1 \input texinfo @c -*-texinfo-*-
2 @comment %**start of header (This is for running Texinfo on a region.)
4 @setfilename ../../info/calc
6 @settitle GNU Emacs Calc Manual
8 @comment %**end of header (This is for running Texinfo on a region.)
10 @include emacsver.texi
12 @c The following macros are used for conditional output for single lines.
14 @c `foo' will appear only in TeX output
16 @c `foo' will appear only in non-TeX output
18 @c @expr{expr} will typeset an expression;
19 @c $x$ in TeX, @samp{x} otherwise.
24 @alias infoline=comment
38 @alias texline=comment
39 @macro infoline{stuff}
58 % Suggested by Karl Berry <karl@@freefriends.org>
59 \gdef\!{\mskip-\thinmuskip}
62 @c Fix some other things specifically for this manual.
65 @mathcode`@:=`@: @c Make Calc fractions come out right in math mode
67 \gdef\coloneq{\mathrel{\mathord:\mathord=}}
69 \gdef\beforedisplay{\vskip-10pt}
70 \gdef\afterdisplay{\vskip-5pt}
71 \gdef\beforedisplayh{\vskip-25pt}
72 \gdef\afterdisplayh{\vskip-10pt}
74 @newdimen@kyvpos @kyvpos=0pt
75 @newdimen@kyhpos @kyhpos=0pt
76 @newcount@calcclubpenalty @calcclubpenalty=1000
79 @newtoks@calcoldeverypar @calcoldeverypar=@everypar
80 @everypar={@calceverypar@the@calcoldeverypar}
81 @ifx@ninett@undefinedzzz@font@ninett=cmtt9@fi
82 @catcode`@\=0 \catcode`\@=11
84 \catcode`\@=0 @catcode`@\=@active
90 This file documents Calc, the GNU Emacs calculator.
93 This file documents Calc, the GNU Emacs calculator, included with
94 GNU Emacs @value{EMACSVER}.
97 Copyright @copyright{} 1990-1991, 2001-2012 Free Software Foundation, Inc.
100 Permission is granted to copy, distribute and/or modify this document
101 under the terms of the GNU Free Documentation License, Version 1.3 or
102 any later version published by the Free Software Foundation; with the
103 Invariant Sections being just ``GNU GENERAL PUBLIC LICENSE'', with the
104 Front-Cover texts being ``A GNU Manual,'' and with the Back-Cover
105 Texts as in (a) below. A copy of the license is included in the section
106 entitled ``GNU Free Documentation License.''
108 (a) The FSF's Back-Cover Text is: ``You have the freedom to copy and
109 modify this GNU manual.''
113 @dircategory Emacs misc features
115 * Calc: (calc). Advanced desk calculator and mathematical tool.
120 @center @titlefont{Calc Manual}
122 @center GNU Emacs Calc
125 @center Dave Gillespie
126 @center daveg@@synaptics.com
129 @vskip 0pt plus 1filll
142 @node Top, Getting Started, (dir), (dir)
143 @chapter The GNU Emacs Calculator
146 @dfn{Calc} is an advanced desk calculator and mathematical tool
147 written by Dave Gillespie that runs as part of the GNU Emacs environment.
149 This manual, also written (mostly) by Dave Gillespie, is divided into
150 three major parts: ``Getting Started,'' the ``Calc Tutorial,'' and the
151 ``Calc Reference.'' The Tutorial introduces all the major aspects of
152 Calculator use in an easy, hands-on way. The remainder of the manual is
153 a complete reference to the features of the Calculator.
157 For help in the Emacs Info system (which you are using to read this
158 file), type @kbd{?}. (You can also type @kbd{h} to run through a
159 longer Info tutorial.)
165 * Getting Started:: General description and overview.
167 * Interactive Tutorial::
169 * Tutorial:: A step-by-step introduction for beginners.
171 * Introduction:: Introduction to the Calc reference manual.
172 * Data Types:: Types of objects manipulated by Calc.
173 * Stack and Trail:: Manipulating the stack and trail buffers.
174 * Mode Settings:: Adjusting display format and other modes.
175 * Arithmetic:: Basic arithmetic functions.
176 * Scientific Functions:: Transcendentals and other scientific functions.
177 * Matrix Functions:: Operations on vectors and matrices.
178 * Algebra:: Manipulating expressions algebraically.
179 * Units:: Operations on numbers with units.
180 * Store and Recall:: Storing and recalling variables.
181 * Graphics:: Commands for making graphs of data.
182 * Kill and Yank:: Moving data into and out of Calc.
183 * Keypad Mode:: Operating Calc from a keypad.
184 * Embedded Mode:: Working with formulas embedded in a file.
185 * Programming:: Calc as a programmable calculator.
187 * Copying:: How you can copy and share Calc.
188 * GNU Free Documentation License:: The license for this documentation.
189 * Customizing Calc:: Customizing Calc.
190 * Reporting Bugs:: How to report bugs and make suggestions.
192 * Summary:: Summary of Calc commands and functions.
194 * Key Index:: The standard Calc key sequences.
195 * Command Index:: The interactive Calc commands.
196 * Function Index:: Functions (in algebraic formulas).
197 * Concept Index:: General concepts.
198 * Variable Index:: Variables used by Calc (both user and internal).
199 * Lisp Function Index:: Internal Lisp math functions.
203 @node Getting Started, Interactive Tutorial, Top, Top
206 @node Getting Started, Tutorial, Top, Top
208 @chapter Getting Started
210 This chapter provides a general overview of Calc, the GNU Emacs
211 Calculator: What it is, how to start it and how to exit from it,
212 and what are the various ways that it can be used.
216 * About This Manual::
217 * Notations Used in This Manual::
218 * Demonstration of Calc::
220 * History and Acknowledgments::
223 @node What is Calc, About This Manual, Getting Started, Getting Started
224 @section What is Calc?
227 @dfn{Calc} is an advanced calculator and mathematical tool that runs as
228 part of the GNU Emacs environment. Very roughly based on the HP-28/48
229 series of calculators, its many features include:
233 Choice of algebraic or RPN (stack-based) entry of calculations.
236 Arbitrary precision integers and floating-point numbers.
239 Arithmetic on rational numbers, complex numbers (rectangular and polar),
240 error forms with standard deviations, open and closed intervals, vectors
241 and matrices, dates and times, infinities, sets, quantities with units,
242 and algebraic formulas.
245 Mathematical operations such as logarithms and trigonometric functions.
248 Programmer's features (bitwise operations, non-decimal numbers).
251 Financial functions such as future value and internal rate of return.
254 Number theoretical features such as prime factorization and arithmetic
255 modulo @var{m} for any @var{m}.
258 Algebraic manipulation features, including symbolic calculus.
261 Moving data to and from regular editing buffers.
264 Embedded mode for manipulating Calc formulas and data directly
265 inside any editing buffer.
268 Graphics using GNUPLOT, a versatile (and free) plotting program.
271 Easy programming using keyboard macros, algebraic formulas,
272 algebraic rewrite rules, or extended Emacs Lisp.
275 Calc tries to include a little something for everyone; as a result it is
276 large and might be intimidating to the first-time user. If you plan to
277 use Calc only as a traditional desk calculator, all you really need to
278 read is the ``Getting Started'' chapter of this manual and possibly the
279 first few sections of the tutorial. As you become more comfortable with
280 the program you can learn its additional features. Calc does not
281 have the scope and depth of a fully-functional symbolic math package,
282 but Calc has the advantages of convenience, portability, and freedom.
284 @node About This Manual, Notations Used in This Manual, What is Calc, Getting Started
285 @section About This Manual
288 This document serves as a complete description of the GNU Emacs
289 Calculator. It works both as an introduction for novices and as
290 a reference for experienced users. While it helps to have some
291 experience with GNU Emacs in order to get the most out of Calc,
292 this manual ought to be readable even if you don't know or use Emacs
295 This manual is divided into three major parts: the ``Getting
296 Started'' chapter you are reading now, the Calc tutorial, and the Calc
299 @c This manual has been printed in two volumes, the @dfn{Tutorial} and the
300 @c @dfn{Reference}. Both volumes include a copy of the ``Getting Started''
303 If you are in a hurry to use Calc, there is a brief ``demonstration''
304 below which illustrates the major features of Calc in just a couple of
305 pages. If you don't have time to go through the full tutorial, this
306 will show you everything you need to know to begin.
307 @xref{Demonstration of Calc}.
309 The tutorial chapter walks you through the various parts of Calc
310 with lots of hands-on examples and explanations. If you are new
311 to Calc and you have some time, try going through at least the
312 beginning of the tutorial. The tutorial includes about 70 exercises
313 with answers. These exercises give you some guided practice with
314 Calc, as well as pointing out some interesting and unusual ways
317 The reference section discusses Calc in complete depth. You can read
318 the reference from start to finish if you want to learn every aspect
319 of Calc. Or, you can look in the table of contents or the Concept
320 Index to find the parts of the manual that discuss the things you
323 @c @cindex Marginal notes
324 Every Calc keyboard command is listed in the Calc Summary, and also
325 in the Key Index. Algebraic functions, @kbd{M-x} commands, and
326 variables also have their own indices.
328 @c @infoline In the printed manual, each
329 @c paragraph that is referenced in the Key or Function Index is marked
330 @c in the margin with its index entry.
332 @c [fix-ref Help Commands]
333 You can access this manual on-line at any time within Calc by pressing
334 the @kbd{h i} key sequence. Outside of the Calc window, you can press
335 @kbd{C-x * i} to read the manual on-line. From within Calc the command
336 @kbd{h t} will jump directly to the Tutorial; from outside of Calc the
337 command @kbd{C-x * t} will jump to the Tutorial and start Calc if
338 necessary. Pressing @kbd{h s} or @kbd{C-x * s} will take you directly
339 to the Calc Summary. Within Calc, you can also go to the part of the
340 manual describing any Calc key, function, or variable using
341 @w{@kbd{h k}}, @kbd{h f}, or @kbd{h v}, respectively. @xref{Help Commands}.
344 The Calc manual can be printed, but because the manual is so large, you
345 should only make a printed copy if you really need it. To print the
346 manual, you will need the @TeX{} typesetting program (this is a free
347 program by Donald Knuth at Stanford University) as well as the
348 @file{texindex} program and @file{texinfo.tex} file, both of which can
349 be obtained from the FSF as part of the @code{texinfo} package.
350 To print the Calc manual in one huge tome, you will need the
351 source code to this manual, @file{calc.texi}, available as part of the
352 Emacs source. Once you have this file, type @kbd{texi2dvi calc.texi}.
353 Alternatively, change to the @file{man} subdirectory of the Emacs
354 source distribution, and type @kbd{make calc.dvi}. (Don't worry if you
355 get some ``overfull box'' warnings while @TeX{} runs.)
356 The result will be a device-independent output file called
357 @file{calc.dvi}, which you must print in whatever way is right
358 for your system. On many systems, the command is
371 @c Printed copies of this manual are also available from the Free Software
374 @node Notations Used in This Manual, Demonstration of Calc, About This Manual, Getting Started
375 @section Notations Used in This Manual
378 This section describes the various notations that are used
379 throughout the Calc manual.
381 In keystroke sequences, uppercase letters mean you must hold down
382 the shift key while typing the letter. Keys pressed with Control
383 held down are shown as @kbd{C-x}. Keys pressed with Meta held down
384 are shown as @kbd{M-x}. Other notations are @key{RET} for the
385 Return key, @key{SPC} for the space bar, @key{TAB} for the Tab key,
386 @key{DEL} for the Delete key, and @key{LFD} for the Line-Feed key.
387 The @key{DEL} key is called Backspace on some keyboards, it is
388 whatever key you would use to correct a simple typing error when
389 regularly using Emacs.
391 (If you don't have the @key{LFD} or @key{TAB} keys on your keyboard,
392 the @kbd{C-j} and @kbd{C-i} keys are equivalent to them, respectively.
393 If you don't have a Meta key, look for Alt or Extend Char. You can
394 also press @key{ESC} or @kbd{C-[} first to get the same effect, so
395 that @kbd{M-x}, @kbd{@key{ESC} x}, and @kbd{C-[ x} are all equivalent.)
397 Sometimes the @key{RET} key is not shown when it is ``obvious''
398 that you must press @key{RET} to proceed. For example, the @key{RET}
399 is usually omitted in key sequences like @kbd{M-x calc-keypad @key{RET}}.
401 Commands are generally shown like this: @kbd{p} (@code{calc-precision})
402 or @kbd{C-x * k} (@code{calc-keypad}). This means that the command is
403 normally used by pressing the @kbd{p} key or @kbd{C-x * k} key sequence,
404 but it also has the full-name equivalent shown, e.g., @kbd{M-x calc-precision}.
406 Commands that correspond to functions in algebraic notation
407 are written: @kbd{C} (@code{calc-cos}) [@code{cos}]. This means
408 the @kbd{C} key is equivalent to @kbd{M-x calc-cos}, and that
409 the corresponding function in an algebraic-style formula would
410 be @samp{cos(@var{x})}.
412 A few commands don't have key equivalents: @code{calc-sincos}
415 @node Demonstration of Calc, Using Calc, Notations Used in This Manual, Getting Started
416 @section A Demonstration of Calc
419 @cindex Demonstration of Calc
420 This section will show some typical small problems being solved with
421 Calc. The focus is more on demonstration than explanation, but
422 everything you see here will be covered more thoroughly in the
425 To begin, start Emacs if necessary (usually the command @code{emacs}
426 does this), and type @kbd{C-x * c} to start the
427 Calculator. (You can also use @kbd{M-x calc} if this doesn't work.
428 @xref{Starting Calc}, for various ways of starting the Calculator.)
430 Be sure to type all the sample input exactly, especially noting the
431 difference between lower-case and upper-case letters. Remember,
432 @key{RET}, @key{TAB}, @key{DEL}, and @key{SPC} are the Return, Tab,
433 Delete, and Space keys.
435 @strong{RPN calculation.} In RPN, you type the input number(s) first,
436 then the command to operate on the numbers.
439 Type @kbd{2 @key{RET} 3 + Q} to compute
440 @texline @math{\sqrt{2+3} = 2.2360679775}.
441 @infoline the square root of 2+3, which is 2.2360679775.
444 Type @kbd{P 2 ^} to compute
445 @texline @math{\pi^2 = 9.86960440109}.
446 @infoline the value of `pi' squared, 9.86960440109.
449 Type @key{TAB} to exchange the order of these two results.
452 Type @kbd{- I H S} to subtract these results and compute the Inverse
453 Hyperbolic sine of the difference, 2.72996136574.
456 Type @key{DEL} to erase this result.
458 @strong{Algebraic calculation.} You can also enter calculations using
459 conventional ``algebraic'' notation. To enter an algebraic formula,
460 use the apostrophe key.
463 Type @kbd{' sqrt(2+3) @key{RET}} to compute
464 @texline @math{\sqrt{2+3}}.
465 @infoline the square root of 2+3.
468 Type @kbd{' pi^2 @key{RET}} to enter
469 @texline @math{\pi^2}.
470 @infoline `pi' squared.
471 To evaluate this symbolic formula as a number, type @kbd{=}.
474 Type @kbd{' arcsinh($ - $$) @key{RET}} to subtract the second-most-recent
475 result from the most-recent and compute the Inverse Hyperbolic sine.
477 @strong{Keypad mode.} If you are using the X window system, press
478 @w{@kbd{C-x * k}} to get Keypad mode. (If you don't use X, skip to
482 Click on the @key{2}, @key{ENTER}, @key{3}, @key{+}, and @key{SQRT}
483 ``buttons'' using your left mouse button.
486 Click on @key{PI}, @key{2}, and @tfn{y^x}.
489 Click on @key{INV}, then @key{ENTER} to swap the two results.
492 Click on @key{-}, @key{INV}, @key{HYP}, and @key{SIN}.
495 Click on @key{<-} to erase the result, then click @key{OFF} to turn
496 the Keypad Calculator off.
498 @strong{Grabbing data.} Type @kbd{C-x * x} if necessary to exit Calc.
499 Now select the following numbers as an Emacs region: ``Mark'' the
500 front of the list by typing @kbd{C-@key{SPC}} or @kbd{C-@@} there,
501 then move to the other end of the list. (Either get this list from
502 the on-line copy of this manual, accessed by @w{@kbd{C-x * i}}, or just
503 type these numbers into a scratch file.) Now type @kbd{C-x * g} to
504 ``grab'' these numbers into Calc.
515 The result @samp{[1.23, 1.97, 1.6, 2, 1.19, 1.08]} is a Calc ``vector.''
516 Type @w{@kbd{V R +}} to compute the sum of these numbers.
519 Type @kbd{U} to Undo this command, then type @kbd{V R *} to compute
520 the product of the numbers.
523 You can also grab data as a rectangular matrix. Place the cursor on
524 the upper-leftmost @samp{1} and set the mark, then move to just after
525 the lower-right @samp{8} and press @kbd{C-x * r}.
528 Type @kbd{v t} to transpose this
529 @texline @math{3\times2}
532 @texline @math{2\times3}
534 matrix. Type @w{@kbd{v u}} to unpack the rows into two separate
535 vectors. Now type @w{@kbd{V R + @key{TAB} V R +}} to compute the sums
536 of the two original columns. (There is also a special
537 grab-and-sum-columns command, @kbd{C-x * :}.)
539 @strong{Units conversion.} Units are entered algebraically.
540 Type @w{@kbd{' 43 mi/hr @key{RET}}} to enter the quantity 43 miles-per-hour.
541 Type @w{@kbd{u c km/hr @key{RET}}}. Type @w{@kbd{u c m/s @key{RET}}}.
543 @strong{Date arithmetic.} Type @kbd{t N} to get the current date and
544 time. Type @kbd{90 +} to find the date 90 days from now. Type
545 @kbd{' <25 dec 87> @key{RET}} to enter a date, then @kbd{- 7 /} to see how
546 many weeks have passed since then.
548 @strong{Algebra.} Algebraic entries can also include formulas
549 or equations involving variables. Type @kbd{@w{' [x + y} = a, x y = 1] @key{RET}}
550 to enter a pair of equations involving three variables.
551 (Note the leading apostrophe in this example; also, note that the space
552 in @samp{x y} is required.) Type @w{@kbd{a S x,y @key{RET}}} to solve
553 these equations for the variables @expr{x} and @expr{y}.
556 Type @kbd{d B} to view the solutions in more readable notation.
557 Type @w{@kbd{d C}} to view them in C language notation, @kbd{d T}
558 to view them in the notation for the @TeX{} typesetting system,
559 and @kbd{d L} to view them in the notation for the @LaTeX{} typesetting
560 system. Type @kbd{d N} to return to normal notation.
563 Type @kbd{7.5}, then @kbd{s l a @key{RET}} to let @expr{a = 7.5} in these formulas.
564 (That's the letter @kbd{l}, not the numeral @kbd{1}.)
567 @strong{Help functions.} You can read about any command in the on-line
568 manual. Type @kbd{C-x * c} to return to Calc after each of these
569 commands: @kbd{h k t N} to read about the @kbd{t N} command,
570 @kbd{h f sqrt @key{RET}} to read about the @code{sqrt} function, and
571 @kbd{h s} to read the Calc summary.
574 @strong{Help functions.} You can read about any command in the on-line
575 manual. Remember to type the letter @kbd{l}, then @kbd{C-x * c}, to
576 return here after each of these commands: @w{@kbd{h k t N}} to read
577 about the @w{@kbd{t N}} command, @kbd{h f sqrt @key{RET}} to read about the
578 @code{sqrt} function, and @kbd{h s} to read the Calc summary.
581 Press @key{DEL} repeatedly to remove any leftover results from the stack.
582 To exit from Calc, press @kbd{q} or @kbd{C-x * c} again.
584 @node Using Calc, History and Acknowledgments, Demonstration of Calc, Getting Started
588 Calc has several user interfaces that are specialized for
589 different kinds of tasks. As well as Calc's standard interface,
590 there are Quick mode, Keypad mode, and Embedded mode.
594 * The Standard Interface::
595 * Quick Mode Overview::
596 * Keypad Mode Overview::
597 * Standalone Operation::
598 * Embedded Mode Overview::
599 * Other C-x * Commands::
602 @node Starting Calc, The Standard Interface, Using Calc, Using Calc
603 @subsection Starting Calc
606 On most systems, you can type @kbd{C-x *} to start the Calculator.
607 The key sequence @kbd{C-x *} is bound to the command @code{calc-dispatch},
608 which can be rebound if convenient (@pxref{Customizing Calc}).
610 When you press @kbd{C-x *}, Emacs waits for you to press a second key to
611 complete the command. In this case, you will follow @kbd{C-x *} with a
612 letter (upper- or lower-case, it doesn't matter for @kbd{C-x *}) that says
613 which Calc interface you want to use.
615 To get Calc's standard interface, type @kbd{C-x * c}. To get
616 Keypad mode, type @kbd{C-x * k}. Type @kbd{C-x * ?} to get a brief
617 list of the available options, and type a second @kbd{?} to get
620 To ease typing, @kbd{C-x * *} also works to start Calc. It starts the
621 same interface (either @kbd{C-x * c} or @w{@kbd{C-x * k}}) that you last
622 used, selecting the @kbd{C-x * c} interface by default.
624 If @kbd{C-x *} doesn't work for you, you can always type explicit
625 commands like @kbd{M-x calc} (for the standard user interface) or
626 @w{@kbd{M-x calc-keypad}} (for Keypad mode). First type @kbd{M-x}
627 (that's Meta with the letter @kbd{x}), then, at the prompt,
628 type the full command (like @kbd{calc-keypad}) and press Return.
630 The same commands (like @kbd{C-x * c} or @kbd{C-x * *}) that start
631 the Calculator also turn it off if it is already on.
633 @node The Standard Interface, Quick Mode Overview, Starting Calc, Using Calc
634 @subsection The Standard Calc Interface
637 @cindex Standard user interface
638 Calc's standard interface acts like a traditional RPN calculator,
639 operated by the normal Emacs keyboard. When you type @kbd{C-x * c}
640 to start the Calculator, the Emacs screen splits into two windows
641 with the file you were editing on top and Calc on the bottom.
647 --**-Emacs: myfile (Fundamental)----All----------------------
648 --- Emacs Calculator Mode --- |Emacs Calculator Trail
656 --%*-Calc: 12 Deg (Calculator)----All----- --%*- *Calc Trail*
660 In this figure, the mode-line for @file{myfile} has moved up and the
661 ``Calculator'' window has appeared below it. As you can see, Calc
662 actually makes two windows side-by-side. The lefthand one is
663 called the @dfn{stack window} and the righthand one is called the
664 @dfn{trail window.} The stack holds the numbers involved in the
665 calculation you are currently performing. The trail holds a complete
666 record of all calculations you have done. In a desk calculator with
667 a printer, the trail corresponds to the paper tape that records what
670 In this case, the trail shows that four numbers (17.3, 3, 2, and 4)
671 were first entered into the Calculator, then the 2 and 4 were
672 multiplied to get 8, then the 3 and 8 were subtracted to get @mathit{-5}.
673 (The @samp{>} symbol shows that this was the most recent calculation.)
674 The net result is the two numbers 17.3 and @mathit{-5} sitting on the stack.
676 Most Calculator commands deal explicitly with the stack only, but
677 there is a set of commands that allow you to search back through
678 the trail and retrieve any previous result.
680 Calc commands use the digits, letters, and punctuation keys.
681 Shifted (i.e., upper-case) letters are different from lowercase
682 letters. Some letters are @dfn{prefix} keys that begin two-letter
683 commands. For example, @kbd{e} means ``enter exponent'' and shifted
684 @kbd{E} means @expr{e^x}. With the @kbd{d} (``display modes'') prefix
685 the letter ``e'' takes on very different meanings: @kbd{d e} means
686 ``engineering notation'' and @kbd{d E} means ``@dfn{eqn} language mode.''
688 There is nothing stopping you from switching out of the Calc
689 window and back into your editing window, say by using the Emacs
690 @w{@kbd{C-x o}} (@code{other-window}) command. When the cursor is
691 inside a regular window, Emacs acts just like normal. When the
692 cursor is in the Calc stack or trail windows, keys are interpreted
695 When you quit by pressing @kbd{C-x * c} a second time, the Calculator
696 windows go away but the actual Stack and Trail are not gone, just
697 hidden. When you press @kbd{C-x * c} once again you will get the
698 same stack and trail contents you had when you last used the
701 The Calculator does not remember its state between Emacs sessions.
702 Thus if you quit Emacs and start it again, @kbd{C-x * c} will give you
703 a fresh stack and trail. There is a command (@kbd{m m}) that lets
704 you save your favorite mode settings between sessions, though.
705 One of the things it saves is which user interface (standard or
706 Keypad) you last used; otherwise, a freshly started Emacs will
707 always treat @kbd{C-x * *} the same as @kbd{C-x * c}.
709 The @kbd{q} key is another equivalent way to turn the Calculator off.
711 If you type @kbd{C-x * b} first and then @kbd{C-x * c}, you get a
712 full-screen version of Calc (@code{full-calc}) in which the stack and
713 trail windows are still side-by-side but are now as tall as the whole
714 Emacs screen. When you press @kbd{q} or @kbd{C-x * c} again to quit,
715 the file you were editing before reappears. The @kbd{C-x * b} key
716 switches back and forth between ``big'' full-screen mode and the
717 normal partial-screen mode.
719 Finally, @kbd{C-x * o} (@code{calc-other-window}) is like @kbd{C-x * c}
720 except that the Calc window is not selected. The buffer you were
721 editing before remains selected instead. If you are in a Calc window,
722 then @kbd{C-x * o} will switch you out of it, being careful not to
723 switch you to the Calc Trail window. So @kbd{C-x * o} is a handy
724 way to switch out of Calc momentarily to edit your file; you can then
725 type @kbd{C-x * c} to switch back into Calc when you are done.
727 @node Quick Mode Overview, Keypad Mode Overview, The Standard Interface, Using Calc
728 @subsection Quick Mode (Overview)
731 @dfn{Quick mode} is a quick way to use Calc when you don't need the
732 full complexity of the stack and trail. To use it, type @kbd{C-x * q}
733 (@code{quick-calc}) in any regular editing buffer.
735 Quick mode is very simple: It prompts you to type any formula in
736 standard algebraic notation (like @samp{4 - 2/3}) and then displays
737 the result at the bottom of the Emacs screen (@mathit{3.33333333333}
738 in this case). You are then back in the same editing buffer you
739 were in before, ready to continue editing or to type @kbd{C-x * q}
740 again to do another quick calculation. The result of the calculation
741 will also be in the Emacs ``kill ring'' so that a @kbd{C-y} command
742 at this point will yank the result into your editing buffer.
744 Calc mode settings affect Quick mode, too, though you will have to
745 go into regular Calc (with @kbd{C-x * c}) to change the mode settings.
747 @c [fix-ref Quick Calculator mode]
748 @xref{Quick Calculator}, for further information.
750 @node Keypad Mode Overview, Standalone Operation, Quick Mode Overview, Using Calc
751 @subsection Keypad Mode (Overview)
754 @dfn{Keypad mode} is a mouse-based interface to the Calculator.
755 It is designed for use with terminals that support a mouse. If you
756 don't have a mouse, you will have to operate Keypad mode with your
757 arrow keys (which is probably more trouble than it's worth).
759 Type @kbd{C-x * k} to turn Keypad mode on or off. Once again you
760 get two new windows, this time on the righthand side of the screen
761 instead of at the bottom. The upper window is the familiar Calc
762 Stack; the lower window is a picture of a typical calculator keypad.
766 \advance \dimen0 by 24\baselineskip%
767 \ifdim \dimen0>\pagegoal \vfill\eject \fi%
772 |--- Emacs Calculator Mode ---
776 |--%*-Calc: 12 Deg (Calcul
777 |----+----+--Calc---+----+----1
778 |FLR |CEIL|RND |TRNC|CLN2|FLT |
779 |----+----+----+----+----+----|
780 | LN |EXP | |ABS |IDIV|MOD |
781 |----+----+----+----+----+----|
782 |SIN |COS |TAN |SQRT|y^x |1/x |
783 |----+----+----+----+----+----|
784 | ENTER |+/- |EEX |UNDO| <- |
785 |-----+---+-+--+--+-+---++----|
786 | INV | 7 | 8 | 9 | / |
787 |-----+-----+-----+-----+-----|
788 | HYP | 4 | 5 | 6 | * |
789 |-----+-----+-----+-----+-----|
790 |EXEC | 1 | 2 | 3 | - |
791 |-----+-----+-----+-----+-----|
792 | OFF | 0 | . | PI | + |
793 |-----+-----+-----+-----+-----+
797 Keypad mode is much easier for beginners to learn, because there
798 is no need to memorize lots of obscure key sequences. But not all
799 commands in regular Calc are available on the Keypad. You can
800 always switch the cursor into the Calc stack window to use
801 standard Calc commands if you need. Serious Calc users, though,
802 often find they prefer the standard interface over Keypad mode.
804 To operate the Calculator, just click on the ``buttons'' of the
805 keypad using your left mouse button. To enter the two numbers
806 shown here you would click @w{@kbd{1 7 .@: 3 ENTER 5 +/- ENTER}}; to
807 add them together you would then click @kbd{+} (to get 12.3 on
810 If you click the right mouse button, the top three rows of the
811 keypad change to show other sets of commands, such as advanced
812 math functions, vector operations, and operations on binary
815 Because Keypad mode doesn't use the regular keyboard, Calc leaves
816 the cursor in your original editing buffer. You can type in
817 this buffer in the usual way while also clicking on the Calculator
818 keypad. One advantage of Keypad mode is that you don't need an
819 explicit command to switch between editing and calculating.
821 If you press @kbd{C-x * b} first, you get a full-screen Keypad mode
822 (@code{full-calc-keypad}) with three windows: The keypad in the lower
823 left, the stack in the lower right, and the trail on top.
825 @c [fix-ref Keypad Mode]
826 @xref{Keypad Mode}, for further information.
828 @node Standalone Operation, Embedded Mode Overview, Keypad Mode Overview, Using Calc
829 @subsection Standalone Operation
832 @cindex Standalone Operation
833 If you are not in Emacs at the moment but you wish to use Calc,
834 you must start Emacs first. If all you want is to run Calc, you
835 can give the commands:
845 emacs -f full-calc-keypad
849 which run a full-screen Calculator (as if by @kbd{C-x * b C-x * c}) or
850 a full-screen X-based Calculator (as if by @kbd{C-x * b C-x * k}).
851 In standalone operation, quitting the Calculator (by pressing
852 @kbd{q} or clicking on the keypad @key{EXIT} button) quits Emacs
855 @node Embedded Mode Overview, Other C-x * Commands, Standalone Operation, Using Calc
856 @subsection Embedded Mode (Overview)
859 @dfn{Embedded mode} is a way to use Calc directly from inside an
860 editing buffer. Suppose you have a formula written as part of a
874 and you wish to have Calc compute and format the derivative for
875 you and store this derivative in the buffer automatically. To
876 do this with Embedded mode, first copy the formula down to where
877 you want the result to be, leaving a blank line before and after the
892 Now, move the cursor onto this new formula and press @kbd{C-x * e}.
893 Calc will read the formula (using the surrounding blank lines to tell
894 how much text to read), then push this formula (invisibly) onto the Calc
895 stack. The cursor will stay on the formula in the editing buffer, but
896 the line with the formula will now appear as it would on the Calc stack
897 (in this case, it will be left-aligned) and the buffer's mode line will
898 change to look like the Calc mode line (with mode indicators like
899 @samp{12 Deg} and so on). Even though you are still in your editing
900 buffer, the keyboard now acts like the Calc keyboard, and any new result
901 you get is copied from the stack back into the buffer. To take the
902 derivative, you would type @kbd{a d x @key{RET}}.
916 (Note that by default, Calc gives division lower precedence than multiplication,
917 so that @samp{1 / x ln(x)} is equivalent to @samp{1 / (x ln(x))}.)
919 To make this look nicer, you might want to press @kbd{d =} to center
920 the formula, and even @kbd{d B} to use Big display mode.
929 % [calc-mode: justify: center]
930 % [calc-mode: language: big]
938 Calc has added annotations to the file to help it remember the modes
939 that were used for this formula. They are formatted like comments
940 in the @TeX{} typesetting language, just in case you are using @TeX{} or
941 @LaTeX{}. (In this example @TeX{} is not being used, so you might want
942 to move these comments up to the top of the file or otherwise put them
945 As an extra flourish, we can add an equation number using a
946 righthand label: Type @kbd{d @} (1) @key{RET}}.
950 % [calc-mode: justify: center]
951 % [calc-mode: language: big]
952 % [calc-mode: right-label: " (1)"]
960 To leave Embedded mode, type @kbd{C-x * e} again. The mode line
961 and keyboard will revert to the way they were before.
963 The related command @kbd{C-x * w} operates on a single word, which
964 generally means a single number, inside text. It searches for an
965 expression which ``looks'' like a number containing the point.
966 Here's an example of its use (before you try this, remove the Calc
967 annotations or use a new buffer so that the extra settings in the
968 annotations don't take effect):
971 A slope of one-third corresponds to an angle of 1 degrees.
974 Place the cursor on the @samp{1}, then type @kbd{C-x * w} to enable
975 Embedded mode on that number. Now type @kbd{3 /} (to get one-third),
976 and @kbd{I T} (the Inverse Tangent converts a slope into an angle),
977 then @w{@kbd{C-x * w}} again to exit Embedded mode.
980 A slope of one-third corresponds to an angle of 18.4349488229 degrees.
983 @c [fix-ref Embedded Mode]
984 @xref{Embedded Mode}, for full details.
986 @node Other C-x * Commands, , Embedded Mode Overview, Using Calc
987 @subsection Other @kbd{C-x *} Commands
990 Two more Calc-related commands are @kbd{C-x * g} and @kbd{C-x * r},
991 which ``grab'' data from a selected region of a buffer into the
992 Calculator. The region is defined in the usual Emacs way, by
993 a ``mark'' placed at one end of the region, and the Emacs
994 cursor or ``point'' placed at the other.
996 The @kbd{C-x * g} command reads the region in the usual left-to-right,
997 top-to-bottom order. The result is packaged into a Calc vector
998 of numbers and placed on the stack. Calc (in its standard
999 user interface) is then started. Type @kbd{v u} if you want
1000 to unpack this vector into separate numbers on the stack. Also,
1001 @kbd{C-u C-x * g} interprets the region as a single number or
1004 The @kbd{C-x * r} command reads a rectangle, with the point and
1005 mark defining opposite corners of the rectangle. The result
1006 is a matrix of numbers on the Calculator stack.
1008 Complementary to these is @kbd{C-x * y}, which ``yanks'' the
1009 value at the top of the Calc stack back into an editing buffer.
1010 If you type @w{@kbd{C-x * y}} while in such a buffer, the value is
1011 yanked at the current position. If you type @kbd{C-x * y} while
1012 in the Calc buffer, Calc makes an educated guess as to which
1013 editing buffer you want to use. The Calc window does not have
1014 to be visible in order to use this command, as long as there
1015 is something on the Calc stack.
1017 Here, for reference, is the complete list of @kbd{C-x *} commands.
1018 The shift, control, and meta keys are ignored for the keystroke
1019 following @kbd{C-x *}.
1022 Commands for turning Calc on and off:
1026 Turn Calc on or off, employing the same user interface as last time.
1028 @item =, +, -, /, \, &, #
1029 Alternatives for @kbd{*}.
1032 Turn Calc on or off using its standard bottom-of-the-screen
1033 interface. If Calc is already turned on but the cursor is not
1034 in the Calc window, move the cursor into the window.
1037 Same as @kbd{C}, but don't select the new Calc window. If
1038 Calc is already turned on and the cursor is in the Calc window,
1039 move it out of that window.
1042 Control whether @kbd{C-x * c} and @kbd{C-x * k} use the full screen.
1045 Use Quick mode for a single short calculation.
1048 Turn Calc Keypad mode on or off.
1051 Turn Calc Embedded mode on or off at the current formula.
1054 Turn Calc Embedded mode on or off, select the interesting part.
1057 Turn Calc Embedded mode on or off at the current word (number).
1060 Turn Calc on in a user-defined way, as defined by a @kbd{Z I} command.
1063 Quit Calc; turn off standard, Keypad, or Embedded mode if on.
1064 (This is like @kbd{q} or @key{OFF} inside of Calc.)
1071 Commands for moving data into and out of the Calculator:
1075 Grab the region into the Calculator as a vector.
1078 Grab the rectangular region into the Calculator as a matrix.
1081 Grab the rectangular region and compute the sums of its columns.
1084 Grab the rectangular region and compute the sums of its rows.
1087 Yank a value from the Calculator into the current editing buffer.
1094 Commands for use with Embedded mode:
1098 ``Activate'' the current buffer. Locate all formulas that
1099 contain @samp{:=} or @samp{=>} symbols and record their locations
1100 so that they can be updated automatically as variables are changed.
1103 Duplicate the current formula immediately below and select
1107 Insert a new formula at the current point.
1110 Move the cursor to the next active formula in the buffer.
1113 Move the cursor to the previous active formula in the buffer.
1116 Update (i.e., as if by the @kbd{=} key) the formula at the current point.
1119 Edit (as if by @code{calc-edit}) the formula at the current point.
1126 Miscellaneous commands:
1130 Run the Emacs Info system to read the Calc manual.
1131 (This is the same as @kbd{h i} inside of Calc.)
1134 Run the Emacs Info system to read the Calc Tutorial.
1137 Run the Emacs Info system to read the Calc Summary.
1140 Load Calc entirely into memory. (Normally the various parts
1141 are loaded only as they are needed.)
1144 Read a region of written keystroke names (like @kbd{C-n a b c @key{RET}})
1145 and record them as the current keyboard macro.
1148 (This is the ``zero'' digit key.) Reset the Calculator to
1149 its initial state: Empty stack, and initial mode settings.
1152 @node History and Acknowledgments, , Using Calc, Getting Started
1153 @section History and Acknowledgments
1156 Calc was originally started as a two-week project to occupy a lull
1157 in the author's schedule. Basically, a friend asked if I remembered
1159 @texline @math{2^{32}}.
1160 @infoline @expr{2^32}.
1161 I didn't offhand, but I said, ``that's easy, just call up an
1162 @code{xcalc}.'' @code{Xcalc} duly reported that the answer to our
1163 question was @samp{4.294967e+09}---with no way to see the full ten
1164 digits even though we knew they were there in the program's memory! I
1165 was so annoyed, I vowed to write a calculator of my own, once and for
1168 I chose Emacs Lisp, a) because I had always been curious about it
1169 and b) because, being only a text editor extension language after
1170 all, Emacs Lisp would surely reach its limits long before the project
1171 got too far out of hand.
1173 To make a long story short, Emacs Lisp turned out to be a distressingly
1174 solid implementation of Lisp, and the humble task of calculating
1175 turned out to be more open-ended than one might have expected.
1177 Emacs Lisp didn't have built-in floating point math (now it does), so
1178 this had to be simulated in software. In fact, Emacs integers would
1179 only comfortably fit six decimal digits or so (at the time)---not
1180 enough for a decent calculator. So I had to write my own
1181 high-precision integer code as well, and once I had this I figured
1182 that arbitrary-size integers were just as easy as large integers.
1183 Arbitrary floating-point precision was the logical next step. Also,
1184 since the large integer arithmetic was there anyway it seemed only
1185 fair to give the user direct access to it, which in turn made it
1186 practical to support fractions as well as floats. All these features
1187 inspired me to look around for other data types that might be worth
1190 Around this time, my friend Rick Koshi showed me his nifty new HP-28
1191 calculator. It allowed the user to manipulate formulas as well as
1192 numerical quantities, and it could also operate on matrices. I
1193 decided that these would be good for Calc to have, too. And once
1194 things had gone this far, I figured I might as well take a look at
1195 serious algebra systems for further ideas. Since these systems did
1196 far more than I could ever hope to implement, I decided to focus on
1197 rewrite rules and other programming features so that users could
1198 implement what they needed for themselves.
1200 Rick complained that matrices were hard to read, so I put in code to
1201 format them in a 2D style. Once these routines were in place, Big mode
1202 was obligatory. Gee, what other language modes would be useful?
1204 Scott Hemphill and Allen Knutson, two friends with a strong mathematical
1205 bent, contributed ideas and algorithms for a number of Calc features
1206 including modulo forms, primality testing, and float-to-fraction conversion.
1208 Units were added at the eager insistence of Mass Sivilotti. Later,
1209 Ulrich Mueller at CERN and Przemek Klosowski at NIST provided invaluable
1210 expert assistance with the units table. As far as I can remember, the
1211 idea of using algebraic formulas and variables to represent units dates
1212 back to an ancient article in Byte magazine about muMath, an early
1213 algebra system for microcomputers.
1215 Many people have contributed to Calc by reporting bugs and suggesting
1216 features, large and small. A few deserve special mention: Tim Peters,
1217 who helped develop the ideas that led to the selection commands, rewrite
1218 rules, and many other algebra features;
1219 @texline Fran\c{c}ois
1221 Pinard, who contributed an early prototype of the Calc Summary appendix
1222 as well as providing valuable suggestions in many other areas of Calc;
1223 Carl Witty, whose eagle eyes discovered many typographical and factual
1224 errors in the Calc manual; Tim Kay, who drove the development of
1225 Embedded mode; Ove Ewerlid, who made many suggestions relating to the
1226 algebra commands and contributed some code for polynomial operations;
1227 Randal Schwartz, who suggested the @code{calc-eval} function; Juha
1228 Sarlin, who first worked out how to split Calc into quickly-loading
1229 parts; Bob Weiner, who helped immensely with the Lucid Emacs port; and
1230 Robert J. Chassell, who suggested the Calc Tutorial and exercises as
1231 well as many other things.
1233 @cindex Bibliography
1234 @cindex Knuth, Art of Computer Programming
1235 @cindex Numerical Recipes
1236 @c Should these be expanded into more complete references?
1237 Among the books used in the development of Calc were Knuth's @emph{Art
1238 of Computer Programming} (especially volume II, @emph{Seminumerical
1239 Algorithms}); @emph{Numerical Recipes} by Press, Flannery, Teukolsky,
1240 and Vetterling; Bevington's @emph{Data Reduction and Error Analysis
1241 for the Physical Sciences}; @emph{Concrete Mathematics} by Graham,
1242 Knuth, and Patashnik; Steele's @emph{Common Lisp, the Language}; the
1243 @emph{CRC Standard Math Tables} (William H. Beyer, ed.); and
1244 Abramowitz and Stegun's venerable @emph{Handbook of Mathematical
1245 Functions}. Also, of course, Calc could not have been written without
1246 the excellent @emph{GNU Emacs Lisp Reference Manual}, by Bil Lewis and
1249 Final thanks go to Richard Stallman, without whose fine implementations
1250 of the Emacs editor, language, and environment, Calc would have been
1251 finished in two weeks.
1256 @c This node is accessed by the `C-x * t' command.
1257 @node Interactive Tutorial, Tutorial, Getting Started, Top
1261 Some brief instructions on using the Emacs Info system for this tutorial:
1263 Press the space bar and Delete keys to go forward and backward in a
1264 section by screenfuls (or use the regular Emacs scrolling commands
1267 Press @kbd{n} or @kbd{p} to go to the Next or Previous section.
1268 If the section has a @dfn{menu}, press a digit key like @kbd{1}
1269 or @kbd{2} to go to a sub-section from the menu. Press @kbd{u} to
1270 go back up from a sub-section to the menu it is part of.
1272 Exercises in the tutorial all have cross-references to the
1273 appropriate page of the ``answers'' section. Press @kbd{f}, then
1274 the exercise number, to see the answer to an exercise. After
1275 you have followed a cross-reference, you can press the letter
1276 @kbd{l} to return to where you were before.
1278 You can press @kbd{?} at any time for a brief summary of Info commands.
1280 Press the number @kbd{1} now to enter the first section of the Tutorial.
1286 @node Tutorial, Introduction, Interactive Tutorial, Top
1289 @node Tutorial, Introduction, Getting Started, Top
1294 This chapter explains how to use Calc and its many features, in
1295 a step-by-step, tutorial way. You are encouraged to run Calc and
1296 work along with the examples as you read (@pxref{Starting Calc}).
1297 If you are already familiar with advanced calculators, you may wish
1299 to skip on to the rest of this manual.
1301 @c to skip on to volume II of this manual, the @dfn{Calc Reference}.
1303 @c [fix-ref Embedded Mode]
1304 This tutorial describes the standard user interface of Calc only.
1305 The Quick mode and Keypad mode interfaces are fairly
1306 self-explanatory. @xref{Embedded Mode}, for a description of
1307 the Embedded mode interface.
1309 The easiest way to read this tutorial on-line is to have two windows on
1310 your Emacs screen, one with Calc and one with the Info system. Press
1311 @kbd{C-x * t} to set this up; the on-line tutorial will be opened in the
1312 current window and Calc will be started in another window. From the
1313 Info window, the command @kbd{C-x * c} can be used to switch to the Calc
1314 window and @kbd{C-x * o} can be used to switch back to the Info window.
1315 (If you have a printed copy of the manual you can use that instead; in
1316 that case you only need to press @kbd{C-x * c} to start Calc.)
1318 This tutorial is designed to be done in sequence. But the rest of this
1319 manual does not assume you have gone through the tutorial. The tutorial
1320 does not cover everything in the Calculator, but it touches on most
1324 You may wish to print out a copy of the Calc Summary and keep notes on
1325 it as you learn Calc. @xref{About This Manual}, to see how to make a
1326 printed summary. @xref{Summary}.
1329 The Calc Summary at the end of the reference manual includes some blank
1330 space for your own use. You may wish to keep notes there as you learn
1336 * Arithmetic Tutorial::
1337 * Vector/Matrix Tutorial::
1339 * Algebra Tutorial::
1340 * Programming Tutorial::
1342 * Answers to Exercises::
1345 @node Basic Tutorial, Arithmetic Tutorial, Tutorial, Tutorial
1346 @section Basic Tutorial
1349 In this section, we learn how RPN and algebraic-style calculations
1350 work, how to undo and redo an operation done by mistake, and how
1351 to control various modes of the Calculator.
1354 * RPN Tutorial:: Basic operations with the stack.
1355 * Algebraic Tutorial:: Algebraic entry; variables.
1356 * Undo Tutorial:: If you make a mistake: Undo and the trail.
1357 * Modes Tutorial:: Common mode-setting commands.
1360 @node RPN Tutorial, Algebraic Tutorial, Basic Tutorial, Basic Tutorial
1361 @subsection RPN Calculations and the Stack
1363 @cindex RPN notation
1366 Calc normally uses RPN notation. You may be familiar with the RPN
1367 system from Hewlett-Packard calculators, FORTH, or PostScript.
1368 (Reverse Polish Notation, RPN, is named after the Polish mathematician
1372 Calc normally uses RPN notation. You may be familiar with the RPN
1373 system from Hewlett-Packard calculators, FORTH, or PostScript.
1374 (Reverse Polish Notation, RPN, is named after the Polish mathematician
1378 The central component of an RPN calculator is the @dfn{stack}. A
1379 calculator stack is like a stack of dishes. New dishes (numbers) are
1380 added at the top of the stack, and numbers are normally only removed
1381 from the top of the stack.
1385 In an operation like @expr{2+3}, the 2 and 3 are called the @dfn{operands}
1386 and the @expr{+} is the @dfn{operator}. In an RPN calculator you always
1387 enter the operands first, then the operator. Each time you type a
1388 number, Calc adds or @dfn{pushes} it onto the top of the Stack.
1389 When you press an operator key like @kbd{+}, Calc @dfn{pops} the appropriate
1390 number of operands from the stack and pushes back the result.
1392 Thus we could add the numbers 2 and 3 in an RPN calculator by typing:
1393 @kbd{2 @key{RET} 3 @key{RET} +}. (The @key{RET} key, Return, corresponds to
1394 the @key{ENTER} key on traditional RPN calculators.) Try this now if
1395 you wish; type @kbd{C-x * c} to switch into the Calc window (you can type
1396 @kbd{C-x * c} again or @kbd{C-x * o} to switch back to the Tutorial window).
1397 The first four keystrokes ``push'' the numbers 2 and 3 onto the stack.
1398 The @kbd{+} key ``pops'' the top two numbers from the stack, adds them,
1399 and pushes the result (5) back onto the stack. Here's how the stack
1400 will look at various points throughout the calculation:
1408 C-x * c 2 @key{RET} 3 @key{RET} + @key{DEL}
1412 The @samp{.} symbol is a marker that represents the top of the stack.
1413 Note that the ``top'' of the stack is really shown at the bottom of
1414 the Stack window. This may seem backwards, but it turns out to be
1415 less distracting in regular use.
1417 @cindex Stack levels
1418 @cindex Levels of stack
1419 The numbers @samp{1:} and @samp{2:} on the left are @dfn{stack level
1420 numbers}. Old RPN calculators always had four stack levels called
1421 @expr{x}, @expr{y}, @expr{z}, and @expr{t}. Calc's stack can grow
1422 as large as you like, so it uses numbers instead of letters. Some
1423 stack-manipulation commands accept a numeric argument that says
1424 which stack level to work on. Normal commands like @kbd{+} always
1425 work on the top few levels of the stack.
1427 @c [fix-ref Truncating the Stack]
1428 The Stack buffer is just an Emacs buffer, and you can move around in
1429 it using the regular Emacs motion commands. But no matter where the
1430 cursor is, even if you have scrolled the @samp{.} marker out of
1431 view, most Calc commands always move the cursor back down to level 1
1432 before doing anything. It is possible to move the @samp{.} marker
1433 upwards through the stack, temporarily ``hiding'' some numbers from
1434 commands like @kbd{+}. This is called @dfn{stack truncation} and
1435 we will not cover it in this tutorial; @pxref{Truncating the Stack},
1436 if you are interested.
1438 You don't really need the second @key{RET} in @kbd{2 @key{RET} 3
1439 @key{RET} +}. That's because if you type any operator name or
1440 other non-numeric key when you are entering a number, the Calculator
1441 automatically enters that number and then does the requested command.
1442 Thus @kbd{2 @key{RET} 3 +} will work just as well.
1444 Examples in this tutorial will often omit @key{RET} even when the
1445 stack displays shown would only happen if you did press @key{RET}:
1458 Here, after pressing @kbd{3} the stack would really show @samp{1: 2}
1459 with @samp{Calc:@: 3} in the minibuffer. In these situations, you can
1460 press the optional @key{RET} to see the stack as the figure shows.
1462 (@bullet{}) @strong{Exercise 1.} (This tutorial will include exercises
1463 at various points. Try them if you wish. Answers to all the exercises
1464 are located at the end of the Tutorial chapter. Each exercise will
1465 include a cross-reference to its particular answer. If you are
1466 reading with the Emacs Info system, press @kbd{f} and the
1467 exercise number to go to the answer, then the letter @kbd{l} to
1468 return to where you were.)
1471 Here's the first exercise: What will the keystrokes @kbd{1 @key{RET} 2
1472 @key{RET} 3 @key{RET} 4 + * -} compute? (@samp{*} is the symbol for
1473 multiplication.) Figure it out by hand, then try it with Calc to see
1474 if you're right. @xref{RPN Answer 1, 1}. (@bullet{})
1476 (@bullet{}) @strong{Exercise 2.} Compute
1477 @texline @math{(2\times4) + (7\times9.5) + {5\over4}}
1478 @infoline @expr{2*4 + 7*9.5 + 5/4}
1479 using the stack. @xref{RPN Answer 2, 2}. (@bullet{})
1481 The @key{DEL} key is called Backspace on some keyboards. It is
1482 whatever key you would use to correct a simple typing error when
1483 regularly using Emacs. The @key{DEL} key pops and throws away the
1484 top value on the stack. (You can still get that value back from
1485 the Trail if you should need it later on.) There are many places
1486 in this tutorial where we assume you have used @key{DEL} to erase the
1487 results of the previous example at the beginning of a new example.
1488 In the few places where it is really important to use @key{DEL} to
1489 clear away old results, the text will remind you to do so.
1491 (It won't hurt to let things accumulate on the stack, except that
1492 whenever you give a display-mode-changing command Calc will have to
1493 spend a long time reformatting such a large stack.)
1495 Since the @kbd{-} key is also an operator (it subtracts the top two
1496 stack elements), how does one enter a negative number? Calc uses
1497 the @kbd{_} (underscore) key to act like the minus sign in a number.
1498 So, typing @kbd{-5 @key{RET}} won't work because the @kbd{-} key
1499 will try to do a subtraction, but @kbd{_5 @key{RET}} works just fine.
1501 You can also press @kbd{n}, which means ``change sign.'' It changes
1502 the number at the top of the stack (or the number being entered)
1503 from positive to negative or vice-versa: @kbd{5 n @key{RET}}.
1505 @cindex Duplicating a stack entry
1506 If you press @key{RET} when you're not entering a number, the effect
1507 is to duplicate the top number on the stack. Consider this calculation:
1511 1: 3 2: 3 1: 9 2: 9 1: 81
1515 3 @key{RET} @key{RET} * @key{RET} *
1520 (Of course, an easier way to do this would be @kbd{3 @key{RET} 4 ^},
1521 to raise 3 to the fourth power.)
1523 The space-bar key (denoted @key{SPC} here) performs the same function
1524 as @key{RET}; you could replace all three occurrences of @key{RET} in
1525 the above example with @key{SPC} and the effect would be the same.
1527 @cindex Exchanging stack entries
1528 Another stack manipulation key is @key{TAB}. This exchanges the top
1529 two stack entries. Suppose you have computed @kbd{2 @key{RET} 3 +}
1530 to get 5, and then you realize what you really wanted to compute
1531 was @expr{20 / (2+3)}.
1535 1: 5 2: 5 2: 20 1: 4
1539 2 @key{RET} 3 + 20 @key{TAB} /
1544 Planning ahead, the calculation would have gone like this:
1548 1: 20 2: 20 3: 20 2: 20 1: 4
1553 20 @key{RET} 2 @key{RET} 3 + /
1557 A related stack command is @kbd{M-@key{TAB}} (hold @key{META} and type
1558 @key{TAB}). It rotates the top three elements of the stack upward,
1559 bringing the object in level 3 to the top.
1563 1: 10 2: 10 3: 10 3: 20 3: 30
1564 . 1: 20 2: 20 2: 30 2: 10
1568 10 @key{RET} 20 @key{RET} 30 @key{RET} M-@key{TAB} M-@key{TAB}
1572 (@bullet{}) @strong{Exercise 3.} Suppose the numbers 10, 20, and 30 are
1573 on the stack. Figure out how to add one to the number in level 2
1574 without affecting the rest of the stack. Also figure out how to add
1575 one to the number in level 3. @xref{RPN Answer 3, 3}. (@bullet{})
1577 Operations like @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/}, and @kbd{^} pop two
1578 arguments from the stack and push a result. Operations like @kbd{n} and
1579 @kbd{Q} (square root) pop a single number and push the result. You can
1580 think of them as simply operating on the top element of the stack.
1584 1: 3 1: 9 2: 9 1: 25 1: 5
1588 3 @key{RET} @key{RET} * 4 @key{RET} @key{RET} * + Q
1593 (Note that capital @kbd{Q} means to hold down the Shift key while
1594 typing @kbd{q}. Remember, plain unshifted @kbd{q} is the Quit command.)
1596 @cindex Pythagorean Theorem
1597 Here we've used the Pythagorean Theorem to determine the hypotenuse of a
1598 right triangle. Calc actually has a built-in command for that called
1599 @kbd{f h}, but let's suppose we can't remember the necessary keystrokes.
1600 We can still enter it by its full name using @kbd{M-x} notation:
1608 3 @key{RET} 4 @key{RET} M-x calc-hypot
1612 All Calculator commands begin with the word @samp{calc-}. Since it
1613 gets tiring to type this, Calc provides an @kbd{x} key which is just
1614 like the regular Emacs @kbd{M-x} key except that it types the @samp{calc-}
1623 3 @key{RET} 4 @key{RET} x hypot
1627 What happens if you take the square root of a negative number?
1631 1: 4 1: -4 1: (0, 2)
1639 The notation @expr{(a, b)} represents a complex number.
1640 Complex numbers are more traditionally written @expr{a + b i};
1641 Calc can display in this format, too, but for now we'll stick to the
1642 @expr{(a, b)} notation.
1644 If you don't know how complex numbers work, you can safely ignore this
1645 feature. Complex numbers only arise from operations that would be
1646 errors in a calculator that didn't have complex numbers. (For example,
1647 taking the square root or logarithm of a negative number produces a
1650 Complex numbers are entered in the notation shown. The @kbd{(} and
1651 @kbd{,} and @kbd{)} keys manipulate ``incomplete complex numbers.''
1655 1: ( ... 2: ( ... 1: (2, ... 1: (2, ... 1: (2, 3)
1663 You can perform calculations while entering parts of incomplete objects.
1664 However, an incomplete object cannot actually participate in a calculation:
1668 1: ( ... 2: ( ... 3: ( ... 1: ( ... 1: ( ...
1678 Adding 5 to an incomplete object makes no sense, so the last command
1679 produces an error message and leaves the stack the same.
1681 Incomplete objects can't participate in arithmetic, but they can be
1682 moved around by the regular stack commands.
1686 2: 2 3: 2 3: 3 1: ( ... 1: (2, 3)
1687 1: 3 2: 3 2: ( ... 2 .
1691 2 @key{RET} 3 @key{RET} ( M-@key{TAB} M-@key{TAB} )
1696 Note that the @kbd{,} (comma) key did not have to be used here.
1697 When you press @kbd{)} all the stack entries between the incomplete
1698 entry and the top are collected, so there's never really a reason
1699 to use the comma. It's up to you.
1701 (@bullet{}) @strong{Exercise 4.} To enter the complex number @expr{(2, 3)},
1702 your friend Joe typed @kbd{( 2 , @key{SPC} 3 )}. What happened?
1703 (Joe thought of a clever way to correct his mistake in only two
1704 keystrokes, but it didn't quite work. Try it to find out why.)
1705 @xref{RPN Answer 4, 4}. (@bullet{})
1707 Vectors are entered the same way as complex numbers, but with square
1708 brackets in place of parentheses. We'll meet vectors again later in
1711 Any Emacs command can be given a @dfn{numeric prefix argument} by
1712 typing a series of @key{META}-digits beforehand. If @key{META} is
1713 awkward for you, you can instead type @kbd{C-u} followed by the
1714 necessary digits. Numeric prefix arguments can be negative, as in
1715 @kbd{M-- M-3 M-5} or @w{@kbd{C-u - 3 5}}. Calc commands use numeric
1716 prefix arguments in a variety of ways. For example, a numeric prefix
1717 on the @kbd{+} operator adds any number of stack entries at once:
1721 1: 10 2: 10 3: 10 3: 10 1: 60
1722 . 1: 20 2: 20 2: 20 .
1726 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u 3 +
1730 For stack manipulation commands like @key{RET}, a positive numeric
1731 prefix argument operates on the top @var{n} stack entries at once. A
1732 negative argument operates on the entry in level @var{n} only. An
1733 argument of zero operates on the entire stack. In this example, we copy
1734 the second-to-top element of the stack:
1738 1: 10 2: 10 3: 10 3: 10 4: 10
1739 . 1: 20 2: 20 2: 20 3: 20
1744 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u -2 @key{RET}
1748 @cindex Clearing the stack
1749 @cindex Emptying the stack
1750 Another common idiom is @kbd{M-0 @key{DEL}}, which clears the stack.
1751 (The @kbd{M-0} numeric prefix tells @key{DEL} to operate on the
1754 @node Algebraic Tutorial, Undo Tutorial, RPN Tutorial, Basic Tutorial
1755 @subsection Algebraic-Style Calculations
1758 If you are not used to RPN notation, you may prefer to operate the
1759 Calculator in Algebraic mode, which is closer to the way
1760 non-RPN calculators work. In Algebraic mode, you enter formulas
1761 in traditional @expr{2+3} notation.
1763 @strong{Notice:} Calc gives @samp{/} lower precedence than @samp{*}, so
1764 that @samp{a/b*c} is interpreted as @samp{a/(b*c)}; this is not
1765 standard across all computer languages. See below for details.
1767 You don't really need any special ``mode'' to enter algebraic formulas.
1768 You can enter a formula at any time by pressing the apostrophe (@kbd{'})
1769 key. Answer the prompt with the desired formula, then press @key{RET}.
1770 The formula is evaluated and the result is pushed onto the RPN stack.
1771 If you don't want to think in RPN at all, you can enter your whole
1772 computation as a formula, read the result from the stack, then press
1773 @key{DEL} to delete it from the stack.
1775 Try pressing the apostrophe key, then @kbd{2+3+4}, then @key{RET}.
1776 The result should be the number 9.
1778 Algebraic formulas use the operators @samp{+}, @samp{-}, @samp{*},
1779 @samp{/}, and @samp{^}. You can use parentheses to make the order
1780 of evaluation clear. In the absence of parentheses, @samp{^} is
1781 evaluated first, then @samp{*}, then @samp{/}, then finally
1782 @samp{+} and @samp{-}. For example, the expression
1785 2 + 3*4*5 / 6*7^8 - 9
1792 2 + ((3*4*5) / (6*(7^8)) - 9
1796 or, in large mathematical notation,
1810 $$ 2 + { 3 \times 4 \times 5 \over 6 \times 7^8 } - 9 $$
1815 The result of this expression will be the number @mathit{-6.99999826533}.
1817 Calc's order of evaluation is the same as for most computer languages,
1818 except that @samp{*} binds more strongly than @samp{/}, as the above
1819 example shows. As in normal mathematical notation, the @samp{*} symbol
1820 can often be omitted: @samp{2 a} is the same as @samp{2*a}.
1822 Operators at the same level are evaluated from left to right, except
1823 that @samp{^} is evaluated from right to left. Thus, @samp{2-3-4} is
1824 equivalent to @samp{(2-3)-4} or @mathit{-5}, whereas @samp{2^3^4} is equivalent
1825 to @samp{2^(3^4)} (a very large integer; try it!).
1827 If you tire of typing the apostrophe all the time, there is
1828 Algebraic mode, where Calc automatically senses
1829 when you are about to type an algebraic expression. To enter this
1830 mode, press the two letters @w{@kbd{m a}}. (An @samp{Alg} indicator
1831 should appear in the Calc window's mode line.)
1833 Press @kbd{m a}, then @kbd{2+3+4} with no apostrophe, then @key{RET}.
1835 In Algebraic mode, when you press any key that would normally begin
1836 entering a number (such as a digit, a decimal point, or the @kbd{_}
1837 key), or if you press @kbd{(} or @kbd{[}, Calc automatically begins
1840 Functions which do not have operator symbols like @samp{+} and @samp{*}
1841 must be entered in formulas using function-call notation. For example,
1842 the function name corresponding to the square-root key @kbd{Q} is
1843 @code{sqrt}. To compute a square root in a formula, you would use
1844 the notation @samp{sqrt(@var{x})}.
1846 Press the apostrophe, then type @kbd{sqrt(5*2) - 3}. The result should
1847 be @expr{0.16227766017}.
1849 Note that if the formula begins with a function name, you need to use
1850 the apostrophe even if you are in Algebraic mode. If you type @kbd{arcsin}
1851 out of the blue, the @kbd{a r} will be taken as an Algebraic Rewrite
1852 command, and the @kbd{csin} will be taken as the name of the rewrite
1855 Some people prefer to enter complex numbers and vectors in algebraic
1856 form because they find RPN entry with incomplete objects to be too
1857 distracting, even though they otherwise use Calc as an RPN calculator.
1859 Still in Algebraic mode, type:
1863 1: (2, 3) 2: (2, 3) 1: (8, -1) 2: (8, -1) 1: (9, -1)
1864 . 1: (1, -2) . 1: 1 .
1867 (2,3) @key{RET} (1,-2) @key{RET} * 1 @key{RET} +
1871 Algebraic mode allows us to enter complex numbers without pressing
1872 an apostrophe first, but it also means we need to press @key{RET}
1873 after every entry, even for a simple number like @expr{1}.
1875 (You can type @kbd{C-u m a} to enable a special Incomplete Algebraic
1876 mode in which the @kbd{(} and @kbd{[} keys use algebraic entry even
1877 though regular numeric keys still use RPN numeric entry. There is also
1878 Total Algebraic mode, started by typing @kbd{m t}, in which all
1879 normal keys begin algebraic entry. You must then use the @key{META} key
1880 to type Calc commands: @kbd{M-m t} to get back out of Total Algebraic
1881 mode, @kbd{M-q} to quit, etc.)
1883 If you're still in Algebraic mode, press @kbd{m a} again to turn it off.
1885 Actual non-RPN calculators use a mixture of algebraic and RPN styles.
1886 In general, operators of two numbers (like @kbd{+} and @kbd{*})
1887 use algebraic form, but operators of one number (like @kbd{n} and @kbd{Q})
1888 use RPN form. Also, a non-RPN calculator allows you to see the
1889 intermediate results of a calculation as you go along. You can
1890 accomplish this in Calc by performing your calculation as a series
1891 of algebraic entries, using the @kbd{$} sign to tie them together.
1892 In an algebraic formula, @kbd{$} represents the number on the top
1893 of the stack. Here, we perform the calculation
1894 @texline @math{\sqrt{2\times4+1}},
1895 @infoline @expr{sqrt(2*4+1)},
1896 which on a traditional calculator would be done by pressing
1897 @kbd{2 * 4 + 1 =} and then the square-root key.
1904 ' 2*4 @key{RET} $+1 @key{RET} Q
1909 Notice that we didn't need to press an apostrophe for the @kbd{$+1},
1910 because the dollar sign always begins an algebraic entry.
1912 (@bullet{}) @strong{Exercise 1.} How could you get the same effect as
1913 pressing @kbd{Q} but using an algebraic entry instead? How about
1914 if the @kbd{Q} key on your keyboard were broken?
1915 @xref{Algebraic Answer 1, 1}. (@bullet{})
1917 The notations @kbd{$$}, @kbd{$$$}, and so on stand for higher stack
1918 entries. For example, @kbd{' $$+$ @key{RET}} is just like typing @kbd{+}.
1920 Algebraic formulas can include @dfn{variables}. To store in a
1921 variable, press @kbd{s s}, then type the variable name, then press
1922 @key{RET}. (There are actually two flavors of store command:
1923 @kbd{s s} stores a number in a variable but also leaves the number
1924 on the stack, while @w{@kbd{s t}} removes a number from the stack and
1925 stores it in the variable.) A variable name should consist of one
1926 or more letters or digits, beginning with a letter.
1930 1: 17 . 1: a + a^2 1: 306
1933 17 s t a @key{RET} ' a+a^2 @key{RET} =
1938 The @kbd{=} key @dfn{evaluates} a formula by replacing all its
1939 variables by the values that were stored in them.
1941 For RPN calculations, you can recall a variable's value on the
1942 stack either by entering its name as a formula and pressing @kbd{=},
1943 or by using the @kbd{s r} command.
1947 1: 17 2: 17 3: 17 2: 17 1: 306
1948 . 1: 17 2: 17 1: 289 .
1952 s r a @key{RET} ' a @key{RET} = 2 ^ +
1956 If you press a single digit for a variable name (as in @kbd{s t 3}, you
1957 get one of ten @dfn{quick variables} @code{q0} through @code{q9}.
1958 They are ``quick'' simply because you don't have to type the letter
1959 @code{q} or the @key{RET} after their names. In fact, you can type
1960 simply @kbd{s 3} as a shorthand for @kbd{s s 3}, and likewise for
1961 @kbd{t 3} and @w{@kbd{r 3}}.
1963 Any variables in an algebraic formula for which you have not stored
1964 values are left alone, even when you evaluate the formula.
1968 1: 2 a + 2 b 1: 2 b + 34
1975 Calls to function names which are undefined in Calc are also left
1976 alone, as are calls for which the value is undefined.
1980 1: log10(0) + log10(x) + log10(5, 6) + foo(3) + 2
1983 ' log10(100) + log10(0) + log10(x) + log10(5,6) + foo(3) @key{RET}
1988 In this example, the first call to @code{log10} works, but the other
1989 calls are not evaluated. In the second call, the logarithm is
1990 undefined for that value of the argument; in the third, the argument
1991 is symbolic, and in the fourth, there are too many arguments. In the
1992 fifth case, there is no function called @code{foo}. You will see a
1993 ``Wrong number of arguments'' message referring to @samp{log10(5,6)}.
1994 Press the @kbd{w} (``why'') key to see any other messages that may
1995 have arisen from the last calculation. In this case you will get
1996 ``logarithm of zero,'' then ``number expected: @code{x}''. Calc
1997 automatically displays the first message only if the message is
1998 sufficiently important; for example, Calc considers ``wrong number
1999 of arguments'' and ``logarithm of zero'' to be important enough to
2000 report automatically, while a message like ``number expected: @code{x}''
2001 will only show up if you explicitly press the @kbd{w} key.
2003 (@bullet{}) @strong{Exercise 2.} Joe entered the formula @samp{2 x y},
2004 stored 5 in @code{x}, pressed @kbd{=}, and got the expected result,
2005 @samp{10 y}. He then tried the same for the formula @samp{2 x (1+y)},
2006 expecting @samp{10 (1+y)}, but it didn't work. Why not?
2007 @xref{Algebraic Answer 2, 2}. (@bullet{})
2009 (@bullet{}) @strong{Exercise 3.} What result would you expect
2010 @kbd{1 @key{RET} 0 /} to give? What if you then type @kbd{0 *}?
2011 @xref{Algebraic Answer 3, 3}. (@bullet{})
2013 One interesting way to work with variables is to use the
2014 @dfn{evaluates-to} (@samp{=>}) operator. It works like this:
2015 Enter a formula algebraically in the usual way, but follow
2016 the formula with an @samp{=>} symbol. (There is also an @kbd{s =}
2017 command which builds an @samp{=>} formula using the stack.) On
2018 the stack, you will see two copies of the formula with an @samp{=>}
2019 between them. The lefthand formula is exactly like you typed it;
2020 the righthand formula has been evaluated as if by typing @kbd{=}.
2024 2: 2 + 3 => 5 2: 2 + 3 => 5
2025 1: 2 a + 2 b => 34 + 2 b 1: 2 a + 2 b => 20 + 2 b
2028 ' 2+3 => @key{RET} ' 2a+2b @key{RET} s = 10 s t a @key{RET}
2033 Notice that the instant we stored a new value in @code{a}, all
2034 @samp{=>} operators already on the stack that referred to @expr{a}
2035 were updated to use the new value. With @samp{=>}, you can push a
2036 set of formulas on the stack, then change the variables experimentally
2037 to see the effects on the formulas' values.
2039 You can also ``unstore'' a variable when you are through with it:
2044 1: 2 a + 2 b => 2 a + 2 b
2051 We will encounter formulas involving variables and functions again
2052 when we discuss the algebra and calculus features of the Calculator.
2054 @node Undo Tutorial, Modes Tutorial, Algebraic Tutorial, Basic Tutorial
2055 @subsection Undo and Redo
2058 If you make a mistake, you can usually correct it by pressing shift-@kbd{U},
2059 the ``undo'' command. First, clear the stack (@kbd{M-0 @key{DEL}}) and exit
2060 and restart Calc (@kbd{C-x * * C-x * *}) to make sure things start off
2061 with a clean slate. Now:
2065 1: 2 2: 2 1: 8 2: 2 1: 6
2073 You can undo any number of times. Calc keeps a complete record of
2074 all you have done since you last opened the Calc window. After the
2075 above example, you could type:
2087 You can also type @kbd{D} to ``redo'' a command that you have undone
2092 . 1: 2 2: 2 1: 6 1: 6
2101 It was not possible to redo past the @expr{6}, since that was placed there
2102 by something other than an undo command.
2105 You can think of undo and redo as a sort of ``time machine.'' Press
2106 @kbd{U} to go backward in time, @kbd{D} to go forward. If you go
2107 backward and do something (like @kbd{*}) then, as any science fiction
2108 reader knows, you have changed your future and you cannot go forward
2109 again. Thus, the inability to redo past the @expr{6} even though there
2110 was an earlier undo command.
2112 You can always recall an earlier result using the Trail. We've ignored
2113 the trail so far, but it has been faithfully recording everything we
2114 did since we loaded the Calculator. If the Trail is not displayed,
2115 press @kbd{t d} now to turn it on.
2117 Let's try grabbing an earlier result. The @expr{8} we computed was
2118 undone by a @kbd{U} command, and was lost even to Redo when we pressed
2119 @kbd{*}, but it's still there in the trail. There should be a little
2120 @samp{>} arrow (the @dfn{trail pointer}) resting on the last trail
2121 entry. If there isn't, press @kbd{t ]} to reset the trail pointer.
2122 Now, press @w{@kbd{t p}} to move the arrow onto the line containing
2123 @expr{8}, and press @w{@kbd{t y}} to ``yank'' that number back onto the
2126 If you press @kbd{t ]} again, you will see that even our Yank command
2127 went into the trail.
2129 Let's go further back in time. Earlier in the tutorial we computed
2130 a huge integer using the formula @samp{2^3^4}. We don't remember
2131 what it was, but the first digits were ``241''. Press @kbd{t r}
2132 (which stands for trail-search-reverse), then type @kbd{241}.
2133 The trail cursor will jump back to the next previous occurrence of
2134 the string ``241'' in the trail. This is just a regular Emacs
2135 incremental search; you can now press @kbd{C-s} or @kbd{C-r} to
2136 continue the search forwards or backwards as you like.
2138 To finish the search, press @key{RET}. This halts the incremental
2139 search and leaves the trail pointer at the thing we found. Now we
2140 can type @kbd{t y} to yank that number onto the stack. If we hadn't
2141 remembered the ``241'', we could simply have searched for @kbd{2^3^4},
2142 then pressed @kbd{@key{RET} t n} to halt and then move to the next item.
2144 You may have noticed that all the trail-related commands begin with
2145 the letter @kbd{t}. (The store-and-recall commands, on the other hand,
2146 all began with @kbd{s}.) Calc has so many commands that there aren't
2147 enough keys for all of them, so various commands are grouped into
2148 two-letter sequences where the first letter is called the @dfn{prefix}
2149 key. If you type a prefix key by accident, you can press @kbd{C-g}
2150 to cancel it. (In fact, you can press @kbd{C-g} to cancel almost
2151 anything in Emacs.) To get help on a prefix key, press that key
2152 followed by @kbd{?}. Some prefixes have several lines of help,
2153 so you need to press @kbd{?} repeatedly to see them all.
2154 You can also type @kbd{h h} to see all the help at once.
2156 Try pressing @kbd{t ?} now. You will see a line of the form,
2159 trail/time: Display; Fwd, Back; Next, Prev, Here, [, ]; Yank: [MORE] t-
2163 The word ``trail'' indicates that the @kbd{t} prefix key contains
2164 trail-related commands. Each entry on the line shows one command,
2165 with a single capital letter showing which letter you press to get
2166 that command. We have used @kbd{t n}, @kbd{t p}, @kbd{t ]}, and
2167 @kbd{t y} so far. The @samp{[MORE]} means you can press @kbd{?}
2168 again to see more @kbd{t}-prefix commands. Notice that the commands
2169 are roughly divided (by semicolons) into related groups.
2171 When you are in the help display for a prefix key, the prefix is
2172 still active. If you press another key, like @kbd{y} for example,
2173 it will be interpreted as a @kbd{t y} command. If all you wanted
2174 was to look at the help messages, press @kbd{C-g} afterwards to cancel
2177 One more way to correct an error is by editing the stack entries.
2178 The actual Stack buffer is marked read-only and must not be edited
2179 directly, but you can press @kbd{`} (the backquote or accent grave)
2180 to edit a stack entry.
2182 Try entering @samp{3.141439} now. If this is supposed to represent
2183 @cpi{}, it's got several errors. Press @kbd{`} to edit this number.
2184 Now use the normal Emacs cursor motion and editing keys to change
2185 the second 4 to a 5, and to transpose the 3 and the 9. When you
2186 press @key{RET}, the number on the stack will be replaced by your
2187 new number. This works for formulas, vectors, and all other types
2188 of values you can put on the stack. The @kbd{`} key also works
2189 during entry of a number or algebraic formula.
2191 @node Modes Tutorial, , Undo Tutorial, Basic Tutorial
2192 @subsection Mode-Setting Commands
2195 Calc has many types of @dfn{modes} that affect the way it interprets
2196 your commands or the way it displays data. We have already seen one
2197 mode, namely Algebraic mode. There are many others, too; we'll
2198 try some of the most common ones here.
2200 Perhaps the most fundamental mode in Calc is the current @dfn{precision}.
2201 Notice the @samp{12} on the Calc window's mode line:
2204 --%*-Calc: 12 Deg (Calculator)----All------
2208 Most of the symbols there are Emacs things you don't need to worry
2209 about, but the @samp{12} and the @samp{Deg} are mode indicators.
2210 The @samp{12} means that calculations should always be carried to
2211 12 significant figures. That is why, when we type @kbd{1 @key{RET} 7 /},
2212 we get @expr{0.142857142857} with exactly 12 digits, not counting
2213 leading and trailing zeros.
2215 You can set the precision to anything you like by pressing @kbd{p},
2216 then entering a suitable number. Try pressing @kbd{p 30 @key{RET}},
2217 then doing @kbd{1 @key{RET} 7 /} again:
2222 2: 0.142857142857142857142857142857
2227 Although the precision can be set arbitrarily high, Calc always
2228 has to have @emph{some} value for the current precision. After
2229 all, the true value @expr{1/7} is an infinitely repeating decimal;
2230 Calc has to stop somewhere.
2232 Of course, calculations are slower the more digits you request.
2233 Press @w{@kbd{p 12}} now to set the precision back down to the default.
2235 Calculations always use the current precision. For example, even
2236 though we have a 30-digit value for @expr{1/7} on the stack, if
2237 we use it in a calculation in 12-digit mode it will be rounded
2238 down to 12 digits before it is used. Try it; press @key{RET} to
2239 duplicate the number, then @w{@kbd{1 +}}. Notice that the @key{RET}
2240 key didn't round the number, because it doesn't do any calculation.
2241 But the instant we pressed @kbd{+}, the number was rounded down.
2246 2: 0.142857142857142857142857142857
2253 In fact, since we added a digit on the left, we had to lose one
2254 digit on the right from even the 12-digit value of @expr{1/7}.
2256 How did we get more than 12 digits when we computed @samp{2^3^4}? The
2257 answer is that Calc makes a distinction between @dfn{integers} and
2258 @dfn{floating-point} numbers, or @dfn{floats}. An integer is a number
2259 that does not contain a decimal point. There is no such thing as an
2260 ``infinitely repeating fraction integer,'' so Calc doesn't have to limit
2261 itself. If you asked for @samp{2^10000} (don't try this!), you would
2262 have to wait a long time but you would eventually get an exact answer.
2263 If you ask for @samp{2.^10000}, you will quickly get an answer which is
2264 correct only to 12 places. The decimal point tells Calc that it should
2265 use floating-point arithmetic to get the answer, not exact integer
2268 You can use the @kbd{F} (@code{calc-floor}) command to convert a
2269 floating-point value to an integer, and @kbd{c f} (@code{calc-float})
2270 to convert an integer to floating-point form.
2272 Let's try entering that last calculation:
2276 1: 2. 2: 2. 1: 1.99506311689e3010
2280 2.0 @key{RET} 10000 @key{RET} ^
2285 @cindex Scientific notation, entry of
2286 Notice the letter @samp{e} in there. It represents ``times ten to the
2287 power of,'' and is used by Calc automatically whenever writing the
2288 number out fully would introduce more extra zeros than you probably
2289 want to see. You can enter numbers in this notation, too.
2293 1: 2. 2: 2. 1: 1.99506311678e3010
2297 2.0 @key{RET} 1e4 @key{RET} ^
2301 @cindex Round-off errors
2303 Hey, the answer is different! Look closely at the middle columns
2304 of the two examples. In the first, the stack contained the
2305 exact integer @expr{10000}, but in the second it contained
2306 a floating-point value with a decimal point. When you raise a
2307 number to an integer power, Calc uses repeated squaring and
2308 multiplication to get the answer. When you use a floating-point
2309 power, Calc uses logarithms and exponentials. As you can see,
2310 a slight error crept in during one of these methods. Which
2311 one should we trust? Let's raise the precision a bit and find
2316 . 1: 2. 2: 2. 1: 1.995063116880828e3010
2320 p 16 @key{RET} 2. @key{RET} 1e4 ^ p 12 @key{RET}
2325 @cindex Guard digits
2326 Presumably, it doesn't matter whether we do this higher-precision
2327 calculation using an integer or floating-point power, since we
2328 have added enough ``guard digits'' to trust the first 12 digits
2329 no matter what. And the verdict is@dots{} Integer powers were more
2330 accurate; in fact, the result was only off by one unit in the
2333 @cindex Guard digits
2334 Calc does many of its internal calculations to a slightly higher
2335 precision, but it doesn't always bump the precision up enough.
2336 In each case, Calc added about two digits of precision during
2337 its calculation and then rounded back down to 12 digits
2338 afterward. In one case, it was enough; in the other, it
2339 wasn't. If you really need @var{x} digits of precision, it
2340 never hurts to do the calculation with a few extra guard digits.
2342 What if we want guard digits but don't want to look at them?
2343 We can set the @dfn{float format}. Calc supports four major
2344 formats for floating-point numbers, called @dfn{normal},
2345 @dfn{fixed-point}, @dfn{scientific notation}, and @dfn{engineering
2346 notation}. You get them by pressing @w{@kbd{d n}}, @kbd{d f},
2347 @kbd{d s}, and @kbd{d e}, respectively. In each case, you can
2348 supply a numeric prefix argument which says how many digits
2349 should be displayed. As an example, let's put a few numbers
2350 onto the stack and try some different display modes. First,
2351 use @kbd{M-0 @key{DEL}} to clear the stack, then enter the four
2356 4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
2357 3: 12345. 3: 12300. 3: 1.2345e4 3: 1.23e4 3: 12345.000
2358 2: 123.45 2: 123. 2: 1.2345e2 2: 1.23e2 2: 123.450
2359 1: 12.345 1: 12.3 1: 1.2345e1 1: 1.23e1 1: 12.345
2362 d n M-3 d n d s M-3 d s M-3 d f
2367 Notice that when we typed @kbd{M-3 d n}, the numbers were rounded down
2368 to three significant digits, but then when we typed @kbd{d s} all
2369 five significant figures reappeared. The float format does not
2370 affect how numbers are stored, it only affects how they are
2371 displayed. Only the current precision governs the actual rounding
2372 of numbers in the Calculator's memory.
2374 Engineering notation, not shown here, is like scientific notation
2375 except the exponent (the power-of-ten part) is always adjusted to be
2376 a multiple of three (as in ``kilo,'' ``micro,'' etc.). As a result
2377 there will be one, two, or three digits before the decimal point.
2379 Whenever you change a display-related mode, Calc redraws everything
2380 in the stack. This may be slow if there are many things on the stack,
2381 so Calc allows you to type shift-@kbd{H} before any mode command to
2382 prevent it from updating the stack. Anything Calc displays after the
2383 mode-changing command will appear in the new format.
2387 4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
2388 3: 12345.000 3: 12345.000 3: 12345.000 3: 1.2345e4 3: 12345.
2389 2: 123.450 2: 123.450 2: 1.2345e1 2: 1.2345e1 2: 123.45
2390 1: 12.345 1: 1.2345e1 1: 1.2345e2 1: 1.2345e2 1: 12.345
2393 H d s @key{DEL} U @key{TAB} d @key{SPC} d n
2398 Here the @kbd{H d s} command changes to scientific notation but without
2399 updating the screen. Deleting the top stack entry and undoing it back
2400 causes it to show up in the new format; swapping the top two stack
2401 entries reformats both entries. The @kbd{d @key{SPC}} command refreshes the
2402 whole stack. The @kbd{d n} command changes back to the normal float
2403 format; since it doesn't have an @kbd{H} prefix, it also updates all
2404 the stack entries to be in @kbd{d n} format.
2406 Notice that the integer @expr{12345} was not affected by any
2407 of the float formats. Integers are integers, and are always
2410 @cindex Large numbers, readability
2411 Large integers have their own problems. Let's look back at
2412 the result of @kbd{2^3^4}.
2415 2417851639229258349412352
2419 Quick---how many digits does this have? Try typing @kbd{d g}:
2422 2,417,851,639,229,258,349,412,352
2426 Now how many digits does this have? It's much easier to tell!
2427 We can actually group digits into clumps of any size. Some
2428 people prefer @kbd{M-5 d g}:
2431 24178,51639,22925,83494,12352
2434 Let's see what happens to floating-point numbers when they are grouped.
2435 First, type @kbd{p 25 @key{RET}} to make sure we have enough precision
2436 to get ourselves into trouble. Now, type @kbd{1e13 /}:
2439 24,17851,63922.9258349412352
2443 The integer part is grouped but the fractional part isn't. Now try
2444 @kbd{M-- M-5 d g} (that's meta-minus-sign, meta-five):
2447 24,17851,63922.92583,49412,352
2450 If you find it hard to tell the decimal point from the commas, try
2451 changing the grouping character to a space with @kbd{d , @key{SPC}}:
2454 24 17851 63922.92583 49412 352
2457 Type @kbd{d , ,} to restore the normal grouping character, then
2458 @kbd{d g} again to turn grouping off. Also, press @kbd{p 12} to
2459 restore the default precision.
2461 Press @kbd{U} enough times to get the original big integer back.
2462 (Notice that @kbd{U} does not undo each mode-setting command; if
2463 you want to undo a mode-setting command, you have to do it yourself.)
2464 Now, type @kbd{d r 16 @key{RET}}:
2467 16#200000000000000000000
2471 The number is now displayed in @dfn{hexadecimal}, or ``base-16'' form.
2472 Suddenly it looks pretty simple; this should be no surprise, since we
2473 got this number by computing a power of two, and 16 is a power of 2.
2474 In fact, we can use @w{@kbd{d r 2 @key{RET}}} to see it in actual binary
2478 2#1000000000000000000000000000000000000000000000000000000 @dots{}
2482 We don't have enough space here to show all the zeros! They won't
2483 fit on a typical screen, either, so you will have to use horizontal
2484 scrolling to see them all. Press @kbd{<} and @kbd{>} to scroll the
2485 stack window left and right by half its width. Another way to view
2486 something large is to press @kbd{`} (back-quote) to edit the top of
2487 stack in a separate window. (Press @kbd{C-c C-c} when you are done.)
2489 You can enter non-decimal numbers using the @kbd{#} symbol, too.
2490 Let's see what the hexadecimal number @samp{5FE} looks like in
2491 binary. Type @kbd{16#5FE} (the letters can be typed in upper or
2492 lower case; they will always appear in upper case). It will also
2493 help to turn grouping on with @kbd{d g}:
2499 Notice that @kbd{d g} groups by fours by default if the display radix
2500 is binary or hexadecimal, but by threes if it is decimal, octal, or any
2503 Now let's see that number in decimal; type @kbd{d r 10}:
2509 Numbers are not @emph{stored} with any particular radix attached. They're
2510 just numbers; they can be entered in any radix, and are always displayed
2511 in whatever radix you've chosen with @kbd{d r}. The current radix applies
2512 to integers, fractions, and floats.
2514 @cindex Roundoff errors, in non-decimal numbers
2515 (@bullet{}) @strong{Exercise 1.} Your friend Joe tried to enter one-third
2516 as @samp{3#0.1} in @kbd{d r 3} mode with a precision of 12. He got
2517 @samp{3#0.0222222...} (with 25 2's) in the display. When he multiplied
2518 that by three, he got @samp{3#0.222222...} instead of the expected
2519 @samp{3#1}. Next, Joe entered @samp{3#0.2} and, to his great relief,
2520 saw @samp{3#0.2} on the screen. But when he typed @kbd{2 /}, he got
2521 @samp{3#0.10000001} (some zeros omitted). What's going on here?
2522 @xref{Modes Answer 1, 1}. (@bullet{})
2524 @cindex Scientific notation, in non-decimal numbers
2525 (@bullet{}) @strong{Exercise 2.} Scientific notation works in non-decimal
2526 modes in the natural way (the exponent is a power of the radix instead of
2527 a power of ten, although the exponent itself is always written in decimal).
2528 Thus @samp{8#1.23e3 = 8#1230.0}. Suppose we have the hexadecimal number
2529 @samp{f.e8f} times 16 to the 15th power: We write @samp{16#f.e8fe15}.
2530 What is wrong with this picture? What could we write instead that would
2531 work better? @xref{Modes Answer 2, 2}. (@bullet{})
2533 The @kbd{m} prefix key has another set of modes, relating to the way
2534 Calc interprets your inputs and does computations. Whereas @kbd{d}-prefix
2535 modes generally affect the way things look, @kbd{m}-prefix modes affect
2536 the way they are actually computed.
2538 The most popular @kbd{m}-prefix mode is the @dfn{angular mode}. Notice
2539 the @samp{Deg} indicator in the mode line. This means that if you use
2540 a command that interprets a number as an angle, it will assume the
2541 angle is measured in degrees. For example,
2545 1: 45 1: 0.707106781187 1: 0.500000000001 1: 0.5
2553 The shift-@kbd{S} command computes the sine of an angle. The sine
2555 @texline @math{\sqrt{2}/2};
2556 @infoline @expr{sqrt(2)/2};
2557 squaring this yields @expr{2/4 = 0.5}. However, there has been a slight
2558 roundoff error because the representation of
2559 @texline @math{\sqrt{2}/2}
2560 @infoline @expr{sqrt(2)/2}
2561 wasn't exact. The @kbd{c 1} command is a handy way to clean up numbers
2562 in this case; it temporarily reduces the precision by one digit while it
2563 re-rounds the number on the top of the stack.
2565 @cindex Roundoff errors, examples
2566 (@bullet{}) @strong{Exercise 3.} Your friend Joe computed the sine
2567 of 45 degrees as shown above, then, hoping to avoid an inexact
2568 result, he increased the precision to 16 digits before squaring.
2569 What happened? @xref{Modes Answer 3, 3}. (@bullet{})
2571 To do this calculation in radians, we would type @kbd{m r} first.
2572 (The indicator changes to @samp{Rad}.) 45 degrees corresponds to
2573 @cpiover{4} radians. To get @cpi{}, press the @kbd{P} key. (Once
2574 again, this is a shifted capital @kbd{P}. Remember, unshifted
2575 @kbd{p} sets the precision.)
2579 1: 3.14159265359 1: 0.785398163398 1: 0.707106781187
2586 Likewise, inverse trigonometric functions generate results in
2587 either radians or degrees, depending on the current angular mode.
2591 1: 0.707106781187 1: 0.785398163398 1: 45.
2594 .5 Q m r I S m d U I S
2599 Here we compute the Inverse Sine of
2600 @texline @math{\sqrt{0.5}},
2601 @infoline @expr{sqrt(0.5)},
2602 first in radians, then in degrees.
2604 Use @kbd{c d} and @kbd{c r} to convert a number from radians to degrees
2609 1: 45 1: 0.785398163397 1: 45.
2616 Another interesting mode is @dfn{Fraction mode}. Normally,
2617 dividing two integers produces a floating-point result if the
2618 quotient can't be expressed as an exact integer. Fraction mode
2619 causes integer division to produce a fraction, i.e., a rational
2624 2: 12 1: 1.33333333333 1: 4:3
2628 12 @key{RET} 9 / m f U / m f
2633 In the first case, we get an approximate floating-point result.
2634 In the second case, we get an exact fractional result (four-thirds).
2636 You can enter a fraction at any time using @kbd{:} notation.
2637 (Calc uses @kbd{:} instead of @kbd{/} as the fraction separator
2638 because @kbd{/} is already used to divide the top two stack
2639 elements.) Calculations involving fractions will always
2640 produce exact fractional results; Fraction mode only says
2641 what to do when dividing two integers.
2643 @cindex Fractions vs. floats
2644 @cindex Floats vs. fractions
2645 (@bullet{}) @strong{Exercise 4.} If fractional arithmetic is exact,
2646 why would you ever use floating-point numbers instead?
2647 @xref{Modes Answer 4, 4}. (@bullet{})
2649 Typing @kbd{m f} doesn't change any existing values in the stack.
2650 In the above example, we had to Undo the division and do it over
2651 again when we changed to Fraction mode. But if you use the
2652 evaluates-to operator you can get commands like @kbd{m f} to
2657 1: 12 / 9 => 1.33333333333 1: 12 / 9 => 1.333 1: 12 / 9 => 4:3
2660 ' 12/9 => @key{RET} p 4 @key{RET} m f
2665 In this example, the righthand side of the @samp{=>} operator
2666 on the stack is recomputed when we change the precision, then
2667 again when we change to Fraction mode. All @samp{=>} expressions
2668 on the stack are recomputed every time you change any mode that
2669 might affect their values.
2671 @node Arithmetic Tutorial, Vector/Matrix Tutorial, Basic Tutorial, Tutorial
2672 @section Arithmetic Tutorial
2675 In this section, we explore the arithmetic and scientific functions
2676 available in the Calculator.
2678 The standard arithmetic commands are @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/},
2679 and @kbd{^}. Each normally takes two numbers from the top of the stack
2680 and pushes back a result. The @kbd{n} and @kbd{&} keys perform
2681 change-sign and reciprocal operations, respectively.
2685 1: 5 1: 0.2 1: 5. 1: -5. 1: 5.
2692 @cindex Binary operators
2693 You can apply a ``binary operator'' like @kbd{+} across any number of
2694 stack entries by giving it a numeric prefix. You can also apply it
2695 pairwise to several stack elements along with the top one if you use
2700 3: 2 1: 9 3: 2 4: 2 3: 12
2701 2: 3 . 2: 3 3: 3 2: 13
2702 1: 4 1: 4 2: 4 1: 14
2706 2 @key{RET} 3 @key{RET} 4 M-3 + U 10 M-- M-3 +
2710 @cindex Unary operators
2711 You can apply a ``unary operator'' like @kbd{&} to the top @var{n}
2712 stack entries with a numeric prefix, too.
2717 2: 3 2: 0.333333333333 2: 3.
