7 JB |
34678 | |
34679 | @c | |
d7b8e6c6 EZ |
34680 | @r{ @: d 0 @: @: 50 @:calc-decimal-radix@:} |
34681 | @r{ @: d 2 @: @: 50 @:calc-binary-radix@:} | |
34682 | @r{ @: d 6 @: @: 50 @:calc-hex-radix@:} | |
34683 | @r{ @: d 8 @: @: 50 @:calc-octal-radix@:} | |
177c0ea7 JB |
34684 | |
34685 | @c | |
d7b8e6c6 EZ |
34686 | @r{ @: d b @: @:12,13,50 @:calc-line-breaking@:} |
34687 | @r{ @: d c @: @: 50 @:calc-complex-notation@:} | |
34688 | @r{ @: d d @:format @: 50 @:calc-date-notation@:} | |
34689 | @r{ @: d e @: @: 5,50 @:calc-eng-notation@:} | |
34690 | @r{ @: d f @:num @: 31,50 @:calc-fix-notation@:} | |
34691 | @r{ @: d g @: @:12,13,50 @:calc-group-digits@:} | |
34692 | @r{ @: d h @:format @: 50 @:calc-hms-notation@:} | |
34693 | @r{ @: d i @: @: 50 @:calc-i-notation@:} | |
34694 | @r{ @: d j @: @: 50 @:calc-j-notation@:} | |
34695 | @r{ @: d l @: @: 12,50 @:calc-line-numbering@:} | |
34696 | @r{ @: d n @: @: 5,50 @:calc-normal-notation@:} | |
34697 | @r{ @: d o @:format @: 50 @:calc-over-notation@:} | |
34698 | @r{ @: d p @: @: 12,50 @:calc-show-plain@:} | |
34699 | @r{ @: d r @:radix @: 31,50 @:calc-radix@:} | |
34700 | @r{ @: d s @: @: 5,50 @:calc-sci-notation@:} | |
34701 | @r{ @: d t @: @: 27 @:calc-truncate-stack@:} | |
34702 | @r{ @: d w @: @: 12,13 @:calc-auto-why@:} | |
34703 | @r{ @: d z @: @: 12,50 @:calc-leading-zeros@:} | |
177c0ea7 JB |
34704 | |
34705 | @c | |
d7b8e6c6 EZ |
34706 | @r{ @: d B @: @: 50 @:calc-big-language@:} |
34707 | @r{ @: d C @: @: 50 @:calc-c-language@:} | |
34708 | @r{ @: d E @: @: 50 @:calc-eqn-language@:} | |
34709 | @r{ @: d F @: @: 50 @:calc-fortran-language@:} | |
34710 | @r{ @: d M @: @: 50 @:calc-mathematica-language@:} | |
34711 | @r{ @: d N @: @: 50 @:calc-normal-language@:} | |
34712 | @r{ @: d O @: @: 50 @:calc-flat-language@:} | |
34713 | @r{ @: d P @: @: 50 @:calc-pascal-language@:} | |
34714 | @r{ @: d T @: @: 50 @:calc-tex-language@:} | |
34715 | @r{ @: d U @: @: 50 @:calc-unformatted-language@:} | |
34716 | @r{ @: d W @: @: 50 @:calc-maple-language@:} | |
177c0ea7 JB |
34717 | |
34718 | @c | |
d7b8e6c6 EZ |
34719 | @r{ a@: f [ @: @: 4 @:decr@:(a,n)} |
34720 | @r{ a@: f ] @: @: 4 @:incr@:(a,n)} | |
177c0ea7 JB |
34721 | |
34722 | @c | |
d7b8e6c6 EZ |
34723 | @r{ a b@: f b @: @: 2 @:beta@:(a,b)} |
34724 | @r{ a@: f e @: @: 1 @:erf@:(a)} | |
34725 | @r{ a@: I f e @: @: 1 @:erfc@:(a)} | |
34726 | @r{ a@: f g @: @: 1 @:gamma@:(a)} | |
34727 | @r{ a b@: f h @: @: 2 @:hypot@:(a,b)} | |
34728 | @r{ a@: f i @: @: 1 @:im@:(a)} | |
34729 | @r{ n a@: f j @: @: 2 @:besJ@:(n,a)} | |
34730 | @r{ a b@: f n @: @: 2 @:min@:(a,b)} | |
34731 | @r{ a@: f r @: @: 1 @:re@:(a)} | |
34732 | @r{ a@: f s @: @: 1 @:sign@:(a)} | |
34733 | @r{ a b@: f x @: @: 2 @:max@:(a,b)} | |
34734 | @r{ n a@: f y @: @: 2 @:besY@:(n,a)} | |
177c0ea7 JB |
34735 | |
34736 | @c | |
d7b8e6c6 EZ |
34737 | @r{ a@: f A @: @: 1 @:abssqr@:(a)} |
34738 | @r{ x a b@: f B @: @: @:betaI@:(x,a,b)} | |
34739 | @r{ x a b@: H f B @: @: @:betaB@:(x,a,b)} | |
34740 | @r{ a@: f E @: @: 1 @:expm1@:(a)} | |
34741 | @r{ a x@: f G @: @: 2 @:gammaP@:(a,x)} | |
34742 | @r{ a x@: I f G @: @: 2 @:gammaQ@:(a,x)} | |
34743 | @r{ a x@: H f G @: @: 2 @:gammag@:(a,x)} | |
34744 | @r{ a x@: I H f G @: @: 2 @:gammaG@:(a,x)} | |
34745 | @r{ a b@: f I @: @: 2 @:ilog@:(a,b)} | |
34746 | @r{ a b@: I f I @: @: 2 @:alog@:(a,b) b^a} | |
34747 | @r{ a@: f L @: @: 1 @:lnp1@:(a)} | |
34748 | @r{ a@: f M @: @: 1 @:mant@:(a)} | |
34749 | @r{ a@: f Q @: @: 1 @:isqrt@:(a)} | |
34750 | @r{ a@: I f Q @: @: 1 @:sqr@:(a) a^2} | |
34751 | @r{ a n@: f S @: @: 2 @:scf@:(a,n)} | |
34752 | @r{ y x@: f T @: @: @:arctan2@:(y,x)} | |
34753 | @r{ a@: f X @: @: 1 @:xpon@:(a)} | |
177c0ea7 JB |
34754 | |
34755 | @c | |
d7b8e6c6 EZ |
34756 | @r{ x y@: g a @: @: 28,40 @:calc-graph-add@:} |
34757 | @r{ @: g b @: @: 12 @:calc-graph-border@:} | |
34758 | @r{ @: g c @: @: @:calc-graph-clear@:} | |
34759 | @r{ @: g d @: @: 41 @:calc-graph-delete@:} | |
34760 | @r{ x y@: g f @: @: 28,40 @:calc-graph-fast@:} | |
34761 | @r{ @: g g @: @: 12 @:calc-graph-grid@:} | |
34762 | @r{ @: g h @:title @: @:calc-graph-header@:} | |
34763 | @r{ @: g j @: @: 4 @:calc-graph-juggle@:} | |
34764 | @r{ @: g k @: @: 12 @:calc-graph-key@:} | |
34765 | @r{ @: g l @: @: 12 @:calc-graph-log-x@:} | |
34766 | @r{ @: g n @:name @: @:calc-graph-name@:} | |
34767 | @r{ @: g p @: @: 42 @:calc-graph-plot@:} | |
34768 | @r{ @: g q @: @: @:calc-graph-quit@:} | |
34769 | @r{ @: g r @:range @: @:calc-graph-range-x@:} | |
34770 | @r{ @: g s @: @: 12,13 @:calc-graph-line-style@:} | |
34771 | @r{ @: g t @:title @: @:calc-graph-title-x@:} | |
34772 | @r{ @: g v @: @: @:calc-graph-view-commands@:} | |
34773 | @r{ @: g x @:display @: @:calc-graph-display@:} | |
34774 | @r{ @: g z @: @: 12 @:calc-graph-zero-x@:} | |
177c0ea7 JB |
34775 | |
34776 | @c | |
d7b8e6c6 EZ |
34777 | @r{ x y z@: g A @: @: 28,40 @:calc-graph-add-3d@:} |
34778 | @r{ @: g C @:command @: @:calc-graph-command@:} | |
34779 | @r{ @: g D @:device @: 43,44 @:calc-graph-device@:} | |
34780 | @r{ x y z@: g F @: @: 28,40 @:calc-graph-fast-3d@:} | |
34781 | @r{ @: g H @: @: 12 @:calc-graph-hide@:} | |
34782 | @r{ @: g K @: @: @:calc-graph-kill@:} | |
34783 | @r{ @: g L @: @: 12 @:calc-graph-log-y@:} | |
34784 | @r{ @: g N @:number @: 43,51 @:calc-graph-num-points@:} | |
34785 | @r{ @: g O @:filename @: 43,44 @:calc-graph-output@:} | |