2721 2 @key{RET} 3 @key{RET} 4 M-3 & M-2 &
2725 Notice that the results here are left in floating-point form.
2726 We can convert them back to integers by pressing @kbd{F}, the
2727 ``floor'' function. This function rounds down to the next lower
2728 integer. There is also @kbd{R}, which rounds to the nearest
2746 Since dividing-and-flooring (i.e., ``integer quotient'') is such a
2747 common operation, Calc provides a special command for that purpose, the
2748 backslash @kbd{\}. Another common arithmetic operator is @kbd{%}, which
2749 computes the remainder that would arise from a @kbd{\} operation, i.e.,
2750 the ``modulo'' of two numbers. For example,
2754 2: 1234 1: 12 2: 1234 1: 34
2758 1234 @key{RET} 100 \ U %
2762 These commands actually work for any real numbers, not just integers.
2766 2: 3.1415 1: 3 2: 3.1415 1: 0.1415
2770 3.1415 @key{RET} 1 \ U %
2774 (@bullet{}) @strong{Exercise 1.} The @kbd{\} command would appear to be a
2775 frill, since you could always do the same thing with @kbd{/ F}. Think
2776 of a situation where this is not true---@kbd{/ F} would be inadequate.
2777 Now think of a way you could get around the problem if Calc didn't
2778 provide a @kbd{\} command. @xref{Arithmetic Answer 1, 1}. (@bullet{})
2780 We've already seen the @kbd{Q} (square root) and @kbd{S} (sine)
2781 commands. Other commands along those lines are @kbd{C} (cosine),
2782 @kbd{T} (tangent), @kbd{E} (@expr{e^x}) and @kbd{L} (natural
2783 logarithm). These can be modified by the @kbd{I} (inverse) and
2784 @kbd{H} (hyperbolic) prefix keys.
2786 Let's compute the sine and cosine of an angle, and verify the
2788 @texline @math{\sin^2x + \cos^2x = 1}.
2789 @infoline @expr{sin(x)^2 + cos(x)^2 = 1}.
2790 We'll arbitrarily pick @mathit{-64} degrees as a good value for @expr{x}.
2791 With the angular mode set to degrees (type @w{@kbd{m d}}), do:
2795 2: -64 2: -64 2: -0.89879 2: -0.89879 1: 1.
2796 1: -64 1: -0.89879 1: -64 1: 0.43837 .
2799 64 n @key{RET} @key{RET} S @key{TAB} C f h
2804 (For brevity, we're showing only five digits of the results here.
2805 You can of course do these calculations to any precision you like.)
2807 Remember, @kbd{f h} is the @code{calc-hypot}, or square-root of sum
2808 of squares, command.
2811 @texline @math{\displaystyle\tan x = {\sin x \over \cos x}}.
2812 @infoline @expr{tan(x) = sin(x) / cos(x)}.
2816 2: -0.89879 1: -2.0503 1: -64.
2824 A physical interpretation of this calculation is that if you move
2825 @expr{0.89879} units downward and @expr{0.43837} units to the right,
2826 your direction of motion is @mathit{-64} degrees from horizontal. Suppose
2827 we move in the opposite direction, up and to the left:
2831 2: -0.89879 2: 0.89879 1: -2.0503 1: -64.
2832 1: 0.43837 1: -0.43837 . .
2840 How can the angle be the same? The answer is that the @kbd{/} operation
2841 loses information about the signs of its inputs. Because the quotient
2842 is negative, we know exactly one of the inputs was negative, but we
2843 can't tell which one. There is an @kbd{f T} [@code{arctan2}] function which
2844 computes the inverse tangent of the quotient of a pair of numbers.
2845 Since you feed it the two original numbers, it has enough information
2846 to give you a full 360-degree answer.
2850 2: 0.89879 1: 116. 3: 116. 2: 116. 1: 180.
2851 1: -0.43837 . 2: -0.89879 1: -64. .
2855 U U f T M-@key{RET} M-2 n f T -
2860 The resulting angles differ by 180 degrees; in other words, they
2861 point in opposite directions, just as we would expect.
2863 The @key{META}-@key{RET} we used in the third step is the
2864 ``last-arguments'' command. It is sort of like Undo, except that it
2865 restores the arguments of the last command to the stack without removing
2866 the command's result. It is useful in situations like this one,
2867 where we need to do several operations on the same inputs. We could
2868 have accomplished the same thing by using @kbd{M-2 @key{RET}} to duplicate
2869 the top two stack elements right after the @kbd{U U}, then a pair of
2870 @kbd{M-@key{TAB}} commands to cycle the 116 up around the duplicates.
2872 A similar identity is supposed to hold for hyperbolic sines and cosines,
2873 except that it is the @emph{difference}
2874 @texline @math{\cosh^2x - \sinh^2x}
2875 @infoline @expr{cosh(x)^2 - sinh(x)^2}
2876 that always equals one. Let's try to verify this identity.
2880 2: -64 2: -64 2: -64 2: 9.7192e54 2: 9.7192e54
2881 1: -64 1: -3.1175e27 1: 9.7192e54 1: -64 1: 9.7192e54
2884 64 n @key{RET} @key{RET} H C 2 ^ @key{TAB} H S 2 ^
2889 @cindex Roundoff errors, examples
2890 Something's obviously wrong, because when we subtract these numbers
2891 the answer will clearly be zero! But if you think about it, if these
2892 numbers @emph{did} differ by one, it would be in the 55th decimal
2893 place. The difference we seek has been lost entirely to roundoff
2896 We could verify this hypothesis by doing the actual calculation with,
2897 say, 60 decimal places of precision. This will be slow, but not
2898 enormously so. Try it if you wish; sure enough, the answer is
2899 0.99999, reasonably close to 1.
2901 Of course, a more reasonable way to verify the identity is to use
2902 a more reasonable value for @expr{x}!
2904 @cindex Common logarithm
2905 Some Calculator commands use the Hyperbolic prefix for other purposes.
2906 The logarithm and exponential functions, for example, work to the base
2907 @expr{e} normally but use base-10 instead if you use the Hyperbolic
2912 1: 1000 1: 6.9077 1: 1000 1: 3
2920 First, we mistakenly compute a natural logarithm. Then we undo
2921 and compute a common logarithm instead.
2923 The @kbd{B} key computes a general base-@var{b} logarithm for any
2928 2: 1000 1: 3 1: 1000. 2: 1000. 1: 6.9077
2929 1: 10 . . 1: 2.71828 .
2932 1000 @key{RET} 10 B H E H P B
2937 Here we first use @kbd{B} to compute the base-10 logarithm, then use
2938 the ``hyperbolic'' exponential as a cheap hack to recover the number
2939 1000, then use @kbd{B} again to compute the natural logarithm. Note
2940 that @kbd{P} with the hyperbolic prefix pushes the constant @expr{e}
2943 You may have noticed that both times we took the base-10 logarithm
2944 of 1000, we got an exact integer result. Calc always tries to give
2945 an exact rational result for calculations involving rational numbers
2946 where possible. But when we used @kbd{H E}, the result was a
2947 floating-point number for no apparent reason. In fact, if we had
2948 computed @kbd{10 @key{RET} 3 ^} we @emph{would} have gotten an
2949 exact integer 1000. But the @kbd{H E} command is rigged to generate
2950 a floating-point result all of the time so that @kbd{1000 H E} will
2951 not waste time computing a thousand-digit integer when all you
2952 probably wanted was @samp{1e1000}.
2954 (@bullet{}) @strong{Exercise 2.} Find a pair of integer inputs to
2955 the @kbd{B} command for which Calc could find an exact rational
2956 result but doesn't. @xref{Arithmetic Answer 2, 2}. (@bullet{})
2958 The Calculator also has a set of functions relating to combinatorics
2959 and statistics. You may be familiar with the @dfn{factorial} function,
2960 which computes the product of all the integers up to a given number.
2964 1: 100 1: 93326215443... 1: 100. 1: 9.3326e157
2972 Recall, the @kbd{c f} command converts the integer or fraction at the
2973 top of the stack to floating-point format. If you take the factorial
2974 of a floating-point number, you get a floating-point result
2975 accurate to the current precision. But if you give @kbd{!} an
2976 exact integer, you get an exact integer result (158 digits long
2979 If you take the factorial of a non-integer, Calc uses a generalized
2980 factorial function defined in terms of Euler's Gamma function
2981 @texline @math{\Gamma(n)}
2982 @infoline @expr{gamma(n)}
2983 (which is itself available as the @kbd{f g} command).
2987 3: 4. 3: 24. 1: 5.5 1: 52.342777847
2988 2: 4.5 2: 52.3427777847 . .
2992 M-3 ! M-0 @key{DEL} 5.5 f g
2997 Here we verify the identity
2998 @texline @math{n! = \Gamma(n+1)}.
2999 @infoline @expr{@var{n}!@: = gamma(@var{n}+1)}.
3001 The binomial coefficient @var{n}-choose-@var{m}
3002 @texline or @math{\displaystyle {n \choose m}}
3004 @texline @math{\displaystyle {n! \over m! \, (n-m)!}}
3005 @infoline @expr{n!@: / m!@: (n-m)!}
3006 for all reals @expr{n} and @expr{m}. The intermediate results in this
3007 formula can become quite large even if the final result is small; the
3008 @kbd{k c} command computes a binomial coefficient in a way that avoids
3009 large intermediate values.
3011 The @kbd{k} prefix key defines several common functions out of
3012 combinatorics and number theory. Here we compute the binomial
3013 coefficient 30-choose-20, then determine its prime factorization.
3017 2: 30 1: 30045015 1: [3, 3, 5, 7, 11, 13, 23, 29]
3021 30 @key{RET} 20 k c k f
3026 You can verify these prime factors by using @kbd{V R *} to multiply
3027 together the elements of this vector. The result is the original
3031 Suppose a program you are writing needs a hash table with at least
3032 10000 entries. It's best to use a prime number as the actual size
3033 of a hash table. Calc can compute the next prime number after 10000:
3037 1: 10000 1: 10007 1: 9973
3045 Just for kicks we've also computed the next prime @emph{less} than
3048 @c [fix-ref Financial Functions]
3049 @xref{Financial Functions}, for a description of the Calculator
3050 commands that deal with business and financial calculations (functions
3051 like @code{pv}, @code{rate}, and @code{sln}).
3053 @c [fix-ref Binary Number Functions]
3054 @xref{Binary Functions}, to read about the commands for operating
3055 on binary numbers (like @code{and}, @code{xor}, and @code{lsh}).
3057 @node Vector/Matrix Tutorial, Types Tutorial, Arithmetic Tutorial, Tutorial
3058 @section Vector/Matrix Tutorial
3061 A @dfn{vector} is a list of numbers or other Calc data objects.
3062 Calc provides a large set of commands that operate on vectors. Some
3063 are familiar operations from vector analysis. Others simply treat
3064 a vector as a list of objects.
3067 * Vector Analysis Tutorial::
3072 @node Vector Analysis Tutorial, Matrix Tutorial, Vector/Matrix Tutorial, Vector/Matrix Tutorial
3073 @subsection Vector Analysis
3076 If you add two vectors, the result is a vector of the sums of the
3077 elements, taken pairwise.
3081 1: [1, 2, 3] 2: [1, 2, 3] 1: [8, 8, 3]
3085 [1,2,3] s 1 [7 6 0] s 2 +
3090 Note that we can separate the vector elements with either commas or
3091 spaces. This is true whether we are using incomplete vectors or
3092 algebraic entry. The @kbd{s 1} and @kbd{s 2} commands save these
3093 vectors so we can easily reuse them later.
3095 If you multiply two vectors, the result is the sum of the products
3096 of the elements taken pairwise. This is called the @dfn{dot product}
3110 The dot product of two vectors is equal to the product of their
3111 lengths times the cosine of the angle between them. (Here the vector
3112 is interpreted as a line from the origin @expr{(0,0,0)} to the
3113 specified point in three-dimensional space.) The @kbd{A}
3114 (absolute value) command can be used to compute the length of a
3119 3: 19 3: 19 1: 0.550782 1: 56.579
3120 2: [1, 2, 3] 2: 3.741657 . .
3121 1: [7, 6, 0] 1: 9.219544
3124 M-@key{RET} M-2 A * / I C
3129 First we recall the arguments to the dot product command, then
3130 we compute the absolute values of the top two stack entries to
3131 obtain the lengths of the vectors, then we divide the dot product
3132 by the product of the lengths to get the cosine of the angle.
3133 The inverse cosine finds that the angle between the vectors
3134 is about 56 degrees.
3136 @cindex Cross product
3137 @cindex Perpendicular vectors
3138 The @dfn{cross product} of two vectors is a vector whose length
3139 is the product of the lengths of the inputs times the sine of the
3140 angle between them, and whose direction is perpendicular to both
3141 input vectors. Unlike the dot product, the cross product is
3142 defined only for three-dimensional vectors. Let's double-check
3143 our computation of the angle using the cross product.
3147 2: [1, 2, 3] 3: [-18, 21, -8] 1: [-0.52, 0.61, -0.23] 1: 56.579
3148 1: [7, 6, 0] 2: [1, 2, 3] . .
3152 r 1 r 2 V C s 3 M-@key{RET} M-2 A * / A I S
3157 First we recall the original vectors and compute their cross product,
3158 which we also store for later reference. Now we divide the vector
3159 by the product of the lengths of the original vectors. The length of
3160 this vector should be the sine of the angle; sure enough, it is!
3162 @c [fix-ref General Mode Commands]
3163 Vector-related commands generally begin with the @kbd{v} prefix key.
3164 Some are uppercase letters and some are lowercase. To make it easier
3165 to type these commands, the shift-@kbd{V} prefix key acts the same as
3166 the @kbd{v} key. (@xref{General Mode Commands}, for a way to make all
3167 prefix keys have this property.)
3169 If we take the dot product of two perpendicular vectors we expect
3170 to get zero, since the cosine of 90 degrees is zero. Let's check
3171 that the cross product is indeed perpendicular to both inputs:
3175 2: [1, 2, 3] 1: 0 2: [7, 6, 0] 1: 0
3176 1: [-18, 21, -8] . 1: [-18, 21, -8] .
3179 r 1 r 3 * @key{DEL} r 2 r 3 *
3183 @cindex Normalizing a vector
3184 @cindex Unit vectors
3185 (@bullet{}) @strong{Exercise 1.} Given a vector on the top of the
3186 stack, what keystrokes would you use to @dfn{normalize} the
3187 vector, i.e., to reduce its length to one without changing its
3188 direction? @xref{Vector Answer 1, 1}. (@bullet{})
3190 (@bullet{}) @strong{Exercise 2.} Suppose a certain particle can be
3191 at any of several positions along a ruler. You have a list of
3192 those positions in the form of a vector, and another list of the
3193 probabilities for the particle to be at the corresponding positions.
3194 Find the average position of the particle.
3195 @xref{Vector Answer 2, 2}. (@bullet{})
3197 @node Matrix Tutorial, List Tutorial, Vector Analysis Tutorial, Vector/Matrix Tutorial
3198 @subsection Matrices
3201 A @dfn{matrix} is just a vector of vectors, all the same length.
3202 This means you can enter a matrix using nested brackets. You can
3203 also use the semicolon character to enter a matrix. We'll show
3208 1: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ]
3209 [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
3212 [[1 2 3] [4 5 6]] ' [1 2 3; 4 5 6] @key{RET}
3217 We'll be using this matrix again, so type @kbd{s 4} to save it now.
3219 Note that semicolons work with incomplete vectors, but they work
3220 better in algebraic entry. That's why we use the apostrophe in
3223 When two matrices are multiplied, the lefthand matrix must have
3224 the same number of columns as the righthand matrix has rows.
3225 Row @expr{i}, column @expr{j} of the result is effectively the
3226 dot product of row @expr{i} of the left matrix by column @expr{j}
3227 of the right matrix.
3229 If we try to duplicate this matrix and multiply it by itself,
3230 the dimensions are wrong and the multiplication cannot take place:
3234 1: [ [ 1, 2, 3 ] * [ [ 1, 2, 3 ]
3235 [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
3243 Though rather hard to read, this is a formula which shows the product
3244 of two matrices. The @samp{*} function, having invalid arguments, has
3245 been left in symbolic form.
3247 We can multiply the matrices if we @dfn{transpose} one of them first.
3251 2: [ [ 1, 2, 3 ] 1: [ [ 14, 32 ] 1: [ [ 17, 22, 27 ]
3252 [ 4, 5, 6 ] ] [ 32, 77 ] ] [ 22, 29, 36 ]
3253 1: [ [ 1, 4 ] . [ 27, 36, 45 ] ]
3258 U v t * U @key{TAB} *
3262 Matrix multiplication is not commutative; indeed, switching the
3263 order of the operands can even change the dimensions of the result
3264 matrix, as happened here!
3266 If you multiply a plain vector by a matrix, it is treated as a
3267 single row or column depending on which side of the matrix it is
3268 on. The result is a plain vector which should also be interpreted
3269 as a row or column as appropriate.
3273 2: [ [ 1, 2, 3 ] 1: [14, 32]
3282 Multiplying in the other order wouldn't work because the number of
3283 rows in the matrix is different from the number of elements in the
3286 (@bullet{}) @strong{Exercise 1.} Use @samp{*} to sum along the rows
3288 @texline @math{2\times3}
3290 matrix to get @expr{[6, 15]}. Now use @samp{*} to sum along the columns
3291 to get @expr{[5, 7, 9]}.
3292 @xref{Matrix Answer 1, 1}. (@bullet{})
3294 @cindex Identity matrix
3295 An @dfn{identity matrix} is a square matrix with ones along the
3296 diagonal and zeros elsewhere. It has the property that multiplication
3297 by an identity matrix, on the left or on the right, always produces
3298 the original matrix.
3302 1: [ [ 1, 2, 3 ] 2: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ]
3303 [ 4, 5, 6 ] ] [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
3304 . 1: [ [ 1, 0, 0 ] .
3309 r 4 v i 3 @key{RET} *
3313 If a matrix is square, it is often possible to find its @dfn{inverse},
3314 that is, a matrix which, when multiplied by the original matrix, yields
3315 an identity matrix. The @kbd{&} (reciprocal) key also computes the
3316 inverse of a matrix.
3320 1: [ [ 1, 2, 3 ] 1: [ [ -2.4, 1.2, -0.2 ]
3321 [ 4, 5, 6 ] [ 2.8, -1.4, 0.4 ]
3322 [ 7, 6, 0 ] ] [ -0.73333, 0.53333, -0.2 ] ]
3330 The vertical bar @kbd{|} @dfn{concatenates} numbers, vectors, and
3331 matrices together. Here we have used it to add a new row onto
3332 our matrix to make it square.
3334 We can multiply these two matrices in either order to get an identity.
3338 1: [ [ 1., 0., 0. ] 1: [ [ 1., 0., 0. ]
3339 [ 0., 1., 0. ] [ 0., 1., 0. ]
3340 [ 0., 0., 1. ] ] [ 0., 0., 1. ] ]
3343 M-@key{RET} * U @key{TAB} *
3347 @cindex Systems of linear equations
3348 @cindex Linear equations, systems of
3349 Matrix inverses are related to systems of linear equations in algebra.
3350 Suppose we had the following set of equations:
3363 $$ \openup1\jot \tabskip=0pt plus1fil
3364 \halign to\displaywidth{\tabskip=0pt
3365 $\hfil#$&$\hfil{}#{}$&
3366 $\hfil#$&$\hfil{}#{}$&
3367 $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
3376 This can be cast into the matrix equation,
3381 [ [ 1, 2, 3 ] [ [ a ] [ [ 6 ]
3382 [ 4, 5, 6 ] * [ b ] = [ 2 ]
3383 [ 7, 6, 0 ] ] [ c ] ] [ 3 ] ]
3389 $$ \pmatrix{ 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 6 & 0 }
3391 \pmatrix{ a \cr b \cr c } = \pmatrix{ 6 \cr 2 \cr 3 }
3396 We can solve this system of equations by multiplying both sides by the
3397 inverse of the matrix. Calc can do this all in one step:
3401 2: [6, 2, 3] 1: [-12.6, 15.2, -3.93333]
3412 The result is the @expr{[a, b, c]} vector that solves the equations.
3413 (Dividing by a square matrix is equivalent to multiplying by its
3416 Let's verify this solution:
3420 2: [ [ 1, 2, 3 ] 1: [6., 2., 3.]
3423 1: [-12.6, 15.2, -3.93333]
3431 Note that we had to be careful about the order in which we multiplied
3432 the matrix and vector. If we multiplied in the other order, Calc would
3433 assume the vector was a row vector in order to make the dimensions
3434 come out right, and the answer would be incorrect. If you
3435 don't feel safe letting Calc take either interpretation of your
3436 vectors, use explicit
3437 @texline @math{N\times1}
3440 @texline @math{1\times N}
3442 matrices instead. In this case, you would enter the original column
3443 vector as @samp{[[6], [2], [3]]} or @samp{[6; 2; 3]}.
3445 (@bullet{}) @strong{Exercise 2.} Algebraic entry allows you to make
3446 vectors and matrices that include variables. Solve the following
3447 system of equations to get expressions for @expr{x} and @expr{y}
3448 in terms of @expr{a} and @expr{b}.
3460 $$ \eqalign{ x &+ a y = 6 \cr
3467 @xref{Matrix Answer 2, 2}. (@bullet{})
3469 @cindex Least-squares for over-determined systems
3470 @cindex Over-determined systems of equations
3471 (@bullet{}) @strong{Exercise 3.} A system of equations is ``over-determined''
3472 if it has more equations than variables. It is often the case that
3473 there are no values for the variables that will satisfy all the
3474 equations at once, but it is still useful to find a set of values
3475 which ``nearly'' satisfy all the equations. In terms of matrix equations,
3476 you can't solve @expr{A X = B} directly because the matrix @expr{A}
3477 is not square for an over-determined system. Matrix inversion works
3478 only for square matrices. One common trick is to multiply both sides
3479 on the left by the transpose of @expr{A}:
3481 @samp{trn(A)*A*X = trn(A)*B}.
3484 $A^T A \, X = A^T B$, where $A^T$ is the transpose \samp{trn(A)}.
3487 @texline @math{A^T A}
3488 @infoline @expr{trn(A)*A}
3489 is a square matrix so a solution is possible. It turns out that the
3490 @expr{X} vector you compute in this way will be a ``least-squares''
3491 solution, which can be regarded as the ``closest'' solution to the set
3492 of equations. Use Calc to solve the following over-determined
3507 $$ \openup1\jot \tabskip=0pt plus1fil
3508 \halign to\displaywidth{\tabskip=0pt
3509 $\hfil#$&$\hfil{}#{}$&
3510 $\hfil#$&$\hfil{}#{}$&
3511 $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
3515 2a&+&4b&+&6c&=11 \cr}
3521 @xref{Matrix Answer 3, 3}. (@bullet{})
3523 @node List Tutorial, , Matrix Tutorial, Vector/Matrix Tutorial
3524 @subsection Vectors as Lists
3528 Although Calc has a number of features for manipulating vectors and
3529 matrices as mathematical objects, you can also treat vectors as
3530 simple lists of values. For example, we saw that the @kbd{k f}
3531 command returns a vector which is a list of the prime factors of a
3534 You can pack and unpack stack entries into vectors:
3538 3: 10 1: [10, 20, 30] 3: 10
3547 You can also build vectors out of consecutive integers, or out
3548 of many copies of a given value:
3552 1: [1, 2, 3, 4] 2: [1, 2, 3, 4] 2: [1, 2, 3, 4]
3553 . 1: 17 1: [17, 17, 17, 17]
3556 v x 4 @key{RET} 17 v b 4 @key{RET}
3560 You can apply an operator to every element of a vector using the
3565 1: [17, 34, 51, 68] 1: [289, 1156, 2601, 4624] 1: [17, 34, 51, 68]
3573 In the first step, we multiply the vector of integers by the vector
3574 of 17's elementwise. In the second step, we raise each element to
3575 the power two. (The general rule is that both operands must be
3576 vectors of the same length, or else one must be a vector and the
3577 other a plain number.) In the final step, we take the square root
3580 (@bullet{}) @strong{Exercise 1.} Compute a vector of powers of two
3582 @texline @math{2^{-4}}
3583 @infoline @expr{2^-4}
3584 to @expr{2^4}. @xref{List Answer 1, 1}. (@bullet{})
3586 You can also @dfn{reduce} a binary operator across a vector.
3587 For example, reducing @samp{*} computes the product of all the
3588 elements in the vector:
3592 1: 123123 1: [3, 7, 11, 13, 41] 1: 123123
3600 In this example, we decompose 123123 into its prime factors, then
3601 multiply those factors together again to yield the original number.
3603 We could compute a dot product ``by hand'' using mapping and
3608 2: [1, 2, 3] 1: [7, 12, 0] 1: 19
3617 Recalling two vectors from the previous section, we compute the
3618 sum of pairwise products of the elements to get the same answer
3619 for the dot product as before.
3621 A slight variant of vector reduction is the @dfn{accumulate} operation,
3622 @kbd{V U}. This produces a vector of the intermediate results from
3623 a corresponding reduction. Here we compute a table of factorials:
3627 1: [1, 2, 3, 4, 5, 6] 1: [1, 2, 6, 24, 120, 720]
3630 v x 6 @key{RET} V U *
3634 Calc allows vectors to grow as large as you like, although it gets
3635 rather slow if vectors have more than about a hundred elements.
3636 Actually, most of the time is spent formatting these large vectors
3637 for display, not calculating on them. Try the following experiment
3638 (if your computer is very fast you may need to substitute a larger
3643 1: [1, 2, 3, 4, ... 1: [2, 3, 4, 5, ...
3646 v x 500 @key{RET} 1 V M +
3650 Now press @kbd{v .} (the letter @kbd{v}, then a period) and try the
3651 experiment again. In @kbd{v .} mode, long vectors are displayed
3652 ``abbreviated'' like this:
3656 1: [1, 2, 3, ..., 500] 1: [2, 3, 4, ..., 501]
3659 v x 500 @key{RET} 1 V M +
3664 (where now the @samp{...} is actually part of the Calc display).
3665 You will find both operations are now much faster. But notice that
3666 even in @w{@kbd{v .}} mode, the full vectors are still shown in the Trail.
3667 Type @w{@kbd{t .}} to cause the trail to abbreviate as well, and try the
3668 experiment one more time. Operations on long vectors are now quite
3669 fast! (But of course if you use @kbd{t .} you will lose the ability
3670 to get old vectors back using the @kbd{t y} command.)
3672 An easy way to view a full vector when @kbd{v .} mode is active is
3673 to press @kbd{`} (back-quote) to edit the vector; editing always works
3674 with the full, unabbreviated value.
3676 @cindex Least-squares for fitting a straight line
3677 @cindex Fitting data to a line
3678 @cindex Line, fitting data to
3679 @cindex Data, extracting from buffers
3680 @cindex Columns of data, extracting
3681 As a larger example, let's try to fit a straight line to some data,
3682 using the method of least squares. (Calc has a built-in command for
3683 least-squares curve fitting, but we'll do it by hand here just to
3684 practice working with vectors.) Suppose we have the following list
3685 of values in a file we have loaded into Emacs:
3712 If you are reading this tutorial in printed form, you will find it
3713 easiest to press @kbd{C-x * i} to enter the on-line Info version of
3714 the manual and find this table there. (Press @kbd{g}, then type
3715 @kbd{List Tutorial}, to jump straight to this section.)
3717 Position the cursor at the upper-left corner of this table, just
3718 to the left of the @expr{1.34}. Press @kbd{C-@@} to set the mark.
3719 (On your system this may be @kbd{C-2}, @kbd{C-@key{SPC}}, or @kbd{NUL}.)
3720 Now position the cursor to the lower-right, just after the @expr{1.354}.
3721 You have now defined this region as an Emacs ``rectangle.'' Still
3722 in the Info buffer, type @kbd{C-x * r}. This command
3723 (@code{calc-grab-rectangle}) will pop you back into the Calculator, with
3724 the contents of the rectangle you specified in the form of a matrix.
3728 1: [ [ 1.34, 0.234 ]
3735 (You may wish to use @kbd{v .} mode to abbreviate the display of this
3738 We want to treat this as a pair of lists. The first step is to
3739 transpose this matrix into a pair of rows. Remember, a matrix is
3740 just a vector of vectors. So we can unpack the matrix into a pair
3741 of row vectors on the stack.
3745 1: [ [ 1.34, 1.41, 1.49, ... ] 2: [1.34, 1.41, 1.49, ... ]
3746 [ 0.234, 0.298, 0.402, ... ] ] 1: [0.234, 0.298, 0.402, ... ]
3754 Let's store these in quick variables 1 and 2, respectively.
3758 1: [1.34, 1.41, 1.49, ... ] .
3766 (Recall that @kbd{t 2} is a variant of @kbd{s 2} that removes the
3767 stored value from the stack.)
3769 In a least squares fit, the slope @expr{m} is given by the formula
3773 m = (N sum(x y) - sum(x) sum(y)) / (N sum(x^2) - sum(x)^2)
3778 $$ m = {N \sum x y - \sum x \sum y \over
3779 N \sum x^2 - \left( \sum x \right)^2} $$
3785 @texline @math{\sum x}
3786 @infoline @expr{sum(x)}
3787 represents the sum of all the values of @expr{x}. While there is an
3788 actual @code{sum} function in Calc, it's easier to sum a vector using a
3789 simple reduction. First, let's compute the four different sums that
3797 r 1 V R + t 3 r 1 2 V M ^ V R + t 4
3804 1: 13.613 1: 33.36554
3807 r 2 V R + t 5 r 1 r 2 V M * V R + t 6
3813 These are @samp{sum(x)}, @samp{sum(x^2)}, @samp{sum(y)}, and @samp{sum(x y)},
3814 respectively. (We could have used @kbd{*} to compute @samp{sum(x^2)} and
3818 These are $\sum x$, $\sum x^2$, $\sum y$, and $\sum x y$,
3819 respectively. (We could have used \kbd{*} to compute $\sum x^2$ and
3823 Finally, we also need @expr{N}, the number of data points. This is just
3824 the length of either of our lists.
3836 (That's @kbd{v} followed by a lower-case @kbd{l}.)
3838 Now we grind through the formula:
3842 1: 633.94526 2: 633.94526 1: 67.23607
3846 r 7 r 6 * r 3 r 5 * -
3853 2: 67.23607 3: 67.23607 2: 67.23607 1: 0.52141679
3854 1: 1862.0057 2: 1862.0057 1: 128.9488 .
3858 r 7 r 4 * r 3 2 ^ - / t 8
3862 That gives us the slope @expr{m}. The y-intercept @expr{b} can now
3863 be found with the simple formula,
3867 b = (sum(y) - m sum(x)) / N
3872 $$ b = {\sum y - m \sum x \over N} $$
3879 1: 13.613 2: 13.613 1: -8.09358 1: -0.425978
3883 r 5 r 8 r 3 * - r 7 / t 9
3887 Let's ``plot'' this straight line approximation,
3888 @texline @math{y \approx m x + b},
3889 @infoline @expr{m x + b},
3890 and compare it with the original data.
3894 1: [0.699, 0.735, ... ] 1: [0.273, 0.309, ... ]
3902 Notice that multiplying a vector by a constant, and adding a constant
3903 to a vector, can be done without mapping commands since these are
3904 common operations from vector algebra. As far as Calc is concerned,
3905 we've just been doing geometry in 19-dimensional space!
3907 We can subtract this vector from our original @expr{y} vector to get
3908 a feel for the error of our fit. Let's find the maximum error:
3912 1: [0.0387, 0.0112, ... ] 1: [0.0387, 0.0112, ... ] 1: 0.0897
3920 First we compute a vector of differences, then we take the absolute
3921 values of these differences, then we reduce the @code{max} function
3922 across the vector. (The @code{max} function is on the two-key sequence
3923 @kbd{f x}; because it is so common to use @code{max} in a vector
3924 operation, the letters @kbd{X} and @kbd{N} are also accepted for
3925 @code{max} and @code{min} in this context. In general, you answer
3926 the @kbd{V M} or @kbd{V R} prompt with the actual key sequence that
3927 invokes the function you want. You could have typed @kbd{V R f x} or
3928 even @kbd{V R x max @key{RET}} if you had preferred.)
3930 If your system has the GNUPLOT program, you can see graphs of your
3931 data and your straight line to see how well they match. (If you have
3932 GNUPLOT 3.0 or higher, the following instructions will work regardless
3933 of the kind of display you have. Some GNUPLOT 2.0, non-X-windows systems
3934 may require additional steps to view the graphs.)
3936 Let's start by plotting the original data. Recall the ``@var{x}'' and ``@var{y}''
3937 vectors onto the stack and press @kbd{g f}. This ``fast'' graphing
3938 command does everything you need to do for simple, straightforward
3943 2: [1.34, 1.41, 1.49, ... ]
3944 1: [0.234, 0.298, 0.402, ... ]
3951 If all goes well, you will shortly get a new window containing a graph
3952 of the data. (If not, contact your GNUPLOT or Calc installer to find
3953 out what went wrong.) In the X window system, this will be a separate
3954 graphics window. For other kinds of displays, the default is to
3955 display the graph in Emacs itself using rough character graphics.
3956 Press @kbd{q} when you are done viewing the character graphics.
3958 Next, let's add the line we got from our least-squares fit.
3960 (If you are reading this tutorial on-line while running Calc, typing
3961 @kbd{g a} may cause the tutorial to disappear from its window and be
3962 replaced by a buffer named @samp{*Gnuplot Commands*}. The tutorial
3963 will reappear when you terminate GNUPLOT by typing @kbd{g q}.)
3968 2: [1.34, 1.41, 1.49, ... ]
3969 1: [0.273, 0.309, 0.351, ... ]
3972 @key{DEL} r 0 g a g p
3976 It's not very useful to get symbols to mark the data points on this
3977 second curve; you can type @kbd{g S g p} to remove them. Type @kbd{g q}
3978 when you are done to remove the X graphics window and terminate GNUPLOT.
3980 (@bullet{}) @strong{Exercise 2.} An earlier exercise showed how to do
3981 least squares fitting to a general system of equations. Our 19 data
3982 points are really 19 equations of the form @expr{y_i = m x_i + b} for
3983 different pairs of @expr{(x_i,y_i)}. Use the matrix-transpose method
3984 to solve for @expr{m} and @expr{b}, duplicating the above result.
3985 @xref{List Answer 2, 2}. (@bullet{})
3987 @cindex Geometric mean
3988 (@bullet{}) @strong{Exercise 3.} If the input data do not form a
3989 rectangle, you can use @w{@kbd{C-x * g}} (@code{calc-grab-region})
3990 to grab the data the way Emacs normally works with regions---it reads
3991 left-to-right, top-to-bottom, treating line breaks the same as spaces.
3992 Use this command to find the geometric mean of the following numbers.
3993 (The geometric mean is the @var{n}th root of the product of @var{n} numbers.)
4002 The @kbd{C-x * g} command accepts numbers separated by spaces or commas,
4003 with or without surrounding vector brackets.
4004 @xref{List Answer 3, 3}. (@bullet{})
4007 As another example, a theorem about binomial coefficients tells
4008 us that the alternating sum of binomial coefficients
4009 @var{n}-choose-0 minus @var{n}-choose-1 plus @var{n}-choose-2, and so
4010 on up to @var{n}-choose-@var{n},
4011 always comes out to zero. Let's verify this
4015 As another example, a theorem about binomial coefficients tells
4016 us that the alternating sum of binomial coefficients
4017 ${n \choose 0} - {n \choose 1} + {n \choose 2} - \cdots \pm {n \choose n}$
4018 always comes out to zero. Let's verify this
4024 1: [1, 2, 3, 4, 5, 6, 7] 1: [0, 1, 2, 3, 4, 5, 6]
4034 1: [1, -6, 15, -20, 15, -6, 1] 1: 0
4037 V M ' (-1)^$ choose(6,$) @key{RET} V R +
4041 The @kbd{V M '} command prompts you to enter any algebraic expression
4042 to define the function to map over the vector. The symbol @samp{$}
4043 inside this expression represents the argument to the function.
4044 The Calculator applies this formula to each element of the vector,
4045 substituting each element's value for the @samp{$} sign(s) in turn.
4047 To define a two-argument function, use @samp{$$} for the first
4048 argument and @samp{$} for the second: @kbd{V M ' $$-$ @key{RET}} is
4049 equivalent to @kbd{V M -}. This is analogous to regular algebraic
4050 entry, where @samp{$$} would refer to the next-to-top stack entry
4051 and @samp{$} would refer to the top stack entry, and @kbd{' $$-$ @key{RET}}
4052 would act exactly like @kbd{-}.
4054 Notice that the @kbd{V M '} command has recorded two things in the
4055 trail: The result, as usual, and also a funny-looking thing marked
4056 @samp{oper} that represents the operator function you typed in.
4057 The function is enclosed in @samp{< >} brackets, and the argument is
4058 denoted by a @samp{#} sign. If there were several arguments, they
4059 would be shown as @samp{#1}, @samp{#2}, and so on. (For example,
4060 @kbd{V M ' $$-$} will put the function @samp{<#1 - #2>} on the
4061 trail.) This object is a ``nameless function''; you can use nameless
4062 @w{@samp{< >}} notation to answer the @kbd{V M '} prompt if you like.
4063 Nameless function notation has the interesting, occasionally useful
4064 property that a nameless function is not actually evaluated until
4065 it is used. For example, @kbd{V M ' $+random(2.0)} evaluates
4066 @samp{random(2.0)} once and adds that random number to all elements
4067 of the vector, but @kbd{V M ' <#+random(2.0)>} evaluates the
4068 @samp{random(2.0)} separately for each vector element.
4070 Another group of operators that are often useful with @kbd{V M} are
4071 the relational operators: @kbd{a =}, for example, compares two numbers
4072 and gives the result 1 if they are equal, or 0 if not. Similarly,
4073 @w{@kbd{a <}} checks for one number being less than another.
4075 Other useful vector operations include @kbd{v v}, to reverse a
4076 vector end-for-end; @kbd{V S}, to sort the elements of a vector
4077 into increasing order; and @kbd{v r} and @w{@kbd{v c}}, to extract
4078 one row or column of a matrix, or (in both cases) to extract one
4079 element of a plain vector. With a negative argument, @kbd{v r}
4080 and @kbd{v c} instead delete one row, column, or vector element.
4082 @cindex Divisor functions
4083 (@bullet{}) @strong{Exercise 4.} The @expr{k}th @dfn{divisor function}
4087 is the sum of the @expr{k}th powers of all the divisors of an
4088 integer @expr{n}. Figure out a method for computing the divisor
4089 function for reasonably small values of @expr{n}. As a test,
4090 the 0th and 1st divisor functions of 30 are 8 and 72, respectively.
4091 @xref{List Answer 4, 4}. (@bullet{})
4093 @cindex Square-free numbers
4094 @cindex Duplicate values in a list
4095 (@bullet{}) @strong{Exercise 5.} The @kbd{k f} command produces a
4096 list of prime factors for a number. Sometimes it is important to
4097 know that a number is @dfn{square-free}, i.e., that no prime occurs
4098 more than once in its list of prime factors. Find a sequence of
4099 keystrokes to tell if a number is square-free; your method should
4100 leave 1 on the stack if it is, or 0 if it isn't.
4101 @xref{List Answer 5, 5}. (@bullet{})
4103 @cindex Triangular lists
4104 (@bullet{}) @strong{Exercise 6.} Build a list of lists that looks
4105 like the following diagram. (You may wish to use the @kbd{v /}
4106 command to enable multi-line display of vectors.)
4115 [1, 2, 3, 4, 5, 6] ]
4120 @xref{List Answer 6, 6}. (@bullet{})
4122 (@bullet{}) @strong{Exercise 7.} Build the following list of lists.
4130 [10, 11, 12, 13, 14],
4131 [15, 16, 17, 18, 19, 20] ]
4136 @xref{List Answer 7, 7}. (@bullet{})
4138 @cindex Maximizing a function over a list of values
4139 @c [fix-ref Numerical Solutions]
4140 (@bullet{}) @strong{Exercise 8.} Compute a list of values of Bessel's
4141 @texline @math{J_1(x)}
4143 function @samp{besJ(1,x)} for @expr{x} from 0 to 5 in steps of 0.25.
4144 Find the value of @expr{x} (from among the above set of values) for
4145 which @samp{besJ(1,x)} is a maximum. Use an ``automatic'' method,
4146 i.e., just reading along the list by hand to find the largest value
4147 is not allowed! (There is an @kbd{a X} command which does this kind
4148 of thing automatically; @pxref{Numerical Solutions}.)
4149 @xref{List Answer 8, 8}. (@bullet{})
4151 @cindex Digits, vectors of
4152 (@bullet{}) @strong{Exercise 9.} You are given an integer in the range
4153 @texline @math{0 \le N < 10^m}
4154 @infoline @expr{0 <= N < 10^m}
4155 for @expr{m=12} (i.e., an integer of less than
4156 twelve digits). Convert this integer into a vector of @expr{m}
4157 digits, each in the range from 0 to 9. In vector-of-digits notation,
4158 add one to this integer to produce a vector of @expr{m+1} digits
4159 (since there could be a carry out of the most significant digit).
4160 Convert this vector back into a regular integer. A good integer
4161 to try is 25129925999. @xref{List Answer 9, 9}. (@bullet{})
4163 (@bullet{}) @strong{Exercise 10.} Your friend Joe tried to use
4164 @kbd{V R a =} to test if all numbers in a list were equal. What
4165 happened? How would you do this test? @xref{List Answer 10, 10}. (@bullet{})
4167 (@bullet{}) @strong{Exercise 11.} The area of a circle of radius one
4168 is @cpi{}. The area of the
4169 @texline @math{2\times2}
4171 square that encloses that circle is 4. So if we throw @var{n} darts at
4172 random points in the square, about @cpiover{4} of them will land inside
4173 the circle. This gives us an entertaining way to estimate the value of
4174 @cpi{}. The @w{@kbd{k r}}
4175 command picks a random number between zero and the value on the stack.
4176 We could get a random floating-point number between @mathit{-1} and 1 by typing
4177 @w{@kbd{2.0 k r 1 -}}. Build a vector of 100 random @expr{(x,y)} points in
4178 this square, then use vector mapping and reduction to count how many
4179 points lie inside the unit circle. Hint: Use the @kbd{v b} command.
4180 @xref{List Answer 11, 11}. (@bullet{})
4182 @cindex Matchstick problem
4183 (@bullet{}) @strong{Exercise 12.} The @dfn{matchstick problem} provides
4184 another way to calculate @cpi{}. Say you have an infinite field
4185 of vertical lines with a spacing of one inch. Toss a one-inch matchstick
4186 onto the field. The probability that the matchstick will land crossing
4187 a line turns out to be
4188 @texline @math{2/\pi}.
4189 @infoline @expr{2/pi}.
4190 Toss 100 matchsticks to estimate @cpi{}. (If you want still more fun,
4191 the probability that the GCD (@w{@kbd{k g}}) of two large integers is
4193 @texline @math{6/\pi^2}.
4194 @infoline @expr{6/pi^2}.
4195 That provides yet another way to estimate @cpi{}.)
4196 @xref{List Answer 12, 12}. (@bullet{})
4198 (@bullet{}) @strong{Exercise 13.} An algebraic entry of a string in
4199 double-quote marks, @samp{"hello"}, creates a vector of the numerical
4200 (ASCII) codes of the characters (here, @expr{[104, 101, 108, 108, 111]}).
4201 Sometimes it is convenient to compute a @dfn{hash code} of a string,
4202 which is just an integer that represents the value of that string.
4203 Two equal strings have the same hash code; two different strings
4204 @dfn{probably} have different hash codes. (For example, Calc has
4205 over 400 function names, but Emacs can quickly find the definition for
4206 any given name because it has sorted the functions into ``buckets'' by
4207 their hash codes. Sometimes a few names will hash into the same bucket,
4208 but it is easier to search among a few names than among all the names.)
4209 One popular hash function is computed as follows: First set @expr{h = 0}.
4210 Then, for each character from the string in turn, set @expr{h = 3h + c_i}
4211 where @expr{c_i} is the character's ASCII code. If we have 511 buckets,
4212 we then take the hash code modulo 511 to get the bucket number. Develop a
4213 simple command or commands for converting string vectors into hash codes.
4214 The hash code for @samp{"Testing, 1, 2, 3"} is 1960915098, which modulo
4215 511 is 121. @xref{List Answer 13, 13}. (@bullet{})
4217 (@bullet{}) @strong{Exercise 14.} The @kbd{H V R} and @kbd{H V U}
4218 commands do nested function evaluations. @kbd{H V U} takes a starting
4219 value and a number of steps @var{n} from the stack; it then applies the
4220 function you give to the starting value 0, 1, 2, up to @var{n} times
4221 and returns a vector of the results. Use this command to create a
4222 ``random walk'' of 50 steps. Start with the two-dimensional point
4223 @expr{(0,0)}; then take one step a random distance between @mathit{-1} and 1
4224 in both @expr{x} and @expr{y}; then take another step, and so on. Use the
4225 @kbd{g f} command to display this random walk. Now modify your random
4226 walk to walk a unit distance, but in a random direction, at each step.
4227 (Hint: The @code{sincos} function returns a vector of the cosine and
4228 sine of an angle.) @xref{List Answer 14, 14}. (@bullet{})
4230 @node Types Tutorial, Algebra Tutorial, Vector/Matrix Tutorial, Tutorial
4231 @section Types Tutorial
4234 Calc understands a variety of data types as well as simple numbers.
4235 In this section, we'll experiment with each of these types in turn.
4237 The numbers we've been using so far have mainly been either @dfn{integers}
4238 or @dfn{floats}. We saw that floats are usually a good approximation to
4239 the mathematical concept of real numbers, but they are only approximations
4240 and are susceptible to roundoff error. Calc also supports @dfn{fractions},
4241 which can exactly represent any rational number.
4245 1: 3628800 2: 3628800 1: 518400:7 1: 518414:7 1: 7:518414
4249 10 ! 49 @key{RET} : 2 + &
4254 The @kbd{:} command divides two integers to get a fraction; @kbd{/}
4255 would normally divide integers to get a floating-point result.
4256 Notice we had to type @key{RET} between the @kbd{49} and the @kbd{:}
4257 since the @kbd{:} would otherwise be interpreted as part of a
4258 fraction beginning with 49.
4260 You can convert between floating-point and fractional format using
4261 @kbd{c f} and @kbd{c F}:
4265 1: 1.35027217629e-5 1: 7:518414
4272 The @kbd{c F} command replaces a floating-point number with the
4273 ``simplest'' fraction whose floating-point representation is the
4274 same, to within the current precision.
4278 1: 3.14159265359 1: 1146408:364913 1: 3.1416 1: 355:113
4281 P c F @key{DEL} p 5 @key{RET} P c F
4285 (@bullet{}) @strong{Exercise 1.} A calculation has produced the
4286 result 1.26508260337. You suspect it is the square root of the
4287 product of @cpi{} and some rational number. Is it? (Be sure
4288 to allow for roundoff error!) @xref{Types Answer 1, 1}. (@bullet{})
4290 @dfn{Complex numbers} can be stored in both rectangular and polar form.
4294 1: -9 1: (0, 3) 1: (3; 90.) 1: (6; 90.) 1: (2.4495; 45.)
4302 The square root of @mathit{-9} is by default rendered in rectangular form
4303 (@w{@expr{0 + 3i}}), but we can convert it to polar form (3 with a
4304 phase angle of 90 degrees). All the usual arithmetic and scientific
4305 operations are defined on both types of complex numbers.
4307 Another generalized kind of number is @dfn{infinity}. Infinity
4308 isn't really a number, but it can sometimes be treated like one.
4309 Calc uses the symbol @code{inf} to represent positive infinity,
4310 i.e., a value greater than any real number. Naturally, you can
4311 also write @samp{-inf} for minus infinity, a value less than any
4312 real number. The word @code{inf} can only be input using
4317 2: inf 2: -inf 2: -inf 2: -inf 1: nan
4318 1: -17 1: -inf 1: -inf 1: inf .
4321 ' inf @key{RET} 17 n * @key{RET} 72 + A +
4326 Since infinity is infinitely large, multiplying it by any finite
4327 number (like @mathit{-17}) has no effect, except that since @mathit{-17}
4328 is negative, it changes a plus infinity to a minus infinity.
4329 (``A huge positive number, multiplied by @mathit{-17}, yields a huge
4330 negative number.'') Adding any finite number to infinity also
4331 leaves it unchanged. Taking an absolute value gives us plus
4332 infinity again. Finally, we add this plus infinity to the minus
4333 infinity we had earlier. If you work it out, you might expect
4334 the answer to be @mathit{-72} for this. But the 72 has been completely
4335 lost next to the infinities; by the time we compute @w{@samp{inf - inf}}
4336 the finite difference between them, if any, is undetectable.
4337 So we say the result is @dfn{indeterminate}, which Calc writes
4338 with the symbol @code{nan} (for Not A Number).
4340 Dividing by zero is normally treated as an error, but you can get
4341 Calc to write an answer in terms of infinity by pressing @kbd{m i}
4342 to turn on Infinite mode.
4346 3: nan 2: nan 2: nan 2: nan 1: nan
4347 2: 1 1: 1 / 0 1: uinf 1: uinf .
4351 1 @key{RET} 0 / m i U / 17 n * +
4356 Dividing by zero normally is left unevaluated, but after @kbd{m i}
4357 it instead gives an infinite result. The answer is actually
4358 @code{uinf}, ``undirected infinity.'' If you look at a graph of
4359 @expr{1 / x} around @w{@expr{x = 0}}, you'll see that it goes toward
4360 plus infinity as you approach zero from above, but toward minus
4361 infinity as you approach from below. Since we said only @expr{1 / 0},
4362 Calc knows that the answer is infinite but not in which direction.
4363 That's what @code{uinf} means. Notice that multiplying @code{uinf}
4364 by a negative number still leaves plain @code{uinf}; there's no
4365 point in saying @samp{-uinf} because the sign of @code{uinf} is
4366 unknown anyway. Finally, we add @code{uinf} to our @code{nan},
4367 yielding @code{nan} again. It's easy to see that, because
4368 @code{nan} means ``totally unknown'' while @code{uinf} means
4369 ``unknown sign but known to be infinite,'' the more mysterious
4370 @code{nan} wins out when it is combined with @code{uinf}, or, for
4371 that matter, with anything else.
4373 (@bullet{}) @strong{Exercise 2.} Predict what Calc will answer
4374 for each of these formulas: @samp{inf / inf}, @samp{exp(inf)},
4375 @samp{exp(-inf)}, @samp{sqrt(-inf)}, @samp{sqrt(uinf)},
4376 @samp{abs(uinf)}, @samp{ln(0)}.
4377 @xref{Types Answer 2, 2}. (@bullet{})
4379 (@bullet{}) @strong{Exercise 3.} We saw that @samp{inf - inf = nan},
4380 which stands for an unknown value. Can @code{nan} stand for
4381 a complex number? Can it stand for infinity?
4382 @xref{Types Answer 3, 3}. (@bullet{})
4384 @dfn{HMS forms} represent a value in terms of hours, minutes, and
4389 1: 2@@ 30' 0" 1: 3@@ 30' 0" 2: 3@@ 30' 0" 1: 2.
4390 . . 1: 1@@ 45' 0." .
4393 2@@ 30' @key{RET} 1 + @key{RET} 2 / /
4397 HMS forms can also be used to hold angles in degrees, minutes, and
4402 1: 0.5 1: 26.56505 1: 26@@ 33' 54.18" 1: 0.44721
4410 First we convert the inverse tangent of 0.5 to degrees-minutes-seconds
4411 form, then we take the sine of that angle. Note that the trigonometric
4412 functions will accept HMS forms directly as input.
4415 (@bullet{}) @strong{Exercise 4.} The Beatles' @emph{Abbey Road} is
4416 47 minutes and 26 seconds long, and contains 17 songs. What is the
4417 average length of a song on @emph{Abbey Road}? If the Extended Disco
4418 Version of @emph{Abbey Road} added 20 seconds to the length of each
4419 song, how long would the album be? @xref{Types Answer 4, 4}. (@bullet{})
4421 A @dfn{date form} represents a date, or a date and time. Dates must
4422 be entered using algebraic entry. Date forms are surrounded by
4423 @samp{< >} symbols; most standard formats for dates are recognized.
4427 2: <Sun Jan 13, 1991> 1: 2.25
4428 1: <6:00pm Thu Jan 10, 1991> .
4431 ' <13 Jan 1991>, <1/10/91, 6pm> @key{RET} -
4436 In this example, we enter two dates, then subtract to find the
4437 number of days between them. It is also possible to add an
4438 HMS form or a number (of days) to a date form to get another
4443 1: <4:45:59pm Mon Jan 14, 1991> 1: <2:50:59am Thu Jan 17, 1991>
4450 @c [fix-ref Date Arithmetic]
4452 The @kbd{t N} (``now'') command pushes the current date and time on the
4453 stack; then we add two days, ten hours and five minutes to the date and
4454 time. Other date-and-time related commands include @kbd{t J}, which
4455 does Julian day conversions, @kbd{t W}, which finds the beginning of
4456 the week in which a date form lies, and @kbd{t I}, which increments a
4457 date by one or several months. @xref{Date Arithmetic}, for more.
4459 (@bullet{}) @strong{Exercise 5.} How many days until the next
4460 Friday the 13th? @xref{Types Answer 5, 5}. (@bullet{})
4462 (@bullet{}) @strong{Exercise 6.} How many leap years will there be
4463 between now and the year 10001 AD@? @xref{Types Answer 6, 6}. (@bullet{})
4465 @cindex Slope and angle of a line
4466 @cindex Angle and slope of a line
4467 An @dfn{error form} represents a mean value with an attached standard
4468 deviation, or error estimate. Suppose our measurements indicate that
4469 a certain telephone pole is about 30 meters away, with an estimated
4470 error of 1 meter, and 8 meters tall, with an estimated error of 0.2
4471 meters. What is the slope of a line from here to the top of the
4472 pole, and what is the equivalent angle in degrees?
4476 1: 8 +/- 0.2 2: 8 +/- 0.2 1: 0.266 +/- 0.011 1: 14.93 +/- 0.594
4480 8 p .2 @key{RET} 30 p 1 / I T
4485 This means that the angle is about 15 degrees, and, assuming our
4486 original error estimates were valid standard deviations, there is about
4487 a 60% chance that the result is correct within 0.59 degrees.
4489 @cindex Torus, volume of
4490 (@bullet{}) @strong{Exercise 7.} The volume of a torus (a donut shape) is
4491 @texline @math{2 \pi^2 R r^2}
4492 @infoline @w{@expr{2 pi^2 R r^2}}
4493 where @expr{R} is the radius of the circle that
4494 defines the center of the tube and @expr{r} is the radius of the tube
4495 itself. Suppose @expr{R} is 20 cm and @expr{r} is 4 cm, each known to
4496 within 5 percent. What is the volume and the relative uncertainty of
4497 the volume? @xref{Types Answer 7, 7}. (@bullet{})
4499 An @dfn{interval form} represents a range of values. While an
4500 error form is best for making statistical estimates, intervals give
4501 you exact bounds on an answer. Suppose we additionally know that
4502 our telephone pole is definitely between 28 and 31 meters away,
4503 and that it is between 7.7 and 8.1 meters tall.
4507 1: [7.7 .. 8.1] 2: [7.7 .. 8.1] 1: [0.24 .. 0.28] 1: [13.9 .. 16.1]
4511 [ 7.7 .. 8.1 ] [ 28 .. 31 ] / I T
4516 If our bounds were correct, then the angle to the top of the pole
4517 is sure to lie in the range shown.
4519 The square brackets around these intervals indicate that the endpoints
4520 themselves are allowable values. In other words, the distance to the
4521 telephone pole is between 28 and 31, @emph{inclusive}. You can also
4522 make an interval that is exclusive of its endpoints by writing
4523 parentheses instead of square brackets. You can even make an interval
4524 which is inclusive (``closed'') on one end and exclusive (``open'') on
4529 1: [1 .. 10) 1: (0.1 .. 1] 2: (0.1 .. 1] 1: (0.2 .. 3)
4533 [ 1 .. 10 ) & [ 2 .. 3 ) *
4538 The Calculator automatically keeps track of which end values should
4539 be open and which should be closed. You can also make infinite or
4540 semi-infinite intervals by using @samp{-inf} or @samp{inf} for one
4543 (@bullet{}) @strong{Exercise 8.} What answer would you expect from
4544 @samp{@w{1 /} @w{(0 .. 10)}}? What about @samp{@w{1 /} @w{(-10 .. 0)}}? What
4545 about @samp{@w{1 /} @w{[0 .. 10]}} (where the interval actually includes
4546 zero)? What about @samp{@w{1 /} @w{(-10 .. 10)}}?
4547 @xref{Types Answer 8, 8}. (@bullet{})
4549 (@bullet{}) @strong{Exercise 9.} Two easy ways of squaring a number
4550 are @kbd{@key{RET} *} and @w{@kbd{2 ^}}. Normally these produce the same
4551 answer. Would you expect this still to hold true for interval forms?
4552 If not, which of these will result in a larger interval?
4553 @xref{Types Answer 9, 9}. (@bullet{})
4555 A @dfn{modulo form} is used for performing arithmetic modulo @var{m}.
4556 For example, arithmetic involving time is generally done modulo 12
4561 1: 17 mod 24 1: 3 mod 24 1: 21 mod 24 1: 9 mod 24
4564 17 M 24 @key{RET} 10 + n 5 /
4569 In this last step, Calc has divided by 5 modulo 24; i.e., it has found a
4570 new number which, when multiplied by 5 modulo 24, produces the original
4571 number, 21. If @var{m} is prime and the divisor is not a multiple of
4572 @var{m}, it is always possible to find such a number. For non-prime
4573 @var{m} like 24, it is only sometimes possible.
4577 1: 10 mod 24 1: 16 mod 24 1: 1000000... 1: 16
4580 10 M 24 @key{RET} 100 ^ 10 @key{RET} 100 ^ 24 %
4585 These two calculations get the same answer, but the first one is
4586 much more efficient because it avoids the huge intermediate value
4587 that arises in the second one.
4589 @cindex Fermat, primality test of
4590 (@bullet{}) @strong{Exercise 10.} A theorem of Pierre de Fermat
4592 @texline @math{x^{n-1} \bmod n = 1}
4593 @infoline @expr{x^(n-1) mod n = 1}
4594 if @expr{n} is a prime number and @expr{x} is an integer less than
4595 @expr{n}. If @expr{n} is @emph{not} a prime number, this will
4596 @emph{not} be true for most values of @expr{x}. Thus we can test
4597 informally if a number is prime by trying this formula for several
4598 values of @expr{x}. Use this test to tell whether the following numbers
4599 are prime: 811749613, 15485863. @xref{Types Answer 10, 10}. (@bullet{})
4601 It is possible to use HMS forms as parts of error forms, intervals,
4602 modulo forms, or as the phase part of a polar complex number.
4603 For example, the @code{calc-time} command pushes the current time
4604 of day on the stack as an HMS/modulo form.
4608 1: 17@@ 34' 45" mod 24@@ 0' 0" 1: 6@@ 22' 15" mod 24@@ 0' 0"
4616 This calculation tells me it is six hours and 22 minutes until midnight.
4618 (@bullet{}) @strong{Exercise 11.} A rule of thumb is that one year
4620 @texline @math{\pi \times 10^7}
4621 @infoline @w{@expr{pi * 10^7}}
4622 seconds. What time will it be that many seconds from right now?
4623 @xref{Types Answer 11, 11}. (@bullet{})
4625 (@bullet{}) @strong{Exercise 12.} You are preparing to order packaging
4626 for the CD release of the Extended Disco Version of @emph{Abbey Road}.
4627 You are told that the songs will actually be anywhere from 20 to 60
4628 seconds longer than the originals. One CD can hold about 75 minutes
4629 of music. Should you order single or double packages?
4630 @xref{Types Answer 12, 12}. (@bullet{})
4632 Another kind of data the Calculator can manipulate is numbers with
4633 @dfn{units}. This isn't strictly a new data type; it's simply an
4634 application of algebraic expressions, where we use variables with
4635 suggestive names like @samp{cm} and @samp{in} to represent units
4636 like centimeters and inches.
4640 1: 2 in 1: 5.08 cm 1: 0.027778 fath 1: 0.0508 m
4643 ' 2in @key{RET} u c cm @key{RET} u c fath @key{RET} u b
4648 We enter the quantity ``2 inches'' (actually an algebraic expression
4649 which means two times the variable @samp{in}), then we convert it
4650 first to centimeters, then to fathoms, then finally to ``base'' units,
4651 which in this case means meters.
4655 1: 9 acre 1: 3 sqrt(acre) 1: 190.84 m 1: 190.84 m + 30 cm
4658 ' 9 acre @key{RET} Q u s ' $+30 cm @key{RET}
4665 1: 191.14 m 1: 36536.3046 m^2 1: 365363046 cm^2
4673 Since units expressions are really just formulas, taking the square
4674 root of @samp{acre} is undefined. After all, @code{acre} might be an
4675 algebraic variable that you will someday assign a value. We use the
4676 ``units-simplify'' command to simplify the expression with variables
4677 being interpreted as unit names.
4679 In the final step, we have converted not to a particular unit, but to a
4680 units system. The ``cgs'' system uses centimeters instead of meters
4681 as its standard unit of length.
4683 There is a wide variety of units defined in the Calculator.
4687 1: 55 mph 1: 88.5139 kph 1: 88.5139 km / hr 1: 8.201407e-8 c
4690 ' 55 mph @key{RET} u c kph @key{RET} u c km/hr @key{RET} u c c @key{RET}
4695 We express a speed first in miles per hour, then in kilometers per
4696 hour, then again using a slightly more explicit notation, then
4697 finally in terms of fractions of the speed of light.
4699 Temperature conversions are a bit more tricky. There are two ways to
4700 interpret ``20 degrees Fahrenheit''---it could mean an actual
4701 temperature, or it could mean a change in temperature. For normal
4702 units there is no difference, but temperature units have an offset
4703 as well as a scale factor and so there must be two explicit commands
4708 1: 20 degF 1: 11.1111 degC 1: -6.666 degC
4711 ' 20 degF @key{RET} u c degC @key{RET} U u t degC @key{RET}
4716 First we convert a change of 20 degrees Fahrenheit into an equivalent
4717 change in degrees Celsius (or Centigrade). Then, we convert the
4718 absolute temperature 20 degrees Fahrenheit into Celsius.
4720 For simple unit conversions, you can put a plain number on the stack.
4721 Then @kbd{u c} and @kbd{u t} will prompt for both old and new units.
4722 When you use this method, you're responsible for remembering which
4723 numbers are in which units:
4727 1: 55 1: 88.5139 1: 8.201407e-8
4730 55 u c mph @key{RET} kph @key{RET} u c km/hr @key{RET} c @key{RET}
4734 To see a complete list of built-in units, type @kbd{u v}. Press
4735 @w{@kbd{C-x * c}} again to re-enter the Calculator when you're done looking
4738 (@bullet{}) @strong{Exercise 13.} How many seconds are there really
4739 in a year? @xref{Types Answer 13, 13}. (@bullet{})
4741 @cindex Speed of light
4742 (@bullet{}) @strong{Exercise 14.} Supercomputer designs are limited by
4743 the speed of light (and of electricity, which is nearly as fast).
4744 Suppose a computer has a 4.1 ns (nanosecond) clock cycle, and its
4745 cabinet is one meter across. Is speed of light going to be a
4746 significant factor in its design? @xref{Types Answer 14, 14}. (@bullet{})
4748 (@bullet{}) @strong{Exercise 15.} Sam the Slug normally travels about
4749 five yards in an hour. He has obtained a supply of Power Pills; each
4750 Power Pill he eats doubles his speed. How many Power Pills can he
4751 swallow and still travel legally on most US highways?
4752 @xref{Types Answer 15, 15}. (@bullet{})
4754 @node Algebra Tutorial, Programming Tutorial, Types Tutorial, Tutorial
4755 @section Algebra and Calculus Tutorial
4758 This section shows how to use Calc's algebra facilities to solve
4759 equations, do simple calculus problems, and manipulate algebraic
4763 * Basic Algebra Tutorial::
4764 * Rewrites Tutorial::
4767 @node Basic Algebra Tutorial, Rewrites Tutorial, Algebra Tutorial, Algebra Tutorial
4768 @subsection Basic Algebra
4771 If you enter a formula in Algebraic mode that refers to variables,
4772 the formula itself is pushed onto the stack. You can manipulate
4773 formulas as regular data objects.
4777 1: 2 x^2 - 6 1: 6 - 2 x^2 1: (3 x^2 + y) (6 - 2 x^2)
4780 ' 2x^2-6 @key{RET} n ' 3x^2+y @key{RET} *
4784 (@bullet{}) @strong{Exercise 1.} Do @kbd{' x @key{RET} Q 2 ^} and
4785 @kbd{' x @key{RET} 2 ^ Q} both wind up with the same result (@samp{x})?
4786 Why or why not? @xref{Algebra Answer 1, 1}. (@bullet{})
4788 There are also commands for doing common algebraic operations on
4789 formulas. Continuing with the formula from the last example,
4793 1: 18 x^2 - 6 x^4 + 6 y - 2 y x^2 1: (18 - 2 y) x^2 - 6 x^4 + 6 y
4801 First we ``expand'' using the distributive law, then we ``collect''
4802 terms involving like powers of @expr{x}.
4804 Let's find the value of this expression when @expr{x} is 2 and @expr{y}
4809 1: 17 x^2 - 6 x^4 + 3 1: -25
4812 1:2 s l y @key{RET} 2 s l x @key{RET}
4817 The @kbd{s l} command means ``let''; it takes a number from the top of
4818 the stack and temporarily assigns it as the value of the variable
4819 you specify. It then evaluates (as if by the @kbd{=} key) the
4820 next expression on the stack. After this command, the variable goes
4821 back to its original value, if any.
4823 (An earlier exercise in this tutorial involved storing a value in the
4824 variable @code{x}; if this value is still there, you will have to
4825 unstore it with @kbd{s u x @key{RET}} before the above example will work
4828 @cindex Maximum of a function using Calculus
4829 Let's find the maximum value of our original expression when @expr{y}
4830 is one-half and @expr{x} ranges over all possible values. We can
4831 do this by taking the derivative with respect to @expr{x} and examining
4832 values of @expr{x} for which the derivative is zero. If the second
4833 derivative of the function at that value of @expr{x} is negative,
4834 the function has a local maximum there.
4838 1: 17 x^2 - 6 x^4 + 3 1: 34 x - 24 x^3
4841 U @key{DEL} s 1 a d x @key{RET} s 2
4846 Well, the derivative is clearly zero when @expr{x} is zero. To find
4847 the other root(s), let's divide through by @expr{x} and then solve:
4851 1: (34 x - 24 x^3) / x 1: 34 - 24 x^2
4861 1: 0.70588 x^2 = 1 1: x = 1.19023
4864 0 a = s 3 a S x @key{RET}
4869 Now we compute the second derivative and plug in our values of @expr{x}:
4873 1: 1.19023 2: 1.19023 2: 1.19023
4874 . 1: 34 x - 24 x^3 1: 34 - 72 x^2
4877 a . r 2 a d x @key{RET} s 4
4882 (The @kbd{a .} command extracts just the righthand side of an equation.
4883 Another method would have been to use @kbd{v u} to unpack the equation
4884 @w{@samp{x = 1.19}} to @samp{x} and @samp{1.19}, then use @kbd{M-- M-2 @key{DEL}}
4885 to delete the @samp{x}.)
4889 2: 34 - 72 x^2 1: -68. 2: 34 - 72 x^2 1: 34
4893 @key{TAB} s l x @key{RET} U @key{DEL} 0 s l x @key{RET}
4898 The first of these second derivatives is negative, so we know the function
4899 has a maximum value at @expr{x = 1.19023}. (The function also has a
4900 local @emph{minimum} at @expr{x = 0}.)
4902 When we solved for @expr{x}, we got only one value even though
4903 @expr{0.70588 x^2 = 1} is a quadratic equation that ought to have
4904 two solutions. The reason is that @w{@kbd{a S}} normally returns a
4905 single ``principal'' solution. If it needs to come up with an
4906 arbitrary sign (as occurs in the quadratic formula) it picks @expr{+}.
4907 If it needs an arbitrary integer, it picks zero. We can get a full
4908 solution by pressing @kbd{H} (the Hyperbolic flag) before @kbd{a S}.
4912 1: 0.70588 x^2 = 1 1: x = 1.19023 s1 1: x = -1.19023
4915 r 3 H a S x @key{RET} s 5 1 n s l s1 @key{RET}
4920 Calc has invented the variable @samp{s1} to represent an unknown sign;
4921 it is supposed to be either @mathit{+1} or @mathit{-1}. Here we have used
4922 the ``let'' command to evaluate the expression when the sign is negative.
4923 If we plugged this into our second derivative we would get the same,
4924 negative, answer, so @expr{x = -1.19023} is also a maximum.
4926 To find the actual maximum value, we must plug our two values of @expr{x}
4927 into the original formula.
4931 2: 17 x^2 - 6 x^4 + 3 1: 24.08333 s1^2 - 12.04166 s1^4 + 3
4935 r 1 r 5 s l @key{RET}
4940 (Here we see another way to use @kbd{s l}; if its input is an equation
4941 with a variable on the lefthand side, then @kbd{s l} treats the equation
4942 like an assignment to that variable if you don't give a variable name.)
4944 It's clear that this will have the same value for either sign of
4945 @code{s1}, but let's work it out anyway, just for the exercise:
4949 2: [-1, 1] 1: [15.04166, 15.04166]
4950 1: 24.08333 s1^2 ... .
4953 [ 1 n , 1 ] @key{TAB} V M $ @key{RET}
4958 Here we have used a vector mapping operation to evaluate the function
4959 at several values of @samp{s1} at once. @kbd{V M $} is like @kbd{V M '}
4960 except that it takes the formula from the top of the stack. The
4961 formula is interpreted as a function to apply across the vector at the
4962 next-to-top stack level. Since a formula on the stack can't contain
4963 @samp{$} signs, Calc assumes the variables in the formula stand for
4964 different arguments. It prompts you for an @dfn{argument list}, giving
4965 the list of all variables in the formula in alphabetical order as the
4966 default list. In this case the default is @samp{(s1)}, which is just
4967 what we want so we simply press @key{RET} at the prompt.
4969 If there had been several different values, we could have used
4970 @w{@kbd{V R X}} to find the global maximum.
4972 Calc has a built-in @kbd{a P} command that solves an equation using
4973 @w{@kbd{H a S}} and returns a vector of all the solutions. It simply
4974 automates the job we just did by hand. Applied to our original
4975 cubic polynomial, it would produce the vector of solutions
4976 @expr{[1.19023, -1.19023, 0]}. (There is also an @kbd{a X} command
4977 which finds a local maximum of a function. It uses a numerical search
4978 method rather than examining the derivatives, and thus requires you
4979 to provide some kind of initial guess to show it where to look.)
4981 (@bullet{}) @strong{Exercise 2.} Given a vector of the roots of a
4982 polynomial (such as the output of an @kbd{a P} command), what
4983 sequence of commands would you use to reconstruct the original
4984 polynomial? (The answer will be unique to within a constant
4985 multiple; choose the solution where the leading coefficient is one.)
4986 @xref{Algebra Answer 2, 2}. (@bullet{})
4988 The @kbd{m s} command enables Symbolic mode, in which formulas
4989 like @samp{sqrt(5)} that can't be evaluated exactly are left in
4990 symbolic form rather than giving a floating-point approximate answer.
4991 Fraction mode (@kbd{m f}) is also useful when doing algebra.
4995 2: 34 x - 24 x^3 2: 34 x - 24 x^3
4996 1: 34 x - 24 x^3 1: [sqrt(51) / 6, sqrt(51) / -6, 0]
4999 r 2 @key{RET} m s m f a P x @key{RET}
5003 One more mode that makes reading formulas easier is Big mode.
5012 1: [-----, -----, 0]
5021 Here things like powers, square roots, and quotients and fractions
5022 are displayed in a two-dimensional pictorial form. Calc has other
5023 language modes as well, such as C mode, FORTRAN mode, @TeX{} mode
5028 2: 34*x - 24*pow(x, 3) 2: 34*x - 24*x**3
5029 1: @{sqrt(51) / 6, sqrt(51) / -6, 0@} 1: /sqrt(51) / 6, sqrt(51) / -6, 0/
5040 2: [@{\sqrt@{51@} \over 6@}, @{\sqrt@{51@} \over -6@}, 0]
5041 1: @{2 \over 3@} \sqrt@{5@}
5044 d T ' 2 \sqrt@{5@} \over 3 @key{RET}
5049 As you can see, language modes affect both entry and display of
5050 formulas. They affect such things as the names used for built-in
5051 functions, the set of arithmetic operators and their precedences,
5052 and notations for vectors and matrices.
5054 Notice that @samp{sqrt(51)} may cause problems with older
5055 implementations of C and FORTRAN, which would require something more
5056 like @samp{sqrt(51.0)}. It is always wise to check over the formulas
5057 produced by the various language modes to make sure they are fully
5060 Type @kbd{m s}, @kbd{m f}, and @kbd{d N} to reset these modes. (You
5061 may prefer to remain in Big mode, but all the examples in the tutorial
5062 are shown in normal mode.)
5064 @cindex Area under a curve
5065 What is the area under the portion of this curve from @expr{x = 1} to @expr{2}?
5066 This is simply the integral of the function:
5070 1: 17 x^2 - 6 x^4 + 3 1: 5.6666 x^3 - 1.2 x^5 + 3 x
5078 We want to evaluate this at our two values for @expr{x} and subtract.
5079 One way to do it is again with vector mapping and reduction:
5083 2: [2, 1] 1: [12.93333, 7.46666] 1: 5.46666
5084 1: 5.6666 x^3 ... . .
5086 [ 2 , 1 ] @key{TAB} V M $ @key{RET} V R -
5090 (@bullet{}) @strong{Exercise 3.} Find the integral from 1 to @expr{y}
5092 @texline @math{x \sin \pi x}
5093 @infoline @w{@expr{x sin(pi x)}}
5094 (where the sine is calculated in radians). Find the values of the
5095 integral for integers @expr{y} from 1 to 5. @xref{Algebra Answer 3,
5098 Calc's integrator can do many simple integrals symbolically, but many
5099 others are beyond its capabilities. Suppose we wish to find the area
5101 @texline @math{\sin x \ln x}
5102 @infoline @expr{sin(x) ln(x)}
5103 over the same range of @expr{x}. If you entered this formula and typed
5104 @kbd{a i x @key{RET}} (don't bother to try this), Calc would work for a
5105 long time but would be unable to find a solution. In fact, there is no
5106 closed-form solution to this integral. Now what do we do?
5108 @cindex Integration, numerical
5109 @cindex Numerical integration
5110 One approach would be to do the integral numerically. It is not hard
5111 to do this by hand using vector mapping and reduction. It is rather
5112 slow, though, since the sine and logarithm functions take a long time.
5113 We can save some time by reducing the working precision.
5117 3: 10 1: [1, 1.1, 1.2, ... , 1.8, 1.9]
5122 10 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x
5127 (Note that we have used the extended version of @kbd{v x}; we could
5128 also have used plain @kbd{v x} as follows: @kbd{v x 10 @key{RET} 9 + .1 *}.)
5132 2: [1, 1.1, ... ] 1: [0., 0.084941, 0.16993, ... ]
5136 ' sin(x) ln(x) @key{RET} s 1 m r p 5 @key{RET} V M $ @key{RET}
5151 (If you got wildly different results, did you remember to switch
5154 Here we have divided the curve into ten segments of equal width;
5155 approximating these segments as rectangular boxes (i.e., assuming
5156 the curve is nearly flat at that resolution), we compute the areas
5157 of the boxes (height times width), then sum the areas. (It is
5158 faster to sum first, then multiply by the width, since the width
5159 is the same for every box.)
5161 The true value of this integral turns out to be about 0.374, so
5162 we're not doing too well. Let's try another approach.
5166 1: ln(x) sin(x) 1: 0.84147 x + 0.11957 (x - 1)^2 - ...
5169 r 1 a t x=1 @key{RET} 4 @key{RET}
5174 Here we have computed the Taylor series expansion of the function
5175 about the point @expr{x=1}. We can now integrate this polynomial
5176 approximation, since polynomials are easy to integrate.
5180 1: 0.42074 x^2 + ... 1: [-0.0446, -0.42073] 1: 0.3761
5183 a i x @key{RET} [ 2 , 1 ] @key{TAB} V M $ @key{RET} V R -
5188 Better! By increasing the precision and/or asking for more terms
5189 in the Taylor series, we can get a result as accurate as we like.
5190 (Taylor series converge better away from singularities in the
5191 function such as the one at @code{ln(0)}, so it would also help to
5192 expand the series about the points @expr{x=2} or @expr{x=1.5} instead
5195 @cindex Simpson's rule
5196 @cindex Integration by Simpson's rule
5197 (@bullet{}) @strong{Exercise 4.} Our first method approximated the
5198 curve by stairsteps of width 0.1; the total area was then the sum
5199 of the areas of the rectangles under these stairsteps. Our second
5200 method approximated the function by a polynomial, which turned out
5201 to be a better approximation than stairsteps. A third method is
5202 @dfn{Simpson's rule}, which is like the stairstep method except
5203 that the steps are not required to be flat. Simpson's rule boils
5204 down to the formula,
5208 (h/3) * (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + ...
5209 + 2 f(a+(n-2)*h) + 4 f(a+(n-1)*h) + f(a+n*h))
5215 \qquad {h \over 3} (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + \cdots
5216 \hfill \cr \hfill {} + 2 f(a+(n-2)h) + 4 f(a+(n-1)h) + f(a+n h)) \qquad
5222 where @expr{n} (which must be even) is the number of slices and @expr{h}
5223 is the width of each slice. These are 10 and 0.1 in our example.
5224 For reference, here is the corresponding formula for the stairstep
5229 h * (f(a) + f(a+h) + f(a+2h) + f(a+3h) + ...
5230 + f(a+(n-2)*h) + f(a+(n-1)*h))
5235 $$ h (f(a) + f(a+h) + f(a+2h) + f(a+3h) + \cdots
5236 + f(a+(n-2)h) + f(a+(n-1)h)) $$
5240 Compute the integral from 1 to 2 of
5241 @texline @math{\sin x \ln x}
5242 @infoline @expr{sin(x) ln(x)}
5243 using Simpson's rule with 10 slices.
5244 @xref{Algebra Answer 4, 4}. (@bullet{})
5246 Calc has a built-in @kbd{a I} command for doing numerical integration.
5247 It uses @dfn{Romberg's method}, which is a more sophisticated cousin
5248 of Simpson's rule. In particular, it knows how to keep refining the
5249 result until the current precision is satisfied.
5251 @c [fix-ref Selecting Sub-Formulas]
5252 Aside from the commands we've seen so far, Calc also provides a
5253 large set of commands for operating on parts of formulas. You
5254 indicate the desired sub-formula by placing the cursor on any part
5255 of the formula before giving a @dfn{selection} command. Selections won't
5256 be covered in the tutorial; @pxref{Selecting Subformulas}, for
5257 details and examples.
5259 @c hard exercise: simplify (2^(n r) - 2^(r*(n - 1))) / (2^r - 1) 2^(n - 1)
5260 @c to 2^((n-1)*(r-1)).
5262 @node Rewrites Tutorial, , Basic Algebra Tutorial, Algebra Tutorial
5263 @subsection Rewrite Rules
5266 No matter how many built-in commands Calc provided for doing algebra,
5267 there would always be something you wanted to do that Calc didn't have
5268 in its repertoire. So Calc also provides a @dfn{rewrite rule} system
5269 that you can use to define your own algebraic manipulations.
5271 Suppose we want to simplify this trigonometric formula:
5275 1: 2 sec(x)^2 / tan(x)^2 - 2 / tan(x)^2
5278 ' 2sec(x)^2/tan(x)^2 - 2/tan(x)^2 @key{RET} s 1
5283 If we were simplifying this by hand, we'd probably combine over the common
5284 denominator. The @kbd{a n} algebra command will do this, but we'll do
5285 it with a rewrite rule just for practice.
5287 Rewrite rules are written with the @samp{:=} symbol.
5291 1: (2 sec(x)^2 - 2) / tan(x)^2
5294 a r a/x + b/x := (a+b)/x @key{RET}
5299 (The ``assignment operator'' @samp{:=} has several uses in Calc. All
5300 by itself the formula @samp{a/x + b/x := (a+b)/x} doesn't do anything,
5301 but when it is given to the @kbd{a r} command, that command interprets
5302 it as a rewrite rule.)
5304 The lefthand side, @samp{a/x + b/x}, is called the @dfn{pattern} of the
5305 rewrite rule. Calc searches the formula on the stack for parts that
5306 match the pattern. Variables in a rewrite pattern are called
5307 @dfn{meta-variables}, and when matching the pattern each meta-variable
5308 can match any sub-formula. Here, the meta-variable @samp{a} matched
5309 the expression @samp{2 sec(x)^2}, the meta-variable @samp{b} matched
5310 the constant @samp{-2} and the meta-variable @samp{x} matched
5311 the expression @samp{tan(x)^2}.
5313 This rule points out several interesting features of rewrite patterns.
5314 First, if a meta-variable appears several times in a pattern, it must
5315 match the same thing everywhere. This rule detects common denominators
5316 because the same meta-variable @samp{x} is used in both of the
5319 Second, meta-variable names are independent from variables in the
5320 target formula. Notice that the meta-variable @samp{x} here matches
5321 the subformula @samp{tan(x)^2}; Calc never confuses the two meanings of
5324 And third, rewrite patterns know a little bit about the algebraic
5325 properties of formulas. The pattern called for a sum of two quotients;
5326 Calc was able to match a difference of two quotients by matching
5327 @samp{a = 2 sec(x)^2}, @samp{b = -2}, and @samp{x = tan(x)^2}.
5329 When the pattern part of a rewrite rule matches a part of the formula,
5330 that part is replaced by the righthand side with all the meta-variables
5331 substituted with the things they matched. So the result is
5332 @samp{(2 sec(x)^2 - 2) / tan(x)^2}.
5334 @c [fix-ref Algebraic Properties of Rewrite Rules]
5335 We could just as easily have written @samp{a/x - b/x := (a-b)/x} for
5336 the rule. It would have worked just the same in all cases. (If we
5337 really wanted the rule to apply only to @samp{+} or only to @samp{-},
5338 we could have used the @code{plain} symbol. @xref{Algebraic Properties
5339 of Rewrite Rules}, for some examples of this.)
5341 One more rewrite will complete the job. We want to use the identity
5342 @samp{tan(x)^2 + 1 = sec(x)^2}, but of course we must first rearrange
5343 the identity in a way that matches our formula. The obvious rule
5344 would be @samp{@w{2 sec(x)^2 - 2} := 2 tan(x)^2}, but a little thought shows
5345 that the rule @samp{sec(x)^2 := 1 + tan(x)^2} will also work. The
5346 latter rule has a more general pattern so it will work in many other
5354 a r sec(x)^2 := 1 + tan(x)^2 @key{RET}
5358 You may ask, what's the point of using the most general rule if you
5359 have to type it in every time anyway? The answer is that Calc allows
5360 you to store a rewrite rule in a variable, then give the variable
5361 name in the @kbd{a r} command. In fact, this is the preferred way to
5362 use rewrites. For one, if you need a rule once you'll most likely
5363 need it again later. Also, if the rule doesn't work quite right you
5364 can simply Undo, edit the variable, and run the rule again without
5365 having to retype it.
5369 ' a/x + b/x := (a+b)/x @key{RET} s t merge @key{RET}
5370 ' sec(x)^2 := 1 + tan(x)^2 @key{RET} s t secsqr @key{RET}
5372 1: 2 sec(x)^2 / tan(x)^2 - 2 / tan(x)^2 1: 2
5375 r 1 a r merge @key{RET} a r secsqr @key{RET}
5379 To edit a variable, type @kbd{s e} and the variable name, use regular
5380 Emacs editing commands as necessary, then type @kbd{C-c C-c} to store
5381 the edited value back into the variable.
5382 You can also use @w{@kbd{s e}} to create a new variable if you wish.
5384 Notice that the first time you use each rule, Calc puts up a ``compiling''
5385 message briefly. The pattern matcher converts rules into a special
5386 optimized pattern-matching language rather than using them directly.
5387 This allows @kbd{a r} to apply even rather complicated rules very
5388 efficiently. If the rule is stored in a variable, Calc compiles it
5389 only once and stores the compiled form along with the variable. That's
5390 another good reason to store your rules in variables rather than
5391 entering them on the fly.
5393 (@bullet{}) @strong{Exercise 1.} Type @kbd{m s} to get Symbolic
5394 mode, then enter the formula @samp{@w{(2 + sqrt(2))} / @w{(1 + sqrt(2))}}.
5395 Using a rewrite rule, simplify this formula by multiplying the top and
5396 bottom by the conjugate @w{@samp{1 - sqrt(2)}}. The result will have
5397 to be expanded by the distributive law; do this with another
5398 rewrite. @xref{Rewrites Answer 1, 1}. (@bullet{})
5400 The @kbd{a r} command can also accept a vector of rewrite rules, or
5401 a variable containing a vector of rules.
5405 1: [merge, secsqr] 1: [a/x + b/x := (a + b)/x, ... ]
5408 ' [merge,sinsqr] @key{RET} =
5415 1: 2 sec(x)^2 / tan(x)^2 - 2 / tan(x)^2 1: 2
5418 s t trig @key{RET} r 1 a r trig @key{RET}
5422 @c [fix-ref Nested Formulas with Rewrite Rules]
5423 Calc tries all the rules you give against all parts of the formula,
5424 repeating until no further change is possible. (The exact order in
5425 which things are tried is rather complex, but for simple rules like
5426 the ones we've used here the order doesn't really matter.
5427 @xref{Nested Formulas with Rewrite Rules}.)
5429 Calc actually repeats only up to 100 times, just in case your rule set
5430 has gotten into an infinite loop. You can give a numeric prefix argument
5431 to @kbd{a r} to specify any limit. In particular, @kbd{M-1 a r} does
5432 only one rewrite at a time.
5436 1: (2 sec(x)^2 - 2) / tan(x)^2 1: 2
5439 r 1 M-1 a r trig @key{RET} M-1 a r trig @key{RET}
5443 You can type @kbd{M-0 a r} if you want no limit at all on the number
5444 of rewrites that occur.
5446 Rewrite rules can also be @dfn{conditional}. Simply follow the rule
5447 with a @samp{::} symbol and the desired condition. For example,
5451 1: sin(x + 2 pi) + sin(x + 3 pi) + sin(x + 4 pi)
5454 ' sin(x+2pi) + sin(x+3pi) + sin(x+4pi) @key{RET}
5461 1: sin(x + 3 pi) + 2 sin(x)
5464 a r sin(a + k pi) := sin(a) :: k % 2 = 0 @key{RET}
5469 (Recall, @samp{k % 2} is the remainder from dividing @samp{k} by 2,
5470 which will be zero only when @samp{k} is an even integer.)
5472 An interesting point is that the variable @samp{pi} was matched
5473 literally rather than acting as a meta-variable.
5474 This is because it is a special-constant variable. The special
5475 constants @samp{e}, @samp{i}, @samp{phi}, and so on also match literally.
5476 A common error with rewrite
5477 rules is to write, say, @samp{f(a,b,c,d,e) := g(a+b+c+d+e)}, expecting
5478 to match any @samp{f} with five arguments but in fact matching
5479 only when the fifth argument is literally @samp{e}!
5481 @cindex Fibonacci numbers
5486 Rewrite rules provide an interesting way to define your own functions.
5487 Suppose we want to define @samp{fib(n)} to produce the @var{n}th
5488 Fibonacci number. The first two Fibonacci numbers are each 1;
5489 later numbers are formed by summing the two preceding numbers in
5490 the sequence. This is easy to express in a set of three rules:
5494 ' [fib(1) := 1, fib(2) := 1, fib(n) := fib(n-1) + fib(n-2)] @key{RET} s t fib
5499 ' fib(7) @key{RET} a r fib @key{RET}
5503 One thing that is guaranteed about the order that rewrites are tried
5504 is that, for any given subformula, earlier rules in the rule set will
5505 be tried for that subformula before later ones. So even though the
5506 first and third rules both match @samp{fib(1)}, we know the first will
5507 be used preferentially.
5509 This rule set has one dangerous bug: Suppose we apply it to the
5510 formula @samp{fib(x)}? (Don't actually try this.) The third rule
5511 will match @samp{fib(x)} and replace it with @w{@samp{fib(x-1) + fib(x-2)}}.
5512 Each of these will then be replaced to get @samp{fib(x-2) + 2 fib(x-3) +
5513 fib(x-4)}, and so on, expanding forever. What we really want is to apply
5514 the third rule only when @samp{n} is an integer greater than two. Type
5515 @w{@kbd{s e fib @key{RET}}}, then edit the third rule to:
5518 fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2
5526 1: fib(6) + fib(x) + fib(0) 1: fib(x) + fib(0) + 8
5529 ' fib(6)+fib(x)+fib(0) @key{RET} a r fib @key{RET}
5534 We've created a new function, @code{fib}, and a new command,
5535 @w{@kbd{a r fib @key{RET}}}, which means ``evaluate all @code{fib} calls in
5536 this formula.'' To make things easier still, we can tell Calc to
5537 apply these rules automatically by storing them in the special
5538 variable @code{EvalRules}.
5542 1: [fib(1) := ...] . 1: [8, 13]
5545 s r fib @key{RET} s t EvalRules @key{RET} ' [fib(6), fib(7)] @key{RET}
5549 It turns out that this rule set has the problem that it does far
5550 more work than it needs to when @samp{n} is large. Consider the
5551 first few steps of the computation of @samp{fib(6)}:
5557 fib(4) + fib(3) + fib(3) + fib(2) =
5558 fib(3) + fib(2) + fib(2) + fib(1) + fib(2) + fib(1) + 1 = ...
5563 Note that @samp{fib(3)} appears three times here. Unless Calc's
5564 algebraic simplifier notices the multiple @samp{fib(3)}s and combines
5565 them (and, as it happens, it doesn't), this rule set does lots of
5566 needless recomputation. To cure the problem, type @code{s e EvalRules}
5567 to edit the rules (or just @kbd{s E}, a shorthand command for editing
5568 @code{EvalRules}) and add another condition:
5571 fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2 :: remember
5575 If a @samp{:: remember} condition appears anywhere in a rule, then if
5576 that rule succeeds Calc will add another rule that describes that match
5577 to the front of the rule set. (Remembering works in any rule set, but
5578 for technical reasons it is most effective in @code{EvalRules}.) For
5579 example, if the rule rewrites @samp{fib(7)} to something that evaluates
5580 to 13, then the rule @samp{fib(7) := 13} will be added to the rule set.
5582 Type @kbd{' fib(8) @key{RET}} to compute the eighth Fibonacci number, then
5583 type @kbd{s E} again to see what has happened to the rule set.
5585 With the @code{remember} feature, our rule set can now compute
5586 @samp{fib(@var{n})} in just @var{n} steps. In the process it builds
5587 up a table of all Fibonacci numbers up to @var{n}. After we have
5588 computed the result for a particular @var{n}, we can get it back
5589 (and the results for all smaller @var{n}) later in just one step.
5591 All Calc operations will run somewhat slower whenever @code{EvalRules}
5592 contains any rules. You should type @kbd{s u EvalRules @key{RET}} now to
5593 un-store the variable.
5595 (@bullet{}) @strong{Exercise 2.} Sometimes it is possible to reformulate
5596 a problem to reduce the amount of recursion necessary to solve it.
5597 Create a rule that, in about @var{n} simple steps and without recourse
5598 to the @code{remember} option, replaces @samp{fib(@var{n}, 1, 1)} with
5599 @samp{fib(1, @var{x}, @var{y})} where @var{x} and @var{y} are the
5600 @var{n}th and @var{n+1}st Fibonacci numbers, respectively. This rule is
5601 rather clunky to use, so add a couple more rules to make the ``user
5602 interface'' the same as for our first version: enter @samp{fib(@var{n})},
5603 get back a plain number. @xref{Rewrites Answer 2, 2}. (@bullet{})
5605 There are many more things that rewrites can do. For example, there
5606 are @samp{&&&} and @samp{|||} pattern operators that create ``and''
5607 and ``or'' combinations of rules. As one really simple example, we
5608 could combine our first two Fibonacci rules thusly:
5611 [fib(1 ||| 2) := 1, fib(n) := ... ]
5615 That means ``@code{fib} of something matching either 1 or 2 rewrites
5618 You can also make meta-variables optional by enclosing them in @code{opt}.
5619 For example, the pattern @samp{a + b x} matches @samp{2 + 3 x} but not
5620 @samp{2 + x} or @samp{3 x} or @samp{x}. The pattern @samp{opt(a) + opt(b) x}
5621 matches all of these forms, filling in a default of zero for @samp{a}
5622 and one for @samp{b}.
5624 (@bullet{}) @strong{Exercise 3.} Your friend Joe had @samp{2 + 3 x}
5625 on the stack and tried to use the rule
5626 @samp{opt(a) + opt(b) x := f(a, b, x)}. What happened?
5627 @xref{Rewrites Answer 3, 3}. (@bullet{})
5629 (@bullet{}) @strong{Exercise 4.} Starting with a positive integer @expr{a},
5630 divide @expr{a} by two if it is even, otherwise compute @expr{3 a + 1}.
5631 Now repeat this step over and over. A famous unproved conjecture
5632 is that for any starting @expr{a}, the sequence always eventually
5633 reaches 1. Given the formula @samp{seq(@var{a}, 0)}, write a set of
5634 rules that convert this into @samp{seq(1, @var{n})} where @var{n}
5635 is the number of steps it took the sequence to reach the value 1.
5636 Now enhance the rules to accept @samp{seq(@var{a})} as a starting
5637 configuration, and to stop with just the number @var{n} by itself.
5638 Now make the result be a vector of values in the sequence, from @var{a}
5639 to 1. (The formula @samp{@var{x}|@var{y}} appends the vectors @var{x}
5640 and @var{y}.) For example, rewriting @samp{seq(6)} should yield the
5641 vector @expr{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.
5642 @xref{Rewrites Answer 4, 4}. (@bullet{})
5644 (@bullet{}) @strong{Exercise 5.} Define, using rewrite rules, a function
5645 @samp{nterms(@var{x})} that returns the number of terms in the sum
5646 @var{x}, or 1 if @var{x} is not a sum. (A @dfn{sum} for our purposes
5647 is one or more non-sum terms separated by @samp{+} or @samp{-} signs,
5648 so that @expr{2 - 3 (x + y) + x y} is a sum of three terms.)
5649 @xref{Rewrites Answer 5, 5}. (@bullet{})
5651 (@bullet{}) @strong{Exercise 6.} A Taylor series for a function is an
5652 infinite series that exactly equals the value of that function at
5653 values of @expr{x} near zero.
5657 cos(x) = 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ...
5662 $$ \cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - {x^6 \over 6!} + \cdots $$
5666 The @kbd{a t} command produces a @dfn{truncated Taylor series} which
5667 is obtained by dropping all the terms higher than, say, @expr{x^2}.
5668 Calc represents the truncated Taylor series as a polynomial in @expr{x}.
5669 Mathematicians often write a truncated series using a ``big-O'' notation
5670 that records what was the lowest term that was truncated.
5674 cos(x) = 1 - x^2 / 2! + O(x^3)
5679 $$ \cos x = 1 - {x^2 \over 2!} + O(x^3) $$
5684 The meaning of @expr{O(x^3)} is ``a quantity which is negligibly small
5685 if @expr{x^3} is considered negligibly small as @expr{x} goes to zero.''
5687 The exercise is to create rewrite rules that simplify sums and products of
5688 power series represented as @samp{@var{polynomial} + O(@var{var}^@var{n})}.
5689 For example, given @samp{1 - x^2 / 2 + O(x^3)} and @samp{x - x^3 / 6 + O(x^4)}
5690 on the stack, we want to be able to type @kbd{*} and get the result
5691 @samp{x - 2:3 x^3 + O(x^4)}. Don't worry if the terms of the sum are
5692 rearranged. (This one is rather tricky; the solution at the end of
5693 this chapter uses 6 rewrite rules. Hint: The @samp{constant(x)}
5694 condition tests whether @samp{x} is a number.) @xref{Rewrites Answer
5697 Just for kicks, try adding the rule @code{2+3 := 6} to @code{EvalRules}.
5698 What happens? (Be sure to remove this rule afterward, or you might get
5699 a nasty surprise when you use Calc to balance your checkbook!)
5701 @xref{Rewrite Rules}, for the whole story on rewrite rules.
5703 @node Programming Tutorial, Answers to Exercises, Algebra Tutorial, Tutorial
5704 @section Programming Tutorial
5707 The Calculator is written entirely in Emacs Lisp, a highly extensible
5708 language. If you know Lisp, you can program the Calculator to do
5709 anything you like. Rewrite rules also work as a powerful programming
5710 system. But Lisp and rewrite rules take a while to master, and often
5711 all you want to do is define a new function or repeat a command a few
5712 times. Calc has features that allow you to do these things easily.
5714 One very limited form of programming is defining your own functions.
5715 Calc's @kbd{Z F} command allows you to define a function name and
5716 key sequence to correspond to any formula. Programming commands use
5717 the shift-@kbd{Z} prefix; the user commands they create use the lower
5718 case @kbd{z} prefix.
5722 1: x + x^2 / 2 + x^3 / 6 + 1 1: x + x^2 / 2 + x^3 / 6 + 1
5725 ' 1 + x + x^2/2! + x^3/3! @key{RET} Z F e myexp @key{RET} @key{RET} @key{RET} y
5729 This polynomial is a Taylor series approximation to @samp{exp(x)}.
5730 The @kbd{Z F} command asks a number of questions. The above answers
5731 say that the key sequence for our function should be @kbd{z e}; the
5732 @kbd{M-x} equivalent should be @code{calc-myexp}; the name of the
5733 function in algebraic formulas should also be @code{myexp}; the
5734 default argument list @samp{(x)} is acceptable; and finally @kbd{y}
5735 answers the question ``leave it in symbolic form for non-constant
5740 1: 1.3495 2: 1.3495 3: 1.3495
5741 . 1: 1.34986 2: 1.34986
5745 .3 z e .3 E ' a+1 @key{RET} z e
5750 First we call our new @code{exp} approximation with 0.3 as an
5751 argument, and compare it with the true @code{exp} function. Then
5752 we note that, as requested, if we try to give @kbd{z e} an
5753 argument that isn't a plain number, it leaves the @code{myexp}
5754 function call in symbolic form. If we had answered @kbd{n} to the
5755 final question, @samp{myexp(a + 1)} would have evaluated by plugging
5756 in @samp{a + 1} for @samp{x} in the defining formula.
5758 @cindex Sine integral Si(x)
5763 (@bullet{}) @strong{Exercise 1.} The ``sine integral'' function
5764 @texline @math{{\rm Si}(x)}
5765 @infoline @expr{Si(x)}
5766 is defined as the integral of @samp{sin(t)/t} for
5767 @expr{t = 0} to @expr{x} in radians. (It was invented because this
5768 integral has no solution in terms of basic functions; if you give it
5769 to Calc's @kbd{a i} command, it will ponder it for a long time and then
5770 give up.) We can use the numerical integration command, however,
5771 which in algebraic notation is written like @samp{ninteg(f(t), t, 0, x)}
5772 with any integrand @samp{f(t)}. Define a @kbd{z s} command and
5773 @code{Si} function that implement this. You will need to edit the
5774 default argument list a bit. As a test, @samp{Si(1)} should return
5775 0.946083. (If you don't get this answer, you might want to check that
5776 Calc is in Radians mode. Also, @code{ninteg} will run a lot faster if
5777 you reduce the precision to, say, six digits beforehand.)
5778 @xref{Programming Answer 1, 1}. (@bullet{})
5780 The simplest way to do real ``programming'' of Emacs is to define a
5781 @dfn{keyboard macro}. A keyboard macro is simply a sequence of
5782 keystrokes which Emacs has stored away and can play back on demand.
5783 For example, if you find yourself typing @kbd{H a S x @key{RET}} often,
5784 you may wish to program a keyboard macro to type this for you.
5788 1: y = sqrt(x) 1: x = y^2
5791 ' y=sqrt(x) @key{RET} C-x ( H a S x @key{RET} C-x )
5793 1: y = cos(x) 1: x = s1 arccos(y) + 2 n1 pi
5796 ' y=cos(x) @key{RET} X
5801 When you type @kbd{C-x (}, Emacs begins recording. But it is also
5802 still ready to execute your keystrokes, so you're really ``training''
5803 Emacs by walking it through the procedure once. When you type
5804 @w{@kbd{C-x )}}, the macro is recorded. You can now type @kbd{X} to
5805 re-execute the same keystrokes.
5807 You can give a name to your macro by typing @kbd{Z K}.
5811 1: . 1: y = x^4 1: x = s2 sqrt(s1 sqrt(y))
5814 Z K x @key{RET} ' y=x^4 @key{RET} z x
5819 Notice that we use shift-@kbd{Z} to define the command, and lower-case
5820 @kbd{z} to call it up.
5822 Keyboard macros can call other macros.
5826 1: abs(x) 1: x = s1 y 1: 2 / x 1: x = 2 / y
5829 ' abs(x) @key{RET} C-x ( ' y @key{RET} a = z x C-x ) ' 2/x @key{RET} X
5833 (@bullet{}) @strong{Exercise 2.} Define a keyboard macro to negate
5834 the item in level 3 of the stack, without disturbing the rest of
5835 the stack. @xref{Programming Answer 2, 2}. (@bullet{})
5837 (@bullet{}) @strong{Exercise 3.} Define keyboard macros to compute
5838 the following functions:
5843 @texline @math{\displaystyle{\sin x \over x}},
5844 @infoline @expr{sin(x) / x},
5845 where @expr{x} is the number on the top of the stack.
5848 Compute the base-@expr{b} logarithm, just like the @kbd{B} key except
5849 the arguments are taken in the opposite order.
5852 Produce a vector of integers from 1 to the integer on the top of
5856 @xref{Programming Answer 3, 3}. (@bullet{})
5858 (@bullet{}) @strong{Exercise 4.} Define a keyboard macro to compute
5859 the average (mean) value of a list of numbers.
5860 @xref{Programming Answer 4, 4}. (@bullet{})
5862 In many programs, some of the steps must execute several times.
5863 Calc has @dfn{looping} commands that allow this. Loops are useful
5864 inside keyboard macros, but actually work at any time.
5868 1: x^6 2: x^6 1: 360 x^2
5872 ' x^6 @key{RET} 4 Z < a d x @key{RET} Z >
5877 Here we have computed the fourth derivative of @expr{x^6} by
5878 enclosing a derivative command in a ``repeat loop'' structure.
5879 This structure pops a repeat count from the stack, then
5880 executes the body of the loop that many times.
5882 If you make a mistake while entering the body of the loop,
5883 type @w{@kbd{Z C-g}} to cancel the loop command.
5885 @cindex Fibonacci numbers
5886 Here's another example:
5895 1 @key{RET} @key{RET} 20 Z < @key{TAB} C-j + Z >
5900 The numbers in levels 2 and 1 should be the 21st and 22nd Fibonacci
5901 numbers, respectively. (To see what's going on, try a few repetitions
5902 of the loop body by hand; @kbd{C-j}, also on the Line-Feed or @key{LFD}
5903 key if you have one, makes a copy of the number in level 2.)
5905 @cindex Golden ratio
5906 @cindex Phi, golden ratio
5907 A fascinating property of the Fibonacci numbers is that the @expr{n}th
5908 Fibonacci number can be found directly by computing
5909 @texline @math{\phi^n / \sqrt{5}}
5910 @infoline @expr{phi^n / sqrt(5)}
5911 and then rounding to the nearest integer, where
5912 @texline @math{\phi} (``phi''),
5913 @infoline @expr{phi},
5914 the ``golden ratio,'' is
5915 @texline @math{(1 + \sqrt{5}) / 2}.
5916 @infoline @expr{(1 + sqrt(5)) / 2}.
5917 (For convenience, this constant is available from the @code{phi}
5918 variable, or the @kbd{I H P} command.)
5922 1: 1.61803 1: 24476.0000409 1: 10945.9999817 1: 10946
5929 @cindex Continued fractions
5930 (@bullet{}) @strong{Exercise 5.} The @dfn{continued fraction}
5932 @texline @math{\phi}
5933 @infoline @expr{phi}
5935 @texline @math{1 + 1/(1 + 1/(1 + 1/( \ldots )))}.
5936 @infoline @expr{1 + 1/(1 + 1/(1 + 1/( ...@: )))}.
5937 We can compute an approximate value by carrying this however far
5938 and then replacing the innermost
5939 @texline @math{1/( \ldots )}
5940 @infoline @expr{1/( ...@: )}
5942 @texline @math{\phi}
5943 @infoline @expr{phi}
5944 using a twenty-term continued fraction.
5945 @xref{Programming Answer 5, 5}. (@bullet{})
5947 (@bullet{}) @strong{Exercise 6.} Linear recurrences like the one for
5948 Fibonacci numbers can be expressed in terms of matrices. Given a
5949 vector @w{@expr{[a, b]}} determine a matrix which, when multiplied by this
5950 vector, produces the vector @expr{[b, c]}, where @expr{a}, @expr{b} and
5951 @expr{c} are three successive Fibonacci numbers. Now write a program
5952 that, given an integer @expr{n}, computes the @expr{n}th Fibonacci number
5953 using matrix arithmetic. @xref{Programming Answer 6, 6}. (@bullet{})
5955 @cindex Harmonic numbers
5956 A more sophisticated kind of loop is the @dfn{for} loop. Suppose
5957 we wish to compute the 20th ``harmonic'' number, which is equal to
5958 the sum of the reciprocals of the integers from 1 to 20.
5967 0 @key{RET} 1 @key{RET} 20 Z ( & + 1 Z )
5972 The ``for'' loop pops two numbers, the lower and upper limits, then
5973 repeats the body of the loop as an internal counter increases from
5974 the lower limit to the upper one. Just before executing the loop
5975 body, it pushes the current loop counter. When the loop body
5976 finishes, it pops the ``step,'' i.e., the amount by which to
5977 increment the loop counter. As you can see, our loop always
5980 This harmonic number function uses the stack to hold the running
5981 total as well as for the various loop housekeeping functions. If
5982 you find this disorienting, you can sum in a variable instead:
5986 1: 0 2: 1 . 1: 3.597739
5990 0 t 7 1 @key{RET} 20 Z ( & s + 7 1 Z ) r 7
5995 The @kbd{s +} command adds the top-of-stack into the value in a
5996 variable (and removes that value from the stack).
5998 It's worth noting that many jobs that call for a ``for'' loop can
5999 also be done more easily by Calc's high-level operations. Two
6000 other ways to compute harmonic numbers are to use vector mapping
6001 and reduction (@kbd{v x 20}, then @w{@kbd{V M &}}, then @kbd{V R +}),
6002 or to use the summation command @kbd{a +}. Both of these are
6003 probably easier than using loops. However, there are some
6004 situations where loops really are the way to go:
6006 (@bullet{}) @strong{Exercise 7.} Use a ``for'' loop to find the first
6007 harmonic number which is greater than 4.0.
6008 @xref{Programming Answer 7, 7}. (@bullet{})
6010 Of course, if we're going to be using variables in our programs,
6011 we have to worry about the programs clobbering values that the
6012 caller was keeping in those same variables. This is easy to
6017 . 1: 0.6667 1: 0.6667 3: 0.6667
6022 Z ` p 4 @key{RET} 2 @key{RET} 3 / s 7 s s a @key{RET} Z ' r 7 s r a @key{RET}
6027 When we type @kbd{Z `} (that's a back-quote character), Calc saves
6028 its mode settings and the contents of the ten ``quick variables''
6029 for later reference. When we type @kbd{Z '} (that's an apostrophe
6030 now), Calc restores those saved values. Thus the @kbd{p 4} and
6031 @kbd{s 7} commands have no effect outside this sequence. Wrapping
6032 this around the body of a keyboard macro ensures that it doesn't
6033 interfere with what the user of the macro was doing. Notice that
6034 the contents of the stack, and the values of named variables,
6035 survive past the @kbd{Z '} command.
6037 @cindex Bernoulli numbers, approximate
6038 The @dfn{Bernoulli numbers} are a sequence with the interesting
6039 property that all of the odd Bernoulli numbers are zero, and the
6040 even ones, while difficult to compute, can be roughly approximated
6042 @texline @math{\displaystyle{2 n! \over (2 \pi)^n}}.
6043 @infoline @expr{2 n!@: / (2 pi)^n}.
6044 Let's write a keyboard macro to compute (approximate) Bernoulli numbers.
6045 (Calc has a command, @kbd{k b}, to compute exact Bernoulli numbers, but
6046 this command is very slow for large @expr{n} since the higher Bernoulli
6047 numbers are very large fractions.)
6054 10 C-x ( @key{RET} 2 % Z [ @key{DEL} 0 Z : ' 2 $! / (2 pi)^$ @key{RET} = Z ] C-x )
6059 You can read @kbd{Z [} as ``then,'' @kbd{Z :} as ``else,'' and
6060 @kbd{Z ]} as ``end-if.'' There is no need for an explicit ``if''
6061 command. For the purposes of @w{@kbd{Z [}}, the condition is ``true''
6062 if the value it pops from the stack is a nonzero number, or ``false''
6063 if it pops zero or something that is not a number (like a formula).
6064 Here we take our integer argument modulo 2; this will be nonzero
6065 if we're asking for an odd Bernoulli number.
6067 The actual tenth Bernoulli number is @expr{5/66}.
6071 3: 0.0756823 1: 0 1: 0.25305 1: 0 1: 1.16659
6076 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
6080 Just to exercise loops a bit more, let's compute a table of even
6085 3: [] 1: [0.10132, 0.03079, 0.02340, 0.033197, ...]
6090 [ ] 2 @key{RET} 30 Z ( X | 2 Z )
6095 The vertical-bar @kbd{|} is the vector-concatenation command. When
6096 we execute it, the list we are building will be in stack level 2
6097 (initially this is an empty list), and the next Bernoulli number
6098 will be in level 1. The effect is to append the Bernoulli number
6099 onto the end of the list. (To create a table of exact fractional
6100 Bernoulli numbers, just replace @kbd{X} with @kbd{k b} in the above
6101 sequence of keystrokes.)
6103 With loops and conditionals, you can program essentially anything
6104 in Calc. One other command that makes looping easier is @kbd{Z /},
6105 which takes a condition from the stack and breaks out of the enclosing
6106 loop if the condition is true (non-zero). You can use this to make
6107 ``while'' and ``until'' style loops.
6109 If you make a mistake when entering a keyboard macro, you can edit
6110 it using @kbd{Z E}. First, you must attach it to a key with @kbd{Z K}.
6111 One technique is to enter a throwaway dummy definition for the macro,
6112 then enter the real one in the edit command.
6116 1: 3 1: 3 Calc Macro Edit Mode.
6117 . . Original keys: 1 <return> 2 +
6124 C-x ( 1 @key{RET} 2 + C-x ) Z K h @key{RET} Z E h
6129 A keyboard macro is stored as a pure keystroke sequence. The
6130 @file{edmacro} package (invoked by @kbd{Z E}) scans along the
6131 macro and tries to decode it back into human-readable steps.
6132 Descriptions of the keystrokes are given as comments, which begin with
6133 @samp{;;}, and which are ignored when the edited macro is saved.
6134 Spaces and line breaks are also ignored when the edited macro is saved.
6135 To enter a space into the macro, type @code{SPC}. All the special
6136 characters @code{RET}, @code{LFD}, @code{TAB}, @code{SPC}, @code{DEL},
6137 and @code{NUL} must be written in all uppercase, as must the prefixes
6138 @code{C-} and @code{M-}.
6140 Let's edit in a new definition, for computing harmonic numbers.
6141 First, erase the four lines of the old definition. Then, type
6142 in the new definition (or use Emacs @kbd{M-w} and @kbd{C-y} commands
6143 to copy it from this page of the Info file; you can of course skip
6144 typing the comments, which begin with @samp{;;}).
6147 Z` ;; calc-kbd-push (Save local values)
6148 0 ;; calc digits (Push a zero onto the stack)
6149 st ;; calc-store-into (Store it in the following variable)
6150 1 ;; calc quick variable (Quick variable q1)
6151 1 ;; calc digits (Initial value for the loop)
6152 TAB ;; calc-roll-down (Swap initial and final)
6153 Z( ;; calc-kbd-for (Begin the "for" loop)
6154 & ;; calc-inv (Take the reciprocal)
6155 s+ ;; calc-store-plus (Add to the following variable)
6156 1 ;; calc quick variable (Quick variable q1)
6157 1 ;; calc digits (The loop step is 1)
6158 Z) ;; calc-kbd-end-for (End the "for" loop)
6159 sr ;; calc-recall (Recall the final accumulated value)
6160 1 ;; calc quick variable (Quick variable q1)
6161 Z' ;; calc-kbd-pop (Restore values)
6165 Press @kbd{C-c C-c} to finish editing and return to the Calculator.
6176 The @file{edmacro} package defines a handy @code{read-kbd-macro} command
6177 which reads the current region of the current buffer as a sequence of
6178 keystroke names, and defines that sequence on the @kbd{X}
6179 (and @kbd{C-x e}) key. Because this is so useful, Calc puts this
6180 command on the @kbd{C-x * m} key. Try reading in this macro in the
6181 following form: Press @kbd{C-@@} (or @kbd{C-@key{SPC}}) at
6182 one end of the text below, then type @kbd{C-x * m} at the other.
6194 (@bullet{}) @strong{Exercise 8.} A general algorithm for solving
6195 equations numerically is @dfn{Newton's Method}. Given the equation
6196 @expr{f(x) = 0} for any function @expr{f}, and an initial guess
6197 @expr{x_0} which is reasonably close to the desired solution, apply
6198 this formula over and over:
6202 new_x = x - f(x)/f'(x)
6207 $$ x_{\rm new} = x - {f(x) \over f^{\prime}(x)} $$
6212 where @expr{f'(x)} is the derivative of @expr{f}. The @expr{x}
6213 values will quickly converge to a solution, i.e., eventually
6214 @texline @math{x_{\rm new}}
6215 @infoline @expr{new_x}
6216 and @expr{x} will be equal to within the limits
6217 of the current precision. Write a program which takes a formula
6218 involving the variable @expr{x}, and an initial guess @expr{x_0},
6219 on the stack, and produces a value of @expr{x} for which the formula
6220 is zero. Use it to find a solution of
6221 @texline @math{\sin(\cos x) = 0.5}
6222 @infoline @expr{sin(cos(x)) = 0.5}
6223 near @expr{x = 4.5}. (Use angles measured in radians.) Note that
6224 the built-in @w{@kbd{a R}} (@code{calc-find-root}) command uses Newton's
6225 method when it is able. @xref{Programming Answer 8, 8}. (@bullet{})
6227 @cindex Digamma function
6228 @cindex Gamma constant, Euler's
6229 @cindex Euler's gamma constant
6230 (@bullet{}) @strong{Exercise 9.} The @dfn{digamma} function
6231 @texline @math{\psi(z) (``psi'')}
6232 @infoline @expr{psi(z)}
6233 is defined as the derivative of
6234 @texline @math{\ln \Gamma(z)}.
6235 @infoline @expr{ln(gamma(z))}.
6236 For large values of @expr{z}, it can be approximated by the infinite sum
6240 psi(z) ~= ln(z) - 1/2z - sum(bern(2 n) / 2 n z^(2 n), n, 1, inf)
6245 $$ \psi(z) \approx \ln z - {1\over2z} -
6246 \sum_{n=1}^\infty {\code{bern}(2 n) \over 2 n z^{2n}}
6253 @texline @math{\sum}
6254 @infoline @expr{sum}
6255 represents the sum over @expr{n} from 1 to infinity
6256 (or to some limit high enough to give the desired accuracy), and
6257 the @code{bern} function produces (exact) Bernoulli numbers.
6258 While this sum is not guaranteed to converge, in practice it is safe.
6259 An interesting mathematical constant is Euler's gamma, which is equal
6260 to about 0.5772. One way to compute it is by the formula,
6261 @texline @math{\gamma = -\psi(1)}.
6262 @infoline @expr{gamma = -psi(1)}.
6263 Unfortunately, 1 isn't a large enough argument
6264 for the above formula to work (5 is a much safer value for @expr{z}).
6265 Fortunately, we can compute
6266 @texline @math{\psi(1)}
6267 @infoline @expr{psi(1)}
6269 @texline @math{\psi(5)}
6270 @infoline @expr{psi(5)}
6271 using the recurrence
6272 @texline @math{\psi(z+1) = \psi(z) + {1 \over z}}.
6273 @infoline @expr{psi(z+1) = psi(z) + 1/z}.
6274 Your task: Develop a program to compute
6275 @texline @math{\psi(z)};
6276 @infoline @expr{psi(z)};
6277 it should ``pump up'' @expr{z}
6278 if necessary to be greater than 5, then use the above summation
6279 formula. Use looping commands to compute the sum. Use your function
6281 @texline @math{\gamma}
6282 @infoline @expr{gamma}
6283 to twelve decimal places. (Calc has a built-in command
6284 for Euler's constant, @kbd{I P}, which you can use to check your answer.)
6285 @xref{Programming Answer 9, 9}. (@bullet{})
6287 @cindex Polynomial, list of coefficients
6288 (@bullet{}) @strong{Exercise 10.} Given a polynomial in @expr{x} and
6289 a number @expr{m} on the stack, where the polynomial is of degree
6290 @expr{m} or less (i.e., does not have any terms higher than @expr{x^m}),
6291 write a program to convert the polynomial into a list-of-coefficients
6292 notation. For example, @expr{5 x^4 + (x + 1)^2} with @expr{m = 6}
6293 should produce the list @expr{[1, 2, 1, 0, 5, 0, 0]}. Also develop
6294 a way to convert from this form back to the standard algebraic form.
6295 @xref{Programming Answer 10, 10}. (@bullet{})
6298 (@bullet{}) @strong{Exercise 11.} The @dfn{Stirling numbers of the
6299 first kind} are defined by the recurrences,
6303 s(n,n) = 1 for n >= 0,
6304 s(n,0) = 0 for n > 0,
6305 s(n+1,m) = s(n,m-1) - n s(n,m) for n >= m >= 1.
6310 $$ \eqalign{ s(n,n) &= 1 \qquad \hbox{for } n \ge 0, \cr
6311 s(n,0) &= 0 \qquad \hbox{for } n > 0, \cr
6312 s(n+1,m) &= s(n,m-1) - n \, s(n,m) \qquad
6313 \hbox{for } n \ge m \ge 1.}
6317 (These numbers are also sometimes written $\displaystyle{n \brack m}$.)
6320 This can be implemented using a @dfn{recursive} program in Calc; the
6321 program must invoke itself in order to calculate the two righthand
6322 terms in the general formula. Since it always invokes itself with
6323 ``simpler'' arguments, it's easy to see that it must eventually finish
6324 the computation. Recursion is a little difficult with Emacs keyboard
6325 macros since the macro is executed before its definition is complete.
6326 So here's the recommended strategy: Create a ``dummy macro'' and assign
6327 it to a key with, e.g., @kbd{Z K s}. Now enter the true definition,
6328 using the @kbd{z s} command to call itself recursively, then assign it
6329 to the same key with @kbd{Z K s}. Now the @kbd{z s} command will run
6330 the complete recursive program. (Another way is to use @w{@kbd{Z E}}
6331 or @kbd{C-x * m} (@code{read-kbd-macro}) to read the whole macro at once,
6332 thus avoiding the ``training'' phase.) The task: Write a program
6333 that computes Stirling numbers of the first kind, given @expr{n} and
6334 @expr{m} on the stack. Test it with @emph{small} inputs like
6335 @expr{s(4,2)}. (There is a built-in command for Stirling numbers,
6336 @kbd{k s}, which you can use to check your answers.)
6337 @xref{Programming Answer 11, 11}. (@bullet{})
6339 The programming commands we've seen in this part of the tutorial
6340 are low-level, general-purpose operations. Often you will find
6341 that a higher-level function, such as vector mapping or rewrite
6342 rules, will do the job much more easily than a detailed, step-by-step
6345 (@bullet{}) @strong{Exercise 12.} Write another program for
6346 computing Stirling numbers of the first kind, this time using
6347 rewrite rules. Once again, @expr{n} and @expr{m} should be taken
6348 from the stack. @xref{Programming Answer 12, 12}. (@bullet{})
6353 This ends the tutorial section of the Calc manual. Now you know enough
6354 about Calc to use it effectively for many kinds of calculations. But
6355 Calc has many features that were not even touched upon in this tutorial.
6357 The rest of this manual tells the whole story.
6359 @c Volume II of this manual, the @dfn{Calc Reference}, tells the whole story.
6362 @node Answers to Exercises, , Programming Tutorial, Tutorial
6363 @section Answers to Exercises
6366 This section includes answers to all the exercises in the Calc tutorial.
6369 * RPN Answer 1:: 1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -
6370 * RPN Answer 2:: 2*4 + 7*9.5 + 5/4
6371 * RPN Answer 3:: Operating on levels 2 and 3
6372 * RPN Answer 4:: Joe's complex problems
6373 * Algebraic Answer 1:: Simulating Q command
6374 * Algebraic Answer 2:: Joe's algebraic woes
6375 * Algebraic Answer 3:: 1 / 0
6376 * Modes Answer 1:: 3#0.1 = 3#0.0222222?
6377 * Modes Answer 2:: 16#f.e8fe15
6378 * Modes Answer 3:: Joe's rounding bug
6379 * Modes Answer 4:: Why floating point?
6380 * Arithmetic Answer 1:: Why the \ command?
6381 * Arithmetic Answer 2:: Tripping up the B command
6382 * Vector Answer 1:: Normalizing a vector
6383 * Vector Answer 2:: Average position
6384 * Matrix Answer 1:: Row and column sums
6385 * Matrix Answer 2:: Symbolic system of equations
6386 * Matrix Answer 3:: Over-determined system
6387 * List Answer 1:: Powers of two
6388 * List Answer 2:: Least-squares fit with matrices
6389 * List Answer 3:: Geometric mean
6390 * List Answer 4:: Divisor function
6391 * List Answer 5:: Duplicate factors
6392 * List Answer 6:: Triangular list
6393 * List Answer 7:: Another triangular list
6394 * List Answer 8:: Maximum of Bessel function
6395 * List Answer 9:: Integers the hard way
6396 * List Answer 10:: All elements equal
6397 * List Answer 11:: Estimating pi with darts
6398 * List Answer 12:: Estimating pi with matchsticks
6399 * List Answer 13:: Hash codes
6400 * List Answer 14:: Random walk
6401 * Types Answer 1:: Square root of pi times rational
6402 * Types Answer 2:: Infinities
6403 * Types Answer 3:: What can "nan" be?
6404 * Types Answer 4:: Abbey Road
6405 * Types Answer 5:: Friday the 13th
6406 * Types Answer 6:: Leap years
6407 * Types Answer 7:: Erroneous donut
6408 * Types Answer 8:: Dividing intervals
6409 * Types Answer 9:: Squaring intervals
6410 * Types Answer 10:: Fermat's primality test
6411 * Types Answer 11:: pi * 10^7 seconds
6412 * Types Answer 12:: Abbey Road on CD
6413 * Types Answer 13:: Not quite pi * 10^7 seconds
6414 * Types Answer 14:: Supercomputers and c
6415 * Types Answer 15:: Sam the Slug
6416 * Algebra Answer 1:: Squares and square roots
6417 * Algebra Answer 2:: Building polynomial from roots
6418 * Algebra Answer 3:: Integral of x sin(pi x)
6419 * Algebra Answer 4:: Simpson's rule
6420 * Rewrites Answer 1:: Multiplying by conjugate
6421 * Rewrites Answer 2:: Alternative fib rule
6422 * Rewrites Answer 3:: Rewriting opt(a) + opt(b) x
6423 * Rewrites Answer 4:: Sequence of integers
6424 * Rewrites Answer 5:: Number of terms in sum
6425 * Rewrites Answer 6:: Truncated Taylor series
6426 * Programming Answer 1:: Fresnel's C(x)
6427 * Programming Answer 2:: Negate third stack element
6428 * Programming Answer 3:: Compute sin(x) / x, etc.
6429 * Programming Answer 4:: Average value of a list
6430 * Programming Answer 5:: Continued fraction phi
6431 * Programming Answer 6:: Matrix Fibonacci numbers
6432 * Programming Answer 7:: Harmonic number greater than 4
6433 * Programming Answer 8:: Newton's method
6434 * Programming Answer 9:: Digamma function
6435 * Programming Answer 10:: Unpacking a polynomial
6436 * Programming Answer 11:: Recursive Stirling numbers
6437 * Programming Answer 12:: Stirling numbers with rewrites
6440 @c The following kludgery prevents the individual answers from
6441 @c being entered on the table of contents.
6443 \global\let\oldwrite=\write
6444 \gdef\skipwrite#1#2{\let\write=\oldwrite}
6445 \global\let\oldchapternofonts=\chapternofonts
6446 \gdef\chapternofonts{\let\write=\skipwrite\oldchapternofonts}
6449 @node RPN Answer 1, RPN Answer 2, Answers to Exercises, Answers to Exercises
6450 @subsection RPN Tutorial Exercise 1
6453 @kbd{1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -}
6456 @texline @math{1 - (2 \times (3 + 4)) = -13}.
6457 @infoline @expr{1 - (2 * (3 + 4)) = -13}.
6459 @node RPN Answer 2, RPN Answer 3, RPN Answer 1, Answers to Exercises
6460 @subsection RPN Tutorial Exercise 2
6463 @texline @math{2\times4 + 7\times9.5 + {5\over4} = 75.75}
6464 @infoline @expr{2*4 + 7*9.5 + 5/4 = 75.75}
6466 After computing the intermediate term
6467 @texline @math{2\times4 = 8},
6468 @infoline @expr{2*4 = 8},
6469 you can leave that result on the stack while you compute the second
6470 term. With both of these results waiting on the stack you can then
6471 compute the final term, then press @kbd{+ +} to add everything up.
6480 2 @key{RET} 4 * 7 @key{RET} 9.5 *
6487 4: 8 3: 8 2: 8 1: 75.75
6488 3: 66.5 2: 66.5 1: 67.75 .
6497 Alternatively, you could add the first two terms before going on
6498 with the third term.
6502 2: 8 1: 74.5 3: 74.5 2: 74.5 1: 75.75
6503 1: 66.5 . 2: 5 1: 1.25 .
6507 ... + 5 @key{RET} 4 / +
6511 On an old-style RPN calculator this second method would have the
6512 advantage of using only three stack levels. But since Calc's stack
6513 can grow arbitrarily large this isn't really an issue. Which method
6514 you choose is purely a matter of taste.
6516 @node RPN Answer 3, RPN Answer 4, RPN Answer 2, Answers to Exercises
6517 @subsection RPN Tutorial Exercise 3
6520 The @key{TAB} key provides a way to operate on the number in level 2.
6524 3: 10 3: 10 4: 10 3: 10 3: 10
6525 2: 20 2: 30 3: 30 2: 30 2: 21
6526 1: 30 1: 20 2: 20 1: 21 1: 30
6530 @key{TAB} 1 + @key{TAB}
6534 Similarly, @kbd{M-@key{TAB}} gives you access to the number in level 3.
6538 3: 10 3: 21 3: 21 3: 30 3: 11
6539 2: 21 2: 30 2: 30 2: 11 2: 21
6540 1: 30 1: 10 1: 11 1: 21 1: 30
6543 M-@key{TAB} 1 + M-@key{TAB} M-@key{TAB}
6547 @node RPN Answer 4, Algebraic Answer 1, RPN Answer 3, Answers to Exercises
6548 @subsection RPN Tutorial Exercise 4
6551 Either @kbd{( 2 , 3 )} or @kbd{( 2 @key{SPC} 3 )} would have worked,
6552 but using both the comma and the space at once yields:
6556 1: ( ... 2: ( ... 1: (2, ... 2: (2, ... 2: (2, ...
6557 . 1: 2 . 1: (2, ... 1: (2, 3)
6564 Joe probably tried to type @kbd{@key{TAB} @key{DEL}} to swap the
6565 extra incomplete object to the top of the stack and delete it.
6566 But a feature of Calc is that @key{DEL} on an incomplete object
6567 deletes just one component out of that object, so he had to press
6568 @key{DEL} twice to finish the job.
6572 2: (2, ... 2: (2, 3) 2: (2, 3) 1: (2, 3)
6573 1: (2, 3) 1: (2, ... 1: ( ... .
6576 @key{TAB} @key{DEL} @key{DEL}
6580 (As it turns out, deleting the second-to-top stack entry happens often
6581 enough that Calc provides a special key, @kbd{M-@key{DEL}}, to do just that.
6582 @kbd{M-@key{DEL}} is just like @kbd{@key{TAB} @key{DEL}}, except that it doesn't exhibit
6583 the ``feature'' that tripped poor Joe.)
6585 @node Algebraic Answer 1, Algebraic Answer 2, RPN Answer 4, Answers to Exercises
6586 @subsection Algebraic Entry Tutorial Exercise 1
6589 Type @kbd{' sqrt($) @key{RET}}.
6591 If the @kbd{Q} key is broken, you could use @kbd{' $^0.5 @key{RET}}.
6592 Or, RPN style, @kbd{0.5 ^}.
6594 (Actually, @samp{$^1:2}, using the fraction one-half as the power, is
6595 a closer equivalent, since @samp{9^0.5} yields @expr{3.0} whereas
6596 @samp{sqrt(9)} and @samp{9^1:2} yield the exact integer @expr{3}.)
6598 @node Algebraic Answer 2, Algebraic Answer 3, Algebraic Answer 1, Answers to Exercises
6599 @subsection Algebraic Entry Tutorial Exercise 2
6602 In the formula @samp{2 x (1+y)}, @samp{x} was interpreted as a function
6603 name with @samp{1+y} as its argument. Assigning a value to a variable
6604 has no relation to a function by the same name. Joe needed to use an
6605 explicit @samp{*} symbol here: @samp{2 x*(1+y)}.
6607 @node Algebraic Answer 3, Modes Answer 1, Algebraic Answer 2, Answers to Exercises
6608 @subsection Algebraic Entry Tutorial Exercise 3
6611 The result from @kbd{1 @key{RET} 0 /} will be the formula @expr{1 / 0}.
6612 The ``function'' @samp{/} cannot be evaluated when its second argument
6613 is zero, so it is left in symbolic form. When you now type @kbd{0 *},
6614 the result will be zero because Calc uses the general rule that ``zero
6615 times anything is zero.''
6617 @c [fix-ref Infinities]
6618 The @kbd{m i} command enables an @dfn{Infinite mode} in which @expr{1 / 0}
6619 results in a special symbol that represents ``infinity.'' If you
6620 multiply infinity by zero, Calc uses another special new symbol to
6621 show that the answer is ``indeterminate.'' @xref{Infinities}, for
6622 further discussion of infinite and indeterminate values.
6624 @node Modes Answer 1, Modes Answer 2, Algebraic Answer 3, Answers to Exercises
6625 @subsection Modes Tutorial Exercise 1
6628 Calc always stores its numbers in decimal, so even though one-third has
6629 an exact base-3 representation (@samp{3#0.1}), it is still stored as
6630 0.3333333 (chopped off after 12 or however many decimal digits) inside
6631 the calculator's memory. When this inexact number is converted back
6632 to base 3 for display, it may still be slightly inexact. When we
6633 multiply this number by 3, we get 0.999999, also an inexact value.
6635 When Calc displays a number in base 3, it has to decide how many digits
6636 to show. If the current precision is 12 (decimal) digits, that corresponds
6637 to @samp{12 / log10(3) = 25.15} base-3 digits. Because 25.15 is not an
6638 exact integer, Calc shows only 25 digits, with the result that stored
6639 numbers carry a little bit of extra information that may not show up on
6640 the screen. When Joe entered @samp{3#0.2}, the stored number 0.666666
6641 happened to round to a pleasing value when it lost that last 0.15 of a
6642 digit, but it was still inexact in Calc's memory. When he divided by 2,
6643 he still got the dreaded inexact value 0.333333. (Actually, he divided
6644 0.666667 by 2 to get 0.333334, which is why he got something a little
6645 higher than @code{3#0.1} instead of a little lower.)
6647 If Joe didn't want to be bothered with all this, he could have typed
6648 @kbd{M-24 d n} to display with one less digit than the default. (If
6649 you give @kbd{d n} a negative argument, it uses default-minus-that,
6650 so @kbd{M-- d n} would be an easier way to get the same effect.) Those
6651 inexact results would still be lurking there, but they would now be
6652 rounded to nice, natural-looking values for display purposes. (Remember,
6653 @samp{0.022222} in base 3 is like @samp{0.099999} in base 10; rounding
6654 off one digit will round the number up to @samp{0.1}.) Depending on the
6655 nature of your work, this hiding of the inexactness may be a benefit or
6656 a danger. With the @kbd{d n} command, Calc gives you the choice.
6658 Incidentally, another consequence of all this is that if you type
6659 @kbd{M-30 d n} to display more digits than are ``really there,''
6660 you'll see garbage digits at the end of the number. (In decimal
6661 display mode, with decimally-stored numbers, these garbage digits are
6662 always zero so they vanish and you don't notice them.) Because Calc
6663 rounds off that 0.15 digit, there is the danger that two numbers could
6664 be slightly different internally but still look the same. If you feel
6665 uneasy about this, set the @kbd{d n} precision to be a little higher
6666 than normal; you'll get ugly garbage digits, but you'll always be able
6667 to tell two distinct numbers apart.
6669 An interesting side note is that most computers store their
6670 floating-point numbers in binary, and convert to decimal for display.
6671 Thus everyday programs have the same problem: Decimal 0.1 cannot be
6672 represented exactly in binary (try it: @kbd{0.1 d 2}), so @samp{0.1 * 10}
6673 comes out as an inexact approximation to 1 on some machines (though
6674 they generally arrange to hide it from you by rounding off one digit as
6675 we did above). Because Calc works in decimal instead of binary, you can
6676 be sure that numbers that look exact @emph{are} exact as long as you stay
6677 in decimal display mode.
6679 It's not hard to show that any number that can be represented exactly
6680 in binary, octal, or hexadecimal is also exact in decimal, so the kinds
6681 of problems we saw in this exercise are likely to be severe only when
6682 you use a relatively unusual radix like 3.
6684 @node Modes Answer 2, Modes Answer 3, Modes Answer 1, Answers to Exercises
6685 @subsection Modes Tutorial Exercise 2
6687 If the radix is 15 or higher, we can't use the letter @samp{e} to mark
6688 the exponent because @samp{e} is interpreted as a digit. When Calc
6689 needs to display scientific notation in a high radix, it writes
6690 @samp{16#F.E8F*16.^15}. You can enter a number like this as an
6691 algebraic entry. Also, pressing @kbd{e} without any digits before it
6692 normally types @kbd{1e}, but in a high radix it types @kbd{16.^} and
6693 puts you in algebraic entry: @kbd{16#f.e8f @key{RET} e 15 @key{RET} *} is another
6694 way to enter this number.
6696 The reason Calc puts a decimal point in the @samp{16.^} is to prevent
6697 huge integers from being generated if the exponent is large (consider
6698 @samp{16#1.23*16^1000}, where we compute @samp{16^1000} as a giant
6699 exact integer and then throw away most of the digits when we multiply
6700 it by the floating-point @samp{16#1.23}). While this wouldn't normally
6701 matter for display purposes, it could give you a nasty surprise if you
6702 copied that number into a file and later moved it back into Calc.
6704 @node Modes Answer 3, Modes Answer 4, Modes Answer 2, Answers to Exercises
6705 @subsection Modes Tutorial Exercise 3
6708 The answer he got was @expr{0.5000000000006399}.
6710 The problem is not that the square operation is inexact, but that the
6711 sine of 45 that was already on the stack was accurate to only 12 places.
6712 Arbitrary-precision calculations still only give answers as good as
6715 The real problem is that there is no 12-digit number which, when
6716 squared, comes out to 0.5 exactly. The @kbd{f [} and @kbd{f ]}
6717 commands decrease or increase a number by one unit in the last
6718 place (according to the current precision). They are useful for
6719 determining facts like this.
6723 1: 0.707106781187 1: 0.500000000001
6733 1: 0.707106781187 1: 0.707106781186 1: 0.499999999999
6740 A high-precision calculation must be carried out in high precision
6741 all the way. The only number in the original problem which was known
6742 exactly was the quantity 45 degrees, so the precision must be raised
6743 before anything is done after the number 45 has been entered in order
6744 for the higher precision to be meaningful.
6746 @node Modes Answer 4, Arithmetic Answer 1, Modes Answer 3, Answers to Exercises
6747 @subsection Modes Tutorial Exercise 4
6750 Many calculations involve real-world quantities, like the width and
6751 height of a piece of wood or the volume of a jar. Such quantities
6752 can't be measured exactly anyway, and if the data that is input to
6753 a calculation is inexact, doing exact arithmetic on it is a waste
6756 Fractions become unwieldy after too many calculations have been
6757 done with them. For example, the sum of the reciprocals of the
6758 integers from 1 to 10 is 7381:2520. The sum from 1 to 30 is
6759 9304682830147:2329089562800. After a point it will take a long
6760 time to add even one more term to this sum, but a floating-point
6761 calculation of the sum will not have this problem.
6763 Also, rational numbers cannot express the results of all calculations.
6764 There is no fractional form for the square root of two, so if you type
6765 @w{@kbd{2 Q}}, Calc has no choice but to give you a floating-point answer.
6767 @node Arithmetic Answer 1, Arithmetic Answer 2, Modes Answer 4, Answers to Exercises
6768 @subsection Arithmetic Tutorial Exercise 1
6771 Dividing two integers that are larger than the current precision may
6772 give a floating-point result that is inaccurate even when rounded
6773 down to an integer. Consider @expr{123456789 / 2} when the current
6774 precision is 6 digits. The true answer is @expr{61728394.5}, but
6775 with a precision of 6 this will be rounded to
6776 @texline @math{12345700.0/2.0 = 61728500.0}.
6777 @infoline @expr{12345700.@: / 2.@: = 61728500.}.
6778 The result, when converted to an integer, will be off by 106.
6780 Here are two solutions: Raise the precision enough that the
6781 floating-point round-off error is strictly to the right of the
6782 decimal point. Or, convert to Fraction mode so that @expr{123456789 / 2}
6783 produces the exact fraction @expr{123456789:2}, which can be rounded
6784 down by the @kbd{F} command without ever switching to floating-point
6787 @node Arithmetic Answer 2, Vector Answer 1, Arithmetic Answer 1, Answers to Exercises
6788 @subsection Arithmetic Tutorial Exercise 2
6791 @kbd{27 @key{RET} 9 B} could give the exact result @expr{3:2}, but it
6792 does a floating-point calculation instead and produces @expr{1.5}.
6794 Calc will find an exact result for a logarithm if the result is an integer
6795 or (when in Fraction mode) the reciprocal of an integer. But there is
6796 no efficient way to search the space of all possible rational numbers
6797 for an exact answer, so Calc doesn't try.
6799 @node Vector Answer 1, Vector Answer 2, Arithmetic Answer 2, Answers to Exercises
6800 @subsection Vector Tutorial Exercise 1
6803 Duplicate the vector, compute its length, then divide the vector
6804 by its length: @kbd{@key{RET} A /}.
6808 1: [1, 2, 3] 2: [1, 2, 3] 1: [0.27, 0.53, 0.80] 1: 1.
6809 . 1: 3.74165738677 . .
6816 The final @kbd{A} command shows that the normalized vector does
6817 indeed have unit length.
6819 @node Vector Answer 2, Matrix Answer 1, Vector Answer 1, Answers to Exercises
6820 @subsection Vector Tutorial Exercise 2
6823 The average position is equal to the sum of the products of the
6824 positions times their corresponding probabilities. This is the
6825 definition of the dot product operation. So all you need to do
6826 is to put the two vectors on the stack and press @kbd{*}.
6828 @node Matrix Answer 1, Matrix Answer 2, Vector Answer 2, Answers to Exercises
6829 @subsection Matrix Tutorial Exercise 1
6832 The trick is to multiply by a vector of ones. Use @kbd{r 4 [1 1 1] *} to
6833 get the row sum. Similarly, use @kbd{[1 1] r 4 *} to get the column sum.
6835 @node Matrix Answer 2, Matrix Answer 3, Matrix Answer 1, Answers to Exercises
6836 @subsection Matrix Tutorial Exercise 2
6848 $$ \eqalign{ x &+ a y = 6 \cr
6854 Just enter the righthand side vector, then divide by the lefthand side
6859 1: [6, 10] 2: [6, 10] 1: [4 a / (a - b) + 6, 4 / (b - a) ]
6864 ' [6 10] @key{RET} ' [1 a; 1 b] @key{RET} /
6868 This can be made more readable using @kbd{d B} to enable Big display
6874 1: [----- + 6, -----]
6879 Type @kbd{d N} to return to Normal display mode afterwards.
6881 @node Matrix Answer 3, List Answer 1, Matrix Answer 2, Answers to Exercises
6882 @subsection Matrix Tutorial Exercise 3
6886 @texline @math{A^T A \, X = A^T B},
6887 @infoline @expr{trn(A) * A * X = trn(A) * B},
6889 @texline @math{A' = A^T A}
6890 @infoline @expr{A2 = trn(A) * A}
6892 @texline @math{B' = A^T B};
6893 @infoline @expr{B2 = trn(A) * B};
6894 now, we have a system
6895 @texline @math{A' X = B'}
6896 @infoline @expr{A2 * X = B2}
6897 which we can solve using Calc's @samp{/} command.
6911 $$ \openup1\jot \tabskip=0pt plus1fil
6912 \halign to\displaywidth{\tabskip=0pt
6913 $\hfil#$&$\hfil{}#{}$&
6914 $\hfil#$&$\hfil{}#{}$&
6915 $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
6919 2a&+&4b&+&6c&=11 \cr}
6924 The first step is to enter the coefficient matrix. We'll store it in
6925 quick variable number 7 for later reference. Next, we compute the
6932 1: [ [ 1, 2, 3 ] 2: [ [ 1, 4, 7, 2 ] 1: [57, 84, 96]
6933 [ 4, 5, 6 ] [ 2, 5, 6, 4 ] .
6934 [ 7, 6, 0 ] [ 3, 6, 0, 6 ] ]
6935 [ 2, 4, 6 ] ] 1: [6, 2, 3, 11]
6938 ' [1 2 3; 4 5 6; 7 6 0; 2 4 6] @key{RET} s 7 v t [6 2 3 11] *
6943 Now we compute the matrix
6950 2: [57, 84, 96] 1: [-11.64, 14.08, -3.64]
6951 1: [ [ 70, 72, 39 ] .
6961 (The actual computed answer will be slightly inexact due to
6964 Notice that the answers are similar to those for the
6965 @texline @math{3\times3}
6967 system solved in the text. That's because the fourth equation that was
6968 added to the system is almost identical to the first one multiplied
6969 by two. (If it were identical, we would have gotten the exact same
6971 @texline @math{4\times3}
6973 system would be equivalent to the original
6974 @texline @math{3\times3}
6978 Since the first and fourth equations aren't quite equivalent, they
6979 can't both be satisfied at once. Let's plug our answers back into
6980 the original system of equations to see how well they match.
6984 2: [-11.64, 14.08, -3.64] 1: [5.6, 2., 3., 11.2]
6996 This is reasonably close to our original @expr{B} vector,
6997 @expr{[6, 2, 3, 11]}.
6999 @node List Answer 1, List Answer 2, Matrix Answer 3, Answers to Exercises
7000 @subsection List Tutorial Exercise 1
7003 We can use @kbd{v x} to build a vector of integers. This needs to be
7004 adjusted to get the range of integers we desire. Mapping @samp{-}
7005 across the vector will accomplish this, although it turns out the
7006 plain @samp{-} key will work just as well.
7011 1: [1, 2, 3, 4, 5, 6, 7, 8, 9] 1: [-4, -3, -2, -1, 0, 1, 2, 3, 4]
7014 2 v x 9 @key{RET} 5 V M - or 5 -
7019 Now we use @kbd{V M ^} to map the exponentiation operator across the
7024 1: [0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8, 16]
7031 @node List Answer 2, List Answer 3, List Answer 1, Answers to Exercises
7032 @subsection List Tutorial Exercise 2
7035 Given @expr{x} and @expr{y} vectors in quick variables 1 and 2 as before,
7036 the first job is to form the matrix that describes the problem.
7045 $$ m \times x + b \times 1 = y $$
7050 @texline @math{19\times2}
7052 matrix with our @expr{x} vector as one column and
7053 ones as the other column. So, first we build the column of ones, then
7054 we combine the two columns to form our @expr{A} matrix.
7058 2: [1.34, 1.41, 1.49, ... ] 1: [ [ 1.34, 1 ]
7059 1: [1, 1, 1, ...] [ 1.41, 1 ]
7063 r 1 1 v b 19 @key{RET} M-2 v p v t s 3
7069 @texline @math{A^T y}
7070 @infoline @expr{trn(A) * y}
7072 @texline @math{A^T A}
7073 @infoline @expr{trn(A) * A}
7078 1: [33.36554, 13.613] 2: [33.36554, 13.613]
7079 . 1: [ [ 98.0003, 41.63 ]
7083 v t r 2 * r 3 v t r 3 *
7088 (Hey, those numbers look familiar!)
7092 1: [0.52141679, -0.425978]
7099 Since we were solving equations of the form
7100 @texline @math{m \times x + b \times 1 = y},
7101 @infoline @expr{m*x + b*1 = y},
7102 these numbers should be @expr{m} and @expr{b}, respectively. Sure
7103 enough, they agree exactly with the result computed using @kbd{V M} and
7106 The moral of this story: @kbd{V M} and @kbd{V R} will probably solve
7107 your problem, but there is often an easier way using the higher-level
7108 arithmetic functions!
7110 @c [fix-ref Curve Fitting]
7111 In fact, there is a built-in @kbd{a F} command that does least-squares
7112 fits. @xref{Curve Fitting}.
7114 @node List Answer 3, List Answer 4, List Answer 2, Answers to Exercises
7115 @subsection List Tutorial Exercise 3
7118 Move to one end of the list and press @kbd{C-@@} (or @kbd{C-@key{SPC}} or
7119 whatever) to set the mark, then move to the other end of the list
7120 and type @w{@kbd{C-x * g}}.
7124 1: [2.3, 6, 22, 15.1, 7, 15, 14, 7.5, 2.5]
7129 To make things interesting, let's assume we don't know at a glance
7130 how many numbers are in this list. Then we could type:
7134 2: [2.3, 6, 22, ... ] 2: [2.3, 6, 22, ... ]
7135 1: [2.3, 6, 22, ... ] 1: 126356422.5
7145 2: 126356422.5 2: 126356422.5 1: 7.94652913734
7146 1: [2.3, 6, 22, ... ] 1: 9 .
7154 (The @kbd{I ^} command computes the @var{n}th root of a number.
7155 You could also type @kbd{& ^} to take the reciprocal of 9 and
7156 then raise the number to that power.)
7158 @node List Answer 4, List Answer 5, List Answer 3, Answers to Exercises
7159 @subsection List Tutorial Exercise 4
7162 A number @expr{j} is a divisor of @expr{n} if
7163 @texline @math{n \mathbin{\hbox{\code{\%}}} j = 0}.
7164 @infoline @samp{n % j = 0}.
7165 The first step is to get a vector that identifies the divisors.
7169 2: 30 2: [0, 0, 0, 2, ...] 1: [1, 1, 1, 0, ...]
7170 1: [1, 2, 3, 4, ...] 1: 0 .
7173 30 @key{RET} v x 30 @key{RET} s 1 V M % 0 V M a = s 2
7178 This vector has 1's marking divisors of 30 and 0's marking non-divisors.
7180 The zeroth divisor function is just the total number of divisors.
7181 The first divisor function is the sum of the divisors.
7186 2: [1, 2, 3, 4, ...] 1: [1, 2, 3, 0, ...] 1: 72
7187 1: [1, 1, 1, 0, ...] . .
7190 V R + r 1 r 2 V M * V R +
7195 Once again, the last two steps just compute a dot product for which
7196 a simple @kbd{*} would have worked equally well.
7198 @node List Answer 5, List Answer 6, List Answer 4, Answers to Exercises
7199 @subsection List Tutorial Exercise 5
7202 The obvious first step is to obtain the list of factors with @kbd{k f}.
7203 This list will always be in sorted order, so if there are duplicates
7204 they will be right next to each other. A suitable method is to compare
7205 the list with a copy of itself shifted over by one.
7209 1: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19, 0]
7210 . 1: [3, 7, 7, 7, 19, 0] 1: [0, 3, 7, 7, 7, 19]
7213 19551 k f @key{RET} 0 | @key{TAB} 0 @key{TAB} |
7220 1: [0, 0, 1, 1, 0, 0] 1: 2 1: 0
7228 Note that we have to arrange for both vectors to have the same length
7229 so that the mapping operation works; no prime factor will ever be
7230 zero, so adding zeros on the left and right is safe. From then on
7231 the job is pretty straightforward.
7233 Incidentally, Calc provides the
7234 @texline @dfn{M@"obius} @math{\mu}
7235 @infoline @dfn{Moebius mu}
7236 function which is zero if and only if its argument is square-free. It
7237 would be a much more convenient way to do the above test in practice.
7239 @node List Answer 6, List Answer 7, List Answer 5, Answers to Exercises
7240 @subsection List Tutorial Exercise 6
7243 First use @kbd{v x 6 @key{RET}} to get a list of integers, then @kbd{V M v x}
7244 to get a list of lists of integers!
7246 @node List Answer 7, List Answer 8, List Answer 6, Answers to Exercises
7247 @subsection List Tutorial Exercise 7
7250 Here's one solution. First, compute the triangular list from the previous
7251 exercise and type @kbd{1 -} to subtract one from all the elements.
7264 The numbers down the lefthand edge of the list we desire are called
7265 the ``triangular numbers'' (now you know why!). The @expr{n}th
7266 triangular number is the sum of the integers from 1 to @expr{n}, and
7267 can be computed directly by the formula
7268 @texline @math{n (n+1) \over 2}.
7269 @infoline @expr{n * (n+1) / 2}.
7273 2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ]
7274 1: [0, 1, 2, 3, 4, 5] 1: [0, 1, 3, 6, 10, 15]
7277 v x 6 @key{RET} 1 - V M ' $ ($+1)/2 @key{RET}
7282 Adding this list to the above list of lists produces the desired
7291 [10, 11, 12, 13, 14],
7292 [15, 16, 17, 18, 19, 20] ]
7299 If we did not know the formula for triangular numbers, we could have
7300 computed them using a @kbd{V U +} command. We could also have
7301 gotten them the hard way by mapping a reduction across the original
7306 2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ]
7307 1: [ [0], [0, 1], ... ] 1: [0, 1, 3, 6, 10, 15]
7315 (This means ``map a @kbd{V R +} command across the vector,'' and
7316 since each element of the main vector is itself a small vector,
7317 @kbd{V R +} computes the sum of its elements.)
7319 @node List Answer 8, List Answer 9, List Answer 7, Answers to Exercises
7320 @subsection List Tutorial Exercise 8
7323 The first step is to build a list of values of @expr{x}.
7327 1: [1, 2, 3, ..., 21] 1: [0, 1, 2, ..., 20] 1: [0, 0.25, 0.5, ..., 5]
7330 v x 21 @key{RET} 1 - 4 / s 1
7334 Next, we compute the Bessel function values.
7338 1: [0., 0.124, 0.242, ..., -0.328]
7341 V M ' besJ(1,$) @key{RET}
7346 (Another way to do this would be @kbd{1 @key{TAB} V M f j}.)
7348 A way to isolate the maximum value is to compute the maximum using
7349 @kbd{V R X}, then compare all the Bessel values with that maximum.
7353 2: [0., 0.124, 0.242, ... ] 1: [0, 0, 0, ... ] 2: [0, 0, 0, ... ]
7357 @key{RET} V R X V M a = @key{RET} V R + @key{DEL}
7362 It's a good idea to verify, as in the last step above, that only
7363 one value is equal to the maximum. (After all, a plot of
7364 @texline @math{\sin x}
7365 @infoline @expr{sin(x)}
7366 might have many points all equal to the maximum value, 1.)
7368 The vector we have now has a single 1 in the position that indicates
7369 the maximum value of @expr{x}. Now it is a simple matter to convert
7370 this back into the corresponding value itself.
7374 2: [0, 0, 0, ... ] 1: [0, 0., 0., ... ] 1: 1.75
7375 1: [0, 0.25, 0.5, ... ] . .
7382 If @kbd{a =} had produced more than one @expr{1} value, this method
7383 would have given the sum of all maximum @expr{x} values; not very
7384 useful! In this case we could have used @kbd{v m} (@code{calc-mask-vector})
7385 instead. This command deletes all elements of a ``data'' vector that
7386 correspond to zeros in a ``mask'' vector, leaving us with, in this
7387 example, a vector of maximum @expr{x} values.
7389 The built-in @kbd{a X} command maximizes a function using more
7390 efficient methods. Just for illustration, let's use @kbd{a X}
7391 to maximize @samp{besJ(1,x)} over this same interval.
7395 2: besJ(1, x) 1: [1.84115, 0.581865]
7399 ' besJ(1,x), [0..5] @key{RET} a X x @key{RET}
7404 The output from @kbd{a X} is a vector containing the value of @expr{x}
7405 that maximizes the function, and the function's value at that maximum.
7406 As you can see, our simple search got quite close to the right answer.
7408 @node List Answer 9, List Answer 10, List Answer 8, Answers to Exercises
7409 @subsection List Tutorial Exercise 9
7412 Step one is to convert our integer into vector notation.
7416 1: 25129925999 3: 25129925999
7418 1: [11, 10, 9, ..., 1, 0]
7421 25129925999 @key{RET} 10 @key{RET} 12 @key{RET} v x 12 @key{RET} -
7428 1: 25129925999 1: [0, 2, 25, 251, 2512, ... ]
7429 2: [100000000000, ... ] .
7437 (Recall, the @kbd{\} command computes an integer quotient.)
7441 1: [0, 2, 5, 1, 2, 9, 9, 2, 5, 9, 9, 9]
7448 Next we must increment this number. This involves adding one to
7449 the last digit, plus handling carries. There is a carry to the
7450 left out of a digit if that digit is a nine and all the digits to
7451 the right of it are nines.
7455 1: [0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1] 1: [1, 1, 1, 0, 0, 1, ... ]
7465 1: [1, 1, 1, 0, 0, 0, ... ] 1: [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]
7473 Accumulating @kbd{*} across a vector of ones and zeros will preserve
7474 only the initial run of ones. These are the carries into all digits
7475 except the rightmost digit. Concatenating a one on the right takes
7476 care of aligning the carries properly, and also adding one to the
7481 2: [0, 0, 0, 0, ... ] 1: [0, 0, 2, 5, 1, 2, 9, 9, 2, 6, 0, 0, 0]
7482 1: [0, 0, 2, 5, ... ] .
7485 0 r 2 | V M + 10 V M %
7490 Here we have concatenated 0 to the @emph{left} of the original number;
7491 this takes care of shifting the carries by one with respect to the
7492 digits that generated them.
7494 Finally, we must convert this list back into an integer.
7498 3: [0, 0, 2, 5, ... ] 2: [0, 0, 2, 5, ... ]
7499 2: 1000000000000 1: [1000000000000, 100000000000, ... ]
7500 1: [100000000000, ... ] .
7503 10 @key{RET} 12 ^ r 1 |
7510 1: [0, 0, 20000000000, 5000000000, ... ] 1: 25129926000
7518 Another way to do this final step would be to reduce the formula
7519 @w{@samp{10 $$ + $}} across the vector of digits.
7523 1: [0, 0, 2, 5, ... ] 1: 25129926000
7526 V R ' 10 $$ + $ @key{RET}
7530 @node List Answer 10, List Answer 11, List Answer 9, Answers to Exercises
7531 @subsection List Tutorial Exercise 10
7534 For the list @expr{[a, b, c, d]}, the result is @expr{((a = b) = c) = d},
7535 which will compare @expr{a} and @expr{b} to produce a 1 or 0, which is
7536 then compared with @expr{c} to produce another 1 or 0, which is then
7537 compared with @expr{d}. This is not at all what Joe wanted.
7539 Here's a more correct method:
7543 1: [7, 7, 7, 8, 7] 2: [7, 7, 7, 8, 7]
7547 ' [7,7,7,8,7] @key{RET} @key{RET} v r 1 @key{RET}
7554 1: [1, 1, 1, 0, 1] 1: 0
7561 @node List Answer 11, List Answer 12, List Answer 10, Answers to Exercises
7562 @subsection List Tutorial Exercise 11
7565 The circle of unit radius consists of those points @expr{(x,y)} for which
7566 @expr{x^2 + y^2 < 1}. We start by generating a vector of @expr{x^2}
7567 and a vector of @expr{y^2}.
7569 We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
7574 2: [2., 2., ..., 2.] 2: [2., 2., ..., 2.]
7575 1: [2., 2., ..., 2.] 1: [1.16, 1.98, ..., 0.81]
7578 v . t . 2. v b 100 @key{RET} @key{RET} V M k r
7585 2: [2., 2., ..., 2.] 1: [0.026, 0.96, ..., 0.036]
7586 1: [0.026, 0.96, ..., 0.036] 2: [0.53, 0.81, ..., 0.094]
7589 1 - 2 V M ^ @key{TAB} V M k r 1 - 2 V M ^
7593 Now we sum the @expr{x^2} and @expr{y^2} values, compare with 1 to
7594 get a vector of 1/0 truth values, then sum the truth values.
7598 1: [0.56, 1.78, ..., 0.13] 1: [1, 0, ..., 1] 1: 84
7606 The ratio @expr{84/100} should approximate the ratio @cpiover{4}.
7610 1: 0.84 1: 3.36 2: 3.36 1: 1.0695
7618 Our estimate, 3.36, is off by about 7%. We could get a better estimate
7619 by taking more points (say, 1000), but it's clear that this method is
7622 (Naturally, since this example uses random numbers your own answer
7623 will be slightly different from the one shown here!)
7625 If you typed @kbd{v .} and @kbd{t .} before, type them again to
7626 return to full-sized display of vectors.
7628 @node List Answer 12, List Answer 13, List Answer 11, Answers to Exercises
7629 @subsection List Tutorial Exercise 12
7632 This problem can be made a lot easier by taking advantage of some
7633 symmetries. First of all, after some thought it's clear that the
7634 @expr{y} axis can be ignored altogether. Just pick a random @expr{x}
7635 component for one end of the match, pick a random direction
7636 @texline @math{\theta},
7637 @infoline @expr{theta},
7638 and see if @expr{x} and
7639 @texline @math{x + \cos \theta}
7640 @infoline @expr{x + cos(theta)}
7641 (which is the @expr{x} coordinate of the other endpoint) cross a line.
7642 The lines are at integer coordinates, so this happens when the two
7643 numbers surround an integer.
7645 Since the two endpoints are equivalent, we may as well choose the leftmost
7646 of the two endpoints as @expr{x}. Then @expr{theta} is an angle pointing
7647 to the right, in the range -90 to 90 degrees. (We could use radians, but
7648 it would feel like cheating to refer to @cpiover{2} radians while trying
7649 to estimate @cpi{}!)
7651 In fact, since the field of lines is infinite we can choose the
7652 coordinates 0 and 1 for the lines on either side of the leftmost
7653 endpoint. The rightmost endpoint will be between 0 and 1 if the
7654 match does not cross a line, or between 1 and 2 if it does. So:
7655 Pick random @expr{x} and
7656 @texline @math{\theta},
7657 @infoline @expr{theta},
7659 @texline @math{x + \cos \theta},
7660 @infoline @expr{x + cos(theta)},
7661 and count how many of the results are greater than one. Simple!
7663 We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
7668 1: [0.52, 0.71, ..., 0.72] 2: [0.52, 0.71, ..., 0.72]
7669 . 1: [78.4, 64.5, ..., -42.9]
7672 v . t . 1. v b 100 @key{RET} V M k r 180. v b 100 @key{RET} V M k r 90 -
7677 (The next step may be slow, depending on the speed of your computer.)
7681 2: [0.52, 0.71, ..., 0.72] 1: [0.72, 1.14, ..., 1.45]
7682 1: [0.20, 0.43, ..., 0.73] .
7692 1: [0, 1, ..., 1] 1: 0.64 1: 3.125
7695 1 V M a > V R + 100 / 2 @key{TAB} /
7699 Let's try the third method, too. We'll use random integers up to
7700 one million. The @kbd{k r} command with an integer argument picks
7705 2: [1000000, 1000000, ..., 1000000] 2: [78489, 527587, ..., 814975]
7706 1: [1000000, 1000000, ..., 1000000] 1: [324014, 358783, ..., 955450]
7709 1000000 v b 100 @key{RET} @key{RET} V M k r @key{TAB} V M k r
7716 1: [1, 1, ..., 25] 1: [1, 1, ..., 0] 1: 0.56
7719 V M k g 1 V M a = V R + 100 /
7733 For a proof of this property of the GCD function, see section 4.5.2,
7734 exercise 10, of Knuth's @emph{Art of Computer Programming}, volume II.
7736 If you typed @kbd{v .} and @kbd{t .} before, type them again to
7737 return to full-sized display of vectors.
7739 @node List Answer 13, List Answer 14, List Answer 12, Answers to Exercises
7740 @subsection List Tutorial Exercise 13
7743 First, we put the string on the stack as a vector of ASCII codes.
7747 1: [84, 101, 115, ..., 51]
7750 "Testing, 1, 2, 3 @key{RET}
7755 Note that the @kbd{"} key, like @kbd{$}, initiates algebraic entry so
7756 there was no need to type an apostrophe. Also, Calc didn't mind that
7757 we omitted the closing @kbd{"}. (The same goes for all closing delimiters
7758 like @kbd{)} and @kbd{]} at the end of a formula.
7760 We'll show two different approaches here. In the first, we note that
7761 if the input vector is @expr{[a, b, c, d]}, then the hash code is
7762 @expr{3 (3 (3a + b) + c) + d = 27a + 9b + 3c + d}. In other words,
7763 it's a sum of descending powers of three times the ASCII codes.
7767 2: [84, 101, 115, ..., 51] 2: [84, 101, 115, ..., 51]
7768 1: 16 1: [15, 14, 13, ..., 0]
7771 @key{RET} v l v x 16 @key{RET} -
7778 2: [84, 101, 115, ..., 51] 1: 1960915098 1: 121
7779 1: [14348907, ..., 1] . .
7782 3 @key{TAB} V M ^ * 511 %
7787 Once again, @kbd{*} elegantly summarizes most of the computation.
7788 But there's an even more elegant approach: Reduce the formula
7789 @kbd{3 $$ + $} across the vector. Recall that this represents a
7790 function of two arguments that computes its first argument times three
7791 plus its second argument.
7795 1: [84, 101, 115, ..., 51] 1: 1960915098
7798 "Testing, 1, 2, 3 @key{RET} V R ' 3$$+$ @key{RET}
7803 If you did the decimal arithmetic exercise, this will be familiar.
7804 Basically, we're turning a base-3 vector of digits into an integer,
7805 except that our ``digits'' are much larger than real digits.
7807 Instead of typing @kbd{511 %} again to reduce the result, we can be
7808 cleverer still and notice that rather than computing a huge integer
7809 and taking the modulo at the end, we can take the modulo at each step
7810 without affecting the result. While this means there are more
7811 arithmetic operations, the numbers we operate on remain small so
7812 the operations are faster.
7816 1: [84, 101, 115, ..., 51] 1: 121
7819 "Testing, 1, 2, 3 @key{RET} V R ' (3$$+$)%511 @key{RET}
7823 Why does this work? Think about a two-step computation:
7824 @w{@expr{3 (3a + b) + c}}. Taking a result modulo 511 basically means
7825 subtracting off enough 511's to put the result in the desired range.
7826 So the result when we take the modulo after every step is,
7830 3 (3 a + b - 511 m) + c - 511 n
7835 $$ 3 (3 a + b - 511 m) + c - 511 n $$
7840 for some suitable integers @expr{m} and @expr{n}. Expanding out by
7841 the distributive law yields
7845 9 a + 3 b + c - 511*3 m - 511 n
7850 $$ 9 a + 3 b + c - 511\times3 m - 511 n $$
7855 The @expr{m} term in the latter formula is redundant because any
7856 contribution it makes could just as easily be made by the @expr{n}
7857 term. So we can take it out to get an equivalent formula with
7862 9 a + 3 b + c - 511 n'
7867 $$ 9 a + 3 b + c - 511 n^{\prime} $$
7872 which is just the formula for taking the modulo only at the end of
7873 the calculation. Therefore the two methods are essentially the same.
7875 Later in the tutorial we will encounter @dfn{modulo forms}, which
7876 basically automate the idea of reducing every intermediate result
7877 modulo some value @var{m}.
7879 @node List Answer 14, Types Answer 1, List Answer 13, Answers to Exercises
7880 @subsection List Tutorial Exercise 14
7882 We want to use @kbd{H V U} to nest a function which adds a random
7883 step to an @expr{(x,y)} coordinate. The function is a bit long, but
7884 otherwise the problem is quite straightforward.
7888 2: [0, 0] 1: [ [ 0, 0 ]
7889 1: 50 [ 0.4288, -0.1695 ]
7890 . [ -0.4787, -0.9027 ]
7893 [0,0] 50 H V U ' <# + [random(2.0)-1, random(2.0)-1]> @key{RET}
7897 Just as the text recommended, we used @samp{< >} nameless function
7898 notation to keep the two @code{random} calls from being evaluated
7899 before nesting even begins.
7901 We now have a vector of @expr{[x, y]} sub-vectors, which by Calc's
7902 rules acts like a matrix. We can transpose this matrix and unpack
7903 to get a pair of vectors, @expr{x} and @expr{y}, suitable for graphing.
7907 2: [ 0, 0.4288, -0.4787, ... ]
7908 1: [ 0, -0.1696, -0.9027, ... ]
7915 Incidentally, because the @expr{x} and @expr{y} are completely
7916 independent in this case, we could have done two separate commands
7917 to create our @expr{x} and @expr{y} vectors of numbers directly.
7919 To make a random walk of unit steps, we note that @code{sincos} of
7920 a random direction exactly gives us an @expr{[x, y]} step of unit
7921 length; in fact, the new nesting function is even briefer, though
7922 we might want to lower the precision a bit for it.
7926 2: [0, 0] 1: [ [ 0, 0 ]
7927 1: 50 [ 0.1318, 0.9912 ]
7928 . [ -0.5965, 0.3061 ]
7931 [0,0] 50 m d p 6 @key{RET} H V U ' <# + sincos(random(360.0))> @key{RET}
7935 Another @kbd{v t v u g f} sequence will graph this new random walk.
7937 An interesting twist on these random walk functions would be to use
7938 complex numbers instead of 2-vectors to represent points on the plane.
7939 In the first example, we'd use something like @samp{random + random*(0,1)},
7940 and in the second we could use polar complex numbers with random phase
7941 angles. (This exercise was first suggested in this form by Randal
7944 @node Types Answer 1, Types Answer 2, List Answer 14, Answers to Exercises
7945 @subsection Types Tutorial Exercise 1
7948 If the number is the square root of @cpi{} times a rational number,
7949 then its square, divided by @cpi{}, should be a rational number.
7953 1: 1.26508260337 1: 0.509433962268 1: 2486645810:4881193627
7961 Technically speaking this is a rational number, but not one that is
7962 likely to have arisen in the original problem. More likely, it just
7963 happens to be the fraction which most closely represents some
7964 irrational number to within 12 digits.
7966 But perhaps our result was not quite exact. Let's reduce the
7967 precision slightly and try again:
7971 1: 0.509433962268 1: 27:53
7974 U p 10 @key{RET} c F
7979 Aha! It's unlikely that an irrational number would equal a fraction
7980 this simple to within ten digits, so our original number was probably
7981 @texline @math{\sqrt{27 \pi / 53}}.
7982 @infoline @expr{sqrt(27 pi / 53)}.
7984 Notice that we didn't need to re-round the number when we reduced the
7985 precision. Remember, arithmetic operations always round their inputs
7986 to the current precision before they begin.
7988 @node Types Answer 2, Types Answer 3, Types Answer 1, Answers to Exercises
7989 @subsection Types Tutorial Exercise 2
7992 @samp{inf / inf = nan}. Perhaps @samp{1} is the ``obvious'' answer.
7993 But if @w{@samp{17 inf = inf}}, then @samp{17 inf / inf = inf / inf = 17}, too.
7995 @samp{exp(inf) = inf}. It's tempting to say that the exponential
7996 of infinity must be ``bigger'' than ``regular'' infinity, but as
7997 far as Calc is concerned all infinities are the same size.
7998 In other words, as @expr{x} goes to infinity, @expr{e^x} also goes
7999 to infinity, but the fact the @expr{e^x} grows much faster than
8000 @expr{x} is not relevant here.
8002 @samp{exp(-inf) = 0}. Here we have a finite answer even though
8003 the input is infinite.
8005 @samp{sqrt(-inf) = (0, 1) inf}. Remember that @expr{(0, 1)}
8006 represents the imaginary number @expr{i}. Here's a derivation:
8007 @samp{sqrt(-inf) = @w{sqrt((-1) * inf)} = sqrt(-1) * sqrt(inf)}.
8008 The first part is, by definition, @expr{i}; the second is @code{inf}
8009 because, once again, all infinities are the same size.
8011 @samp{sqrt(uinf) = uinf}. In fact, we do know something about the
8012 direction because @code{sqrt} is defined to return a value in the
8013 right half of the complex plane. But Calc has no notation for this,
8014 so it settles for the conservative answer @code{uinf}.
8016 @samp{abs(uinf) = inf}. No matter which direction @expr{x} points,
8017 @samp{abs(x)} always points along the positive real axis.
8019 @samp{ln(0) = -inf}. Here we have an infinite answer to a finite
8020 input. As in the @expr{1 / 0} case, Calc will only use infinities
8021 here if you have turned on Infinite mode. Otherwise, it will
8022 treat @samp{ln(0)} as an error.
8024 @node Types Answer 3, Types Answer 4, Types Answer 2, Answers to Exercises
8025 @subsection Types Tutorial Exercise 3
8028 We can make @samp{inf - inf} be any real number we like, say,
8029 @expr{a}, just by claiming that we added @expr{a} to the first
8030 infinity but not to the second. This is just as true for complex
8031 values of @expr{a}, so @code{nan} can stand for a complex number.
8032 (And, similarly, @code{uinf} can stand for an infinity that points
8033 in any direction in the complex plane, such as @samp{(0, 1) inf}).
8035 In fact, we can multiply the first @code{inf} by two. Surely
8036 @w{@samp{2 inf - inf = inf}}, but also @samp{2 inf - inf = inf - inf = nan}.
8037 So @code{nan} can even stand for infinity. Obviously it's just
8038 as easy to make it stand for minus infinity as for plus infinity.
8040 The moral of this story is that ``infinity'' is a slippery fish
8041 indeed, and Calc tries to handle it by having a very simple model
8042 for infinities (only the direction counts, not the ``size''); but
8043 Calc is careful to write @code{nan} any time this simple model is
8044 unable to tell what the true answer is.
8046 @node Types Answer 4, Types Answer 5, Types Answer 3, Answers to Exercises
8047 @subsection Types Tutorial Exercise 4
8051 2: 0@@ 47' 26" 1: 0@@ 2' 47.411765"
8055 0@@ 47' 26" @key{RET} 17 /
8060 The average song length is two minutes and 47.4 seconds.
8064 2: 0@@ 2' 47.411765" 1: 0@@ 3' 7.411765" 1: 0@@ 53' 6.000005"
8073 The album would be 53 minutes and 6 seconds long.
8075 @node Types Answer 5, Types Answer 6, Types Answer 4, Answers to Exercises
8076 @subsection Types Tutorial Exercise 5
8079 Let's suppose it's January 14, 1991. The easiest thing to do is
8080 to keep trying 13ths of months until Calc reports a Friday.
8081 We can do this by manually entering dates, or by using @kbd{t I}:
8085 1: <Wed Feb 13, 1991> 1: <Wed Mar 13, 1991> 1: <Sat Apr 13, 1991>
8088 ' <2/13> @key{RET} @key{DEL} ' <3/13> @key{RET} t I
8093 (Calc assumes the current year if you don't say otherwise.)
8095 This is getting tedious---we can keep advancing the date by typing
8096 @kbd{t I} over and over again, but let's automate the job by using
8097 vector mapping. The @kbd{t I} command actually takes a second
8098 ``how-many-months'' argument, which defaults to one. This
8099 argument is exactly what we want to map over:
8103 2: <Sat Apr 13, 1991> 1: [<Mon May 13, 1991>, <Thu Jun 13, 1991>,
8104 1: [1, 2, 3, 4, 5, 6] <Sat Jul 13, 1991>, <Tue Aug 13, 1991>,
8105 . <Fri Sep 13, 1991>, <Sun Oct 13, 1991>]
8108 v x 6 @key{RET} V M t I
8113 Et voil@`a, September 13, 1991 is a Friday.
8120 ' <sep 13> - <jan 14> @key{RET}
8125 And the answer to our original question: 242 days to go.
8127 @node Types Answer 6, Types Answer 7, Types Answer 5, Answers to Exercises
8128 @subsection Types Tutorial Exercise 6
8131 The full rule for leap years is that they occur in every year divisible
8132 by four, except that they don't occur in years divisible by 100, except
8133 that they @emph{do} in years divisible by 400. We could work out the
8134 answer by carefully counting the years divisible by four and the
8135 exceptions, but there is a much simpler way that works even if we
8136 don't know the leap year rule.
8138 Let's assume the present year is 1991. Years have 365 days, except
8139 that leap years (whenever they occur) have 366 days. So let's count
8140 the number of days between now and then, and compare that to the
8141 number of years times 365. The number of extra days we find must be
8142 equal to the number of leap years there were.
8146 1: <Mon Jan 1, 10001> 2: <Mon Jan 1, 10001> 1: 2925593
8147 . 1: <Tue Jan 1, 1991> .
8150 ' <jan 1 10001> @key{RET} ' <jan 1 1991> @key{RET} -
8157 3: 2925593 2: 2925593 2: 2925593 1: 1943
8158 2: 10001 1: 8010 1: 2923650 .
8162 10001 @key{RET} 1991 - 365 * -
8166 @c [fix-ref Date Forms]
8168 There will be 1943 leap years before the year 10001. (Assuming,
8169 of course, that the algorithm for computing leap years remains
8170 unchanged for that long. @xref{Date Forms}, for some interesting
8171 background information in that regard.)
8173 @node Types Answer 7, Types Answer 8, Types Answer 6, Answers to Exercises
8174 @subsection Types Tutorial Exercise 7
8177 The relative errors must be converted to absolute errors so that
8178 @samp{+/-} notation may be used.
8186 20 @key{RET} .05 * 4 @key{RET} .05 *
8190 Now we simply chug through the formula.
8194 1: 19.7392088022 1: 394.78 +/- 19.739 1: 6316.5 +/- 706.21
8197 2 P 2 ^ * 20 p 1 * 4 p .2 @key{RET} 2 ^ *
8201 It turns out the @kbd{v u} command will unpack an error form as
8202 well as a vector. This saves us some retyping of numbers.
8206 3: 6316.5 +/- 706.21 2: 6316.5 +/- 706.21
8211 @key{RET} v u @key{TAB} /
8216 Thus the volume is 6316 cubic centimeters, within about 11 percent.
8218 @node Types Answer 8, Types Answer 9, Types Answer 7, Answers to Exercises
8219 @subsection Types Tutorial Exercise 8
8222 The first answer is pretty simple: @samp{1 / (0 .. 10) = (0.1 .. inf)}.
8223 Since a number in the interval @samp{(0 .. 10)} can get arbitrarily
8224 close to zero, its reciprocal can get arbitrarily large, so the answer
8225 is an interval that effectively means, ``any number greater than 0.1''
8226 but with no upper bound.
8228 The second answer, similarly, is @samp{1 / (-10 .. 0) = (-inf .. -0.1)}.
8230 Calc normally treats division by zero as an error, so that the formula
8231 @w{@samp{1 / 0}} is left unsimplified. Our third problem,
8232 @w{@samp{1 / [0 .. 10]}}, also (potentially) divides by zero because zero
8233 is now a member of the interval. So Calc leaves this one unevaluated, too.
8235 If you turn on Infinite mode by pressing @kbd{m i}, you will
8236 instead get the answer @samp{[0.1 .. inf]}, which includes infinity
8237 as a possible value.
8239 The fourth calculation, @samp{1 / (-10 .. 10)}, has the same problem.
8240 Zero is buried inside the interval, but it's still a possible value.
8241 It's not hard to see that the actual result of @samp{1 / (-10 .. 10)}
8242 will be either greater than @mathit{0.1}, or less than @mathit{-0.1}. Thus
8243 the interval goes from minus infinity to plus infinity, with a ``hole''
8244 in it from @mathit{-0.1} to @mathit{0.1}. Calc doesn't have any way to
8245 represent this, so it just reports @samp{[-inf .. inf]} as the answer.
8246 It may be disappointing to hear ``the answer lies somewhere between
8247 minus infinity and plus infinity, inclusive,'' but that's the best
8248 that interval arithmetic can do in this case.
8250 @node Types Answer 9, Types Answer 10, Types Answer 8, Answers to Exercises
8251 @subsection Types Tutorial Exercise 9
8255 1: [-3 .. 3] 2: [-3 .. 3] 2: [0 .. 9]
8256 . 1: [0 .. 9] 1: [-9 .. 9]
8259 [ 3 n .. 3 ] @key{RET} 2 ^ @key{TAB} @key{RET} *
8264 In the first case the result says, ``if a number is between @mathit{-3} and
8265 3, its square is between 0 and 9.'' The second case says, ``the product
8266 of two numbers each between @mathit{-3} and 3 is between @mathit{-9} and 9.''
8268 An interval form is not a number; it is a symbol that can stand for
8269 many different numbers. Two identical-looking interval forms can stand
8270 for different numbers.
8272 The same issue arises when you try to square an error form.
8274 @node Types Answer 10, Types Answer 11, Types Answer 9, Answers to Exercises
8275 @subsection Types Tutorial Exercise 10
8278 Testing the first number, we might arbitrarily choose 17 for @expr{x}.
8282 1: 17 mod 811749613 2: 17 mod 811749613 1: 533694123 mod 811749613
8286 17 M 811749613 @key{RET} 811749612 ^
8291 Since 533694123 is (considerably) different from 1, the number 811749613
8294 It's awkward to type the number in twice as we did above. There are
8295 various ways to avoid this, and algebraic entry is one. In fact, using
8296 a vector mapping operation we can perform several tests at once. Let's
8297 use this method to test the second number.
8301 2: [17, 42, 100000] 1: [1 mod 15485863, 1 mod ... ]
8305 [17 42 100000] 15485863 @key{RET} V M ' ($$ mod $)^($-1) @key{RET}
8310 The result is three ones (modulo @expr{n}), so it's very probable that
8311 15485863 is prime. (In fact, this number is the millionth prime.)
8313 Note that the functions @samp{($$^($-1)) mod $} or @samp{$$^($-1) % $}
8314 would have been hopelessly inefficient, since they would have calculated
8315 the power using full integer arithmetic.
8317 Calc has a @kbd{k p} command that does primality testing. For small
8318 numbers it does an exact test; for large numbers it uses a variant
8319 of the Fermat test we used here. You can use @kbd{k p} repeatedly
8320 to prove that a large integer is prime with any desired probability.
8322 @node Types Answer 11, Types Answer 12, Types Answer 10, Answers to Exercises
8323 @subsection Types Tutorial Exercise 11
8326 There are several ways to insert a calculated number into an HMS form.
8327 One way to convert a number of seconds to an HMS form is simply to
8328 multiply the number by an HMS form representing one second:
8332 1: 31415926.5359 2: 31415926.5359 1: 8726@@ 38' 46.5359"
8343 2: 8726@@ 38' 46.5359" 1: 6@@ 6' 2.5359" mod 24@@ 0' 0"
8344 1: 15@@ 27' 16" mod 24@@ 0' 0" .
8352 It will be just after six in the morning.
8354 The algebraic @code{hms} function can also be used to build an
8359 1: hms(0, 0, 10000000. pi) 1: 8726@@ 38' 46.5359"
8362 ' hms(0, 0, 1e7 pi) @key{RET} =
8367 The @kbd{=} key is necessary to evaluate the symbol @samp{pi} to
8368 the actual number 3.14159...
8370 @node Types Answer 12, Types Answer 13, Types Answer 11, Answers to Exercises
8371 @subsection Types Tutorial Exercise 12
8374 As we recall, there are 17 songs of about 2 minutes and 47 seconds
8379 2: 0@@ 2' 47" 1: [0@@ 3' 7" .. 0@@ 3' 47"]
8380 1: [0@@ 0' 20" .. 0@@ 1' 0"] .
8383 [ 0@@ 20" .. 0@@ 1' ] +
8390 1: [0@@ 52' 59." .. 1@@ 4' 19."]
8398 No matter how long it is, the album will fit nicely on one CD.
8400 @node Types Answer 13, Types Answer 14, Types Answer 12, Answers to Exercises
8401 @subsection Types Tutorial Exercise 13
8404 Type @kbd{' 1 yr @key{RET} u c s @key{RET}}. The answer is 31557600 seconds.
8406 @node Types Answer 14, Types Answer 15, Types Answer 13, Answers to Exercises
8407 @subsection Types Tutorial Exercise 14
8410 How long will it take for a signal to get from one end of the computer
8415 1: m / c 1: 3.3356 ns
8418 ' 1 m / c @key{RET} u c ns @key{RET}
8423 (Recall, @samp{c} is a ``unit'' corresponding to the speed of light.)
8427 1: 3.3356 ns 1: 0.81356
8431 ' 4.1 ns @key{RET} /
8436 Thus a signal could take up to 81 percent of a clock cycle just to
8437 go from one place to another inside the computer, assuming the signal
8438 could actually attain the full speed of light. Pretty tight!
8440 @node Types Answer 15, Algebra Answer 1, Types Answer 14, Answers to Exercises
8441 @subsection Types Tutorial Exercise 15
8444 The speed limit is 55 miles per hour on most highways. We want to
8445 find the ratio of Sam's speed to the US speed limit.
8449 1: 55 mph 2: 55 mph 3: 11 hr mph / yd
8453 ' 55 mph @key{RET} ' 5 yd/hr @key{RET} /
8457 The @kbd{u s} command cancels out these units to get a plain
8458 number. Now we take the logarithm base two to find the final
8459 answer, assuming that each successive pill doubles his speed.
8463 1: 19360. 2: 19360. 1: 14.24
8472 Thus Sam can take up to 14 pills without a worry.
8474 @node Algebra Answer 1, Algebra Answer 2, Types Answer 15, Answers to Exercises
8475 @subsection Algebra Tutorial Exercise 1
8478 @c [fix-ref Declarations]
8479 The result @samp{sqrt(x)^2} is simplified back to @expr{x} by the
8480 Calculator, but @samp{sqrt(x^2)} is not. (Consider what happens
8481 if @w{@expr{x = -4}}.) If @expr{x} is real, this formula could be
8482 simplified to @samp{abs(x)}, but for general complex arguments even
8483 that is not safe. (@xref{Declarations}, for a way to tell Calc
8484 that @expr{x} is known to be real.)
8486 @node Algebra Answer 2, Algebra Answer 3, Algebra Answer 1, Answers to Exercises
8487 @subsection Algebra Tutorial Exercise 2
8490 Suppose our roots are @expr{[a, b, c]}. We want a polynomial which
8491 is zero when @expr{x} is any of these values. The trivial polynomial
8492 @expr{x-a} is zero when @expr{x=a}, so the product @expr{(x-a)(x-b)(x-c)}
8493 will do the job. We can use @kbd{a c x} to write this in a more
8498 1: 34 x - 24 x^3 1: [1.19023, -1.19023, 0]
8508 1: [x - 1.19023, x + 1.19023, x] 1: x*(x + 1.19023) (x - 1.19023)
8511 V M ' x-$ @key{RET} V R *
8518 1: x^3 - 1.41666 x 1: 34 x - 24 x^3
8521 a c x @key{RET} 24 n * a x
8526 Sure enough, our answer (multiplied by a suitable constant) is the
8527 same as the original polynomial.
8529 @node Algebra Answer 3, Algebra Answer 4, Algebra Answer 2, Answers to Exercises
8530 @subsection Algebra Tutorial Exercise 3
8534 1: x sin(pi x) 1: sin(pi x) / pi^2 - x cos(pi x) / pi
8537 ' x sin(pi x) @key{RET} m r a i x @key{RET}
8545 2: sin(pi x) / pi^2 - x cos(pi x) / pi
8548 ' [y,1] @key{RET} @key{TAB}
8555 1: [sin(pi y) / pi^2 - y cos(pi y) / pi, 1 / pi]
8565 1: sin(pi y) / pi^2 - y cos(pi y) / pi - 1 / pi
8575 1: sin(3.14159 y) / 9.8696 - y cos(3.14159 y) / 3.14159 - 0.3183
8585 1: [0., -0.95493, 0.63662, -1.5915, 1.2732]
8588 v x 5 @key{RET} @key{TAB} V M $ @key{RET}
8592 @node Algebra Answer 4, Rewrites Answer 1, Algebra Answer 3, Answers to Exercises
8593 @subsection Algebra Tutorial Exercise 4
8596 The hard part is that @kbd{V R +} is no longer sufficient to add up all
8597 the contributions from the slices, since the slices have varying
8598 coefficients. So first we must come up with a vector of these
8599 coefficients. Here's one way:
8603 2: -1 2: 3 1: [4, 2, ..., 4]
8604 1: [1, 2, ..., 9] 1: [-1, 1, ..., -1] .
8607 1 n v x 9 @key{RET} V M ^ 3 @key{TAB} -
8614 1: [4, 2, ..., 4, 1] 1: [1, 4, 2, ..., 4, 1]
8622 Now we compute the function values. Note that for this method we need
8623 eleven values, including both endpoints of the desired interval.
8627 2: [1, 4, 2, ..., 4, 1]
8628 1: [1, 1.1, 1.2, ... , 1.8, 1.9, 2.]
8631 11 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x
8638 2: [1, 4, 2, ..., 4, 1]
8639 1: [0., 0.084941, 0.16993, ... ]
8642 ' sin(x) ln(x) @key{RET} m r p 5 @key{RET} V M $ @key{RET}
8647 Once again this calls for @kbd{V M * V R +}; a simple @kbd{*} does the
8652 1: 11.22 1: 1.122 1: 0.374
8660 Wow! That's even better than the result from the Taylor series method.
8662 @node Rewrites Answer 1, Rewrites Answer 2, Algebra Answer 4, Answers to Exercises
8663 @subsection Rewrites Tutorial Exercise 1
8666 We'll use Big mode to make the formulas more readable.
8672 1: (2 + sqrt(2)) / (1 + sqrt(2)) 1: ---------
8678 ' (2+sqrt(2)) / (1+sqrt(2)) @key{RET} d B
8683 Multiplying by the conjugate helps because @expr{(a+b) (a-b) = a^2 - b^2}.
8688 1: (2 + V 2 ) (V 2 - 1)
8691 a r a/(b+c) := a*(b-c) / (b^2-c^2) @key{RET}
8702 a r a*(b+c) := a*b + a*c
8707 (We could have used @kbd{a x} instead of a rewrite rule for the
8710 The multiply-by-conjugate rule turns out to be useful in many
8711 different circumstances, such as when the denominator involves
8712 sines and cosines or the imaginary constant @code{i}.