34786 | @r{ @: g P @: @: 42 @:calc-graph-print@:} | |
34787 | @r{ @: g R @:range @: @:calc-graph-range-y@:} | |
34788 | @r{ @: g S @: @: 12,13 @:calc-graph-point-style@:} | |
34789 | @r{ @: g T @:title @: @:calc-graph-title-y@:} | |
34790 | @r{ @: g V @: @: @:calc-graph-view-trail@:} | |
34791 | @r{ @: g X @:format @: @:calc-graph-geometry@:} | |
34792 | @r{ @: g Z @: @: 12 @:calc-graph-zero-y@:} | |
177c0ea7 JB |
34793 | |
34794 | @c | |
d7b8e6c6 EZ |
34795 | @r{ @: g C-l @: @: 12 @:calc-graph-log-z@:} |
34796 | @r{ @: g C-r @:range @: @:calc-graph-range-z@:} | |
34797 | @r{ @: g C-t @:title @: @:calc-graph-title-z@:} | |
177c0ea7 JB |
34798 | |
34799 | @c | |
d7b8e6c6 EZ |
34800 | @r{ @: h b @: @: @:calc-describe-bindings@:} |
34801 | @r{ @: h c @:key @: @:calc-describe-key-briefly@:} | |
34802 | @r{ @: h f @:function @: @:calc-describe-function@:} | |
34803 | @r{ @: h h @: @: @:calc-full-help@:} | |
34804 | @r{ @: h i @: @: @:calc-info@:} | |
34805 | @r{ @: h k @:key @: @:calc-describe-key@:} | |
34806 | @r{ @: h n @: @: @:calc-view-news@:} | |
34807 | @r{ @: h s @: @: @:calc-info-summary@:} | |
34808 | @r{ @: h t @: @: @:calc-tutorial@:} | |
34809 | @r{ @: h v @:var @: @:calc-describe-variable@:} | |
177c0ea7 JB |
34810 | |
34811 | @c | |
d7b8e6c6 | 34812 | @r{ @: j 1-9 @: @: @:calc-select-part@:} |
5d67986c RS |
34813 | @r{ @: j @key{RET} @: @: 27 @:calc-copy-selection@:} |
34814 | @r{ @: j @key{DEL} @: @: 27 @:calc-del-selection@:} | |
d7b8e6c6 EZ |
34815 | @r{ @: j ' @:formula @: 27 @:calc-enter-selection@:} |
34816 | @r{ @: j ` @:editing @: 27,30 @:calc-edit-selection@:} | |
34817 | @r{ @: j " @: @: 7,27 @:calc-sel-expand-formula@:} | |
177c0ea7 JB |
34818 | |
34819 | @c | |
d7b8e6c6 EZ |
34820 | @r{ @: j + @:formula @: 27 @:calc-sel-add-both-sides@:} |
34821 | @r{ @: j - @:formula @: 27 @:calc-sel-sub-both-sides@:} | |
34822 | @r{ @: j * @:formula @: 27 @:calc-sel-mul-both-sides@:} | |
34823 | @r{ @: j / @:formula @: 27 @:calc-sel-div-both-sides@:} | |
34824 | @r{ @: j & @: @: 27 @:calc-sel-invert@:} | |
177c0ea7 JB |
34825 | |
34826 | @c | |
d7b8e6c6 EZ |
34827 | @r{ @: j a @: @: 27 @:calc-select-additional@:} |
34828 | @r{ @: j b @: @: 12 @:calc-break-selections@:} | |
34829 | @r{ @: j c @: @: @:calc-clear-selections@:} | |
34830 | @r{ @: j d @: @: 12,50 @:calc-show-selections@:} | |
34831 | @r{ @: j e @: @: 12 @:calc-enable-selections@:} | |
34832 | @r{ @: j l @: @: 4,27 @:calc-select-less@:} | |
34833 | @r{ @: j m @: @: 4,27 @:calc-select-more@:} | |
34834 | @r{ @: j n @: @: 4 @:calc-select-next@:} | |
34835 | @r{ @: j o @: @: 4,27 @:calc-select-once@:} | |
34836 | @r{ @: j p @: @: 4 @:calc-select-previous@:} | |
34837 | @r{ @: j r @:rules @:4,8,27 @:calc-rewrite-selection@:} | |
34838 | @r{ @: j s @: @: 4,27 @:calc-select-here@:} | |
34839 | @r{ @: j u @: @: 27 @:calc-unselect@:} | |
34840 | @r{ @: j v @: @: 7,27 @:calc-sel-evaluate@:} | |
177c0ea7 JB |
34841 | |
34842 | @c | |
d7b8e6c6 EZ |
34843 | @r{ @: j C @: @: 27 @:calc-sel-commute@:} |
34844 | @r{ @: j D @: @: 4,27 @:calc-sel-distribute@:} | |
34845 | @r{ @: j E @: @: 27 @:calc-sel-jump-equals@:} | |
34846 | @r{ @: j I @: @: 27 @:calc-sel-isolate@:} | |
34847 | @r{ @: H j I @: @: 27 @:calc-sel-isolate@: (full)} | |
34848 | @r{ @: j L @: @: 4,27 @:calc-commute-left@:} | |
34849 | @r{ @: j M @: @: 27 @:calc-sel-merge@:} | |
34850 | @r{ @: j N @: @: 27 @:calc-sel-negate@:} | |
34851 | @r{ @: j O @: @: 4,27 @:calc-select-once-maybe@:} | |
34852 | @r{ @: j R @: @: 4,27 @:calc-commute-right@:} | |
34853 | @r{ @: j S @: @: 4,27 @:calc-select-here-maybe@:} | |
34854 | @r{ @: j U @: @: 27 @:calc-sel-unpack@:} | |
177c0ea7 JB |
34855 | |
34856 | @c | |
d7b8e6c6 EZ |
34857 | @r{ @: k a @: @: @:calc-random-again@:} |
34858 | @r{ n@: k b @: @: 1 @:bern@:(n)} | |
34859 | @r{ n x@: H k b @: @: 2 @:bern@:(n,x)} | |
34860 | @r{ n m@: k c @: @: 2 @:choose@:(n,m)} | |
34861 | @r{ n m@: H k c @: @: 2 @:perm@:(n,m)} | |
34862 | @r{ n@: k d @: @: 1 @:dfact@:(n) n!!} | |
34863 | @r{ n@: k e @: @: 1 @:euler@:(n)} | |
34864 | @r{ n x@: H k e @: @: 2 @:euler@:(n,x)} | |
34865 | @r{ n@: k f @: @: 4 @:prfac@:(n)} | |
34866 | @r{ n m@: k g @: @: 2 @:gcd@:(n,m)} | |
34867 | @r{ m n@: k h @: @: 14 @:shuffle@:(n,m)} | |
34868 | @r{ n m@: k l @: @: 2 @:lcm@:(n,m)} | |
34869 | @r{ n@: k m @: @: 1 @:moebius@:(n)} | |
34870 | @r{ n@: k n @: @: 4 @:nextprime@:(n)} | |
34871 | @r{ n@: I k n @: @: 4 @:prevprime@:(n)} | |
34872 | @r{ n@: k p @: @: 4,28 @:calc-prime-test@:} | |
34873 | @r{ m@: k r @: @: 14 @:random@:(m)} | |
34874 | @r{ n m@: k s @: @: 2 @:stir1@:(n,m)} | |
34875 | @r{ n m@: H k s @: @: 2 @:stir2@:(n,m)} | |
34876 | @r{ n@: k t @: @: 1 @:totient@:(n)} | |
177c0ea7 JB |
34877 | |
34878 | @c | |
d7b8e6c6 EZ |
34879 | @r{ n p x@: k B @: @: @:utpb@:(x,n,p)} |
34880 | @r{ n p x@: I k B @: @: @:ltpb@:(x,n,p)} | |
34881 | @r{ v x@: k C @: @: @:utpc@:(x,v)} | |
34882 | @r{ v x@: I k C @: @: @:ltpc@:(x,v)} | |
34883 | @r{ n m@: k E @: @: @:egcd@:(n,m)} | |
34884 | @r{v1 v2 x@: k F @: @: @:utpf@:(x,v1,v2)} | |
34885 | @r{v1 v2 x@: I k F @: @: @:ltpf@:(x,v1,v2)} | |
34886 | @r{ m s x@: k N @: @: @:utpn@:(x,m,s)} | |
34887 | @r{ m s x@: I k N @: @: @:ltpn@:(x,m,s)} | |
34888 | @r{ m x@: k P @: @: @:utpp@:(x,m)} | |
34889 | @r{ m x@: I k P @: @: @:ltpp@:(x,m)} | |
34890 | @r{ v x@: k T @: @: @:utpt@:(x,v)} | |