8714 @node Rewrites Answer 2, Rewrites Answer 3, Rewrites Answer 1, Answers to Exercises
8715 @subsection Rewrites Tutorial Exercise 2
8718 Here is the rule set:
8722 [ fib(n) := fib(n, 1, 1) :: integer(n) :: n >= 1,
8724 fib(n, x, y) := fib(n-1, y, x+y) ]
8729 The first rule turns a one-argument @code{fib} that people like to write
8730 into a three-argument @code{fib} that makes computation easier. The
8731 second rule converts back from three-argument form once the computation
8732 is done. The third rule does the computation itself. It basically
8733 says that if @expr{x} and @expr{y} are two consecutive Fibonacci numbers,
8734 then @expr{y} and @expr{x+y} are the next (overlapping) pair of Fibonacci
8737 Notice that because the number @expr{n} was ``validated'' by the
8738 conditions on the first rule, there is no need to put conditions on
8739 the other rules because the rule set would never get that far unless
8740 the input were valid. That further speeds computation, since no
8741 extra conditions need to be checked at every step.
8743 Actually, a user with a nasty sense of humor could enter a bad
8744 three-argument @code{fib} call directly, say, @samp{fib(0, 1, 1)},
8745 which would get the rules into an infinite loop. One thing that would
8746 help keep this from happening by accident would be to use something like
8747 @samp{ZzFib} instead of @code{fib} as the name of the three-argument
8750 @node Rewrites Answer 3, Rewrites Answer 4, Rewrites Answer 2, Answers to Exercises
8751 @subsection Rewrites Tutorial Exercise 3
8754 He got an infinite loop. First, Calc did as expected and rewrote
8755 @w{@samp{2 + 3 x}} to @samp{f(2, 3, x)}. Then it looked for ways to
8756 apply the rule again, and found that @samp{f(2, 3, x)} looks like
8757 @samp{a + b x} with @w{@samp{a = 0}} and @samp{b = 1}, so it rewrote to
8758 @samp{f(0, 1, f(2, 3, x))}. It then wrapped another @samp{f(0, 1, ...)}
8759 around that, and so on, ad infinitum. Joe should have used @kbd{M-1 a r}
8760 to make sure the rule applied only once.
8762 (Actually, even the first step didn't work as he expected. What Calc
8763 really gives for @kbd{M-1 a r} in this situation is @samp{f(3 x, 1, 2)},
8764 treating 2 as the ``variable,'' and @samp{3 x} as a constant being added
8765 to it. While this may seem odd, it's just as valid a solution as the
8766 ``obvious'' one. One way to fix this would be to add the condition
8767 @samp{:: variable(x)} to the rule, to make sure the thing that matches
8768 @samp{x} is indeed a variable, or to change @samp{x} to @samp{quote(x)}
8769 on the lefthand side, so that the rule matches the actual variable
8770 @samp{x} rather than letting @samp{x} stand for something else.)
8772 @node Rewrites Answer 4, Rewrites Answer 5, Rewrites Answer 3, Answers to Exercises
8773 @subsection Rewrites Tutorial Exercise 4
8780 Here is a suitable set of rules to solve the first part of the problem:
8784 [ seq(n, c) := seq(n/2, c+1) :: n%2 = 0,
8785 seq(n, c) := seq(3n+1, c+1) :: n%2 = 1 :: n > 1 ]
8789 Given the initial formula @samp{seq(6, 0)}, application of these
8790 rules produces the following sequence of formulas:
8804 whereupon neither of the rules match, and rewriting stops.
8806 We can pretty this up a bit with a couple more rules:
8810 [ seq(n) := seq(n, 0),
8817 Now, given @samp{seq(6)} as the starting configuration, we get 8
8820 The change to return a vector is quite simple:
8824 [ seq(n) := seq(n, []) :: integer(n) :: n > 0,
8826 seq(n, v) := seq(n/2, v | n) :: n%2 = 0,
8827 seq(n, v) := seq(3n+1, v | n) :: n%2 = 1 ]
8832 Given @samp{seq(6)}, the result is @samp{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.
8834 Notice that the @expr{n > 1} guard is no longer necessary on the last
8835 rule since the @expr{n = 1} case is now detected by another rule.
8836 But a guard has been added to the initial rule to make sure the
8837 initial value is suitable before the computation begins.
8839 While still a good idea, this guard is not as vitally important as it
8840 was for the @code{fib} function, since calling, say, @samp{seq(x, [])}
8841 will not get into an infinite loop. Calc will not be able to prove
8842 the symbol @samp{x} is either even or odd, so none of the rules will
8843 apply and the rewrites will stop right away.
8845 @node Rewrites Answer 5, Rewrites Answer 6, Rewrites Answer 4, Answers to Exercises
8846 @subsection Rewrites Tutorial Exercise 5
8853 If @expr{x} is the sum @expr{a + b}, then `@tfn{nterms(}@var{x}@tfn{)}' must
8854 be `@tfn{nterms(}@var{a}@tfn{)}' plus `@tfn{nterms(}@var{b}@tfn{)}'. If @expr{x}
8855 is not a sum, then `@tfn{nterms(}@var{x}@tfn{)}' = 1.
8859 [ nterms(a + b) := nterms(a) + nterms(b),
8865 Here we have taken advantage of the fact that earlier rules always
8866 match before later rules; @samp{nterms(x)} will only be tried if we
8867 already know that @samp{x} is not a sum.
8869 @node Rewrites Answer 6, Programming Answer 1, Rewrites Answer 5, Answers to Exercises
8870 @subsection Rewrites Tutorial Exercise 6
8873 Here is a rule set that will do the job:
8877 [ a*(b + c) := a*b + a*c,
8878 opt(a) O(x^n) + opt(b) O(x^m) := O(x^n) :: n <= m
8879 :: constant(a) :: constant(b),
8880 opt(a) O(x^n) + opt(b) x^m := O(x^n) :: n <= m
8881 :: constant(a) :: constant(b),
8882 a O(x^n) := O(x^n) :: constant(a),
8883 x^opt(m) O(x^n) := O(x^(n+m)),
8884 O(x^n) O(x^m) := O(x^(n+m)) ]
8888 If we really want the @kbd{+} and @kbd{*} keys to operate naturally
8889 on power series, we should put these rules in @code{EvalRules}. For
8890 testing purposes, it is better to put them in a different variable,
8891 say, @code{O}, first.
8893 The first rule just expands products of sums so that the rest of the
8894 rules can assume they have an expanded-out polynomial to work with.
8895 Note that this rule does not mention @samp{O} at all, so it will
8896 apply to any product-of-sum it encounters---this rule may surprise
8897 you if you put it into @code{EvalRules}!
8899 In the second rule, the sum of two O's is changed to the smaller O@.
8900 The optional constant coefficients are there mostly so that
8901 @samp{O(x^2) - O(x^3)} and @samp{O(x^3) - O(x^2)} are handled
8902 as well as @samp{O(x^2) + O(x^3)}.
8904 The third rule absorbs higher powers of @samp{x} into O's.
8906 The fourth rule says that a constant times a negligible quantity
8907 is still negligible. (This rule will also match @samp{O(x^3) / 4},
8908 with @samp{a = 1/4}.)
8910 The fifth rule rewrites, for example, @samp{x^2 O(x^3)} to @samp{O(x^5)}.
8911 (It is easy to see that if one of these forms is negligible, the other
8912 is, too.) Notice the @samp{x^opt(m)} to pick up terms like
8913 @w{@samp{x O(x^3)}}. Optional powers will match @samp{x} as @samp{x^1}
8914 but not 1 as @samp{x^0}. This turns out to be exactly what we want here.
8916 The sixth rule is the corresponding rule for products of two O's.
8918 Another way to solve this problem would be to create a new ``data type''
8919 that represents truncated power series. We might represent these as
8920 function calls @samp{series(@var{coefs}, @var{x})} where @var{coefs} is
8921 a vector of coefficients for @expr{x^0}, @expr{x^1}, @expr{x^2}, and so
8922 on. Rules would exist for sums and products of such @code{series}
8923 objects, and as an optional convenience could also know how to combine a
8924 @code{series} object with a normal polynomial. (With this, and with a
8925 rule that rewrites @samp{O(x^n)} to the equivalent @code{series} form,
8926 you could still enter power series in exactly the same notation as
8927 before.) Operations on such objects would probably be more efficient,
8928 although the objects would be a bit harder to read.
8930 @c [fix-ref Compositions]
8931 Some other symbolic math programs provide a power series data type
8932 similar to this. Mathematica, for example, has an object that looks
8933 like @samp{PowerSeries[@var{x}, @var{x0}, @var{coefs}, @var{nmin},
8934 @var{nmax}, @var{den}]}, where @var{x0} is the point about which the
8935 power series is taken (we've been assuming this was always zero),
8936 and @var{nmin}, @var{nmax}, and @var{den} allow pseudo-power-series
8937 with fractional or negative powers. Also, the @code{PowerSeries}
8938 objects have a special display format that makes them look like
8939 @samp{2 x^2 + O(x^4)} when they are printed out. (@xref{Compositions},
8940 for a way to do this in Calc, although for something as involved as
8941 this it would probably be better to write the formatting routine
8944 @node Programming Answer 1, Programming Answer 2, Rewrites Answer 6, Answers to Exercises
8945 @subsection Programming Tutorial Exercise 1
8948 Just enter the formula @samp{ninteg(sin(t)/t, t, 0, x)}, type
8949 @kbd{Z F}, and answer the questions. Since this formula contains two
8950 variables, the default argument list will be @samp{(t x)}. We want to
8951 change this to @samp{(x)} since @expr{t} is really a dummy variable
8952 to be used within @code{ninteg}.
8954 The exact keystrokes are @kbd{Z F s Si @key{RET} @key{RET} C-b C-b @key{DEL} @key{DEL} @key{RET} y}.
8955 (The @kbd{C-b C-b @key{DEL} @key{DEL}} are what fix the argument list.)
8957 @node Programming Answer 2, Programming Answer 3, Programming Answer 1, Answers to Exercises
8958 @subsection Programming Tutorial Exercise 2
8961 One way is to move the number to the top of the stack, operate on
8962 it, then move it back: @kbd{C-x ( M-@key{TAB} n M-@key{TAB} M-@key{TAB} C-x )}.
8964 Another way is to negate the top three stack entries, then negate
8965 again the top two stack entries: @kbd{C-x ( M-3 n M-2 n C-x )}.
8967 Finally, it turns out that a negative prefix argument causes a
8968 command like @kbd{n} to operate on the specified stack entry only,
8969 which is just what we want: @kbd{C-x ( M-- 3 n C-x )}.
8971 Just for kicks, let's also do it algebraically:
8972 @w{@kbd{C-x ( ' -$$$, $$, $ @key{RET} C-x )}}.
8974 @node Programming Answer 3, Programming Answer 4, Programming Answer 2, Answers to Exercises
8975 @subsection Programming Tutorial Exercise 3
8978 Each of these functions can be computed using the stack, or using
8979 algebraic entry, whichever way you prefer:
8983 @texline @math{\displaystyle{\sin x \over x}}:
8984 @infoline @expr{sin(x) / x}:
8986 Using the stack: @kbd{C-x ( @key{RET} S @key{TAB} / C-x )}.
8988 Using algebraic entry: @kbd{C-x ( ' sin($)/$ @key{RET} C-x )}.
8991 Computing the logarithm:
8993 Using the stack: @kbd{C-x ( @key{TAB} B C-x )}
8995 Using algebraic entry: @kbd{C-x ( ' log($,$$) @key{RET} C-x )}.
8998 Computing the vector of integers:
9000 Using the stack: @kbd{C-x ( 1 @key{RET} 1 C-u v x C-x )}. (Recall that
9001 @kbd{C-u v x} takes the vector size, starting value, and increment
9004 Alternatively: @kbd{C-x ( ~ v x C-x )}. (The @kbd{~} key pops a
9005 number from the stack and uses it as the prefix argument for the
9008 Using algebraic entry: @kbd{C-x ( ' index($) @key{RET} C-x )}.
9010 @node Programming Answer 4, Programming Answer 5, Programming Answer 3, Answers to Exercises
9011 @subsection Programming Tutorial Exercise 4
9014 Here's one way: @kbd{C-x ( @key{RET} V R + @key{TAB} v l / C-x )}.
9016 @node Programming Answer 5, Programming Answer 6, Programming Answer 4, Answers to Exercises
9017 @subsection Programming Tutorial Exercise 5
9021 2: 1 1: 1.61803398502 2: 1.61803398502
9022 1: 20 . 1: 1.61803398875
9025 1 @key{RET} 20 Z < & 1 + Z > I H P
9030 This answer is quite accurate.
9032 @node Programming Answer 6, Programming Answer 7, Programming Answer 5, Answers to Exercises
9033 @subsection Programming Tutorial Exercise 6
9039 [ [ 0, 1 ] * [a, b] = [b, a + b]
9044 Thus @samp{[0, 1; 1, 1]^n * [1, 1]} computes Fibonacci numbers @expr{n+1}
9045 and @expr{n+2}. Here's one program that does the job:
9048 C-x ( ' [0, 1; 1, 1] ^ ($-1) * [1, 1] @key{RET} v u @key{DEL} C-x )
9052 This program is quite efficient because Calc knows how to raise a
9053 matrix (or other value) to the power @expr{n} in only
9054 @texline @math{\log_2 n}
9055 @infoline @expr{log(n,2)}
9056 steps. For example, this program can compute the 1000th Fibonacci
9057 number (a 209-digit integer!) in about 10 steps; even though the
9058 @kbd{Z < ... Z >} solution had much simpler steps, it would have
9059 required so many steps that it would not have been practical.
9061 @node Programming Answer 7, Programming Answer 8, Programming Answer 6, Answers to Exercises
9062 @subsection Programming Tutorial Exercise 7
9065 The trick here is to compute the harmonic numbers differently, so that
9066 the loop counter itself accumulates the sum of reciprocals. We use
9067 a separate variable to hold the integer counter.
9075 1 t 1 1 @key{RET} 4 Z ( t 2 r 1 1 + s 1 & Z )
9080 The body of the loop goes as follows: First save the harmonic sum
9081 so far in variable 2. Then delete it from the stack; the for loop
9082 itself will take care of remembering it for us. Next, recall the
9083 count from variable 1, add one to it, and feed its reciprocal to
9084 the for loop to use as the step value. The for loop will increase
9085 the ``loop counter'' by that amount and keep going until the
9086 loop counter exceeds 4.
9091 1: 3.99498713092 2: 3.99498713092
9095 r 1 r 2 @key{RET} 31 & +
9099 Thus we find that the 30th harmonic number is 3.99, and the 31st
9100 harmonic number is 4.02.
9102 @node Programming Answer 8, Programming Answer 9, Programming Answer 7, Answers to Exercises
9103 @subsection Programming Tutorial Exercise 8
9106 The first step is to compute the derivative @expr{f'(x)} and thus
9108 @texline @math{\displaystyle{x - {f(x) \over f'(x)}}}.
9109 @infoline @expr{x - f(x)/f'(x)}.
9111 (Because this definition is long, it will be repeated in concise form
9112 below. You can use @w{@kbd{C-x * m}} to load it from there. While you are
9113 entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
9114 keystrokes without executing them. In the following diagrams we'll
9115 pretend Calc actually executed the keystrokes as you typed them,
9116 just for purposes of illustration.)
9120 2: sin(cos(x)) - 0.5 3: 4.5
9121 1: 4.5 2: sin(cos(x)) - 0.5
9122 . 1: -(sin(x) cos(cos(x)))
9125 ' sin(cos(x))-0.5 @key{RET} 4.5 m r C-x ( Z ` @key{TAB} @key{RET} a d x @key{RET}
9133 1: x + (sin(cos(x)) - 0.5) / sin(x) cos(cos(x))
9136 / ' x @key{RET} @key{TAB} - t 1
9140 Now, we enter the loop. We'll use a repeat loop with a 20-repetition
9141 limit just in case the method fails to converge for some reason.
9142 (Normally, the @w{@kbd{Z /}} command will stop the loop before all 20
9143 repetitions are done.)
9147 1: 4.5 3: 4.5 2: 4.5
9148 . 2: x + (sin(cos(x)) ... 1: 5.24196456928
9152 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET}
9156 This is the new guess for @expr{x}. Now we compare it with the
9157 old one to see if we've converged.
9161 3: 5.24196 2: 5.24196 1: 5.24196 1: 5.26345856348
9166 @key{RET} M-@key{TAB} a = Z / Z > Z ' C-x )
9170 The loop converges in just a few steps to this value. To check
9171 the result, we can simply substitute it back into the equation.
9179 @key{RET} ' sin(cos($)) @key{RET}
9183 Let's test the new definition again:
9191 ' x^2-9 @key{RET} 1 X
9195 Once again, here's the full Newton's Method definition:
9199 C-x ( Z ` @key{TAB} @key{RET} a d x @key{RET} / ' x @key{RET} @key{TAB} - t 1
9200 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET}
9201 @key{RET} M-@key{TAB} a = Z /
9208 @c [fix-ref Nesting and Fixed Points]
9209 It turns out that Calc has a built-in command for applying a formula
9210 repeatedly until it converges to a number. @xref{Nesting and Fixed Points},
9211 to see how to use it.
9213 @c [fix-ref Root Finding]
9214 Also, of course, @kbd{a R} is a built-in command that uses Newton's
9215 method (among others) to look for numerical solutions to any equation.
9216 @xref{Root Finding}.
9218 @node Programming Answer 9, Programming Answer 10, Programming Answer 8, Answers to Exercises
9219 @subsection Programming Tutorial Exercise 9
9222 The first step is to adjust @expr{z} to be greater than 5. A simple
9223 ``for'' loop will do the job here. If @expr{z} is less than 5, we
9224 reduce the problem using
9225 @texline @math{\psi(z) = \psi(z+1) - 1/z}.
9226 @infoline @expr{psi(z) = psi(z+1) - 1/z}. We go
9228 @texline @math{\psi(z+1)},
9229 @infoline @expr{psi(z+1)},
9230 and remember to add back a factor of @expr{-1/z} when we're done. This
9231 step is repeated until @expr{z > 5}.
9233 (Because this definition is long, it will be repeated in concise form
9234 below. You can use @w{@kbd{C-x * m}} to load it from there. While you are
9235 entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
9236 keystrokes without executing them. In the following diagrams we'll
9237 pretend Calc actually executed the keystrokes as you typed them,
9238 just for purposes of illustration.)
9245 1.0 @key{RET} C-x ( Z ` s 1 0 t 2
9249 Here, variable 1 holds @expr{z} and variable 2 holds the adjustment
9250 factor. If @expr{z < 5}, we use a loop to increase it.
9252 (By the way, we started with @samp{1.0} instead of the integer 1 because
9253 otherwise the calculation below will try to do exact fractional arithmetic,
9254 and will never converge because fractions compare equal only if they
9255 are exactly equal, not just equal to within the current precision.)
9264 @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ]
9268 Now we compute the initial part of the sum:
9269 @texline @math{\ln z - {1 \over 2z}}
9270 @infoline @expr{ln(z) - 1/2z}
9271 minus the adjustment factor.
9275 2: 1.79175946923 2: 1.7084261359 1: -0.57490719743
9276 1: 0.0833333333333 1: 2.28333333333 .
9283 Now we evaluate the series. We'll use another ``for'' loop counting
9284 up the value of @expr{2 n}. (Calc does have a summation command,
9285 @kbd{a +}, but we'll use loops just to get more practice with them.)
9289 3: -0.5749 3: -0.5749 4: -0.5749 2: -0.5749
9290 2: 2 2: 1:6 3: 1:6 1: 2.3148e-3
9295 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1 @key{TAB} ^ * /
9302 3: -0.5749 3: -0.5772 2: -0.5772 1: -0.577215664892
9303 2: -0.5749 2: -0.5772 1: 0 .
9304 1: 2.3148e-3 1: -0.5749 .
9307 @key{TAB} @key{RET} M-@key{TAB} - @key{RET} M-@key{TAB} a = Z / 2 Z ) Z ' C-x )
9311 This is the value of
9312 @texline @math{-\gamma},
9313 @infoline @expr{- gamma},
9314 with a slight bit of roundoff error. To get a full 12 digits, let's use
9319 2: -0.577215664892 2: -0.577215664892
9320 1: 1. 1: -0.577215664901532
9322 1. @key{RET} p 16 @key{RET} X
9326 Here's the complete sequence of keystrokes:
9331 @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ]
9333 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1 @key{TAB} ^ * /
9334 @key{TAB} @key{RET} M-@key{TAB} - @key{RET} M-@key{TAB} a = Z /
9341 @node Programming Answer 10, Programming Answer 11, Programming Answer 9, Answers to Exercises
9342 @subsection Programming Tutorial Exercise 10
9345 Taking the derivative of a term of the form @expr{x^n} will produce
9347 @texline @math{n x^{n-1}}.
9348 @infoline @expr{n x^(n-1)}.
9349 Taking the derivative of a constant
9350 produces zero. From this it is easy to see that the @expr{n}th
9351 derivative of a polynomial, evaluated at @expr{x = 0}, will equal the
9352 coefficient on the @expr{x^n} term times @expr{n!}.
9354 (Because this definition is long, it will be repeated in concise form
9355 below. You can use @w{@kbd{C-x * m}} to load it from there. While you are
9356 entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
9357 keystrokes without executing them. In the following diagrams we'll
9358 pretend Calc actually executed the keystrokes as you typed them,
9359 just for purposes of illustration.)
9363 2: 5 x^4 + (x + 1)^2 3: 5 x^4 + (x + 1)^2
9368 ' 5 x^4 + (x+1)^2 @key{RET} 6 C-x ( Z ` [ ] t 1 0 @key{TAB}
9373 Variable 1 will accumulate the vector of coefficients.
9377 2: 0 3: 0 2: 5 x^4 + ...
9378 1: 5 x^4 + ... 2: 5 x^4 + ... 1: 1
9382 Z ( @key{TAB} @key{RET} 0 s l x @key{RET} M-@key{TAB} ! / s | 1
9387 Note that @kbd{s | 1} appends the top-of-stack value to the vector
9388 in a variable; it is completely analogous to @kbd{s + 1}. We could
9389 have written instead, @kbd{r 1 @key{TAB} | t 1}.
9393 1: 20 x^3 + 2 x + 2 1: 0 1: [1, 2, 1, 0, 5, 0, 0]
9396 a d x @key{RET} 1 Z ) @key{DEL} r 1 Z ' C-x )
9400 To convert back, a simple method is just to map the coefficients
9401 against a table of powers of @expr{x}.
9405 2: [1, 2, 1, 0, 5, 0, 0] 2: [1, 2, 1, 0, 5, 0, 0]
9406 1: 6 1: [0, 1, 2, 3, 4, 5, 6]
9409 6 @key{RET} 1 + 0 @key{RET} 1 C-u v x
9416 2: [1, 2, 1, 0, 5, 0, 0] 2: 1 + 2 x + x^2 + 5 x^4
9417 1: [1, x, x^2, x^3, ... ] .
9420 ' x @key{RET} @key{TAB} V M ^ *
9424 Once again, here are the whole polynomial to/from vector programs:
9428 C-x ( Z ` [ ] t 1 0 @key{TAB}
9429 Z ( @key{TAB} @key{RET} 0 s l x @key{RET} M-@key{TAB} ! / s | 1
9435 C-x ( 1 + 0 @key{RET} 1 C-u v x ' x @key{RET} @key{TAB} V M ^ * C-x )
9439 @node Programming Answer 11, Programming Answer 12, Programming Answer 10, Answers to Exercises
9440 @subsection Programming Tutorial Exercise 11
9443 First we define a dummy program to go on the @kbd{z s} key. The true
9444 @w{@kbd{z s}} key is supposed to take two numbers from the stack and
9445 return one number, so @key{DEL} as a dummy definition will make
9446 sure the stack comes out right.
9454 4 @key{RET} 2 C-x ( @key{DEL} C-x ) Z K s @key{RET} 2
9458 The last step replaces the 2 that was eaten during the creation
9459 of the dummy @kbd{z s} command. Now we move on to the real
9460 definition. The recurrence needs to be rewritten slightly,
9461 to the form @expr{s(n,m) = s(n-1,m-1) - (n-1) s(n-1,m)}.
9463 (Because this definition is long, it will be repeated in concise form
9464 below. You can use @kbd{C-x * m} to load it from there.)
9474 C-x ( M-2 @key{RET} a = Z [ @key{DEL} @key{DEL} 1 Z :
9481 4: 4 2: 4 2: 3 4: 3 4: 3 3: 3
9482 3: 2 1: 2 1: 2 3: 2 3: 2 2: 2
9483 2: 2 . . 2: 3 2: 3 1: 3
9487 @key{RET} 0 a = Z [ @key{DEL} @key{DEL} 0 Z : @key{TAB} 1 - @key{TAB} M-2 @key{RET} 1 - z s
9492 (Note that the value 3 that our dummy @kbd{z s} produces is not correct;
9493 it is merely a placeholder that will do just as well for now.)
9497 3: 3 4: 3 3: 3 2: 3 1: -6
9498 2: 3 3: 3 2: 3 1: 9 .
9503 M-@key{TAB} M-@key{TAB} @key{TAB} @key{RET} M-@key{TAB} z s * -
9510 1: -6 2: 4 1: 11 2: 11
9514 Z ] Z ] C-x ) Z K s @key{RET} @key{DEL} 4 @key{RET} 2 z s M-@key{RET} k s
9518 Even though the result that we got during the definition was highly
9519 bogus, once the definition is complete the @kbd{z s} command gets
9522 Here's the full program once again:
9526 C-x ( M-2 @key{RET} a =
9527 Z [ @key{DEL} @key{DEL} 1
9529 Z [ @key{DEL} @key{DEL} 0
9530 Z : @key{TAB} 1 - @key{TAB} M-2 @key{RET} 1 - z s
9531 M-@key{TAB} M-@key{TAB} @key{TAB} @key{RET} M-@key{TAB} z s * -
9538 You can read this definition using @kbd{C-x * m} (@code{read-kbd-macro})
9539 followed by @kbd{Z K s}, without having to make a dummy definition
9540 first, because @code{read-kbd-macro} doesn't need to execute the
9541 definition as it reads it in. For this reason, @code{C-x * m} is often
9542 the easiest way to create recursive programs in Calc.
9544 @node Programming Answer 12, , Programming Answer 11, Answers to Exercises
9545 @subsection Programming Tutorial Exercise 12
9548 This turns out to be a much easier way to solve the problem. Let's
9549 denote Stirling numbers as calls of the function @samp{s}.
9551 First, we store the rewrite rules corresponding to the definition of
9552 Stirling numbers in a convenient variable:
9555 s e StirlingRules @key{RET}
9556 [ s(n,n) := 1 :: n >= 0,
9557 s(n,0) := 0 :: n > 0,
9558 s(n,m) := s(n-1,m-1) - (n-1) s(n-1,m) :: n >= m :: m >= 1 ]
9562 Now, it's just a matter of applying the rules:
9566 2: 4 1: s(4, 2) 1: 11
9570 4 @key{RET} 2 C-x ( ' s($$,$) @key{RET} a r StirlingRules @key{RET} C-x )
9574 As in the case of the @code{fib} rules, it would be useful to put these
9575 rules in @code{EvalRules} and to add a @samp{:: remember} condition to
9578 @c This ends the table-of-contents kludge from above:
9580 \global\let\chapternofonts=\oldchapternofonts
9585 @node Introduction, Data Types, Tutorial, Top
9586 @chapter Introduction
9589 This chapter is the beginning of the Calc reference manual.
9590 It covers basic concepts such as the stack, algebraic and
9591 numeric entry, undo, numeric prefix arguments, etc.
9594 @c (Chapter 2, the Tutorial, has been printed in a separate volume.)
9602 * Quick Calculator::
9603 * Prefix Arguments::
9606 * Multiple Calculators::
9607 * Troubleshooting Commands::
9610 @node Basic Commands, Help Commands, Introduction, Introduction
9611 @section Basic Commands
9616 @cindex Starting the Calculator
9617 @cindex Running the Calculator
9618 To start the Calculator in its standard interface, type @kbd{M-x calc}.
9619 By default this creates a pair of small windows, @samp{*Calculator*}
9620 and @samp{*Calc Trail*}. The former displays the contents of the
9621 Calculator stack and is manipulated exclusively through Calc commands.
9622 It is possible (though not usually necessary) to create several Calc
9623 mode buffers each of which has an independent stack, undo list, and
9624 mode settings. There is exactly one Calc Trail buffer; it records a
9625 list of the results of all calculations that have been done. The
9626 Calc Trail buffer uses a variant of Calc mode, so Calculator commands
9627 still work when the trail buffer's window is selected. It is possible
9628 to turn the trail window off, but the @samp{*Calc Trail*} buffer itself
9629 still exists and is updated silently. @xref{Trail Commands}.
9636 In most installations, the @kbd{C-x * c} key sequence is a more
9637 convenient way to start the Calculator. Also, @kbd{C-x * *}
9638 is a synonym for @kbd{C-x * c} unless you last used Calc
9643 @pindex calc-execute-extended-command
9644 Most Calc commands use one or two keystrokes. Lower- and upper-case
9645 letters are distinct. Commands may also be entered in full @kbd{M-x} form;
9646 for some commands this is the only form. As a convenience, the @kbd{x}
9647 key (@code{calc-execute-extended-command})
9648 is like @kbd{M-x} except that it enters the initial string @samp{calc-}
9649 for you. For example, the following key sequences are equivalent:
9650 @kbd{S}, @kbd{M-x calc-sin @key{RET}}, @kbd{x sin @key{RET}}.
9652 Although Calc is designed to be used from the keyboard, some of
9653 Calc's more common commands are available from a menu. In the menu, the
9654 arguments to the functions are given by referring to their stack level
9657 @cindex Extensions module
9658 @cindex @file{calc-ext} module
9659 The Calculator exists in many parts. When you type @kbd{C-x * c}, the
9660 Emacs ``auto-load'' mechanism will bring in only the first part, which
9661 contains the basic arithmetic functions. The other parts will be
9662 auto-loaded the first time you use the more advanced commands like trig
9663 functions or matrix operations. This is done to improve the response time
9664 of the Calculator in the common case when all you need to do is a
9665 little arithmetic. If for some reason the Calculator fails to load an
9666 extension module automatically, you can force it to load all the
9667 extensions by using the @kbd{C-x * L} (@code{calc-load-everything})
9668 command. @xref{Mode Settings}.
9670 If you type @kbd{M-x calc} or @kbd{C-x * c} with any numeric prefix argument,
9671 the Calculator is loaded if necessary, but it is not actually started.
9672 If the argument is positive, the @file{calc-ext} extensions are also
9673 loaded if necessary. User-written Lisp code that wishes to make use
9674 of Calc's arithmetic routines can use @samp{(calc 0)} or @samp{(calc 1)}
9675 to auto-load the Calculator.
9679 If you type @kbd{C-x * b}, then next time you use @kbd{C-x * c} you
9680 will get a Calculator that uses the full height of the Emacs screen.
9681 When full-screen mode is on, @kbd{C-x * c} runs the @code{full-calc}
9682 command instead of @code{calc}. From the Unix shell you can type
9683 @samp{emacs -f full-calc} to start a new Emacs specifically for use
9684 as a calculator. When Calc is started from the Emacs command line
9685 like this, Calc's normal ``quit'' commands actually quit Emacs itself.
9688 @pindex calc-other-window
9689 The @kbd{C-x * o} command is like @kbd{C-x * c} except that the Calc
9690 window is not actually selected. If you are already in the Calc
9691 window, @kbd{C-x * o} switches you out of it. (The regular Emacs
9692 @kbd{C-x o} command would also work for this, but it has a
9693 tendency to drop you into the Calc Trail window instead, which
9694 @kbd{C-x * o} takes care not to do.)
9699 For one quick calculation, you can type @kbd{C-x * q} (@code{quick-calc})
9700 which prompts you for a formula (like @samp{2+3/4}). The result is
9701 displayed at the bottom of the Emacs screen without ever creating
9702 any special Calculator windows. @xref{Quick Calculator}.
9707 Finally, if you are using the X window system you may want to try
9708 @kbd{C-x * k} (@code{calc-keypad}) which runs Calc with a
9709 ``calculator keypad'' picture as well as a stack display. Click on
9710 the keys with the mouse to operate the calculator. @xref{Keypad Mode}.
9714 @cindex Quitting the Calculator
9715 @cindex Exiting the Calculator
9716 The @kbd{q} key (@code{calc-quit}) exits Calc mode and closes the
9717 Calculator's window(s). It does not delete the Calculator buffers.
9718 If you type @kbd{M-x calc} again, the Calculator will reappear with the
9719 contents of the stack intact. Typing @kbd{C-x * c} or @kbd{C-x * *}
9720 again from inside the Calculator buffer is equivalent to executing
9721 @code{calc-quit}; you can think of @kbd{C-x * *} as toggling the
9722 Calculator on and off.
9725 The @kbd{C-x * x} command also turns the Calculator off, no matter which
9726 user interface (standard, Keypad, or Embedded) is currently active.
9727 It also cancels @code{calc-edit} mode if used from there.
9730 @pindex calc-refresh
9731 @cindex Refreshing a garbled display
9732 @cindex Garbled displays, refreshing
9733 The @kbd{d @key{SPC}} key sequence (@code{calc-refresh}) redraws the contents
9734 of the Calculator buffer from memory. Use this if the contents of the
9735 buffer have been damaged somehow.
9740 The @kbd{o} key (@code{calc-realign}) moves the cursor back to its
9741 ``home'' position at the bottom of the Calculator buffer.
9745 @pindex calc-scroll-left
9746 @pindex calc-scroll-right
9747 @cindex Horizontal scrolling
9749 @cindex Wide text, scrolling
9750 The @kbd{<} and @kbd{>} keys are bound to @code{calc-scroll-left} and
9751 @code{calc-scroll-right}. These are just like the normal horizontal
9752 scrolling commands except that they scroll one half-screen at a time by
9753 default. (Calc formats its output to fit within the bounds of the
9754 window whenever it can.)
9758 @pindex calc-scroll-down
9759 @pindex calc-scroll-up
9760 @cindex Vertical scrolling
9761 The @kbd{@{} and @kbd{@}} keys are bound to @code{calc-scroll-down}
9762 and @code{calc-scroll-up}. They scroll up or down by one-half the
9763 height of the Calc window.
9767 The @kbd{C-x * 0} command (@code{calc-reset}; that's @kbd{C-x *} followed
9768 by a zero) resets the Calculator to its initial state. This clears
9769 the stack, resets all the modes to their initial values (the values
9770 that were saved with @kbd{m m} (@code{calc-save-modes})), clears the
9771 caches (@pxref{Caches}), and so on. (It does @emph{not} erase the
9772 values of any variables.) With an argument of 0, Calc will be reset to
9773 its default state; namely, the modes will be given their default values.
9774 With a positive prefix argument, @kbd{C-x * 0} preserves the contents of
9775 the stack but resets everything else to its initial state; with a
9776 negative prefix argument, @kbd{C-x * 0} preserves the contents of the
9777 stack but resets everything else to its default state.
9779 @node Help Commands, Stack Basics, Basic Commands, Introduction
9780 @section Help Commands
9783 @cindex Help commands
9803 The @kbd{?} key (@code{calc-help}) displays a series of brief help messages.
9804 Some keys (such as @kbd{b} and @kbd{d}) are prefix keys, like Emacs's
9805 @key{ESC} and @kbd{C-x} prefixes. You can type
9806 @kbd{?} after a prefix to see a list of commands beginning with that
9807 prefix. (If the message includes @samp{[MORE]}, press @kbd{?} again
9808 to see additional commands for that prefix.)
9811 @pindex calc-full-help
9812 The @kbd{h h} (@code{calc-full-help}) command displays all the @kbd{?}
9813 responses at once. When printed, this makes a nice, compact (three pages)
9814 summary of Calc keystrokes.
9816 In general, the @kbd{h} key prefix introduces various commands that
9817 provide help within Calc. Many of the @kbd{h} key functions are
9818 Calc-specific analogues to the @kbd{C-h} functions for Emacs help.
9824 The @kbd{h i} (@code{calc-info}) command runs the Emacs Info system
9825 to read this manual on-line. This is basically the same as typing
9826 @kbd{C-h i} (the regular way to run the Info system), then, if Info
9827 is not already in the Calc manual, selecting the beginning of the
9828 manual. The @kbd{C-x * i} command is another way to read the Calc
9829 manual; it is different from @kbd{h i} in that it works any time,
9830 not just inside Calc. The plain @kbd{i} key is also equivalent to
9831 @kbd{h i}, though this key is obsolete and may be replaced with a
9832 different command in a future version of Calc.
9836 @pindex calc-tutorial
9837 The @kbd{h t} (@code{calc-tutorial}) command runs the Info system on
9838 the Tutorial section of the Calc manual. It is like @kbd{h i},
9839 except that it selects the starting node of the tutorial rather
9840 than the beginning of the whole manual. (It actually selects the
9841 node ``Interactive Tutorial'' which tells a few things about
9842 using the Info system before going on to the actual tutorial.)
9843 The @kbd{C-x * t} key is equivalent to @kbd{h t} (but it works at
9848 @pindex calc-info-summary
9849 The @kbd{h s} (@code{calc-info-summary}) command runs the Info system
9850 on the Summary node of the Calc manual. @xref{Summary}. The @kbd{C-x * s}
9851 key is equivalent to @kbd{h s}.
9854 @pindex calc-describe-key
9855 The @kbd{h k} (@code{calc-describe-key}) command looks up a key
9856 sequence in the Calc manual. For example, @kbd{h k H a S} looks
9857 up the documentation on the @kbd{H a S} (@code{calc-solve-for})
9858 command. This works by looking up the textual description of
9859 the key(s) in the Key Index of the manual, then jumping to the
9860 node indicated by the index.
9862 Most Calc commands do not have traditional Emacs documentation
9863 strings, since the @kbd{h k} command is both more convenient and
9864 more instructive. This means the regular Emacs @kbd{C-h k}
9865 (@code{describe-key}) command will not be useful for Calc keystrokes.
9868 @pindex calc-describe-key-briefly
9869 The @kbd{h c} (@code{calc-describe-key-briefly}) command reads a
9870 key sequence and displays a brief one-line description of it at
9871 the bottom of the screen. It looks for the key sequence in the
9872 Summary node of the Calc manual; if it doesn't find the sequence
9873 there, it acts just like its regular Emacs counterpart @kbd{C-h c}
9874 (@code{describe-key-briefly}). For example, @kbd{h c H a S}
9875 gives the description:
9878 H a S runs calc-solve-for: a `H a S' v => fsolve(a,v) (?=notes)
9882 which means the command @kbd{H a S} or @kbd{H M-x calc-solve-for}
9883 takes a value @expr{a} from the stack, prompts for a value @expr{v},
9884 then applies the algebraic function @code{fsolve} to these values.
9885 The @samp{?=notes} message means you can now type @kbd{?} to see
9886 additional notes from the summary that apply to this command.
9889 @pindex calc-describe-function
9890 The @kbd{h f} (@code{calc-describe-function}) command looks up an
9891 algebraic function or a command name in the Calc manual. Enter an
9892 algebraic function name to look up that function in the Function
9893 Index or enter a command name beginning with @samp{calc-} to look it
9894 up in the Command Index. This command will also look up operator
9895 symbols that can appear in algebraic formulas, like @samp{%} and
9899 @pindex calc-describe-variable
9900 The @kbd{h v} (@code{calc-describe-variable}) command looks up a
9901 variable in the Calc manual. Enter a variable name like @code{pi} or
9905 @pindex describe-bindings
9906 The @kbd{h b} (@code{calc-describe-bindings}) command is just like
9907 @kbd{C-h b}, except that only local (Calc-related) key bindings are
9911 The @kbd{h n} or @kbd{h C-n} (@code{calc-view-news}) command displays
9912 the ``news'' or change history of Calc. This is kept in the file
9913 @file{README}, which Calc looks for in the same directory as the Calc
9919 The @kbd{h C-c}, @kbd{h C-d}, and @kbd{h C-w} keys display copying,
9920 distribution, and warranty information about Calc. These work by
9921 pulling up the appropriate parts of the ``Copying'' or ``Reporting
9922 Bugs'' sections of the manual.
9924 @node Stack Basics, Numeric Entry, Help Commands, Introduction
9925 @section Stack Basics
9928 @cindex Stack basics
9929 @c [fix-tut RPN Calculations and the Stack]
9930 Calc uses RPN notation. If you are not familiar with RPN, @pxref{RPN
9933 To add the numbers 1 and 2 in Calc you would type the keys:
9934 @kbd{1 @key{RET} 2 +}.
9935 (@key{RET} corresponds to the @key{ENTER} key on most calculators.)
9936 The first three keystrokes ``push'' the numbers 1 and 2 onto the stack. The
9937 @kbd{+} key always ``pops'' the top two numbers from the stack, adds them,
9938 and pushes the result (3) back onto the stack. This number is ready for
9939 further calculations: @kbd{5 -} pushes 5 onto the stack, then pops the
9940 3 and 5, subtracts them, and pushes the result (@mathit{-2}).
9942 Note that the ``top'' of the stack actually appears at the @emph{bottom}
9943 of the buffer. A line containing a single @samp{.} character signifies
9944 the end of the buffer; Calculator commands operate on the number(s)
9945 directly above this line. The @kbd{d t} (@code{calc-truncate-stack})
9946 command allows you to move the @samp{.} marker up and down in the stack;
9947 @pxref{Truncating the Stack}.
9950 @pindex calc-line-numbering
9951 Stack elements are numbered consecutively, with number 1 being the top of
9952 the stack. These line numbers are ordinarily displayed on the lefthand side
9953 of the window. The @kbd{d l} (@code{calc-line-numbering}) command controls
9954 whether these numbers appear. (Line numbers may be turned off since they
9955 slow the Calculator down a bit and also clutter the display.)
9958 @pindex calc-realign
9959 The unshifted letter @kbd{o} (@code{calc-realign}) command repositions
9960 the cursor to its top-of-stack ``home'' position. It also undoes any
9961 horizontal scrolling in the window. If you give it a numeric prefix
9962 argument, it instead moves the cursor to the specified stack element.
9964 The @key{RET} (or equivalent @key{SPC}) key is only required to separate
9965 two consecutive numbers.
9966 (After all, if you typed @kbd{1 2} by themselves the Calculator
9967 would enter the number 12.) If you press @key{RET} or @key{SPC} @emph{not}
9968 right after typing a number, the key duplicates the number on the top of
9969 the stack. @kbd{@key{RET} *} is thus a handy way to square a number.
9971 The @key{DEL} key pops and throws away the top number on the stack.
9972 The @key{TAB} key swaps the top two objects on the stack.
9973 @xref{Stack and Trail}, for descriptions of these and other stack-related
9976 @node Numeric Entry, Algebraic Entry, Stack Basics, Introduction
9977 @section Numeric Entry
9983 @cindex Numeric entry
9984 @cindex Entering numbers
9985 Pressing a digit or other numeric key begins numeric entry using the
9986 minibuffer. The number is pushed on the stack when you press the @key{RET}
9987 or @key{SPC} keys. If you press any other non-numeric key, the number is
9988 pushed onto the stack and the appropriate operation is performed. If
9989 you press a numeric key which is not valid, the key is ignored.
9992 @cindex Negative numbers, entering
9994 There are three different concepts corresponding to the word ``minus,''
9995 typified by @expr{a-b} (subtraction), @expr{-x}
9996 (change-sign), and @expr{-5} (negative number). Calc uses three
9997 different keys for these operations, respectively:
9998 @kbd{-}, @kbd{n}, and @kbd{_} (the underscore). The @kbd{-} key subtracts
9999 the two numbers on the top of the stack. The @kbd{n} key changes the sign
10000 of the number on the top of the stack or the number currently being entered.
10001 The @kbd{_} key begins entry of a negative number or changes the sign of
10002 the number currently being entered. The following sequences all enter the
10003 number @mathit{-5} onto the stack: @kbd{0 @key{RET} 5 -}, @kbd{5 n @key{RET}},
10004 @kbd{5 @key{RET} n}, @kbd{_ 5 @key{RET}}, @kbd{5 _ @key{RET}}.
10006 Some other keys are active during numeric entry, such as @kbd{#} for
10007 non-decimal numbers, @kbd{:} for fractions, and @kbd{@@} for HMS forms.
10008 These notations are described later in this manual with the corresponding
10009 data types. @xref{Data Types}.
10011 During numeric entry, the only editing key available is @key{DEL}.
10013 @node Algebraic Entry, Quick Calculator, Numeric Entry, Introduction
10014 @section Algebraic Entry
10018 @pindex calc-algebraic-entry
10019 @cindex Algebraic notation
10020 @cindex Formulas, entering
10021 The @kbd{'} (@code{calc-algebraic-entry}) command can be used to enter
10022 calculations in algebraic form. This is accomplished by typing the
10023 apostrophe key, ', followed by the expression in standard format:
10031 @texline @math{2+(3\times4) = 14}
10032 @infoline @expr{2+(3*4) = 14}
10033 and push it on the stack. If you wish you can
10034 ignore the RPN aspect of Calc altogether and simply enter algebraic
10035 expressions in this way. You may want to use @key{DEL} every so often to
10036 clear previous results off the stack.
10038 You can press the apostrophe key during normal numeric entry to switch
10039 the half-entered number into Algebraic entry mode. One reason to do
10040 this would be to fix a typo, as the full Emacs cursor motion and editing
10041 keys are available during algebraic entry but not during numeric entry.
10043 In the same vein, during either numeric or algebraic entry you can
10044 press @kbd{`} (backquote) to switch to @code{calc-edit} mode, where
10045 you complete your half-finished entry in a separate buffer.
10046 @xref{Editing Stack Entries}.
10049 @pindex calc-algebraic-mode
10050 @cindex Algebraic Mode
10051 If you prefer algebraic entry, you can use the command @kbd{m a}
10052 (@code{calc-algebraic-mode}) to set Algebraic mode. In this mode,
10053 digits and other keys that would normally start numeric entry instead
10054 start full algebraic entry; as long as your formula begins with a digit
10055 you can omit the apostrophe. Open parentheses and square brackets also
10056 begin algebraic entry. You can still do RPN calculations in this mode,
10057 but you will have to press @key{RET} to terminate every number:
10058 @kbd{2 @key{RET} 3 @key{RET} * 4 @key{RET} +} would accomplish the same
10059 thing as @kbd{2*3+4 @key{RET}}.
10061 @cindex Incomplete Algebraic Mode
10062 If you give a numeric prefix argument like @kbd{C-u} to the @kbd{m a}
10063 command, it enables Incomplete Algebraic mode; this is like regular
10064 Algebraic mode except that it applies to the @kbd{(} and @kbd{[} keys
10065 only. Numeric keys still begin a numeric entry in this mode.
10068 @pindex calc-total-algebraic-mode
10069 @cindex Total Algebraic Mode
10070 The @kbd{m t} (@code{calc-total-algebraic-mode}) gives you an even
10071 stronger algebraic-entry mode, in which @emph{all} regular letter and
10072 punctuation keys begin algebraic entry. Use this if you prefer typing
10073 @w{@kbd{sqrt( )}} instead of @kbd{Q}, @w{@kbd{factor( )}} instead of
10074 @kbd{a f}, and so on. To type regular Calc commands when you are in
10075 Total Algebraic mode, hold down the @key{META} key. Thus @kbd{M-q}
10076 is the command to quit Calc, @kbd{M-p} sets the precision, and
10077 @kbd{M-m t} (or @kbd{M-m M-t}, if you prefer) turns Total Algebraic
10078 mode back off again. Meta keys also terminate algebraic entry, so
10079 that @kbd{2+3 M-S} is equivalent to @kbd{2+3 @key{RET} M-S}. The symbol
10080 @samp{Alg*} will appear in the mode line whenever you are in this mode.
10082 Pressing @kbd{'} (the apostrophe) a second time re-enters the previous
10083 algebraic formula. You can then use the normal Emacs editing keys to
10084 modify this formula to your liking before pressing @key{RET}.
10087 @cindex Formulas, referring to stack
10088 Within a formula entered from the keyboard, the symbol @kbd{$}
10089 represents the number on the top of the stack. If an entered formula
10090 contains any @kbd{$} characters, the Calculator replaces the top of
10091 stack with that formula rather than simply pushing the formula onto the
10092 stack. Thus, @kbd{' 1+2 @key{RET}} pushes 3 on the stack, and @kbd{$*2
10093 @key{RET}} replaces it with 6. Note that the @kbd{$} key always
10094 initiates algebraic entry; the @kbd{'} is unnecessary if @kbd{$} is the
10095 first character in the new formula.
10097 Higher stack elements can be accessed from an entered formula with the
10098 symbols @kbd{$$}, @kbd{$$$}, and so on. The number of stack elements
10099 removed (to be replaced by the entered values) equals the number of dollar
10100 signs in the longest such symbol in the formula. For example, @samp{$$+$$$}
10101 adds the second and third stack elements, replacing the top three elements
10102 with the answer. (All information about the top stack element is thus lost
10103 since no single @samp{$} appears in this formula.)
10105 A slightly different way to refer to stack elements is with a dollar
10106 sign followed by a number: @samp{$1}, @samp{$2}, and so on are much
10107 like @samp{$}, @samp{$$}, etc., except that stack entries referred
10108 to numerically are not replaced by the algebraic entry. That is, while
10109 @samp{$+1} replaces 5 on the stack with 6, @samp{$1+1} leaves the 5
10110 on the stack and pushes an additional 6.
10112 If a sequence of formulas are entered separated by commas, each formula
10113 is pushed onto the stack in turn. For example, @samp{1,2,3} pushes
10114 those three numbers onto the stack (leaving the 3 at the top), and
10115 @samp{$+1,$-1} replaces a 5 on the stack with 4 followed by 6. Also,
10116 @samp{$,$$} exchanges the top two elements of the stack, just like the
10119 You can finish an algebraic entry with @kbd{M-=} or @kbd{M-@key{RET}} instead
10120 of @key{RET}. This uses @kbd{=} to evaluate the variables in each
10121 formula that goes onto the stack. (Thus @kbd{' pi @key{RET}} pushes
10122 the variable @samp{pi}, but @kbd{' pi M-@key{RET}} pushes 3.1415.)
10124 If you finish your algebraic entry by pressing @key{LFD} (or @kbd{C-j})
10125 instead of @key{RET}, Calc disables simplification
10126 (as if by @kbd{m O}; @pxref{Simplification Modes}) while the entry
10127 is being pushed on the stack. Thus @kbd{' 1+2 @key{RET}} pushes 3
10128 on the stack, but @kbd{' 1+2 @key{LFD}} pushes the formula @expr{1+2};
10129 you might then press @kbd{=} when it is time to evaluate this formula.
10131 @node Quick Calculator, Prefix Arguments, Algebraic Entry, Introduction
10132 @section ``Quick Calculator'' Mode
10137 @cindex Quick Calculator
10138 There is another way to invoke the Calculator if all you need to do
10139 is make one or two quick calculations. Type @kbd{C-x * q} (or
10140 @kbd{M-x quick-calc}), then type any formula as an algebraic entry.
10141 The Calculator will compute the result and display it in the echo
10142 area, without ever actually putting up a Calc window.
10144 You can use the @kbd{$} character in a Quick Calculator formula to
10145 refer to the previous Quick Calculator result. Older results are
10146 not retained; the Quick Calculator has no effect on the full
10147 Calculator's stack or trail. If you compute a result and then
10148 forget what it was, just run @code{C-x * q} again and enter
10149 @samp{$} as the formula.
10151 If this is the first time you have used the Calculator in this Emacs
10152 session, the @kbd{C-x * q} command will create the @code{*Calculator*}
10153 buffer and perform all the usual initializations; it simply will
10154 refrain from putting that buffer up in a new window. The Quick
10155 Calculator refers to the @code{*Calculator*} buffer for all mode
10156 settings. Thus, for example, to set the precision that the Quick
10157 Calculator uses, simply run the full Calculator momentarily and use
10158 the regular @kbd{p} command.
10160 If you use @code{C-x * q} from inside the Calculator buffer, the
10161 effect is the same as pressing the apostrophe key (algebraic entry).
10163 The result of a Quick calculation is placed in the Emacs ``kill ring''
10164 as well as being displayed. A subsequent @kbd{C-y} command will
10165 yank the result into the editing buffer. You can also use this
10166 to yank the result into the next @kbd{C-x * q} input line as a more
10167 explicit alternative to @kbd{$} notation, or to yank the result
10168 into the Calculator stack after typing @kbd{C-x * c}.
10170 If you finish your formula by typing @key{LFD} (or @kbd{C-j}) instead
10171 of @key{RET}, the result is inserted immediately into the current
10172 buffer rather than going into the kill ring.
10174 Quick Calculator results are actually evaluated as if by the @kbd{=}
10175 key (which replaces variable names by their stored values, if any).
10176 If the formula you enter is an assignment to a variable using the
10177 @samp{:=} operator, say, @samp{foo := 2 + 3} or @samp{foo := foo + 1},
10178 then the result of the evaluation is stored in that Calc variable.
10179 @xref{Store and Recall}.
10181 If the result is an integer and the current display radix is decimal,
10182 the number will also be displayed in hex, octal and binary formats. If
10183 the integer is in the range from 1 to 126, it will also be displayed as
10184 an ASCII character.
10186 For example, the quoted character @samp{"x"} produces the vector
10187 result @samp{[120]} (because 120 is the ASCII code of the lower-case
10188 `x'; @pxref{Strings}). Since this is a vector, not an integer, it
10189 is displayed only according to the current mode settings. But
10190 running Quick Calc again and entering @samp{120} will produce the
10191 result @samp{120 (16#78, 8#170, x)} which shows the number in its
10192 decimal, hexadecimal, octal, and ASCII forms.
10194 Please note that the Quick Calculator is not any faster at loading
10195 or computing the answer than the full Calculator; the name ``quick''
10196 merely refers to the fact that it's much less hassle to use for
10197 small calculations.
10199 @node Prefix Arguments, Undo, Quick Calculator, Introduction
10200 @section Numeric Prefix Arguments
10203 Many Calculator commands use numeric prefix arguments. Some, such as
10204 @kbd{d s} (@code{calc-sci-notation}), set a parameter to the value of
10205 the prefix argument or use a default if you don't use a prefix.
10206 Others (like @kbd{d f} (@code{calc-fix-notation})) require an argument
10207 and prompt for a number if you don't give one as a prefix.
10209 As a rule, stack-manipulation commands accept a numeric prefix argument
10210 which is interpreted as an index into the stack. A positive argument
10211 operates on the top @var{n} stack entries; a negative argument operates
10212 on the @var{n}th stack entry in isolation; and a zero argument operates
10213 on the entire stack.
10215 Most commands that perform computations (such as the arithmetic and
10216 scientific functions) accept a numeric prefix argument that allows the
10217 operation to be applied across many stack elements. For unary operations
10218 (that is, functions of one argument like absolute value or complex
10219 conjugate), a positive prefix argument applies that function to the top
10220 @var{n} stack entries simultaneously, and a negative argument applies it
10221 to the @var{n}th stack entry only. For binary operations (functions of
10222 two arguments like addition, GCD, and vector concatenation), a positive
10223 prefix argument ``reduces'' the function across the top @var{n}
10224 stack elements (for example, @kbd{C-u 5 +} sums the top 5 stack entries;
10225 @pxref{Reducing and Mapping}), and a negative argument maps the next-to-top
10226 @var{n} stack elements with the top stack element as a second argument
10227 (for example, @kbd{7 c-u -5 +} adds 7 to the top 5 stack elements).
10228 This feature is not available for operations which use the numeric prefix
10229 argument for some other purpose.
10231 Numeric prefixes are specified the same way as always in Emacs: Press
10232 a sequence of @key{META}-digits, or press @key{ESC} followed by digits,
10233 or press @kbd{C-u} followed by digits. Some commands treat plain
10234 @kbd{C-u} (without any actual digits) specially.
10237 @pindex calc-num-prefix
10238 You can type @kbd{~} (@code{calc-num-prefix}) to pop an integer from the
10239 top of the stack and enter it as the numeric prefix for the next command.
10240 For example, @kbd{C-u 16 p} sets the precision to 16 digits; an alternate
10241 (silly) way to do this would be @kbd{2 @key{RET} 4 ^ ~ p}, i.e., compute 2
10242 to the fourth power and set the precision to that value.
10244 Conversely, if you have typed a numeric prefix argument the @kbd{~} key
10245 pushes it onto the stack in the form of an integer.
10247 @node Undo, Error Messages, Prefix Arguments, Introduction
10248 @section Undoing Mistakes
10254 @cindex Mistakes, undoing
10255 @cindex Undoing mistakes
10256 @cindex Errors, undoing
10257 The shift-@kbd{U} key (@code{calc-undo}) undoes the most recent operation.
10258 If that operation added or dropped objects from the stack, those objects
10259 are removed or restored. If it was a ``store'' operation, you are
10260 queried whether or not to restore the variable to its original value.
10261 The @kbd{U} key may be pressed any number of times to undo successively
10262 farther back in time; with a numeric prefix argument it undoes a
10263 specified number of operations. When the Calculator is quit, as with
10264 the @kbd{q} (@code{calc-quit}) command, the undo history will be
10265 truncated to the length of the customizable variable
10266 @code{calc-undo-length} (@pxref{Customizing Calc}), which by default
10267 is @expr{100}. (Recall that @kbd{C-x * c} is synonymous with
10268 @code{calc-quit} while inside the Calculator; this also truncates the
10271 Currently the mode-setting commands (like @code{calc-precision}) are not
10272 undoable. You can undo past a point where you changed a mode, but you
10273 will need to reset the mode yourself.
10277 @cindex Redoing after an Undo
10278 The shift-@kbd{D} key (@code{calc-redo}) redoes an operation that was
10279 mistakenly undone. Pressing @kbd{U} with a negative prefix argument is
10280 equivalent to executing @code{calc-redo}. You can redo any number of
10281 times, up to the number of recent consecutive undo commands. Redo
10282 information is cleared whenever you give any command that adds new undo
10283 information, i.e., if you undo, then enter a number on the stack or make
10284 any other change, then it will be too late to redo.
10286 @kindex M-@key{RET}
10287 @pindex calc-last-args
10288 @cindex Last-arguments feature
10289 @cindex Arguments, restoring
10290 The @kbd{M-@key{RET}} key (@code{calc-last-args}) is like undo in that
10291 it restores the arguments of the most recent command onto the stack;
10292 however, it does not remove the result of that command. Given a numeric
10293 prefix argument, this command applies to the @expr{n}th most recent
10294 command which removed items from the stack; it pushes those items back
10297 The @kbd{K} (@code{calc-keep-args}) command provides a related function
10298 to @kbd{M-@key{RET}}. @xref{Stack and Trail}.
10300 It is also possible to recall previous results or inputs using the trail.
10301 @xref{Trail Commands}.
10303 The standard Emacs @kbd{C-_} undo key is recognized as a synonym for @kbd{U}.
10305 @node Error Messages, Multiple Calculators, Undo, Introduction
10306 @section Error Messages
10311 @cindex Errors, messages
10312 @cindex Why did an error occur?
10313 Many situations that would produce an error message in other calculators
10314 simply create unsimplified formulas in the Emacs Calculator. For example,
10315 @kbd{1 @key{RET} 0 /} pushes the formula @expr{1 / 0}; @w{@kbd{0 L}} pushes
10316 the formula @samp{ln(0)}. Floating-point overflow and underflow are also
10317 reasons for this to happen.
10319 When a function call must be left in symbolic form, Calc usually
10320 produces a message explaining why. Messages that are probably
10321 surprising or indicative of user errors are displayed automatically.
10322 Other messages are simply kept in Calc's memory and are displayed only
10323 if you type @kbd{w} (@code{calc-why}). You can also press @kbd{w} if
10324 the same computation results in several messages. (The first message
10325 will end with @samp{[w=more]} in this case.)
10328 @pindex calc-auto-why
10329 The @kbd{d w} (@code{calc-auto-why}) command controls when error messages
10330 are displayed automatically. (Calc effectively presses @kbd{w} for you
10331 after your computation finishes.) By default, this occurs only for
10332 ``important'' messages. The other possible modes are to report
10333 @emph{all} messages automatically, or to report none automatically (so
10334 that you must always press @kbd{w} yourself to see the messages).
10336 @node Multiple Calculators, Troubleshooting Commands, Error Messages, Introduction
10337 @section Multiple Calculators
10340 @pindex another-calc
10341 It is possible to have any number of Calc mode buffers at once.
10342 Usually this is done by executing @kbd{M-x another-calc}, which
10343 is similar to @kbd{C-x * c} except that if a @samp{*Calculator*}
10344 buffer already exists, a new, independent one with a name of the
10345 form @samp{*Calculator*<@var{n}>} is created. You can also use the
10346 command @code{calc-mode} to put any buffer into Calculator mode, but
10347 this would ordinarily never be done.
10349 The @kbd{q} (@code{calc-quit}) command does not destroy a Calculator buffer;
10350 it only closes its window. Use @kbd{M-x kill-buffer} to destroy a
10353 Each Calculator buffer keeps its own stack, undo list, and mode settings
10354 such as precision, angular mode, and display formats. In Emacs terms,
10355 variables such as @code{calc-stack} are buffer-local variables. The
10356 global default values of these variables are used only when a new
10357 Calculator buffer is created. The @code{calc-quit} command saves
10358 the stack and mode settings of the buffer being quit as the new defaults.
10360 There is only one trail buffer, @samp{*Calc Trail*}, used by all
10361 Calculator buffers.
10363 @node Troubleshooting Commands, , Multiple Calculators, Introduction
10364 @section Troubleshooting Commands
10367 This section describes commands you can use in case a computation
10368 incorrectly fails or gives the wrong answer.
10370 @xref{Reporting Bugs}, if you find a problem that appears to be due
10371 to a bug or deficiency in Calc.
10374 * Autoloading Problems::
10375 * Recursion Depth::
10380 @node Autoloading Problems, Recursion Depth, Troubleshooting Commands, Troubleshooting Commands
10381 @subsection Autoloading Problems
10384 The Calc program is split into many component files; components are
10385 loaded automatically as you use various commands that require them.
10386 Occasionally Calc may lose track of when a certain component is
10387 necessary; typically this means you will type a command and it won't
10388 work because some function you've never heard of was undefined.
10391 @pindex calc-load-everything
10392 If this happens, the easiest workaround is to type @kbd{C-x * L}
10393 (@code{calc-load-everything}) to force all the parts of Calc to be
10394 loaded right away. This will cause Emacs to take up a lot more
10395 memory than it would otherwise, but it's guaranteed to fix the problem.
10397 @node Recursion Depth, Caches, Autoloading Problems, Troubleshooting Commands
10398 @subsection Recursion Depth
10403 @pindex calc-more-recursion-depth
10404 @pindex calc-less-recursion-depth
10405 @cindex Recursion depth
10406 @cindex ``Computation got stuck'' message
10407 @cindex @code{max-lisp-eval-depth}
10408 @cindex @code{max-specpdl-size}
10409 Calc uses recursion in many of its calculations. Emacs Lisp keeps a
10410 variable @code{max-lisp-eval-depth} which limits the amount of recursion
10411 possible in an attempt to recover from program bugs. If a calculation
10412 ever halts incorrectly with the message ``Computation got stuck or
10413 ran too long,'' use the @kbd{M} command (@code{calc-more-recursion-depth})
10414 to increase this limit. (Of course, this will not help if the
10415 calculation really did get stuck due to some problem inside Calc.)
10417 The limit is always increased (multiplied) by a factor of two. There
10418 is also an @kbd{I M} (@code{calc-less-recursion-depth}) command which
10419 decreases this limit by a factor of two, down to a minimum value of 200.
10420 The default value is 1000.
10422 These commands also double or halve @code{max-specpdl-size}, another
10423 internal Lisp recursion limit. The minimum value for this limit is 600.
10425 @node Caches, Debugging Calc, Recursion Depth, Troubleshooting Commands
10430 @cindex Flushing caches
10431 Calc saves certain values after they have been computed once. For
10432 example, the @kbd{P} (@code{calc-pi}) command initially ``knows'' the
10433 constant @cpi{} to about 20 decimal places; if the current precision
10434 is greater than this, it will recompute @cpi{} using a series
10435 approximation. This value will not need to be recomputed ever again
10436 unless you raise the precision still further. Many operations such as
10437 logarithms and sines make use of similarly cached values such as
10439 @texline @math{\ln 2}.
10440 @infoline @expr{ln(2)}.
10441 The visible effect of caching is that
10442 high-precision computations may seem to do extra work the first time.
10443 Other things cached include powers of two (for the binary arithmetic
10444 functions), matrix inverses and determinants, symbolic integrals, and
10445 data points computed by the graphing commands.
10447 @pindex calc-flush-caches
10448 If you suspect a Calculator cache has become corrupt, you can use the
10449 @code{calc-flush-caches} command to reset all caches to the empty state.
10450 (This should only be necessary in the event of bugs in the Calculator.)
10451 The @kbd{C-x * 0} (with the zero key) command also resets caches along
10452 with all other aspects of the Calculator's state.
10454 @node Debugging Calc, , Caches, Troubleshooting Commands
10455 @subsection Debugging Calc
10458 A few commands exist to help in the debugging of Calc commands.
10459 @xref{Programming}, to see the various ways that you can write
10460 your own Calc commands.
10463 @pindex calc-timing
10464 The @kbd{Z T} (@code{calc-timing}) command turns on and off a mode
10465 in which the timing of slow commands is reported in the Trail.
10466 Any Calc command that takes two seconds or longer writes a line
10467 to the Trail showing how many seconds it took. This value is
10468 accurate only to within one second.
10470 All steps of executing a command are included; in particular, time
10471 taken to format the result for display in the stack and trail is
10472 counted. Some prompts also count time taken waiting for them to
10473 be answered, while others do not; this depends on the exact
10474 implementation of the command. For best results, if you are timing
10475 a sequence that includes prompts or multiple commands, define a
10476 keyboard macro to run the whole sequence at once. Calc's @kbd{X}
10477 command (@pxref{Keyboard Macros}) will then report the time taken
10478 to execute the whole macro.
10480 Another advantage of the @kbd{X} command is that while it is
10481 executing, the stack and trail are not updated from step to step.
10482 So if you expect the output of your test sequence to leave a result
10483 that may take a long time to format and you don't wish to count
10484 this formatting time, end your sequence with a @key{DEL} keystroke
10485 to clear the result from the stack. When you run the sequence with
10486 @kbd{X}, Calc will never bother to format the large result.
10488 Another thing @kbd{Z T} does is to increase the Emacs variable
10489 @code{gc-cons-threshold} to a much higher value (two million; the
10490 usual default in Calc is 250,000) for the duration of each command.
10491 This generally prevents garbage collection during the timing of
10492 the command, though it may cause your Emacs process to grow
10493 abnormally large. (Garbage collection time is a major unpredictable
10494 factor in the timing of Emacs operations.)
10496 Another command that is useful when debugging your own Lisp
10497 extensions to Calc is @kbd{M-x calc-pass-errors}, which disables
10498 the error handler that changes the ``@code{max-lisp-eval-depth}
10499 exceeded'' message to the much more friendly ``Computation got
10500 stuck or ran too long.'' This handler interferes with the Emacs
10501 Lisp debugger's @code{debug-on-error} mode. Errors are reported
10502 in the handler itself rather than at the true location of the
10503 error. After you have executed @code{calc-pass-errors}, Lisp
10504 errors will be reported correctly but the user-friendly message
10507 @node Data Types, Stack and Trail, Introduction, Top
10508 @chapter Data Types
10511 This chapter discusses the various types of objects that can be placed
10512 on the Calculator stack, how they are displayed, and how they are
10513 entered. (@xref{Data Type Formats}, for information on how these data
10514 types are represented as underlying Lisp objects.)
10516 Integers, fractions, and floats are various ways of describing real
10517 numbers. HMS forms also for many purposes act as real numbers. These
10518 types can be combined to form complex numbers, modulo forms, error forms,
10519 or interval forms. (But these last four types cannot be combined
10520 arbitrarily: error forms may not contain modulo forms, for example.)
10521 Finally, all these types of numbers may be combined into vectors,
10522 matrices, or algebraic formulas.
10525 * Integers:: The most basic data type.
10526 * Fractions:: This and above are called @dfn{rationals}.
10527 * Floats:: This and above are called @dfn{reals}.
10528 * Complex Numbers:: This and above are called @dfn{numbers}.
10530 * Vectors and Matrices::
10537 * Incomplete Objects::
10542 @node Integers, Fractions, Data Types, Data Types
10547 The Calculator stores integers to arbitrary precision. Addition,
10548 subtraction, and multiplication of integers always yields an exact
10549 integer result. (If the result of a division or exponentiation of
10550 integers is not an integer, it is expressed in fractional or
10551 floating-point form according to the current Fraction mode.
10552 @xref{Fraction Mode}.)
10554 A decimal integer is represented as an optional sign followed by a
10555 sequence of digits. Grouping (@pxref{Grouping Digits}) can be used to
10556 insert a comma at every third digit for display purposes, but you
10557 must not type commas during the entry of numbers.
10560 A non-decimal integer is represented as an optional sign, a radix
10561 between 2 and 36, a @samp{#} symbol, and one or more digits. For radix 11
10562 and above, the letters A through Z (upper- or lower-case) count as
10563 digits and do not terminate numeric entry mode. @xref{Radix Modes}, for how
10564 to set the default radix for display of integers. Numbers of any radix
10565 may be entered at any time. If you press @kbd{#} at the beginning of a
10566 number, the current display radix is used.
10568 @node Fractions, Floats, Integers, Data Types
10573 A @dfn{fraction} is a ratio of two integers. Fractions are traditionally
10574 written ``2/3'' but Calc uses the notation @samp{2:3}. (The @kbd{/} key
10575 performs RPN division; the following two sequences push the number
10576 @samp{2:3} on the stack: @kbd{2 :@: 3 @key{RET}}, or @kbd{2 @key{RET} 3 /}
10577 assuming Fraction mode has been enabled.)
10578 When the Calculator produces a fractional result it always reduces it to
10579 simplest form, which may in fact be an integer.
10581 Fractions may also be entered in a three-part form, where @samp{2:3:4}
10582 represents two-and-three-quarters. @xref{Fraction Formats}, for fraction
10585 Non-decimal fractions are entered and displayed as
10586 @samp{@var{radix}#@var{num}:@var{denom}} (or in the analogous three-part
10587 form). The numerator and denominator always use the same radix.
10589 @node Floats, Complex Numbers, Fractions, Data Types
10593 @cindex Floating-point numbers
10594 A floating-point number or @dfn{float} is a number stored in scientific
10595 notation. The number of significant digits in the fractional part is
10596 governed by the current floating precision (@pxref{Precision}). The
10597 range of acceptable values is from
10598 @texline @math{10^{-3999999}}
10599 @infoline @expr{10^-3999999}
10601 @texline @math{10^{4000000}}
10602 @infoline @expr{10^4000000}
10603 (exclusive), plus the corresponding negative values and zero.
10605 Calculations that would exceed the allowable range of values (such
10606 as @samp{exp(exp(20))}) are left in symbolic form by Calc. The
10607 messages ``floating-point overflow'' or ``floating-point underflow''
10608 indicate that during the calculation a number would have been produced
10609 that was too large or too close to zero, respectively, to be represented
10610 by Calc. This does not necessarily mean the final result would have
10611 overflowed, just that an overflow occurred while computing the result.
10612 (In fact, it could report an underflow even though the final result
10613 would have overflowed!)
10615 If a rational number and a float are mixed in a calculation, the result
10616 will in general be expressed as a float. Commands that require an integer
10617 value (such as @kbd{k g} [@code{gcd}]) will also accept integer-valued
10618 floats, i.e., floating-point numbers with nothing after the decimal point.
10620 Floats are identified by the presence of a decimal point and/or an
10621 exponent. In general a float consists of an optional sign, digits
10622 including an optional decimal point, and an optional exponent consisting
10623 of an @samp{e}, an optional sign, and up to seven exponent digits.
10624 For example, @samp{23.5e-2} is 23.5 times ten to the minus-second power,
10627 Floating-point numbers are normally displayed in decimal notation with
10628 all significant figures shown. Exceedingly large or small numbers are
10629 displayed in scientific notation. Various other display options are
10630 available. @xref{Float Formats}.
10632 @cindex Accuracy of calculations
10633 Floating-point numbers are stored in decimal, not binary. The result
10634 of each operation is rounded to the nearest value representable in the
10635 number of significant digits specified by the current precision,
10636 rounding away from zero in the case of a tie. Thus (in the default
10637 display mode) what you see is exactly what you get. Some operations such
10638 as square roots and transcendental functions are performed with several
10639 digits of extra precision and then rounded down, in an effort to make the
10640 final result accurate to the full requested precision. However,
10641 accuracy is not rigorously guaranteed. If you suspect the validity of a
10642 result, try doing the same calculation in a higher precision. The
10643 Calculator's arithmetic is not intended to be IEEE-conformant in any
10646 While floats are always @emph{stored} in decimal, they can be entered
10647 and displayed in any radix just like integers and fractions. Since a
10648 float that is entered in a radix other that 10 will be converted to
10649 decimal, the number that Calc stores may not be exactly the number that
10650 was entered, it will be the closest decimal approximation given the
10651 current precision. The notation @samp{@var{radix}#@var{ddd}.@var{ddd}}
10652 is a floating-point number whose digits are in the specified radix.
10653 Note that the @samp{.} is more aptly referred to as a ``radix point''
10654 than as a decimal point in this case. The number @samp{8#123.4567} is
10655 defined as @samp{8#1234567 * 8^-4}. If the radix is 14 or less, you can
10656 use @samp{e} notation to write a non-decimal number in scientific
10657 notation. The exponent is written in decimal, and is considered to be a
10658 power of the radix: @samp{8#1234567e-4}. If the radix is 15 or above,
10659 the letter @samp{e} is a digit, so scientific notation must be written
10660 out, e.g., @samp{16#123.4567*16^2}. The first two exercises of the
10661 Modes Tutorial explore some of the properties of non-decimal floats.
10663 @node Complex Numbers, Infinities, Floats, Data Types
10664 @section Complex Numbers
10667 @cindex Complex numbers
10668 There are two supported formats for complex numbers: rectangular and
10669 polar. The default format is rectangular, displayed in the form
10670 @samp{(@var{real},@var{imag})} where @var{real} is the real part and
10671 @var{imag} is the imaginary part, each of which may be any real number.
10672 Rectangular complex numbers can also be displayed in @samp{@var{a}+@var{b}i}
10673 notation; @pxref{Complex Formats}.
10675 Polar complex numbers are displayed in the form
10676 @texline `@tfn{(}@var{r}@tfn{;}@math{\theta}@tfn{)}'
10677 @infoline `@tfn{(}@var{r}@tfn{;}@var{theta}@tfn{)}'
10678 where @var{r} is the nonnegative magnitude and
10679 @texline @math{\theta}
10680 @infoline @var{theta}
10681 is the argument or phase angle. The range of
10682 @texline @math{\theta}
10683 @infoline @var{theta}
10684 depends on the current angular mode (@pxref{Angular Modes}); it is
10685 generally between @mathit{-180} and @mathit{+180} degrees or the equivalent range
10688 Complex numbers are entered in stages using incomplete objects.
10689 @xref{Incomplete Objects}.
10691 Operations on rectangular complex numbers yield rectangular complex
10692 results, and similarly for polar complex numbers. Where the two types
10693 are mixed, or where new complex numbers arise (as for the square root of
10694 a negative real), the current @dfn{Polar mode} is used to determine the
10695 type. @xref{Polar Mode}.
10697 A complex result in which the imaginary part is zero (or the phase angle
10698 is 0 or 180 degrees or @cpi{} radians) is automatically converted to a real
10701 @node Infinities, Vectors and Matrices, Complex Numbers, Data Types
10702 @section Infinities
10706 @cindex @code{inf} variable
10707 @cindex @code{uinf} variable
10708 @cindex @code{nan} variable
10712 The word @code{inf} represents the mathematical concept of @dfn{infinity}.
10713 Calc actually has three slightly different infinity-like values:
10714 @code{inf}, @code{uinf}, and @code{nan}. These are just regular
10715 variable names (@pxref{Variables}); you should avoid using these
10716 names for your own variables because Calc gives them special
10717 treatment. Infinities, like all variable names, are normally
10718 entered using algebraic entry.
10720 Mathematically speaking, it is not rigorously correct to treat
10721 ``infinity'' as if it were a number, but mathematicians often do
10722 so informally. When they say that @samp{1 / inf = 0}, what they
10723 really mean is that @expr{1 / x}, as @expr{x} becomes larger and
10724 larger, becomes arbitrarily close to zero. So you can imagine
10725 that if @expr{x} got ``all the way to infinity,'' then @expr{1 / x}
10726 would go all the way to zero. Similarly, when they say that
10727 @samp{exp(inf) = inf}, they mean that
10728 @texline @math{e^x}
10729 @infoline @expr{exp(x)}
10730 grows without bound as @expr{x} grows. The symbol @samp{-inf} likewise
10731 stands for an infinitely negative real value; for example, we say that
10732 @samp{exp(-inf) = 0}. You can have an infinity pointing in any
10733 direction on the complex plane: @samp{sqrt(-inf) = i inf}.
10735 The same concept of limits can be used to define @expr{1 / 0}. We
10736 really want the value that @expr{1 / x} approaches as @expr{x}
10737 approaches zero. But if all we have is @expr{1 / 0}, we can't
10738 tell which direction @expr{x} was coming from. If @expr{x} was
10739 positive and decreasing toward zero, then we should say that
10740 @samp{1 / 0 = inf}. But if @expr{x} was negative and increasing
10741 toward zero, the answer is @samp{1 / 0 = -inf}. In fact, @expr{x}
10742 could be an imaginary number, giving the answer @samp{i inf} or
10743 @samp{-i inf}. Calc uses the special symbol @samp{uinf} to mean
10744 @dfn{undirected infinity}, i.e., a value which is infinitely
10745 large but with an unknown sign (or direction on the complex plane).
10747 Calc actually has three modes that say how infinities are handled.
10748 Normally, infinities never arise from calculations that didn't
10749 already have them. Thus, @expr{1 / 0} is treated simply as an
10750 error and left unevaluated. The @kbd{m i} (@code{calc-infinite-mode})
10751 command (@pxref{Infinite Mode}) enables a mode in which
10752 @expr{1 / 0} evaluates to @code{uinf} instead. There is also
10753 an alternative type of infinite mode which says to treat zeros
10754 as if they were positive, so that @samp{1 / 0 = inf}. While this
10755 is less mathematically correct, it may be the answer you want in
10758 Since all infinities are ``as large'' as all others, Calc simplifies,
10759 e.g., @samp{5 inf} to @samp{inf}. Another example is
10760 @samp{5 - inf = -inf}, where the @samp{-inf} is so large that
10761 adding a finite number like five to it does not affect it.
10762 Note that @samp{a - inf} also results in @samp{-inf}; Calc assumes
10763 that variables like @code{a} always stand for finite quantities.
10764 Just to show that infinities really are all the same size,
10765 note that @samp{sqrt(inf) = inf^2 = exp(inf) = inf} in Calc's
10768 It's not so easy to define certain formulas like @samp{0 * inf} and
10769 @samp{inf / inf}. Depending on where these zeros and infinities
10770 came from, the answer could be literally anything. The latter
10771 formula could be the limit of @expr{x / x} (giving a result of one),
10772 or @expr{2 x / x} (giving two), or @expr{x^2 / x} (giving @code{inf}),
10773 or @expr{x / x^2} (giving zero). Calc uses the symbol @code{nan}
10774 to represent such an @dfn{indeterminate} value. (The name ``nan''
10775 comes from analogy with the ``NAN'' concept of IEEE standard
10776 arithmetic; it stands for ``Not A Number.'' This is somewhat of a
10777 misnomer, since @code{nan} @emph{does} stand for some number or
10778 infinity, it's just that @emph{which} number it stands for
10779 cannot be determined.) In Calc's notation, @samp{0 * inf = nan}
10780 and @samp{inf / inf = nan}. A few other common indeterminate
10781 expressions are @samp{inf - inf} and @samp{inf ^ 0}. Also,
10782 @samp{0 / 0 = nan} if you have turned on Infinite mode
10783 (as described above).
10785 Infinities are especially useful as parts of @dfn{intervals}.
10786 @xref{Interval Forms}.
10788 @node Vectors and Matrices, Strings, Infinities, Data Types
10789 @section Vectors and Matrices
10793 @cindex Plain vectors
10795 The @dfn{vector} data type is flexible and general. A vector is simply a
10796 list of zero or more data objects. When these objects are numbers, the
10797 whole is a vector in the mathematical sense. When these objects are
10798 themselves vectors of equal (nonzero) length, the whole is a @dfn{matrix}.
10799 A vector which is not a matrix is referred to here as a @dfn{plain vector}.
10801 A vector is displayed as a list of values separated by commas and enclosed
10802 in square brackets: @samp{[1, 2, 3]}. Thus the following is a 2 row by
10803 3 column matrix: @samp{[[1, 2, 3], [4, 5, 6]]}. Vectors, like complex
10804 numbers, are entered as incomplete objects. @xref{Incomplete Objects}.
10805 During algebraic entry, vectors are entered all at once in the usual
10806 brackets-and-commas form. Matrices may be entered algebraically as nested
10807 vectors, or using the shortcut notation @w{@samp{[1, 2, 3; 4, 5, 6]}},
10808 with rows separated by semicolons. The commas may usually be omitted
10809 when entering vectors: @samp{[1 2 3]}. Curly braces may be used in
10810 place of brackets: @samp{@{1, 2, 3@}}, but the commas are required in
10813 Traditional vector and matrix arithmetic is also supported;
10814 @pxref{Basic Arithmetic} and @pxref{Matrix Functions}.
10815 Many other operations are applied to vectors element-wise. For example,
10816 the complex conjugate of a vector is a vector of the complex conjugates
10823 Algebraic functions for building vectors include @samp{vec(a, b, c)}
10824 to build @samp{[a, b, c]}, @samp{cvec(a, n, m)} to build an
10825 @texline @math{n\times m}
10826 @infoline @var{n}x@var{m}
10827 matrix of @samp{a}s, and @samp{index(n)} to build a vector of integers
10828 from 1 to @samp{n}.
10830 @node Strings, HMS Forms, Vectors and Matrices, Data Types
10836 @cindex Character strings
10837 Character strings are not a special data type in the Calculator.
10838 Rather, a string is represented simply as a vector all of whose
10839 elements are integers in the range 0 to 255 (ASCII codes). You can
10840 enter a string at any time by pressing the @kbd{"} key. Quotation
10841 marks and backslashes are written @samp{\"} and @samp{\\}, respectively,
10842 inside strings. Other notations introduced by backslashes are:
10858 Finally, a backslash followed by three octal digits produces any
10859 character from its ASCII code.
10862 @pindex calc-display-strings
10863 Strings are normally displayed in vector-of-integers form. The
10864 @w{@kbd{d "}} (@code{calc-display-strings}) command toggles a mode in
10865 which any vectors of small integers are displayed as quoted strings
10868 The backslash notations shown above are also used for displaying
10869 strings. Characters 128 and above are not translated by Calc; unless
10870 you have an Emacs modified for 8-bit fonts, these will show up in
10871 backslash-octal-digits notation. For characters below 32, and
10872 for character 127, Calc uses the backslash-letter combination if
10873 there is one, or otherwise uses a @samp{\^} sequence.
10875 The only Calc feature that uses strings is @dfn{compositions};
10876 @pxref{Compositions}. Strings also provide a convenient
10877 way to do conversions between ASCII characters and integers.
10883 There is a @code{string} function which provides a different display
10884 format for strings. Basically, @samp{string(@var{s})}, where @var{s}
10885 is a vector of integers in the proper range, is displayed as the
10886 corresponding string of characters with no surrounding quotation
10887 marks or other modifications. Thus @samp{string("ABC")} (or
10888 @samp{string([65 66 67])}) will look like @samp{ABC} on the stack.
10889 This happens regardless of whether @w{@kbd{d "}} has been used. The
10890 only way to turn it off is to use @kbd{d U} (unformatted language
10891 mode) which will display @samp{string("ABC")} instead.
10893 Control characters are displayed somewhat differently by @code{string}.
10894 Characters below 32, and character 127, are shown using @samp{^} notation
10895 (same as shown above, but without the backslash). The quote and
10896 backslash characters are left alone, as are characters 128 and above.
10902 The @code{bstring} function is just like @code{string} except that
10903 the resulting string is breakable across multiple lines if it doesn't
10904 fit all on one line. Potential break points occur at every space
10905 character in the string.
10907 @node HMS Forms, Date Forms, Strings, Data Types
10911 @cindex Hours-minutes-seconds forms
10912 @cindex Degrees-minutes-seconds forms
10913 @dfn{HMS} stands for Hours-Minutes-Seconds; when used as an angular
10914 argument, the interpretation is Degrees-Minutes-Seconds. All functions
10915 that operate on angles accept HMS forms. These are interpreted as
10916 degrees regardless of the current angular mode. It is also possible to
10917 use HMS as the angular mode so that calculated angles are expressed in
10918 degrees, minutes, and seconds.
10924 @kindex ' (HMS forms)
10928 @kindex " (HMS forms)
10932 @kindex h (HMS forms)
10936 @kindex o (HMS forms)
10940 @kindex m (HMS forms)
10944 @kindex s (HMS forms)
10945 The default format for HMS values is
10946 @samp{@var{hours}@@ @var{mins}' @var{secs}"}. During entry, the letters
10947 @samp{h} (for ``hours'') or
10948 @samp{o} (approximating the ``degrees'' symbol) are accepted as well as
10949 @samp{@@}, @samp{m} is accepted in place of @samp{'}, and @samp{s} is
10950 accepted in place of @samp{"}.
10951 The @var{hours} value is an integer (or integer-valued float).
10952 The @var{mins} value is an integer or integer-valued float between 0 and 59.
10953 The @var{secs} value is a real number between 0 (inclusive) and 60
10954 (exclusive). A positive HMS form is interpreted as @var{hours} +
10955 @var{mins}/60 + @var{secs}/3600. A negative HMS form is interpreted
10956 as @mathit{- @var{hours}} @mathit{-} @var{mins}/60 @mathit{-} @var{secs}/3600.
10957 Display format for HMS forms is quite flexible. @xref{HMS Formats}.
10959 HMS forms can be added and subtracted. When they are added to numbers,
10960 the numbers are interpreted according to the current angular mode. HMS
10961 forms can also be multiplied and divided by real numbers. Dividing
10962 two HMS forms produces a real-valued ratio of the two angles.
10965 @cindex Time of day
10966 Just for kicks, @kbd{M-x calc-time} pushes the current time of day on
10967 the stack as an HMS form.
10969 @node Date Forms, Modulo Forms, HMS Forms, Data Types
10970 @section Date Forms
10974 A @dfn{date form} represents a date and possibly an associated time.
10975 Simple date arithmetic is supported: Adding a number to a date
10976 produces a new date shifted by that many days; adding an HMS form to
10977 a date shifts it by that many hours. Subtracting two date forms
10978 computes the number of days between them (represented as a simple
10979 number). Many other operations, such as multiplying two date forms,
10980 are nonsensical and are not allowed by Calc.
10982 Date forms are entered and displayed enclosed in @samp{< >} brackets.
10983 The default format is, e.g., @samp{<Wed Jan 9, 1991>} for dates,
10984 or @samp{<3:32:20pm Wed Jan 9, 1991>} for dates with times.
10985 Input is flexible; date forms can be entered in any of the usual
10986 notations for dates and times. @xref{Date Formats}.
10988 Date forms are stored internally as numbers, specifically the number
10989 of days since midnight on the morning of December 31 of the year 1 BC@.
10990 If the internal number is an integer, the form represents a date only;
10991 if the internal number is a fraction or float, the form represents
10992 a date and time. For example, @samp{<6:00am Thu Jan 10, 1991>}
10993 is represented by the number 726842.25. The standard precision of
10994 12 decimal digits is enough to ensure that a (reasonable) date and
10995 time can be stored without roundoff error.
10997 If the current precision is greater than 12, date forms will keep
10998 additional digits in the seconds position. For example, if the
10999 precision is 15, the seconds will keep three digits after the
11000 decimal point. Decreasing the precision below 12 may cause the
11001 time part of a date form to become inaccurate. This can also happen
11002 if astronomically high years are used, though this will not be an
11003 issue in everyday (or even everymillennium) use. Note that date
11004 forms without times are stored as exact integers, so roundoff is
11005 never an issue for them.
11007 You can use the @kbd{v p} (@code{calc-pack}) and @kbd{v u}
11008 (@code{calc-unpack}) commands to get at the numerical representation
11009 of a date form. @xref{Packing and Unpacking}.
11011 Date forms can go arbitrarily far into the future or past. Negative
11012 year numbers represent years BC@. There is no ``year 0''; the day
11013 before @samp{<Mon Jan 1, +1>} is @samp{<Sun Dec 31, -1>}. These are
11014 days 1 and 0 respectively in Calc's internal numbering scheme. The
11015 Gregorian calendar is used for all dates, including dates before the
11016 Gregorian calendar was invented (although that can be configured; see
11017 below). Thus Calc's use of the day number @mathit{-10000} to
11018 represent August 15, 28 BC should be taken with a grain of salt.
11020 @cindex Julian calendar
11021 @cindex Gregorian calendar
11022 Some historical background: The Julian calendar was created by
11023 Julius Caesar in the year 46 BC as an attempt to fix the confusion
11024 caused by the irregular Roman calendar that was used before that time.
11025 The Julian calendar introduced an extra day in all years divisible by
11026 four. After some initial confusion, the calendar was adopted around
11027 the year we call 8 AD@. Some centuries later it became
11028 apparent that the Julian year of 365.25 days was itself not quite
11029 right. In 1582 Pope Gregory XIII introduced the Gregorian calendar,
11030 which added the new rule that years divisible by 100, but not by 400,
11031 were not to be considered leap years despite being divisible by four.
11032 Many countries delayed adoption of the Gregorian calendar
11033 because of religious differences. For example, Great Britain and the
11034 British colonies switched to the Gregorian calendar in September
11035 1752, when the Julian calendar was eleven days behind the
11036 Gregorian calendar. That year in Britain, the day after September 2
11037 was September 14. To take another example, Russia did not adopt the
11038 Gregorian calendar until 1918, and that year in Russia the day after
11039 January 31 was February 14. Calc's reckoning therefore matches English
11040 practice starting in 1752 and Russian practice starting in 1918, but
11041 disagrees with earlier dates in both countries.
11043 When the Julian calendar was introduced, it had January 1 as the first
11044 day of the year. By the Middle Ages, many European countries
11045 had changed the beginning of a new year to a different date, often to
11046 a religious festival. Almost all countries reverted to using January 1
11047 as the beginning of the year by the time they adopted the Gregorian
11050 Some calendars attempt to mimic the historical situation by using the
11051 Gregorian calendar for recent dates and the Julian calendar for older
11052 dates. The @code{cal} program in most Unix implementations does this,
11053 for example. While January 1 wasn't always the beginning of a calendar
11054 year, these hybrid calendars still use January 1 as the beginning of
11055 the year even for older dates. The customizable variable
11056 @code{calc-gregorian-switch} (@pxref{Customizing Calc}) can be set to
11057 have Calc's date forms switch from the Julian to Gregorian calendar at
11058 any specified date.
11060 Today's timekeepers introduce an occasional ``leap second''.
11061 These do not occur regularly and Calc does not take these minor
11062 effects into account. (If it did, it would have to report a
11063 non-integer number of days between, say,
11064 @samp{<12:00am Mon Jan 1, 1900>} and
11065 @samp{<12:00am Sat Jan 1, 2000>}.)
11067 @cindex Julian day counting
11068 Another day counting system in common use is, confusingly, also called
11069 ``Julian.'' Julian days go from noon to noon. The Julian day number
11070 is the numbers of days since 12:00 noon (GMT) on November 24, 4714 BC
11071 in the Gregorian calendar (i.e., January 1, 4713 BC in the Julian
11072 calendar). In Calc's scheme (in GMT) the Julian day origin is
11073 @mathit{-1721422.5}, because Calc starts at midnight instead of noon.
11074 Thus to convert a Calc date code obtained by unpacking a
11075 date form into a Julian day number, simply add 1721422.5 after
11076 compensating for the time zone difference. The built-in @kbd{t J}
11077 command performs this conversion for you.
11079 The Julian day number is based on the Julian cycle, which was invented
11080 in 1583 by Joseph Justus Scaliger. Scaliger named it the Julian cycle
11081 since it involves the Julian calendar, but some have suggested that
11082 Scaliger named it in honor of his father, Julius Caesar Scaliger. The
11083 Julian cycle is based on three other cycles: the indiction cycle, the
11084 Metonic cycle, and the solar cycle. The indiction cycle is a 15 year
11085 cycle originally used by the Romans for tax purposes but later used to
11086 date medieval documents. The Metonic cycle is a 19 year cycle; 19
11087 years is close to being a common multiple of a solar year and a lunar
11088 month, and so every 19 years the phases of the moon will occur on the
11089 same days of the year. The solar cycle is a 28 year cycle; the Julian
11090 calendar repeats itself every 28 years. The smallest time period
11091 which contains multiples of all three cycles is the least common
11092 multiple of 15 years, 19 years and 28 years, which (since they're
11093 pairwise relatively prime) is
11094 @texline @math{15\times 19\times 28 = 7980} years.
11095 @infoline 15*19*28 = 7980 years.
11096 This is the length of a Julian cycle. Working backwards, the previous
11097 year in which all three cycles began was 4713 BC, and so Scaliger
11098 chose that year as the beginning of a Julian cycle. Since at the time
11099 there were no historical records from before 4713 BC, using this year
11100 as a starting point had the advantage of avoiding negative year
11101 numbers. In 1849, the astronomer John Herschel (son of William
11102 Herschel) suggested using the number of days since the beginning of
11103 the Julian cycle as an astronomical dating system; this idea was taken
11104 up by other astronomers. (At the time, noon was the start of the
11105 astronomical day. Herschel originally suggested counting the days
11106 since Jan 1, 4713 BC at noon Alexandria time; this was later amended to
11107 noon GMT@.) Julian day numbering is largely used in astronomy.
11109 @cindex Unix time format
11110 The Unix operating system measures time as an integer number of
11111 seconds since midnight, Jan 1, 1970. To convert a Calc date
11112 value into a Unix time stamp, first subtract 719164 (the code
11113 for @samp{<Jan 1, 1970>}), then multiply by 86400 (the number of
11114 seconds in a day) and press @kbd{R} to round to the nearest
11115 integer. If you have a date form, you can simply subtract the
11116 day @samp{<Jan 1, 1970>} instead of unpacking and subtracting
11117 719164. Likewise, divide by 86400 and add @samp{<Jan 1, 1970>}
11118 to convert from Unix time to a Calc date form. (Note that
11119 Unix normally maintains the time in the GMT time zone; you may
11120 need to subtract five hours to get New York time, or eight hours
11121 for California time. The same is usually true of Julian day
11122 counts.) The built-in @kbd{t U} command performs these
11125 @node Modulo Forms, Error Forms, Date Forms, Data Types
11126 @section Modulo Forms
11129 @cindex Modulo forms
11130 A @dfn{modulo form} is a real number which is taken modulo (i.e., within
11131 an integer multiple of) some value @var{M}. Arithmetic modulo @var{M}
11132 often arises in number theory. Modulo forms are written
11133 `@var{a} @tfn{mod} @var{M}',
11134 where @var{a} and @var{M} are real numbers or HMS forms, and
11135 @texline @math{0 \le a < M}.
11136 @infoline @expr{0 <= a < @var{M}}.
11137 In many applications @expr{a} and @expr{M} will be
11138 integers but this is not required.
11143 @kindex M (modulo forms)
11147 @tindex mod (operator)
11148 To create a modulo form during numeric entry, press the shift-@kbd{M}
11149 key to enter the word @samp{mod}. As a special convenience, pressing
11150 shift-@kbd{M} a second time automatically enters the value of @expr{M}
11151 that was most recently used before. During algebraic entry, either
11152 type @samp{mod} by hand or press @kbd{M-m} (that's @kbd{@key{META}-m}).
11153 Once again, pressing this a second time enters the current modulo.
11155 Modulo forms are not to be confused with the modulo operator @samp{%}.
11156 The expression @samp{27 % 10} means to compute 27 modulo 10 to produce
11157 the result 7. Further computations treat this 7 as just a regular integer.
11158 The expression @samp{27 mod 10} produces the result @samp{7 mod 10};
11159 further computations with this value are again reduced modulo 10 so that
11160 the result always lies in the desired range.
11162 When two modulo forms with identical @expr{M}'s are added or multiplied,
11163 the Calculator simply adds or multiplies the values, then reduces modulo
11164 @expr{M}. If one argument is a modulo form and the other a plain number,
11165 the plain number is treated like a compatible modulo form. It is also
11166 possible to raise modulo forms to powers; the result is the value raised
11167 to the power, then reduced modulo @expr{M}. (When all values involved
11168 are integers, this calculation is done much more efficiently than
11169 actually computing the power and then reducing.)
11171 @cindex Modulo division
11172 Two modulo forms `@var{a} @tfn{mod} @var{M}' and `@var{b} @tfn{mod} @var{M}'
11173 can be divided if @expr{a}, @expr{b}, and @expr{M} are all
11174 integers. The result is the modulo form which, when multiplied by
11175 `@var{b} @tfn{mod} @var{M}', produces `@var{a} @tfn{mod} @var{M}'. If
11176 there is no solution to this equation (which can happen only when
11177 @expr{M} is non-prime), or if any of the arguments are non-integers, the
11178 division is left in symbolic form. Other operations, such as square
11179 roots, are not yet supported for modulo forms. (Note that, although
11180 @w{`@tfn{(}@var{a} @tfn{mod} @var{M}@tfn{)^.5}'} will compute a ``modulo square root''
11181 in the sense of reducing
11182 @texline @math{\sqrt a}
11183 @infoline @expr{sqrt(a)}
11184 modulo @expr{M}, this is not a useful definition from the
11185 number-theoretical point of view.)
11187 It is possible to mix HMS forms and modulo forms. For example, an
11188 HMS form modulo 24 could be used to manipulate clock times; an HMS
11189 form modulo 360 would be suitable for angles. Making the modulo @expr{M}
11190 also be an HMS form eliminates troubles that would arise if the angular
11191 mode were inadvertently set to Radians, in which case
11192 @w{@samp{2@@ 0' 0" mod 24}} would be interpreted as two degrees modulo
11195 Modulo forms cannot have variables or formulas for components. If you
11196 enter the formula @samp{(x + 2) mod 5}, Calc propagates the modulus
11197 to each of the coefficients: @samp{(1 mod 5) x + (2 mod 5)}.
11199 You can use @kbd{v p} and @kbd{%} to modify modulo forms.
11200 @xref{Packing and Unpacking}. @xref{Basic Arithmetic}.
11206 The algebraic function @samp{makemod(a, m)} builds the modulo form
11207 @w{@samp{a mod m}}.
11209 @node Error Forms, Interval Forms, Modulo Forms, Data Types
11210 @section Error Forms
11213 @cindex Error forms
11214 @cindex Standard deviations
11215 An @dfn{error form} is a number with an associated standard
11216 deviation, as in @samp{2.3 +/- 0.12}. The notation
11217 @texline `@var{x} @tfn{+/-} @math{\sigma}'
11218 @infoline `@var{x} @tfn{+/-} sigma'
11219 stands for an uncertain value which follows
11220 a normal or Gaussian distribution of mean @expr{x} and standard
11221 deviation or ``error''
11222 @texline @math{\sigma}.
11223 @infoline @expr{sigma}.
11224 Both the mean and the error can be either numbers or
11225 formulas. Generally these are real numbers but the mean may also be
11226 complex. If the error is negative or complex, it is changed to its
11227 absolute value. An error form with zero error is converted to a
11228 regular number by the Calculator.
11230 All arithmetic and transcendental functions accept error forms as input.
11231 Operations on the mean-value part work just like operations on regular
11232 numbers. The error part for any function @expr{f(x)} (such as
11233 @texline @math{\sin x}
11234 @infoline @expr{sin(x)})
11235 is defined by the error of @expr{x} times the derivative of @expr{f}
11236 evaluated at the mean value of @expr{x}. For a two-argument function
11237 @expr{f(x,y)} (such as addition) the error is the square root of the sum
11238 of the squares of the errors due to @expr{x} and @expr{y}.
11241 f(x \hbox{\code{ +/- }} \sigma)
11242 &= f(x) \hbox{\code{ +/- }} \sigma \left| {df(x) \over dx} \right| \cr
11243 f(x \hbox{\code{ +/- }} \sigma_x, y \hbox{\code{ +/- }} \sigma_y)
11244 &= f(x,y) \hbox{\code{ +/- }}
11245 \sqrt{\left(\sigma_x \left| {\partial f(x,y) \over \partial x}
11247 +\left(\sigma_y \left| {\partial f(x,y) \over \partial y}
11248 \right| \right)^2 } \cr
11252 definition assumes the errors in @expr{x} and @expr{y} are uncorrelated.
11253 A side effect of this definition is that @samp{(2 +/- 1) * (2 +/- 1)}
11254 is not the same as @samp{(2 +/- 1)^2}; the former represents the product
11255 of two independent values which happen to have the same probability
11256 distributions, and the latter is the product of one random value with itself.
11257 The former will produce an answer with less error, since on the average
11258 the two independent errors can be expected to cancel out.
11260 Consult a good text on error analysis for a discussion of the proper use
11261 of standard deviations. Actual errors often are neither Gaussian-distributed
11262 nor uncorrelated, and the above formulas are valid only when errors
11263 are small. As an example, the error arising from
11264 @texline `@tfn{sin(}@var{x} @tfn{+/-} @math{\sigma}@tfn{)}'
11265 @infoline `@tfn{sin(}@var{x} @tfn{+/-} @var{sigma}@tfn{)}'
11267 @texline `@math{\sigma} @tfn{abs(cos(}@var{x}@tfn{))}'.
11268 @infoline `@var{sigma} @tfn{abs(cos(}@var{x}@tfn{))}'.
11269 When @expr{x} is close to zero,
11270 @texline @math{\cos x}
11271 @infoline @expr{cos(x)}
11272 is close to one so the error in the sine is close to
11273 @texline @math{\sigma};
11274 @infoline @expr{sigma};
11275 this makes sense, since
11276 @texline @math{\sin x}
11277 @infoline @expr{sin(x)}
11278 is approximately @expr{x} near zero, so a given error in @expr{x} will
11279 produce about the same error in the sine. Likewise, near 90 degrees
11280 @texline @math{\cos x}
11281 @infoline @expr{cos(x)}
11282 is nearly zero and so the computed error is
11283 small: The sine curve is nearly flat in that region, so an error in @expr{x}
11284 has relatively little effect on the value of
11285 @texline @math{\sin x}.
11286 @infoline @expr{sin(x)}.
11287 However, consider @samp{sin(90 +/- 1000)}. The cosine of 90 is zero, so
11288 Calc will report zero error! We get an obviously wrong result because
11289 we have violated the small-error approximation underlying the error
11290 analysis. If the error in @expr{x} had been small, the error in
11291 @texline @math{\sin x}
11292 @infoline @expr{sin(x)}
11293 would indeed have been negligible.
11298 @kindex p (error forms)
11300 To enter an error form during regular numeric entry, use the @kbd{p}
11301 (``plus-or-minus'') key to type the @samp{+/-} symbol. (If you try actually
11302 typing @samp{+/-} the @kbd{+} key will be interpreted as the Calculator's
11303 @kbd{+} command!) Within an algebraic formula, you can press @kbd{M-+} to
11304 type the @samp{+/-} symbol, or type it out by hand.
11306 Error forms and complex numbers can be mixed; the formulas shown above
11307 are used for complex numbers, too; note that if the error part evaluates
11308 to a complex number its absolute value (or the square root of the sum of
11309 the squares of the absolute values of the two error contributions) is
11310 used. Mathematically, this corresponds to a radially symmetric Gaussian
11311 distribution of numbers on the complex plane. However, note that Calc
11312 considers an error form with real components to represent a real number,
11313 not a complex distribution around a real mean.
11315 Error forms may also be composed of HMS forms. For best results, both
11316 the mean and the error should be HMS forms if either one is.
11322 The algebraic function @samp{sdev(a, b)} builds the error form @samp{a +/- b}.
11324 @node Interval Forms, Incomplete Objects, Error Forms, Data Types
11325 @section Interval Forms
11328 @cindex Interval forms
11329 An @dfn{interval} is a subset of consecutive real numbers. For example,
11330 the interval @samp{[2 ..@: 4]} represents all the numbers from 2 to 4,
11331 inclusive. If you multiply it by the interval @samp{[0.5 ..@: 2]} you
11332 obtain @samp{[1 ..@: 8]}. This calculation represents the fact that if
11333 you multiply some number in the range @samp{[2 ..@: 4]} by some other
11334 number in the range @samp{[0.5 ..@: 2]}, your result will lie in the range
11335 from 1 to 8. Interval arithmetic is used to get a worst-case estimate
11336 of the possible range of values a computation will produce, given the
11337 set of possible values of the input.
11340 Calc supports several varieties of intervals, including @dfn{closed}
11341 intervals of the type shown above, @dfn{open} intervals such as
11342 @samp{(2 ..@: 4)}, which represents the range of numbers from 2 to 4
11343 @emph{exclusive}, and @dfn{semi-open} intervals in which one end
11344 uses a round parenthesis and the other a square bracket. In mathematical
11346 @samp{[2 ..@: 4]} means @expr{2 <= x <= 4}, whereas
11347 @samp{[2 ..@: 4)} represents @expr{2 <= x < 4},
11348 @samp{(2 ..@: 4]} represents @expr{2 < x <= 4}, and
11349 @samp{(2 ..@: 4)} represents @expr{2 < x < 4}.
11352 Calc supports several varieties of intervals, including \dfn{closed}
11353 intervals of the type shown above, \dfn{open} intervals such as
11354 \samp{(2 ..\: 4)}, which represents the range of numbers from 2 to 4
11355 \emph{exclusive}, and \dfn{semi-open} intervals in which one end
11356 uses a round parenthesis and the other a square bracket. In mathematical
11359 [2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 \le x \le 4 \cr
11360 [2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 \le x < 4 \cr
11361 (2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 < x \le 4 \cr
11362 (2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 < x < 4 \cr
11366 The lower and upper limits of an interval must be either real numbers
11367 (or HMS or date forms), or symbolic expressions which are assumed to be
11368 real-valued, or @samp{-inf} and @samp{inf}. In general the lower limit
11369 must be less than the upper limit. A closed interval containing only
11370 one value, @samp{[3 ..@: 3]}, is converted to a plain number (3)
11371 automatically. An interval containing no values at all (such as
11372 @samp{[3 ..@: 2]} or @samp{[2 ..@: 2)}) can be represented but is not
11373 guaranteed to behave well when used in arithmetic. Note that the
11374 interval @samp{[3 .. inf)} represents all real numbers greater than
11375 or equal to 3, and @samp{(-inf .. inf)} represents all real numbers.
11376 In fact, @samp{[-inf .. inf]} represents all real numbers including
11377 the real infinities.
11379 Intervals are entered in the notation shown here, either as algebraic
11380 formulas, or using incomplete forms. (@xref{Incomplete Objects}.)
11381 In algebraic formulas, multiple periods in a row are collected from
11382 left to right, so that @samp{1...1e2} is interpreted as @samp{1.0 ..@: 1e2}
11383 rather than @samp{1 ..@: 0.1e2}. Add spaces or zeros if you want to
11384 get the other interpretation. If you omit the lower or upper limit,
11385 a default of @samp{-inf} or @samp{inf} (respectively) is furnished.
11387 Infinite mode also affects operations on intervals
11388 (@pxref{Infinities}). Calc will always introduce an open infinity,
11389 as in @samp{1 / (0 .. 2] = [0.5 .. inf)}. But closed infinities,
11390 @w{@samp{1 / [0 .. 2] = [0.5 .. inf]}}, arise only in Infinite mode;
11391 otherwise they are left unevaluated. Note that the ``direction'' of
11392 a zero is not an issue in this case since the zero is always assumed
11393 to be continuous with the rest of the interval. For intervals that
11394 contain zero inside them Calc is forced to give the result,
11395 @samp{1 / (-2 .. 2) = [-inf .. inf]}.
11397 While it may seem that intervals and error forms are similar, they are
11398 based on entirely different concepts of inexact quantities. An error
11400 @texline `@var{x} @tfn{+/-} @math{\sigma}'
11401 @infoline `@var{x} @tfn{+/-} @var{sigma}'
11402 means a variable is random, and its value could
11403 be anything but is ``probably'' within one
11404 @texline @math{\sigma}
11405 @infoline @var{sigma}
11406 of the mean value @expr{x}. An interval
11407 `@tfn{[}@var{a} @tfn{..@:} @var{b}@tfn{]}' means a
11408 variable's value is unknown, but guaranteed to lie in the specified
11409 range. Error forms are statistical or ``average case'' approximations;
11410 interval arithmetic tends to produce ``worst case'' bounds on an
11413 Intervals may not contain complex numbers, but they may contain
11414 HMS forms or date forms.
11416 @xref{Set Operations}, for commands that interpret interval forms
11417 as subsets of the set of real numbers.
11423 The algebraic function @samp{intv(n, a, b)} builds an interval form
11424 from @samp{a} to @samp{b}; @samp{n} is an integer code which must
11425 be 0 for @samp{(..)}, 1 for @samp{(..]}, 2 for @samp{[..)}, or
11428 Please note that in fully rigorous interval arithmetic, care would be
11429 taken to make sure that the computation of the lower bound rounds toward
11430 minus infinity, while upper bound computations round toward plus
11431 infinity. Calc's arithmetic always uses a round-to-nearest mode,
11432 which means that roundoff errors could creep into an interval
11433 calculation to produce intervals slightly smaller than they ought to
11434 be. For example, entering @samp{[1..2]} and pressing @kbd{Q 2 ^}
11435 should yield the interval @samp{[1..2]} again, but in fact it yields the
11436 (slightly too small) interval @samp{[1..1.9999999]} due to roundoff
11439 @node Incomplete Objects, Variables, Interval Forms, Data Types
11440 @section Incomplete Objects
11460 @cindex Incomplete vectors
11461 @cindex Incomplete complex numbers
11462 @cindex Incomplete interval forms
11463 When @kbd{(} or @kbd{[} is typed to begin entering a complex number or
11464 vector, respectively, the effect is to push an @dfn{incomplete} complex
11465 number or vector onto the stack. The @kbd{,} key adds the value(s) at
11466 the top of the stack onto the current incomplete object. The @kbd{)}
11467 and @kbd{]} keys ``close'' the incomplete object after adding any values
11468 on the top of the stack in front of the incomplete object.
11470 As a result, the sequence of keystrokes @kbd{[ 2 , 3 @key{RET} 2 * , 9 ]}
11471 pushes the vector @samp{[2, 6, 9]} onto the stack. Likewise, @kbd{( 1 , 2 Q )}
11472 pushes the complex number @samp{(1, 1.414)} (approximately).
11474 If several values lie on the stack in front of the incomplete object,
11475 all are collected and appended to the object. Thus the @kbd{,} key
11476 is redundant: @kbd{[ 2 @key{RET} 3 @key{RET} 2 * 9 ]}. Some people
11477 prefer the equivalent @key{SPC} key to @key{RET}.
11479 As a special case, typing @kbd{,} immediately after @kbd{(}, @kbd{[}, or
11480 @kbd{,} adds a zero or duplicates the preceding value in the list being
11481 formed. Typing @key{DEL} during incomplete entry removes the last item
11485 The @kbd{;} key is used in the same way as @kbd{,} to create polar complex
11486 numbers: @kbd{( 1 ; 2 )}. When entering a vector, @kbd{;} is useful for
11487 creating a matrix. In particular, @kbd{[ [ 1 , 2 ; 3 , 4 ; 5 , 6 ] ]} is
11488 equivalent to @kbd{[ [ 1 , 2 ] , [ 3 , 4 ] , [ 5 , 6 ] ]}.
11492 Incomplete entry is also used to enter intervals. For example,
11493 @kbd{[ 2 ..@: 4 )} enters a semi-open interval. Note that when you type
11494 the first period, it will be interpreted as a decimal point, but when
11495 you type a second period immediately afterward, it is re-interpreted as
11496 part of the interval symbol. Typing @kbd{..} corresponds to executing
11497 the @code{calc-dots} command.
11499 If you find incomplete entry distracting, you may wish to enter vectors
11500 and complex numbers as algebraic formulas by pressing the apostrophe key.
11502 @node Variables, Formulas, Incomplete Objects, Data Types
11506 @cindex Variables, in formulas
11507 A @dfn{variable} is somewhere between a storage register on a conventional
11508 calculator, and a variable in a programming language. (In fact, a Calc
11509 variable is really just an Emacs Lisp variable that contains a Calc number
11510 or formula.) A variable's name is normally composed of letters and digits.
11511 Calc also allows apostrophes and @code{#} signs in variable names.
11512 (The Calc variable @code{foo} corresponds to the Emacs Lisp variable
11513 @code{var-foo}, but unless you access the variable from within Emacs
11514 Lisp, you don't need to worry about it. Variable names in algebraic
11515 formulas implicitly have @samp{var-} prefixed to their names. The
11516 @samp{#} character in variable names used in algebraic formulas
11517 corresponds to a dash @samp{-} in the Lisp variable name. If the name
11518 contains any dashes, the prefix @samp{var-} is @emph{not} automatically
11519 added. Thus the two formulas @samp{foo + 1} and @samp{var#foo + 1} both
11520 refer to the same variable.)
11522 In a command that takes a variable name, you can either type the full
11523 name of a variable, or type a single digit to use one of the special
11524 convenience variables @code{q0} through @code{q9}. For example,
11525 @kbd{3 s s 2} stores the number 3 in variable @code{q2}, and
11526 @w{@kbd{3 s s foo @key{RET}}} stores that number in variable
11529 To push a variable itself (as opposed to the variable's value) on the
11530 stack, enter its name as an algebraic expression using the apostrophe
11534 @pindex calc-evaluate
11535 @cindex Evaluation of variables in a formula
11536 @cindex Variables, evaluation
11537 @cindex Formulas, evaluation
11538 The @kbd{=} (@code{calc-evaluate}) key ``evaluates'' a formula by
11539 replacing all variables in the formula which have been given values by a
11540 @code{calc-store} or @code{calc-let} command by their stored values.
11541 Other variables are left alone. Thus a variable that has not been
11542 stored acts like an abstract variable in algebra; a variable that has
11543 been stored acts more like a register in a traditional calculator.
11544 With a positive numeric prefix argument, @kbd{=} evaluates the top
11545 @var{n} stack entries; with a negative argument, @kbd{=} evaluates
11546 the @var{n}th stack entry.
11548 @cindex @code{e} variable
11549 @cindex @code{pi} variable
11550 @cindex @code{i} variable
11551 @cindex @code{phi} variable
11552 @cindex @code{gamma} variable
11558 A few variables are called @dfn{special constants}. Their names are
11559 @samp{e}, @samp{pi}, @samp{i}, @samp{phi}, and @samp{gamma}.
11560 (@xref{Scientific Functions}.) When they are evaluated with @kbd{=},
11561 their values are calculated if necessary according to the current precision
11562 or complex polar mode. If you wish to use these symbols for other purposes,
11563 simply undefine or redefine them using @code{calc-store}.
11565 The variables @samp{inf}, @samp{uinf}, and @samp{nan} stand for
11566 infinite or indeterminate values. It's best not to use them as
11567 regular variables, since Calc uses special algebraic rules when
11568 it manipulates them. Calc displays a warning message if you store
11569 a value into any of these special variables.
11571 @xref{Store and Recall}, for a discussion of commands dealing with variables.
11573 @node Formulas, , Variables, Data Types
11578 @cindex Expressions
11579 @cindex Operators in formulas
11580 @cindex Precedence of operators
11581 When you press the apostrophe key you may enter any expression or formula
11582 in algebraic form. (Calc uses the terms ``expression'' and ``formula''
11583 interchangeably.) An expression is built up of numbers, variable names,
11584 and function calls, combined with various arithmetic operators.
11586 be used to indicate grouping. Spaces are ignored within formulas, except
11587 that spaces are not permitted within variable names or numbers.
11588 Arithmetic operators, in order from highest to lowest precedence, and
11589 with their equivalent function names, are:
11591 @samp{_} [@code{subscr}] (subscripts);
11593 postfix @samp{%} [@code{percent}] (as in @samp{25% = 0.25});
11595 prefix @samp{!} [@code{lnot}] (logical ``not,'' as in @samp{!x});
11597 @samp{+/-} [@code{sdev}] (the standard deviation symbol) and
11598 @samp{mod} [@code{makemod}] (the symbol for modulo forms);
11600 postfix @samp{!} [@code{fact}] (factorial, as in @samp{n!})
11601 and postfix @samp{!!} [@code{dfact}] (double factorial);
11603 @samp{^} [@code{pow}] (raised-to-the-power-of);
11605 prefix @samp{+} and @samp{-} [@code{neg}] (as in @samp{-x});
11607 @samp{*} [@code{mul}];
11609 @samp{/} [@code{div}], @samp{%} [@code{mod}] (modulo), and
11610 @samp{\} [@code{idiv}] (integer division);
11612 infix @samp{+} [@code{add}] and @samp{-} [@code{sub}] (as in @samp{x-y});
11614 @samp{|} [@code{vconcat}] (vector concatenation);
11616 relations @samp{=} [@code{eq}], @samp{!=} [@code{neq}], @samp{<} [@code{lt}],
11617 @samp{>} [@code{gt}], @samp{<=} [@code{leq}], and @samp{>=} [@code{geq}];
11619 @samp{&&} [@code{land}] (logical ``and'');
11621 @samp{||} [@code{lor}] (logical ``or'');
11623 the C-style ``if'' operator @samp{a?b:c} [@code{if}];
11625 @samp{!!!} [@code{pnot}] (rewrite pattern ``not'');
11627 @samp{&&&} [@code{pand}] (rewrite pattern ``and'');
11629 @samp{|||} [@code{por}] (rewrite pattern ``or'');
11631 @samp{:=} [@code{assign}] (for assignments and rewrite rules);
11633 @samp{::} [@code{condition}] (rewrite pattern condition);
11635 @samp{=>} [@code{evalto}].
11637 Note that, unlike in usual computer notation, multiplication binds more
11638 strongly than division: @samp{a*b/c*d} is equivalent to
11639 @texline @math{a b \over c d}.
11640 @infoline @expr{(a*b)/(c*d)}.
11642 @cindex Multiplication, implicit
11643 @cindex Implicit multiplication
11644 The multiplication sign @samp{*} may be omitted in many cases. In particular,
11645 if the righthand side is a number, variable name, or parenthesized
11646 expression, the @samp{*} may be omitted. Implicit multiplication has the
11647 same precedence as the explicit @samp{*} operator. The one exception to
11648 the rule is that a variable name followed by a parenthesized expression,
11650 is interpreted as a function call, not an implicit @samp{*}. In many
11651 cases you must use a space if you omit the @samp{*}: @samp{2a} is the
11652 same as @samp{2*a}, and @samp{a b} is the same as @samp{a*b}, but @samp{ab}
11653 is a variable called @code{ab}, @emph{not} the product of @samp{a} and
11654 @samp{b}! Also note that @samp{f (x)} is still a function call.
11656 @cindex Implicit comma in vectors
11657 The rules are slightly different for vectors written with square brackets.
11658 In vectors, the space character is interpreted (like the comma) as a
11659 separator of elements of the vector. Thus @w{@samp{[ 2a b+c d ]}} is
11660 equivalent to @samp{[2*a, b+c, d]}, whereas @samp{2a b+c d} is equivalent
11661 to @samp{2*a*b + c*d}.
11662 Note that spaces around the brackets, and around explicit commas, are
11663 ignored. To force spaces to be interpreted as multiplication you can
11664 enclose a formula in parentheses as in @samp{[(a b) 2(c d)]}, which is
11665 interpreted as @samp{[a*b, 2*c*d]}. An implicit comma is also inserted
11666 between @samp{][}, as in the matrix @samp{[[1 2][3 4]]}.
11668 Vectors that contain commas (not embedded within nested parentheses or
11669 brackets) do not treat spaces specially: @samp{[a b, 2 c d]} is a vector
11670 of two elements. Also, if it would be an error to treat spaces as
11671 separators, but not otherwise, then Calc will ignore spaces:
11672 @w{@samp{[a - b]}} is a vector of one element, but @w{@samp{[a -b]}} is
11673 a vector of two elements. Finally, vectors entered with curly braces
11674 instead of square brackets do not give spaces any special treatment.
11675 When Calc displays a vector that does not contain any commas, it will
11676 insert parentheses if necessary to make the meaning clear:
11677 @w{@samp{[(a b)]}}.
11679 The expression @samp{5%-2} is ambiguous; is this five-percent minus two,
11680 or five modulo minus-two? Calc always interprets the leftmost symbol as
11681 an infix operator preferentially (modulo, in this case), so you would
11682 need to write @samp{(5%)-2} to get the former interpretation.
11684 @cindex Function call notation
11685 A function call is, e.g., @samp{sin(1+x)}. (The Calc algebraic function
11686 @code{foo} corresponds to the Emacs Lisp function @code{calcFunc-foo},
11687 but unless you access the function from within Emacs Lisp, you don't
11688 need to worry about it.) Most mathematical Calculator commands like
11689 @code{calc-sin} have function equivalents like @code{sin}.
11690 If no Lisp function is defined for a function called by a formula, the
11691 call is left as it is during algebraic manipulation: @samp{f(x+y)} is
11692 left alone. Beware that many innocent-looking short names like @code{in}
11693 and @code{re} have predefined meanings which could surprise you; however,
11694 single letters or single letters followed by digits are always safe to
11695 use for your own function names. @xref{Function Index}.
11697 In the documentation for particular commands, the notation @kbd{H S}
11698 (@code{calc-sinh}) [@code{sinh}] means that the key sequence @kbd{H S}, the
11699 command @kbd{M-x calc-sinh}, and the algebraic function @code{sinh(x)} all
11700 represent the same operation.
11702 Commands that interpret (``parse'') text as algebraic formulas include
11703 algebraic entry (@kbd{'}), editing commands like @kbd{`} which parse
11704 the contents of the editing buffer when you finish, the @kbd{C-x * g}
11705 and @w{@kbd{C-x * r}} commands, the @kbd{C-y} command, the X window system
11706 ``paste'' mouse operation, and Embedded mode. All of these operations
11707 use the same rules for parsing formulas; in particular, language modes
11708 (@pxref{Language Modes}) affect them all in the same way.
11710 When you read a large amount of text into the Calculator (say a vector
11711 which represents a big set of rewrite rules; @pxref{Rewrite Rules}),
11712 you may wish to include comments in the text. Calc's formula parser
11713 ignores the symbol @samp{%%} and anything following it on a line:
11716 [ a + b, %% the sum of "a" and "b"
11718 %% last line is coming up:
11723 This is parsed exactly the same as @samp{[ a + b, c + d, e + f ]}.
11725 @xref{Syntax Tables}, for a way to create your own operators and other
11726 input notations. @xref{Compositions}, for a way to create new display
11729 @xref{Algebra}, for commands for manipulating formulas symbolically.
11731 @node Stack and Trail, Mode Settings, Data Types, Top
11732 @chapter Stack and Trail Commands
11735 This chapter describes the Calc commands for manipulating objects on the
11736 stack and in the trail buffer. (These commands operate on objects of any
11737 type, such as numbers, vectors, formulas, and incomplete objects.)
11740 * Stack Manipulation::
11741 * Editing Stack Entries::
11746 @node Stack Manipulation, Editing Stack Entries, Stack and Trail, Stack and Trail
11747 @section Stack Manipulation Commands
11753 @cindex Duplicating stack entries
11754 To duplicate the top object on the stack, press @key{RET} or @key{SPC}
11755 (two equivalent keys for the @code{calc-enter} command).
11756 Given a positive numeric prefix argument, these commands duplicate
11757 several elements at the top of the stack.
11758 Given a negative argument,
11759 these commands duplicate the specified element of the stack.
11760 Given an argument of zero, they duplicate the entire stack.
11761 For example, with @samp{10 20 30} on the stack,
11762 @key{RET} creates @samp{10 20 30 30},
11763 @kbd{C-u 2 @key{RET}} creates @samp{10 20 30 20 30},
11764 @kbd{C-u - 2 @key{RET}} creates @samp{10 20 30 20}, and
11765 @kbd{C-u 0 @key{RET}} creates @samp{10 20 30 10 20 30}.
11769 The @key{LFD} (@code{calc-over}) command (on a key marked Line-Feed if you
11770 have it, else on @kbd{C-j}) is like @code{calc-enter}
11771 except that the sign of the numeric prefix argument is interpreted
11772 oppositely. Also, with no prefix argument the default argument is 2.
11773 Thus with @samp{10 20 30} on the stack, @key{LFD} and @kbd{C-u 2 @key{LFD}}
11774 are both equivalent to @kbd{C-u - 2 @key{RET}}, producing
11775 @samp{10 20 30 20}.
11780 @cindex Removing stack entries
11781 @cindex Deleting stack entries
11782 To remove the top element from the stack, press @key{DEL} (@code{calc-pop}).
11783 The @kbd{C-d} key is a synonym for @key{DEL}.
11784 (If the top element is an incomplete object with at least one element, the
11785 last element is removed from it.) Given a positive numeric prefix argument,
11786 several elements are removed. Given a negative argument, the specified
11787 element of the stack is deleted. Given an argument of zero, the entire
11789 For example, with @samp{10 20 30} on the stack,
11790 @key{DEL} leaves @samp{10 20},
11791 @kbd{C-u 2 @key{DEL}} leaves @samp{10},
11792 @kbd{C-u - 2 @key{DEL}} leaves @samp{10 30}, and
11793 @kbd{C-u 0 @key{DEL}} leaves an empty stack.
11795 @kindex M-@key{DEL}
11796 @pindex calc-pop-above
11797 The @kbd{M-@key{DEL}} (@code{calc-pop-above}) command is to @key{DEL} what
11798 @key{LFD} is to @key{RET}: It interprets the sign of the numeric
11799 prefix argument in the opposite way, and the default argument is 2.
11800 Thus @kbd{M-@key{DEL}} by itself removes the second-from-top stack element,
11801 leaving the first, third, fourth, and so on; @kbd{M-3 M-@key{DEL}} deletes
11802 the third stack element.
11805 @pindex calc-roll-down
11806 To exchange the top two elements of the stack, press @key{TAB}
11807 (@code{calc-roll-down}). Given a positive numeric prefix argument, the
11808 specified number of elements at the top of the stack are rotated downward.
11809 Given a negative argument, the entire stack is rotated downward the specified
11810 number of times. Given an argument of zero, the entire stack is reversed
11812 For example, with @samp{10 20 30 40 50} on the stack,
11813 @key{TAB} creates @samp{10 20 30 50 40},
11814 @kbd{C-u 3 @key{TAB}} creates @samp{10 20 50 30 40},
11815 @kbd{C-u - 2 @key{TAB}} creates @samp{40 50 10 20 30}, and
11816 @kbd{C-u 0 @key{TAB}} creates @samp{50 40 30 20 10}.
11818 @kindex M-@key{TAB}
11819 @pindex calc-roll-up
11820 The command @kbd{M-@key{TAB}} (@code{calc-roll-up}) is analogous to @key{TAB}
11821 except that it rotates upward instead of downward. Also, the default
11822 with no prefix argument is to rotate the top 3 elements.
11823 For example, with @samp{10 20 30 40 50} on the stack,
11824 @kbd{M-@key{TAB}} creates @samp{10 20 40 50 30},
11825 @kbd{C-u 4 M-@key{TAB}} creates @samp{10 30 40 50 20},
11826 @kbd{C-u - 2 M-@key{TAB}} creates @samp{30 40 50 10 20}, and
11827 @kbd{C-u 0 M-@key{TAB}} creates @samp{50 40 30 20 10}.
11829 A good way to view the operation of @key{TAB} and @kbd{M-@key{TAB}} is in
11830 terms of moving a particular element to a new position in the stack.
11831 With a positive argument @var{n}, @key{TAB} moves the top stack
11832 element down to level @var{n}, making room for it by pulling all the
11833 intervening stack elements toward the top. @kbd{M-@key{TAB}} moves the
11834 element at level @var{n} up to the top. (Compare with @key{LFD},
11835 which copies instead of moving the element in level @var{n}.)
11837 With a negative argument @mathit{-@var{n}}, @key{TAB} rotates the stack
11838 to move the object in level @var{n} to the deepest place in the
11839 stack, and the object in level @mathit{@var{n}+1} to the top. @kbd{M-@key{TAB}}
11840 rotates the deepest stack element to be in level @var{n}, also
11841 putting the top stack element in level @mathit{@var{n}+1}.
11843 @xref{Selecting Subformulas}, for a way to apply these commands to
11844 any portion of a vector or formula on the stack.
11847 @pindex calc-transpose-lines
11848 @cindex Moving stack entries
11849 The command @kbd{C-x C-t} (@code{calc-transpose-lines}) will transpose
11850 the stack object determined by the point with the stack object at the
11851 next higher level. For example, with @samp{10 20 30 40 50} on the
11852 stack and the point on the line containing @samp{30}, @kbd{C-x C-t}
11853 creates @samp{10 20 40 30 50}. More generally, @kbd{C-x C-t} acts on
11854 the stack objects determined by the current point (and mark) similar
11855 to how the text-mode command @code{transpose-lines} acts on
11856 lines. With argument @var{n}, @kbd{C-x C-t} will move the stack object
11857 at the level above the current point and move it past N other objects;
11858 for example, with @samp{10 20 30 40 50} on the stack and the point on
11859 the line containing @samp{30}, @kbd{C-u 2 C-x C-t} creates
11860 @samp{10 40 20 30 50}. With an argument of 0, @kbd{C-x C-t} will switch
11861 the stack objects at the levels determined by the point and the mark.
11863 @node Editing Stack Entries, Trail Commands, Stack Manipulation, Stack and Trail
11864 @section Editing Stack Entries
11869 @pindex calc-edit-finish
11870 @cindex Editing the stack with Emacs
11871 The @kbd{`} (@code{calc-edit}) command creates a temporary buffer
11872 (@samp{*Calc Edit*}) for editing the top-of-stack value using regular
11873 Emacs commands. Note that @kbd{`} is a backquote, not a quote. With a
11874 numeric prefix argument, it edits the specified number of stack entries
11875 at once. (An argument of zero edits the entire stack; a negative
11876 argument edits one specific stack entry.)
11878 When you are done editing, press @kbd{C-c C-c} to finish and return
11879 to Calc. The @key{RET} and @key{LFD} keys also work to finish most
11880 sorts of editing, though in some cases Calc leaves @key{RET} with its
11881 usual meaning (``insert a newline'') if it's a situation where you
11882 might want to insert new lines into the editing buffer.
11884 When you finish editing, the Calculator parses the lines of text in
11885 the @samp{*Calc Edit*} buffer as numbers or formulas, replaces the
11886 original stack elements in the original buffer with these new values,
11887 then kills the @samp{*Calc Edit*} buffer. The original Calculator buffer
11888 continues to exist during editing, but for best results you should be
11889 careful not to change it until you have finished the edit. You can
11890 also cancel the edit by killing the buffer with @kbd{C-x k}.
11892 The formula is normally reevaluated as it is put onto the stack.
11893 For example, editing @samp{a + 2} to @samp{3 + 2} and pressing
11894 @kbd{C-c C-c} will push 5 on the stack. If you use @key{LFD} to
11895 finish, Calc will put the result on the stack without evaluating it.
11897 If you give a prefix argument to @kbd{C-c C-c},
11898 Calc will not kill the @samp{*Calc Edit*} buffer. You can switch
11899 back to that buffer and continue editing if you wish. However, you
11900 should understand that if you initiated the edit with @kbd{`}, the
11901 @kbd{C-c C-c} operation will be programmed to replace the top of the
11902 stack with the new edited value, and it will do this even if you have
11903 rearranged the stack in the meanwhile. This is not so much of a problem
11904 with other editing commands, though, such as @kbd{s e}
11905 (@code{calc-edit-variable}; @pxref{Operations on Variables}).
11907 If the @code{calc-edit} command involves more than one stack entry,
11908 each line of the @samp{*Calc Edit*} buffer is interpreted as a
11909 separate formula. Otherwise, the entire buffer is interpreted as
11910 one formula, with line breaks ignored. (You can use @kbd{C-o} or
11911 @kbd{C-q C-j} to insert a newline in the buffer without pressing @key{RET}.)
11913 The @kbd{`} key also works during numeric or algebraic entry. The
11914 text entered so far is moved to the @code{*Calc Edit*} buffer for
11915 more extensive editing than is convenient in the minibuffer.
11917 @node Trail Commands, Keep Arguments, Editing Stack Entries, Stack and Trail
11918 @section Trail Commands
11921 @cindex Trail buffer
11922 The commands for manipulating the Calc Trail buffer are two-key sequences
11923 beginning with the @kbd{t} prefix.
11926 @pindex calc-trail-display
11927 The @kbd{t d} (@code{calc-trail-display}) command turns display of the
11928 trail on and off. Normally the trail display is toggled on if it was off,
11929 off if it was on. With a numeric prefix of zero, this command always
11930 turns the trail off; with a prefix of one, it always turns the trail on.
11931 The other trail-manipulation commands described here automatically turn
11932 the trail on. Note that when the trail is off values are still recorded
11933 there; they are simply not displayed. To set Emacs to turn the trail
11934 off by default, type @kbd{t d} and then save the mode settings with
11935 @kbd{m m} (@code{calc-save-modes}).
11938 @pindex calc-trail-in
11940 @pindex calc-trail-out
11941 The @kbd{t i} (@code{calc-trail-in}) and @kbd{t o}
11942 (@code{calc-trail-out}) commands switch the cursor into and out of the
11943 Calc Trail window. In practice they are rarely used, since the commands
11944 shown below are a more convenient way to move around in the
11945 trail, and they work ``by remote control'' when the cursor is still
11946 in the Calculator window.
11948 @cindex Trail pointer
11949 There is a @dfn{trail pointer} which selects some entry of the trail at
11950 any given time. The trail pointer looks like a @samp{>} symbol right
11951 before the selected number. The following commands operate on the
11952 trail pointer in various ways.
11955 @pindex calc-trail-yank
11956 @cindex Retrieving previous results
11957 The @kbd{t y} (@code{calc-trail-yank}) command reads the selected value in
11958 the trail and pushes it onto the Calculator stack. It allows you to
11959 re-use any previously computed value without retyping. With a numeric
11960 prefix argument @var{n}, it yanks the value @var{n} lines above the current
11964 @pindex calc-trail-scroll-left
11966 @pindex calc-trail-scroll-right
11967 The @kbd{t <} (@code{calc-trail-scroll-left}) and @kbd{t >}
11968 (@code{calc-trail-scroll-right}) commands horizontally scroll the trail
11969 window left or right by one half of its width.
11972 @pindex calc-trail-next
11974 @pindex calc-trail-previous
11976 @pindex calc-trail-forward
11978 @pindex calc-trail-backward
11979 The @kbd{t n} (@code{calc-trail-next}) and @kbd{t p}
11980 (@code{calc-trail-previous)} commands move the trail pointer down or up
11981 one line. The @kbd{t f} (@code{calc-trail-forward}) and @kbd{t b}
11982 (@code{calc-trail-backward}) commands move the trail pointer down or up
11983 one screenful at a time. All of these commands accept numeric prefix
11984 arguments to move several lines or screenfuls at a time.
11987 @pindex calc-trail-first
11989 @pindex calc-trail-last
11991 @pindex calc-trail-here
11992 The @kbd{t [} (@code{calc-trail-first}) and @kbd{t ]}
11993 (@code{calc-trail-last}) commands move the trail pointer to the first or
11994 last line of the trail. The @kbd{t h} (@code{calc-trail-here}) command
11995 moves the trail pointer to the cursor position; unlike the other trail
11996 commands, @kbd{t h} works only when Calc Trail is the selected window.
11999 @pindex calc-trail-isearch-forward
12001 @pindex calc-trail-isearch-backward
12003 The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
12004 (@code{calc-trail-isearch-backward}) commands perform an incremental
12005 search forward or backward through the trail. You can press @key{RET}
12006 to terminate the search; the trail pointer moves to the current line.
12007 If you cancel the search with @kbd{C-g}, the trail pointer stays where
12008 it was when the search began.
12011 The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
12012 (@code{calc-trail-isearch-backward}) com\-mands perform an incremental
12013 search forward or backward through the trail. You can press @key{RET}
12014 to terminate the search; the trail pointer moves to the current line.
12015 If you cancel the search with @kbd{C-g}, the trail pointer stays where
12016 it was when the search began.
12020 @pindex calc-trail-marker
12021 The @kbd{t m} (@code{calc-trail-marker}) command allows you to enter a
12022 line of text of your own choosing into the trail. The text is inserted
12023 after the line containing the trail pointer; this usually means it is
12024 added to the end of the trail. Trail markers are useful mainly as the
12025 targets for later incremental searches in the trail.
12028 @pindex calc-trail-kill
12029 The @kbd{t k} (@code{calc-trail-kill}) command removes the selected line
12030 from the trail. The line is saved in the Emacs kill ring suitable for
12031 yanking into another buffer, but it is not easy to yank the text back
12032 into the trail buffer. With a numeric prefix argument, this command
12033 kills the @var{n} lines below or above the selected one.
12035 The @kbd{t .} (@code{calc-full-trail-vectors}) command is described
12036 elsewhere; @pxref{Vector and Matrix Formats}.
12038 @node Keep Arguments, , Trail Commands, Stack and Trail
12039 @section Keep Arguments
12043 @pindex calc-keep-args
12044 The @kbd{K} (@code{calc-keep-args}) command acts like a prefix for
12045 the following command. It prevents that command from removing its
12046 arguments from the stack. For example, after @kbd{2 @key{RET} 3 +},
12047 the stack contains the sole number 5, but after @kbd{2 @key{RET} 3 K +},
12048 the stack contains the arguments and the result: @samp{2 3 5}.
12050 With the exception of keyboard macros, this works for all commands that
12051 take arguments off the stack. (To avoid potentially unpleasant behavior,
12052 a @kbd{K} prefix before a keyboard macro will be ignored. A @kbd{K}
12053 prefix called @emph{within} the keyboard macro will still take effect.)
12054 As another example, @kbd{K a s} simplifies a formula, pushing the
12055 simplified version of the formula onto the stack after the original
12056 formula (rather than replacing the original formula). Note that you
12057 could get the same effect by typing @kbd{@key{RET} a s}, copying the
12058 formula and then simplifying the copy. One difference is that for a very
12059 large formula the time taken to format the intermediate copy in
12060 @kbd{@key{RET} a s} could be noticeable; @kbd{K a s} would avoid this
12063 Even stack manipulation commands are affected. @key{TAB} works by
12064 popping two values and pushing them back in the opposite order,
12065 so @kbd{2 @key{RET} 3 K @key{TAB}} produces @samp{2 3 3 2}.
12067 A few Calc commands provide other ways of doing the same thing.
12068 For example, @kbd{' sin($)} replaces the number on the stack with
12069 its sine using algebraic entry; to push the sine and keep the
12070 original argument you could use either @kbd{' sin($1)} or
12071 @kbd{K ' sin($)}. @xref{Algebraic Entry}. Also, the @kbd{s s}
12072 command is effectively the same as @kbd{K s t}. @xref{Storing Variables}.
12074 If you execute a command and then decide you really wanted to keep
12075 the argument, you can press @kbd{M-@key{RET}} (@code{calc-last-args}).
12076 This command pushes the last arguments that were popped by any command
12077 onto the stack. Note that the order of things on the stack will be
12078 different than with @kbd{K}: @kbd{2 @key{RET} 3 + M-@key{RET}} leaves
12079 @samp{5 2 3} on the stack instead of @samp{2 3 5}. @xref{Undo}.
12081 @node Mode Settings, Arithmetic, Stack and Trail, Top
12082 @chapter Mode Settings
12085 This chapter describes commands that set modes in the Calculator.
12086 They do not affect the contents of the stack, although they may change
12087 the @emph{appearance} or @emph{interpretation} of the stack's contents.
12090 * General Mode Commands::
12092 * Inverse and Hyperbolic::
12093 * Calculation Modes::
12094 * Simplification Modes::
12102 @node General Mode Commands, Precision, Mode Settings, Mode Settings
12103 @section General Mode Commands
12107 @pindex calc-save-modes
12108 @cindex Continuous memory
12109 @cindex Saving mode settings
12110 @cindex Permanent mode settings
12111 @cindex Calc init file, mode settings
12112 You can save all of the current mode settings in your Calc init file
12113 (the file given by the variable @code{calc-settings-file}, typically
12114 @file{~/.emacs.d/calc.el}) with the @kbd{m m} (@code{calc-save-modes})
12115 command. This will cause Emacs to reestablish these modes each time
12116 it starts up. The modes saved in the file include everything
12117 controlled by the @kbd{m} and @kbd{d} prefix keys, the current
12118 precision and binary word size, whether or not the trail is displayed,
12119 the current height of the Calc window, and more. The current
12120 interface (used when you type @kbd{C-x * *}) is also saved. If there
12121 were already saved mode settings in the file, they are replaced.
12122 Otherwise, the new mode information is appended to the end of the
12126 @pindex calc-mode-record-mode
12127 The @kbd{m R} (@code{calc-mode-record-mode}) command tells Calc to
12128 record all the mode settings (as if by pressing @kbd{m m}) every
12129 time a mode setting changes. If the modes are saved this way, then this
12130 ``automatic mode recording'' mode is also saved.
12131 Type @kbd{m R} again to disable this method of recording the mode
12132 settings. To turn it off permanently, the @kbd{m m} command will also be
12133 necessary. (If Embedded mode is enabled, other options for recording
12134 the modes are available; @pxref{Mode Settings in Embedded Mode}.)
12137 @pindex calc-settings-file-name
12138 The @kbd{m F} (@code{calc-settings-file-name}) command allows you to
12139 choose a different file than the current value of @code{calc-settings-file}
12140 for @kbd{m m}, @kbd{Z P}, and similar commands to save permanent information.
12141 You are prompted for a file name. All Calc modes are then reset to
12142 their default values, then settings from the file you named are loaded
12143 if this file exists, and this file becomes the one that Calc will
12144 use in the future for commands like @kbd{m m}. The default settings
12145 file name is @file{~/.emacs.d/calc.el}. You can see the current file name by
12146 giving a blank response to the @kbd{m F} prompt. See also the
12147 discussion of the @code{calc-settings-file} variable; @pxref{Customizing Calc}.
12149 If the file name you give is your user init file (typically
12150 @file{~/.emacs}), @kbd{m F} will not automatically load the new file. This
12151 is because your user init file may contain other things you don't want
12152 to reread. You can give
12153 a numeric prefix argument of 1 to @kbd{m F} to force it to read the
12154 file no matter what. Conversely, an argument of @mathit{-1} tells
12155 @kbd{m F} @emph{not} to read the new file. An argument of 2 or @mathit{-2}
12156 tells @kbd{m F} not to reset the modes to their defaults beforehand,
12157 which is useful if you intend your new file to have a variant of the
12158 modes present in the file you were using before.
12161 @pindex calc-always-load-extensions
12162 The @kbd{m x} (@code{calc-always-load-extensions}) command enables a mode
12163 in which the first use of Calc loads the entire program, including all
12164 extensions modules. Otherwise, the extensions modules will not be loaded
12165 until the various advanced Calc features are used. Since this mode only
12166 has effect when Calc is first loaded, @kbd{m x} is usually followed by
12167 @kbd{m m} to make the mode-setting permanent. To load all of Calc just
12168 once, rather than always in the future, you can press @kbd{C-x * L}.
12171 @pindex calc-shift-prefix
12172 The @kbd{m S} (@code{calc-shift-prefix}) command enables a mode in which
12173 all of Calc's letter prefix keys may be typed shifted as well as unshifted.
12174 If you are typing, say, @kbd{a S} (@code{calc-solve-for}) quite often
12175 you might find it easier to turn this mode on so that you can type
12176 @kbd{A S} instead. When this mode is enabled, the commands that used to
12177 be on those single shifted letters (e.g., @kbd{A} (@code{calc-abs})) can
12178 now be invoked by pressing the shifted letter twice: @kbd{A A}. Note
12179 that the @kbd{v} prefix key always works both shifted and unshifted, and
12180 the @kbd{z} and @kbd{Z} prefix keys are always distinct. Also, the @kbd{h}
12181 prefix is not affected by this mode. Press @kbd{m S} again to disable
12182 shifted-prefix mode.
12184 @node Precision, Inverse and Hyperbolic, General Mode Commands, Mode Settings
12189 @pindex calc-precision
12190 @cindex Precision of calculations
12191 The @kbd{p} (@code{calc-precision}) command controls the precision to
12192 which floating-point calculations are carried. The precision must be
12193 at least 3 digits and may be arbitrarily high, within the limits of
12194 memory and time. This affects only floats: Integer and rational
12195 calculations are always carried out with as many digits as necessary.
12197 The @kbd{p} key prompts for the current precision. If you wish you
12198 can instead give the precision as a numeric prefix argument.
12200 Many internal calculations are carried to one or two digits higher
12201 precision than normal. Results are rounded down afterward to the
12202 current precision. Unless a special display mode has been selected,
12203 floats are always displayed with their full stored precision, i.e.,
12204 what you see is what you get. Reducing the current precision does not
12205 round values already on the stack, but those values will be rounded
12206 down before being used in any calculation. The @kbd{c 0} through
12207 @kbd{c 9} commands (@pxref{Conversions}) can be used to round an
12208 existing value to a new precision.
12210 @cindex Accuracy of calculations
12211 It is important to distinguish the concepts of @dfn{precision} and
12212 @dfn{accuracy}. In the normal usage of these words, the number
12213 123.4567 has a precision of 7 digits but an accuracy of 4 digits.
12214 The precision is the total number of digits not counting leading
12215 or trailing zeros (regardless of the position of the decimal point).
12216 The accuracy is simply the number of digits after the decimal point
12217 (again not counting trailing zeros). In Calc you control the precision,
12218 not the accuracy of computations. If you were to set the accuracy
12219 instead, then calculations like @samp{exp(100)} would generate many
12220 more digits than you would typically need, while @samp{exp(-100)} would
12221 probably round to zero! In Calc, both these computations give you
12222 exactly 12 (or the requested number of) significant digits.
12224 The only Calc features that deal with accuracy instead of precision
12225 are fixed-point display mode for floats (@kbd{d f}; @pxref{Float Formats}),
12226 and the rounding functions like @code{floor} and @code{round}
12227 (@pxref{Integer Truncation}). Also, @kbd{c 0} through @kbd{c 9}
12228 deal with both precision and accuracy depending on the magnitudes
12229 of the numbers involved.
12231 If you need to work with a particular fixed accuracy (say, dollars and
12232 cents with two digits after the decimal point), one solution is to work
12233 with integers and an ``implied'' decimal point. For example, $8.99
12234 divided by 6 would be entered @kbd{899 @key{RET} 6 /}, yielding 149.833
12235 (actually $1.49833 with our implied decimal point); pressing @kbd{R}
12236 would round this to 150 cents, i.e., $1.50.
12238 @xref{Floats}, for still more on floating-point precision and related
12241 @node Inverse and Hyperbolic, Calculation Modes, Precision, Mode Settings
12242 @section Inverse and Hyperbolic Flags
12246 @pindex calc-inverse
12247 There is no single-key equivalent to the @code{calc-arcsin} function.
12248 Instead, you must first press @kbd{I} (@code{calc-inverse}) to set
12249 the @dfn{Inverse Flag}, then press @kbd{S} (@code{calc-sin}).
12250 The @kbd{I} key actually toggles the Inverse Flag. When this flag
12251 is set, the word @samp{Inv} appears in the mode line.
12254 @pindex calc-hyperbolic
12255 Likewise, the @kbd{H} key (@code{calc-hyperbolic}) sets or clears the
12256 Hyperbolic Flag, which transforms @code{calc-sin} into @code{calc-sinh}.
12257 If both of these flags are set at once, the effect will be
12258 @code{calc-arcsinh}. (The Hyperbolic flag is also used by some
12259 non-trigonometric commands; for example @kbd{H L} computes a base-10,
12260 instead of base-@mathit{e}, logarithm.)
12262 Command names like @code{calc-arcsin} are provided for completeness, and
12263 may be executed with @kbd{x} or @kbd{M-x}. Their effect is simply to
12264 toggle the Inverse and/or Hyperbolic flags and then execute the
12265 corresponding base command (@code{calc-sin} in this case).
12268 @pindex calc-option
12269 The @kbd{O} key (@code{calc-option}) sets another flag, the
12270 @dfn{Option Flag}, which also can alter the subsequent Calc command in
12273 The Inverse, Hyperbolic and Option flags apply only to the next
12274 Calculator command, after which they are automatically cleared. (They
12275 are also cleared if the next keystroke is not a Calc command.) Digits
12276 you type after @kbd{I}, @kbd{H} or @kbd{O} (or @kbd{K}) are treated as
12277 prefix arguments for the next command, not as numeric entries. The
12278 same is true of @kbd{C-u}, but not of the minus sign (@kbd{K -} means
12279 to subtract and keep arguments).
12281 Another Calc prefix flag, @kbd{K} (keep-arguments), is discussed
12282 elsewhere. @xref{Keep Arguments}.
12284 @node Calculation Modes, Simplification Modes, Inverse and Hyperbolic, Mode Settings
12285 @section Calculation Modes
12288 The commands in this section are two-key sequences beginning with
12289 the @kbd{m} prefix. (That's the letter @kbd{m}, not the @key{META} key.)
12290 The @samp{m a} (@code{calc-algebraic-mode}) command is described elsewhere
12291 (@pxref{Algebraic Entry}).
12300 * Automatic Recomputation::
12301 * Working Message::
12304 @node Angular Modes, Polar Mode, Calculation Modes, Calculation Modes
12305 @subsection Angular Modes
12308 @cindex Angular mode
12309 The Calculator supports three notations for angles: radians, degrees,
12310 and degrees-minutes-seconds. When a number is presented to a function
12311 like @code{sin} that requires an angle, the current angular mode is
12312 used to interpret the number as either radians or degrees. If an HMS
12313 form is presented to @code{sin}, it is always interpreted as
12314 degrees-minutes-seconds.
12316 Functions that compute angles produce a number in radians, a number in
12317 degrees, or an HMS form depending on the current angular mode. If the
12318 result is a complex number and the current mode is HMS, the number is
12319 instead expressed in degrees. (Complex-number calculations would
12320 normally be done in Radians mode, though. Complex numbers are converted
12321 to degrees by calculating the complex result in radians and then
12322 multiplying by 180 over @cpi{}.)
12325 @pindex calc-radians-mode
12327 @pindex calc-degrees-mode
12329 @pindex calc-hms-mode
12330 The @kbd{m r} (@code{calc-radians-mode}), @kbd{m d} (@code{calc-degrees-mode}),
12331 and @kbd{m h} (@code{calc-hms-mode}) commands control the angular mode.
12332 The current angular mode is displayed on the Emacs mode line.
12333 The default angular mode is Degrees.
12335 @node Polar Mode, Fraction Mode, Angular Modes, Calculation Modes
12336 @subsection Polar Mode
12340 The Calculator normally ``prefers'' rectangular complex numbers in the
12341 sense that rectangular form is used when the proper form can not be
12342 decided from the input. This might happen by multiplying a rectangular
12343 number by a polar one, by taking the square root of a negative real
12344 number, or by entering @kbd{( 2 @key{SPC} 3 )}.
12347 @pindex calc-polar-mode
12348 The @kbd{m p} (@code{calc-polar-mode}) command toggles complex-number
12349 preference between rectangular and polar forms. In Polar mode, all
12350 of the above example situations would produce polar complex numbers.
12352 @node Fraction Mode, Infinite Mode, Polar Mode, Calculation Modes
12353 @subsection Fraction Mode
12356 @cindex Fraction mode
12357 @cindex Division of integers
12358 Division of two integers normally yields a floating-point number if the
12359 result cannot be expressed as an integer. In some cases you would
12360 rather get an exact fractional answer. One way to accomplish this is
12361 to use the @kbd{:} (@code{calc-fdiv}) [@code{fdiv}] command, which
12362 divides the two integers on the top of the stack to produce a fraction:
12363 @kbd{6 @key{RET} 4 :} produces @expr{3:2} even though
12364 @kbd{6 @key{RET} 4 /} produces @expr{1.5}.
12367 @pindex calc-frac-mode
12368 To set the Calculator to produce fractional results for normal integer
12369 divisions, use the @kbd{m f} (@code{calc-frac-mode}) command.
12370 For example, @expr{8/4} produces @expr{2} in either mode,
12371 but @expr{6/4} produces @expr{3:2} in Fraction mode, @expr{1.5} in
12374 At any time you can use @kbd{c f} (@code{calc-float}) to convert a
12375 fraction to a float, or @kbd{c F} (@code{calc-fraction}) to convert a
12376 float to a fraction. @xref{Conversions}.
12378 @node Infinite Mode, Symbolic Mode, Fraction Mode, Calculation Modes
12379 @subsection Infinite Mode
12382 @cindex Infinite mode
12383 The Calculator normally treats results like @expr{1 / 0} as errors;
12384 formulas like this are left in unsimplified form. But Calc can be
12385 put into a mode where such calculations instead produce ``infinite''
12389 @pindex calc-infinite-mode
12390 The @kbd{m i} (@code{calc-infinite-mode}) command turns this mode
12391 on and off. When the mode is off, infinities do not arise except
12392 in calculations that already had infinities as inputs. (One exception
12393 is that infinite open intervals like @samp{[0 .. inf)} can be
12394 generated; however, intervals closed at infinity (@samp{[0 .. inf]})
12395 will not be generated when Infinite mode is off.)
12397 With Infinite mode turned on, @samp{1 / 0} will generate @code{uinf},
12398 an undirected infinity. @xref{Infinities}, for a discussion of the
12399 difference between @code{inf} and @code{uinf}. Also, @expr{0 / 0}
12400 evaluates to @code{nan}, the ``indeterminate'' symbol. Various other
12401 functions can also return infinities in this mode; for example,
12402 @samp{ln(0) = -inf}, and @samp{gamma(-7) = uinf}. Once again,
12403 note that @samp{exp(inf) = inf} regardless of Infinite mode because
12404 this calculation has infinity as an input.
12406 @cindex Positive Infinite mode
12407 The @kbd{m i} command with a numeric prefix argument of zero,
12408 i.e., @kbd{C-u 0 m i}, turns on a Positive Infinite mode in
12409 which zero is treated as positive instead of being directionless.
12410 Thus, @samp{1 / 0 = inf} and @samp{-1 / 0 = -inf} in this mode.
12411 Note that zero never actually has a sign in Calc; there are no
12412 separate representations for @mathit{+0} and @mathit{-0}. Positive
12413 Infinite mode merely changes the interpretation given to the
12414 single symbol, @samp{0}. One consequence of this is that, while
12415 you might expect @samp{1 / -0 = -inf}, actually @samp{1 / -0}
12416 is equivalent to @samp{1 / 0}, which is equal to positive @code{inf}.
12418 @node Symbolic Mode, Matrix Mode, Infinite Mode, Calculation Modes
12419 @subsection Symbolic Mode
12422 @cindex Symbolic mode
12423 @cindex Inexact results
12424 Calculations are normally performed numerically wherever possible.
12425 For example, the @code{calc-sqrt} command, or @code{sqrt} function in an
12426 algebraic expression, produces a numeric answer if the argument is a
12427 number or a symbolic expression if the argument is an expression:
12428 @kbd{2 Q} pushes 1.4142 but @kbd{@key{'} x+1 @key{RET} Q} pushes @samp{sqrt(x+1)}.
12431 @pindex calc-symbolic-mode
12432 In @dfn{Symbolic mode}, controlled by the @kbd{m s} (@code{calc-symbolic-mode})
12433 command, functions which would produce inexact, irrational results are
12434 left in symbolic form. Thus @kbd{16 Q} pushes 4, but @kbd{2 Q} pushes
12438 @pindex calc-eval-num
12439 The shift-@kbd{N} (@code{calc-eval-num}) command evaluates numerically
12440 the expression at the top of the stack, by temporarily disabling
12441 @code{calc-symbolic-mode} and executing @kbd{=} (@code{calc-evaluate}).
12442 Given a numeric prefix argument, it also
12443 sets the floating-point precision to the specified value for the duration
12446 To evaluate a formula numerically without expanding the variables it
12447 contains, you can use the key sequence @kbd{m s a v m s} (this uses
12448 @code{calc-alg-evaluate}, which resimplifies but doesn't evaluate
12451 @node Matrix Mode, Automatic Recomputation, Symbolic Mode, Calculation Modes
12452 @subsection Matrix and Scalar Modes
12455 @cindex Matrix mode
12456 @cindex Scalar mode
12457 Calc sometimes makes assumptions during algebraic manipulation that
12458 are awkward or incorrect when vectors and matrices are involved.
12459 Calc has two modes, @dfn{Matrix mode} and @dfn{Scalar mode}, which
12460 modify its behavior around vectors in useful ways.
12463 @pindex calc-matrix-mode
12464 Press @kbd{m v} (@code{calc-matrix-mode}) once to enter Matrix mode.
12465 In this mode, all objects are assumed to be matrices unless provably
12466 otherwise. One major effect is that Calc will no longer consider
12467 multiplication to be commutative. (Recall that in matrix arithmetic,
12468 @samp{A*B} is not the same as @samp{B*A}.) This assumption affects
12469 rewrite rules and algebraic simplification. Another effect of this
12470 mode is that calculations that would normally produce constants like
12471 0 and 1 (e.g., @expr{a - a} and @expr{a / a}, respectively) will now
12472 produce function calls that represent ``generic'' zero or identity
12473 matrices: @samp{idn(0)}, @samp{idn(1)}. The @code{idn} function
12474 @samp{idn(@var{a},@var{n})} returns @var{a} times an @var{n}x@var{n}
12475 identity matrix; if @var{n} is omitted, it doesn't know what
12476 dimension to use and so the @code{idn} call remains in symbolic
12477 form. However, if this generic identity matrix is later combined
12478 with a matrix whose size is known, it will be converted into
12479 a true identity matrix of the appropriate size. On the other hand,
12480 if it is combined with a scalar (as in @samp{idn(1) + 2}), Calc
12481 will assume it really was a scalar after all and produce, e.g., 3.
12483 Press @kbd{m v} a second time to get Scalar mode. Here, objects are
12484 assumed @emph{not} to be vectors or matrices unless provably so.
12485 For example, normally adding a variable to a vector, as in
12486 @samp{[x, y, z] + a}, will leave the sum in symbolic form because
12487 as far as Calc knows, @samp{a} could represent either a number or
12488 another 3-vector. In Scalar mode, @samp{a} is assumed to be a
12489 non-vector, and the addition is evaluated to @samp{[x+a, y+a, z+a]}.
12491 Press @kbd{m v} a third time to return to the normal mode of operation.
12493 If you press @kbd{m v} with a numeric prefix argument @var{n}, you
12494 get a special ``dimensioned'' Matrix mode in which matrices of
12495 unknown size are assumed to be @var{n}x@var{n} square matrices.
12496 Then, the function call @samp{idn(1)} will expand into an actual
12497 matrix rather than representing a ``generic'' matrix. Simply typing
12498 @kbd{C-u m v} will get you a square Matrix mode, in which matrices of
12499 unknown size are assumed to be square matrices of unspecified size.
12501 @cindex Declaring scalar variables
12502 Of course these modes are approximations to the true state of
12503 affairs, which is probably that some quantities will be matrices
12504 and others will be scalars. One solution is to ``declare''
12505 certain variables or functions to be scalar-valued.
12506 @xref{Declarations}, to see how to make declarations in Calc.
12508 There is nothing stopping you from declaring a variable to be
12509 scalar and then storing a matrix in it; however, if you do, the
12510 results you get from Calc may not be valid. Suppose you let Calc
12511 get the result @samp{[x+a, y+a, z+a]} shown above, and then stored
12512 @samp{[1, 2, 3]} in @samp{a}. The result would not be the same as
12513 for @samp{[x, y, z] + [1, 2, 3]}, but that's because you have broken
12514 your earlier promise to Calc that @samp{a} would be scalar.
12516 Another way to mix scalars and matrices is to use selections
12517 (@pxref{Selecting Subformulas}). Use Matrix mode when operating on
12518 your formula normally; then, to apply Scalar mode to a certain part
12519 of the formula without affecting the rest just select that part,
12520 change into Scalar mode and press @kbd{=} to resimplify the part
12521 under this mode, then change back to Matrix mode before deselecting.
12523 @node Automatic Recomputation, Working Message, Matrix Mode, Calculation Modes
12524 @subsection Automatic Recomputation
12527 The @dfn{evaluates-to} operator, @samp{=>}, has the special
12528 property that any @samp{=>} formulas on the stack are recomputed
12529 whenever variable values or mode settings that might affect them
12530 are changed. @xref{Evaluates-To Operator}.
12533 @pindex calc-auto-recompute
12534 The @kbd{m C} (@code{calc-auto-recompute}) command turns this
12535 automatic recomputation on and off. If you turn it off, Calc will
12536 not update @samp{=>} operators on the stack (nor those in the
12537 attached Embedded mode buffer, if there is one). They will not
12538 be updated unless you explicitly do so by pressing @kbd{=} or until
12539 you press @kbd{m C} to turn recomputation back on. (While automatic
12540 recomputation is off, you can think of @kbd{m C m C} as a command
12541 to update all @samp{=>} operators while leaving recomputation off.)
12543 To update @samp{=>} operators in an Embedded buffer while
12544 automatic recomputation is off, use @w{@kbd{C-x * u}}.
12545 @xref{Embedded Mode}.
12547 @node Working Message, , Automatic Recomputation, Calculation Modes
12548 @subsection Working Messages
12551 @cindex Performance
12552 @cindex Working messages
12553 Since the Calculator is written entirely in Emacs Lisp, which is not
12554 designed for heavy numerical work, many operations are quite slow.
12555 The Calculator normally displays the message @samp{Working...} in the
12556 echo area during any command that may be slow. In addition, iterative
12557 operations such as square roots and trigonometric functions display the
12558 intermediate result at each step. Both of these types of messages can
12559 be disabled if you find them distracting.
12562 @pindex calc-working
12563 Type @kbd{m w} (@code{calc-working}) with a numeric prefix of 0 to
12564 disable all ``working'' messages. Use a numeric prefix of 1 to enable
12565 only the plain @samp{Working...} message. Use a numeric prefix of 2 to
12566 see intermediate results as well. With no numeric prefix this displays
12569 While it may seem that the ``working'' messages will slow Calc down
12570 considerably, experiments have shown that their impact is actually
12571 quite small. But if your terminal is slow you may find that it helps
12572 to turn the messages off.
12574 @node Simplification Modes, Declarations, Calculation Modes, Mode Settings
12575 @section Simplification Modes
12578 The current @dfn{simplification mode} controls how numbers and formulas
12579 are ``normalized'' when being taken from or pushed onto the stack.
12580 Some normalizations are unavoidable, such as rounding floating-point
12581 results to the current precision, and reducing fractions to simplest
12582 form. Others, such as simplifying a formula like @expr{a+a} (or @expr{2+3}),
12583 are done automatically but can be turned off when necessary.
12585 When you press a key like @kbd{+} when @expr{2} and @expr{3} are on the
12586 stack, Calc pops these numbers, normalizes them, creates the formula
12587 @expr{2+3}, normalizes it, and pushes the result. Of course the standard
12588 rules for normalizing @expr{2+3} will produce the result @expr{5}.
12590 Simplification mode commands consist of the lower-case @kbd{m} prefix key
12591 followed by a shifted letter.
12594 @pindex calc-no-simplify-mode
12595 The @kbd{m O} (@code{calc-no-simplify-mode}) command turns off all optional
12596 simplifications. These would leave a formula like @expr{2+3} alone. In
12597 fact, nothing except simple numbers are ever affected by normalization
12598 in this mode. Explicit simplification commands, such as @kbd{=} or
12599 @kbd{a s}, can still be given to simplify any formulas.
12600 @xref{Algebraic Definitions}, for a sample use of
12601 No-Simplification mode.
12605 @pindex calc-num-simplify-mode
12606 The @kbd{m N} (@code{calc-num-simplify-mode}) command turns off simplification
12607 of any formulas except those for which all arguments are constants. For
12608 example, @expr{1+2} is simplified to @expr{3}, and @expr{a+(2-2)} is
12609 simplified to @expr{a+0} but no further, since one argument of the sum
12610 is not a constant. Unfortunately, @expr{(a+2)-2} is @emph{not} simplified
12611 because the top-level @samp{-} operator's arguments are not both
12612 constant numbers (one of them is the formula @expr{a+2}).
12613 A constant is a number or other numeric object (such as a constant
12614 error form or modulo form), or a vector all of whose
12615 elements are constant.
12618 @pindex calc-basic-simplify-mode
12619 The @kbd{m I} (@code{calc-basic-simplify-mode}) command does some basic
12620 simplifications for all formulas. This includes many easy and
12621 fast algebraic simplifications such as @expr{a+0} to @expr{a}, and
12622 @expr{a + 2 a} to @expr{3 a}, as well as evaluating functions like
12623 @expr{@tfn{deriv}(x^2, x)} to @expr{2 x}.
12626 @pindex calc-bin-simplify-mode
12627 The @kbd{m B} (@code{calc-bin-simplify-mode}) mode applies the basic
12628 simplifications to a result and then, if the result is an integer,
12629 uses the @kbd{b c} (@code{calc-clip}) command to clip the integer according
12630 to the current binary word size. @xref{Binary Functions}. Real numbers
12631 are rounded to the nearest integer and then clipped; other kinds of
12632 results (after the basic simplifications) are left alone.
12635 @pindex calc-alg-simplify-mode
12636 The @kbd{m A} (@code{calc-alg-simplify-mode}) mode does standard
12637 algebraic simplifications. @xref{Algebraic Simplifications}.
12640 @pindex calc-ext-simplify-mode
12641 The @kbd{m E} (@code{calc-ext-simplify-mode}) mode does ``extended'', or
12642 ``unsafe'', algebraic simplification. @xref{Unsafe Simplifications}.
12645 @pindex calc-units-simplify-mode
12646 The @kbd{m U} (@code{calc-units-simplify-mode}) mode does units
12647 simplification. @xref{Simplification of Units}. These include the
12648 algebraic simplifications, plus variable names which
12649 are identifiable as unit names (like @samp{mm} for ``millimeters'')
12650 are simplified with their unit definitions in mind.
12652 A common technique is to set the simplification mode down to the lowest
12653 amount of simplification you will allow to be applied automatically, then
12654 use manual commands like @kbd{a s} and @kbd{c c} (@code{calc-clean}) to
12655 perform higher types of simplifications on demand.
12656 @node Declarations, Display Modes, Simplification Modes, Mode Settings
12657 @section Declarations
12660 A @dfn{declaration} is a statement you make that promises you will
12661 use a certain variable or function in a restricted way. This may
12662 give Calc the freedom to do things that it couldn't do if it had to
12663 take the fully general situation into account.
12666 * Declaration Basics::
12667 * Kinds of Declarations::
12668 * Functions for Declarations::
12671 @node Declaration Basics, Kinds of Declarations, Declarations, Declarations
12672 @subsection Declaration Basics
12676 @pindex calc-declare-variable
12677 The @kbd{s d} (@code{calc-declare-variable}) command is the easiest
12678 way to make a declaration for a variable. This command prompts for
12679 the variable name, then prompts for the declaration. The default
12680 at the declaration prompt is the previous declaration, if any.
12681 You can edit this declaration, or press @kbd{C-k} to erase it and
12682 type a new declaration. (Or, erase it and press @key{RET} to clear
12683 the declaration, effectively ``undeclaring'' the variable.)
12685 A declaration is in general a vector of @dfn{type symbols} and
12686 @dfn{range} values. If there is only one type symbol or range value,
12687 you can write it directly rather than enclosing it in a vector.
12688 For example, @kbd{s d foo @key{RET} real @key{RET}} declares @code{foo} to
12689 be a real number, and @kbd{s d bar @key{RET} [int, const, [1..6]] @key{RET}}
12690 declares @code{bar} to be a constant integer between 1 and 6.
12691 (Actually, you can omit the outermost brackets and Calc will
12692 provide them for you: @kbd{s d bar @key{RET} int, const, [1..6] @key{RET}}.)
12694 @cindex @code{Decls} variable
12696 Declarations in Calc are kept in a special variable called @code{Decls}.
12697 This variable encodes the set of all outstanding declarations in
12698 the form of a matrix. Each row has two elements: A variable or
12699 vector of variables declared by that row, and the declaration
12700 specifier as described above. You can use the @kbd{s D} command to
12701 edit this variable if you wish to see all the declarations at once.
12702 @xref{Operations on Variables}, for a description of this command
12703 and the @kbd{s p} command that allows you to save your declarations
12704 permanently if you wish.
12706 Items being declared can also be function calls. The arguments in
12707 the call are ignored; the effect is to say that this function returns
12708 values of the declared type for any valid arguments. The @kbd{s d}
12709 command declares only variables, so if you wish to make a function
12710 declaration you will have to edit the @code{Decls} matrix yourself.
12712 For example, the declaration matrix
12718 [ f(1,2,3), [0 .. inf) ] ]
12723 declares that @code{foo} represents a real number, @code{j}, @code{k}
12724 and @code{n} represent integers, and the function @code{f} always
12725 returns a real number in the interval shown.
12728 If there is a declaration for the variable @code{All}, then that
12729 declaration applies to all variables that are not otherwise declared.
12730 It does not apply to function names. For example, using the row
12731 @samp{[All, real]} says that all your variables are real unless they
12732 are explicitly declared without @code{real} in some other row.
12733 The @kbd{s d} command declares @code{All} if you give a blank
12734 response to the variable-name prompt.
12736 @node Kinds of Declarations, Functions for Declarations, Declaration Basics, Declarations
12737 @subsection Kinds of Declarations
12740 The type-specifier part of a declaration (that is, the second prompt
12741 in the @kbd{s d} command) can be a type symbol, an interval, or a
12742 vector consisting of zero or more type symbols followed by zero or
12743 more intervals or numbers that represent the set of possible values
12748 [ [ a, [1, 2, 3, 4, 5] ]
12750 [ c, [int, 1 .. 5] ] ]
12754 Here @code{a} is declared to contain one of the five integers shown;
12755 @code{b} is any number in the interval from 1 to 5 (any real number
12756 since we haven't specified), and @code{c} is any integer in that
12757 interval. Thus the declarations for @code{a} and @code{c} are
12758 nearly equivalent (see below).
12760 The type-specifier can be the empty vector @samp{[]} to say that
12761 nothing is known about a given variable's value. This is the same
12762 as not declaring the variable at all except that it overrides any
12763 @code{All} declaration which would otherwise apply.
12765 The initial value of @code{Decls} is the empty vector @samp{[]}.
12766 If @code{Decls} has no stored value or if the value stored in it
12767 is not valid, it is ignored and there are no declarations as far
12768 as Calc is concerned. (The @kbd{s d} command will replace such a
12769 malformed value with a fresh empty matrix, @samp{[]}, before recording
12770 the new declaration.) Unrecognized type symbols are ignored.
12772 The following type symbols describe what sorts of numbers will be
12773 stored in a variable:
12779 Numerical integers. (Integers or integer-valued floats.)
12781 Fractions. (Rational numbers which are not integers.)
12783 Rational numbers. (Either integers or fractions.)
12785 Floating-point numbers.
12787 Real numbers. (Integers, fractions, or floats. Actually,
12788 intervals and error forms with real components also count as
12791 Positive real numbers. (Strictly greater than zero.)
12793 Nonnegative real numbers. (Greater than or equal to zero.)
12795 Numbers. (Real or complex.)
12798 Calc uses this information to determine when certain simplifications
12799 of formulas are safe. For example, @samp{(x^y)^z} cannot be
12800 simplified to @samp{x^(y z)} in general; for example,
12801 @samp{((-3)^2)^1:2} is 3, but @samp{(-3)^(2*1:2) = (-3)^1} is @mathit{-3}.
12802 However, this simplification @emph{is} safe if @code{z} is known
12803 to be an integer, or if @code{x} is known to be a nonnegative
12804 real number. If you have given declarations that allow Calc to
12805 deduce either of these facts, Calc will perform this simplification
12808 Calc can apply a certain amount of logic when using declarations.
12809 For example, @samp{(x^y)^(2n+1)} will be simplified if @code{n}
12810 has been declared @code{int}; Calc knows that an integer times an
12811 integer, plus an integer, must always be an integer. (In fact,
12812 Calc would simplify @samp{(-x)^(2n+1)} to @samp{-(x^(2n+1))} since
12813 it is able to determine that @samp{2n+1} must be an odd integer.)
12815 Similarly, @samp{(abs(x)^y)^z} will be simplified to @samp{abs(x)^(y z)}
12816 because Calc knows that the @code{abs} function always returns a
12817 nonnegative real. If you had a @code{myabs} function that also had
12818 this property, you could get Calc to recognize it by adding the row
12819 @samp{[myabs(), nonneg]} to the @code{Decls} matrix.
12821 One instance of this simplification is @samp{sqrt(x^2)} (since the
12822 @code{sqrt} function is effectively a one-half power). Normally
12823 Calc leaves this formula alone. After the command
12824 @kbd{s d x @key{RET} real @key{RET}}, however, it can simplify the formula to
12825 @samp{abs(x)}. And after @kbd{s d x @key{RET} nonneg @key{RET}}, Calc can
12826 simplify this formula all the way to @samp{x}.
12828 If there are any intervals or real numbers in the type specifier,
12829 they comprise the set of possible values that the variable or
12830 function being declared can have. In particular, the type symbol
12831 @code{real} is effectively the same as the range @samp{[-inf .. inf]}
12832 (note that infinity is included in the range of possible values);
12833 @code{pos} is the same as @samp{(0 .. inf]}, and @code{nonneg} is
12834 the same as @samp{[0 .. inf]}. Saying @samp{[real, [-5 .. 5]]} is
12835 redundant because the fact that the variable is real can be
12836 deduced just from the interval, but @samp{[int, [-5 .. 5]]} and
12837 @samp{[rat, [-5 .. 5]]} are useful combinations.
12839 Note that the vector of intervals or numbers is in the same format
12840 used by Calc's set-manipulation commands. @xref{Set Operations}.
12842 The type specifier @samp{[1, 2, 3]} is equivalent to
12843 @samp{[numint, 1, 2, 3]}, @emph{not} to @samp{[int, 1, 2, 3]}.
12844 In other words, the range of possible values means only that
12845 the variable's value must be numerically equal to a number in
12846 that range, but not that it must be equal in type as well.
12847 Calc's set operations act the same way; @samp{in(2, [1., 2., 3.])}
12848 and @samp{in(1.5, [1:2, 3:2, 5:2])} both report ``true.''
12850 If you use a conflicting combination of type specifiers, the
12851 results are unpredictable. An example is @samp{[pos, [0 .. 5]]},
12852 where the interval does not lie in the range described by the
12855 ``Real'' declarations mostly affect simplifications involving powers
12856 like the one described above. Another case where they are used
12857 is in the @kbd{a P} command which returns a list of all roots of a
12858 polynomial; if the variable has been declared real, only the real
12859 roots (if any) will be included in the list.
12861 ``Integer'' declarations are used for simplifications which are valid
12862 only when certain values are integers (such as @samp{(x^y)^z}
12865 Calc's algebraic simplifications also make use of declarations when
12866 simplifying equations and inequalities. They will cancel @code{x}
12867 from both sides of @samp{a x = b x} only if it is sure @code{x}
12868 is non-zero, say, because it has a @code{pos} declaration.
12869 To declare specifically that @code{x} is real and non-zero,
12870 use @samp{[[-inf .. 0), (0 .. inf]]}. (There is no way in the
12871 current notation to say that @code{x} is nonzero but not necessarily
12872 real.) The @kbd{a e} command does ``unsafe'' simplifications,
12873 including canceling @samp{x} from the equation when @samp{x} is
12874 not known to be nonzero.
12876 Another set of type symbols distinguish between scalars and vectors.
12880 The value is not a vector.
12882 The value is a vector.
12884 The value is a matrix (a rectangular vector of vectors).
12886 The value is a square matrix.
12889 These type symbols can be combined with the other type symbols
12890 described above; @samp{[int, matrix]} describes an object which
12891 is a matrix of integers.
12893 Scalar/vector declarations are used to determine whether certain
12894 algebraic operations are safe. For example, @samp{[a, b, c] + x}
12895 is normally not simplified to @samp{[a + x, b + x, c + x]}, but
12896 it will be if @code{x} has been declared @code{scalar}. On the
12897 other hand, multiplication is usually assumed to be commutative,
12898 but the terms in @samp{x y} will never be exchanged if both @code{x}
12899 and @code{y} are known to be vectors or matrices. (Calc currently
12900 never distinguishes between @code{vector} and @code{matrix}
12903 @xref{Matrix Mode}, for a discussion of Matrix mode and
12904 Scalar mode, which are similar to declaring @samp{[All, matrix]}
12905 or @samp{[All, scalar]} but much more convenient.
12907 One more type symbol that is recognized is used with the @kbd{H a d}
12908 command for taking total derivatives of a formula. @xref{Calculus}.
12912 The value is a constant with respect to other variables.
12915 Calc does not check the declarations for a variable when you store
12916 a value in it. However, storing @mathit{-3.5} in a variable that has
12917 been declared @code{pos}, @code{int}, or @code{matrix} may have
12918 unexpected effects; Calc may evaluate @samp{sqrt(x^2)} to @expr{3.5}
12919 if it substitutes the value first, or to @expr{-3.5} if @code{x}
12920 was declared @code{pos} and the formula @samp{sqrt(x^2)} is
12921 simplified to @samp{x} before the value is substituted. Before
12922 using a variable for a new purpose, it is best to use @kbd{s d}
12923 or @kbd{s D} to check to make sure you don't still have an old
12924 declaration for the variable that will conflict with its new meaning.
12926 @node Functions for Declarations, , Kinds of Declarations, Declarations
12927 @subsection Functions for Declarations
12930 Calc has a set of functions for accessing the current declarations
12931 in a convenient manner. These functions return 1 if the argument
12932 can be shown to have the specified property, or 0 if the argument
12933 can be shown @emph{not} to have that property; otherwise they are
12934 left unevaluated. These functions are suitable for use with rewrite
12935 rules (@pxref{Conditional Rewrite Rules}) or programming constructs
12936 (@pxref{Conditionals in Macros}). They can be entered only using
12937 algebraic notation. @xref{Logical Operations}, for functions
12938 that perform other tests not related to declarations.
12940 For example, @samp{dint(17)} returns 1 because 17 is an integer, as
12941 do @samp{dint(n)} and @samp{dint(2 n - 3)} if @code{n} has been declared
12942 @code{int}, but @samp{dint(2.5)} and @samp{dint(n + 0.5)} return 0.
12943 Calc consults knowledge of its own built-in functions as well as your
12944 own declarations: @samp{dint(floor(x))} returns 1.
12958 The @code{dint} function checks if its argument is an integer.
12959 The @code{dnatnum} function checks if its argument is a natural
12960 number, i.e., a nonnegative integer. The @code{dnumint} function
12961 checks if its argument is numerically an integer, i.e., either an
12962 integer or an integer-valued float. Note that these and the other
12963 data type functions also accept vectors or matrices composed of
12964 suitable elements, and that real infinities @samp{inf} and @samp{-inf}
12965 are considered to be integers for the purposes of these functions.
12971 The @code{drat} function checks if its argument is rational, i.e.,
12972 an integer or fraction. Infinities count as rational, but intervals
12973 and error forms do not.
12979 The @code{dreal} function checks if its argument is real. This
12980 includes integers, fractions, floats, real error forms, and intervals.
12986 The @code{dimag} function checks if its argument is imaginary,
12987 i.e., is mathematically equal to a real number times @expr{i}.
13001 The @code{dpos} function checks for positive (but nonzero) reals.
13002 The @code{dneg} function checks for negative reals. The @code{dnonneg}
13003 function checks for nonnegative reals, i.e., reals greater than or
13004 equal to zero. Note that Calc's algebraic simplifications, which are
13005 effectively applied to all conditions in rewrite rules, can simplify
13006 an expression like @expr{x > 0} to 1 or 0 using @code{dpos}.
13007 So the actual functions @code{dpos}, @code{dneg}, and @code{dnonneg}
13008 are rarely necessary.
13014 The @code{dnonzero} function checks that its argument is nonzero.
13015 This includes all nonzero real or complex numbers, all intervals that
13016 do not include zero, all nonzero modulo forms, vectors all of whose
13017 elements are nonzero, and variables or formulas whose values can be
13018 deduced to be nonzero. It does not include error forms, since they
13019 represent values which could be anything including zero. (This is
13020 also the set of objects considered ``true'' in conditional contexts.)
13030 The @code{deven} function returns 1 if its argument is known to be
13031 an even integer (or integer-valued float); it returns 0 if its argument
13032 is known not to be even (because it is known to be odd or a non-integer).
13033 Calc's algebraic simplifications use this to simplify a test of the form
13034 @samp{x % 2 = 0}. There is also an analogous @code{dodd} function.
13040 The @code{drange} function returns a set (an interval or a vector
13041 of intervals and/or numbers; @pxref{Set Operations}) that describes
13042 the set of possible values of its argument. If the argument is
13043 a variable or a function with a declaration, the range is copied
13044 from the declaration. Otherwise, the possible signs of the
13045 expression are determined using a method similar to @code{dpos},
13046 etc., and a suitable set like @samp{[0 .. inf]} is returned. If
13047 the expression is not provably real, the @code{drange} function
13048 remains unevaluated.
13054 The @code{dscalar} function returns 1 if its argument is provably
13055 scalar, or 0 if its argument is provably non-scalar. It is left
13056 unevaluated if this cannot be determined. (If Matrix mode or Scalar
13057 mode is in effect, this function returns 1 or 0, respectively,
13058 if it has no other information.) When Calc interprets a condition
13059 (say, in a rewrite rule) it considers an unevaluated formula to be
13060 ``false.'' Thus, @samp{dscalar(a)} is ``true'' only if @code{a} is
13061 provably scalar, and @samp{!dscalar(a)} is ``true'' only if @code{a}
13062 is provably non-scalar; both are ``false'' if there is insufficient
13063 information to tell.
13065 @node Display Modes, Language Modes, Declarations, Mode Settings
13066 @section Display Modes
13069 The commands in this section are two-key sequences beginning with the
13070 @kbd{d} prefix. The @kbd{d l} (@code{calc-line-numbering}) and @kbd{d b}
13071 (@code{calc-line-breaking}) commands are described elsewhere;
13072 @pxref{Stack Basics} and @pxref{Normal Language Modes}, respectively.
13073 Display formats for vectors and matrices are also covered elsewhere;
13074 @pxref{Vector and Matrix Formats}.
13076 One thing all display modes have in common is their treatment of the
13077 @kbd{H} prefix. This prefix causes any mode command that would normally
13078 refresh the stack to leave the stack display alone. The word ``Dirty''
13079 will appear in the mode line when Calc thinks the stack display may not
13080 reflect the latest mode settings.
13082 @kindex d @key{RET}
13083 @pindex calc-refresh-top
13084 The @kbd{d @key{RET}} (@code{calc-refresh-top}) command reformats the
13085 top stack entry according to all the current modes. Positive prefix
13086 arguments reformat the top @var{n} entries; negative prefix arguments
13087 reformat the specified entry, and a prefix of zero is equivalent to
13088 @kbd{d @key{SPC}} (@code{calc-refresh}), which reformats the entire stack.
13089 For example, @kbd{H d s M-2 d @key{RET}} changes to scientific notation
13090 but reformats only the top two stack entries in the new mode.
13092 The @kbd{I} prefix has another effect on the display modes. The mode
13093 is set only temporarily; the top stack entry is reformatted according
13094 to that mode, then the original mode setting is restored. In other
13095 words, @kbd{I d s} is equivalent to @kbd{H d s d @key{RET} H d (@var{old mode})}.
13099 * Grouping Digits::
13101 * Complex Formats::
13102 * Fraction Formats::
13105 * Truncating the Stack::
13110 @node Radix Modes, Grouping Digits, Display Modes, Display Modes
13111 @subsection Radix Modes
13114 @cindex Radix display
13115 @cindex Non-decimal numbers
13116 @cindex Decimal and non-decimal numbers
13117 Calc normally displays numbers in decimal (@dfn{base-10} or @dfn{radix-10})
13118 notation. Calc can actually display in any radix from two (binary) to 36.
13119 When the radix is above 10, the letters @code{A} to @code{Z} are used as
13120 digits. When entering such a number, letter keys are interpreted as
13121 potential digits rather than terminating numeric entry mode.
13127 @cindex Hexadecimal integers
13128 @cindex Octal integers
13129 The key sequences @kbd{d 2}, @kbd{d 8}, @kbd{d 6}, and @kbd{d 0} select
13130 binary, octal, hexadecimal, and decimal as the current display radix,
13131 respectively. Numbers can always be entered in any radix, though the
13132 current radix is used as a default if you press @kbd{#} without any initial
13133 digits. A number entered without a @kbd{#} is @emph{always} interpreted
13138 To set the radix generally, use @kbd{d r} (@code{calc-radix}) and enter
13139 an integer from 2 to 36. You can specify the radix as a numeric prefix
13140 argument; otherwise you will be prompted for it.
13143 @pindex calc-leading-zeros
13144 @cindex Leading zeros
13145 Integers normally are displayed with however many digits are necessary to
13146 represent the integer and no more. The @kbd{d z} (@code{calc-leading-zeros})
13147 command causes integers to be padded out with leading zeros according to the
13148 current binary word size. (@xref{Binary Functions}, for a discussion of
13149 word size.) If the absolute value of the word size is @expr{w}, all integers
13150 are displayed with at least enough digits to represent
13151 @texline @math{2^w-1}
13152 @infoline @expr{(2^w)-1}
13153 in the current radix. (Larger integers will still be displayed in their
13156 @cindex Two's complements
13157 Calc can display @expr{w}-bit integers using two's complement
13158 notation, although this is most useful with the binary, octal and
13159 hexadecimal display modes. This option is selected by using the
13160 @kbd{O} option prefix before setting the display radix, and a negative word
13161 size might be appropriate (@pxref{Binary Functions}). In two's
13162 complement notation, the integers in the (nearly) symmetric interval
13164 @texline @math{-2^{w-1}}
13165 @infoline @expr{-2^(w-1)}
13167 @texline @math{2^{w-1}-1}
13168 @infoline @expr{2^(w-1)-1}
13169 are represented by the integers from @expr{0} to @expr{2^w-1}:
13170 the integers from @expr{0} to
13171 @texline @math{2^{w-1}-1}
13172 @infoline @expr{2^(w-1)-1}
13173 are represented by themselves and the integers from
13174 @texline @math{-2^{w-1}}
13175 @infoline @expr{-2^(w-1)}
13176 to @expr{-1} are represented by the integers from
13177 @texline @math{2^{w-1}}
13178 @infoline @expr{2^(w-1)}
13179 to @expr{2^w-1} (the integer @expr{k} is represented by @expr{k+2^w}).
13180 Calc will display a two's complement integer by the radix (either
13181 @expr{2}, @expr{8} or @expr{16}), two @kbd{#} symbols, and then its
13182 representation (including any leading zeros necessary to include all
13183 @expr{w} bits). In a two's complement display mode, numbers that
13184 are not displayed in two's complement notation (i.e., that aren't
13186 @texline @math{-2^{w-1}}
13187 @infoline @expr{-2^(w-1)}
13190 @texline @math{2^{w-1}-1})
13191 @infoline @expr{2^(w-1)-1})
13192 will be represented using Calc's usual notation (in the appropriate
13195 @node Grouping Digits, Float Formats, Radix Modes, Display Modes
13196 @subsection Grouping Digits
13200 @pindex calc-group-digits
13201 @cindex Grouping digits
13202 @cindex Digit grouping
13203 Long numbers can be hard to read if they have too many digits. For
13204 example, the factorial of 30 is 33 digits long! Press @kbd{d g}
13205 (@code{calc-group-digits}) to enable @dfn{Grouping} mode, in which digits
13206 are displayed in clumps of 3 or 4 (depending on the current radix)
13207 separated by commas.