34891 | @r{ v x@: I k T @: @: @:ltpt@:(x,v)} | |
177c0ea7 JB |
34892 | |
34893 | @c | |
d7b8e6c6 EZ |
34894 | @r{ @: m a @: @: 12,13 @:calc-algebraic-mode@:} |
34895 | @r{ @: m d @: @: @:calc-degrees-mode@:} | |
34896 | @r{ @: m f @: @: 12 @:calc-frac-mode@:} | |
34897 | @r{ @: m g @: @: 52 @:calc-get-modes@:} | |
34898 | @r{ @: m h @: @: @:calc-hms-mode@:} | |
34899 | @r{ @: m i @: @: 12,13 @:calc-infinite-mode@:} | |
34900 | @r{ @: m m @: @: @:calc-save-modes@:} | |
34901 | @r{ @: m p @: @: 12 @:calc-polar-mode@:} | |
34902 | @r{ @: m r @: @: @:calc-radians-mode@:} | |
34903 | @r{ @: m s @: @: 12 @:calc-symbolic-mode@:} | |
34904 | @r{ @: m t @: @: 12 @:calc-total-algebraic-mode@:} | |
34905 | @r{ @: m v @: @: 12,13 @:calc-matrix-mode@:} | |
34906 | @r{ @: m w @: @: 13 @:calc-working@:} | |
34907 | @r{ @: m x @: @: @:calc-always-load-extensions@:} | |
177c0ea7 JB |
34908 | |
34909 | @c | |
d7b8e6c6 EZ |
34910 | @r{ @: m A @: @: 12 @:calc-alg-simplify-mode@:} |
34911 | @r{ @: m B @: @: 12 @:calc-bin-simplify-mode@:} | |
34912 | @r{ @: m C @: @: 12 @:calc-auto-recompute@:} | |
34913 | @r{ @: m D @: @: @:calc-default-simplify-mode@:} | |
34914 | @r{ @: m E @: @: 12 @:calc-ext-simplify-mode@:} | |
34915 | @r{ @: m F @:filename @: 13 @:calc-settings-file-name@:} | |
34916 | @r{ @: m N @: @: 12 @:calc-num-simplify-mode@:} | |
34917 | @r{ @: m O @: @: 12 @:calc-no-simplify-mode@:} | |
34918 | @r{ @: m R @: @: 12,13 @:calc-mode-record-mode@:} | |
34919 | @r{ @: m S @: @: 12 @:calc-shift-prefix@:} | |
34920 | @r{ @: m U @: @: 12 @:calc-units-simplify-mode@:} | |
177c0ea7 JB |
34921 | |
34922 | @c | |
d7b8e6c6 EZ |
34923 | @r{ @: s c @:var1, var2 @: 29 @:calc-copy-variable@:} |
34924 | @r{ @: s d @:var, decl @: @:calc-declare-variable@:} | |
34925 | @r{ @: s e @:var, editing @: 29,30 @:calc-edit-variable@:} | |
34926 | @r{ @: s i @:buffer @: @:calc-insert-variables@:} | |
34927 | @r{ a b@: s l @:var @: 29 @:@:a (letting var=b)} | |
34928 | @r{ a ...@: s m @:op, var @: 22,29 @:calc-store-map@:} | |
34929 | @r{ @: s n @:var @: 29,47 @:calc-store-neg@: (v/-1)} | |
34930 | @r{ @: s p @:var @: 29 @:calc-permanent-variable@:} | |
34931 | @r{ @: s r @:var @: 29 @:@:v (recalled value)} | |
34932 | @r{ @: r 0-9 @: @: @:calc-recall-quick@:} | |
34933 | @r{ a@: s s @:var @: 28,29 @:calc-store@:} | |
34934 | @r{ a@: s 0-9 @: @: @:calc-store-quick@:} | |
34935 | @r{ a@: s t @:var @: 29 @:calc-store-into@:} | |
34936 | @r{ a@: t 0-9 @: @: @:calc-store-into-quick@:} | |
34937 | @r{ @: s u @:var @: 29 @:calc-unstore@:} | |
34938 | @r{ a@: s x @:var @: 29 @:calc-store-exchange@:} | |
177c0ea7 JB |
34939 | |
34940 | @c | |
d7b8e6c6 EZ |
34941 | @r{ @: s A @:editing @: 30 @:calc-edit-AlgSimpRules@:} |
34942 | @r{ @: s D @:editing @: 30 @:calc-edit-Decls@:} | |
34943 | @r{ @: s E @:editing @: 30 @:calc-edit-EvalRules@:} | |
34944 | @r{ @: s F @:editing @: 30 @:calc-edit-FitRules@:} | |
34945 | @r{ @: s G @:editing @: 30 @:calc-edit-GenCount@:} | |
34946 | @r{ @: s H @:editing @: 30 @:calc-edit-Holidays@:} | |
34947 | @r{ @: s I @:editing @: 30 @:calc-edit-IntegLimit@:} | |
34948 | @r{ @: s L @:editing @: 30 @:calc-edit-LineStyles@:} | |
34949 | @r{ @: s P @:editing @: 30 @:calc-edit-PointStyles@:} | |
34950 | @r{ @: s R @:editing @: 30 @:calc-edit-PlotRejects@:} | |
34951 | @r{ @: s T @:editing @: 30 @:calc-edit-TimeZone@:} | |
34952 | @r{ @: s U @:editing @: 30 @:calc-edit-Units@:} | |
34953 | @r{ @: s X @:editing @: 30 @:calc-edit-ExtSimpRules@:} | |
177c0ea7 JB |
34954 | |
34955 | @c | |
d7b8e6c6 EZ |
34956 | @r{ a@: s + @:var @: 29,47 @:calc-store-plus@: (v+a)} |
34957 | @r{ a@: s - @:var @: 29,47 @:calc-store-minus@: (v-a)} | |
34958 | @r{ a@: s * @:var @: 29,47 @:calc-store-times@: (v*a)} | |
34959 | @r{ a@: s / @:var @: 29,47 @:calc-store-div@: (v/a)} | |
34960 | @r{ a@: s ^ @:var @: 29,47 @:calc-store-power@: (v^a)} | |
34961 | @r{ a@: s | @:var @: 29,47 @:calc-store-concat@: (v|a)} | |
34962 | @r{ @: s & @:var @: 29,47 @:calc-store-inv@: (v^-1)} | |
34963 | @r{ @: s [ @:var @: 29,47 @:calc-store-decr@: (v-1)} | |
34964 | @r{ @: s ] @:var @: 29,47 @:calc-store-incr@: (v-(-1))} | |
34965 | @r{ a b@: s : @: @: 2 @:assign@:(a,b) a @t{:=} b} | |
34966 | @r{ a@: s = @: @: 1 @:evalto@:(a,b) a @t{=>}} | |
177c0ea7 JB |
34967 | |
34968 | @c | |
d7b8e6c6 EZ |
34969 | @r{ @: t [ @: @: 4 @:calc-trail-first@:} |
34970 | @r{ @: t ] @: @: 4 @:calc-trail-last@:} | |
34971 | @r{ @: t < @: @: 4 @:calc-trail-scroll-left@:} | |
34972 | @r{ @: t > @: @: 4 @:calc-trail-scroll-right@:} | |
34973 | @r{ @: t . @: @: 12 @:calc-full-trail-vectors@:} | |
177c0ea7 JB |
34974 | |
34975 | @c | |
d7b8e6c6 EZ |
34976 | @r{ @: t b @: @: 4 @:calc-trail-backward@:} |
34977 | @r{ @: t d @: @: 12,50 @:calc-trail-display@:} | |
34978 | @r{ @: t f @: @: 4 @:calc-trail-forward@:} | |
34979 | @r{ @: t h @: @: @:calc-trail-here@:} | |
34980 | @r{ @: t i @: @: @:calc-trail-in@:} | |
34981 | @r{ @: t k @: @: 4 @:calc-trail-kill@:} | |
34982 | @r{ @: t m @:string @: @:calc-trail-marker@:} | |
34983 | @r{ @: t n @: @: 4 @:calc-trail-next@:} | |
34984 | @r{ @: t o @: @: @:calc-trail-out@:} | |
34985 | @r{ @: t p @: @: 4 @:calc-trail-previous@:} | |
34986 | @r{ @: t r @:string @: @:calc-trail-isearch-backward@:} | |
34987 | @r{ @: t s @:string @: @:calc-trail-isearch-forward@:} | |
34988 | @r{ @: t y @: @: 4 @:calc-trail-yank@:} | |