13209 The @kbd{d g} command toggles grouping on and off.
13210 With a numeric prefix of 0, this command displays the current state of
13211 the grouping flag; with an argument of minus one it disables grouping;
13212 with a positive argument @expr{N} it enables grouping on every @expr{N}
13213 digits. For floating-point numbers, grouping normally occurs only
13214 before the decimal point. A negative prefix argument @expr{-N} enables
13215 grouping every @expr{N} digits both before and after the decimal point.
13218 @pindex calc-group-char
13219 The @kbd{d ,} (@code{calc-group-char}) command allows you to choose any
13220 character as the grouping separator. The default is the comma character.
13221 If you find it difficult to read vectors of large integers grouped with
13222 commas, you may wish to use spaces or some other character instead.
13223 This command takes the next character you type, whatever it is, and
13224 uses it as the digit separator. As a special case, @kbd{d , \} selects
13225 @samp{\,} (@TeX{}'s thin-space symbol) as the digit separator.
13227 Please note that grouped numbers will not generally be parsed correctly
13228 if re-read in textual form, say by the use of @kbd{C-x * y} and @kbd{C-x * g}.
13229 (@xref{Kill and Yank}, for details on these commands.) One exception is
13230 the @samp{\,} separator, which doesn't interfere with parsing because it
13231 is ignored by @TeX{} language mode.
13233 @node Float Formats, Complex Formats, Grouping Digits, Display Modes
13234 @subsection Float Formats
13237 Floating-point quantities are normally displayed in standard decimal
13238 form, with scientific notation used if the exponent is especially high
13239 or low. All significant digits are normally displayed. The commands
13240 in this section allow you to choose among several alternative display
13241 formats for floats.
13244 @pindex calc-normal-notation
13245 The @kbd{d n} (@code{calc-normal-notation}) command selects the normal
13246 display format. All significant figures in a number are displayed.
13247 With a positive numeric prefix, numbers are rounded if necessary to
13248 that number of significant digits. With a negative numerix prefix,
13249 the specified number of significant digits less than the current
13250 precision is used. (Thus @kbd{C-u -2 d n} displays 10 digits if the
13251 current precision is 12.)
13254 @pindex calc-fix-notation
13255 The @kbd{d f} (@code{calc-fix-notation}) command selects fixed-point
13256 notation. The numeric argument is the number of digits after the
13257 decimal point, zero or more. This format will relax into scientific
13258 notation if a nonzero number would otherwise have been rounded all the
13259 way to zero. Specifying a negative number of digits is the same as
13260 for a positive number, except that small nonzero numbers will be rounded
13261 to zero rather than switching to scientific notation.
13264 @pindex calc-sci-notation
13265 @cindex Scientific notation, display of
13266 The @kbd{d s} (@code{calc-sci-notation}) command selects scientific
13267 notation. A positive argument sets the number of significant figures
13268 displayed, of which one will be before and the rest after the decimal
13269 point. A negative argument works the same as for @kbd{d n} format.
13270 The default is to display all significant digits.
13273 @pindex calc-eng-notation
13274 @cindex Engineering notation, display of
13275 The @kbd{d e} (@code{calc-eng-notation}) command selects engineering
13276 notation. This is similar to scientific notation except that the
13277 exponent is rounded down to a multiple of three, with from one to three
13278 digits before the decimal point. An optional numeric prefix sets the
13279 number of significant digits to display, as for @kbd{d s}.
13281 It is important to distinguish between the current @emph{precision} and
13282 the current @emph{display format}. After the commands @kbd{C-u 10 p}
13283 and @kbd{C-u 6 d n} the Calculator computes all results to ten
13284 significant figures but displays only six. (In fact, intermediate
13285 calculations are often carried to one or two more significant figures,
13286 but values placed on the stack will be rounded down to ten figures.)
13287 Numbers are never actually rounded to the display precision for storage,
13288 except by commands like @kbd{C-k} and @kbd{C-x * y} which operate on the
13289 actual displayed text in the Calculator buffer.
13292 @pindex calc-point-char
13293 The @kbd{d .} (@code{calc-point-char}) command selects the character used
13294 as a decimal point. Normally this is a period; users in some countries
13295 may wish to change this to a comma. Note that this is only a display
13296 style; on entry, periods must always be used to denote floating-point
13297 numbers, and commas to separate elements in a list.
13299 @node Complex Formats, Fraction Formats, Float Formats, Display Modes
13300 @subsection Complex Formats
13304 @pindex calc-complex-notation
13305 There are three supported notations for complex numbers in rectangular
13306 form. The default is as a pair of real numbers enclosed in parentheses
13307 and separated by a comma: @samp{(a,b)}. The @kbd{d c}
13308 (@code{calc-complex-notation}) command selects this style.
13311 @pindex calc-i-notation
13313 @pindex calc-j-notation
13314 The other notations are @kbd{d i} (@code{calc-i-notation}), in which
13315 numbers are displayed in @samp{a+bi} form, and @kbd{d j}
13316 (@code{calc-j-notation}) which displays the form @samp{a+bj} preferred
13317 in some disciplines.
13319 @cindex @code{i} variable
13321 Complex numbers are normally entered in @samp{(a,b)} format.
13322 If you enter @samp{2+3i} as an algebraic formula, it will be stored as
13323 the formula @samp{2 + 3 * i}. However, if you use @kbd{=} to evaluate
13324 this formula and you have not changed the variable @samp{i}, the @samp{i}
13325 will be interpreted as @samp{(0,1)} and the formula will be simplified
13326 to @samp{(2,3)}. Other commands (like @code{calc-sin}) will @emph{not}
13327 interpret the formula @samp{2 + 3 * i} as a complex number.
13328 @xref{Variables}, under ``special constants.''
13330 @node Fraction Formats, HMS Formats, Complex Formats, Display Modes
13331 @subsection Fraction Formats
13335 @pindex calc-over-notation
13336 Display of fractional numbers is controlled by the @kbd{d o}
13337 (@code{calc-over-notation}) command. By default, a number like
13338 eight thirds is displayed in the form @samp{8:3}. The @kbd{d o} command
13339 prompts for a one- or two-character format. If you give one character,
13340 that character is used as the fraction separator. Common separators are
13341 @samp{:} and @samp{/}. (During input of numbers, the @kbd{:} key must be
13342 used regardless of the display format; in particular, the @kbd{/} is used
13343 for RPN-style division, @emph{not} for entering fractions.)
13345 If you give two characters, fractions use ``integer-plus-fractional-part''
13346 notation. For example, the format @samp{+/} would display eight thirds
13347 as @samp{2+2/3}. If two colons are present in a number being entered,
13348 the number is interpreted in this form (so that the entries @kbd{2:2:3}
13349 and @kbd{8:3} are equivalent).
13351 It is also possible to follow the one- or two-character format with
13352 a number. For example: @samp{:10} or @samp{+/3}. In this case,
13353 Calc adjusts all fractions that are displayed to have the specified
13354 denominator, if possible. Otherwise it adjusts the denominator to
13355 be a multiple of the specified value. For example, in @samp{:6} mode
13356 the fraction @expr{1:6} will be unaffected, but @expr{2:3} will be
13357 displayed as @expr{4:6}, @expr{1:2} will be displayed as @expr{3:6},
13358 and @expr{1:8} will be displayed as @expr{3:24}. Integers are also
13359 affected by this mode: 3 is displayed as @expr{18:6}. Note that the
13360 format @samp{:1} writes fractions the same as @samp{:}, but it writes
13361 integers as @expr{n:1}.
13363 The fraction format does not affect the way fractions or integers are
13364 stored, only the way they appear on the screen. The fraction format
13365 never affects floats.
13367 @node HMS Formats, Date Formats, Fraction Formats, Display Modes
13368 @subsection HMS Formats
13372 @pindex calc-hms-notation
13373 The @kbd{d h} (@code{calc-hms-notation}) command controls the display of
13374 HMS (hours-minutes-seconds) forms. It prompts for a string which
13375 consists basically of an ``hours'' marker, optional punctuation, a
13376 ``minutes'' marker, more optional punctuation, and a ``seconds'' marker.
13377 Punctuation is zero or more spaces, commas, or semicolons. The hours
13378 marker is one or more non-punctuation characters. The minutes and
13379 seconds markers must be single non-punctuation characters.
13381 The default HMS format is @samp{@@ ' "}, producing HMS values of the form
13382 @samp{23@@ 30' 15.75"}. The format @samp{deg, ms} would display this same
13383 value as @samp{23deg, 30m15.75s}. During numeric entry, the @kbd{h} or @kbd{o}
13384 keys are recognized as synonyms for @kbd{@@} regardless of display format.
13385 The @kbd{m} and @kbd{s} keys are recognized as synonyms for @kbd{'} and
13386 @kbd{"}, respectively, but only if an @kbd{@@} (or @kbd{h} or @kbd{o}) has
13387 already been typed; otherwise, they have their usual meanings
13388 (@kbd{m-} prefix and @kbd{s-} prefix). Thus, @kbd{5 "}, @kbd{0 @@ 5 "}, and
13389 @kbd{0 h 5 s} are some of the ways to enter the quantity ``five seconds.''
13390 The @kbd{'} key is recognized as ``minutes'' only if @kbd{@@} (or @kbd{h} or
13391 @kbd{o}) has already been pressed; otherwise it means to switch to algebraic
13394 @node Date Formats, Truncating the Stack, HMS Formats, Display Modes
13395 @subsection Date Formats
13399 @pindex calc-date-notation
13400 The @kbd{d d} (@code{calc-date-notation}) command controls the display
13401 of date forms (@pxref{Date Forms}). It prompts for a string which
13402 contains letters that represent the various parts of a date and time.
13403 To show which parts should be omitted when the form represents a pure
13404 date with no time, parts of the string can be enclosed in @samp{< >}
13405 marks. If you don't include @samp{< >} markers in the format, Calc
13406 guesses at which parts, if any, should be omitted when formatting
13409 The default format is: @samp{<H:mm:SSpp >Www Mmm D, YYYY}.
13410 An example string in this format is @samp{3:32pm Wed Jan 9, 1991}.
13411 If you enter a blank format string, this default format is
13414 Calc uses @samp{< >} notation for nameless functions as well as for
13415 dates. @xref{Specifying Operators}. To avoid confusion with nameless
13416 functions, your date formats should avoid using the @samp{#} character.
13420 * Date Formatting Codes::
13421 * Free-Form Dates::
13422 * Standard Date Formats::
13425 @node ISO 8601, Date Formatting Codes, Date Formats, Date Formats
13426 @subsubsection ISO 8601
13430 The same date can be written down in different formats and Calc tries
13431 to allow you to choose your preferred format. Some common formats are
13432 ambiguous, however; for example, 10/11/2012 means October 11,
13433 2012 in the United States but it means November 10, 2012 in
13434 Europe. To help avoid such ambiguities, the International Organization
13435 for Standardization (ISO) provides the ISO 8601 standard, which
13436 provides three different but easily distinguishable and unambiguous
13437 ways to represent a date.
13439 The ISO 8601 calendar date representation is
13442 @var{YYYY}-@var{MM}-@var{DD}
13446 where @var{YYYY} is the four digit year, @var{MM} is the two-digit month
13447 number (01 for January to 12 for December), and @var{DD} is the
13448 two-digit day of the month (01 to 31). (Note that @var{YYYY} does not
13449 correspond to Calc's date formatting code, which will be introduced
13450 later.) The year, which should be padded with zeros to ensure it has at
13451 least four digits, is the Gregorian year, except that the year before
13452 0001 (1 AD) is the year 0000 (1 BC). The date October 11, 2012 is
13453 written 2012-10-11 in this representation and November 10, 2012 is
13454 written 2012-11-10.
13456 The ISO 8601 ordinal date representation is
13459 @var{YYYY}-@var{DDD}
13463 where @var{YYYY} is the year, as above, and @var{DDD} is the day of the year.
13464 The date December 31, 2011 is written 2011-365 in this representation
13465 and January 1, 2012 is written 2012-001.
13467 The ISO 8601 week date representation is
13470 @var{YYYY}-W@var{ww}-@var{D}
13474 where @var{YYYY} is the ISO week-numbering year, @var{ww} is the two
13475 digit week number (preceded by a literal ``W''), and @var{D} is the day
13476 of the week (1 for Monday through 7 for Sunday). The ISO week-numbering
13477 year is based on the Gregorian year but can differ slightly. The first
13478 week of an ISO week-numbering year is the week with the Gregorian year's
13479 first Thursday in it (equivalently, the week containing January 4);
13480 any day of that week (Monday through Sunday) is part of the same ISO
13481 week-numbering year, any day from the previous week is part of the
13482 previous year. For example, January 4, 2013 is on a Friday, and so
13483 the first week for the ISO week-numbering year 2013 starts on
13484 Monday, December 31, 2012. The day December 31, 2012 is then part of the
13485 Gregorian year 2012 but ISO week-numbering year 2013. In the week
13486 date representation, this week goes from 2013-W01-1 (December 31,
13487 2012) to 2013-W01-7 (January 6, 2013).
13489 All three ISO 8601 representations arrange the numbers from most
13490 significant to least significant; as well as being unambiguous
13491 representations, they are easy to sort since chronological order in
13492 this formats corresponds to lexicographical order. The hyphens are
13495 The ISO 8601 standard uses a 24 hour clock; a particular time is
13496 represented by @var{hh}:@var{mm}:@var{ss} where @var{hh} is the
13497 two-digit hour (from 00 to 24), @var{mm} is the two-digit minute (from
13498 00 to 59) and @var{ss} is the two-digit second. The seconds or minutes
13499 and seconds can be omitted, and decimals can be added. If a date with a
13500 time is represented, they should be separated by a literal ``T'', so noon
13501 on December 13, 2012 can be represented as 2012-12-13T12:00.
13503 @node Date Formatting Codes, Free-Form Dates, ISO 8601, Date Formats
13504 @subsubsection Date Formatting Codes
13507 When displaying a date, the current date format is used. All
13508 characters except for letters and @samp{<} and @samp{>} are
13509 copied literally when dates are formatted. The portion between
13510 @samp{< >} markers is omitted for pure dates, or included for
13511 date/time forms. Letters are interpreted according to the table
13514 When dates are read in during algebraic entry, Calc first tries to
13515 match the input string to the current format either with or without
13516 the time part. The punctuation characters (including spaces) must
13517 match exactly; letter fields must correspond to suitable text in
13518 the input. If this doesn't work, Calc checks if the input is a
13519 simple number; if so, the number is interpreted as a number of days
13520 since Dec 31, 1 BC@. Otherwise, Calc tries a much more relaxed and
13521 flexible algorithm which is described in the next section.
13523 Weekday names are ignored during reading.
13525 Two-digit year numbers are interpreted as lying in the range
13526 from 1941 to 2039. Years outside that range are always
13527 entered and displayed in full. Year numbers with a leading
13528 @samp{+} sign are always interpreted exactly, allowing the
13529 entry and display of the years 1 through 99 AD.
13531 Here is a complete list of the formatting codes for dates:
13535 Year: ``91'' for 1991, ``7'' for 2007, ``+23'' for 23 AD.
13537 Year: ``91'' for 1991, ``07'' for 2007, ``+23'' for 23 AD.
13539 Year: ``91'' for 1991, `` 7'' for 2007, ``+23'' for 23 AD.
13541 Year: ``1991'' for 1991, ``23'' for 23 AD.
13543 Year: ``1991'' for 1991, ``+23'' for 23 AD.
13545 Year: ``1991'' for 1991, ``0023'' for 23 AD, ``0000'' for 1 BC.
13547 Year: ISO 8601 week-numbering year.
13549 Year: ``ad'' or blank.
13551 Year: ``AD'' or blank.
13553 Year: ``ad '' or blank. (Note trailing space.)
13555 Year: ``AD '' or blank.
13557 Year: ``a.d.@:'' or blank.
13559 Year: ``A.D.'' or blank.
13561 Year: ``bc'' or blank.
13563 Year: ``BC'' or blank.
13565 Year: `` bc'' or blank. (Note leading space.)
13567 Year: `` BC'' or blank.
13569 Year: ``b.c.@:'' or blank.
13571 Year: ``B.C.'' or blank.
13573 Month: ``8'' for August.
13575 Month: ``08'' for August.
13577 Month: `` 8'' for August.
13579 Month: ``AUG'' for August.
13581 Month: ``Aug'' for August.
13583 Month: ``aug'' for August.
13585 Month: ``AUGUST'' for August.
13587 Month: ``August'' for August.
13589 Day: ``7'' for 7th day of month.
13591 Day: ``07'' for 7th day of month.
13593 Day: `` 7'' for 7th day of month.
13595 Weekday: ``0'' for Sunday, ``6'' for Saturday.
13597 Weekday: ``1'' for Monday, ``7'' for Sunday.
13599 Weekday: ``SUN'' for Sunday.
13601 Weekday: ``Sun'' for Sunday.
13603 Weekday: ``sun'' for Sunday.
13605 Weekday: ``SUNDAY'' for Sunday.
13607 Weekday: ``Sunday'' for Sunday.
13609 Week number: ISO 8601 week number, ``W01'' for week 1.
13611 Day of year: ``34'' for Feb. 3.
13613 Day of year: ``034'' for Feb. 3.
13615 Day of year: `` 34'' for Feb. 3.
13617 Letter: Literal ``T''.
13619 Hour: ``5'' for 5 AM; ``17'' for 5 PM.
13621 Hour: ``05'' for 5 AM; ``17'' for 5 PM.
13623 Hour: `` 5'' for 5 AM; ``17'' for 5 PM.
13625 Hour: ``5'' for 5 AM and 5 PM.
13627 Hour: ``05'' for 5 AM and 5 PM.
13629 Hour: `` 5'' for 5 AM and 5 PM.
13631 AM/PM: ``a'' or ``p''.
13633 AM/PM: ``A'' or ``P''.
13635 AM/PM: ``am'' or ``pm''.
13637 AM/PM: ``AM'' or ``PM''.
13639 AM/PM: ``a.m.@:'' or ``p.m.''.
13641 AM/PM: ``A.M.'' or ``P.M.''.
13643 Minutes: ``7'' for 7.
13645 Minutes: ``07'' for 7.
13647 Minutes: `` 7'' for 7.
13649 Seconds: ``7'' for 7; ``7.23'' for 7.23.
13651 Seconds: ``07'' for 7; ``07.23'' for 7.23.
13653 Seconds: `` 7'' for 7; `` 7.23'' for 7.23.
13655 Optional seconds: ``07'' for 7; blank for 0.
13657 Optional seconds: `` 7'' for 7; blank for 0.
13659 Numeric date/time: ``726842.25'' for 6:00am Wed Jan 9, 1991.
13661 Numeric date: ``726842'' for any time on Wed Jan 9, 1991.
13663 Julian date/time: ``2448265.75'' for 6:00am Wed Jan 9, 1991.
13665 Julian date: ``2448266'' for any time on Wed Jan 9, 1991.
13667 Unix time: ``663400800'' for 6:00am Wed Jan 9, 1991.
13669 Brackets suppression. An ``X'' at the front of the format
13670 causes the surrounding @w{@samp{< >}} delimiters to be omitted
13671 when formatting dates. Note that the brackets are still
13672 required for algebraic entry.
13675 If ``SS'' or ``BS'' (optional seconds) is preceded by a colon, the
13676 colon is also omitted if the seconds part is zero.
13678 If ``bb,'' ``bbb'' or ``bbbb'' or their upper-case equivalents
13679 appear in the format, then negative year numbers are displayed
13680 without a minus sign. Note that ``aa'' and ``bb'' are mutually
13681 exclusive. Some typical usages would be @samp{YYYY AABB};
13682 @samp{AAAYYYYBBB}; @samp{YYYYBBB}.
13684 The formats ``YY,'' ``YYYY,'' ``MM,'' ``DD,'' ``ddd,'' ``hh,'' ``HH,''
13685 ``mm,'' ``ss,'' and ``SS'' actually match any number of digits during
13686 reading unless several of these codes are strung together with no
13687 punctuation in between, in which case the input must have exactly as
13688 many digits as there are letters in the format.
13690 The ``j,'' ``J,'' and ``U'' formats do not make any time zone
13691 adjustment. They effectively use @samp{julian(x,0)} and
13692 @samp{unixtime(x,0)} to make the conversion; @pxref{Date Arithmetic}.
13694 @node Free-Form Dates, Standard Date Formats, Date Formatting Codes, Date Formats
13695 @subsubsection Free-Form Dates
13698 When reading a date form during algebraic entry, Calc falls back
13699 on the algorithm described here if the input does not exactly
13700 match the current date format. This algorithm generally
13701 ``does the right thing'' and you don't have to worry about it,
13702 but it is described here in full detail for the curious.
13704 Calc does not distinguish between upper- and lower-case letters
13705 while interpreting dates.
13707 First, the time portion, if present, is located somewhere in the
13708 text and then removed. The remaining text is then interpreted as
13711 A time is of the form @samp{hh:mm:ss}, possibly with the seconds
13712 part omitted and possibly with an AM/PM indicator added to indicate
13713 12-hour time. If the AM/PM is present, the minutes may also be
13714 omitted. The AM/PM part may be any of the words @samp{am},
13715 @samp{pm}, @samp{noon}, or @samp{midnight}; each of these may be
13716 abbreviated to one letter, and the alternate forms @samp{a.m.},
13717 @samp{p.m.}, and @samp{mid} are also understood. Obviously
13718 @samp{noon} and @samp{midnight} are allowed only on 12:00:00.
13719 The words @samp{noon}, @samp{mid}, and @samp{midnight} are also
13720 recognized with no number attached.
13722 If there is no AM/PM indicator, the time is interpreted in 24-hour
13725 To read the date portion, all words and numbers are isolated
13726 from the string; other characters are ignored. All words must
13727 be either month names or day-of-week names (the latter of which
13728 are ignored). Names can be written in full or as three-letter
13731 Large numbers, or numbers with @samp{+} or @samp{-} signs,
13732 are interpreted as years. If one of the other numbers is
13733 greater than 12, then that must be the day and the remaining
13734 number in the input is therefore the month. Otherwise, Calc
13735 assumes the month, day and year are in the same order that they
13736 appear in the current date format. If the year is omitted, the
13737 current year is taken from the system clock.
13739 If there are too many or too few numbers, or any unrecognizable
13740 words, then the input is rejected.
13742 If there are any large numbers (of five digits or more) other than
13743 the year, they are ignored on the assumption that they are something
13744 like Julian dates that were included along with the traditional
13745 date components when the date was formatted.
13747 One of the words @samp{ad}, @samp{a.d.}, @samp{bc}, or @samp{b.c.}
13748 may optionally be used; the latter two are equivalent to a
13749 minus sign on the year value.
13751 If you always enter a four-digit year, and use a name instead
13752 of a number for the month, there is no danger of ambiguity.
13754 @node Standard Date Formats, , Free-Form Dates, Date Formats
13755 @subsubsection Standard Date Formats
13758 There are actually ten standard date formats, numbered 0 through 9.
13759 Entering a blank line at the @kbd{d d} command's prompt gives
13760 you format number 1, Calc's usual format. You can enter any digit
13761 to select the other formats.
13763 To create your own standard date formats, give a numeric prefix
13764 argument from 0 to 9 to the @w{@kbd{d d}} command. The format you
13765 enter will be recorded as the new standard format of that
13766 number, as well as becoming the new current date format.
13767 You can save your formats permanently with the @w{@kbd{m m}}
13768 command (@pxref{Mode Settings}).
13772 @samp{N} (Numerical format)
13774 @samp{<H:mm:SSpp >Www Mmm D, YYYY} (American format)
13776 @samp{D Mmm YYYY<, h:mm:SS>} (European format)
13778 @samp{Www Mmm BD< hh:mm:ss> YYYY} (Unix written date format)
13780 @samp{M/D/Y< H:mm:SSpp>} (American slashed format)
13782 @samp{D.M.Y< h:mm:SS>} (European dotted format)
13784 @samp{M-D-Y< H:mm:SSpp>} (American dashed format)
13786 @samp{D-M-Y< h:mm:SS>} (European dashed format)
13788 @samp{j<, h:mm:ss>} (Julian day plus time)
13790 @samp{YYddd< hh:mm:ss>} (Year-day format)
13792 @samp{ZYYY-MM-DD Www< hh:mm>} (Org mode format)
13794 @samp{IYYY-Iww-w<Thh:mm:ss>} (ISO 8601 week numbering format)
13797 @node Truncating the Stack, Justification, Date Formats, Display Modes
13798 @subsection Truncating the Stack
13802 @pindex calc-truncate-stack
13803 @cindex Truncating the stack
13804 @cindex Narrowing the stack
13805 The @kbd{d t} (@code{calc-truncate-stack}) command moves the @samp{.}@:
13806 line that marks the top-of-stack up or down in the Calculator buffer.
13807 The number right above that line is considered to the be at the top of
13808 the stack. Any numbers below that line are ``hidden'' from all stack
13809 operations (although still visible to the user). This is similar to the
13810 Emacs ``narrowing'' feature, except that the values below the @samp{.}
13811 are @emph{visible}, just temporarily frozen. This feature allows you to
13812 keep several independent calculations running at once in different parts
13813 of the stack, or to apply a certain command to an element buried deep in
13816 Pressing @kbd{d t} by itself moves the @samp{.} to the line the cursor
13817 is on. Thus, this line and all those below it become hidden. To un-hide
13818 these lines, move down to the end of the buffer and press @w{@kbd{d t}}.
13819 With a positive numeric prefix argument @expr{n}, @kbd{d t} hides the
13820 bottom @expr{n} values in the buffer. With a negative argument, it hides
13821 all but the top @expr{n} values. With an argument of zero, it hides zero
13822 values, i.e., moves the @samp{.} all the way down to the bottom.
13825 @pindex calc-truncate-up
13827 @pindex calc-truncate-down
13828 The @kbd{d [} (@code{calc-truncate-up}) and @kbd{d ]}
13829 (@code{calc-truncate-down}) commands move the @samp{.} up or down one
13830 line at a time (or several lines with a prefix argument).
13832 @node Justification, Labels, Truncating the Stack, Display Modes
13833 @subsection Justification
13837 @pindex calc-left-justify
13839 @pindex calc-center-justify
13841 @pindex calc-right-justify
13842 Values on the stack are normally left-justified in the window. You can
13843 control this arrangement by typing @kbd{d <} (@code{calc-left-justify}),
13844 @kbd{d >} (@code{calc-right-justify}), or @kbd{d =}
13845 (@code{calc-center-justify}). For example, in Right-Justification mode,
13846 stack entries are displayed flush-right against the right edge of the
13849 If you change the width of the Calculator window you may have to type
13850 @kbd{d @key{SPC}} (@code{calc-refresh}) to re-align right-justified or centered
13853 Right-justification is especially useful together with fixed-point
13854 notation (see @code{d f}; @code{calc-fix-notation}). With these modes
13855 together, the decimal points on numbers will always line up.
13857 With a numeric prefix argument, the justification commands give you
13858 a little extra control over the display. The argument specifies the
13859 horizontal ``origin'' of a display line. It is also possible to
13860 specify a maximum line width using the @kbd{d b} command (@pxref{Normal
13861 Language Modes}). For reference, the precise rules for formatting and
13862 breaking lines are given below. Notice that the interaction between
13863 origin and line width is slightly different in each justification
13866 In Left-Justified mode, the line is indented by a number of spaces
13867 given by the origin (default zero). If the result is longer than the
13868 maximum line width, if given, or too wide to fit in the Calc window
13869 otherwise, then it is broken into lines which will fit; each broken
13870 line is indented to the origin.
13872 In Right-Justified mode, lines are shifted right so that the rightmost
13873 character is just before the origin, or just before the current
13874 window width if no origin was specified. If the line is too long
13875 for this, then it is broken; the current line width is used, if
13876 specified, or else the origin is used as a width if that is
13877 specified, or else the line is broken to fit in the window.
13879 In Centering mode, the origin is the column number of the center of
13880 each stack entry. If a line width is specified, lines will not be
13881 allowed to go past that width; Calc will either indent less or
13882 break the lines if necessary. If no origin is specified, half the
13883 line width or Calc window width is used.
13885 Note that, in each case, if line numbering is enabled the display
13886 is indented an additional four spaces to make room for the line
13887 number. The width of the line number is taken into account when
13888 positioning according to the current Calc window width, but not
13889 when positioning by explicit origins and widths. In the latter
13890 case, the display is formatted as specified, and then uniformly
13891 shifted over four spaces to fit the line numbers.
13893 @node Labels, , Justification, Display Modes
13898 @pindex calc-left-label
13899 The @kbd{d @{} (@code{calc-left-label}) command prompts for a string,
13900 then displays that string to the left of every stack entry. If the
13901 entries are left-justified (@pxref{Justification}), then they will
13902 appear immediately after the label (unless you specified an origin
13903 greater than the length of the label). If the entries are centered
13904 or right-justified, the label appears on the far left and does not
13905 affect the horizontal position of the stack entry.
13907 Give a blank string (with @kbd{d @{ @key{RET}}) to turn the label off.
13910 @pindex calc-right-label
13911 The @kbd{d @}} (@code{calc-right-label}) command similarly adds a
13912 label on the righthand side. It does not affect positioning of
13913 the stack entries unless they are right-justified. Also, if both
13914 a line width and an origin are given in Right-Justified mode, the
13915 stack entry is justified to the origin and the righthand label is
13916 justified to the line width.
13918 One application of labels would be to add equation numbers to
13919 formulas you are manipulating in Calc and then copying into a
13920 document (possibly using Embedded mode). The equations would
13921 typically be centered, and the equation numbers would be on the
13922 left or right as you prefer.
13924 @node Language Modes, Modes Variable, Display Modes, Mode Settings
13925 @section Language Modes
13928 The commands in this section change Calc to use a different notation for
13929 entry and display of formulas, corresponding to the conventions of some
13930 other common language such as Pascal or @LaTeX{}. Objects displayed on the
13931 stack or yanked from the Calculator to an editing buffer will be formatted
13932 in the current language; objects entered in algebraic entry or yanked from
13933 another buffer will be interpreted according to the current language.
13935 The current language has no effect on things written to or read from the
13936 trail buffer, nor does it affect numeric entry. Only algebraic entry is
13937 affected. You can make even algebraic entry ignore the current language
13938 and use the standard notation by giving a numeric prefix, e.g., @kbd{C-u '}.
13940 For example, suppose the formula @samp{2*a[1] + atan(a[2])} occurs in a C
13941 program; elsewhere in the program you need the derivatives of this formula
13942 with respect to @samp{a[1]} and @samp{a[2]}. First, type @kbd{d C}
13943 to switch to C notation. Now use @code{C-u C-x * g} to grab the formula
13944 into the Calculator, @kbd{a d a[1] @key{RET}} to differentiate with respect
13945 to the first variable, and @kbd{C-x * y} to yank the formula for the derivative
13946 back into your C program. Press @kbd{U} to undo the differentiation and
13947 repeat with @kbd{a d a[2] @key{RET}} for the other derivative.
13949 Without being switched into C mode first, Calc would have misinterpreted
13950 the brackets in @samp{a[1]} and @samp{a[2]}, would not have known that
13951 @code{atan} was equivalent to Calc's built-in @code{arctan} function,
13952 and would have written the formula back with notations (like implicit
13953 multiplication) which would not have been valid for a C program.
13955 As another example, suppose you are maintaining a C program and a @LaTeX{}
13956 document, each of which needs a copy of the same formula. You can grab the
13957 formula from the program in C mode, switch to @LaTeX{} mode, and yank the
13958 formula into the document in @LaTeX{} math-mode format.
13960 Language modes are selected by typing the letter @kbd{d} followed by a
13961 shifted letter key.
13964 * Normal Language Modes::
13965 * C FORTRAN Pascal::
13966 * TeX and LaTeX Language Modes::
13967 * Eqn Language Mode::
13968 * Yacas Language Mode::
13969 * Maxima Language Mode::
13970 * Giac Language Mode::
13971 * Mathematica Language Mode::
13972 * Maple Language Mode::
13977 @node Normal Language Modes, C FORTRAN Pascal, Language Modes, Language Modes
13978 @subsection Normal Language Modes
13982 @pindex calc-normal-language
13983 The @kbd{d N} (@code{calc-normal-language}) command selects the usual
13984 notation for Calc formulas, as described in the rest of this manual.
13985 Matrices are displayed in a multi-line tabular format, but all other
13986 objects are written in linear form, as they would be typed from the
13990 @pindex calc-flat-language
13991 @cindex Matrix display
13992 The @kbd{d O} (@code{calc-flat-language}) command selects a language
13993 identical with the normal one, except that matrices are written in
13994 one-line form along with everything else. In some applications this
13995 form may be more suitable for yanking data into other buffers.
13998 @pindex calc-line-breaking
13999 @cindex Line breaking
14000 @cindex Breaking up long lines
14001 Even in one-line mode, long formulas or vectors will still be split
14002 across multiple lines if they exceed the width of the Calculator window.
14003 The @kbd{d b} (@code{calc-line-breaking}) command turns this line-breaking
14004 feature on and off. (It works independently of the current language.)
14005 If you give a numeric prefix argument of five or greater to the @kbd{d b}
14006 command, that argument will specify the line width used when breaking
14010 @pindex calc-big-language
14011 The @kbd{d B} (@code{calc-big-language}) command selects a language
14012 which uses textual approximations to various mathematical notations,
14013 such as powers, quotients, and square roots:
14023 in place of @samp{sqrt((a+1)/b + c^2)}.
14025 Subscripts like @samp{a_i} are displayed as actual subscripts in Big
14026 mode. Double subscripts, @samp{a_i_j} (@samp{subscr(subscr(a, i), j)})
14027 are displayed as @samp{a} with subscripts separated by commas:
14028 @samp{i, j}. They must still be entered in the usual underscore
14031 One slight ambiguity of Big notation is that
14040 can represent either the negative rational number @expr{-3:4}, or the
14041 actual expression @samp{-(3/4)}; but the latter formula would normally
14042 never be displayed because it would immediately be evaluated to
14043 @expr{-3:4} or @expr{-0.75}, so this ambiguity is not a problem in
14046 Non-decimal numbers are displayed with subscripts. Thus there is no
14047 way to tell the difference between @samp{16#C2} and @samp{C2_16},
14048 though generally you will know which interpretation is correct.
14049 Logarithms @samp{log(x,b)} and @samp{log10(x)} also use subscripts
14052 In Big mode, stack entries often take up several lines. To aid
14053 readability, stack entries are separated by a blank line in this mode.
14054 You may find it useful to expand the Calc window's height using
14055 @kbd{C-x ^} (@code{enlarge-window}) or to make the Calc window the only
14056 one on the screen with @kbd{C-x 1} (@code{delete-other-windows}).
14058 Long lines are currently not rearranged to fit the window width in
14059 Big mode, so you may need to use the @kbd{<} and @kbd{>} keys
14060 to scroll across a wide formula. For really big formulas, you may
14061 even need to use @kbd{@{} and @kbd{@}} to scroll up and down.
14064 @pindex calc-unformatted-language
14065 The @kbd{d U} (@code{calc-unformatted-language}) command altogether disables
14066 the use of operator notation in formulas. In this mode, the formula
14067 shown above would be displayed:
14070 sqrt(add(div(add(a, 1), b), pow(c, 2)))
14073 These four modes differ only in display format, not in the format
14074 expected for algebraic entry. The standard Calc operators work in
14075 all four modes, and unformatted notation works in any language mode
14076 (except that Mathematica mode expects square brackets instead of
14079 @node C FORTRAN Pascal, TeX and LaTeX Language Modes, Normal Language Modes, Language Modes
14080 @subsection C, FORTRAN, and Pascal Modes
14084 @pindex calc-c-language
14086 The @kbd{d C} (@code{calc-c-language}) command selects the conventions
14087 of the C language for display and entry of formulas. This differs from
14088 the normal language mode in a variety of (mostly minor) ways. In
14089 particular, C language operators and operator precedences are used in
14090 place of Calc's usual ones. For example, @samp{a^b} means @samp{xor(a,b)}
14091 in C mode; a value raised to a power is written as a function call,
14094 In C mode, vectors and matrices use curly braces instead of brackets.
14095 Octal and hexadecimal values are written with leading @samp{0} or @samp{0x}
14096 rather than using the @samp{#} symbol. Array subscripting is
14097 translated into @code{subscr} calls, so that @samp{a[i]} in C
14098 mode is the same as @samp{a_i} in Normal mode. Assignments
14099 turn into the @code{assign} function, which Calc normally displays
14100 using the @samp{:=} symbol.
14102 The variables @code{pi} and @code{e} would be displayed @samp{pi}
14103 and @samp{e} in Normal mode, but in C mode they are displayed as
14104 @samp{M_PI} and @samp{M_E}, corresponding to the names of constants
14105 typically provided in the @file{<math.h>} header. Functions whose
14106 names are different in C are translated automatically for entry and
14107 display purposes. For example, entering @samp{asin(x)} will push the
14108 formula @samp{arcsin(x)} onto the stack; this formula will be displayed
14109 as @samp{asin(x)} as long as C mode is in effect.
14112 @pindex calc-pascal-language
14113 @cindex Pascal language
14114 The @kbd{d P} (@code{calc-pascal-language}) command selects Pascal
14115 conventions. Like C mode, Pascal mode interprets array brackets and uses
14116 a different table of operators. Hexadecimal numbers are entered and
14117 displayed with a preceding dollar sign. (Thus the regular meaning of
14118 @kbd{$2} during algebraic entry does not work in Pascal mode, though
14119 @kbd{$} (and @kbd{$$}, etc.) not followed by digits works the same as
14120 always.) No special provisions are made for other non-decimal numbers,
14121 vectors, and so on, since there is no universally accepted standard way
14122 of handling these in Pascal.
14125 @pindex calc-fortran-language
14126 @cindex FORTRAN language
14127 The @kbd{d F} (@code{calc-fortran-language}) command selects FORTRAN
14128 conventions. Various function names are transformed into FORTRAN
14129 equivalents. Vectors are written as @samp{/1, 2, 3/}, and may be
14130 entered this way or using square brackets. Since FORTRAN uses round
14131 parentheses for both function calls and array subscripts, Calc displays
14132 both in the same way; @samp{a(i)} is interpreted as a function call
14133 upon reading, and subscripts must be entered as @samp{subscr(a, i)}.
14134 If the variable @code{a} has been declared to have type
14135 @code{vector} or @code{matrix}, however, then @samp{a(i)} will be
14136 parsed as a subscript. (@xref{Declarations}.) Usually it doesn't
14137 matter, though; if you enter the subscript expression @samp{a(i)} and
14138 Calc interprets it as a function call, you'll never know the difference
14139 unless you switch to another language mode or replace @code{a} with an
14140 actual vector (or unless @code{a} happens to be the name of a built-in
14143 Underscores are allowed in variable and function names in all of these
14144 language modes. The underscore here is equivalent to the @samp{#} in
14145 Normal mode, or to hyphens in the underlying Emacs Lisp variable names.
14147 FORTRAN and Pascal modes normally do not adjust the case of letters in
14148 formulas. Most built-in Calc names use lower-case letters. If you use a
14149 positive numeric prefix argument with @kbd{d P} or @kbd{d F}, these
14150 modes will use upper-case letters exclusively for display, and will
14151 convert to lower-case on input. With a negative prefix, these modes
14152 convert to lower-case for display and input.
14154 @node TeX and LaTeX Language Modes, Eqn Language Mode, C FORTRAN Pascal, Language Modes
14155 @subsection @TeX{} and @LaTeX{} Language Modes
14159 @pindex calc-tex-language
14160 @cindex TeX language
14162 @pindex calc-latex-language
14163 @cindex LaTeX language
14164 The @kbd{d T} (@code{calc-tex-language}) command selects the conventions
14165 of ``math mode'' in Donald Knuth's @TeX{} typesetting language,
14166 and the @kbd{d L} (@code{calc-latex-language}) command selects the
14167 conventions of ``math mode'' in @LaTeX{}, a typesetting language that
14168 uses @TeX{} as its formatting engine. Calc's @LaTeX{} language mode can
14169 read any formula that the @TeX{} language mode can, although @LaTeX{}
14170 mode may display it differently.
14172 Formulas are entered and displayed in the appropriate notation;
14173 @texline @math{\sin(a/b)}
14174 @infoline @expr{sin(a/b)}
14175 will appear as @samp{\sin\left( @{a \over b@} \right)} in @TeX{} mode and
14176 @samp{\sin\left(\frac@{a@}@{b@}\right)} in @LaTeX{} mode.
14177 Math formulas are often enclosed by @samp{$ $} signs in @TeX{} and
14178 @LaTeX{}; these should be omitted when interfacing with Calc. To Calc,
14179 the @samp{$} sign has the same meaning it always does in algebraic
14180 formulas (a reference to an existing entry on the stack).
14182 Complex numbers are displayed as in @samp{3 + 4i}. Fractions and
14183 quotients are written using @code{\over} in @TeX{} mode (as in
14184 @code{@{a \over b@}}) and @code{\frac} in @LaTeX{} mode (as in
14185 @code{\frac@{a@}@{b@}}); binomial coefficients are written with
14186 @code{\choose} in @TeX{} mode (as in @code{@{a \choose b@}}) and
14187 @code{\binom} in @LaTeX{} mode (as in @code{\binom@{a@}@{b@}}).
14188 Interval forms are written with @code{\ldots}, and error forms are
14189 written with @code{\pm}. Absolute values are written as in
14190 @samp{|x + 1|}, and the floor and ceiling functions are written with
14191 @code{\lfloor}, @code{\rfloor}, etc. The words @code{\left} and
14192 @code{\right} are ignored when reading formulas in @TeX{} and @LaTeX{}
14193 modes. Both @code{inf} and @code{uinf} are written as @code{\infty};
14194 when read, @code{\infty} always translates to @code{inf}.
14196 Function calls are written the usual way, with the function name followed
14197 by the arguments in parentheses. However, functions for which @TeX{}
14198 and @LaTeX{} have special names (like @code{\sin}) will use curly braces
14199 instead of parentheses for very simple arguments. During input, curly
14200 braces and parentheses work equally well for grouping, but when the
14201 document is formatted the curly braces will be invisible. Thus the
14203 @texline @math{\sin{2 x}}
14204 @infoline @expr{sin 2x}
14206 @texline @math{\sin(2 + x)}.
14207 @infoline @expr{sin(2 + x)}.
14209 The @TeX{} specific unit names (@pxref{Predefined Units}) will not use
14210 the @samp{tex} prefix; the unit name for a @TeX{} point will be
14211 @samp{pt} instead of @samp{texpt}, for example.
14213 Function and variable names not treated specially by @TeX{} and @LaTeX{}
14214 are simply written out as-is, which will cause them to come out in
14215 italic letters in the printed document. If you invoke @kbd{d T} or
14216 @kbd{d L} with a positive numeric prefix argument, names of more than
14217 one character will instead be enclosed in a protective commands that
14218 will prevent them from being typeset in the math italics; they will be
14219 written @samp{\hbox@{@var{name}@}} in @TeX{} mode and
14220 @samp{\text@{@var{name}@}} in @LaTeX{} mode. The
14221 @samp{\hbox@{ @}} and @samp{\text@{ @}} notations are ignored during
14222 reading. If you use a negative prefix argument, such function names are
14223 written @samp{\@var{name}}, and function names that begin with @code{\} during
14224 reading have the @code{\} removed. (Note that in this mode, long
14225 variable names are still written with @code{\hbox} or @code{\text}.
14226 However, you can always make an actual variable name like @code{\bar} in
14229 During reading, text of the form @samp{\matrix@{ ...@: @}} is replaced
14230 by @samp{[ ...@: ]}. The same also applies to @code{\pmatrix} and
14231 @code{\bmatrix}. In @LaTeX{} mode this also applies to
14232 @samp{\begin@{matrix@} ... \end@{matrix@}},
14233 @samp{\begin@{bmatrix@} ... \end@{bmatrix@}},
14234 @samp{\begin@{pmatrix@} ... \end@{pmatrix@}}, as well as
14235 @samp{\begin@{smallmatrix@} ... \end@{smallmatrix@}}.
14236 The symbol @samp{&} is interpreted as a comma,
14237 and the symbols @samp{\cr} and @samp{\\} are interpreted as semicolons.
14238 During output, matrices are displayed in @samp{\matrix@{ a & b \\ c & d@}}
14239 format in @TeX{} mode and in
14240 @samp{\begin@{pmatrix@} a & b \\ c & d \end@{pmatrix@}} format in
14241 @LaTeX{} mode; you may need to edit this afterwards to change to your
14242 preferred matrix form. If you invoke @kbd{d T} or @kbd{d L} with an
14243 argument of 2 or -2, then matrices will be displayed in two-dimensional
14254 This may be convenient for isolated matrices, but could lead to
14255 expressions being displayed like
14258 \begin@{pmatrix@} \times x
14265 While this wouldn't bother Calc, it is incorrect @LaTeX{}.
14266 (Similarly for @TeX{}.)
14268 Accents like @code{\tilde} and @code{\bar} translate into function
14269 calls internally (@samp{tilde(x)}, @samp{bar(x)}). The @code{\underline}
14270 sequence is treated as an accent. The @code{\vec} accent corresponds
14271 to the function name @code{Vec}, because @code{vec} is the name of
14272 a built-in Calc function. The following table shows the accents
14273 in Calc, @TeX{}, @LaTeX{} and @dfn{eqn} (described in the next section):
14278 @let@calcindexershow=@calcindexernoshow @c Suppress marginal notes
14279 @let@calcindexersh=@calcindexernoshow
14388 acute \acute \acute
14392 breve \breve \breve
14394 check \check \check
14400 dotdot \ddot \ddot dotdot
14403 grave \grave \grave
14408 tilde \tilde \tilde tilde
14410 under \underline \underline under
14415 The @samp{=>} (evaluates-to) operator appears as a @code{\to} symbol:
14416 @samp{@{@var{a} \to @var{b}@}}. @TeX{} defines @code{\to} as an
14417 alias for @code{\rightarrow}. However, if the @samp{=>} is the
14418 top-level expression being formatted, a slightly different notation
14419 is used: @samp{\evalto @var{a} \to @var{b}}. The @code{\evalto}
14420 word is ignored by Calc's input routines, and is undefined in @TeX{}.
14421 You will typically want to include one of the following definitions
14422 at the top of a @TeX{} file that uses @code{\evalto}:
14426 \def\evalto#1\to@{@}
14429 The first definition formats evaluates-to operators in the usual
14430 way. The second causes only the @var{b} part to appear in the
14431 printed document; the @var{a} part and the arrow are hidden.
14432 Another definition you may wish to use is @samp{\let\to=\Rightarrow}
14433 which causes @code{\to} to appear more like Calc's @samp{=>} symbol.
14434 @xref{Evaluates-To Operator}, for a discussion of @code{evalto}.
14436 The complete set of @TeX{} control sequences that are ignored during
14440 \hbox \mbox \text \left \right
14441 \, \> \: \; \! \quad \qquad \hfil \hfill
14442 \displaystyle \textstyle \dsize \tsize
14443 \scriptstyle \scriptscriptstyle \ssize \ssize
14444 \rm \bf \it \sl \roman \bold \italic \slanted
14445 \cal \mit \Cal \Bbb \frak \goth
14449 Note that, because these symbols are ignored, reading a @TeX{} or
14450 @LaTeX{} formula into Calc and writing it back out may lose spacing and
14453 Also, the ``discretionary multiplication sign'' @samp{\*} is read
14454 the same as @samp{*}.
14457 The @TeX{} version of this manual includes some printed examples at the
14458 end of this section.
14461 Here are some examples of how various Calc formulas are formatted in @TeX{}:
14466 \sin\left( {a^2 \over b_i} \right)
14470 $$ \sin\left( a^2 \over b_i \right) $$
14476 [(3, 4), 3:4, 3 +/- 4, [3 .. inf)]
14477 [3 + 4i, @{3 \over 4@}, 3 \pm 4, [3 \ldots \infty)]
14481 $$ [3 + 4i, {3 \over 4}, 3 \pm 4, [ 3 \ldots \infty)] $$
14487 [abs(a), abs(a / b), floor(a), ceil(a / b)]
14488 [|a|, \left| a \over b \right|,
14489 \lfloor a \rfloor, \left\lceil a \over b \right\rceil]
14493 $$ [|a|, \left| a \over b \right|,
14494 \lfloor a \rfloor, \left\lceil a \over b \right\rceil] $$
14500 [sin(a), sin(2 a), sin(2 + a), sin(a / b)]
14501 [\sin@{a@}, \sin@{2 a@}, \sin(2 + a),
14502 \sin\left( @{a \over b@} \right)]
14506 $$ [\sin{a}, \sin{2 a}, \sin(2 + a), \sin\left( {a \over b} \right)] $$
14510 First with plain @kbd{d T}, then with @kbd{C-u d T}, then finally with
14511 @kbd{C-u - d T} (using the example definition
14512 @samp{\def\foo#1@{\tilde F(#1)@}}:
14516 [f(a), foo(bar), sin(pi)]
14517 [f(a), foo(bar), \sin{\pi}]
14518 [f(a), \hbox@{foo@}(\hbox@{bar@}), \sin@{\pi@}]
14519 [f(a), \foo@{\hbox@{bar@}@}, \sin@{\pi@}]
14523 $$ [f(a), foo(bar), \sin{\pi}] $$
14524 $$ [f(a), \hbox{foo}(\hbox{bar}), \sin{\pi}] $$
14525 $$ [f(a), \tilde F(\hbox{bar}), \sin{\pi}] $$
14529 First with @samp{\def\evalto@{@}}, then with @samp{\def\evalto#1\to@{@}}:
14534 \evalto 2 + 3 \to 5
14543 First with standard @code{\to}, then with @samp{\let\to\Rightarrow}:
14547 [2 + 3 => 5, a / 2 => (b + c) / 2]
14548 [@{2 + 3 \to 5@}, @{@{a \over 2@} \to @{b + c \over 2@}@}]
14552 $$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$
14553 {\let\to\Rightarrow
14554 $$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$}
14558 Matrices normally, then changing @code{\matrix} to @code{\pmatrix}:
14562 [ [ a / b, 0 ], [ 0, 2^(x + 1) ] ]
14563 \matrix@{ @{a \over b@} & 0 \\ 0 & 2^@{(x + 1)@} @}
14564 \pmatrix@{ @{a \over b@} & 0 \\ 0 & 2^@{(x + 1)@} @}
14568 $$ \matrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
14569 $$ \pmatrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
14574 @node Eqn Language Mode, Yacas Language Mode, TeX and LaTeX Language Modes, Language Modes
14575 @subsection Eqn Language Mode
14579 @pindex calc-eqn-language
14580 @dfn{Eqn} is another popular formatter for math formulas. It is
14581 designed for use with the TROFF text formatter, and comes standard
14582 with many versions of Unix. The @kbd{d E} (@code{calc-eqn-language})
14583 command selects @dfn{eqn} notation.
14585 The @dfn{eqn} language's main idiosyncrasy is that whitespace plays
14586 a significant part in the parsing of the language. For example,
14587 @samp{sqrt x+1 + y} treats @samp{x+1} as the argument of the
14588 @code{sqrt} operator. @dfn{Eqn} also understands more conventional
14589 grouping using curly braces: @samp{sqrt@{x+1@} + y}. Braces are
14590 required only when the argument contains spaces.
14592 In Calc's @dfn{eqn} mode, however, curly braces are required to
14593 delimit arguments of operators like @code{sqrt}. The first of the
14594 above examples would treat only the @samp{x} as the argument of
14595 @code{sqrt}, and in fact @samp{sin x+1} would be interpreted as
14596 @samp{sin * x + 1}, because @code{sin} is not a special operator
14597 in the @dfn{eqn} language. If you always surround the argument
14598 with curly braces, Calc will never misunderstand.
14600 Calc also understands parentheses as grouping characters. Another
14601 peculiarity of @dfn{eqn}'s syntax makes it advisable to separate
14602 words with spaces from any surrounding characters that aren't curly
14603 braces, so Calc writes @samp{sin ( x + y )} in @dfn{eqn} mode.
14604 (The spaces around @code{sin} are important to make @dfn{eqn}
14605 recognize that @code{sin} should be typeset in a roman font, and
14606 the spaces around @code{x} and @code{y} are a good idea just in
14607 case the @dfn{eqn} document has defined special meanings for these
14610 Powers and subscripts are written with the @code{sub} and @code{sup}
14611 operators, respectively. Note that the caret symbol @samp{^} is
14612 treated the same as a space in @dfn{eqn} mode, as is the @samp{~}
14613 symbol (these are used to introduce spaces of various widths into
14614 the typeset output of @dfn{eqn}).
14616 As in @LaTeX{} mode, Calc's formatter omits parentheses around the
14617 arguments of functions like @code{ln} and @code{sin} if they are
14618 ``simple-looking''; in this case Calc surrounds the argument with
14619 braces, separated by a @samp{~} from the function name: @samp{sin~@{x@}}.
14621 Font change codes (like @samp{roman @var{x}}) and positioning codes
14622 (like @samp{~} and @samp{down @var{n} @var{x}}) are ignored by the
14623 @dfn{eqn} reader. Also ignored are the words @code{left}, @code{right},
14624 @code{mark}, and @code{lineup}. Quotation marks in @dfn{eqn} mode input
14625 are treated the same as curly braces: @samp{sqrt "1+x"} is equivalent to
14626 @samp{sqrt @{1+x@}}; this is only an approximation to the true meaning
14627 of quotes in @dfn{eqn}, but it is good enough for most uses.
14629 Accent codes (@samp{@var{x} dot}) are handled by treating them as
14630 function calls (@samp{dot(@var{x})}) internally.
14631 @xref{TeX and LaTeX Language Modes}, for a table of these accent
14632 functions. The @code{prime} accent is treated specially if it occurs on
14633 a variable or function name: @samp{f prime prime @w{( x prime )}} is
14634 stored internally as @samp{f'@w{'}(x')}. For example, taking the
14635 derivative of @samp{f(2 x)} with @kbd{a d x} will produce @samp{2 f'(2
14636 x)}, which @dfn{eqn} mode will display as @samp{2 f prime ( 2 x )}.
14638 Assignments are written with the @samp{<-} (left-arrow) symbol,
14639 and @code{evalto} operators are written with @samp{->} or
14640 @samp{evalto ... ->} (@pxref{TeX and LaTeX Language Modes}, for a discussion
14641 of this). The regular Calc symbols @samp{:=} and @samp{=>} are also
14642 recognized for these operators during reading.
14644 Vectors in @dfn{eqn} mode use regular Calc square brackets, but
14645 matrices are formatted as @samp{matrix @{ ccol @{ a above b @} ... @}}.
14646 The words @code{lcol} and @code{rcol} are recognized as synonyms
14647 for @code{ccol} during input, and are generated instead of @code{ccol}
14648 if the matrix justification mode so specifies.
14650 @node Yacas Language Mode, Maxima Language Mode, Eqn Language Mode, Language Modes
14651 @subsection Yacas Language Mode
14655 @pindex calc-yacas-language
14656 @cindex Yacas language
14657 The @kbd{d Y} (@code{calc-yacas-language}) command selects the
14658 conventions of Yacas, a free computer algebra system. While the
14659 operators and functions in Yacas are similar to those of Calc, the names
14660 of built-in functions in Yacas are capitalized. The Calc formula
14661 @samp{sin(2 x)}, for example, is entered and displayed @samp{Sin(2 x)}
14662 in Yacas mode, and `@samp{arcsin(x^2)} is @samp{ArcSin(x^2)} in Yacas
14663 mode. Complex numbers are written are written @samp{3 + 4 I}.
14664 The standard special constants are written @code{Pi}, @code{E},
14665 @code{I}, @code{GoldenRatio} and @code{Gamma}. @code{Infinity}
14666 represents both @code{inf} and @code{uinf}, and @code{Undefined}
14667 represents @code{nan}.
14669 Certain operators on functions, such as @code{D} for differentiation
14670 and @code{Integrate} for integration, take a prefix form in Yacas. For
14671 example, the derivative of @w{@samp{e^x sin(x)}} can be computed with
14672 @w{@samp{D(x) Exp(x)*Sin(x)}}.
14674 Other notable differences between Yacas and standard Calc expressions
14675 are that vectors and matrices use curly braces in Yacas, and subscripts
14676 use square brackets. If, for example, @samp{A} represents the list
14677 @samp{@{a,2,c,4@}}, then @samp{A[3]} would equal @samp{c}.
14680 @node Maxima Language Mode, Giac Language Mode, Yacas Language Mode, Language Modes
14681 @subsection Maxima Language Mode
14685 @pindex calc-maxima-language
14686 @cindex Maxima language
14687 The @kbd{d X} (@code{calc-maxima-language}) command selects the
14688 conventions of Maxima, another free computer algebra system. The
14689 function names in Maxima are similar, but not always identical, to Calc.
14690 For example, instead of @samp{arcsin(x)}, Maxima will use
14691 @samp{asin(x)}. Complex numbers are written @samp{3 + 4 %i}. The
14692 standard special constants are written @code{%pi}, @code{%e},
14693 @code{%i}, @code{%phi} and @code{%gamma}. In Maxima, @code{inf} means
14694 the same as in Calc, but @code{infinity} represents Calc's @code{uinf}.
14696 Underscores as well as percent signs are allowed in function and
14697 variable names in Maxima mode. The underscore again is equivalent to
14698 the @samp{#} in Normal mode, and the percent sign is equivalent to
14701 Maxima uses square brackets for lists and vectors, and matrices are
14702 written as calls to the function @code{matrix}, given the row vectors of
14703 the matrix as arguments. Square brackets are also used as subscripts.
14705 @node Giac Language Mode, Mathematica Language Mode, Maxima Language Mode, Language Modes
14706 @subsection Giac Language Mode
14710 @pindex calc-giac-language
14711 @cindex Giac language
14712 The @kbd{d A} (@code{calc-giac-language}) command selects the
14713 conventions of Giac, another free computer algebra system. The function
14714 names in Giac are similar to Maxima. Complex numbers are written
14715 @samp{3 + 4 i}. The standard special constants in Giac are the same as
14716 in Calc, except that @code{infinity} represents both Calc's @code{inf}
14719 Underscores are allowed in function and variable names in Giac mode.
14720 Brackets are used for subscripts. In Giac, indexing of lists begins at
14721 0, instead of 1 as in Calc. So if @samp{A} represents the list
14722 @samp{[a,2,c,4]}, then @samp{A[2]} would equal @samp{c}. In general,
14723 @samp{A[n]} in Giac mode corresponds to @samp{A_(n+1)} in Normal mode.
14725 The Giac interval notation @samp{2 .. 3} has no surrounding brackets;
14726 Calc reads @samp{2 .. 3} as the closed interval @samp{[2 .. 3]} and
14727 writes any kind of interval as @samp{2 .. 3}. This means you cannot see
14728 the difference between an open and a closed interval while in Giac mode.
14730 @node Mathematica Language Mode, Maple Language Mode, Giac Language Mode, Language Modes
14731 @subsection Mathematica Language Mode
14735 @pindex calc-mathematica-language
14736 @cindex Mathematica language
14737 The @kbd{d M} (@code{calc-mathematica-language}) command selects the
14738 conventions of Mathematica. Notable differences in Mathematica mode
14739 are that the names of built-in functions are capitalized, and function
14740 calls use square brackets instead of parentheses. Thus the Calc
14741 formula @samp{sin(2 x)} is entered and displayed @w{@samp{Sin[2 x]}} in
14744 Vectors and matrices use curly braces in Mathematica. Complex numbers
14745 are written @samp{3 + 4 I}. The standard special constants in Calc are
14746 written @code{Pi}, @code{E}, @code{I}, @code{GoldenRatio}, @code{EulerGamma},
14747 @code{Infinity}, @code{ComplexInfinity}, and @code{Indeterminate} in
14749 Non-decimal numbers are written, e.g., @samp{16^^7fff}. Floating-point
14750 numbers in scientific notation are written @samp{1.23*10.^3}.
14751 Subscripts use double square brackets: @samp{a[[i]]}.
14753 @node Maple Language Mode, Compositions, Mathematica Language Mode, Language Modes
14754 @subsection Maple Language Mode
14758 @pindex calc-maple-language
14759 @cindex Maple language
14760 The @kbd{d W} (@code{calc-maple-language}) command selects the
14761 conventions of Maple.
14763 Maple's language is much like C@. Underscores are allowed in symbol
14764 names; square brackets are used for subscripts; explicit @samp{*}s for
14765 multiplications are required. Use either @samp{^} or @samp{**} to
14768 Maple uses square brackets for lists and curly braces for sets. Calc
14769 interprets both notations as vectors, and displays vectors with square
14770 brackets. This means Maple sets will be converted to lists when they
14771 pass through Calc. As a special case, matrices are written as calls
14772 to the function @code{matrix}, given a list of lists as the argument,
14773 and can be read in this form or with all-capitals @code{MATRIX}.
14775 The Maple interval notation @samp{2 .. 3} is like Giac's interval
14776 notation, and is handled the same by Calc.
14778 Maple writes complex numbers as @samp{3 + 4*I}. Its special constants
14779 are @code{Pi}, @code{E}, @code{I}, and @code{infinity} (all three of
14780 @code{inf}, @code{uinf}, and @code{nan} display as @code{infinity}).
14781 Floating-point numbers are written @samp{1.23*10.^3}.
14783 Among things not currently handled by Calc's Maple mode are the
14784 various quote symbols, procedures and functional operators, and
14785 inert (@samp{&}) operators.
14787 @node Compositions, Syntax Tables, Maple Language Mode, Language Modes
14788 @subsection Compositions
14791 @cindex Compositions
14792 There are several @dfn{composition functions} which allow you to get
14793 displays in a variety of formats similar to those in Big language
14794 mode. Most of these functions do not evaluate to anything; they are
14795 placeholders which are left in symbolic form by Calc's evaluator but
14796 are recognized by Calc's display formatting routines.
14798 Two of these, @code{string} and @code{bstring}, are described elsewhere.
14799 @xref{Strings}. For example, @samp{string("ABC")} is displayed as
14800 @samp{ABC}. When viewed on the stack it will be indistinguishable from
14801 the variable @code{ABC}, but internally it will be stored as
14802 @samp{string([65, 66, 67])} and can still be manipulated this way; for
14803 example, the selection and vector commands @kbd{j 1 v v j u} would
14804 select the vector portion of this object and reverse the elements, then
14805 deselect to reveal a string whose characters had been reversed.
14807 The composition functions do the same thing in all language modes
14808 (although their components will of course be formatted in the current
14809 language mode). The one exception is Unformatted mode (@kbd{d U}),
14810 which does not give the composition functions any special treatment.
14811 The functions are discussed here because of their relationship to
14812 the language modes.
14815 * Composition Basics::
14816 * Horizontal Compositions::
14817 * Vertical Compositions::
14818 * Other Compositions::
14819 * Information about Compositions::
14820 * User-Defined Compositions::
14823 @node Composition Basics, Horizontal Compositions, Compositions, Compositions
14824 @subsubsection Composition Basics
14827 Compositions are generally formed by stacking formulas together
14828 horizontally or vertically in various ways. Those formulas are
14829 themselves compositions. @TeX{} users will find this analogous
14830 to @TeX{}'s ``boxes.'' Each multi-line composition has a
14831 @dfn{baseline}; horizontal compositions use the baselines to
14832 decide how formulas should be positioned relative to one another.
14833 For example, in the Big mode formula
14845 the second term of the sum is four lines tall and has line three as
14846 its baseline. Thus when the term is combined with 17, line three
14847 is placed on the same level as the baseline of 17.
14853 Another important composition concept is @dfn{precedence}. This is
14854 an integer that represents the binding strength of various operators.
14855 For example, @samp{*} has higher precedence (195) than @samp{+} (180),
14856 which means that @samp{(a * b) + c} will be formatted without the
14857 parentheses, but @samp{a * (b + c)} will keep the parentheses.
14859 The operator table used by normal and Big language modes has the
14860 following precedences:
14863 _ 1200 @r{(subscripts)}
14864 % 1100 @r{(as in n}%@r{)}
14865 ! 1000 @r{(as in }!@r{n)}
14868 !! 210 @r{(as in n}!!@r{)}
14869 ! 210 @r{(as in n}!@r{)}
14871 - 197 @r{(as in }-@r{n)}
14872 * 195 @r{(or implicit multiplication)}
14874 + - 180 @r{(as in a}+@r{b)}
14876 < = 160 @r{(and other relations)}
14888 The general rule is that if an operator with precedence @expr{n}
14889 occurs as an argument to an operator with precedence @expr{m}, then
14890 the argument is enclosed in parentheses if @expr{n < m}. Top-level
14891 expressions and expressions which are function arguments, vector
14892 components, etc., are formatted with precedence zero (so that they
14893 normally never get additional parentheses).
14895 For binary left-associative operators like @samp{+}, the righthand
14896 argument is actually formatted with one-higher precedence than shown
14897 in the table. This makes sure @samp{(a + b) + c} omits the parentheses,
14898 but the unnatural form @samp{a + (b + c)} keeps its parentheses.
14899 Right-associative operators like @samp{^} format the lefthand argument
14900 with one-higher precedence.
14906 The @code{cprec} function formats an expression with an arbitrary
14907 precedence. For example, @samp{cprec(abc, 185)} will combine into
14908 sums and products as follows: @samp{7 + abc}, @samp{7 (abc)} (because
14909 this @code{cprec} form has higher precedence than addition, but lower
14910 precedence than multiplication).
14916 A final composition issue is @dfn{line breaking}. Calc uses two
14917 different strategies for ``flat'' and ``non-flat'' compositions.
14918 A non-flat composition is anything that appears on multiple lines
14919 (not counting line breaking). Examples would be matrices and Big
14920 mode powers and quotients. Non-flat compositions are displayed
14921 exactly as specified. If they come out wider than the current
14922 window, you must use horizontal scrolling (@kbd{<} and @kbd{>}) to
14925 Flat compositions, on the other hand, will be broken across several
14926 lines if they are too wide to fit the window. Certain points in a
14927 composition are noted internally as @dfn{break points}. Calc's
14928 general strategy is to fill each line as much as possible, then to
14929 move down to the next line starting at the first break point that
14930 didn't fit. However, the line breaker understands the hierarchical
14931 structure of formulas. It will not break an ``inner'' formula if
14932 it can use an earlier break point from an ``outer'' formula instead.
14933 For example, a vector of sums might be formatted as:
14937 [ a + b + c, d + e + f,
14938 g + h + i, j + k + l, m ]
14943 If the @samp{m} can fit, then so, it seems, could the @samp{g}.
14944 But Calc prefers to break at the comma since the comma is part
14945 of a ``more outer'' formula. Calc would break at a plus sign
14946 only if it had to, say, if the very first sum in the vector had
14947 itself been too large to fit.
14949 Of the composition functions described below, only @code{choriz}
14950 generates break points. The @code{bstring} function (@pxref{Strings})
14951 also generates breakable items: A break point is added after every
14952 space (or group of spaces) except for spaces at the very beginning or
14955 Composition functions themselves count as levels in the formula
14956 hierarchy, so a @code{choriz} that is a component of a larger
14957 @code{choriz} will be less likely to be broken. As a special case,
14958 if a @code{bstring} occurs as a component of a @code{choriz} or
14959 @code{choriz}-like object (such as a vector or a list of arguments
14960 in a function call), then the break points in that @code{bstring}
14961 will be on the same level as the break points of the surrounding
14964 @node Horizontal Compositions, Vertical Compositions, Composition Basics, Compositions
14965 @subsubsection Horizontal Compositions
14972 The @code{choriz} function takes a vector of objects and composes
14973 them horizontally. For example, @samp{choriz([17, a b/c, d])} formats
14974 as @w{@samp{17a b / cd}} in Normal language mode, or as
14985 in Big language mode. This is actually one case of the general
14986 function @samp{choriz(@var{vec}, @var{sep}, @var{prec})}, where
14987 either or both of @var{sep} and @var{prec} may be omitted.
14988 @var{Prec} gives the @dfn{precedence} to use when formatting
14989 each of the components of @var{vec}. The default precedence is
14990 the precedence from the surrounding environment.
14992 @var{Sep} is a string (i.e., a vector of character codes as might
14993 be entered with @code{" "} notation) which should separate components
14994 of the composition. Also, if @var{sep} is given, the line breaker
14995 will allow lines to be broken after each occurrence of @var{sep}.
14996 If @var{sep} is omitted, the composition will not be breakable
14997 (unless any of its component compositions are breakable).
14999 For example, @samp{2 choriz([a, b c, d = e], " + ", 180)} is
15000 formatted as @samp{2 a + b c + (d = e)}. To get the @code{choriz}
15001 to have precedence 180 ``outwards'' as well as ``inwards,''
15002 enclose it in a @code{cprec} form: @samp{2 cprec(choriz(...), 180)}
15003 formats as @samp{2 (a + b c + (d = e))}.
15005 The baseline of a horizontal composition is the same as the
15006 baselines of the component compositions, which are all aligned.
15008 @node Vertical Compositions, Other Compositions, Horizontal Compositions, Compositions
15009 @subsubsection Vertical Compositions
15016 The @code{cvert} function makes a vertical composition. Each
15017 component of the vector is centered in a column. The baseline of
15018 the result is by default the top line of the resulting composition.
15019 For example, @samp{f(cvert([a, bb, ccc]), cvert([a^2 + 1, b^2]))}
15020 formats in Big mode as
15035 There are several special composition functions that work only as
15036 components of a vertical composition. The @code{cbase} function
15037 controls the baseline of the vertical composition; the baseline
15038 will be the same as the baseline of whatever component is enclosed
15039 in @code{cbase}. Thus @samp{f(cvert([a, cbase(bb), ccc]),
15040 cvert([a^2 + 1, cbase(b^2)]))} displays as
15060 There are also @code{ctbase} and @code{cbbase} functions which
15061 make the baseline of the vertical composition equal to the top
15062 or bottom line (rather than the baseline) of that component.
15063 Thus @samp{cvert([cbase(a / b)]) + cvert([ctbase(a / b)]) +
15064 cvert([cbbase(a / b)])} gives
15076 There should be only one @code{cbase}, @code{ctbase}, or @code{cbbase}
15077 function in a given vertical composition. These functions can also
15078 be written with no arguments: @samp{ctbase()} is a zero-height object
15079 which means the baseline is the top line of the following item, and
15080 @samp{cbbase()} means the baseline is the bottom line of the preceding
15087 The @code{crule} function builds a ``rule,'' or horizontal line,
15088 across a vertical composition. By itself @samp{crule()} uses @samp{-}
15089 characters to build the rule. You can specify any other character,
15090 e.g., @samp{crule("=")}. The argument must be a character code or
15091 vector of exactly one character code. It is repeated to match the
15092 width of the widest item in the stack. For example, a quotient
15093 with a thick line is @samp{cvert([a + 1, cbase(crule("=")), b^2])}:
15112 Finally, the functions @code{clvert} and @code{crvert} act exactly
15113 like @code{cvert} except that the items are left- or right-justified
15114 in the stack. Thus @samp{clvert([a, bb, ccc]) + crvert([a, bb, ccc])}
15125 Like @code{choriz}, the vertical compositions accept a second argument
15126 which gives the precedence to use when formatting the components.
15127 Vertical compositions do not support separator strings.
15129 @node Other Compositions, Information about Compositions, Vertical Compositions, Compositions
15130 @subsubsection Other Compositions
15137 The @code{csup} function builds a superscripted expression. For
15138 example, @samp{csup(a, b)} looks the same as @samp{a^b} does in Big
15139 language mode. This is essentially a horizontal composition of
15140 @samp{a} and @samp{b}, where @samp{b} is shifted up so that its
15141 bottom line is one above the baseline.
15147 Likewise, the @code{csub} function builds a subscripted expression.
15148 This shifts @samp{b} down so that its top line is one below the
15149 bottom line of @samp{a} (note that this is not quite analogous to
15150 @code{csup}). Other arrangements can be obtained by using
15151 @code{choriz} and @code{cvert} directly.
15157 The @code{cflat} function formats its argument in ``flat'' mode,
15158 as obtained by @samp{d O}, if the current language mode is normal
15159 or Big. It has no effect in other language modes. For example,
15160 @samp{a^(b/c)} is formatted by Big mode like @samp{csup(a, cflat(b/c))}
15161 to improve its readability.
15167 The @code{cspace} function creates horizontal space. For example,
15168 @samp{cspace(4)} is effectively the same as @samp{string(" ")}.
15169 A second string (i.e., vector of characters) argument is repeated
15170 instead of the space character. For example, @samp{cspace(4, "ab")}
15171 looks like @samp{abababab}. If the second argument is not a string,
15172 it is formatted in the normal way and then several copies of that
15173 are composed together: @samp{cspace(4, a^2)} yields
15183 If the number argument is zero, this is a zero-width object.
15189 The @code{cvspace} function creates vertical space, or a vertical
15190 stack of copies of a certain string or formatted object. The
15191 baseline is the center line of the resulting stack. A numerical
15192 argument of zero will produce an object which contributes zero
15193 height if used in a vertical composition.
15203 There are also @code{ctspace} and @code{cbspace} functions which
15204 create vertical space with the baseline the same as the baseline
15205 of the top or bottom copy, respectively, of the second argument.
15206 Thus @samp{cvspace(2, a/b) + ctspace(2, a/b) + cbspace(2, a/b)}
15223 @node Information about Compositions, User-Defined Compositions, Other Compositions, Compositions
15224 @subsubsection Information about Compositions
15227 The functions in this section are actual functions; they compose their
15228 arguments according to the current language and other display modes,
15229 then return a certain measurement of the composition as an integer.
15235 The @code{cwidth} function measures the width, in characters, of a
15236 composition. For example, @samp{cwidth(a + b)} is 5, and
15237 @samp{cwidth(a / b)} is 5 in Normal mode, 1 in Big mode, and 11 in
15238 @TeX{} mode (for @samp{@{a \over b@}}). The argument may involve
15239 the composition functions described in this section.
15245 The @code{cheight} function measures the height of a composition.
15246 This is the total number of lines in the argument's printed form.
15256 The functions @code{cascent} and @code{cdescent} measure the amount
15257 of the height that is above (and including) the baseline, or below
15258 the baseline, respectively. Thus @samp{cascent(@var{x}) + cdescent(@var{x})}
15259 always equals @samp{cheight(@var{x})}. For a one-line formula like
15260 @samp{a + b}, @code{cascent} returns 1 and @code{cdescent} returns 0.
15261 For @samp{a / b} in Big mode, @code{cascent} returns 2 and @code{cdescent}
15262 returns 1. The only formula for which @code{cascent} will return zero
15263 is @samp{cvspace(0)} or equivalents.
15265 @node User-Defined Compositions, , Information about Compositions, Compositions
15266 @subsubsection User-Defined Compositions
15270 @pindex calc-user-define-composition
15271 The @kbd{Z C} (@code{calc-user-define-composition}) command lets you
15272 define the display format for any algebraic function. You provide a
15273 formula containing a certain number of argument variables on the stack.
15274 Any time Calc formats a call to the specified function in the current
15275 language mode and with that number of arguments, Calc effectively
15276 replaces the function call with that formula with the arguments
15279 Calc builds the default argument list by sorting all the variable names
15280 that appear in the formula into alphabetical order. You can edit this
15281 argument list before pressing @key{RET} if you wish. Any variables in
15282 the formula that do not appear in the argument list will be displayed
15283 literally; any arguments that do not appear in the formula will not
15284 affect the display at all.
15286 You can define formats for built-in functions, for functions you have
15287 defined with @kbd{Z F} (@pxref{Algebraic Definitions}), or for functions
15288 which have no definitions but are being used as purely syntactic objects.
15289 You can define different formats for each language mode, and for each
15290 number of arguments, using a succession of @kbd{Z C} commands. When
15291 Calc formats a function call, it first searches for a format defined
15292 for the current language mode (and number of arguments); if there is
15293 none, it uses the format defined for the Normal language mode. If
15294 neither format exists, Calc uses its built-in standard format for that
15295 function (usually just @samp{@var{func}(@var{args})}).
15297 If you execute @kbd{Z C} with the number 0 on the stack instead of a
15298 formula, any defined formats for the function in the current language
15299 mode will be removed. The function will revert to its standard format.
15301 For example, the default format for the binomial coefficient function
15302 @samp{choose(n, m)} in the Big language mode is
15313 You might prefer the notation,
15323 To define this notation, first make sure you are in Big mode,
15324 then put the formula
15327 choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])])
15331 on the stack and type @kbd{Z C}. Answer the first prompt with
15332 @code{choose}. The second prompt will be the default argument list
15333 of @samp{(C m n)}. Edit this list to be @samp{(n m)} and press
15334 @key{RET}. Now, try it out: For example, turn simplification
15335 off with @kbd{m O} and enter @samp{choose(a,b) + choose(7,3)}
15336 as an algebraic entry.
15345 As another example, let's define the usual notation for Stirling
15346 numbers of the first kind, @samp{stir1(n, m)}. This is just like
15347 the regular format for binomial coefficients but with square brackets
15348 instead of parentheses.
15351 choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")])
15354 Now type @kbd{Z C stir1 @key{RET}}, edit the argument list to
15355 @samp{(n m)}, and type @key{RET}.
15357 The formula provided to @kbd{Z C} usually will involve composition
15358 functions, but it doesn't have to. Putting the formula @samp{a + b + c}
15359 onto the stack and typing @kbd{Z C foo @key{RET} @key{RET}} would define
15360 the function @samp{foo(x,y,z)} to display like @samp{x + y + z}.
15361 This ``sum'' will act exactly like a real sum for all formatting
15362 purposes (it will be parenthesized the same, and so on). However
15363 it will be computationally unrelated to a sum. For example, the
15364 formula @samp{2 * foo(1, 2, 3)} will display as @samp{2 (1 + 2 + 3)}.
15365 Operator precedences have caused the ``sum'' to be written in
15366 parentheses, but the arguments have not actually been summed.
15367 (Generally a display format like this would be undesirable, since
15368 it can easily be confused with a real sum.)
15370 The special function @code{eval} can be used inside a @kbd{Z C}
15371 composition formula to cause all or part of the formula to be
15372 evaluated at display time. For example, if the formula is
15373 @samp{a + eval(b + c)}, then @samp{foo(1, 2, 3)} will be displayed
15374 as @samp{1 + 5}. Evaluation will use the default simplifications,
15375 regardless of the current simplification mode. There are also
15376 @code{evalsimp} and @code{evalextsimp} which simplify as if by
15377 @kbd{a s} and @kbd{a e} (respectively). Note that these ``functions''
15378 operate only in the context of composition formulas (and also in
15379 rewrite rules, where they serve a similar purpose; @pxref{Rewrite
15380 Rules}). On the stack, a call to @code{eval} will be left in
15383 It is not a good idea to use @code{eval} except as a last resort.
15384 It can cause the display of formulas to be extremely slow. For
15385 example, while @samp{eval(a + b)} might seem quite fast and simple,
15386 there are several situations where it could be slow. For example,
15387 @samp{a} and/or @samp{b} could be polar complex numbers, in which
15388 case doing the sum requires trigonometry. Or, @samp{a} could be
15389 the factorial @samp{fact(100)} which is unevaluated because you
15390 have typed @kbd{m O}; @code{eval} will evaluate it anyway to
15391 produce a large, unwieldy integer.
15393 You can save your display formats permanently using the @kbd{Z P}
15394 command (@pxref{Creating User Keys}).
15396 @node Syntax Tables, , Compositions, Language Modes
15397 @subsection Syntax Tables
15400 @cindex Syntax tables
15401 @cindex Parsing formulas, customized
15402 Syntax tables do for input what compositions do for output: They
15403 allow you to teach custom notations to Calc's formula parser.
15404 Calc keeps a separate syntax table for each language mode.
15406 (Note that the Calc ``syntax tables'' discussed here are completely
15407 unrelated to the syntax tables described in the Emacs manual.)
15410 @pindex calc-edit-user-syntax
15411 The @kbd{Z S} (@code{calc-edit-user-syntax}) command edits the
15412 syntax table for the current language mode. If you want your
15413 syntax to work in any language, define it in the Normal language
15414 mode. Type @kbd{C-c C-c} to finish editing the syntax table, or
15415 @kbd{C-x k} to cancel the edit. The @kbd{m m} command saves all
15416 the syntax tables along with the other mode settings;
15417 @pxref{General Mode Commands}.
15420 * Syntax Table Basics::
15421 * Precedence in Syntax Tables::
15422 * Advanced Syntax Patterns::
15423 * Conditional Syntax Rules::
15426 @node Syntax Table Basics, Precedence in Syntax Tables, Syntax Tables, Syntax Tables
15427 @subsubsection Syntax Table Basics
15430 @dfn{Parsing} is the process of converting a raw string of characters,
15431 such as you would type in during algebraic entry, into a Calc formula.
15432 Calc's parser works in two stages. First, the input is broken down
15433 into @dfn{tokens}, such as words, numbers, and punctuation symbols
15434 like @samp{+}, @samp{:=}, and @samp{+/-}. Space between tokens is
15435 ignored (except when it serves to separate adjacent words). Next,
15436 the parser matches this string of tokens against various built-in
15437 syntactic patterns, such as ``an expression followed by @samp{+}
15438 followed by another expression'' or ``a name followed by @samp{(},
15439 zero or more expressions separated by commas, and @samp{)}.''
15441 A @dfn{syntax table} is a list of user-defined @dfn{syntax rules},
15442 which allow you to specify new patterns to define your own
15443 favorite input notations. Calc's parser always checks the syntax
15444 table for the current language mode, then the table for the Normal
15445 language mode, before it uses its built-in rules to parse an
15446 algebraic formula you have entered. Each syntax rule should go on
15447 its own line; it consists of a @dfn{pattern}, a @samp{:=} symbol,
15448 and a Calc formula with an optional @dfn{condition}. (Syntax rules
15449 resemble algebraic rewrite rules, but the notation for patterns is
15450 completely different.)
15452 A syntax pattern is a list of tokens, separated by spaces.
15453 Except for a few special symbols, tokens in syntax patterns are
15454 matched literally, from left to right. For example, the rule,
15461 would cause Calc to parse the formula @samp{4+foo()*5} as if it
15462 were @samp{4+(2+3)*5}. Notice that the parentheses were written
15463 as two separate tokens in the rule. As a result, the rule works
15464 for both @samp{foo()} and @w{@samp{foo ( )}}. If we had written
15465 the rule as @samp{foo () := 2+3}, then Calc would treat @samp{()}
15466 as a single, indivisible token, so that @w{@samp{foo( )}} would
15467 not be recognized by the rule. (It would be parsed as a regular
15468 zero-argument function call instead.) In fact, this rule would
15469 also make trouble for the rest of Calc's parser: An unrelated
15470 formula like @samp{bar()} would now be tokenized into @samp{bar ()}
15471 instead of @samp{bar ( )}, so that the standard parser for function
15472 calls would no longer recognize it!
15474 While it is possible to make a token with a mixture of letters
15475 and punctuation symbols, this is not recommended. It is better to
15476 break it into several tokens, as we did with @samp{foo()} above.
15478 The symbol @samp{#} in a syntax pattern matches any Calc expression.
15479 On the righthand side, the things that matched the @samp{#}s can
15480 be referred to as @samp{#1}, @samp{#2}, and so on (where @samp{#1}
15481 matches the leftmost @samp{#} in the pattern). For example, these
15482 rules match a user-defined function, prefix operator, infix operator,
15483 and postfix operator, respectively:
15486 foo ( # ) := myfunc(#1)
15487 foo # := myprefix(#1)
15488 # foo # := myinfix(#1,#2)
15489 # foo := mypostfix(#1)
15492 Thus @samp{foo(3)} will parse as @samp{myfunc(3)}, and @samp{2+3 foo}
15493 will parse as @samp{mypostfix(2+3)}.
15495 It is important to write the first two rules in the order shown,
15496 because Calc tries rules in order from first to last. If the
15497 pattern @samp{foo #} came first, it would match anything that could
15498 match the @samp{foo ( # )} rule, since an expression in parentheses
15499 is itself a valid expression. Thus the @w{@samp{foo ( # )}} rule would
15500 never get to match anything. Likewise, the last two rules must be
15501 written in the order shown or else @samp{3 foo 4} will be parsed as
15502 @samp{mypostfix(3) * 4}. (Of course, the best way to avoid these
15503 ambiguities is not to use the same symbol in more than one way at
15504 the same time! In case you're not convinced, try the following
15505 exercise: How will the above rules parse the input @samp{foo(3,4)},
15506 if at all? Work it out for yourself, then try it in Calc and see.)
15508 Calc is quite flexible about what sorts of patterns are allowed.
15509 The only rule is that every pattern must begin with a literal
15510 token (like @samp{foo} in the first two patterns above), or with
15511 a @samp{#} followed by a literal token (as in the last two
15512 patterns). After that, any mixture is allowed, although putting
15513 two @samp{#}s in a row will not be very useful since two
15514 expressions with nothing between them will be parsed as one
15515 expression that uses implicit multiplication.
15517 As a more practical example, Maple uses the notation
15518 @samp{sum(a(i), i=1..10)} for sums, which Calc's Maple mode doesn't
15519 recognize at present. To handle this syntax, we simply add the
15523 sum ( # , # = # .. # ) := sum(#1,#2,#3,#4)
15527 to the Maple mode syntax table. As another example, C mode can't
15528 read assignment operators like @samp{++} and @samp{*=}. We can
15529 define these operators quite easily:
15532 # *= # := muleq(#1,#2)
15533 # ++ := postinc(#1)
15538 To complete the job, we would use corresponding composition functions
15539 and @kbd{Z C} to cause these functions to display in their respective
15540 Maple and C notations. (Note that the C example ignores issues of
15541 operator precedence, which are discussed in the next section.)
15543 You can enclose any token in quotes to prevent its usual
15544 interpretation in syntax patterns:
15547 # ":=" # := becomes(#1,#2)
15550 Quotes also allow you to include spaces in a token, although once
15551 again it is generally better to use two tokens than one token with
15552 an embedded space. To include an actual quotation mark in a quoted
15553 token, precede it with a backslash. (This also works to include
15554 backslashes in tokens.)
15557 # "bad token" # "/\"\\" # := silly(#1,#2,#3)
15561 This will parse @samp{3 bad token 4 /"\ 5} to @samp{silly(3,4,5)}.
15563 The token @kbd{#} has a predefined meaning in Calc's formula parser;
15564 it is not valid to use @samp{"#"} in a syntax rule. However, longer
15565 tokens that include the @samp{#} character are allowed. Also, while
15566 @samp{"$"} and @samp{"\""} are allowed as tokens, their presence in
15567 the syntax table will prevent those characters from working in their
15568 usual ways (referring to stack entries and quoting strings,
15571 Finally, the notation @samp{%%} anywhere in a syntax table causes
15572 the rest of the line to be ignored as a comment.
15574 @node Precedence in Syntax Tables, Advanced Syntax Patterns, Syntax Table Basics, Syntax Tables
15575 @subsubsection Precedence
15578 Different operators are generally assigned different @dfn{precedences}.
15579 By default, an operator defined by a rule like
15582 # foo # := foo(#1,#2)
15586 will have an extremely low precedence, so that @samp{2*3+4 foo 5 == 6}
15587 will be parsed as @samp{(2*3+4) foo (5 == 6)}. To change the
15588 precedence of an operator, use the notation @samp{#/@var{p}} in
15589 place of @samp{#}, where @var{p} is an integer precedence level.
15590 For example, 185 lies between the precedences for @samp{+} and
15591 @samp{*}, so if we change this rule to
15594 #/185 foo #/186 := foo(#1,#2)
15598 then @samp{2+3 foo 4*5} will be parsed as @samp{2+(3 foo (4*5))}.
15599 Also, because we've given the righthand expression slightly higher
15600 precedence, our new operator will be left-associative:
15601 @samp{1 foo 2 foo 3} will be parsed as @samp{(1 foo 2) foo 3}.
15602 By raising the precedence of the lefthand expression instead, we
15603 can create a right-associative operator.
15605 @xref{Composition Basics}, for a table of precedences of the
15606 standard Calc operators. For the precedences of operators in other
15607 language modes, look in the Calc source file @file{calc-lang.el}.
15609 @node Advanced Syntax Patterns, Conditional Syntax Rules, Precedence in Syntax Tables, Syntax Tables
15610 @subsubsection Advanced Syntax Patterns
15613 To match a function with a variable number of arguments, you could
15617 foo ( # ) := myfunc(#1)
15618 foo ( # , # ) := myfunc(#1,#2)
15619 foo ( # , # , # ) := myfunc(#1,#2,#3)
15623 but this isn't very elegant. To match variable numbers of items,
15624 Calc uses some notations inspired regular expressions and the
15625 ``extended BNF'' style used by some language designers.
15628 foo ( @{ # @}*, ) := apply(myfunc,#1)
15631 The token @samp{@{} introduces a repeated or optional portion.
15632 One of the three tokens @samp{@}*}, @samp{@}+}, or @samp{@}?}
15633 ends the portion. These will match zero or more, one or more,
15634 or zero or one copies of the enclosed pattern, respectively.
15635 In addition, @samp{@}*} and @samp{@}+} can be followed by a
15636 separator token (with no space in between, as shown above).
15637 Thus @samp{@{ # @}*,} matches nothing, or one expression, or
15638 several expressions separated by commas.
15640 A complete @samp{@{ ... @}} item matches as a vector of the
15641 items that matched inside it. For example, the above rule will
15642 match @samp{foo(1,2,3)} to get @samp{apply(myfunc,[1,2,3])}.
15643 The Calc @code{apply} function takes a function name and a vector
15644 of arguments and builds a call to the function with those
15645 arguments, so the net result is the formula @samp{myfunc(1,2,3)}.
15647 If the body of a @samp{@{ ... @}} contains several @samp{#}s
15648 (or nested @samp{@{ ... @}} constructs), then the items will be
15649 strung together into the resulting vector. If the body
15650 does not contain anything but literal tokens, the result will
15651 always be an empty vector.
15654 foo ( @{ # , # @}+, ) := bar(#1)
15655 foo ( @{ @{ # @}*, @}*; ) := matrix(#1)
15659 will parse @samp{foo(1, 2, 3, 4)} as @samp{bar([1, 2, 3, 4])}, and
15660 @samp{foo(1, 2; 3, 4)} as @samp{matrix([[1, 2], [3, 4]])}. Also, after
15661 some thought it's easy to see how this pair of rules will parse
15662 @samp{foo(1, 2, 3)} as @samp{matrix([[1, 2, 3]])}, since the first
15663 rule will only match an even number of arguments. The rule
15666 foo ( # @{ , # , # @}? ) := bar(#1,#2)
15670 will parse @samp{foo(2,3,4)} as @samp{bar(2,[3,4])}, and
15671 @samp{foo(2)} as @samp{bar(2,[])}.
15673 The notation @samp{@{ ... @}?.} (note the trailing period) works
15674 just the same as regular @samp{@{ ... @}?}, except that it does not
15675 count as an argument; the following two rules are equivalent:
15678 foo ( # , @{ also @}? # ) := bar(#1,#3)
15679 foo ( # , @{ also @}?. # ) := bar(#1,#2)
15683 Note that in the first case the optional text counts as @samp{#2},
15684 which will always be an empty vector, but in the second case no
15685 empty vector is produced.
15687 Another variant is @samp{@{ ... @}?$}, which means the body is
15688 optional only at the end of the input formula. All built-in syntax
15689 rules in Calc use this for closing delimiters, so that during
15690 algebraic entry you can type @kbd{[sqrt(2), sqrt(3 @key{RET}}, omitting
15691 the closing parenthesis and bracket. Calc does this automatically
15692 for trailing @samp{)}, @samp{]}, and @samp{>} tokens in syntax
15693 rules, but you can use @samp{@{ ... @}?$} explicitly to get
15694 this effect with any token (such as @samp{"@}"} or @samp{end}).
15695 Like @samp{@{ ... @}?.}, this notation does not count as an
15696 argument. Conversely, you can use quotes, as in @samp{")"}, to
15697 prevent a closing-delimiter token from being automatically treated
15700 Calc's parser does not have full backtracking, which means some
15701 patterns will not work as you might expect:
15704 foo ( @{ # , @}? # , # ) := bar(#1,#2,#3)
15708 Here we are trying to make the first argument optional, so that
15709 @samp{foo(2,3)} parses as @samp{bar([],2,3)}. Unfortunately, Calc
15710 first tries to match @samp{2,} against the optional part of the
15711 pattern, finds a match, and so goes ahead to match the rest of the
15712 pattern. Later on it will fail to match the second comma, but it
15713 doesn't know how to go back and try the other alternative at that
15714 point. One way to get around this would be to use two rules:
15717 foo ( # , # , # ) := bar([#1],#2,#3)
15718 foo ( # , # ) := bar([],#1,#2)
15721 More precisely, when Calc wants to match an optional or repeated
15722 part of a pattern, it scans forward attempting to match that part.
15723 If it reaches the end of the optional part without failing, it
15724 ``finalizes'' its choice and proceeds. If it fails, though, it
15725 backs up and tries the other alternative. Thus Calc has ``partial''
15726 backtracking. A fully backtracking parser would go on to make sure
15727 the rest of the pattern matched before finalizing the choice.
15729 @node Conditional Syntax Rules, , Advanced Syntax Patterns, Syntax Tables
15730 @subsubsection Conditional Syntax Rules
15733 It is possible to attach a @dfn{condition} to a syntax rule. For
15737 foo ( # ) := ifoo(#1) :: integer(#1)
15738 foo ( # ) := gfoo(#1)
15742 will parse @samp{foo(3)} as @samp{ifoo(3)}, but will parse
15743 @samp{foo(3.5)} and @samp{foo(x)} as calls to @code{gfoo}. Any
15744 number of conditions may be attached; all must be true for the
15745 rule to succeed. A condition is ``true'' if it evaluates to a
15746 nonzero number. @xref{Logical Operations}, for a list of Calc
15747 functions like @code{integer} that perform logical tests.
15749 The exact sequence of events is as follows: When Calc tries a
15750 rule, it first matches the pattern as usual. It then substitutes
15751 @samp{#1}, @samp{#2}, etc., in the conditions, if any. Next, the
15752 conditions are simplified and evaluated in order from left to right,
15753 using the algebraic simplifications (@pxref{Simplifying Formulas}).