177c0ea7 JB |
34989 | |
34990 | @c | |
d7b8e6c6 EZ |
34991 | @r{ d@: t C @:oz, nz @: @:tzconv@:(d,oz,nz)} |
34992 | @r{d oz nz@: t C @:$ @: @:tzconv@:(d,oz,nz)} | |
34993 | @r{ d@: t D @: @: 15 @:date@:(d)} | |
34994 | @r{ d@: t I @: @: 4 @:incmonth@:(d,n)} | |
34995 | @r{ d@: t J @: @: 16 @:julian@:(d,z)} | |
34996 | @r{ d@: t M @: @: 17 @:newmonth@:(d,n)} | |
34997 | @r{ @: t N @: @: 16 @:now@:(z)} | |
34998 | @r{ d@: t P @:1 @: 31 @:year@:(d)} | |
34999 | @r{ d@: t P @:2 @: 31 @:month@:(d)} | |
35000 | @r{ d@: t P @:3 @: 31 @:day@:(d)} | |
35001 | @r{ d@: t P @:4 @: 31 @:hour@:(d)} | |
35002 | @r{ d@: t P @:5 @: 31 @:minute@:(d)} | |
35003 | @r{ d@: t P @:6 @: 31 @:second@:(d)} | |
35004 | @r{ d@: t P @:7 @: 31 @:weekday@:(d)} | |
35005 | @r{ d@: t P @:8 @: 31 @:yearday@:(d)} | |
35006 | @r{ d@: t P @:9 @: 31 @:time@:(d)} | |
35007 | @r{ d@: t U @: @: 16 @:unixtime@:(d,z)} | |
35008 | @r{ d@: t W @: @: 17 @:newweek@:(d,w)} | |
35009 | @r{ d@: t Y @: @: 17 @:newyear@:(d,n)} | |
177c0ea7 JB |
35010 | |
35011 | @c | |
d7b8e6c6 EZ |
35012 | @r{ a b@: t + @: @: 2 @:badd@:(a,b)} |
35013 | @r{ a b@: t - @: @: 2 @:bsub@:(a,b)} | |
177c0ea7 JB |
35014 | |
35015 | @c | |
d7b8e6c6 EZ |
35016 | @r{ @: u a @: @: 12 @:calc-autorange-units@:} |
35017 | @r{ a@: u b @: @: @:calc-base-units@:} | |
35018 | @r{ a@: u c @:units @: 18 @:calc-convert-units@:} | |
35019 | @r{ defn@: u d @:unit, descr @: @:calc-define-unit@:} | |
35020 | @r{ @: u e @: @: @:calc-explain-units@:} | |
35021 | @r{ @: u g @:unit @: @:calc-get-unit-definition@:} | |
35022 | @r{ @: u p @: @: @:calc-permanent-units@:} | |
35023 | @r{ a@: u r @: @: @:calc-remove-units@:} | |
35024 | @r{ a@: u s @: @: @:usimplify@:(a)} | |
35025 | @r{ a@: u t @:units @: 18 @:calc-convert-temperature@:} | |
35026 | @r{ @: u u @:unit @: @:calc-undefine-unit@:} | |
35027 | @r{ @: u v @: @: @:calc-enter-units-table@:} | |
35028 | @r{ a@: u x @: @: @:calc-extract-units@:} | |
35029 | @r{ a@: u 0-9 @: @: @:calc-quick-units@:} | |
177c0ea7 JB |
35030 | |
35031 | @c | |
d7b8e6c6 EZ |
35032 | @r{ v1 v2@: u C @: @: 20 @:vcov@:(v1,v2)} |
35033 | @r{ v1 v2@: I u C @: @: 20 @:vpcov@:(v1,v2)} | |
35034 | @r{ v1 v2@: H u C @: @: 20 @:vcorr@:(v1,v2)} | |
35035 | @r{ v@: u G @: @: 19 @:vgmean@:(v)} | |
35036 | @r{ a b@: H u G @: @: 2 @:agmean@:(a,b)} | |
35037 | @r{ v@: u M @: @: 19 @:vmean@:(v)} | |
35038 | @r{ v@: I u M @: @: 19 @:vmeane@:(v)} | |
35039 | @r{ v@: H u M @: @: 19 @:vmedian@:(v)} | |
35040 | @r{ v@: I H u M @: @: 19 @:vhmean@:(v)} | |
35041 | @r{ v@: u N @: @: 19 @:vmin@:(v)} | |
35042 | @r{ v@: u S @: @: 19 @:vsdev@:(v)} | |
35043 | @r{ v@: I u S @: @: 19 @:vpsdev@:(v)} | |
35044 | @r{ v@: H u S @: @: 19 @:vvar@:(v)} | |
35045 | @r{ v@: I H u S @: @: 19 @:vpvar@:(v)} | |
35046 | @r{ @: u V @: @: @:calc-view-units-table@:} | |
35047 | @r{ v@: u X @: @: 19 @:vmax@:(v)} | |
177c0ea7 JB |
35048 | |
35049 | @c | |
d7b8e6c6 EZ |
35050 | @r{ v@: u + @: @: 19 @:vsum@:(v)} |
35051 | @r{ v@: u * @: @: 19 @:vprod@:(v)} | |
35052 | @r{ v@: u # @: @: 19 @:vcount@:(v)} | |
177c0ea7 JB |
35053 | |
35054 | @c | |
d7b8e6c6 EZ |
35055 | @r{ @: V ( @: @: 50 @:calc-vector-parens@:} |
35056 | @r{ @: V @{ @: @: 50 @:calc-vector-braces@:} | |
35057 | @r{ @: V [ @: @: 50 @:calc-vector-brackets@:} | |
35058 | @r{ @: V ] @:ROCP @: 50 @:calc-matrix-brackets@:} | |
35059 | @r{ @: V , @: @: 50 @:calc-vector-commas@:} | |
35060 | @r{ @: V < @: @: 50 @:calc-matrix-left-justify@:} | |
35061 | @r{ @: V = @: @: 50 @:calc-matrix-center-justify@:} | |
35062 | @r{ @: V > @: @: 50 @:calc-matrix-right-justify@:} | |
35063 | @r{ @: V / @: @: 12,50 @:calc-break-vectors@:} | |
35064 | @r{ @: V . @: @: 12,50 @:calc-full-vectors@:} | |
177c0ea7 JB |
35065 | |
35066 | @c | |
d7b8e6c6 EZ |
35067 | @r{ s t@: V ^ @: @: 2 @:vint@:(s,t)} |
35068 | @r{ s t@: V - @: @: 2 @:vdiff@:(s,t)} | |
35069 | @r{ s@: V ~ @: @: 1 @:vcompl@:(s)} | |
35070 | @r{ s@: V # @: @: 1 @:vcard@:(s)} | |
35071 | @r{ s@: V : @: @: 1 @:vspan@:(s)} | |
35072 | @r{ s@: V + @: @: 1 @:rdup@:(s)} | |
177c0ea7 JB |
35073 | |
35074 | @c | |
d7b8e6c6 | 35075 | @r{ m@: V & @: @: 1 @:inv@:(m) 1/m} |
177c0ea7 JB |
35076 | |
35077 | @c | |
d7b8e6c6 EZ |
35078 | @r{ v@: v a @:n @: @:arrange@:(v,n)} |
35079 | @r{ a@: v b @:n @: @:cvec@:(a,n)} | |
35080 | @r{ v@: v c @:n >0 @: 21,31 @:mcol@:(v,n)} | |
35081 | @r{ v@: v c @:n <0 @: 31 @:mrcol@:(v,-n)} | |
35082 | @r{ m@: v c @:0 @: 31 @:getdiag@:(m)} | |
35083 | @r{ v@: v d @: @: 25 @:diag@:(v,n)} | |
35084 | @r{ v m@: v e @: @: 2 @:vexp@:(v,m)} | |
35085 | @r{ v m f@: H v e @: @: 2 @:vexp@:(v,m,f)} | |
35086 | @r{ v a@: v f @: @: 26 @:find@:(v,a,n)} | |
35087 | @r{ v@: v h @: @: 1 @:head@:(v)} | |
35088 | @r{ v@: I v h @: @: 1 @:tail@:(v)} | |
35089 | @r{ v@: H v h @: @: 1 @:rhead@:(v)} | |
35090 | @r{ v@: I H v h @: @: 1 @:rtail@:(v)} | |
35091 | @r{ @: v i @:n @: 31 @:idn@:(1,n)} | |
35092 | @r{ @: v i @:0 @: 31 @:idn@:(1)} | |
35093 | @r{ h t@: v k @: @: 2 @:cons@:(h,t)} | |
35094 | @r{ h t@: H v k @: @: 2 @:rcons@:(h,t)} | |
35095 | @r{ v@: v l @: @: 1 @:vlen@:(v)} | |
35096 | @r{ v@: H v l @: @: 1 @:mdims@:(v)} | |
35097 | @r{ v m@: v m @: @: 2 @:vmask@:(v,m)} | |
35098 | @r{ v@: v n @: @: 1 @:rnorm@:(v)} | |
35099 | @r{ a b c@: v p @: @: 24 @:calc-pack@:} | |
35100 | @r{ v@: v r @:n >0 @: 21,31 @:mrow@:(v,n)} | |
35101 | @r{ v@: v r @:n <0 @: 31 @:mrrow@:(v,-n)} | |
35102 | @r{ m@: v r @:0 @: 31 @:getdiag@:(m)} | |
35103 | @r{ v i j@: v s @: @: @:subvec@:(v,i,j)} | |
35104 | @r{ v i j@: I v s @: @: @:rsubvec@:(v,i,j)} | |
35105 | @r{ m@: v t @: @: 1 @:trn@:(m)} | |
35106 | @r{ v@: v u @: @: 24 @:calc-unpack@:} | |
35107 | @r{ v@: v v @: @: 1 @:rev@:(v)} | |
35108 | @r{ @: v x @:n @: 31 @:index@:(n)} | |
35109 | @r{ n s i@: C-u v x @: @: @:index@:(n,s,i)} | |
177c0ea7 JB |
35110 | |
35111 | @c | |
d7b8e6c6 EZ |
35112 | @r{ v@: V A @:op @: 22 @:apply@:(op,v)} |
35113 | @r{ v1 v2@: V C @: @: 2 @:cross@:(v1,v2)} | |
35114 | @r{ m@: V D @: @: 1 @:det@:(m)} | |
35115 | @r{ s@: V E @: @: 1 @:venum@:(s)} | |
35116 | @r{ s@: V F @: @: 1 @:vfloor@:(s)} | |
35117 | @r{ v@: V G @: @: @:grade@:(v)} | |
35118 | @r{ v@: I V G @: @: @:rgrade@:(v)} | |
35119 | @r{ v@: V H @:n @: 31 @:histogram@:(v,n)} | |
35120 | @r{ v w@: H V H @:n @: 31 @:histogram@:(v,w,n)} | |
35121 | @r{ v1 v2@: V I @:mop aop @: 22 @:inner@:(mop,aop,v1,v2)} | |
35122 | @r{ m@: V J @: @: 1 @:ctrn@:(m)} | |
35123 | @r{ m@: V L @: @: 1 @:lud@:(m)} | |
35124 | @r{ v@: V M @:op @: 22,23 @:map@:(op,v)} | |
35125 | @r{ v@: V N @: @: 1 @:cnorm@:(v)} | |
35126 | @r{ v1 v2@: V O @:op @: 22 @:outer@:(op,v1,v2)} | |
35127 | @r{ v@: V R @:op @: 22,23 @:reduce@:(op,v)} | |
35128 | @r{ v@: I V R @:op @: 22,23 @:rreduce@:(op,v)} | |
35129 | @r{ a n@: H V R @:op @: 22 @:nest@:(op,a,n)} | |
35130 | @r{ a@: I H V R @:op @: 22 @:fixp@:(op,a)} | |
35131 | @r{ v@: V S @: @: @:sort@:(v)} | |
35132 | @r{ v@: I V S @: @: @:rsort@:(v)} | |
35133 | @r{ m@: V T @: @: 1 @:tr@:(m)} | |
35134 | @r{ v@: V U @:op @: 22 @:accum@:(op,v)} | |
35135 | @r{ v@: I V U @:op @: 22 @:raccum@:(op,v)} | |
35136 | @r{ a n@: H V U @:op @: 22 @:anest@:(op,a,n)} | |
35137 | @r{ a@: I H V U @:op @: 22 @:afixp@:(op,a)} | |
35138 | @r{ s t@: V V @: @: 2 @:vunion@:(s,t)} | |
35139 | @r{ s t@: V X @: @: 2 @:vxor@:(s,t)} | |
177c0ea7 JB |
35140 | |
35141 | @c | |
d7b8e6c6 | 35142 | @r{ @: Y @: @: @:@:user commands} |
177c0ea7 JB |
35143 | |
35144 | @c | |
d7b8e6c6 | 35145 | @r{ @: z @: @: @:@:user commands} |
177c0ea7 JB |
35146 | |
35147 | @c | |
d7b8e6c6 EZ |
35148 | @r{ c@: Z [ @: @: 45 @:calc-kbd-if@:} |
35149 | @r{ c@: Z | @: @: 45 @:calc-kbd-else-if@:} | |
35150 | @r{ @: Z : @: @: @:calc-kbd-else@:} | |
35151 | @r{ @: Z ] @: @: @:calc-kbd-end-if@:} | |
177c0ea7 JB |
35152 | |
35153 | @c | |
d7b8e6c6 EZ |
35154 | @r{ @: Z @{ @: @: 4 @:calc-kbd-loop@:} |
35155 | @r{ c@: Z / @: @: 45 @:calc-kbd-break@:} | |
35156 | @r{ @: Z @} @: @: @:calc-kbd-end-loop@:} | |
35157 | @r{ n@: Z < @: @: @:calc-kbd-repeat@:} | |
35158 | @r{ @: Z > @: @: @:calc-kbd-end-repeat@:} | |
35159 | @r{ n m@: Z ( @: @: @:calc-kbd-for@:} | |
35160 | @r{ s@: Z ) @: @: @:calc-kbd-end-for@:} | |
177c0ea7 JB |
35161 | |
35162 | @c | |
d7b8e6c6 | 35163 | @r{ @: Z C-g @: @: @:@:cancel if/loop command} |
177c0ea7 JB |
35164 | |
35165 | @c | |
d7b8e6c6 EZ |
35166 | @r{ @: Z ` @: @: @:calc-kbd-push@:} |
35167 | @r{ @: Z ' @: @: @:calc-kbd-pop@:} | |
35168 | @r{ a@: Z = @:message @: 28 @:calc-kbd-report@:} | |
35169 | @r{ @: Z # @:prompt @: @:calc-kbd-query@:} | |
177c0ea7 JB |
35170 | |
35171 | @c | |
d7b8e6c6 EZ |
35172 | @r{ comp@: Z C @:func, args @: 50 @:calc-user-define-composition@:} |
35173 | @r{ @: Z D @:key, command @: @:calc-user-define@:} | |
35174 | @r{ @: Z E @:key, editing @: 30 @:calc-user-define-edit@:} | |
35175 | @r{ defn@: Z F @:k, c, f, a, n@: 28 @:calc-user-define-formula@:} | |
35176 | @r{ @: Z G @:key @: @:calc-get-user-defn@:} | |
35177 | @r{ @: Z I @: @: @:calc-user-define-invocation@:} | |
35178 | @r{ @: Z K @:key, command @: @:calc-user-define-kbd-macro@:} | |
35179 | @r{ @: Z P @:key @: @:calc-user-define-permanent@:} | |
35180 | @r{ @: Z S @: @: 30 @:calc-edit-user-syntax@:} | |
35181 | @r{ @: Z T @: @: 12 @:calc-timing@:} | |
35182 | @r{ @: Z U @:key @: @:calc-user-undefine@:} | |
35183 | ||
35184 | @end format | |
35185 | ||
35186 | @noindent | |
35187 | NOTES | |
35188 | ||
35189 | @enumerate | |
35190 | @c 1 | |
35191 | @item | |
35192 | Positive prefix arguments apply to @cite{n} stack entries. | |
35193 | Negative prefix arguments apply to the @cite{-n}th stack entry. | |
35194 | A prefix of zero applies to the entire stack. (For @key{LFD} and | |
5d67986c | 35195 | @kbd{M-@key{DEL}}, the meaning of the sign is reversed.) |
d7b8e6c6 EZ |
35196 | |
35197 | @c 2 | |
35198 | @item | |
35199 | Positive prefix arguments apply to @cite{n} stack entries. | |
35200 | Negative prefix arguments apply to the top stack entry | |
35201 | and the next @cite{-n} stack entries. | |
35202 | ||
35203 | @c 3 | |
35204 | @item | |
35205 | Positive prefix arguments rotate top @cite{n} stack entries by one. | |
35206 | Negative prefix arguments rotate the entire stack by @cite{-n}. | |
35207 | A prefix of zero reverses the entire stack. | |
35208 | ||
35209 | @c 4 | |
35210 | @item | |
35211 | Prefix argument specifies a repeat count or distance. | |
35212 | ||
35213 | @c 5 | |
35214 | @item | |
35215 | Positive prefix arguments specify a precision @cite{p}. | |
35216 | Negative prefix arguments reduce the current precision by @cite{-p}. | |
35217 | ||
35218 | @c 6 | |
35219 | @item | |
35220 | A prefix argument is interpreted as an additional step-size parameter. | |
35221 | A plain @kbd{C-u} prefix means to prompt for the step size. | |