15754 Each result is true if it is a nonzero number, or an expression
15755 that can be proven to be nonzero (@pxref{Declarations}). If the
15756 results of all conditions are true, the expression (such as
15757 @samp{ifoo(#1)}) has its @samp{#}s substituted, and that is the
15758 result of the parse. If the result of any condition is false, Calc
15759 goes on to try the next rule in the syntax table.
15761 Syntax rules also support @code{let} conditions, which operate in
15762 exactly the same way as they do in algebraic rewrite rules.
15763 @xref{Other Features of Rewrite Rules}, for details. A @code{let}
15764 condition is always true, but as a side effect it defines a
15765 variable which can be used in later conditions, and also in the
15766 expression after the @samp{:=} sign:
15769 foo ( # ) := hifoo(x) :: let(x := #1 + 0.5) :: dnumint(x)
15773 The @code{dnumint} function tests if a value is numerically an
15774 integer, i.e., either a true integer or an integer-valued float.
15775 This rule will parse @code{foo} with a half-integer argument,
15776 like @samp{foo(3.5)}, to a call like @samp{hifoo(4.)}.
15778 The lefthand side of a syntax rule @code{let} must be a simple
15779 variable, not the arbitrary pattern that is allowed in rewrite
15782 The @code{matches} function is also treated specially in syntax
15783 rule conditions (again, in the same way as in rewrite rules).
15784 @xref{Matching Commands}. If the matching pattern contains
15785 meta-variables, then those meta-variables may be used in later
15786 conditions and in the result expression. The arguments to
15787 @code{matches} are not evaluated in this situation.
15790 sum ( # , # ) := sum(#1,a,b,c) :: matches(#2, a=[b..c])
15794 This is another way to implement the Maple mode @code{sum} notation.
15795 In this approach, we allow @samp{#2} to equal the whole expression
15796 @samp{i=1..10}. Then, we use @code{matches} to break it apart into
15797 its components. If the expression turns out not to match the pattern,
15798 the syntax rule will fail. Note that @kbd{Z S} always uses Calc's
15799 Normal language mode for editing expressions in syntax rules, so we
15800 must use regular Calc notation for the interval @samp{[b..c]} that
15801 will correspond to the Maple mode interval @samp{1..10}.
15803 @node Modes Variable, Calc Mode Line, Language Modes, Mode Settings
15804 @section The @code{Modes} Variable
15808 @pindex calc-get-modes
15809 The @kbd{m g} (@code{calc-get-modes}) command pushes onto the stack
15810 a vector of numbers that describes the various mode settings that
15811 are in effect. With a numeric prefix argument, it pushes only the
15812 @var{n}th mode, i.e., the @var{n}th element of this vector. Keyboard
15813 macros can use the @kbd{m g} command to modify their behavior based
15814 on the current mode settings.
15816 @cindex @code{Modes} variable
15818 The modes vector is also available in the special variable
15819 @code{Modes}. In other words, @kbd{m g} is like @kbd{s r Modes @key{RET}}.
15820 It will not work to store into this variable; in fact, if you do,
15821 @code{Modes} will cease to track the current modes. (The @kbd{m g}
15822 command will continue to work, however.)
15824 In general, each number in this vector is suitable as a numeric
15825 prefix argument to the associated mode-setting command. (Recall
15826 that the @kbd{~} key takes a number from the stack and gives it as
15827 a numeric prefix to the next command.)
15829 The elements of the modes vector are as follows:
15833 Current precision. Default is 12; associated command is @kbd{p}.
15836 Binary word size. Default is 32; associated command is @kbd{b w}.
15839 Stack size (not counting the value about to be pushed by @kbd{m g}).
15840 This is zero if @kbd{m g} is executed with an empty stack.
15843 Number radix. Default is 10; command is @kbd{d r}.
15846 Floating-point format. This is the number of digits, plus the
15847 constant 0 for normal notation, 10000 for scientific notation,
15848 20000 for engineering notation, or 30000 for fixed-point notation.
15849 These codes are acceptable as prefix arguments to the @kbd{d n}
15850 command, but note that this may lose information: For example,
15851 @kbd{d s} and @kbd{C-u 12 d s} have similar (but not quite
15852 identical) effects if the current precision is 12, but they both
15853 produce a code of 10012, which will be treated by @kbd{d n} as
15854 @kbd{C-u 12 d s}. If the precision then changes, the float format
15855 will still be frozen at 12 significant figures.
15858 Angular mode. Default is 1 (degrees). Other values are 2 (radians)
15859 and 3 (HMS). The @kbd{m d} command accepts these prefixes.
15862 Symbolic mode. Value is 0 or 1; default is 0. Command is @kbd{m s}.
15865 Fraction mode. Value is 0 or 1; default is 0. Command is @kbd{m f}.
15868 Polar mode. Value is 0 (rectangular) or 1 (polar); default is 0.
15869 Command is @kbd{m p}.
15872 Matrix/Scalar mode. Default value is @mathit{-1}. Value is 0 for Scalar
15873 mode, @mathit{-2} for Matrix mode, @mathit{-3} for square Matrix mode,
15875 @texline @math{N\times N}
15876 @infoline @var{N}x@var{N}
15877 Matrix mode. Command is @kbd{m v}.
15880 Simplification mode. Default is 1. Value is @mathit{-1} for off (@kbd{m O}),
15881 0 for @kbd{m N}, 2 for @kbd{m B}, 3 for @kbd{m A}, 4 for @kbd{m E},
15882 or 5 for @w{@kbd{m U}}. The @kbd{m D} command accepts these prefixes.
15885 Infinite mode. Default is @mathit{-1} (off). Value is 1 if the mode is on,
15886 or 0 if the mode is on with positive zeros. Command is @kbd{m i}.
15889 For example, the sequence @kbd{M-1 m g @key{RET} 2 + ~ p} increases the
15890 precision by two, leaving a copy of the old precision on the stack.
15891 Later, @kbd{~ p} will restore the original precision using that
15892 stack value. (This sequence might be especially useful inside a
15895 As another example, @kbd{M-3 m g 1 - ~ @key{DEL}} deletes all but the
15896 oldest (bottommost) stack entry.
15898 Yet another example: The HP-48 ``round'' command rounds a number
15899 to the current displayed precision. You could roughly emulate this
15900 in Calc with the sequence @kbd{M-5 m g 10000 % ~ c c}. (This
15901 would not work for fixed-point mode, but it wouldn't be hard to
15902 do a full emulation with the help of the @kbd{Z [} and @kbd{Z ]}
15903 programming commands. @xref{Conditionals in Macros}.)
15905 @node Calc Mode Line, , Modes Variable, Mode Settings
15906 @section The Calc Mode Line
15909 @cindex Mode line indicators
15910 This section is a summary of all symbols that can appear on the
15911 Calc mode line, the highlighted bar that appears under the Calc
15912 stack window (or under an editing window in Embedded mode).
15914 The basic mode line format is:
15917 --%*-Calc: 12 Deg @var{other modes} (Calculator)
15920 The @samp{%*} indicates that the buffer is ``read-only''; it shows that
15921 regular Emacs commands are not allowed to edit the stack buffer
15922 as if it were text.
15924 The word @samp{Calc:} changes to @samp{CalcEmbed:} if Embedded mode
15925 is enabled. The words after this describe the various Calc modes
15926 that are in effect.
15928 The first mode is always the current precision, an integer.
15929 The second mode is always the angular mode, either @code{Deg},
15930 @code{Rad}, or @code{Hms}.
15932 Here is a complete list of the remaining symbols that can appear
15937 Algebraic mode (@kbd{m a}; @pxref{Algebraic Entry}).
15940 Incomplete algebraic mode (@kbd{C-u m a}).
15943 Total algebraic mode (@kbd{m t}).
15946 Symbolic mode (@kbd{m s}; @pxref{Symbolic Mode}).
15949 Matrix mode (@kbd{m v}; @pxref{Matrix Mode}).
15951 @item Matrix@var{n}
15952 Dimensioned Matrix mode (@kbd{C-u @var{n} m v}; @pxref{Matrix Mode}).
15955 Square Matrix mode (@kbd{C-u m v}; @pxref{Matrix Mode}).
15958 Scalar mode (@kbd{m v}; @pxref{Matrix Mode}).
15961 Polar complex mode (@kbd{m p}; @pxref{Polar Mode}).
15964 Fraction mode (@kbd{m f}; @pxref{Fraction Mode}).
15967 Infinite mode (@kbd{m i}; @pxref{Infinite Mode}).
15970 Positive Infinite mode (@kbd{C-u 0 m i}).
15973 Default simplifications off (@kbd{m O}; @pxref{Simplification Modes}).
15976 Default simplifications for numeric arguments only (@kbd{m N}).
15978 @item BinSimp@var{w}
15979 Binary-integer simplification mode; word size @var{w} (@kbd{m B}, @kbd{b w}).
15982 Basic simplification mode (@kbd{m I}).
15985 Extended algebraic simplification mode (@kbd{m E}).
15988 Units simplification mode (@kbd{m U}).
15991 Current radix is 2 (@kbd{d 2}; @pxref{Radix Modes}).
15994 Current radix is 8 (@kbd{d 8}).
15997 Current radix is 16 (@kbd{d 6}).
16000 Current radix is @var{n} (@kbd{d r}).
16003 Leading zeros (@kbd{d z}; @pxref{Radix Modes}).
16006 Big language mode (@kbd{d B}; @pxref{Normal Language Modes}).
16009 One-line normal language mode (@kbd{d O}).
16012 Unformatted language mode (@kbd{d U}).
16015 C language mode (@kbd{d C}; @pxref{C FORTRAN Pascal}).
16018 Pascal language mode (@kbd{d P}).
16021 FORTRAN language mode (@kbd{d F}).
16024 @TeX{} language mode (@kbd{d T}; @pxref{TeX and LaTeX Language Modes}).
16027 @LaTeX{} language mode (@kbd{d L}; @pxref{TeX and LaTeX Language Modes}).
16030 @dfn{Eqn} language mode (@kbd{d E}; @pxref{Eqn Language Mode}).
16033 Mathematica language mode (@kbd{d M}; @pxref{Mathematica Language Mode}).
16036 Maple language mode (@kbd{d W}; @pxref{Maple Language Mode}).
16039 Normal float mode with @var{n} digits (@kbd{d n}; @pxref{Float Formats}).
16042 Fixed point mode with @var{n} digits after the point (@kbd{d f}).
16045 Scientific notation mode (@kbd{d s}).
16048 Scientific notation with @var{n} digits (@kbd{d s}).
16051 Engineering notation mode (@kbd{d e}).
16054 Engineering notation with @var{n} digits (@kbd{d e}).
16057 Left-justified display indented by @var{n} (@kbd{d <}; @pxref{Justification}).
16060 Right-justified display (@kbd{d >}).
16063 Right-justified display with width @var{n} (@kbd{d >}).
16066 Centered display (@kbd{d =}).
16068 @item Center@var{n}
16069 Centered display with center column @var{n} (@kbd{d =}).
16072 Line breaking with width @var{n} (@kbd{d b}; @pxref{Normal Language Modes}).
16075 No line breaking (@kbd{d b}).
16078 Selections show deep structure (@kbd{j b}; @pxref{Making Selections}).
16081 Record modes in @file{~/.emacs.d/calc.el} (@kbd{m R}; @pxref{General Mode Commands}).
16084 Record modes in Embedded buffer (@kbd{m R}).
16087 Record modes as editing-only in Embedded buffer (@kbd{m R}).
16090 Record modes as permanent-only in Embedded buffer (@kbd{m R}).
16093 Record modes as global in Embedded buffer (@kbd{m R}).
16096 Automatic recomputation turned off (@kbd{m C}; @pxref{Automatic
16100 GNUPLOT process is alive in background (@pxref{Graphics}).
16103 Top-of-stack has a selection (Embedded only; @pxref{Making Selections}).
16106 The stack display may not be up-to-date (@pxref{Display Modes}).
16109 ``Inverse'' prefix was pressed (@kbd{I}; @pxref{Inverse and Hyperbolic}).
16112 ``Hyperbolic'' prefix was pressed (@kbd{H}).
16115 ``Keep-arguments'' prefix was pressed (@kbd{K}).
16118 Stack is truncated (@kbd{d t}; @pxref{Truncating the Stack}).
16121 In addition, the symbols @code{Active} and @code{~Active} can appear
16122 as minor modes on an Embedded buffer's mode line. @xref{Embedded Mode}.
16124 @node Arithmetic, Scientific Functions, Mode Settings, Top
16125 @chapter Arithmetic Functions
16128 This chapter describes the Calc commands for doing simple calculations
16129 on numbers, such as addition, absolute value, and square roots. These
16130 commands work by removing the top one or two values from the stack,
16131 performing the desired operation, and pushing the result back onto the
16132 stack. If the operation cannot be performed, the result pushed is a
16133 formula instead of a number, such as @samp{2/0} (because division by zero
16134 is invalid) or @samp{sqrt(x)} (because the argument @samp{x} is a formula).
16136 Most of the commands described here can be invoked by a single keystroke.
16137 Some of the more obscure ones are two-letter sequences beginning with
16138 the @kbd{f} (``functions'') prefix key.
16140 @xref{Prefix Arguments}, for a discussion of the effect of numeric
16141 prefix arguments on commands in this chapter which do not otherwise
16142 interpret a prefix argument.
16145 * Basic Arithmetic::
16146 * Integer Truncation::
16147 * Complex Number Functions::
16149 * Date Arithmetic::
16150 * Financial Functions::
16151 * Binary Functions::
16154 @node Basic Arithmetic, Integer Truncation, Arithmetic, Arithmetic
16155 @section Basic Arithmetic
16164 The @kbd{+} (@code{calc-plus}) command adds two numbers. The numbers may
16165 be any of the standard Calc data types. The resulting sum is pushed back
16168 If both arguments of @kbd{+} are vectors or matrices (of matching dimensions),
16169 the result is a vector or matrix sum. If one argument is a vector and the
16170 other a scalar (i.e., a non-vector), the scalar is added to each of the
16171 elements of the vector to form a new vector. If the scalar is not a
16172 number, the operation is left in symbolic form: Suppose you added @samp{x}
16173 to the vector @samp{[1,2]}. You may want the result @samp{[1+x,2+x]}, or
16174 you may plan to substitute a 2-vector for @samp{x} in the future. Since
16175 the Calculator can't tell which interpretation you want, it makes the
16176 safest assumption. @xref{Reducing and Mapping}, for a way to add @samp{x}
16177 to every element of a vector.
16179 If either argument of @kbd{+} is a complex number, the result will in general
16180 be complex. If one argument is in rectangular form and the other polar,
16181 the current Polar mode determines the form of the result. If Symbolic
16182 mode is enabled, the sum may be left as a formula if the necessary
16183 conversions for polar addition are non-trivial.
16185 If both arguments of @kbd{+} are HMS forms, the forms are added according to
16186 the usual conventions of hours-minutes-seconds notation. If one argument
16187 is an HMS form and the other is a number, that number is converted from
16188 degrees or radians (depending on the current Angular mode) to HMS format
16189 and then the two HMS forms are added.
16191 If one argument of @kbd{+} is a date form, the other can be either a
16192 real number, which advances the date by a certain number of days, or
16193 an HMS form, which advances the date by a certain amount of time.
16194 Subtracting two date forms yields the number of days between them.
16195 Adding two date forms is meaningless, but Calc interprets it as the
16196 subtraction of one date form and the negative of the other. (The
16197 negative of a date form can be understood by remembering that dates
16198 are stored as the number of days before or after Jan 1, 1 AD.)
16200 If both arguments of @kbd{+} are error forms, the result is an error form
16201 with an appropriately computed standard deviation. If one argument is an
16202 error form and the other is a number, the number is taken to have zero error.
16203 Error forms may have symbolic formulas as their mean and/or error parts;
16204 adding these will produce a symbolic error form result. However, adding an
16205 error form to a plain symbolic formula (as in @samp{(a +/- b) + c}) will not
16206 work, for the same reasons just mentioned for vectors. Instead you must
16207 write @samp{(a +/- b) + (c +/- 0)}.
16209 If both arguments of @kbd{+} are modulo forms with equal values of @expr{M},
16210 or if one argument is a modulo form and the other a plain number, the
16211 result is a modulo form which represents the sum, modulo @expr{M}, of
16214 If both arguments of @kbd{+} are intervals, the result is an interval
16215 which describes all possible sums of the possible input values. If
16216 one argument is a plain number, it is treated as the interval
16217 @w{@samp{[x ..@: x]}}.
16219 If one argument of @kbd{+} is an infinity and the other is not, the
16220 result is that same infinity. If both arguments are infinite and in
16221 the same direction, the result is the same infinity, but if they are
16222 infinite in different directions the result is @code{nan}.
16230 The @kbd{-} (@code{calc-minus}) command subtracts two values. The top
16231 number on the stack is subtracted from the one behind it, so that the
16232 computation @kbd{5 @key{RET} 2 -} produces 3, not @mathit{-3}. All options
16233 available for @kbd{+} are available for @kbd{-} as well.
16241 The @kbd{*} (@code{calc-times}) command multiplies two numbers. If one
16242 argument is a vector and the other a scalar, the scalar is multiplied by
16243 the elements of the vector to produce a new vector. If both arguments
16244 are vectors, the interpretation depends on the dimensions of the
16245 vectors: If both arguments are matrices, a matrix multiplication is
16246 done. If one argument is a matrix and the other a plain vector, the
16247 vector is interpreted as a row vector or column vector, whichever is
16248 dimensionally correct. If both arguments are plain vectors, the result
16249 is a single scalar number which is the dot product of the two vectors.
16251 If one argument of @kbd{*} is an HMS form and the other a number, the
16252 HMS form is multiplied by that amount. It is an error to multiply two
16253 HMS forms together, or to attempt any multiplication involving date
16254 forms. Error forms, modulo forms, and intervals can be multiplied;
16255 see the comments for addition of those forms. When two error forms
16256 or intervals are multiplied they are considered to be statistically
16257 independent; thus, @samp{[-2 ..@: 3] * [-2 ..@: 3]} is @samp{[-6 ..@: 9]},
16258 whereas @w{@samp{[-2 ..@: 3] ^ 2}} is @samp{[0 ..@: 9]}.
16261 @pindex calc-divide
16266 The @kbd{/} (@code{calc-divide}) command divides two numbers.
16268 When combining multiplication and division in an algebraic formula, it
16269 is good style to use parentheses to distinguish between possible
16270 interpretations; the expression @samp{a/b*c} should be written
16271 @samp{(a/b)*c} or @samp{a/(b*c)}, as appropriate. Without the
16272 parentheses, Calc will interpret @samp{a/b*c} as @samp{a/(b*c)}, since
16273 in algebraic entry Calc gives division a lower precedence than
16274 multiplication. (This is not standard across all computer languages, and
16275 Calc may change the precedence depending on the language mode being used.
16276 @xref{Language Modes}.) This default ordering can be changed by setting
16277 the customizable variable @code{calc-multiplication-has-precedence} to
16278 @code{nil} (@pxref{Customizing Calc}); this will give multiplication and
16279 division equal precedences. Note that Calc's default choice of
16280 precedence allows @samp{a b / c d} to be used as a shortcut for
16289 When dividing a scalar @expr{B} by a square matrix @expr{A}, the
16290 computation performed is @expr{B} times the inverse of @expr{A}. This
16291 also occurs if @expr{B} is itself a vector or matrix, in which case the
16292 effect is to solve the set of linear equations represented by @expr{B}.
16293 If @expr{B} is a matrix with the same number of rows as @expr{A}, or a
16294 plain vector (which is interpreted here as a column vector), then the
16295 equation @expr{A X = B} is solved for the vector or matrix @expr{X}.
16296 Otherwise, if @expr{B} is a non-square matrix with the same number of
16297 @emph{columns} as @expr{A}, the equation @expr{X A = B} is solved. If
16298 you wish a vector @expr{B} to be interpreted as a row vector to be
16299 solved as @expr{X A = B}, make it into a one-row matrix with @kbd{C-u 1
16300 v p} first. To force a left-handed solution with a square matrix
16301 @expr{B}, transpose @expr{A} and @expr{B} before dividing, then
16302 transpose the result.
16304 HMS forms can be divided by real numbers or by other HMS forms. Error
16305 forms can be divided in any combination of ways. Modulo forms where both
16306 values and the modulo are integers can be divided to get an integer modulo
16307 form result. Intervals can be divided; dividing by an interval that
16308 encompasses zero or has zero as a limit will result in an infinite
16317 The @kbd{^} (@code{calc-power}) command raises a number to a power. If
16318 the power is an integer, an exact result is computed using repeated
16319 multiplications. For non-integer powers, Calc uses Newton's method or
16320 logarithms and exponentials. Square matrices can be raised to integer
16321 powers. If either argument is an error (or interval or modulo) form,
16322 the result is also an error (or interval or modulo) form.
16326 If you press the @kbd{I} (inverse) key first, the @kbd{I ^} command
16327 computes an Nth root: @kbd{125 @key{RET} 3 I ^} computes the number 5.
16328 (This is entirely equivalent to @kbd{125 @key{RET} 1:3 ^}.)
16337 The @kbd{\} (@code{calc-idiv}) command divides two numbers on the stack
16338 to produce an integer result. It is equivalent to dividing with
16339 @key{/}, then rounding down with @kbd{F} (@code{calc-floor}), only a bit
16340 more convenient and efficient. Also, since it is an all-integer
16341 operation when the arguments are integers, it avoids problems that
16342 @kbd{/ F} would have with floating-point roundoff.
16350 The @kbd{%} (@code{calc-mod}) command performs a ``modulo'' (or ``remainder'')
16351 operation. Mathematically, @samp{a%b = a - (a\b)*b}, and is defined
16352 for all real numbers @expr{a} and @expr{b} (except @expr{b=0}). For
16353 positive @expr{b}, the result will always be between 0 (inclusive) and
16354 @expr{b} (exclusive). Modulo does not work for HMS forms and error forms.
16355 If @expr{a} is a modulo form, its modulo is changed to @expr{b}, which
16356 must be positive real number.
16361 The @kbd{:} (@code{calc-fdiv}) [@code{fdiv}] command
16362 divides the two integers on the top of the stack to produce a fractional
16363 result. This is a convenient shorthand for enabling Fraction mode (with
16364 @kbd{m f}) temporarily and using @samp{/}. Note that during numeric entry
16365 the @kbd{:} key is interpreted as a fraction separator, so to divide 8 by 6
16366 you would have to type @kbd{8 @key{RET} 6 @key{RET} :}. (Of course, in
16367 this case, it would be much easier simply to enter the fraction directly
16368 as @kbd{8:6 @key{RET}}!)
16371 @pindex calc-change-sign
16372 The @kbd{n} (@code{calc-change-sign}) command negates the number on the top
16373 of the stack. It works on numbers, vectors and matrices, HMS forms, date
16374 forms, error forms, intervals, and modulo forms.
16379 The @kbd{A} (@code{calc-abs}) [@code{abs}] command computes the absolute
16380 value of a number. The result of @code{abs} is always a nonnegative
16381 real number: With a complex argument, it computes the complex magnitude.
16382 With a vector or matrix argument, it computes the Frobenius norm, i.e.,
16383 the square root of the sum of the squares of the absolute values of the
16384 elements. The absolute value of an error form is defined by replacing
16385 the mean part with its absolute value and leaving the error part the same.
16386 The absolute value of a modulo form is undefined. The absolute value of
16387 an interval is defined in the obvious way.
16390 @pindex calc-abssqr
16392 The @kbd{f A} (@code{calc-abssqr}) [@code{abssqr}] command computes the
16393 absolute value squared of a number, vector or matrix, or error form.
16398 The @kbd{f s} (@code{calc-sign}) [@code{sign}] command returns 1 if its
16399 argument is positive, @mathit{-1} if its argument is negative, or 0 if its
16400 argument is zero. In algebraic form, you can also write @samp{sign(a,x)}
16401 which evaluates to @samp{x * sign(a)}, i.e., either @samp{x}, @samp{-x}, or
16402 zero depending on the sign of @samp{a}.
16408 The @kbd{&} (@code{calc-inv}) [@code{inv}] command computes the
16409 reciprocal of a number, i.e., @expr{1 / x}. Operating on a square
16410 matrix, it computes the inverse of that matrix.
16415 The @kbd{Q} (@code{calc-sqrt}) [@code{sqrt}] command computes the square
16416 root of a number. For a negative real argument, the result will be a
16417 complex number whose form is determined by the current Polar mode.
16422 The @kbd{f h} (@code{calc-hypot}) [@code{hypot}] command computes the square
16423 root of the sum of the squares of two numbers. That is, @samp{hypot(a,b)}
16424 is the length of the hypotenuse of a right triangle with sides @expr{a}
16425 and @expr{b}. If the arguments are complex numbers, their squared
16426 magnitudes are used.
16431 The @kbd{f Q} (@code{calc-isqrt}) [@code{isqrt}] command computes the
16432 integer square root of an integer. This is the true square root of the
16433 number, rounded down to an integer. For example, @samp{isqrt(10)}
16434 produces 3. Note that, like @kbd{\} [@code{idiv}], this uses exact
16435 integer arithmetic throughout to avoid roundoff problems. If the input
16436 is a floating-point number or other non-integer value, this is exactly
16437 the same as @samp{floor(sqrt(x))}.
16445 The @kbd{f n} (@code{calc-min}) [@code{min}] and @kbd{f x} (@code{calc-max})
16446 [@code{max}] commands take the minimum or maximum of two real numbers,
16447 respectively. These commands also work on HMS forms, date forms,
16448 intervals, and infinities. (In algebraic expressions, these functions
16449 take any number of arguments and return the maximum or minimum among
16450 all the arguments.)
16454 @pindex calc-mant-part
16456 @pindex calc-xpon-part
16458 The @kbd{f M} (@code{calc-mant-part}) [@code{mant}] function extracts
16459 the ``mantissa'' part @expr{m} of its floating-point argument; @kbd{f X}
16460 (@code{calc-xpon-part}) [@code{xpon}] extracts the ``exponent'' part
16461 @expr{e}. The original number is equal to
16462 @texline @math{m \times 10^e},
16463 @infoline @expr{m * 10^e},
16464 where @expr{m} is in the interval @samp{[1.0 ..@: 10.0)} except that
16465 @expr{m=e=0} if the original number is zero. For integers
16466 and fractions, @code{mant} returns the number unchanged and @code{xpon}
16467 returns zero. The @kbd{v u} (@code{calc-unpack}) command can also be
16468 used to ``unpack'' a floating-point number; this produces an integer
16469 mantissa and exponent, with the constraint that the mantissa is not
16470 a multiple of ten (again except for the @expr{m=e=0} case).
16473 @pindex calc-scale-float
16475 The @kbd{f S} (@code{calc-scale-float}) [@code{scf}] function scales a number
16476 by a given power of ten. Thus, @samp{scf(mant(x), xpon(x)) = x} for any
16477 real @samp{x}. The second argument must be an integer, but the first
16478 may actually be any numeric value. For example, @samp{scf(5,-2) = 0.05}
16479 or @samp{1:20} depending on the current Fraction mode.
16483 @pindex calc-decrement
16484 @pindex calc-increment
16487 The @kbd{f [} (@code{calc-decrement}) [@code{decr}] and @kbd{f ]}
16488 (@code{calc-increment}) [@code{incr}] functions decrease or increase
16489 a number by one unit. For integers, the effect is obvious. For
16490 floating-point numbers, the change is by one unit in the last place.
16491 For example, incrementing @samp{12.3456} when the current precision
16492 is 6 digits yields @samp{12.3457}. If the current precision had been
16493 8 digits, the result would have been @samp{12.345601}. Incrementing
16494 @samp{0.0} produces
16495 @texline @math{10^{-p}},
16496 @infoline @expr{10^-p},
16497 where @expr{p} is the current
16498 precision. These operations are defined only on integers and floats.
16499 With numeric prefix arguments, they change the number by @expr{n} units.
16501 Note that incrementing followed by decrementing, or vice-versa, will
16502 almost but not quite always cancel out. Suppose the precision is
16503 6 digits and the number @samp{9.99999} is on the stack. Incrementing
16504 will produce @samp{10.0000}; decrementing will produce @samp{9.9999}.
16505 One digit has been dropped. This is an unavoidable consequence of the
16506 way floating-point numbers work.
16508 Incrementing a date/time form adjusts it by a certain number of seconds.
16509 Incrementing a pure date form adjusts it by a certain number of days.
16511 @node Integer Truncation, Complex Number Functions, Basic Arithmetic, Arithmetic
16512 @section Integer Truncation
16515 There are four commands for truncating a real number to an integer,
16516 differing mainly in their treatment of negative numbers. All of these
16517 commands have the property that if the argument is an integer, the result
16518 is the same integer. An integer-valued floating-point argument is converted
16521 If you press @kbd{H} (@code{calc-hyperbolic}) first, the result will be
16522 expressed as an integer-valued floating-point number.
16524 @cindex Integer part of a number
16533 The @kbd{F} (@code{calc-floor}) [@code{floor} or @code{ffloor}] command
16534 truncates a real number to the next lower integer, i.e., toward minus
16535 infinity. Thus @kbd{3.6 F} produces 3, but @kbd{_3.6 F} produces
16539 @pindex calc-ceiling
16546 The @kbd{I F} (@code{calc-ceiling}) [@code{ceil} or @code{fceil}]
16547 command truncates toward positive infinity. Thus @kbd{3.6 I F} produces
16548 4, and @kbd{_3.6 I F} produces @mathit{-3}.
16558 The @kbd{R} (@code{calc-round}) [@code{round} or @code{fround}] command
16559 rounds to the nearest integer. When the fractional part is .5 exactly,
16560 this command rounds away from zero. (All other rounding in the
16561 Calculator uses this convention as well.) Thus @kbd{3.5 R} produces 4
16562 but @kbd{3.4 R} produces 3; @kbd{_3.5 R} produces @mathit{-4}.
16572 The @kbd{I R} (@code{calc-trunc}) [@code{trunc} or @code{ftrunc}]
16573 command truncates toward zero. In other words, it ``chops off''
16574 everything after the decimal point. Thus @kbd{3.6 I R} produces 3 and
16575 @kbd{_3.6 I R} produces @mathit{-3}.
16577 These functions may not be applied meaningfully to error forms, but they
16578 do work for intervals. As a convenience, applying @code{floor} to a
16579 modulo form floors the value part of the form. Applied to a vector,
16580 these functions operate on all elements of the vector one by one.
16581 Applied to a date form, they operate on the internal numerical
16582 representation of dates, converting a date/time form into a pure date.
16600 There are two more rounding functions which can only be entered in
16601 algebraic notation. The @code{roundu} function is like @code{round}
16602 except that it rounds up, toward plus infinity, when the fractional
16603 part is .5. This distinction matters only for negative arguments.
16604 Also, @code{rounde} rounds to an even number in the case of a tie,
16605 rounding up or down as necessary. For example, @samp{rounde(3.5)} and
16606 @samp{rounde(4.5)} both return 4, but @samp{rounde(5.5)} returns 6.
16607 The advantage of round-to-even is that the net error due to rounding
16608 after a long calculation tends to cancel out to zero. An important
16609 subtle point here is that the number being fed to @code{rounde} will
16610 already have been rounded to the current precision before @code{rounde}
16611 begins. For example, @samp{rounde(2.500001)} with a current precision
16612 of 6 will incorrectly, or at least surprisingly, yield 2 because the
16613 argument will first have been rounded down to @expr{2.5} (which
16614 @code{rounde} sees as an exact tie between 2 and 3).
16616 Each of these functions, when written in algebraic formulas, allows
16617 a second argument which specifies the number of digits after the
16618 decimal point to keep. For example, @samp{round(123.4567, 2)} will
16619 produce the answer 123.46, and @samp{round(123.4567, -1)} will
16620 produce 120 (i.e., the cutoff is one digit to the @emph{left} of
16621 the decimal point). A second argument of zero is equivalent to
16622 no second argument at all.
16624 @cindex Fractional part of a number
16625 To compute the fractional part of a number (i.e., the amount which, when
16626 added to `@tfn{floor(}@var{n}@tfn{)}', will produce @var{n}) just take @var{n}
16627 modulo 1 using the @code{%} command.
16629 Note also the @kbd{\} (integer quotient), @kbd{f I} (integer logarithm),
16630 and @kbd{f Q} (integer square root) commands, which are analogous to
16631 @kbd{/}, @kbd{B}, and @kbd{Q}, respectively, except that they take integer
16632 arguments and return the result rounded down to an integer.
16634 @node Complex Number Functions, Conversions, Integer Truncation, Arithmetic
16635 @section Complex Number Functions
16641 The @kbd{J} (@code{calc-conj}) [@code{conj}] command computes the
16642 complex conjugate of a number. For complex number @expr{a+bi}, the
16643 complex conjugate is @expr{a-bi}. If the argument is a real number,
16644 this command leaves it the same. If the argument is a vector or matrix,
16645 this command replaces each element by its complex conjugate.
16648 @pindex calc-argument
16650 The @kbd{G} (@code{calc-argument}) [@code{arg}] command computes the
16651 ``argument'' or polar angle of a complex number. For a number in polar
16652 notation, this is simply the second component of the pair
16653 @texline `@tfn{(}@var{r}@tfn{;}@math{\theta}@tfn{)}'.
16654 @infoline `@tfn{(}@var{r}@tfn{;}@var{theta}@tfn{)}'.
16655 The result is expressed according to the current angular mode and will
16656 be in the range @mathit{-180} degrees (exclusive) to @mathit{+180} degrees
16657 (inclusive), or the equivalent range in radians.
16659 @pindex calc-imaginary
16660 The @code{calc-imaginary} command multiplies the number on the
16661 top of the stack by the imaginary number @expr{i = (0,1)}. This
16662 command is not normally bound to a key in Calc, but it is available
16663 on the @key{IMAG} button in Keypad mode.
16668 The @kbd{f r} (@code{calc-re}) [@code{re}] command replaces a complex number
16669 by its real part. This command has no effect on real numbers. (As an
16670 added convenience, @code{re} applied to a modulo form extracts
16676 The @kbd{f i} (@code{calc-im}) [@code{im}] command replaces a complex number
16677 by its imaginary part; real numbers are converted to zero. With a vector
16678 or matrix argument, these functions operate element-wise.
16683 @kindex v p (complex)
16684 @kindex V p (complex)
16686 The @kbd{v p} (@code{calc-pack}) command can pack the top two numbers on
16687 the stack into a composite object such as a complex number. With
16688 a prefix argument of @mathit{-1}, it produces a rectangular complex number;
16689 with an argument of @mathit{-2}, it produces a polar complex number.
16690 (Also, @pxref{Building Vectors}.)
16695 @kindex v u (complex)
16696 @kindex V u (complex)
16697 @pindex calc-unpack
16698 The @kbd{v u} (@code{calc-unpack}) command takes the complex number
16699 (or other composite object) on the top of the stack and unpacks it
16700 into its separate components.
16702 @node Conversions, Date Arithmetic, Complex Number Functions, Arithmetic
16703 @section Conversions
16706 The commands described in this section convert numbers from one form
16707 to another; they are two-key sequences beginning with the letter @kbd{c}.
16712 The @kbd{c f} (@code{calc-float}) [@code{pfloat}] command converts the
16713 number on the top of the stack to floating-point form. For example,
16714 @expr{23} is converted to @expr{23.0}, @expr{3:2} is converted to
16715 @expr{1.5}, and @expr{2.3} is left the same. If the value is a composite
16716 object such as a complex number or vector, each of the components is
16717 converted to floating-point. If the value is a formula, all numbers
16718 in the formula are converted to floating-point. Note that depending
16719 on the current floating-point precision, conversion to floating-point
16720 format may lose information.
16722 As a special exception, integers which appear as powers or subscripts
16723 are not floated by @kbd{c f}. If you really want to float a power,
16724 you can use a @kbd{j s} command to select the power followed by @kbd{c f}.
16725 Because @kbd{c f} cannot examine the formula outside of the selection,
16726 it does not notice that the thing being floated is a power.
16727 @xref{Selecting Subformulas}.
16729 The normal @kbd{c f} command is ``pervasive'' in the sense that it
16730 applies to all numbers throughout the formula. The @code{pfloat}
16731 algebraic function never stays around in a formula; @samp{pfloat(a + 1)}
16732 changes to @samp{a + 1.0} as soon as it is evaluated.
16736 With the Hyperbolic flag, @kbd{H c f} [@code{float}] operates
16737 only on the number or vector of numbers at the top level of its
16738 argument. Thus, @samp{float(1)} is 1.0, but @samp{float(a + 1)}
16739 is left unevaluated because its argument is not a number.
16741 You should use @kbd{H c f} if you wish to guarantee that the final
16742 value, once all the variables have been assigned, is a float; you
16743 would use @kbd{c f} if you wish to do the conversion on the numbers
16744 that appear right now.
16747 @pindex calc-fraction
16749 The @kbd{c F} (@code{calc-fraction}) [@code{pfrac}] command converts a
16750 floating-point number into a fractional approximation. By default, it
16751 produces a fraction whose decimal representation is the same as the
16752 input number, to within the current precision. You can also give a
16753 numeric prefix argument to specify a tolerance, either directly, or,
16754 if the prefix argument is zero, by using the number on top of the stack
16755 as the tolerance. If the tolerance is a positive integer, the fraction
16756 is correct to within that many significant figures. If the tolerance is
16757 a non-positive integer, it specifies how many digits fewer than the current
16758 precision to use. If the tolerance is a floating-point number, the
16759 fraction is correct to within that absolute amount.
16763 The @code{pfrac} function is pervasive, like @code{pfloat}.
16764 There is also a non-pervasive version, @kbd{H c F} [@code{frac}],
16765 which is analogous to @kbd{H c f} discussed above.
16768 @pindex calc-to-degrees
16770 The @kbd{c d} (@code{calc-to-degrees}) [@code{deg}] command converts a
16771 number into degrees form. The value on the top of the stack may be an
16772 HMS form (interpreted as degrees-minutes-seconds), or a real number which
16773 will be interpreted in radians regardless of the current angular mode.
16776 @pindex calc-to-radians
16778 The @kbd{c r} (@code{calc-to-radians}) [@code{rad}] command converts an
16779 HMS form or angle in degrees into an angle in radians.
16782 @pindex calc-to-hms
16784 The @kbd{c h} (@code{calc-to-hms}) [@code{hms}] command converts a real
16785 number, interpreted according to the current angular mode, to an HMS
16786 form describing the same angle. In algebraic notation, the @code{hms}
16787 function also accepts three arguments: @samp{hms(@var{h}, @var{m}, @var{s})}.
16788 (The three-argument version is independent of the current angular mode.)
16790 @pindex calc-from-hms
16791 The @code{calc-from-hms} command converts the HMS form on the top of the
16792 stack into a real number according to the current angular mode.
16799 The @kbd{c p} (@code{calc-polar}) command converts the complex number on
16800 the top of the stack from polar to rectangular form, or from rectangular
16801 to polar form, whichever is appropriate. Real numbers are left the same.
16802 This command is equivalent to the @code{rect} or @code{polar}
16803 functions in algebraic formulas, depending on the direction of
16804 conversion. (It uses @code{polar}, except that if the argument is
16805 already a polar complex number, it uses @code{rect} instead. The
16806 @kbd{I c p} command always uses @code{rect}.)
16811 The @kbd{c c} (@code{calc-clean}) [@code{pclean}] command ``cleans'' the
16812 number on the top of the stack. Floating point numbers are re-rounded
16813 according to the current precision. Polar numbers whose angular
16814 components have strayed from the @mathit{-180} to @mathit{+180} degree range
16815 are normalized. (Note that results will be undesirable if the current
16816 angular mode is different from the one under which the number was
16817 produced!) Integers and fractions are generally unaffected by this
16818 operation. Vectors and formulas are cleaned by cleaning each component
16819 number (i.e., pervasively).
16821 If the simplification mode is set below basic simplification, it is raised
16822 for the purposes of this command. Thus, @kbd{c c} applies the basic
16823 simplifications even if their automatic application is disabled.
16824 @xref{Simplification Modes}.
16826 @cindex Roundoff errors, correcting
16827 A numeric prefix argument to @kbd{c c} sets the floating-point precision
16828 to that value for the duration of the command. A positive prefix (of at
16829 least 3) sets the precision to the specified value; a negative or zero
16830 prefix decreases the precision by the specified amount.
16833 @pindex calc-clean-num
16834 The keystroke sequences @kbd{c 0} through @kbd{c 9} are equivalent
16835 to @kbd{c c} with the corresponding negative prefix argument. If roundoff
16836 errors have changed 2.0 into 1.999999, typing @kbd{c 1} to clip off one
16837 decimal place often conveniently does the trick.
16839 The @kbd{c c} command with a numeric prefix argument, and the @kbd{c 0}
16840 through @kbd{c 9} commands, also ``clip'' very small floating-point
16841 numbers to zero. If the exponent is less than or equal to the negative
16842 of the specified precision, the number is changed to 0.0. For example,
16843 if the current precision is 12, then @kbd{c 2} changes the vector
16844 @samp{[1e-8, 1e-9, 1e-10, 1e-11]} to @samp{[1e-8, 1e-9, 0, 0]}.
16845 Numbers this small generally arise from roundoff noise.
16847 If the numbers you are using really are legitimately this small,
16848 you should avoid using the @kbd{c 0} through @kbd{c 9} commands.
16849 (The plain @kbd{c c} command rounds to the current precision but
16850 does not clip small numbers.)
16852 One more property of @kbd{c 0} through @kbd{c 9}, and of @kbd{c c} with
16853 a prefix argument, is that integer-valued floats are converted to
16854 plain integers, so that @kbd{c 1} on @samp{[1., 1.5, 2., 2.5, 3.]}
16855 produces @samp{[1, 1.5, 2, 2.5, 3]}. This is not done for huge
16856 numbers (@samp{1e100} is technically an integer-valued float, but
16857 you wouldn't want it automatically converted to a 100-digit integer).
16862 With the Hyperbolic flag, @kbd{H c c} and @kbd{H c 0} through @kbd{H c 9}
16863 operate non-pervasively [@code{clean}].
16865 @node Date Arithmetic, Financial Functions, Conversions, Arithmetic
16866 @section Date Arithmetic
16869 @cindex Date arithmetic, additional functions
16870 The commands described in this section perform various conversions
16871 and calculations involving date forms (@pxref{Date Forms}). They
16872 use the @kbd{t} (for time/date) prefix key followed by shifted
16875 The simplest date arithmetic is done using the regular @kbd{+} and @kbd{-}
16876 commands. In particular, adding a number to a date form advances the
16877 date form by a certain number of days; adding an HMS form to a date
16878 form advances the date by a certain amount of time; and subtracting two
16879 date forms produces a difference measured in days. The commands
16880 described here provide additional, more specialized operations on dates.
16882 Many of these commands accept a numeric prefix argument; if you give
16883 plain @kbd{C-u} as the prefix, these commands will instead take the
16884 additional argument from the top of the stack.
16887 * Date Conversions::
16893 @node Date Conversions, Date Functions, Date Arithmetic, Date Arithmetic
16894 @subsection Date Conversions
16900 The @kbd{t D} (@code{calc-date}) [@code{date}] command converts a
16901 date form into a number, measured in days since Jan 1, 1 AD@. The
16902 result will be an integer if @var{date} is a pure date form, or a
16903 fraction or float if @var{date} is a date/time form. Or, if its
16904 argument is a number, it converts this number into a date form.
16906 With a numeric prefix argument, @kbd{t D} takes that many objects
16907 (up to six) from the top of the stack and interprets them in one
16908 of the following ways:
16910 The @samp{date(@var{year}, @var{month}, @var{day})} function
16911 builds a pure date form out of the specified year, month, and
16912 day, which must all be integers. @var{Year} is a year number,
16913 such as 1991 (@emph{not} the same as 91!). @var{Month} must be
16914 an integer in the range 1 to 12; @var{day} must be in the range
16915 1 to 31. If the specified month has fewer than 31 days and
16916 @var{day} is too large, the equivalent day in the following
16917 month will be used.
16919 The @samp{date(@var{month}, @var{day})} function builds a
16920 pure date form using the current year, as determined by the
16923 The @samp{date(@var{year}, @var{month}, @var{day}, @var{hms})}
16924 function builds a date/time form using an @var{hms} form.
16926 The @samp{date(@var{year}, @var{month}, @var{day}, @var{hour},
16927 @var{minute}, @var{second})} function builds a date/time form.
16928 @var{hour} should be an integer in the range 0 to 23;
16929 @var{minute} should be an integer in the range 0 to 59;
16930 @var{second} should be any real number in the range @samp{[0 .. 60)}.
16931 The last two arguments default to zero if omitted.
16934 @pindex calc-julian
16936 @cindex Julian day counts, conversions
16937 The @kbd{t J} (@code{calc-julian}) [@code{julian}] command converts
16938 a date form into a Julian day count, which is the number of days
16939 since noon (GMT) on Jan 1, 4713 BC@. A pure date is converted to an
16940 integer Julian count representing noon of that day. A date/time form
16941 is converted to an exact floating-point Julian count, adjusted to
16942 interpret the date form in the current time zone but the Julian
16943 day count in Greenwich Mean Time. A numeric prefix argument allows
16944 you to specify the time zone; @pxref{Time Zones}. Use a prefix of
16945 zero to suppress the time zone adjustment. Note that pure date forms
16946 are never time-zone adjusted.
16948 This command can also do the opposite conversion, from a Julian day
16949 count (either an integer day, or a floating-point day and time in
16950 the GMT zone), into a pure date form or a date/time form in the
16951 current or specified time zone.
16954 @pindex calc-unix-time
16956 @cindex Unix time format, conversions
16957 The @kbd{t U} (@code{calc-unix-time}) [@code{unixtime}] command
16958 converts a date form into a Unix time value, which is the number of
16959 seconds since midnight on Jan 1, 1970, or vice-versa. The numeric result
16960 will be an integer if the current precision is 12 or less; for higher
16961 precision, the result may be a float with (@var{precision}@minus{}12)
16962 digits after the decimal. Just as for @kbd{t J}, the numeric time
16963 is interpreted in the GMT time zone and the date form is interpreted
16964 in the current or specified zone. Some systems use Unix-like
16965 numbering but with the local time zone; give a prefix of zero to
16966 suppress the adjustment if so.
16969 @pindex calc-convert-time-zones
16971 @cindex Time Zones, converting between
16972 The @kbd{t C} (@code{calc-convert-time-zones}) [@code{tzconv}]
16973 command converts a date form from one time zone to another. You
16974 are prompted for each time zone name in turn; you can answer with
16975 any suitable Calc time zone expression (@pxref{Time Zones}).
16976 If you answer either prompt with a blank line, the local time
16977 zone is used for that prompt. You can also answer the first
16978 prompt with @kbd{$} to take the two time zone names from the
16979 stack (and the date to be converted from the third stack level).
16981 @node Date Functions, Business Days, Date Conversions, Date Arithmetic
16982 @subsection Date Functions
16988 The @kbd{t N} (@code{calc-now}) [@code{now}] command pushes the
16989 current date and time on the stack as a date form. The time is
16990 reported in terms of the specified time zone; with no numeric prefix
16991 argument, @kbd{t N} reports for the current time zone.
16994 @pindex calc-date-part
16995 The @kbd{t P} (@code{calc-date-part}) command extracts one part
16996 of a date form. The prefix argument specifies the part; with no
16997 argument, this command prompts for a part code from 1 to 9.
16998 The various part codes are described in the following paragraphs.
17001 The @kbd{M-1 t P} [@code{year}] function extracts the year number
17002 from a date form as an integer, e.g., 1991. This and the
17003 following functions will also accept a real number for an
17004 argument, which is interpreted as a standard Calc day number.
17005 Note that this function will never return zero, since the year
17006 1 BC immediately precedes the year 1 AD.
17009 The @kbd{M-2 t P} [@code{month}] function extracts the month number
17010 from a date form as an integer in the range 1 to 12.
17013 The @kbd{M-3 t P} [@code{day}] function extracts the day number
17014 from a date form as an integer in the range 1 to 31.
17017 The @kbd{M-4 t P} [@code{hour}] function extracts the hour from
17018 a date form as an integer in the range 0 (midnight) to 23. Note
17019 that 24-hour time is always used. This returns zero for a pure
17020 date form. This function (and the following two) also accept
17021 HMS forms as input.
17024 The @kbd{M-5 t P} [@code{minute}] function extracts the minute
17025 from a date form as an integer in the range 0 to 59.
17028 The @kbd{M-6 t P} [@code{second}] function extracts the second
17029 from a date form. If the current precision is 12 or less,
17030 the result is an integer in the range 0 to 59. For higher
17031 precision, the result may instead be a floating-point number.
17034 The @kbd{M-7 t P} [@code{weekday}] function extracts the weekday
17035 number from a date form as an integer in the range 0 (Sunday)
17039 The @kbd{M-8 t P} [@code{yearday}] function extracts the day-of-year
17040 number from a date form as an integer in the range 1 (January 1)
17041 to 366 (December 31 of a leap year).
17044 The @kbd{M-9 t P} [@code{time}] function extracts the time portion
17045 of a date form as an HMS form. This returns @samp{0@@ 0' 0"}
17046 for a pure date form.
17049 @pindex calc-new-month
17051 The @kbd{t M} (@code{calc-new-month}) [@code{newmonth}] command
17052 computes a new date form that represents the first day of the month
17053 specified by the input date. The result is always a pure date
17054 form; only the year and month numbers of the input are retained.
17055 With a numeric prefix argument @var{n} in the range from 1 to 31,
17056 @kbd{t M} computes the @var{n}th day of the month. (If @var{n}
17057 is greater than the actual number of days in the month, or if
17058 @var{n} is zero, the last day of the month is used.)
17061 @pindex calc-new-year
17063 The @kbd{t Y} (@code{calc-new-year}) [@code{newyear}] command
17064 computes a new pure date form that represents the first day of
17065 the year specified by the input. The month, day, and time
17066 of the input date form are lost. With a numeric prefix argument
17067 @var{n} in the range from 1 to 366, @kbd{t Y} computes the
17068 @var{n}th day of the year (366 is treated as 365 in non-leap
17069 years). A prefix argument of 0 computes the last day of the
17070 year (December 31). A negative prefix argument from @mathit{-1} to
17071 @mathit{-12} computes the first day of the @var{n}th month of the year.
17074 @pindex calc-new-week
17076 The @kbd{t W} (@code{calc-new-week}) [@code{newweek}] command
17077 computes a new pure date form that represents the Sunday on or before
17078 the input date. With a numeric prefix argument, it can be made to
17079 use any day of the week as the starting day; the argument must be in
17080 the range from 0 (Sunday) to 6 (Saturday). This function always
17081 subtracts between 0 and 6 days from the input date.
17083 Here's an example use of @code{newweek}: Find the date of the next
17084 Wednesday after a given date. Using @kbd{M-3 t W} or @samp{newweek(d, 3)}
17085 will give you the @emph{preceding} Wednesday, so @samp{newweek(d+7, 3)}
17086 will give you the following Wednesday. A further look at the definition
17087 of @code{newweek} shows that if the input date is itself a Wednesday,
17088 this formula will return the Wednesday one week in the future. An
17089 exercise for the reader is to modify this formula to yield the same day
17090 if the input is already a Wednesday. Another interesting exercise is
17091 to preserve the time-of-day portion of the input (@code{newweek} resets
17092 the time to midnight; hint: how can @code{newweek} be defined in terms
17093 of the @code{weekday} function?).
17099 The @samp{pwday(@var{date})} function (not on any key) computes the
17100 day-of-month number of the Sunday on or before @var{date}. With
17101 two arguments, @samp{pwday(@var{date}, @var{day})} computes the day
17102 number of the Sunday on or before day number @var{day} of the month
17103 specified by @var{date}. The @var{day} must be in the range from
17104 7 to 31; if the day number is greater than the actual number of days
17105 in the month, the true number of days is used instead. Thus
17106 @samp{pwday(@var{date}, 7)} finds the first Sunday of the month, and
17107 @samp{pwday(@var{date}, 31)} finds the last Sunday of the month.
17108 With a third @var{weekday} argument, @code{pwday} can be made to look
17109 for any day of the week instead of Sunday.
17112 @pindex calc-inc-month
17114 The @kbd{t I} (@code{calc-inc-month}) [@code{incmonth}] command
17115 increases a date form by one month, or by an arbitrary number of
17116 months specified by a numeric prefix argument. The time portion,
17117 if any, of the date form stays the same. The day also stays the
17118 same, except that if the new month has fewer days the day
17119 number may be reduced to lie in the valid range. For example,
17120 @samp{incmonth(<Jan 31, 1991>)} produces @samp{<Feb 28, 1991>}.
17121 Because of this, @kbd{t I t I} and @kbd{M-2 t I} do not always give
17122 the same results (@samp{<Mar 28, 1991>} versus @samp{<Mar 31, 1991>}
17129 The @samp{incyear(@var{date}, @var{step})} function increases
17130 a date form by the specified number of years, which may be
17131 any positive or negative integer. Note that @samp{incyear(d, n)}
17132 is equivalent to @w{@samp{incmonth(d, 12*n)}}, but these do not have
17133 simple equivalents in terms of day arithmetic because
17134 months and years have varying lengths. If the @var{step}
17135 argument is omitted, 1 year is assumed. There is no keyboard
17136 command for this function; use @kbd{C-u 12 t I} instead.
17138 There is no @code{newday} function at all because @kbd{F} [@code{floor}]
17139 serves this purpose. Similarly, instead of @code{incday} and
17140 @code{incweek} simply use @expr{d + n} or @expr{d + 7 n}.
17142 @xref{Basic Arithmetic}, for the @kbd{f ]} [@code{incr}] command
17143 which can adjust a date/time form by a certain number of seconds.
17145 @node Business Days, Time Zones, Date Functions, Date Arithmetic
17146 @subsection Business Days
17149 Often time is measured in ``business days'' or ``working days,''
17150 where weekends and holidays are skipped. Calc's normal date
17151 arithmetic functions use calendar days, so that subtracting two
17152 consecutive Mondays will yield a difference of 7 days. By contrast,
17153 subtracting two consecutive Mondays would yield 5 business days
17154 (assuming two-day weekends and the absence of holidays).
17160 @pindex calc-business-days-plus
17161 @pindex calc-business-days-minus
17162 The @kbd{t +} (@code{calc-business-days-plus}) [@code{badd}]
17163 and @kbd{t -} (@code{calc-business-days-minus}) [@code{bsub}]
17164 commands perform arithmetic using business days. For @kbd{t +},
17165 one argument must be a date form and the other must be a real
17166 number (positive or negative). If the number is not an integer,
17167 then a certain amount of time is added as well as a number of
17168 days; for example, adding 0.5 business days to a time in Friday
17169 evening will produce a time in Monday morning. It is also
17170 possible to add an HMS form; adding @samp{12@@ 0' 0"} also adds
17171 half a business day. For @kbd{t -}, the arguments are either a
17172 date form and a number or HMS form, or two date forms, in which
17173 case the result is the number of business days between the two
17176 @cindex @code{Holidays} variable
17178 By default, Calc considers any day that is not a Saturday or
17179 Sunday to be a business day. You can define any number of
17180 additional holidays by editing the variable @code{Holidays}.
17181 (There is an @w{@kbd{s H}} convenience command for editing this
17182 variable.) Initially, @code{Holidays} contains the vector
17183 @samp{[sat, sun]}. Entries in the @code{Holidays} vector may
17184 be any of the following kinds of objects:
17188 Date forms (pure dates, not date/time forms). These specify
17189 particular days which are to be treated as holidays.
17192 Intervals of date forms. These specify a range of days, all of
17193 which are holidays (e.g., Christmas week). @xref{Interval Forms}.
17196 Nested vectors of date forms. Each date form in the vector is
17197 considered to be a holiday.
17200 Any Calc formula which evaluates to one of the above three things.
17201 If the formula involves the variable @expr{y}, it stands for a
17202 yearly repeating holiday; @expr{y} will take on various year
17203 numbers like 1992. For example, @samp{date(y, 12, 25)} specifies
17204 Christmas day, and @samp{newweek(date(y, 11, 7), 4) + 21} specifies
17205 Thanksgiving (which is held on the fourth Thursday of November).
17206 If the formula involves the variable @expr{m}, that variable
17207 takes on month numbers from 1 to 12: @samp{date(y, m, 15)} is
17208 a holiday that takes place on the 15th of every month.
17211 A weekday name, such as @code{sat} or @code{sun}. This is really
17212 a variable whose name is a three-letter, lower-case day name.
17215 An interval of year numbers (integers). This specifies the span of
17216 years over which this holiday list is to be considered valid. Any
17217 business-day arithmetic that goes outside this range will result
17218 in an error message. Use this if you are including an explicit
17219 list of holidays, rather than a formula to generate them, and you
17220 want to make sure you don't accidentally go beyond the last point
17221 where the holidays you entered are complete. If there is no
17222 limiting interval in the @code{Holidays} vector, the default
17223 @samp{[1 .. 2737]} is used. (This is the absolute range of years
17224 for which Calc's business-day algorithms will operate.)
17227 An interval of HMS forms. This specifies the span of hours that
17228 are to be considered one business day. For example, if this
17229 range is @samp{[9@@ 0' 0" .. 17@@ 0' 0"]} (i.e., 9am to 5pm), then
17230 the business day is only eight hours long, so that @kbd{1.5 t +}
17231 on @samp{<4:00pm Fri Dec 13, 1991>} will add one business day and
17232 four business hours to produce @samp{<12:00pm Tue Dec 17, 1991>}.
17233 Likewise, @kbd{t -} will now express differences in time as
17234 fractions of an eight-hour day. Times before 9am will be treated
17235 as 9am by business date arithmetic, and times at or after 5pm will
17236 be treated as 4:59:59pm. If there is no HMS interval in @code{Holidays},
17237 the full 24-hour day @samp{[0@ 0' 0" .. 24@ 0' 0"]} is assumed.
17238 (Regardless of the type of bounds you specify, the interval is
17239 treated as inclusive on the low end and exclusive on the high end,
17240 so that the work day goes from 9am up to, but not including, 5pm.)
17243 If the @code{Holidays} vector is empty, then @kbd{t +} and
17244 @kbd{t -} will act just like @kbd{+} and @kbd{-} because there will
17245 then be no difference between business days and calendar days.
17247 Calc expands the intervals and formulas you give into a complete
17248 list of holidays for internal use. This is done mainly to make
17249 sure it can detect multiple holidays. (For example,
17250 @samp{<Jan 1, 1989>} is both New Year's Day and a Sunday, but
17251 Calc's algorithms take care to count it only once when figuring
17252 the number of holidays between two dates.)
17254 Since the complete list of holidays for all the years from 1 to
17255 2737 would be huge, Calc actually computes only the part of the
17256 list between the smallest and largest years that have been involved
17257 in business-day calculations so far. Normally, you won't have to
17258 worry about this. Keep in mind, however, that if you do one
17259 calculation for 1992, and another for 1792, even if both involve
17260 only a small range of years, Calc will still work out all the
17261 holidays that fall in that 200-year span.
17263 If you add a (positive) number of days to a date form that falls on a
17264 weekend or holiday, the date form is treated as if it were the most
17265 recent business day. (Thus adding one business day to a Friday,
17266 Saturday, or Sunday will all yield the following Monday.) If you
17267 subtract a number of days from a weekend or holiday, the date is
17268 effectively on the following business day. (So subtracting one business
17269 day from Saturday, Sunday, or Monday yields the preceding Friday.) The
17270 difference between two dates one or both of which fall on holidays
17271 equals the number of actual business days between them. These
17272 conventions are consistent in the sense that, if you add @var{n}
17273 business days to any date, the difference between the result and the
17274 original date will come out to @var{n} business days. (It can't be
17275 completely consistent though; a subtraction followed by an addition
17276 might come out a bit differently, since @kbd{t +} is incapable of
17277 producing a date that falls on a weekend or holiday.)
17283 There is a @code{holiday} function, not on any keys, that takes
17284 any date form and returns 1 if that date falls on a weekend or
17285 holiday, as defined in @code{Holidays}, or 0 if the date is a
17288 @node Time Zones, , Business Days, Date Arithmetic
17289 @subsection Time Zones
17293 @cindex Daylight saving time
17294 Time zones and daylight saving time are a complicated business.
17295 The conversions to and from Julian and Unix-style dates automatically
17296 compute the correct time zone and daylight saving adjustment to use,
17297 provided they can figure out this information. This section describes
17298 Calc's time zone adjustment algorithm in detail, in case you want to
17299 do conversions in different time zones or in case Calc's algorithms
17300 can't determine the right correction to use.
17302 Adjustments for time zones and daylight saving time are done by
17303 @kbd{t U}, @kbd{t J}, @kbd{t N}, and @kbd{t C}, but not by any other
17304 commands. In particular, @samp{<may 1 1991> - <apr 1 1991>} evaluates
17305 to exactly 30 days even though there is a daylight-saving
17306 transition in between. This is also true for Julian pure dates:
17307 @samp{julian(<may 1 1991>) - julian(<apr 1 1991>)}. But Julian
17308 and Unix date/times will adjust for daylight saving time: using Calc's
17309 default daylight saving time rule (see the explanation below),
17310 @samp{julian(<12am may 1 1991>) - julian(<12am apr 1 1991>)}
17311 evaluates to @samp{29.95833} (that's 29 days and 23 hours)
17312 because one hour was lost when daylight saving commenced on
17315 In brief, the idiom @samp{julian(@var{date1}) - julian(@var{date2})}
17316 computes the actual number of 24-hour periods between two dates, whereas
17317 @samp{@var{date1} - @var{date2}} computes the number of calendar
17318 days between two dates without taking daylight saving into account.
17320 @pindex calc-time-zone
17325 The @code{calc-time-zone} [@code{tzone}] command converts the time
17326 zone specified by its numeric prefix argument into a number of
17327 seconds difference from Greenwich mean time (GMT). If the argument
17328 is a number, the result is simply that value multiplied by 3600.
17329 Typical arguments for North America are 5 (Eastern) or 8 (Pacific). If
17330 Daylight Saving time is in effect, one hour should be subtracted from
17331 the normal difference.
17333 If you give a prefix of plain @kbd{C-u}, @code{calc-time-zone} (like other
17334 date arithmetic commands that include a time zone argument) takes the
17335 zone argument from the top of the stack. (In the case of @kbd{t J}
17336 and @kbd{t U}, the normal argument is then taken from the second-to-top
17337 stack position.) This allows you to give a non-integer time zone
17338 adjustment. The time-zone argument can also be an HMS form, or
17339 it can be a variable which is a time zone name in upper- or lower-case.
17340 For example @samp{tzone(PST) = tzone(8)} and @samp{tzone(pdt) = tzone(7)}
17341 (for Pacific standard and daylight saving times, respectively).
17343 North American and European time zone names are defined as follows;
17344 note that for each time zone there is one name for standard time,
17345 another for daylight saving time, and a third for ``generalized'' time
17346 in which the daylight saving adjustment is computed from context.
17350 YST PST MST CST EST AST NST GMT WET MET MEZ
17351 9 8 7 6 5 4 3.5 0 -1 -2 -2
17353 YDT PDT MDT CDT EDT ADT NDT BST WETDST METDST MESZ
17354 8 7 6 5 4 3 2.5 -1 -2 -3 -3
17356 YGT PGT MGT CGT EGT AGT NGT BGT WEGT MEGT MEGZ
17357 9/8 8/7 7/6 6/5 5/4 4/3 3.5/2.5 0/-1 -1/-2 -2/-3 -2/-3
17361 @vindex math-tzone-names
17362 To define time zone names that do not appear in the above table,
17363 you must modify the Lisp variable @code{math-tzone-names}. This
17364 is a list of lists describing the different time zone names; its
17365 structure is best explained by an example. The three entries for
17366 Pacific Time look like this:
17370 ( ( "PST" 8 0 ) ; Name as an upper-case string, then standard
17371 ( "PDT" 8 -1 ) ; adjustment, then daylight saving adjustment.
17372 ( "PGT" 8 "PST" "PDT" ) ) ; Generalized time zone.
17376 @cindex @code{TimeZone} variable
17378 With no arguments, @code{calc-time-zone} or @samp{tzone()} will by
17379 default get the time zone and daylight saving information from the
17380 calendar (@pxref{Daylight Saving,Calendar/Diary,The Calendar and the Diary,
17381 emacs,The GNU Emacs Manual}). To use a different time zone, or if the
17382 calendar does not give the desired result, you can set the Calc variable
17383 @code{TimeZone} (which is by default @code{nil}) to an appropriate
17384 time zone name. (The easiest way to do this is to edit the
17385 @code{TimeZone} variable using Calc's @kbd{s T} command, then use the
17386 @kbd{s p} (@code{calc-permanent-variable}) command to save the value of
17387 @code{TimeZone} permanently.)
17388 If the time zone given by @code{TimeZone} is a generalized time zone,
17389 e.g., @code{EGT}, Calc examines the date being converted to tell whether
17390 to use standard or daylight saving time. But if the current time zone
17391 is explicit, e.g., @code{EST} or @code{EDT}, then that adjustment is
17392 used exactly and Calc's daylight saving algorithm is not consulted.
17393 The special time zone name @code{local}
17394 is equivalent to no argument; i.e., it uses the information obtained
17397 The @kbd{t J} and @code{t U} commands with no numeric prefix
17398 arguments do the same thing as @samp{tzone()}; namely, use the
17399 information from the calendar if @code{TimeZone} is @code{nil},
17400 otherwise use the time zone given by @code{TimeZone}.
17402 @vindex math-daylight-savings-hook
17403 @findex math-std-daylight-savings
17404 When Calc computes the daylight saving information itself (i.e., when
17405 the @code{TimeZone} variable is set), it will by default consider
17406 daylight saving time to begin at 2 a.m.@: on the second Sunday of March
17407 (for years from 2007 on) or on the last Sunday in April (for years
17408 before 2007), and to end at 2 a.m.@: on the first Sunday of
17409 November. (for years from 2007 on) or the last Sunday in October (for
17410 years before 2007). These are the rules that have been in effect in
17411 much of North America since 1966 and take into account the rule change
17412 that began in 2007. If you are in a country that uses different rules
17413 for computing daylight saving time, you have two choices: Write your own
17414 daylight saving hook, or control time zones explicitly by setting the
17415 @code{TimeZone} variable and/or always giving a time-zone argument for
17416 the conversion functions.
17418 The Lisp variable @code{math-daylight-savings-hook} holds the
17419 name of a function that is used to compute the daylight saving
17420 adjustment for a given date. The default is
17421 @code{math-std-daylight-savings}, which computes an adjustment
17422 (either 0 or @mathit{-1}) using the North American rules given above.
17424 The daylight saving hook function is called with four arguments:
17425 The date, as a floating-point number in standard Calc format;
17426 a six-element list of the date decomposed into year, month, day,
17427 hour, minute, and second, respectively; a string which contains
17428 the generalized time zone name in upper-case, e.g., @code{"WEGT"};
17429 and a special adjustment to be applied to the hour value when
17430 converting into a generalized time zone (see below).
17432 @findex math-prev-weekday-in-month
17433 The Lisp function @code{math-prev-weekday-in-month} is useful for
17434 daylight saving computations. This is an internal version of
17435 the user-level @code{pwday} function described in the previous
17436 section. It takes four arguments: The floating-point date value,
17437 the corresponding six-element date list, the day-of-month number,
17438 and the weekday number (0-6).
17440 The default daylight saving hook ignores the time zone name, but a
17441 more sophisticated hook could use different algorithms for different
17442 time zones. It would also be possible to use different algorithms
17443 depending on the year number, but the default hook always uses the
17444 algorithm for 1987 and later. Here is a listing of the default
17445 daylight saving hook:
17448 (defun math-std-daylight-savings (date dt zone bump)
17449 (cond ((< (nth 1 dt) 4) 0)
17451 (let ((sunday (math-prev-weekday-in-month date dt 7 0)))
17452 (cond ((< (nth 2 dt) sunday) 0)
17453 ((= (nth 2 dt) sunday)
17454 (if (>= (nth 3 dt) (+ 3 bump)) -1 0))
17456 ((< (nth 1 dt) 10) -1)
17458 (let ((sunday (math-prev-weekday-in-month date dt 31 0)))
17459 (cond ((< (nth 2 dt) sunday) -1)
17460 ((= (nth 2 dt) sunday)
17461 (if (>= (nth 3 dt) (+ 2 bump)) 0 -1))
17468 The @code{bump} parameter is equal to zero when Calc is converting
17469 from a date form in a generalized time zone into a GMT date value.
17470 It is @mathit{-1} when Calc is converting in the other direction. The
17471 adjustments shown above ensure that the conversion behaves correctly
17472 and reasonably around the 2 a.m.@: transition in each direction.
17474 There is a ``missing'' hour between 2 a.m.@: and 3 a.m.@: at the
17475 beginning of daylight saving time; converting a date/time form that
17476 falls in this hour results in a time value for the following hour,
17477 from 3 a.m.@: to 4 a.m. At the end of daylight saving time, the
17478 hour from 1 a.m.@: to 2 a.m.@: repeats itself; converting a date/time
17479 form that falls in this hour results in a time value for the first
17480 manifestation of that time (@emph{not} the one that occurs one hour
17483 If @code{math-daylight-savings-hook} is @code{nil}, then the
17484 daylight saving adjustment is always taken to be zero.
17486 In algebraic formulas, @samp{tzone(@var{zone}, @var{date})}
17487 computes the time zone adjustment for a given zone name at a
17488 given date. The @var{date} is ignored unless @var{zone} is a
17489 generalized time zone. If @var{date} is a date form, the
17490 daylight saving computation is applied to it as it appears.
17491 If @var{date} is a numeric date value, it is adjusted for the
17492 daylight-saving version of @var{zone} before being given to
17493 the daylight saving hook. This odd-sounding rule ensures
17494 that the daylight-saving computation is always done in
17495 local time, not in the GMT time that a numeric @var{date}
17496 is typically represented in.
17502 The @samp{dsadj(@var{date}, @var{zone})} function computes the
17503 daylight saving adjustment that is appropriate for @var{date} in
17504 time zone @var{zone}. If @var{zone} is explicitly in or not in
17505 daylight saving time (e.g., @code{PDT} or @code{PST}) the
17506 @var{date} is ignored. If @var{zone} is a generalized time zone,
17507 the algorithms described above are used. If @var{zone} is omitted,
17508 the computation is done for the current time zone.
17510 @node Financial Functions, Binary Functions, Date Arithmetic, Arithmetic
17511 @section Financial Functions
17514 Calc's financial or business functions use the @kbd{b} prefix
17515 key followed by a shifted letter. (The @kbd{b} prefix followed by
17516 a lower-case letter is used for operations on binary numbers.)
17518 Note that the rate and the number of intervals given to these
17519 functions must be on the same time scale, e.g., both months or
17520 both years. Mixing an annual interest rate with a time expressed
17521 in months will give you very wrong answers!
17523 It is wise to compute these functions to a higher precision than
17524 you really need, just to make sure your answer is correct to the
17525 last penny; also, you may wish to check the definitions at the end
17526 of this section to make sure the functions have the meaning you expect.
17532 * Related Financial Functions::
17533 * Depreciation Functions::
17534 * Definitions of Financial Functions::
17537 @node Percentages, Future Value, Financial Functions, Financial Functions
17538 @subsection Percentages
17541 @pindex calc-percent
17544 The @kbd{M-%} (@code{calc-percent}) command takes a percentage value,
17545 say 5.4, and converts it to an equivalent actual number. For example,
17546 @kbd{5.4 M-%} enters 0.054 on the stack. (That's the @key{META} or
17547 @key{ESC} key combined with @kbd{%}.)
17549 Actually, @kbd{M-%} creates a formula of the form @samp{5.4%}.
17550 You can enter @samp{5.4%} yourself during algebraic entry. The
17551 @samp{%} operator simply means, ``the preceding value divided by
17552 100.'' The @samp{%} operator has very high precedence, so that
17553 @samp{1+8%} is interpreted as @samp{1+(8%)}, not as @samp{(1+8)%}.
17554 (The @samp{%} operator is just a postfix notation for the
17555 @code{percent} function, just like @samp{20!} is the notation for
17556 @samp{fact(20)}, or twenty-factorial.)
17558 The formula @samp{5.4%} would normally evaluate immediately to
17559 0.054, but the @kbd{M-%} command suppresses evaluation as it puts
17560 the formula onto the stack. However, the next Calc command that
17561 uses the formula @samp{5.4%} will evaluate it as its first step.
17562 The net effect is that you get to look at @samp{5.4%} on the stack,
17563 but Calc commands see it as @samp{0.054}, which is what they expect.
17565 In particular, @samp{5.4%} and @samp{0.054} are suitable values
17566 for the @var{rate} arguments of the various financial functions,
17567 but the number @samp{5.4} is probably @emph{not} suitable---it
17568 represents a rate of 540 percent!
17570 The key sequence @kbd{M-% *} effectively means ``percent-of.''
17571 For example, @kbd{68 @key{RET} 25 M-% *} computes 17, which is 25% of
17572 68 (and also 68% of 25, which comes out to the same thing).
17575 @pindex calc-convert-percent
17576 The @kbd{c %} (@code{calc-convert-percent}) command converts the
17577 value on the top of the stack from numeric to percentage form.
17578 For example, if 0.08 is on the stack, @kbd{c %} converts it to
17579 @samp{8%}. The quantity is the same, it's just represented
17580 differently. (Contrast this with @kbd{M-%}, which would convert
17581 this number to @samp{0.08%}.) The @kbd{=} key is a convenient way
17582 to convert a formula like @samp{8%} back to numeric form, 0.08.
17584 To compute what percentage one quantity is of another quantity,
17585 use @kbd{/ c %}. For example, @w{@kbd{17 @key{RET} 68 / c %}} displays
17589 @pindex calc-percent-change
17591 The @kbd{b %} (@code{calc-percent-change}) [@code{relch}] command
17592 calculates the percentage change from one number to another.
17593 For example, @kbd{40 @key{RET} 50 b %} produces the answer @samp{25%},
17594 since 50 is 25% larger than 40. A negative result represents a
17595 decrease: @kbd{50 @key{RET} 40 b %} produces @samp{-20%}, since 40 is
17596 20% smaller than 50. (The answers are different in magnitude
17597 because, in the first case, we're increasing by 25% of 40, but
17598 in the second case, we're decreasing by 20% of 50.) The effect
17599 of @kbd{40 @key{RET} 50 b %} is to compute @expr{(50-40)/40}, converting
17600 the answer to percentage form as if by @kbd{c %}.
17602 @node Future Value, Present Value, Percentages, Financial Functions
17603 @subsection Future Value
17607 @pindex calc-fin-fv
17609 The @kbd{b F} (@code{calc-fin-fv}) [@code{fv}] command computes
17610 the future value of an investment. It takes three arguments
17611 from the stack: @samp{fv(@var{rate}, @var{n}, @var{payment})}.
17612 If you give payments of @var{payment} every year for @var{n}
17613 years, and the money you have paid earns interest at @var{rate} per
17614 year, then this function tells you what your investment would be
17615 worth at the end of the period. (The actual interval doesn't
17616 have to be years, as long as @var{n} and @var{rate} are expressed
17617 in terms of the same intervals.) This function assumes payments
17618 occur at the @emph{end} of each interval.
17622 The @kbd{I b F} [@code{fvb}] command does the same computation,
17623 but assuming your payments are at the beginning of each interval.
17624 Suppose you plan to deposit $1000 per year in a savings account
17625 earning 5.4% interest, starting right now. How much will be
17626 in the account after five years? @code{fvb(5.4%, 5, 1000) = 5870.73}.
17627 Thus you will have earned $870 worth of interest over the years.
17628 Using the stack, this calculation would have been
17629 @kbd{5.4 M-% 5 @key{RET} 1000 I b F}. Note that the rate is expressed
17630 as a number between 0 and 1, @emph{not} as a percentage.
17634 The @kbd{H b F} [@code{fvl}] command computes the future value
17635 of an initial lump sum investment. Suppose you could deposit
17636 those five thousand dollars in the bank right now; how much would
17637 they be worth in five years? @code{fvl(5.4%, 5, 5000) = 6503.89}.
17639 The algebraic functions @code{fv} and @code{fvb} accept an optional
17640 fourth argument, which is used as an initial lump sum in the sense
17641 of @code{fvl}. In other words, @code{fv(@var{rate}, @var{n},
17642 @var{payment}, @var{initial}) = fv(@var{rate}, @var{n}, @var{payment})
17643 + fvl(@var{rate}, @var{n}, @var{initial})}.
17645 To illustrate the relationships between these functions, we could
17646 do the @code{fvb} calculation ``by hand'' using @code{fvl}. The
17647 final balance will be the sum of the contributions of our five
17648 deposits at various times. The first deposit earns interest for
17649 five years: @code{fvl(5.4%, 5, 1000) = 1300.78}. The second
17650 deposit only earns interest for four years: @code{fvl(5.4%, 4, 1000) =
17651 1234.13}. And so on down to the last deposit, which earns one
17652 year's interest: @code{fvl(5.4%, 1, 1000) = 1054.00}. The sum of
17653 these five values is, sure enough, $5870.73, just as was computed
17654 by @code{fvb} directly.
17656 What does @code{fv(5.4%, 5, 1000) = 5569.96} mean? The payments
17657 are now at the ends of the periods. The end of one year is the same
17658 as the beginning of the next, so what this really means is that we've
17659 lost the payment at year zero (which contributed $1300.78), but we're
17660 now counting the payment at year five (which, since it didn't have
17661 a chance to earn interest, counts as $1000). Indeed, @expr{5569.96 =
17662 5870.73 - 1300.78 + 1000} (give or take a bit of roundoff error).
17664 @node Present Value, Related Financial Functions, Future Value, Financial Functions
17665 @subsection Present Value
17669 @pindex calc-fin-pv
17671 The @kbd{b P} (@code{calc-fin-pv}) [@code{pv}] command computes
17672 the present value of an investment. Like @code{fv}, it takes
17673 three arguments: @code{pv(@var{rate}, @var{n}, @var{payment})}.
17674 It computes the present value of a series of regular payments.
17675 Suppose you have the chance to make an investment that will
17676 pay $2000 per year over the next four years; as you receive
17677 these payments you can put them in the bank at 9% interest.
17678 You want to know whether it is better to make the investment, or
17679 to keep the money in the bank where it earns 9% interest right
17680 from the start. The calculation @code{pv(9%, 4, 2000)} gives the
17681 result 6479.44. If your initial investment must be less than this,
17682 say, $6000, then the investment is worthwhile. But if you had to
17683 put up $7000, then it would be better just to leave it in the bank.
17685 Here is the interpretation of the result of @code{pv}: You are
17686 trying to compare the return from the investment you are
17687 considering, which is @code{fv(9%, 4, 2000) = 9146.26}, with
17688 the return from leaving the money in the bank, which is
17689 @code{fvl(9%, 4, @var{x})} where @var{x} is the amount of money
17690 you would have to put up in advance. The @code{pv} function
17691 finds the break-even point, @expr{x = 6479.44}, at which
17692 @code{fvl(9%, 4, 6479.44)} is also equal to 9146.26. This is
17693 the largest amount you should be willing to invest.
17697 The @kbd{I b P} [@code{pvb}] command solves the same problem,
17698 but with payments occurring at the beginning of each interval.
17699 It has the same relationship to @code{fvb} as @code{pv} has
17700 to @code{fv}. For example @code{pvb(9%, 4, 2000) = 7062.59},
17701 a larger number than @code{pv} produced because we get to start
17702 earning interest on the return from our investment sooner.
17706 The @kbd{H b P} [@code{pvl}] command computes the present value of
17707 an investment that will pay off in one lump sum at the end of the
17708 period. For example, if we get our $8000 all at the end of the
17709 four years, @code{pvl(9%, 4, 8000) = 5667.40}. This is much
17710 less than @code{pv} reported, because we don't earn any interest
17711 on the return from this investment. Note that @code{pvl} and
17712 @code{fvl} are simple inverses: @code{fvl(9%, 4, 5667.40) = 8000}.
17714 You can give an optional fourth lump-sum argument to @code{pv}
17715 and @code{pvb}; this is handled in exactly the same way as the
17716 fourth argument for @code{fv} and @code{fvb}.
17719 @pindex calc-fin-npv
17721 The @kbd{b N} (@code{calc-fin-npv}) [@code{npv}] command computes
17722 the net present value of a series of irregular investments.
17723 The first argument is the interest rate. The second argument is
17724 a vector which represents the expected return from the investment
17725 at the end of each interval. For example, if the rate represents
17726 a yearly interest rate, then the vector elements are the return
17727 from the first year, second year, and so on.
17729 Thus, @code{npv(9%, [2000,2000,2000,2000]) = pv(9%, 4, 2000) = 6479.44}.
17730 Obviously this function is more interesting when the payments are
17733 The @code{npv} function can actually have two or more arguments.
17734 Multiple arguments are interpreted in the same way as for the
17735 vector statistical functions like @code{vsum}.
17736 @xref{Single-Variable Statistics}. Basically, if there are several
17737 payment arguments, each either a vector or a plain number, all these
17738 values are collected left-to-right into the complete list of payments.
17739 A numeric prefix argument on the @kbd{b N} command says how many
17740 payment values or vectors to take from the stack.
17744 The @kbd{I b N} [@code{npvb}] command computes the net present
17745 value where payments occur at the beginning of each interval
17746 rather than at the end.
17748 @node Related Financial Functions, Depreciation Functions, Present Value, Financial Functions
17749 @subsection Related Financial Functions
17752 The functions in this section are basically inverses of the
17753 present value functions with respect to the various arguments.
17756 @pindex calc-fin-pmt
17758 The @kbd{b M} (@code{calc-fin-pmt}) [@code{pmt}] command computes
17759 the amount of periodic payment necessary to amortize a loan.
17760 Thus @code{pmt(@var{rate}, @var{n}, @var{amount})} equals the
17761 value of @var{payment} such that @code{pv(@var{rate}, @var{n},
17762 @var{payment}) = @var{amount}}.
17766 The @kbd{I b M} [@code{pmtb}] command does the same computation
17767 but using @code{pvb} instead of @code{pv}. Like @code{pv} and
17768 @code{pvb}, these functions can also take a fourth argument which
17769 represents an initial lump-sum investment.
17772 The @kbd{H b M} key just invokes the @code{fvl} function, which is
17773 the inverse of @code{pvl}. There is no explicit @code{pmtl} function.
17776 @pindex calc-fin-nper
17778 The @kbd{b #} (@code{calc-fin-nper}) [@code{nper}] command computes
17779 the number of regular payments necessary to amortize a loan.
17780 Thus @code{nper(@var{rate}, @var{payment}, @var{amount})} equals
17781 the value of @var{n} such that @code{pv(@var{rate}, @var{n},
17782 @var{payment}) = @var{amount}}. If @var{payment} is too small
17783 ever to amortize a loan for @var{amount} at interest rate @var{rate},
17784 the @code{nper} function is left in symbolic form.
17788 The @kbd{I b #} [@code{nperb}] command does the same computation
17789 but using @code{pvb} instead of @code{pv}. You can give a fourth
17790 lump-sum argument to these functions, but the computation will be
17791 rather slow in the four-argument case.
17795 The @kbd{H b #} [@code{nperl}] command does the same computation
17796 using @code{pvl}. By exchanging @var{payment} and @var{amount} you
17797 can also get the solution for @code{fvl}. For example,
17798 @code{nperl(8%, 2000, 1000) = 9.006}, so if you place $1000 in a
17799 bank account earning 8%, it will take nine years to grow to $2000.
17802 @pindex calc-fin-rate
17804 The @kbd{b T} (@code{calc-fin-rate}) [@code{rate}] command computes
17805 the rate of return on an investment. This is also an inverse of @code{pv}:
17806 @code{rate(@var{n}, @var{payment}, @var{amount})} computes the value of
17807 @var{rate} such that @code{pv(@var{rate}, @var{n}, @var{payment}) =
17808 @var{amount}}. The result is expressed as a formula like @samp{6.3%}.
17814 The @kbd{I b T} [@code{rateb}] and @kbd{H b T} [@code{ratel}]
17815 commands solve the analogous equations with @code{pvb} or @code{pvl}
17816 in place of @code{pv}. Also, @code{rate} and @code{rateb} can
17817 accept an optional fourth argument just like @code{pv} and @code{pvb}.
17818 To redo the above example from a different perspective,
17819 @code{ratel(9, 2000, 1000) = 8.00597%}, which says you will need an
17820 interest rate of 8% in order to double your account in nine years.
17823 @pindex calc-fin-irr
17825 The @kbd{b I} (@code{calc-fin-irr}) [@code{irr}] command is the
17826 analogous function to @code{rate} but for net present value.
17827 Its argument is a vector of payments. Thus @code{irr(@var{payments})}
17828 computes the @var{rate} such that @code{npv(@var{rate}, @var{payments}) = 0};
17829 this rate is known as the @dfn{internal rate of return}.
17833 The @kbd{I b I} [@code{irrb}] command computes the internal rate of
17834 return assuming payments occur at the beginning of each period.
17836 @node Depreciation Functions, Definitions of Financial Functions, Related Financial Functions, Financial Functions
17837 @subsection Depreciation Functions
17840 The functions in this section calculate @dfn{depreciation}, which is
17841 the amount of value that a possession loses over time. These functions
17842 are characterized by three parameters: @var{cost}, the original cost
17843 of the asset; @var{salvage}, the value the asset will have at the end
17844 of its expected ``useful life''; and @var{life}, the number of years
17845 (or other periods) of the expected useful life.
17847 There are several methods for calculating depreciation that differ in
17848 the way they spread the depreciation over the lifetime of the asset.
17851 @pindex calc-fin-sln
17853 The @kbd{b S} (@code{calc-fin-sln}) [@code{sln}] command computes the
17854 ``straight-line'' depreciation. In this method, the asset depreciates
17855 by the same amount every year (or period). For example,
17856 @samp{sln(12000, 2000, 5)} returns 2000. The asset costs $12000
17857 initially and will be worth $2000 after five years; it loses $2000
17861 @pindex calc-fin-syd
17863 The @kbd{b Y} (@code{calc-fin-syd}) [@code{syd}] command computes the
17864 accelerated ``sum-of-years'-digits'' depreciation. Here the depreciation
17865 is higher during the early years of the asset's life. Since the
17866 depreciation is different each year, @kbd{b Y} takes a fourth @var{period}
17867 parameter which specifies which year is requested, from 1 to @var{life}.
17868 If @var{period} is outside this range, the @code{syd} function will
17872 @pindex calc-fin-ddb
17874 The @kbd{b D} (@code{calc-fin-ddb}) [@code{ddb}] command computes an
17875 accelerated depreciation using the double-declining balance method.
17876 It also takes a fourth @var{period} parameter.
17878 For symmetry, the @code{sln} function will accept a @var{period}
17879 parameter as well, although it will ignore its value except that the
17880 return value will as usual be zero if @var{period} is out of range.
17882 For example, pushing the vector @expr{[1,2,3,4,5]} (perhaps with @kbd{v x 5})
17883 and then mapping @kbd{V M ' [sln(12000,2000,5,$), syd(12000,2000,5,$),
17884 ddb(12000,2000,5,$)] @key{RET}} produces a matrix that allows us to compare
17885 the three depreciation methods:
17889 [ [ 2000, 3333, 4800 ]
17890 [ 2000, 2667, 2880 ]
17891 [ 2000, 2000, 1728 ]
17892 [ 2000, 1333, 592 ]
17898 (Values have been rounded to nearest integers in this figure.)
17899 We see that @code{sln} depreciates by the same amount each year,
17900 @kbd{syd} depreciates more at the beginning and less at the end,
17901 and @kbd{ddb} weights the depreciation even more toward the beginning.
17903 Summing columns with @kbd{V R : +} yields @expr{[10000, 10000, 10000]};
17904 the total depreciation in any method is (by definition) the
17905 difference between the cost and the salvage value.
17907 @node Definitions of Financial Functions, , Depreciation Functions, Financial Functions
17908 @subsection Definitions
17911 For your reference, here are the actual formulas used to compute
17912 Calc's financial functions.
17914 Calc will not evaluate a financial function unless the @var{rate} or
17915 @var{n} argument is known. However, @var{payment} or @var{amount} can
17916 be a variable. Calc expands these functions according to the
17917 formulas below for symbolic arguments only when you use the @kbd{a "}
17918 (@code{calc-expand-formula}) command, or when taking derivatives or
17919 integrals or solving equations involving the functions.
17922 These formulas are shown using the conventions of Big display
17923 mode (@kbd{d B}); for example, the formula for @code{fv} written
17924 linearly is @samp{pmt * ((1 + rate)^n) - 1) / rate}.
17929 fv(rate, n, pmt) = pmt * ---------------
17933 ((1 + rate) - 1) (1 + rate)
17934 fvb(rate, n, pmt) = pmt * ----------------------------
17938 fvl(rate, n, pmt) = pmt * (1 + rate)
17942 pv(rate, n, pmt) = pmt * ----------------
17946 (1 - (1 + rate) ) (1 + rate)
17947 pvb(rate, n, pmt) = pmt * -----------------------------
17951 pvl(rate, n, pmt) = pmt * (1 + rate)
17954 npv(rate, [a, b, c]) = a*(1 + rate) + b*(1 + rate) + c*(1 + rate)
17957 npvb(rate, [a, b, c]) = a + b*(1 + rate) + c*(1 + rate)
17960 (amt - x * (1 + rate) ) * rate
17961 pmt(rate, n, amt, x) = -------------------------------
17966 (amt - x * (1 + rate) ) * rate
17967 pmtb(rate, n, amt, x) = -------------------------------
17969 (1 - (1 + rate) ) (1 + rate)
17972 nper(rate, pmt, amt) = - log(1 - ------------, 1 + rate)
17976 nperb(rate, pmt, amt) = - log(1 - ---------------, 1 + rate)
17980 nperl(rate, pmt, amt) = - log(---, 1 + rate)
17985 ratel(n, pmt, amt) = ------ - 1
17990 sln(cost, salv, life) = -----------
17993 (cost - salv) * (life - per + 1)
17994 syd(cost, salv, life, per) = --------------------------------
17995 life * (life + 1) / 2
17998 ddb(cost, salv, life, per) = --------, book = cost - depreciation so far
18003 $$ \code{fv}(r, n, p) = p { (1 + r)^n - 1 \over r } $$
18004 $$ \code{fvb}(r, n, p) = p { ((1 + r)^n - 1) (1 + r) \over r } $$
18005 $$ \code{fvl}(r, n, p) = p (1 + r)^n $$
18006 $$ \code{pv}(r, n, p) = p { 1 - (1 + r)^{-n} \over r } $$
18007 $$ \code{pvb}(r, n, p) = p { (1 - (1 + r)^{-n}) (1 + r) \over r } $$
18008 $$ \code{pvl}(r, n, p) = p (1 + r)^{-n} $$
18009 $$ \code{npv}(r, [a,b,c]) = a (1 + r)^{-1} + b (1 + r)^{-2} + c (1 + r)^{-3} $$
18010 $$ \code{npvb}(r, [a,b,c]) = a + b (1 + r)^{-1} + c (1 + r)^{-2} $$
18011 $$ \code{pmt}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over 1 - (1 + r)^{-n} }$$
18012 $$ \code{pmtb}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over
18013 (1 - (1 + r)^{-n}) (1 + r) } $$
18014 $$ \code{nper}(r, p, a) = -\code{log}(1 - { a r \over p }, 1 + r) $$
18015 $$ \code{nperb}(r, p, a) = -\code{log}(1 - { a r \over p (1 + r) }, 1 + r) $$
18016 $$ \code{nperl}(r, p, a) = -\code{log}({a \over p}, 1 + r) $$
18017 $$ \code{ratel}(n, p, a) = { p^{1/n} \over a^{1/n} } - 1 $$
18018 $$ \code{sln}(c, s, l) = { c - s \over l } $$
18019 $$ \code{syd}(c, s, l, p) = { (c - s) (l - p + 1) \over l (l+1) / 2 } $$
18020 $$ \code{ddb}(c, s, l, p) = { 2 (c - \hbox{depreciation so far}) \over l } $$
18024 In @code{pmt} and @code{pmtb}, @expr{x=0} if omitted.
18026 These functions accept any numeric objects, including error forms,
18027 intervals, and even (though not very usefully) complex numbers. The
18028 above formulas specify exactly the behavior of these functions with
18029 all sorts of inputs.
18031 Note that if the first argument to the @code{log} in @code{nper} is
18032 negative, @code{nper} leaves itself in symbolic form rather than
18033 returning a (financially meaningless) complex number.
18035 @samp{rate(num, pmt, amt)} solves the equation
18036 @samp{pv(rate, num, pmt) = amt} for @samp{rate} using @kbd{H a R}
18037 (@code{calc-find-root}), with the interval @samp{[.01% .. 100%]}
18038 for an initial guess. The @code{rateb} function is the same except
18039 that it uses @code{pvb}. Note that @code{ratel} can be solved
18040 directly; its formula is shown in the above list.
18042 Similarly, @samp{irr(pmts)} solves the equation @samp{npv(rate, pmts) = 0}
18045 If you give a fourth argument to @code{nper} or @code{nperb}, Calc
18046 will also use @kbd{H a R} to solve the equation using an initial
18047 guess interval of @samp{[0 .. 100]}.
18049 A fourth argument to @code{fv} simply sums the two components
18050 calculated from the above formulas for @code{fv} and @code{fvl}.
18051 The same is true of @code{fvb}, @code{pv}, and @code{pvb}.
18053 The @kbd{ddb} function is computed iteratively; the ``book'' value
18054 starts out equal to @var{cost}, and decreases according to the above
18055 formula for the specified number of periods. If the book value
18056 would decrease below @var{salvage}, it only decreases to @var{salvage}
18057 and the depreciation is zero for all subsequent periods. The @code{ddb}
18058 function returns the amount the book value decreased in the specified
18061 @node Binary Functions, , Financial Functions, Arithmetic
18062 @section Binary Number Functions
18065 The commands in this chapter all use two-letter sequences beginning with
18066 the @kbd{b} prefix.