35222 | ||
35223 | @c 7 | |
35224 | @item | |
35225 | A prefix argument specifies simplification level and depth. | |
35226 | 1=Default, 2=like @kbd{a s}, 3=like @kbd{a e}. | |
35227 | ||
35228 | @c 8 | |
35229 | @item | |
35230 | A negative prefix operates only on the top level of the input formula. | |
35231 | ||
35232 | @c 9 | |
35233 | @item | |
35234 | Positive prefix arguments specify a word size of @cite{w} bits, unsigned. | |
35235 | Negative prefix arguments specify a word size of @cite{w} bits, signed. | |
35236 | ||
35237 | @c 10 | |
35238 | @item | |
35239 | Prefix arguments specify the shift amount @cite{n}. The @cite{w} argument | |
35240 | cannot be specified in the keyboard version of this command. | |
35241 | ||
35242 | @c 11 | |
35243 | @item | |
35244 | From the keyboard, @cite{d} is omitted and defaults to zero. | |
35245 | ||
35246 | @c 12 | |
35247 | @item | |
35248 | Mode is toggled; a positive prefix always sets the mode, and a negative | |
35249 | prefix always clears the mode. | |
35250 | ||
35251 | @c 13 | |
35252 | @item | |
35253 | Some prefix argument values provide special variations of the mode. | |
35254 | ||
35255 | @c 14 | |
35256 | @item | |
35257 | A prefix argument, if any, is used for @cite{m} instead of taking | |
35258 | @cite{m} from the stack. @cite{M} may take any of these values: | |
35259 | @iftex | |
35260 | {@advance@tableindent10pt | |
35261 | @end iftex | |
35262 | @table @asis | |
35263 | @item Integer | |
35264 | Random integer in the interval @cite{[0 .. m)}. | |
35265 | @item Float | |
35266 | Random floating-point number in the interval @cite{[0 .. m)}. | |
35267 | @item 0.0 | |
35268 | Gaussian with mean 1 and standard deviation 0. | |
35269 | @item Error form | |
35270 | Gaussian with specified mean and standard deviation. | |
35271 | @item Interval | |
35272 | Random integer or floating-point number in that interval. | |
35273 | @item Vector | |
35274 | Random element from the vector. | |
35275 | @end table | |
35276 | @iftex | |
35277 | } | |
35278 | @end iftex | |
35279 | ||
35280 | @c 15 | |
35281 | @item | |
35282 | A prefix argument from 1 to 6 specifies number of date components | |
35283 | to remove from the stack. @xref{Date Conversions}. | |
35284 | ||
35285 | @c 16 | |
35286 | @item | |
35287 | A prefix argument specifies a time zone; @kbd{C-u} says to take the | |
35288 | time zone number or name from the top of the stack. @xref{Time Zones}. | |
35289 | ||
35290 | @c 17 | |
35291 | @item | |
35292 | A prefix argument specifies a day number (0-6, 0-31, or 0-366). | |
35293 | ||
35294 | @c 18 | |
35295 | @item | |
35296 | If the input has no units, you will be prompted for both the old and | |
35297 | the new units. | |
35298 | ||
35299 | @c 19 | |
35300 | @item | |
35301 | With a prefix argument, collect that many stack entries to form the | |
35302 | input data set. Each entry may be a single value or a vector of values. | |
35303 | ||
35304 | @c 20 | |
35305 | @item | |
5d67986c RS |
35306 | With a prefix argument of 1, take a single @c{$@var{n}\times2$} |
35307 | @i{@var{N}x2} matrix from the | |
d7b8e6c6 EZ |
35308 | stack instead of two separate data vectors. |
35309 | ||
35310 | @c 21 | |
35311 | @item | |
35312 | The row or column number @cite{n} may be given as a numeric prefix | |
35313 | argument instead. A plain @kbd{C-u} prefix says to take @cite{n} | |
35314 | from the top of the stack. If @cite{n} is a vector or interval, | |
35315 | a subvector/submatrix of the input is created. | |
35316 | ||
35317 | @c 22 | |
35318 | @item | |
35319 | The @cite{op} prompt can be answered with the key sequence for the | |
35320 | desired function, or with @kbd{x} or @kbd{z} followed by a function name, | |
35321 | or with @kbd{$} to take a formula from the top of the stack, or with | |
35322 | @kbd{'} and a typed formula. In the last two cases, the formula may | |
35323 | be a nameless function like @samp{<#1+#2>} or @samp{<x, y : x+y>}, or it | |
35324 | may include @kbd{$}, @kbd{$$}, etc. (where @kbd{$} will correspond to the | |
35325 | last argument of the created function), or otherwise you will be | |
35326 | prompted for an argument list. The number of vectors popped from the | |
35327 | stack by @kbd{V M} depends on the number of arguments of the function. | |
35328 | ||
35329 | @c 23 | |
35330 | @item | |
35331 | One of the mapping direction keys @kbd{_} (horizontal, i.e., map | |
35332 | by rows or reduce across), @kbd{:} (vertical, i.e., map by columns or | |
35333 | reduce down), or @kbd{=} (map or reduce by rows) may be used before | |
35334 | entering @cite{op}; these modify the function name by adding the letter | |
35335 | @code{r} for ``rows,'' @code{c} for ``columns,'' @code{a} for ``across,'' | |
35336 | or @code{d} for ``down.'' | |
35337 | ||
35338 | @c 24 | |
35339 | @item | |
35340 | The prefix argument specifies a packing mode. A nonnegative mode | |
35341 | is the number of items (for @kbd{v p}) or the number of levels | |
35342 | (for @kbd{v u}). A negative mode is as described below. With no | |
35343 | prefix argument, the mode is taken from the top of the stack and | |
35344 | may be an integer or a vector of integers. | |
35345 | @iftex | |
35346 | {@advance@tableindent-20pt | |
35347 | @end iftex | |
35348 | @table @cite | |
35349 | @item -1 | |
b275eac7 | 35350 | (@var{2}) Rectangular complex number. |
d7b8e6c6 | 35351 | @item -2 |
b275eac7 | 35352 | (@var{2}) Polar complex number. |
d7b8e6c6 | 35353 | @item -3 |
b275eac7 | 35354 | (@var{3}) HMS form. |
d7b8e6c6 | 35355 | @item -4 |
b275eac7 | 35356 | (@var{2}) Error form. |
d7b8e6c6 | 35357 | @item -5 |
b275eac7 | 35358 | (@var{2}) Modulo form. |
d7b8e6c6 | 35359 | @item -6 |
b275eac7 | 35360 | (@var{2}) Closed interval. |
d7b8e6c6 | 35361 | @item -7 |
b275eac7 | 35362 | (@var{2}) Closed .. open interval. |
d7b8e6c6 | 35363 | @item -8 |
b275eac7 | 35364 | (@var{2}) Open .. closed interval. |
d7b8e6c6 | 35365 | @item -9 |
b275eac7 | 35366 | (@var{2}) Open interval. |
d7b8e6c6 | 35367 | @item -10 |
b275eac7 | 35368 | (@var{2}) Fraction. |
d7b8e6c6 | 35369 | @item -11 |
b275eac7 | 35370 | (@var{2}) Float with integer mantissa. |
d7b8e6c6 | 35371 | @item -12 |
b275eac7 | 35372 | (@var{2}) Float with mantissa in @cite{[1 .. 10)}. |
d7b8e6c6 | 35373 | @item -13 |
b275eac7 | 35374 | (@var{1}) Date form (using date numbers). |
d7b8e6c6 | 35375 | @item -14 |
b275eac7 | 35376 | (@var{3}) Date form (using year, month, day). |
d7b8e6c6 | 35377 | @item -15 |
b275eac7 | 35378 | (@var{6}) Date form (using year, month, day, hour, minute, second). |
d7b8e6c6 EZ |
35379 | @end table |
35380 | @iftex | |
35381 | } | |
35382 | @end iftex | |
35383 | ||
35384 | @c 25 | |
35385 | @item | |
35386 | A prefix argument specifies the size @cite{n} of the matrix. With no | |
35387 | prefix argument, @cite{n} is omitted and the size is inferred from | |
35388 | the input vector. | |
35389 | ||
35390 | @c 26 | |
35391 | @item | |
35392 | The prefix argument specifies the starting position @cite{n} (default 1). | |
35393 | ||
35394 | @c 27 | |
35395 | @item | |
35396 | Cursor position within stack buffer affects this command. | |
35397 | ||
35398 | @c 28 | |
35399 | @item | |
35400 | Arguments are not actually removed from the stack by this command. | |
35401 | ||
35402 | @c 29 | |
35403 | @item | |
35404 | Variable name may be a single digit or a full name. | |
35405 | ||
35406 | @c 30 | |
35407 | @item | |
35408 | Editing occurs in a separate buffer. Press @kbd{M-# M-#} (or @kbd{C-c C-c}, | |
35409 | @key{LFD}, or in some cases @key{RET}) to finish the edit, or press | |
35410 | @kbd{M-# x} to cancel the edit. The @key{LFD} key prevents evaluation | |
35411 | of the result of the edit. | |
35412 | ||
35413 | @c 31 | |
35414 | @item | |
35415 | The number prompted for can also be provided as a prefix argument. | |
35416 | ||
35417 | @c 32 | |
35418 | @item | |
35419 | Press this key a second time to cancel the prefix. | |
35420 | ||
35421 | @c 33 | |
35422 | @item | |
35423 | With a negative prefix, deactivate all formulas. With a positive | |
35424 | prefix, deactivate and then reactivate from scratch. | |
35425 | ||
35426 | @c 34 | |
35427 | @item | |
35428 | Default is to scan for nearest formula delimiter symbols. With a | |
35429 | prefix of zero, formula is delimited by mark and point. With a | |
35430 | non-zero prefix, formula is delimited by scanning forward or | |
35431 | backward by that many lines. | |
35432 | ||
35433 | @c 35 | |
35434 | @item | |
35435 | Parse the region between point and mark as a vector. A nonzero prefix | |
35436 | parses @var{n} lines before or after point as a vector. A zero prefix | |
35437 | parses the current line as a vector. A @kbd{C-u} prefix parses the | |
35438 | region between point and mark as a single formula. | |
35439 | ||
35440 | @c 36 | |
35441 | @item | |
35442 | Parse the rectangle defined by point and mark as a matrix. A positive | |
35443 | prefix @var{n} divides the rectangle into columns of width @var{n}. | |
35444 | A zero or @kbd{C-u} prefix parses each line as one formula. A negative | |
35445 | prefix suppresses special treatment of bracketed portions of a line. | |
35446 | ||
35447 | @c 37 | |
35448 | @item | |
35449 | A numeric prefix causes the current language mode to be ignored. | |
35450 | ||
35451 | @c 38 | |
35452 | @item | |
35453 | Responding to a prompt with a blank line answers that and all | |
35454 | later prompts by popping additional stack entries. | |
35455 | ||
35456 | @c 39 | |
35457 | @item | |
35458 | Answer for @cite{v} may also be of the form @cite{v = v_0} or | |
35459 | @cite{v - v_0}. | |
35460 | ||
35461 | @c 40 | |
35462 | @item | |
35463 | With a positive prefix argument, stack contains many @cite{y}'s and one | |
35464 | common @cite{x}. With a zero prefix, stack contains a vector of | |
35465 | @cite{y}s and a common @cite{x}. With a negative prefix, stack | |
35466 | contains many @cite{[x,y]} vectors. (For 3D plots, substitute | |
35467 | @cite{z} for @cite{y} and @cite{x,y} for @cite{x}.) | |
35468 | ||
35469 | @c 41 | |
35470 | @item | |
35471 | With any prefix argument, all curves in the graph are deleted. | |
35472 | ||
35473 | @c 42 | |
35474 | @item | |
35475 | With a positive prefix, refines an existing plot with more data points. | |
35476 | With a negative prefix, forces recomputation of the plot data. | |
35477 | ||
35478 | @c 43 | |
35479 | @item | |
35480 | With any prefix argument, set the default value instead of the | |
35481 | value for this graph. | |
35482 | ||
35483 | @c 44 | |
35484 | @item | |
35485 | With a negative prefix argument, set the value for the printer. | |
35486 | ||
35487 | @c 45 | |
35488 | @item | |
35489 | Condition is considered ``true'' if it is a nonzero real or complex | |
35490 | number, or a formula whose value is known to be nonzero; it is ``false'' | |
35491 | otherwise. | |
35492 | ||
35493 | @c 46 | |
35494 | @item | |
35495 | Several formulas separated by commas are pushed as multiple stack | |
35496 | entries. Trailing @kbd{)}, @kbd{]}, @kbd{@}}, @kbd{>}, and @kbd{"} | |
35497 | delimiters may be omitted. The notation @kbd{$$$} refers to the value | |
35498 | in stack level three, and causes the formula to replace the top three | |
35499 | stack levels. The notation @kbd{$3} refers to stack level three without | |
35500 | causing that value to be removed from the stack. Use @key{LFD} in place | |
35501 | of @key{RET} to prevent evaluation; use @kbd{M-=} in place of @key{RET} | |
35502 | to evaluate variables.@refill | |
35503 | ||
35504 | @c 47 | |
35505 | @item | |
35506 | The variable is replaced by the formula shown on the right. The | |
35507 | Inverse flag reverses the order of the operands, e.g., @kbd{I s - x} | |
35508 | assigns @c{$x \coloneq a-x$} | |
35509 | @cite{x := a-x}. | |
35510 | ||
35511 | @c 48 | |
35512 | @item | |
35513 | Press @kbd{?} repeatedly to see how to choose a model. Answer the | |
35514 | variables prompt with @cite{iv} or @cite{iv;pv} to specify | |
35515 | independent and parameter variables. A positive prefix argument | |
5d67986c | 35516 | takes @i{@var{n}+1} vectors from the stack; a zero prefix takes a matrix |
d7b8e6c6 EZ |
35517 | and a vector from the stack. |
35518 | ||
35519 | @c 49 | |
35520 | @item | |
35521 | With a plain @kbd{C-u} prefix, replace the current region of the | |
35522 | destination buffer with the yanked text instead of inserting. | |
35523 | ||
35524 | @c 50 | |
35525 | @item | |
35526 | All stack entries are reformatted; the @kbd{H} prefix inhibits this. | |
35527 | The @kbd{I} prefix sets the mode temporarily, redraws the top stack | |
35528 | entry, then restores the original setting of the mode. | |
35529 | ||
35530 | @c 51 | |
35531 | @item | |
35532 | A negative prefix sets the default 3D resolution instead of the | |
35533 | default 2D resolution. | |
35534 | ||
35535 | @c 52 | |
35536 | @item | |
35537 | This grabs a vector of the form [@var{prec}, @var{wsize}, @var{ssize}, | |
35538 | @var{radix}, @var{flfmt}, @var{ang}, @var{frac}, @var{symb}, @var{polar}, | |
35539 | @var{matrix}, @var{simp}, @var{inf}]. A prefix argument from 1 to 12 | |
35540 | grabs the @var{n}th mode value only. | |
35541 | @end enumerate | |
35542 | ||
35543 | @iftex | |
35544 | (Space is provided below for you to keep your own written notes.) | |
35545 | @page | |
35546 | @endgroup | |
35547 | @end iftex | |
35548 | ||
35549 | ||
35550 | @c [end-summary] | |
35551 | ||
35552 | @node Key Index, Command Index, Summary, Top | |
35553 | @unnumbered Index of Key Sequences | |
35554 | ||
35555 | @printindex ky | |
35556 | ||
35557 | @node Command Index, Function Index, Key Index, Top | |
35558 | @unnumbered Index of Calculator Commands | |
35559 | ||
35560 | Since all Calculator commands begin with the prefix @samp{calc-}, the | |
35561 | @kbd{x} key has been provided as a variant of @kbd{M-x} which automatically | |
35562 | types @samp{calc-} for you. Thus, @kbd{x last-args} is short for | |
35563 | @kbd{M-x calc-last-args}. | |
35564 | ||
35565 | @printindex pg | |
35566 | ||
35567 | @node Function Index, Concept Index, Command Index, Top | |
35568 | @unnumbered Index of Algebraic Functions | |
35569 | ||
35570 | This is a list of built-in functions and operators usable in algebraic | |
35571 | expressions. Their full Lisp names are derived by adding the prefix | |
35572 | @samp{calcFunc-}, as in @code{calcFunc-sqrt}. | |
35573 | @iftex | |
35574 | All functions except those noted with ``*'' have corresponding | |
35575 | Calc keystrokes and can also be found in the Calc Summary. | |
35576 | @end iftex | |
35577 | ||
35578 | @printindex tp | |
35579 | ||
35580 | @node Concept Index, Variable Index, Function Index, Top | |
35581 | @unnumbered Concept Index | |
35582 | ||
35583 | @printindex cp | |
35584 | ||
35585 | @node Variable Index, Lisp Function Index, Concept Index, Top | |
35586 | @unnumbered Index of Variables | |
35587 | ||
35588 | The variables in this list that do not contain dashes are accessible | |
35589 | as Calc variables. Add a @samp{var-} prefix to get the name of the | |
35590 | corresponding Lisp variable. | |
35591 | ||
35592 | The remaining variables are Lisp variables suitable for @code{setq}ing | |
35593 | in your @file{.emacs} file. | |
35594 | ||
35595 | @printindex vr | |
35596 | ||
35597 | @node Lisp Function Index, , Variable Index, Top | |
35598 | @unnumbered Index of Lisp Math Functions | |
35599 | ||
35600 | The following functions are meant to be used with @code{defmath}, not | |
35601 | @code{defun} definitions. For names that do not start with @samp{calc-}, | |
35602 | the corresponding full Lisp name is derived by adding a prefix of | |
35603 | @samp{math-}. | |
35604 | ||
35605 | @printindex fn | |
35606 | ||
35607 | @summarycontents | |
35608 | ||
35609 | @c [end] | |
35610 | ||
35611 | @contents | |
35612 | @bye | |
35613 | ||
35614 | ||
6b61353c KH |
35615 | @ignore |
35616 | arch-tag: 77a71809-fa4d-40be-b2cc-da3e8fb137c0 | |
35617 | @end ignore |