18068 @cindex Binary numbers
18069 The ``binary'' operations actually work regardless of the currently
18070 displayed radix, although their results make the most sense in a radix
18071 like 2, 8, or 16 (as obtained by the @kbd{d 2}, @kbd{d 8}, or @w{@kbd{d 6}}
18072 commands, respectively). You may also wish to enable display of leading
18073 zeros with @kbd{d z}. @xref{Radix Modes}.
18075 @cindex Word size for binary operations
18076 The Calculator maintains a current @dfn{word size} @expr{w}, an
18077 arbitrary positive or negative integer. For a positive word size, all
18078 of the binary operations described here operate modulo @expr{2^w}. In
18079 particular, negative arguments are converted to positive integers modulo
18080 @expr{2^w} by all binary functions.
18082 If the word size is negative, binary operations produce twos-complement
18084 @texline @math{-2^{-w-1}}
18085 @infoline @expr{-(2^(-w-1))}
18087 @texline @math{2^{-w-1}-1}
18088 @infoline @expr{2^(-w-1)-1}
18089 inclusive. Either mode accepts inputs in any range; the sign of
18090 @expr{w} affects only the results produced.
18095 The @kbd{b c} (@code{calc-clip})
18096 [@code{clip}] command can be used to clip a number by reducing it modulo
18097 @expr{2^w}. The commands described in this chapter automatically clip
18098 their results to the current word size. Note that other operations like
18099 addition do not use the current word size, since integer addition
18100 generally is not ``binary.'' (However, @pxref{Simplification Modes},
18101 @code{calc-bin-simplify-mode}.) For example, with a word size of 8
18102 bits @kbd{b c} converts a number to the range 0 to 255; with a word
18103 size of @mathit{-8} @kbd{b c} converts to the range @mathit{-128} to 127.
18106 @pindex calc-word-size
18107 The default word size is 32 bits. All operations except the shifts and
18108 rotates allow you to specify a different word size for that one
18109 operation by giving a numeric prefix argument: @kbd{C-u 8 b c} clips the
18110 top of stack to the range 0 to 255 regardless of the current word size.
18111 To set the word size permanently, use @kbd{b w} (@code{calc-word-size}).
18112 This command displays a prompt with the current word size; press @key{RET}
18113 immediately to keep this word size, or type a new word size at the prompt.
18115 When the binary operations are written in symbolic form, they take an
18116 optional second (or third) word-size parameter. When a formula like
18117 @samp{and(a,b)} is finally evaluated, the word size current at that time
18118 will be used, but when @samp{and(a,b,-8)} is evaluated, a word size of
18119 @mathit{-8} will always be used. A symbolic binary function will be left
18120 in symbolic form unless the all of its argument(s) are integers or
18121 integer-valued floats.
18123 If either or both arguments are modulo forms for which @expr{M} is a
18124 power of two, that power of two is taken as the word size unless a
18125 numeric prefix argument overrides it. The current word size is never
18126 consulted when modulo-power-of-two forms are involved.
18131 The @kbd{b a} (@code{calc-and}) [@code{and}] command computes the bitwise
18132 AND of the two numbers on the top of the stack. In other words, for each
18133 of the @expr{w} binary digits of the two numbers (pairwise), the corresponding
18134 bit of the result is 1 if and only if both input bits are 1:
18135 @samp{and(2#1100, 2#1010) = 2#1000}.
18140 The @kbd{b o} (@code{calc-or}) [@code{or}] command computes the bitwise
18141 inclusive OR of two numbers. A bit is 1 if either of the input bits, or
18142 both, are 1: @samp{or(2#1100, 2#1010) = 2#1110}.
18147 The @kbd{b x} (@code{calc-xor}) [@code{xor}] command computes the bitwise
18148 exclusive OR of two numbers. A bit is 1 if exactly one of the input bits
18149 is 1: @samp{xor(2#1100, 2#1010) = 2#0110}.
18154 The @kbd{b d} (@code{calc-diff}) [@code{diff}] command computes the bitwise
18155 difference of two numbers; this is defined by @samp{diff(a,b) = and(a,not(b))},
18156 so that @samp{diff(2#1100, 2#1010) = 2#0100}.
18161 The @kbd{b n} (@code{calc-not}) [@code{not}] command computes the bitwise
18162 NOT of a number. A bit is 1 if the input bit is 0 and vice-versa.
18165 @pindex calc-lshift-binary
18167 The @kbd{b l} (@code{calc-lshift-binary}) [@code{lsh}] command shifts a
18168 number left by one bit, or by the number of bits specified in the numeric
18169 prefix argument. A negative prefix argument performs a logical right shift,
18170 in which zeros are shifted in on the left. In symbolic form, @samp{lsh(a)}
18171 is short for @samp{lsh(a,1)}, which in turn is short for @samp{lsh(a,n,w)}.
18172 Bits shifted ``off the end,'' according to the current word size, are lost.
18188 The @kbd{H b l} command also does a left shift, but it takes two arguments
18189 from the stack (the value to shift, and, at top-of-stack, the number of
18190 bits to shift). This version interprets the prefix argument just like
18191 the regular binary operations, i.e., as a word size. The Hyperbolic flag
18192 has a similar effect on the rest of the binary shift and rotate commands.
18195 @pindex calc-rshift-binary
18197 The @kbd{b r} (@code{calc-rshift-binary}) [@code{rsh}] command shifts a
18198 number right by one bit, or by the number of bits specified in the numeric
18199 prefix argument: @samp{rsh(a,n) = lsh(a,-n)}.
18202 @pindex calc-lshift-arith
18204 The @kbd{b L} (@code{calc-lshift-arith}) [@code{ash}] command shifts a
18205 number left. It is analogous to @code{lsh}, except that if the shift
18206 is rightward (the prefix argument is negative), an arithmetic shift
18207 is performed as described below.
18210 @pindex calc-rshift-arith
18212 The @kbd{b R} (@code{calc-rshift-arith}) [@code{rash}] command performs
18213 an ``arithmetic'' shift to the right, in which the leftmost bit (according
18214 to the current word size) is duplicated rather than shifting in zeros.
18215 This corresponds to dividing by a power of two where the input is interpreted
18216 as a signed, twos-complement number. (The distinction between the @samp{rsh}
18217 and @samp{rash} operations is totally independent from whether the word
18218 size is positive or negative.) With a negative prefix argument, this
18219 performs a standard left shift.
18222 @pindex calc-rotate-binary
18224 The @kbd{b t} (@code{calc-rotate-binary}) [@code{rot}] command rotates a
18225 number one bit to the left. The leftmost bit (according to the current
18226 word size) is dropped off the left and shifted in on the right. With a
18227 numeric prefix argument, the number is rotated that many bits to the left
18230 @xref{Set Operations}, for the @kbd{b p} and @kbd{b u} commands that
18231 pack and unpack binary integers into sets. (For example, @kbd{b u}
18232 unpacks the number @samp{2#11001} to the set of bit-numbers
18233 @samp{[0, 3, 4]}.) Type @kbd{b u V #} to count the number of ``1''
18234 bits in a binary integer.
18236 Another interesting use of the set representation of binary integers
18237 is to reverse the bits in, say, a 32-bit integer. Type @kbd{b u} to
18238 unpack; type @kbd{31 @key{TAB} -} to replace each bit-number in the set
18239 with 31 minus that bit-number; type @kbd{b p} to pack the set back
18240 into a binary integer.
18242 @node Scientific Functions, Matrix Functions, Arithmetic, Top
18243 @chapter Scientific Functions
18246 The functions described here perform trigonometric and other transcendental
18247 calculations. They generally produce floating-point answers correct to the
18248 full current precision. The @kbd{H} (Hyperbolic) and @kbd{I} (Inverse)
18249 flag keys must be used to get some of these functions from the keyboard.
18253 @cindex @code{pi} variable
18256 @cindex @code{e} variable
18259 @cindex @code{gamma} variable
18261 @cindex Gamma constant, Euler's
18262 @cindex Euler's gamma constant
18264 @cindex @code{phi} variable
18265 @cindex Phi, golden ratio
18266 @cindex Golden ratio
18267 One miscellaneous command is shift-@kbd{P} (@code{calc-pi}), which pushes
18268 the value of @cpi{} (at the current precision) onto the stack. With the
18269 Hyperbolic flag, it pushes the value @expr{e}, the base of natural logarithms.
18270 With the Inverse flag, it pushes Euler's constant
18271 @texline @math{\gamma}
18272 @infoline @expr{gamma}
18273 (about 0.5772). With both Inverse and Hyperbolic, it
18274 pushes the ``golden ratio''
18275 @texline @math{\phi}
18276 @infoline @expr{phi}
18277 (about 1.618). (At present, Euler's constant is not available
18278 to unlimited precision; Calc knows only the first 100 digits.)
18279 In Symbolic mode, these commands push the
18280 actual variables @samp{pi}, @samp{e}, @samp{gamma}, and @samp{phi},
18281 respectively, instead of their values; @pxref{Symbolic Mode}.
18291 The @kbd{Q} (@code{calc-sqrt}) [@code{sqrt}] function is described elsewhere;
18292 @pxref{Basic Arithmetic}. With the Inverse flag [@code{sqr}], this command
18293 computes the square of the argument.
18295 @xref{Prefix Arguments}, for a discussion of the effect of numeric
18296 prefix arguments on commands in this chapter which do not otherwise
18297 interpret a prefix argument.
18300 * Logarithmic Functions::
18301 * Trigonometric and Hyperbolic Functions::
18302 * Advanced Math Functions::
18305 * Combinatorial Functions::
18306 * Probability Distribution Functions::
18309 @node Logarithmic Functions, Trigonometric and Hyperbolic Functions, Scientific Functions, Scientific Functions
18310 @section Logarithmic Functions
18320 The shift-@kbd{L} (@code{calc-ln}) [@code{ln}] command computes the natural
18321 logarithm of the real or complex number on the top of the stack. With
18322 the Inverse flag it computes the exponential function instead, although
18323 this is redundant with the @kbd{E} command.
18332 The shift-@kbd{E} (@code{calc-exp}) [@code{exp}] command computes the
18333 exponential, i.e., @expr{e} raised to the power of the number on the stack.
18334 The meanings of the Inverse and Hyperbolic flags follow from those for
18335 the @code{calc-ln} command.
18350 The @kbd{H L} (@code{calc-log10}) [@code{log10}] command computes the common
18351 (base-10) logarithm of a number. (With the Inverse flag [@code{exp10}],
18352 it raises ten to a given power.) Note that the common logarithm of a
18353 complex number is computed by taking the natural logarithm and dividing
18355 @texline @math{\ln10}.
18356 @infoline @expr{ln(10)}.
18363 The @kbd{B} (@code{calc-log}) [@code{log}] command computes a logarithm
18364 to any base. For example, @kbd{1024 @key{RET} 2 B} produces 10, since
18365 @texline @math{2^{10} = 1024}.
18366 @infoline @expr{2^10 = 1024}.
18367 In certain cases like @samp{log(3,9)}, the result
18368 will be either @expr{1:2} or @expr{0.5} depending on the current Fraction
18369 mode setting. With the Inverse flag [@code{alog}], this command is
18370 similar to @kbd{^} except that the order of the arguments is reversed.
18375 The @kbd{f I} (@code{calc-ilog}) [@code{ilog}] command computes the
18376 integer logarithm of a number to any base. The number and the base must
18377 themselves be positive integers. This is the true logarithm, rounded
18378 down to an integer. Thus @kbd{ilog(x,10)} is 3 for all @expr{x} in the
18379 range from 1000 to 9999. If both arguments are positive integers, exact
18380 integer arithmetic is used; otherwise, this is equivalent to
18381 @samp{floor(log(x,b))}.
18386 The @kbd{f E} (@code{calc-expm1}) [@code{expm1}] command computes
18387 @texline @math{e^x - 1},
18388 @infoline @expr{exp(x)-1},
18389 but using an algorithm that produces a more accurate
18390 answer when the result is close to zero, i.e., when
18391 @texline @math{e^x}
18392 @infoline @expr{exp(x)}
18398 The @kbd{f L} (@code{calc-lnp1}) [@code{lnp1}] command computes
18399 @texline @math{\ln(x+1)},
18400 @infoline @expr{ln(x+1)},
18401 producing a more accurate answer when @expr{x} is close to zero.
18403 @node Trigonometric and Hyperbolic Functions, Advanced Math Functions, Logarithmic Functions, Scientific Functions
18404 @section Trigonometric/Hyperbolic Functions
18410 The shift-@kbd{S} (@code{calc-sin}) [@code{sin}] command computes the sine
18411 of an angle or complex number. If the input is an HMS form, it is interpreted
18412 as degrees-minutes-seconds; otherwise, the input is interpreted according
18413 to the current angular mode. It is best to use Radians mode when operating
18414 on complex numbers.
18416 Calc's ``units'' mechanism includes angular units like @code{deg},
18417 @code{rad}, and @code{grad}. While @samp{sin(45 deg)} is not evaluated
18418 all the time, the @kbd{u s} (@code{calc-simplify-units}) command will
18419 simplify @samp{sin(45 deg)} by taking the sine of 45 degrees, regardless
18420 of the current angular mode. @xref{Basic Operations on Units}.
18422 Also, the symbolic variable @code{pi} is not ordinarily recognized in
18423 arguments to trigonometric functions, as in @samp{sin(3 pi / 4)}, but
18424 the default algebraic simplifications recognize many such
18425 formulas when the current angular mode is Radians @emph{and} Symbolic
18426 mode is enabled; this example would be replaced by @samp{sqrt(2) / 2}.
18427 @xref{Symbolic Mode}. Beware, this simplification occurs even if you
18428 have stored a different value in the variable @samp{pi}; this is one
18429 reason why changing built-in variables is a bad idea. Arguments of
18430 the form @expr{x} plus a multiple of @cpiover{2} are also simplified.
18431 Calc includes similar formulas for @code{cos} and @code{tan}.
18433 Calc's algebraic simplifications know all angles which are integer multiples of
18434 @cpiover{12}, @cpiover{10}, or @cpiover{8} radians. In Degrees mode,
18435 analogous simplifications occur for integer multiples of 15 or 18
18436 degrees, and for arguments plus multiples of 90 degrees.
18439 @pindex calc-arcsin
18441 With the Inverse flag, @code{calc-sin} computes an arcsine. This is also
18442 available as the @code{calc-arcsin} command or @code{arcsin} algebraic
18443 function. The returned argument is converted to degrees, radians, or HMS
18444 notation depending on the current angular mode.
18450 @pindex calc-arcsinh
18452 With the Hyperbolic flag, @code{calc-sin} computes the hyperbolic
18453 sine, also available as @code{calc-sinh} [@code{sinh}]. With the
18454 Hyperbolic and Inverse flags, it computes the hyperbolic arcsine
18455 (@code{calc-arcsinh}) [@code{arcsinh}].
18464 @pindex calc-arccos
18482 @pindex calc-arccosh
18500 @pindex calc-arctan
18518 @pindex calc-arctanh
18523 The shift-@kbd{C} (@code{calc-cos}) [@code{cos}] command computes the cosine
18524 of an angle or complex number, and shift-@kbd{T} (@code{calc-tan}) [@code{tan}]
18525 computes the tangent, along with all the various inverse and hyperbolic
18526 variants of these functions.
18529 @pindex calc-arctan2
18531 The @kbd{f T} (@code{calc-arctan2}) [@code{arctan2}] command takes two
18532 numbers from the stack and computes the arc tangent of their ratio. The
18533 result is in the full range from @mathit{-180} (exclusive) to @mathit{+180}
18534 (inclusive) degrees, or the analogous range in radians. A similar
18535 result would be obtained with @kbd{/} followed by @kbd{I T}, but the
18536 value would only be in the range from @mathit{-90} to @mathit{+90} degrees
18537 since the division loses information about the signs of the two
18538 components, and an error might result from an explicit division by zero
18539 which @code{arctan2} would avoid. By (arbitrary) definition,
18540 @samp{arctan2(0,0)=0}.
18542 @pindex calc-sincos
18554 The @code{calc-sincos} [@code{sincos}] command computes the sine and
18555 cosine of a number, returning them as a vector of the form
18556 @samp{[@var{cos}, @var{sin}]}.
18557 With the Inverse flag [@code{arcsincos}], this command takes a two-element
18558 vector as an argument and computes @code{arctan2} of the elements.
18559 (This command does not accept the Hyperbolic flag.)
18573 The remaining trigonometric functions, @code{calc-sec} [@code{sec}],
18574 @code{calc-csc} [@code{csc}] and @code{calc-cot} [@code{cot}], are also
18575 available. With the Hyperbolic flag, these compute their hyperbolic
18576 counterparts, which are also available separately as @code{calc-sech}
18577 [@code{sech}], @code{calc-csch} [@code{csch}] and @code{calc-coth}
18578 [@code{coth}]. (These commands do not accept the Inverse flag.)
18580 @node Advanced Math Functions, Branch Cuts, Trigonometric and Hyperbolic Functions, Scientific Functions
18581 @section Advanced Mathematical Functions
18584 Calc can compute a variety of less common functions that arise in
18585 various branches of mathematics. All of the functions described in
18586 this section allow arbitrary complex arguments and, except as noted,
18587 will work to arbitrarily large precision. They can not at present
18588 handle error forms or intervals as arguments.
18590 NOTE: These functions are still experimental. In particular, their
18591 accuracy is not guaranteed in all domains. It is advisable to set the
18592 current precision comfortably higher than you actually need when
18593 using these functions. Also, these functions may be impractically
18594 slow for some values of the arguments.
18599 The @kbd{f g} (@code{calc-gamma}) [@code{gamma}] command computes the Euler
18600 gamma function. For positive integer arguments, this is related to the
18601 factorial function: @samp{gamma(n+1) = fact(n)}. For general complex
18602 arguments the gamma function can be defined by the following definite
18604 @texline @math{\Gamma(a) = \int_0^\infty t^{a-1} e^t dt}.
18605 @infoline @expr{gamma(a) = integ(t^(a-1) exp(t), t, 0, inf)}.
18606 (The actual implementation uses far more efficient computational methods.)
18622 @pindex calc-inc-gamma
18635 The @kbd{f G} (@code{calc-inc-gamma}) [@code{gammaP}] command computes
18636 the incomplete gamma function, denoted @samp{P(a,x)}. This is defined by
18638 @texline @math{P(a,x) = \left( \int_0^x t^{a-1} e^t dt \right) / \Gamma(a)}.
18639 @infoline @expr{gammaP(a,x) = integ(t^(a-1) exp(t), t, 0, x) / gamma(a)}.
18640 This implies that @samp{gammaP(a,inf) = 1} for any @expr{a} (see the
18641 definition of the normal gamma function).
18643 Several other varieties of incomplete gamma function are defined.
18644 The complement of @expr{P(a,x)}, called @expr{Q(a,x) = 1-P(a,x)} by
18645 some authors, is computed by the @kbd{I f G} [@code{gammaQ}] command.
18646 You can think of this as taking the other half of the integral, from
18647 @expr{x} to infinity.
18650 The functions corresponding to the integrals that define @expr{P(a,x)}
18651 and @expr{Q(a,x)} but without the normalizing @expr{1/gamma(a)}
18652 factor are called @expr{g(a,x)} and @expr{G(a,x)}, respectively
18653 (where @expr{g} and @expr{G} represent the lower- and upper-case Greek
18654 letter gamma). You can obtain these using the @kbd{H f G} [@code{gammag}]
18655 and @kbd{H I f G} [@code{gammaG}] commands.
18658 The functions corresponding to the integrals that define $P(a,x)$
18659 and $Q(a,x)$ but without the normalizing $1/\Gamma(a)$
18660 factor are called $\gamma(a,x)$ and $\Gamma(a,x)$, respectively.
18661 You can obtain these using the \kbd{H f G} [\code{gammag}] and
18662 \kbd{I H f G} [\code{gammaG}] commands.
18668 The @kbd{f b} (@code{calc-beta}) [@code{beta}] command computes the
18669 Euler beta function, which is defined in terms of the gamma function as
18670 @texline @math{B(a,b) = \Gamma(a) \Gamma(b) / \Gamma(a+b)},
18671 @infoline @expr{beta(a,b) = gamma(a) gamma(b) / gamma(a+b)},
18673 @texline @math{B(a,b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt}.
18674 @infoline @expr{beta(a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, 1)}.
18678 @pindex calc-inc-beta
18681 The @kbd{f B} (@code{calc-inc-beta}) [@code{betaI}] command computes
18682 the incomplete beta function @expr{I(x,a,b)}. It is defined by
18683 @texline @math{I(x,a,b) = \left( \int_0^x t^{a-1} (1-t)^{b-1} dt \right) / B(a,b)}.
18684 @infoline @expr{betaI(x,a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, x) / beta(a,b)}.
18685 Once again, the @kbd{H} (hyperbolic) prefix gives the corresponding
18686 un-normalized version [@code{betaB}].
18693 The @kbd{f e} (@code{calc-erf}) [@code{erf}] command computes the
18695 @texline @math{\hbox{erf}(x) = {2 \over \sqrt{\pi}} \int_0^x e^{-t^2} dt}.
18696 @infoline @expr{erf(x) = 2 integ(exp(-(t^2)), t, 0, x) / sqrt(pi)}.
18697 The complementary error function @kbd{I f e} (@code{calc-erfc}) [@code{erfc}]
18698 is the corresponding integral from @samp{x} to infinity; the sum
18699 @texline @math{\hbox{erf}(x) + \hbox{erfc}(x) = 1}.
18700 @infoline @expr{erf(x) + erfc(x) = 1}.
18704 @pindex calc-bessel-J
18705 @pindex calc-bessel-Y
18708 The @kbd{f j} (@code{calc-bessel-J}) [@code{besJ}] and @kbd{f y}
18709 (@code{calc-bessel-Y}) [@code{besY}] commands compute the Bessel
18710 functions of the first and second kinds, respectively.
18711 In @samp{besJ(n,x)} and @samp{besY(n,x)} the ``order'' parameter
18712 @expr{n} is often an integer, but is not required to be one.
18713 Calc's implementation of the Bessel functions currently limits the
18714 precision to 8 digits, and may not be exact even to that precision.
18717 @node Branch Cuts, Random Numbers, Advanced Math Functions, Scientific Functions
18718 @section Branch Cuts and Principal Values
18721 @cindex Branch cuts
18722 @cindex Principal values
18723 All of the logarithmic, trigonometric, and other scientific functions are
18724 defined for complex numbers as well as for reals.
18725 This section describes the values
18726 returned in cases where the general result is a family of possible values.
18727 Calc follows section 12.5.3 of Steele's @dfn{Common Lisp, the Language},
18728 second edition, in these matters. This section will describe each
18729 function briefly; for a more detailed discussion (including some nifty
18730 diagrams), consult Steele's book.
18732 Note that the branch cuts for @code{arctan} and @code{arctanh} were
18733 changed between the first and second editions of Steele. Recent
18734 versions of Calc follow the second edition.
18736 The new branch cuts exactly match those of the HP-28/48 calculators.
18737 They also match those of Mathematica 1.2, except that Mathematica's
18738 @code{arctan} cut is always in the right half of the complex plane,
18739 and its @code{arctanh} cut is always in the top half of the plane.
18740 Calc's cuts are continuous with quadrants I and III for @code{arctan},
18741 or II and IV for @code{arctanh}.
18743 Note: The current implementations of these functions with complex arguments
18744 are designed with proper behavior around the branch cuts in mind, @emph{not}
18745 efficiency or accuracy. You may need to increase the floating precision
18746 and wait a while to get suitable answers from them.
18748 For @samp{sqrt(a+bi)}: When @expr{a<0} and @expr{b} is small but positive
18749 or zero, the result is close to the @expr{+i} axis. For @expr{b} small and
18750 negative, the result is close to the @expr{-i} axis. The result always lies
18751 in the right half of the complex plane.
18753 For @samp{ln(a+bi)}: The real part is defined as @samp{ln(abs(a+bi))}.
18754 The imaginary part is defined as @samp{arg(a+bi) = arctan2(b,a)}.
18755 Thus the branch cuts for @code{sqrt} and @code{ln} both lie on the
18756 negative real axis.
18758 The following table describes these branch cuts in another way.
18759 If the real and imaginary parts of @expr{z} are as shown, then
18760 the real and imaginary parts of @expr{f(z)} will be as shown.
18761 Here @code{eps} stands for a small positive value; each
18762 occurrence of @code{eps} may stand for a different small value.
18766 ----------------------------------------
18769 -, +eps +eps, + +eps, +
18770 -, -eps +eps, - +eps, -
18773 For @samp{z1^z2}: This is defined by @samp{exp(ln(z1)*z2)}.
18774 One interesting consequence of this is that @samp{(-8)^1:3} does
18775 not evaluate to @mathit{-2} as you might expect, but to the complex
18776 number @expr{(1., 1.732)}. Both of these are valid cube roots
18777 of @mathit{-8} (as is @expr{(1., -1.732)}); Calc chooses a perhaps
18778 less-obvious root for the sake of mathematical consistency.
18780 For @samp{arcsin(z)}: This is defined by @samp{-i*ln(i*z + sqrt(1-z^2))}.
18781 The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.
18783 For @samp{arccos(z)}: This is defined by @samp{-i*ln(z + i*sqrt(1-z^2))},
18784 or equivalently by @samp{pi/2 - arcsin(z)}. The branch cuts are on
18785 the real axis, less than @mathit{-1} and greater than 1.
18787 For @samp{arctan(z)}: This is defined by
18788 @samp{(ln(1+i*z) - ln(1-i*z)) / (2*i)}. The branch cuts are on the
18789 imaginary axis, below @expr{-i} and above @expr{i}.
18791 For @samp{arcsinh(z)}: This is defined by @samp{ln(z + sqrt(1+z^2))}.
18792 The branch cuts are on the imaginary axis, below @expr{-i} and
18795 For @samp{arccosh(z)}: This is defined by
18796 @samp{ln(z + (z+1)*sqrt((z-1)/(z+1)))}. The branch cut is on the
18797 real axis less than 1.
18799 For @samp{arctanh(z)}: This is defined by @samp{(ln(1+z) - ln(1-z)) / 2}.
18800 The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.
18802 The following tables for @code{arcsin}, @code{arccos}, and
18803 @code{arctan} assume the current angular mode is Radians. The
18804 hyperbolic functions operate independently of the angular mode.
18807 z arcsin(z) arccos(z)
18808 -------------------------------------------------------
18809 (-1..1), 0 (-pi/2..pi/2), 0 (0..pi), 0
18810 (-1..1), +eps (-pi/2..pi/2), +eps (0..pi), -eps
18811 (-1..1), -eps (-pi/2..pi/2), -eps (0..pi), +eps
18812 <-1, 0 -pi/2, + pi, -
18813 <-1, +eps -pi/2 + eps, + pi - eps, -
18814 <-1, -eps -pi/2 + eps, - pi - eps, +
18816 >1, +eps pi/2 - eps, + +eps, -
18817 >1, -eps pi/2 - eps, - +eps, +
18821 z arccosh(z) arctanh(z)
18822 -----------------------------------------------------
18823 (-1..1), 0 0, (0..pi) any, 0
18824 (-1..1), +eps +eps, (0..pi) any, +eps
18825 (-1..1), -eps +eps, (-pi..0) any, -eps
18826 <-1, 0 +, pi -, pi/2
18827 <-1, +eps +, pi - eps -, pi/2 - eps
18828 <-1, -eps +, -pi + eps -, -pi/2 + eps
18829 >1, 0 +, 0 +, -pi/2
18830 >1, +eps +, +eps +, pi/2 - eps
18831 >1, -eps +, -eps +, -pi/2 + eps
18835 z arcsinh(z) arctan(z)
18836 -----------------------------------------------------
18837 0, (-1..1) 0, (-pi/2..pi/2) 0, any
18838 0, <-1 -, -pi/2 -pi/2, -
18839 +eps, <-1 +, -pi/2 + eps pi/2 - eps, -
18840 -eps, <-1 -, -pi/2 + eps -pi/2 + eps, -
18841 0, >1 +, pi/2 pi/2, +
18842 +eps, >1 +, pi/2 - eps pi/2 - eps, +
18843 -eps, >1 -, pi/2 - eps -pi/2 + eps, +
18846 Finally, the following identities help to illustrate the relationship
18847 between the complex trigonometric and hyperbolic functions. They
18848 are valid everywhere, including on the branch cuts.
18851 sin(i*z) = i*sinh(z) arcsin(i*z) = i*arcsinh(z)
18852 cos(i*z) = cosh(z) arcsinh(i*z) = i*arcsin(z)
18853 tan(i*z) = i*tanh(z) arctan(i*z) = i*arctanh(z)
18854 sinh(i*z) = i*sin(z) cosh(i*z) = cos(z)
18857 The ``advanced math'' functions (gamma, Bessel, etc.@:) are also defined
18858 for general complex arguments, but their branch cuts and principal values
18859 are not rigorously specified at present.
18861 @node Random Numbers, Combinatorial Functions, Branch Cuts, Scientific Functions
18862 @section Random Numbers
18866 @pindex calc-random
18868 The @kbd{k r} (@code{calc-random}) [@code{random}] command produces
18869 random numbers of various sorts.
18871 Given a positive numeric prefix argument @expr{M}, it produces a random
18872 integer @expr{N} in the range
18873 @texline @math{0 \le N < M}.
18874 @infoline @expr{0 <= N < M}.
18875 Each possible value @expr{N} appears with equal probability.
18877 With no numeric prefix argument, the @kbd{k r} command takes its argument
18878 from the stack instead. Once again, if this is a positive integer @expr{M}
18879 the result is a random integer less than @expr{M}. However, note that
18880 while numeric prefix arguments are limited to six digits or so, an @expr{M}
18881 taken from the stack can be arbitrarily large. If @expr{M} is negative,
18882 the result is a random integer in the range
18883 @texline @math{M < N \le 0}.
18884 @infoline @expr{M < N <= 0}.
18886 If the value on the stack is a floating-point number @expr{M}, the result
18887 is a random floating-point number @expr{N} in the range
18888 @texline @math{0 \le N < M}
18889 @infoline @expr{0 <= N < M}
18891 @texline @math{M < N \le 0},
18892 @infoline @expr{M < N <= 0},
18893 according to the sign of @expr{M}.
18895 If @expr{M} is zero, the result is a Gaussian-distributed random real
18896 number; the distribution has a mean of zero and a standard deviation
18897 of one. The algorithm used generates random numbers in pairs; thus,
18898 every other call to this function will be especially fast.
18900 If @expr{M} is an error form
18901 @texline @math{m} @code{+/-} @math{\sigma}
18902 @infoline @samp{m +/- s}
18904 @texline @math{\sigma}
18906 are both real numbers, the result uses a Gaussian distribution with mean
18907 @var{m} and standard deviation
18908 @texline @math{\sigma}.
18911 If @expr{M} is an interval form, the lower and upper bounds specify the
18912 acceptable limits of the random numbers. If both bounds are integers,
18913 the result is a random integer in the specified range. If either bound
18914 is floating-point, the result is a random real number in the specified
18915 range. If the interval is open at either end, the result will be sure
18916 not to equal that end value. (This makes a big difference for integer
18917 intervals, but for floating-point intervals it's relatively minor:
18918 with a precision of 6, @samp{random([1.0..2.0))} will return any of one
18919 million numbers from 1.00000 to 1.99999; @samp{random([1.0..2.0])} may
18920 additionally return 2.00000, but the probability of this happening is
18923 If @expr{M} is a vector, the result is one element taken at random from
18924 the vector. All elements of the vector are given equal probabilities.
18927 The sequence of numbers produced by @kbd{k r} is completely random by
18928 default, i.e., the sequence is seeded each time you start Calc using
18929 the current time and other information. You can get a reproducible
18930 sequence by storing a particular ``seed value'' in the Calc variable
18931 @code{RandSeed}. Any integer will do for a seed; integers of from 1
18932 to 12 digits are good. If you later store a different integer into
18933 @code{RandSeed}, Calc will switch to a different pseudo-random
18934 sequence. If you ``unstore'' @code{RandSeed}, Calc will re-seed itself
18935 from the current time. If you store the same integer that you used
18936 before back into @code{RandSeed}, you will get the exact same sequence
18937 of random numbers as before.
18939 @pindex calc-rrandom
18940 The @code{calc-rrandom} command (not on any key) produces a random real
18941 number between zero and one. It is equivalent to @samp{random(1.0)}.
18944 @pindex calc-random-again
18945 The @kbd{k a} (@code{calc-random-again}) command produces another random
18946 number, re-using the most recent value of @expr{M}. With a numeric
18947 prefix argument @var{n}, it produces @var{n} more random numbers using
18948 that value of @expr{M}.
18951 @pindex calc-shuffle
18953 The @kbd{k h} (@code{calc-shuffle}) command produces a vector of several
18954 random values with no duplicates. The value on the top of the stack
18955 specifies the set from which the random values are drawn, and may be any
18956 of the @expr{M} formats described above. The numeric prefix argument
18957 gives the length of the desired list. (If you do not provide a numeric
18958 prefix argument, the length of the list is taken from the top of the
18959 stack, and @expr{M} from second-to-top.)
18961 If @expr{M} is a floating-point number, zero, or an error form (so
18962 that the random values are being drawn from the set of real numbers)
18963 there is little practical difference between using @kbd{k h} and using
18964 @kbd{k r} several times. But if the set of possible values consists
18965 of just a few integers, or the elements of a vector, then there is
18966 a very real chance that multiple @kbd{k r}'s will produce the same
18967 number more than once. The @kbd{k h} command produces a vector whose
18968 elements are always distinct. (Actually, there is a slight exception:
18969 If @expr{M} is a vector, no given vector element will be drawn more
18970 than once, but if several elements of @expr{M} are equal, they may
18971 each make it into the result vector.)
18973 One use of @kbd{k h} is to rearrange a list at random. This happens
18974 if the prefix argument is equal to the number of values in the list:
18975 @kbd{[1, 1.5, 2, 2.5, 3] 5 k h} might produce the permuted list
18976 @samp{[2.5, 1, 1.5, 3, 2]}. As a convenient feature, if the argument
18977 @var{n} is negative it is replaced by the size of the set represented
18978 by @expr{M}. Naturally, this is allowed only when @expr{M} specifies
18979 a small discrete set of possibilities.
18981 To do the equivalent of @kbd{k h} but with duplications allowed,
18982 given @expr{M} on the stack and with @var{n} just entered as a numeric
18983 prefix, use @kbd{v b} to build a vector of copies of @expr{M}, then use
18984 @kbd{V M k r} to ``map'' the normal @kbd{k r} function over the
18985 elements of this vector. @xref{Matrix Functions}.
18988 * Random Number Generator:: (Complete description of Calc's algorithm)
18991 @node Random Number Generator, , Random Numbers, Random Numbers
18992 @subsection Random Number Generator
18994 Calc's random number generator uses several methods to ensure that
18995 the numbers it produces are highly random. Knuth's @emph{Art of
18996 Computer Programming}, Volume II, contains a thorough description
18997 of the theory of random number generators and their measurement and
19000 If @code{RandSeed} has no stored value, Calc calls Emacs's built-in
19001 @code{random} function to get a stream of random numbers, which it
19002 then treats in various ways to avoid problems inherent in the simple
19003 random number generators that many systems use to implement @code{random}.
19005 When Calc's random number generator is first invoked, it ``seeds''
19006 the low-level random sequence using the time of day, so that the
19007 random number sequence will be different every time you use Calc.
19009 Since Emacs Lisp doesn't specify the range of values that will be
19010 returned by its @code{random} function, Calc exercises the function
19011 several times to estimate the range. When Calc subsequently uses
19012 the @code{random} function, it takes only 10 bits of the result
19013 near the most-significant end. (It avoids at least the bottom
19014 four bits, preferably more, and also tries to avoid the top two
19015 bits.) This strategy works well with the linear congruential
19016 generators that are typically used to implement @code{random}.
19018 If @code{RandSeed} contains an integer, Calc uses this integer to
19019 seed an ``additive congruential'' method (Knuth's algorithm 3.2.2A,
19021 @texline @math{X_{n-55} - X_{n-24}}.
19022 @infoline @expr{X_n-55 - X_n-24}).
19023 This method expands the seed
19024 value into a large table which is maintained internally; the variable
19025 @code{RandSeed} is changed from, e.g., 42 to the vector @expr{[42]}
19026 to indicate that the seed has been absorbed into this table. When
19027 @code{RandSeed} contains a vector, @kbd{k r} and related commands
19028 continue to use the same internal table as last time. There is no
19029 way to extract the complete state of the random number generator
19030 so that you can restart it from any point; you can only restart it
19031 from the same initial seed value. A simple way to restart from the
19032 same seed is to type @kbd{s r RandSeed} to get the seed vector,
19033 @kbd{v u} to unpack it back into a number, then @kbd{s t RandSeed}
19034 to reseed the generator with that number.
19036 Calc uses a ``shuffling'' method as described in algorithm 3.2.2B
19037 of Knuth. It fills a table with 13 random 10-bit numbers. Then,
19038 to generate a new random number, it uses the previous number to
19039 index into the table, picks the value it finds there as the new
19040 random number, then replaces that table entry with a new value
19041 obtained from a call to the base random number generator (either
19042 the additive congruential generator or the @code{random} function
19043 supplied by the system). If there are any flaws in the base
19044 generator, shuffling will tend to even them out. But if the system
19045 provides an excellent @code{random} function, shuffling will not
19046 damage its randomness.
19048 To create a random integer of a certain number of digits, Calc
19049 builds the integer three decimal digits at a time. For each group
19050 of three digits, Calc calls its 10-bit shuffling random number generator
19051 (which returns a value from 0 to 1023); if the random value is 1000
19052 or more, Calc throws it out and tries again until it gets a suitable
19055 To create a random floating-point number with precision @var{p}, Calc
19056 simply creates a random @var{p}-digit integer and multiplies by
19057 @texline @math{10^{-p}}.
19058 @infoline @expr{10^-p}.
19059 The resulting random numbers should be very clean, but note
19060 that relatively small numbers will have few significant random digits.
19061 In other words, with a precision of 12, you will occasionally get
19062 numbers on the order of
19063 @texline @math{10^{-9}}
19064 @infoline @expr{10^-9}
19066 @texline @math{10^{-10}},
19067 @infoline @expr{10^-10},
19068 but those numbers will only have two or three random digits since they
19069 correspond to small integers times
19070 @texline @math{10^{-12}}.
19071 @infoline @expr{10^-12}.
19073 To create a random integer in the interval @samp{[0 .. @var{m})}, Calc
19074 counts the digits in @var{m}, creates a random integer with three
19075 additional digits, then reduces modulo @var{m}. Unless @var{m} is a
19076 power of ten the resulting values will be very slightly biased toward
19077 the lower numbers, but this bias will be less than 0.1%. (For example,
19078 if @var{m} is 42, Calc will reduce a random integer less than 100000
19079 modulo 42 to get a result less than 42. It is easy to show that the
19080 numbers 40 and 41 will be only 2380/2381 as likely to result from this
19081 modulo operation as numbers 39 and below.) If @var{m} is a power of
19082 ten, however, the numbers should be completely unbiased.
19084 The Gaussian random numbers generated by @samp{random(0.0)} use the
19085 ``polar'' method described in Knuth section 3.4.1C@. This method
19086 generates a pair of Gaussian random numbers at a time, so only every
19087 other call to @samp{random(0.0)} will require significant calculations.
19089 @node Combinatorial Functions, Probability Distribution Functions, Random Numbers, Scientific Functions
19090 @section Combinatorial Functions
19093 Commands relating to combinatorics and number theory begin with the
19094 @kbd{k} key prefix.
19099 The @kbd{k g} (@code{calc-gcd}) [@code{gcd}] command computes the
19100 Greatest Common Divisor of two integers. It also accepts fractions;
19101 the GCD of two fractions is defined by taking the GCD of the
19102 numerators, and the LCM of the denominators. This definition is
19103 consistent with the idea that @samp{a / gcd(a,x)} should yield an
19104 integer for any @samp{a} and @samp{x}. For other types of arguments,
19105 the operation is left in symbolic form.
19110 The @kbd{k l} (@code{calc-lcm}) [@code{lcm}] command computes the
19111 Least Common Multiple of two integers or fractions. The product of
19112 the LCM and GCD of two numbers is equal to the product of the
19116 @pindex calc-extended-gcd
19118 The @kbd{k E} (@code{calc-extended-gcd}) [@code{egcd}] command computes
19119 the GCD of two integers @expr{x} and @expr{y} and returns a vector
19120 @expr{[g, a, b]} where
19121 @texline @math{g = \gcd(x,y) = a x + b y}.
19122 @infoline @expr{g = gcd(x,y) = a x + b y}.
19125 @pindex calc-factorial
19131 The @kbd{!} (@code{calc-factorial}) [@code{fact}] command computes the
19132 factorial of the number at the top of the stack. If the number is an
19133 integer, the result is an exact integer. If the number is an
19134 integer-valued float, the result is a floating-point approximation. If
19135 the number is a non-integral real number, the generalized factorial is used,
19136 as defined by the Euler Gamma function. Please note that computation of
19137 large factorials can be slow; using floating-point format will help
19138 since fewer digits must be maintained. The same is true of many of
19139 the commands in this section.
19142 @pindex calc-double-factorial
19148 The @kbd{k d} (@code{calc-double-factorial}) [@code{dfact}] command
19149 computes the ``double factorial'' of an integer. For an even integer,
19150 this is the product of even integers from 2 to @expr{N}. For an odd
19151 integer, this is the product of odd integers from 3 to @expr{N}. If
19152 the argument is an integer-valued float, the result is a floating-point
19153 approximation. This function is undefined for negative even integers.
19154 The notation @expr{N!!} is also recognized for double factorials.
19157 @pindex calc-choose
19159 The @kbd{k c} (@code{calc-choose}) [@code{choose}] command computes the
19160 binomial coefficient @expr{N}-choose-@expr{M}, where @expr{M} is the number
19161 on the top of the stack and @expr{N} is second-to-top. If both arguments
19162 are integers, the result is an exact integer. Otherwise, the result is a
19163 floating-point approximation. The binomial coefficient is defined for all
19165 @texline @math{N! \over M! (N-M)!\,}.
19166 @infoline @expr{N! / M! (N-M)!}.
19172 The @kbd{H k c} (@code{calc-perm}) [@code{perm}] command computes the
19173 number-of-permutations function @expr{N! / (N-M)!}.
19176 The \kbd{H k c} (\code{calc-perm}) [\code{perm}] command computes the
19177 number-of-perm\-utations function $N! \over (N-M)!\,$.
19182 @pindex calc-bernoulli-number
19184 The @kbd{k b} (@code{calc-bernoulli-number}) [@code{bern}] command
19185 computes a given Bernoulli number. The value at the top of the stack
19186 is a nonnegative integer @expr{n} that specifies which Bernoulli number
19187 is desired. The @kbd{H k b} command computes a Bernoulli polynomial,
19188 taking @expr{n} from the second-to-top position and @expr{x} from the
19189 top of the stack. If @expr{x} is a variable or formula the result is
19190 a polynomial in @expr{x}; if @expr{x} is a number the result is a number.
19194 @pindex calc-euler-number
19196 The @kbd{k e} (@code{calc-euler-number}) [@code{euler}] command similarly
19197 computes an Euler number, and @w{@kbd{H k e}} computes an Euler polynomial.
19198 Bernoulli and Euler numbers occur in the Taylor expansions of several
19203 @pindex calc-stirling-number
19206 The @kbd{k s} (@code{calc-stirling-number}) [@code{stir1}] command
19207 computes a Stirling number of the first
19208 @texline kind@tie{}@math{n \brack m},
19210 given two integers @expr{n} and @expr{m} on the stack. The @kbd{H k s}
19211 [@code{stir2}] command computes a Stirling number of the second
19212 @texline kind@tie{}@math{n \brace m}.
19214 These are the number of @expr{m}-cycle permutations of @expr{n} objects,
19215 and the number of ways to partition @expr{n} objects into @expr{m}
19216 non-empty sets, respectively.
19219 @pindex calc-prime-test
19221 The @kbd{k p} (@code{calc-prime-test}) command checks if the integer on
19222 the top of the stack is prime. For integers less than eight million, the
19223 answer is always exact and reasonably fast. For larger integers, a
19224 probabilistic method is used (see Knuth vol. II, section 4.5.4, algorithm P).
19225 The number is first checked against small prime factors (up to 13). Then,
19226 any number of iterations of the algorithm are performed. Each step either
19227 discovers that the number is non-prime, or substantially increases the
19228 certainty that the number is prime. After a few steps, the chance that
19229 a number was mistakenly described as prime will be less than one percent.
19230 (Indeed, this is a worst-case estimate of the probability; in practice
19231 even a single iteration is quite reliable.) After the @kbd{k p} command,
19232 the number will be reported as definitely prime or non-prime if possible,
19233 or otherwise ``probably'' prime with a certain probability of error.
19239 The normal @kbd{k p} command performs one iteration of the primality
19240 test. Pressing @kbd{k p} repeatedly for the same integer will perform
19241 additional iterations. Also, @kbd{k p} with a numeric prefix performs
19242 the specified number of iterations. There is also an algebraic function
19243 @samp{prime(n)} or @samp{prime(n,iters)} which returns 1 if @expr{n}
19244 is (probably) prime and 0 if not.
19247 @pindex calc-prime-factors
19249 The @kbd{k f} (@code{calc-prime-factors}) [@code{prfac}] command
19250 attempts to decompose an integer into its prime factors. For numbers up
19251 to 25 million, the answer is exact although it may take some time. The
19252 result is a vector of the prime factors in increasing order. For larger
19253 inputs, prime factors above 5000 may not be found, in which case the
19254 last number in the vector will be an unfactored integer greater than 25
19255 million (with a warning message). For negative integers, the first
19256 element of the list will be @mathit{-1}. For inputs @mathit{-1}, @mathit{0}, and
19257 @mathit{1}, the result is a list of the same number.
19260 @pindex calc-next-prime
19262 @mindex nextpr@idots
19265 The @kbd{k n} (@code{calc-next-prime}) [@code{nextprime}] command finds
19266 the next prime above a given number. Essentially, it searches by calling
19267 @code{calc-prime-test} on successive integers until it finds one that
19268 passes the test. This is quite fast for integers less than eight million,
19269 but once the probabilistic test comes into play the search may be rather
19270 slow. Ordinarily this command stops for any prime that passes one iteration
19271 of the primality test. With a numeric prefix argument, a number must pass
19272 the specified number of iterations before the search stops. (This only
19273 matters when searching above eight million.) You can always use additional
19274 @kbd{k p} commands to increase your certainty that the number is indeed
19278 @pindex calc-prev-prime
19280 @mindex prevpr@idots
19283 The @kbd{I k n} (@code{calc-prev-prime}) [@code{prevprime}] command
19284 analogously finds the next prime less than a given number.
19287 @pindex calc-totient
19289 The @kbd{k t} (@code{calc-totient}) [@code{totient}] command computes the
19291 @texline function@tie{}@math{\phi(n)},
19292 @infoline function,
19293 the number of integers less than @expr{n} which
19294 are relatively prime to @expr{n}.
19297 @pindex calc-moebius
19299 The @kbd{k m} (@code{calc-moebius}) [@code{moebius}] command computes the
19300 @texline M@"obius @math{\mu}
19301 @infoline Moebius ``mu''
19302 function. If the input number is a product of @expr{k}
19303 distinct factors, this is @expr{(-1)^k}. If the input number has any
19304 duplicate factors (i.e., can be divided by the same prime more than once),
19305 the result is zero.
19307 @node Probability Distribution Functions, , Combinatorial Functions, Scientific Functions
19308 @section Probability Distribution Functions
19311 The functions in this section compute various probability distributions.
19312 For continuous distributions, this is the integral of the probability
19313 density function from @expr{x} to infinity. (These are the ``upper
19314 tail'' distribution functions; there are also corresponding ``lower
19315 tail'' functions which integrate from minus infinity to @expr{x}.)
19316 For discrete distributions, the upper tail function gives the sum
19317 from @expr{x} to infinity; the lower tail function gives the sum
19318 from minus infinity up to, but not including,@w{ }@expr{x}.
19320 To integrate from @expr{x} to @expr{y}, just use the distribution
19321 function twice and subtract. For example, the probability that a
19322 Gaussian random variable with mean 2 and standard deviation 1 will
19323 lie in the range from 2.5 to 2.8 is @samp{utpn(2.5,2,1) - utpn(2.8,2,1)}
19324 (``the probability that it is greater than 2.5, but not greater than 2.8''),
19325 or equivalently @samp{ltpn(2.8,2,1) - ltpn(2.5,2,1)}.
19332 The @kbd{k B} (@code{calc-utpb}) [@code{utpb}] function uses the
19333 binomial distribution. Push the parameters @var{n}, @var{p}, and
19334 then @var{x} onto the stack; the result (@samp{utpb(x,n,p)}) is the
19335 probability that an event will occur @var{x} or more times out
19336 of @var{n} trials, if its probability of occurring in any given
19337 trial is @var{p}. The @kbd{I k B} [@code{ltpb}] function is
19338 the probability that the event will occur fewer than @var{x} times.
19340 The other probability distribution functions similarly take the
19341 form @kbd{k @var{X}} (@code{calc-utp@var{x}}) [@code{utp@var{x}}]
19342 and @kbd{I k @var{X}} [@code{ltp@var{x}}], for various letters
19343 @var{x}. The arguments to the algebraic functions are the value of
19344 the random variable first, then whatever other parameters define the
19345 distribution. Note these are among the few Calc functions where the
19346 order of the arguments in algebraic form differs from the order of
19347 arguments as found on the stack. (The random variable comes last on
19348 the stack, so that you can type, e.g., @kbd{2 @key{RET} 1 @key{RET} 2.5
19349 k N M-@key{RET} @key{DEL} 2.8 k N -}, using @kbd{M-@key{RET} @key{DEL}} to
19350 recover the original arguments but substitute a new value for @expr{x}.)
19363 The @samp{utpc(x,v)} function uses the chi-square distribution with
19364 @texline @math{\nu}
19366 degrees of freedom. It is the probability that a model is
19367 correct if its chi-square statistic is @expr{x}.
19380 The @samp{utpf(F,v1,v2)} function uses the F distribution, used in
19381 various statistical tests. The parameters
19382 @texline @math{\nu_1}
19383 @infoline @expr{v1}
19385 @texline @math{\nu_2}
19386 @infoline @expr{v2}
19387 are the degrees of freedom in the numerator and denominator,
19388 respectively, used in computing the statistic @expr{F}.
19401 The @samp{utpn(x,m,s)} function uses a normal (Gaussian) distribution
19402 with mean @expr{m} and standard deviation
19403 @texline @math{\sigma}.
19404 @infoline @expr{s}.
19405 It is the probability that such a normal-distributed random variable
19406 would exceed @expr{x}.
19419 The @samp{utpp(n,x)} function uses a Poisson distribution with
19420 mean @expr{x}. It is the probability that @expr{n} or more such
19421 Poisson random events will occur.
19434 The @samp{utpt(t,v)} function uses the Student's ``t'' distribution
19436 @texline @math{\nu}
19438 degrees of freedom. It is the probability that a
19439 t-distributed random variable will be greater than @expr{t}.
19440 (Note: This computes the distribution function
19441 @texline @math{A(t|\nu)}
19442 @infoline @expr{A(t|v)}
19444 @texline @math{A(0|\nu) = 1}
19445 @infoline @expr{A(0|v) = 1}
19447 @texline @math{A(\infty|\nu) \to 0}.
19448 @infoline @expr{A(inf|v) -> 0}.
19449 The @code{UTPT} operation on the HP-48 uses a different definition which
19450 returns half of Calc's value: @samp{UTPT(t,v) = .5*utpt(t,v)}.)
19452 While Calc does not provide inverses of the probability distribution
19453 functions, the @kbd{a R} command can be used to solve for the inverse.
19454 Since the distribution functions are monotonic, @kbd{a R} is guaranteed
19455 to be able to find a solution given any initial guess.
19456 @xref{Numerical Solutions}.
19458 @node Matrix Functions, Algebra, Scientific Functions, Top
19459 @chapter Vector/Matrix Functions
19462 Many of the commands described here begin with the @kbd{v} prefix.
19463 (For convenience, the shift-@kbd{V} prefix is equivalent to @kbd{v}.)
19464 The commands usually apply to both plain vectors and matrices; some
19465 apply only to matrices or only to square matrices. If the argument
19466 has the wrong dimensions the operation is left in symbolic form.
19468 Vectors are entered and displayed using @samp{[a,b,c]} notation.
19469 Matrices are vectors of which all elements are vectors of equal length.
19470 (Though none of the standard Calc commands use this concept, a
19471 three-dimensional matrix or rank-3 tensor could be defined as a
19472 vector of matrices, and so on.)
19475 * Packing and Unpacking::
19476 * Building Vectors::
19477 * Extracting Elements::
19478 * Manipulating Vectors::
19479 * Vector and Matrix Arithmetic::
19481 * Statistical Operations::
19482 * Reducing and Mapping::
19483 * Vector and Matrix Formats::
19486 @node Packing and Unpacking, Building Vectors, Matrix Functions, Matrix Functions
19487 @section Packing and Unpacking
19490 Calc's ``pack'' and ``unpack'' commands collect stack entries to build
19491 composite objects such as vectors and complex numbers. They are
19492 described in this chapter because they are most often used to build
19498 The @kbd{v p} (@code{calc-pack}) [@code{pack}] command collects several
19499 elements from the stack into a matrix, complex number, HMS form, error
19500 form, etc. It uses a numeric prefix argument to specify the kind of
19501 object to be built; this argument is referred to as the ``packing mode.''
19502 If the packing mode is a nonnegative integer, a vector of that
19503 length is created. For example, @kbd{C-u 5 v p} will pop the top
19504 five stack elements and push back a single vector of those five
19505 elements. (@kbd{C-u 0 v p} simply creates an empty vector.)
19507 The same effect can be had by pressing @kbd{[} to push an incomplete
19508 vector on the stack, using @key{TAB} (@code{calc-roll-down}) to sneak
19509 the incomplete object up past a certain number of elements, and
19510 then pressing @kbd{]} to complete the vector.
19512 Negative packing modes create other kinds of composite objects:
19516 Two values are collected to build a complex number. For example,
19517 @kbd{5 @key{RET} 7 C-u -1 v p} creates the complex number
19518 @expr{(5, 7)}. The result is always a rectangular complex
19519 number. The two input values must both be real numbers,
19520 i.e., integers, fractions, or floats. If they are not, Calc
19521 will instead build a formula like @samp{a + (0, 1) b}. (The
19522 other packing modes also create a symbolic answer if the
19523 components are not suitable.)
19526 Two values are collected to build a polar complex number.
19527 The first is the magnitude; the second is the phase expressed
19528 in either degrees or radians according to the current angular
19532 Three values are collected into an HMS form. The first
19533 two values (hours and minutes) must be integers or
19534 integer-valued floats. The third value may be any real
19538 Two values are collected into an error form. The inputs
19539 may be real numbers or formulas.
19542 Two values are collected into a modulo form. The inputs
19543 must be real numbers.
19546 Two values are collected into the interval @samp{[a .. b]}.
19547 The inputs may be real numbers, HMS or date forms, or formulas.
19550 Two values are collected into the interval @samp{[a .. b)}.
19553 Two values are collected into the interval @samp{(a .. b]}.
19556 Two values are collected into the interval @samp{(a .. b)}.
19559 Two integer values are collected into a fraction.
19562 Two values are collected into a floating-point number.
19563 The first is the mantissa; the second, which must be an
19564 integer, is the exponent. The result is the mantissa
19565 times ten to the power of the exponent.
19568 This is treated the same as @mathit{-11} by the @kbd{v p} command.
19569 When unpacking, @mathit{-12} specifies that a floating-point mantissa
19573 A real number is converted into a date form.
19576 Three numbers (year, month, day) are packed into a pure date form.
19579 Six numbers are packed into a date/time form.
19582 With any of the two-input negative packing modes, either or both
19583 of the inputs may be vectors. If both are vectors of the same
19584 length, the result is another vector made by packing corresponding
19585 elements of the input vectors. If one input is a vector and the
19586 other is a plain number, the number is packed along with each vector
19587 element to produce a new vector. For example, @kbd{C-u -4 v p}
19588 could be used to convert a vector of numbers and a vector of errors
19589 into a single vector of error forms; @kbd{C-u -5 v p} could convert
19590 a vector of numbers and a single number @var{M} into a vector of
19591 numbers modulo @var{M}.
19593 If you don't give a prefix argument to @kbd{v p}, it takes
19594 the packing mode from the top of the stack. The elements to
19595 be packed then begin at stack level 2. Thus
19596 @kbd{1 @key{RET} 2 @key{RET} 4 n v p} is another way to
19597 enter the error form @samp{1 +/- 2}.
19599 If the packing mode taken from the stack is a vector, the result is a
19600 matrix with the dimensions specified by the elements of the vector,
19601 which must each be integers. For example, if the packing mode is
19602 @samp{[2, 3]}, then six numbers will be taken from the stack and
19603 returned in the form @samp{[@w{[a, b, c]}, [d, e, f]]}.
19605 If any elements of the vector are negative, other kinds of
19606 packing are done at that level as described above. For
19607 example, @samp{[2, 3, -4]} takes 12 objects and creates a
19608 @texline @math{2\times3}
19610 matrix of error forms: @samp{[[a +/- b, c +/- d ... ]]}.
19611 Also, @samp{[-4, -10]} will convert four integers into an
19612 error form consisting of two fractions: @samp{a:b +/- c:d}.
19618 There is an equivalent algebraic function,
19619 @samp{pack(@var{mode}, @var{items})} where @var{mode} is a
19620 packing mode (an integer or a vector of integers) and @var{items}
19621 is a vector of objects to be packed (re-packed, really) according
19622 to that mode. For example, @samp{pack([3, -4], [a,b,c,d,e,f])}
19623 yields @samp{[a +/- b, @w{c +/- d}, e +/- f]}. The function is
19624 left in symbolic form if the packing mode is invalid, or if the
19625 number of data items does not match the number of items required
19630 @pindex calc-unpack
19631 The @kbd{v u} (@code{calc-unpack}) command takes the vector, complex
19632 number, HMS form, or other composite object on the top of the stack and
19633 ``unpacks'' it, pushing each of its elements onto the stack as separate
19634 objects. Thus, it is the ``inverse'' of @kbd{v p}. If the value
19635 at the top of the stack is a formula, @kbd{v u} unpacks it by pushing
19636 each of the arguments of the top-level operator onto the stack.
19638 You can optionally give a numeric prefix argument to @kbd{v u}
19639 to specify an explicit (un)packing mode. If the packing mode is
19640 negative and the input is actually a vector or matrix, the result
19641 will be two or more similar vectors or matrices of the elements.
19642 For example, given the vector @samp{[@w{a +/- b}, c^2, d +/- 7]},
19643 the result of @kbd{C-u -4 v u} will be the two vectors
19644 @samp{[a, c^2, d]} and @w{@samp{[b, 0, 7]}}.
19646 Note that the prefix argument can have an effect even when the input is
19647 not a vector. For example, if the input is the number @mathit{-5}, then
19648 @kbd{c-u -1 v u} yields @mathit{-5} and 0 (the components of @mathit{-5}
19649 when viewed as a rectangular complex number); @kbd{C-u -2 v u} yields 5
19650 and 180 (assuming Degrees mode); and @kbd{C-u -10 v u} yields @mathit{-5}
19651 and 1 (the numerator and denominator of @mathit{-5}, viewed as a rational
19652 number). Plain @kbd{v u} with this input would complain that the input
19653 is not a composite object.
19655 Unpacking mode @mathit{-11} converts a float into an integer mantissa and
19656 an integer exponent, where the mantissa is not divisible by 10
19657 (except that 0.0 is represented by a mantissa and exponent of 0).
19658 Unpacking mode @mathit{-12} converts a float into a floating-point mantissa
19659 and integer exponent, where the mantissa (for non-zero numbers)
19660 is guaranteed to lie in the range [1 .. 10). In both cases,
19661 the mantissa is shifted left or right (and the exponent adjusted
19662 to compensate) in order to satisfy these constraints.
19664 Positive unpacking modes are treated differently than for @kbd{v p}.
19665 A mode of 1 is much like plain @kbd{v u} with no prefix argument,
19666 except that in addition to the components of the input object,
19667 a suitable packing mode to re-pack the object is also pushed.
19668 Thus, @kbd{C-u 1 v u} followed by @kbd{v p} will re-build the
19671 A mode of 2 unpacks two levels of the object; the resulting
19672 re-packing mode will be a vector of length 2. This might be used
19673 to unpack a matrix, say, or a vector of error forms. Higher
19674 unpacking modes unpack the input even more deeply.
19680 There are two algebraic functions analogous to @kbd{v u}.
19681 The @samp{unpack(@var{mode}, @var{item})} function unpacks the
19682 @var{item} using the given @var{mode}, returning the result as
19683 a vector of components. Here the @var{mode} must be an
19684 integer, not a vector. For example, @samp{unpack(-4, a +/- b)}
19685 returns @samp{[a, b]}, as does @samp{unpack(1, a +/- b)}.
19691 The @code{unpackt} function is like @code{unpack} but instead
19692 of returning a simple vector of items, it returns a vector of
19693 two things: The mode, and the vector of items. For example,
19694 @samp{unpackt(1, 2:3 +/- 1:4)} returns @samp{[-4, [2:3, 1:4]]},
19695 and @samp{unpackt(2, 2:3 +/- 1:4)} returns @samp{[[-4, -10], [2, 3, 1, 4]]}.
19696 The identity for re-building the original object is
19697 @samp{apply(pack, unpackt(@var{n}, @var{x})) = @var{x}}. (The
19698 @code{apply} function builds a function call given the function
19699 name and a vector of arguments.)
19701 @cindex Numerator of a fraction, extracting
19702 Subscript notation is a useful way to extract a particular part
19703 of an object. For example, to get the numerator of a rational
19704 number, you can use @samp{unpack(-10, @var{x})_1}.
19706 @node Building Vectors, Extracting Elements, Packing and Unpacking, Matrix Functions
19707 @section Building Vectors
19710 Vectors and matrices can be added,
19711 subtracted, multiplied, and divided; @pxref{Basic Arithmetic}.
19714 @pindex calc-concat
19719 The @kbd{|} (@code{calc-concat}) [@code{vconcat}] command ``concatenates'' two vectors
19720 into one. For example, after @kbd{@w{[ 1 , 2 ]} [ 3 , 4 ] |}, the stack
19721 will contain the single vector @samp{[1, 2, 3, 4]}. If the arguments
19722 are matrices, the rows of the first matrix are concatenated with the
19723 rows of the second. (In other words, two matrices are just two vectors
19724 of row-vectors as far as @kbd{|} is concerned.)
19726 If either argument to @kbd{|} is a scalar (a non-vector), it is treated
19727 like a one-element vector for purposes of concatenation: @kbd{1 [ 2 , 3 ] |}
19728 produces the vector @samp{[1, 2, 3]}. Likewise, if one argument is a
19729 matrix and the other is a plain vector, the vector is treated as a
19734 The @kbd{H |} (@code{calc-append}) [@code{append}] command concatenates
19735 two vectors without any special cases. Both inputs must be vectors.
19736 Whether or not they are matrices is not taken into account. If either
19737 argument is a scalar, the @code{append} function is left in symbolic form.
19738 See also @code{cons} and @code{rcons} below.
19742 The @kbd{I |} and @kbd{H I |} commands are similar, but they use their
19743 two stack arguments in the opposite order. Thus @kbd{I |} is equivalent
19744 to @kbd{@key{TAB} |}, but possibly more convenient and also a bit faster.
19750 The @kbd{v d} (@code{calc-diag}) [@code{diag}] function builds a diagonal
19751 square matrix. The optional numeric prefix gives the number of rows
19752 and columns in the matrix. If the value at the top of the stack is a
19753 vector, the elements of the vector are used as the diagonal elements; the
19754 prefix, if specified, must match the size of the vector. If the value on
19755 the stack is a scalar, it is used for each element on the diagonal, and
19756 the prefix argument is required.
19758 To build a constant square matrix, e.g., a
19759 @texline @math{3\times3}
19761 matrix filled with ones, use @kbd{0 M-3 v d 1 +}, i.e., build a zero
19762 matrix first and then add a constant value to that matrix. (Another
19763 alternative would be to use @kbd{v b} and @kbd{v a}; see below.)
19769 The @kbd{v i} (@code{calc-ident}) [@code{idn}] function builds an identity
19770 matrix of the specified size. It is a convenient form of @kbd{v d}
19771 where the diagonal element is always one. If no prefix argument is given,
19772 this command prompts for one.
19774 In algebraic notation, @samp{idn(a,n)} acts much like @samp{diag(a,n)},
19775 except that @expr{a} is required to be a scalar (non-vector) quantity.
19776 If @expr{n} is omitted, @samp{idn(a)} represents @expr{a} times an
19777 identity matrix of unknown size. Calc can operate algebraically on
19778 such generic identity matrices, and if one is combined with a matrix
19779 whose size is known, it is converted automatically to an identity
19780 matrix of a suitable matching size. The @kbd{v i} command with an
19781 argument of zero creates a generic identity matrix, @samp{idn(1)}.
19782 Note that in dimensioned Matrix mode (@pxref{Matrix Mode}), generic
19783 identity matrices are immediately expanded to the current default
19790 The @kbd{v x} (@code{calc-index}) [@code{index}] function builds a vector
19791 of consecutive integers from 1 to @var{n}, where @var{n} is the numeric
19792 prefix argument. If you do not provide a prefix argument, you will be
19793 prompted to enter a suitable number. If @var{n} is negative, the result
19794 is a vector of negative integers from @var{n} to @mathit{-1}.
19796 With a prefix argument of just @kbd{C-u}, the @kbd{v x} command takes
19797 three values from the stack: @var{n}, @var{start}, and @var{incr} (with
19798 @var{incr} at top-of-stack). Counting starts at @var{start} and increases
19799 by @var{incr} for successive vector elements. If @var{start} or @var{n}
19800 is in floating-point format, the resulting vector elements will also be
19801 floats. Note that @var{start} and @var{incr} may in fact be any kind
19802 of numbers or formulas.
19804 When @var{start} and @var{incr} are specified, a negative @var{n} has a
19805 different interpretation: It causes a geometric instead of arithmetic
19806 sequence to be generated. For example, @samp{index(-3, a, b)} produces
19807 @samp{[a, a b, a b^2]}. If you omit @var{incr} in the algebraic form,
19808 @samp{index(@var{n}, @var{start})}, the default value for @var{incr}
19809 is one for positive @var{n} or two for negative @var{n}.
19813 @pindex calc-build-vector
19815 The @kbd{v b} (@code{calc-build-vector}) [@code{cvec}] function builds a
19816 vector of @var{n} copies of the value on the top of the stack, where @var{n}
19817 is the numeric prefix argument. In algebraic formulas, @samp{cvec(x,n,m)}
19818 can also be used to build an @var{n}-by-@var{m} matrix of copies of @var{x}.
19819 (Interactively, just use @kbd{v b} twice: once to build a row, then again
19820 to build a matrix of copies of that row.)
19830 The @kbd{v h} (@code{calc-head}) [@code{head}] function returns the first
19831 element of a vector. The @kbd{I v h} (@code{calc-tail}) [@code{tail}]
19832 function returns the vector with its first element removed. In both
19833 cases, the argument must be a non-empty vector.
19839 The @kbd{v k} (@code{calc-cons}) [@code{cons}] function takes a value @var{h}
19840 and a vector @var{t} from the stack, and produces the vector whose head is
19841 @var{h} and whose tail is @var{t}. This is similar to @kbd{|}, except
19842 if @var{h} is itself a vector, @kbd{|} will concatenate the two vectors
19843 whereas @code{cons} will insert @var{h} at the front of the vector @var{t}.
19866 Each of these three functions also accepts the Hyperbolic flag [@code{rhead},
19867 @code{rtail}, @code{rcons}] in which case @var{t} instead represents
19868 the @emph{last} single element of the vector, with @var{h}
19869 representing the remainder of the vector. Thus the vector
19870 @samp{[a, b, c, d] = cons(a, [b, c, d]) = rcons([a, b, c], d)}.
19871 Also, @samp{head([a, b, c, d]) = a}, @samp{tail([a, b, c, d]) = [b, c, d]},
19872 @samp{rhead([a, b, c, d]) = [a, b, c]}, and @samp{rtail([a, b, c, d]) = d}.
19874 @node Extracting Elements, Manipulating Vectors, Building Vectors, Matrix Functions
19875 @section Extracting Vector Elements
19882 The @kbd{v r} (@code{calc-mrow}) [@code{mrow}] command extracts one row of
19883 the matrix on the top of the stack, or one element of the plain vector on
19884 the top of the stack. The row or element is specified by the numeric
19885 prefix argument; the default is to prompt for the row or element number.
19886 The matrix or vector is replaced by the specified row or element in the
19887 form of a vector or scalar, respectively.
19889 @cindex Permutations, applying
19890 With a prefix argument of @kbd{C-u} only, @kbd{v r} takes the index of
19891 the element or row from the top of the stack, and the vector or matrix
19892 from the second-to-top position. If the index is itself a vector of
19893 integers, the result is a vector of the corresponding elements of the
19894 input vector, or a matrix of the corresponding rows of the input matrix.
19895 This command can be used to obtain any permutation of a vector.
19897 With @kbd{C-u}, if the index is an interval form with integer components,
19898 it is interpreted as a range of indices and the corresponding subvector or
19899 submatrix is returned.
19901 @cindex Subscript notation
19903 @pindex calc-subscript
19906 Subscript notation in algebraic formulas (@samp{a_b}) stands for the
19907 Calc function @code{subscr}, which is synonymous with @code{mrow}.
19908 Thus, @samp{[x, y, z]_k} produces @expr{x}, @expr{y}, or @expr{z} if
19909 @expr{k} is one, two, or three, respectively. A double subscript
19910 (@samp{M_i_j}, equivalent to @samp{subscr(subscr(M, i), j)}) will
19911 access the element at row @expr{i}, column @expr{j} of a matrix.
19912 The @kbd{a _} (@code{calc-subscript}) command creates a subscript
19913 formula @samp{a_b} out of two stack entries. (It is on the @kbd{a}
19914 ``algebra'' prefix because subscripted variables are often used
19915 purely as an algebraic notation.)
19918 Given a negative prefix argument, @kbd{v r} instead deletes one row or
19919 element from the matrix or vector on the top of the stack. Thus
19920 @kbd{C-u 2 v r} replaces a matrix with its second row, but @kbd{C-u -2 v r}
19921 replaces the matrix with the same matrix with its second row removed.
19922 In algebraic form this function is called @code{mrrow}.
19925 Given a prefix argument of zero, @kbd{v r} extracts the diagonal elements
19926 of a square matrix in the form of a vector. In algebraic form this
19927 function is called @code{getdiag}.
19934 The @kbd{v c} (@code{calc-mcol}) [@code{mcol} or @code{mrcol}] command is
19935 the analogous operation on columns of a matrix. Given a plain vector
19936 it extracts (or removes) one element, just like @kbd{v r}. If the
19937 index in @kbd{C-u v c} is an interval or vector and the argument is a
19938 matrix, the result is a submatrix with only the specified columns
19939 retained (and possibly permuted in the case of a vector index).
19941 To extract a matrix element at a given row and column, use @kbd{v r} to
19942 extract the row as a vector, then @kbd{v c} to extract the column element
19943 from that vector. In algebraic formulas, it is often more convenient to
19944 use subscript notation: @samp{m_i_j} gives row @expr{i}, column @expr{j}
19945 of matrix @expr{m}.
19949 @pindex calc-subvector
19951 The @kbd{v s} (@code{calc-subvector}) [@code{subvec}] command extracts
19952 a subvector of a vector. The arguments are the vector, the starting
19953 index, and the ending index, with the ending index in the top-of-stack
19954 position. The starting index indicates the first element of the vector
19955 to take. The ending index indicates the first element @emph{past} the
19956 range to be taken. Thus, @samp{subvec([a, b, c, d, e], 2, 4)} produces
19957 the subvector @samp{[b, c]}. You could get the same result using
19958 @samp{mrow([a, b, c, d, e], @w{[2 .. 4)})}.
19960 If either the start or the end index is zero or negative, it is
19961 interpreted as relative to the end of the vector. Thus
19962 @samp{subvec([a, b, c, d, e], 2, -2)} also produces @samp{[b, c]}. In
19963 the algebraic form, the end index can be omitted in which case it
19964 is taken as zero, i.e., elements from the starting element to the
19965 end of the vector are used. The infinity symbol, @code{inf}, also
19966 has this effect when used as the ending index.
19971 With the Inverse flag, @kbd{I v s} [@code{rsubvec}] removes a subvector
19972 from a vector. The arguments are interpreted the same as for the
19973 normal @kbd{v s} command. Thus, @samp{rsubvec([a, b, c, d, e], 2, 4)}
19974 produces @samp{[a, d, e]}. It is always true that @code{subvec} and
19975 @code{rsubvec} return complementary parts of the input vector.
19977 @xref{Selecting Subformulas}, for an alternative way to operate on
19978 vectors one element at a time.
19980 @node Manipulating Vectors, Vector and Matrix Arithmetic, Extracting Elements, Matrix Functions
19981 @section Manipulating Vectors
19986 @pindex calc-vlength
19988 The @kbd{v l} (@code{calc-vlength}) [@code{vlen}] command computes the
19989 length of a vector. The length of a non-vector is considered to be zero.
19990 Note that matrices are just vectors of vectors for the purposes of this
19996 With the Hyperbolic flag, @kbd{H v l} [@code{mdims}] computes a vector
19997 of the dimensions of a vector, matrix, or higher-order object. For
19998 example, @samp{mdims([[a,b,c],[d,e,f]])} returns @samp{[2, 3]} since
20000 @texline @math{2\times3}
20006 @pindex calc-vector-find
20008 The @kbd{v f} (@code{calc-vector-find}) [@code{find}] command searches
20009 along a vector for the first element equal to a given target. The target
20010 is on the top of the stack; the vector is in the second-to-top position.
20011 If a match is found, the result is the index of the matching element.
20012 Otherwise, the result is zero. The numeric prefix argument, if given,
20013 allows you to select any starting index for the search.
20017 @pindex calc-arrange-vector
20019 @cindex Arranging a matrix
20020 @cindex Reshaping a matrix
20021 @cindex Flattening a matrix
20022 The @kbd{v a} (@code{calc-arrange-vector}) [@code{arrange}] command
20023 rearranges a vector to have a certain number of columns and rows. The
20024 numeric prefix argument specifies the number of columns; if you do not
20025 provide an argument, you will be prompted for the number of columns.
20026 The vector or matrix on the top of the stack is @dfn{flattened} into a
20027 plain vector. If the number of columns is nonzero, this vector is
20028 then formed into a matrix by taking successive groups of @var{n} elements.
20029 If the number of columns does not evenly divide the number of elements
20030 in the vector, the last row will be short and the result will not be
20031 suitable for use as a matrix. For example, with the matrix
20032 @samp{[[1, 2], @w{[3, 4]}]} on the stack, @kbd{v a 4} produces
20033 @samp{[[1, 2, 3, 4]]} (a
20034 @texline @math{1\times4}
20036 matrix), @kbd{v a 1} produces @samp{[[1], [2], [3], [4]]} (a
20037 @texline @math{4\times1}
20039 matrix), @kbd{v a 2} produces @samp{[[1, 2], [3, 4]]} (the original
20040 @texline @math{2\times2}
20042 matrix), @w{@kbd{v a 3}} produces @samp{[[1, 2, 3], [4]]} (not a
20043 matrix), and @kbd{v a 0} produces the flattened list
20044 @samp{[1, 2, @w{3, 4}]}.
20046 @cindex Sorting data
20054 The @kbd{V S} (@code{calc-sort}) [@code{sort}] command sorts the elements of
20055 a vector into increasing order. Real numbers, real infinities, and
20056 constant interval forms come first in this ordering; next come other
20057 kinds of numbers, then variables (in alphabetical order), then finally
20058 come formulas and other kinds of objects; these are sorted according
20059 to a kind of lexicographic ordering with the useful property that
20060 one vector is less or greater than another if the first corresponding
20061 unequal elements are less or greater, respectively. Since quoted strings
20062 are stored by Calc internally as vectors of ASCII character codes
20063 (@pxref{Strings}), this means vectors of strings are also sorted into
20064 alphabetical order by this command.
20066 The @kbd{I V S} [@code{rsort}] command sorts a vector into decreasing order.
20068 @cindex Permutation, inverse of
20069 @cindex Inverse of permutation
20070 @cindex Index tables
20071 @cindex Rank tables
20079 The @kbd{V G} (@code{calc-grade}) [@code{grade}, @code{rgrade}] command
20080 produces an index table or permutation vector which, if applied to the
20081 input vector (as the index of @kbd{C-u v r}, say), would sort the vector.
20082 A permutation vector is just a vector of integers from 1 to @var{n}, where
20083 each integer occurs exactly once. One application of this is to sort a
20084 matrix of data rows using one column as the sort key; extract that column,
20085 grade it with @kbd{V G}, then use the result to reorder the original matrix
20086 with @kbd{C-u v r}. Another interesting property of the @code{V G} command
20087 is that, if the input is itself a permutation vector, the result will
20088 be the inverse of the permutation. The inverse of an index table is
20089 a rank table, whose @var{k}th element says where the @var{k}th original
20090 vector element will rest when the vector is sorted. To get a rank
20091 table, just use @kbd{V G V G}.
20093 With the Inverse flag, @kbd{I V G} produces an index table that would
20094 sort the input into decreasing order. Note that @kbd{V S} and @kbd{V G}
20095 use a ``stable'' sorting algorithm, i.e., any two elements which are equal
20096 will not be moved out of their original order. Generally there is no way
20097 to tell with @kbd{V S}, since two elements which are equal look the same,
20098 but with @kbd{V G} this can be an important issue. In the matrix-of-rows
20099 example, suppose you have names and telephone numbers as two columns and
20100 you wish to sort by phone number primarily, and by name when the numbers
20101 are equal. You can sort the data matrix by names first, and then again
20102 by phone numbers. Because the sort is stable, any two rows with equal
20103 phone numbers will remain sorted by name even after the second sort.
20108 @pindex calc-histogram
20110 @mindex histo@idots
20113 The @kbd{V H} (@code{calc-histogram}) [@code{histogram}] command builds a
20114 histogram of a vector of numbers. Vector elements are assumed to be
20115 integers or real numbers in the range [0..@var{n}) for some ``number of
20116 bins'' @var{n}, which is the numeric prefix argument given to the
20117 command. The result is a vector of @var{n} counts of how many times
20118 each value appeared in the original vector. Non-integers in the input
20119 are rounded down to integers. Any vector elements outside the specified
20120 range are ignored. (You can tell if elements have been ignored by noting
20121 that the counts in the result vector don't add up to the length of the
20124 If no prefix is given, then you will be prompted for a vector which
20125 will be used to determine the bins. (If a positive integer is given at
20126 this prompt, it will be still treated as if it were given as a
20127 prefix.) Each bin will consist of the interval of numbers closest to
20128 the corresponding number of this new vector; if the vector
20129 @expr{[a, b, c, ...]} is entered at the prompt, the bins will be
20130 @expr{(-inf, (a+b)/2]}, @expr{((a+b)/2, (b+c)/2]}, etc. The result of
20131 this command will be a vector counting how many elements of the
20132 original vector are in each bin.
20134 The result will then be a vector with the same length as this new vector;
20135 each element of the new vector will be replaced by the number of
20136 elements of the original vector which are closest to it.
20140 With the Hyperbolic flag, @kbd{H V H} pulls two vectors from the stack.
20141 The second-to-top vector is the list of numbers as before. The top
20142 vector is an equal-sized list of ``weights'' to attach to the elements
20143 of the data vector. For example, if the first data element is 4.2 and
20144 the first weight is 10, then 10 will be added to bin 4 of the result
20145 vector. Without the hyperbolic flag, every element has a weight of one.
20149 @pindex calc-transpose
20151 The @kbd{v t} (@code{calc-transpose}) [@code{trn}] command computes
20152 the transpose of the matrix at the top of the stack. If the argument
20153 is a plain vector, it is treated as a row vector and transposed into
20154 a one-column matrix.
20158 @pindex calc-reverse-vector
20160 The @kbd{v v} (@code{calc-reverse-vector}) [@code{rev}] command reverses
20161 a vector end-for-end. Given a matrix, it reverses the order of the rows.
20162 (To reverse the columns instead, just use @kbd{v t v v v t}. The same
20163 principle can be used to apply other vector commands to the columns of
20168 @pindex calc-mask-vector
20170 The @kbd{v m} (@code{calc-mask-vector}) [@code{vmask}] command uses
20171 one vector as a mask to extract elements of another vector. The mask
20172 is in the second-to-top position; the target vector is on the top of
20173 the stack. These vectors must have the same length. The result is
20174 the same as the target vector, but with all elements which correspond
20175 to zeros in the mask vector deleted. Thus, for example,
20176 @samp{vmask([1, 0, 1, 0, 1], [a, b, c, d, e])} produces @samp{[a, c, e]}.
20177 @xref{Logical Operations}.
20181 @pindex calc-expand-vector
20183 The @kbd{v e} (@code{calc-expand-vector}) [@code{vexp}] command
20184 expands a vector according to another mask vector. The result is a
20185 vector the same length as the mask, but with nonzero elements replaced
20186 by successive elements from the target vector. The length of the target
20187 vector is normally the number of nonzero elements in the mask. If the
20188 target vector is longer, its last few elements are lost. If the target
20189 vector is shorter, the last few nonzero mask elements are left
20190 unreplaced in the result. Thus @samp{vexp([2, 0, 3, 0, 7], [a, b])}
20191 produces @samp{[a, 0, b, 0, 7]}.
20195 With the Hyperbolic flag, @kbd{H v e} takes a filler value from the
20196 top of the stack; the mask and target vectors come from the third and
20197 second elements of the stack. This filler is used where the mask is
20198 zero: @samp{vexp([2, 0, 3, 0, 7], [a, b], z)} produces
20199 @samp{[a, z, c, z, 7]}. If the filler value is itself a vector,
20200 then successive values are taken from it, so that the effect is to
20201 interleave two vectors according to the mask:
20202 @samp{vexp([2, 0, 3, 7, 0, 0], [a, b], [x, y])} produces
20203 @samp{[a, x, b, 7, y, 0]}.
20205 Another variation on the masking idea is to combine @samp{[a, b, c, d, e]}
20206 with the mask @samp{[1, 0, 1, 0, 1]} to produce @samp{[a, 0, c, 0, e]}.
20207 You can accomplish this with @kbd{V M a &}, mapping the logical ``and''
20208 operation across the two vectors. @xref{Logical Operations}. Note that
20209 the @code{? :} operation also discussed there allows other types of
20210 masking using vectors.
20212 @node Vector and Matrix Arithmetic, Set Operations, Manipulating Vectors, Matrix Functions
20213 @section Vector and Matrix Arithmetic
20216 Basic arithmetic operations like addition and multiplication are defined
20217 for vectors and matrices as well as for numbers. Division of matrices, in
20218 the sense of multiplying by the inverse, is supported. (Division by a
20219 matrix actually uses LU-decomposition for greater accuracy and speed.)
20220 @xref{Basic Arithmetic}.
20222 The following functions are applied element-wise if their arguments are
20223 vectors or matrices: @code{change-sign}, @code{conj}, @code{arg},
20224 @code{re}, @code{im}, @code{polar}, @code{rect}, @code{clean},
20225 @code{float}, @code{frac}. @xref{Function Index}.
20229 @pindex calc-conj-transpose
20231 The @kbd{V J} (@code{calc-conj-transpose}) [@code{ctrn}] command computes
20232 the conjugate transpose of its argument, i.e., @samp{conj(trn(x))}.
20237 @kindex A (vectors)
20238 @pindex calc-abs (vectors)
20242 @tindex abs (vectors)
20243 The @kbd{A} (@code{calc-abs}) [@code{abs}] command computes the
20244 Frobenius norm of a vector or matrix argument. This is the square
20245 root of the sum of the squares of the absolute values of the
20246 elements of the vector or matrix. If the vector is interpreted as
20247 a point in two- or three-dimensional space, this is the distance
20248 from that point to the origin.
20254 The @kbd{v n} (@code{calc-rnorm}) [@code{rnorm}] command computes the
20255 infinity-norm of a vector, or the row norm of a matrix. For a plain
20256 vector, this is the maximum of the absolute values of the elements. For
20257 a matrix, this is the maximum of the row-absolute-value-sums, i.e., of
20258 the sums of the absolute values of the elements along the various rows.
20264 The @kbd{V N} (@code{calc-cnorm}) [@code{cnorm}] command computes
20265 the one-norm of a vector, or column norm of a matrix. For a plain
20266 vector, this is the sum of the absolute values of the elements.
20267 For a matrix, this is the maximum of the column-absolute-value-sums.
20268 General @expr{k}-norms for @expr{k} other than one or infinity are
20269 not provided. However, the 2-norm (or Frobenius norm) is provided for
20270 vectors by the @kbd{A} (@code{calc-abs}) command.
20276 The @kbd{V C} (@code{calc-cross}) [@code{cross}] command computes the
20277 right-handed cross product of two vectors, each of which must have
20278 exactly three elements.
20283 @kindex & (matrices)
20284 @pindex calc-inv (matrices)
20288 @tindex inv (matrices)
20289 The @kbd{&} (@code{calc-inv}) [@code{inv}] command computes the
20290 inverse of a square matrix. If the matrix is singular, the inverse
20291 operation is left in symbolic form. Matrix inverses are recorded so
20292 that once an inverse (or determinant) of a particular matrix has been
20293 computed, the inverse and determinant of the matrix can be recomputed
20294 quickly in the future.
20296 If the argument to @kbd{&} is a plain number @expr{x}, this
20297 command simply computes @expr{1/x}. This is okay, because the
20298 @samp{/} operator also does a matrix inversion when dividing one
20305 The @kbd{V D} (@code{calc-mdet}) [@code{det}] command computes the
20306 determinant of a square matrix.
20312 The @kbd{V L} (@code{calc-mlud}) [@code{lud}] command computes the
20313 LU decomposition of a matrix. The result is a list of three matrices
20314 which, when multiplied together left-to-right, form the original matrix.
20315 The first is a permutation matrix that arises from pivoting in the
20316 algorithm, the second is lower-triangular with ones on the diagonal,
20317 and the third is upper-triangular.
20321 @pindex calc-mtrace
20323 The @kbd{V T} (@code{calc-mtrace}) [@code{tr}] command computes the
20324 trace of a square matrix. This is defined as the sum of the diagonal
20325 elements of the matrix.
20331 The @kbd{V K} (@code{calc-kron}) [@code{kron}] command computes
20332 the Kronecker product of two matrices.
20334 @node Set Operations, Statistical Operations, Vector and Matrix Arithmetic, Matrix Functions
20335 @section Set Operations using Vectors
20338 @cindex Sets, as vectors
20339 Calc includes several commands which interpret vectors as @dfn{sets} of
20340 objects. A set is a collection of objects; any given object can appear
20341 only once in the set. Calc stores sets as vectors of objects in
20342 sorted order. Objects in a Calc set can be any of the usual things,
20343 such as numbers, variables, or formulas. Two set elements are considered
20344 equal if they are identical, except that numerically equal numbers like
20345 the integer 4 and the float 4.0 are considered equal even though they
20346 are not ``identical.'' Variables are treated like plain symbols without
20347 attached values by the set operations; subtracting the set @samp{[b]}
20348 from @samp{[a, b]} always yields the set @samp{[a]} even though if
20349 the variables @samp{a} and @samp{b} both equaled 17, you might
20350 expect the answer @samp{[]}.
20352 If a set contains interval forms, then it is assumed to be a set of
20353 real numbers. In this case, all set operations require the elements
20354 of the set to be only things that are allowed in intervals: Real
20355 numbers, plus and minus infinity, HMS forms, and date forms. If
20356 there are variables or other non-real objects present in a real set,
20357 all set operations on it will be left in unevaluated form.
20359 If the input to a set operation is a plain number or interval form
20360 @var{a}, it is treated like the one-element vector @samp{[@var{a}]}.
20361 The result is always a vector, except that if the set consists of a
20362 single interval, the interval itself is returned instead.
20364 @xref{Logical Operations}, for the @code{in} function which tests if
20365 a certain value is a member of a given set. To test if the set @expr{A}
20366 is a subset of the set @expr{B}, use @samp{vdiff(A, B) = []}.
20370 @pindex calc-remove-duplicates
20372 The @kbd{V +} (@code{calc-remove-duplicates}) [@code{rdup}] command
20373 converts an arbitrary vector into set notation. It works by sorting
20374 the vector as if by @kbd{V S}, then removing duplicates. (For example,
20375 @kbd{[a, 5, 4, a, 4.0]} is sorted to @samp{[4, 4.0, 5, a, a]} and then
20376 reduced to @samp{[4, 5, a]}). Overlapping intervals are merged as
20377 necessary. You rarely need to use @kbd{V +} explicitly, since all the
20378 other set-based commands apply @kbd{V +} to their inputs before using
20383 @pindex calc-set-union
20385 The @kbd{V V} (@code{calc-set-union}) [@code{vunion}] command computes
20386 the union of two sets. An object is in the union of two sets if and
20387 only if it is in either (or both) of the input sets. (You could
20388 accomplish the same thing by concatenating the sets with @kbd{|},
20389 then using @kbd{V +}.)
20393 @pindex calc-set-intersect
20395 The @kbd{V ^} (@code{calc-set-intersect}) [@code{vint}] command computes
20396 the intersection of two sets. An object is in the intersection if
20397 and only if it is in both of the input sets. Thus if the input
20398 sets are disjoint, i.e., if they share no common elements, the result
20399 will be the empty vector @samp{[]}. Note that the characters @kbd{V}
20400 and @kbd{^} were chosen to be close to the conventional mathematical
20402 @texline union@tie{}(@math{A \cup B})
20405 @texline intersection@tie{}(@math{A \cap B}).
20406 @infoline intersection.
20410 @pindex calc-set-difference
20412 The @kbd{V -} (@code{calc-set-difference}) [@code{vdiff}] command computes
20413 the difference between two sets. An object is in the difference
20414 @expr{A - B} if and only if it is in @expr{A} but not in @expr{B}.
20415 Thus subtracting @samp{[y,z]} from a set will remove the elements
20416 @samp{y} and @samp{z} if they are present. You can also think of this
20417 as a general @dfn{set complement} operator; if @expr{A} is the set of
20418 all possible values, then @expr{A - B} is the ``complement'' of @expr{B}.
20419 Obviously this is only practical if the set of all possible values in
20420 your problem is small enough to list in a Calc vector (or simple
20421 enough to express in a few intervals).
20425 @pindex calc-set-xor
20427 The @kbd{V X} (@code{calc-set-xor}) [@code{vxor}] command computes
20428 the ``exclusive-or,'' or ``symmetric difference'' of two sets.
20429 An object is in the symmetric difference of two sets if and only
20430 if it is in one, but @emph{not} both, of the sets. Objects that
20431 occur in both sets ``cancel out.''
20435 @pindex calc-set-complement
20437 The @kbd{V ~} (@code{calc-set-complement}) [@code{vcompl}] command
20438 computes the complement of a set with respect to the real numbers.
20439 Thus @samp{vcompl(x)} is equivalent to @samp{vdiff([-inf .. inf], x)}.
20440 For example, @samp{vcompl([2, (3 .. 4]])} evaluates to
20441 @samp{[[-inf .. 2), (2 .. 3], (4 .. inf]]}.
20445 @pindex calc-set-floor
20447 The @kbd{V F} (@code{calc-set-floor}) [@code{vfloor}] command
20448 reinterprets a set as a set of integers. Any non-integer values,
20449 and intervals that do not enclose any integers, are removed. Open
20450 intervals are converted to equivalent closed intervals. Successive
20451 integers are converted into intervals of integers. For example, the
20452 complement of the set @samp{[2, 6, 7, 8]} is messy, but if you wanted
20453 the complement with respect to the set of integers you could type
20454 @kbd{V ~ V F} to get @samp{[[-inf .. 1], [3 .. 5], [9 .. inf]]}.
20458 @pindex calc-set-enumerate
20460 The @kbd{V E} (@code{calc-set-enumerate}) [@code{venum}] command
20461 converts a set of integers into an explicit vector. Intervals in
20462 the set are expanded out to lists of all integers encompassed by
20463 the intervals. This only works for finite sets (i.e., sets which
20464 do not involve @samp{-inf} or @samp{inf}).
20468 @pindex calc-set-span
20470 The @kbd{V :} (@code{calc-set-span}) [@code{vspan}] command converts any
20471 set of reals into an interval form that encompasses all its elements.
20472 The lower limit will be the smallest element in the set; the upper
20473 limit will be the largest element. For an empty set, @samp{vspan([])}
20474 returns the empty interval @w{@samp{[0 .. 0)}}.
20478 @pindex calc-set-cardinality
20480 The @kbd{V #} (@code{calc-set-cardinality}) [@code{vcard}] command counts
20481 the number of integers in a set. The result is the length of the vector
20482 that would be produced by @kbd{V E}, although the computation is much
20483 more efficient than actually producing that vector.
20485 @cindex Sets, as binary numbers
20486 Another representation for sets that may be more appropriate in some
20487 cases is binary numbers. If you are dealing with sets of integers
20488 in the range 0 to 49, you can use a 50-bit binary number where a
20489 particular bit is 1 if the corresponding element is in the set.
20490 @xref{Binary Functions}, for a list of commands that operate on
20491 binary numbers. Note that many of the above set operations have
20492 direct equivalents in binary arithmetic: @kbd{b o} (@code{calc-or}),
20493 @kbd{b a} (@code{calc-and}), @kbd{b d} (@code{calc-diff}),
20494 @kbd{b x} (@code{calc-xor}), and @kbd{b n} (@code{calc-not}),
20495 respectively. You can use whatever representation for sets is most
20500 @pindex calc-pack-bits
20501 @pindex calc-unpack-bits
20504 The @kbd{b u} (@code{calc-unpack-bits}) [@code{vunpack}] command
20505 converts an integer that represents a set in binary into a set
20506 in vector/interval notation. For example, @samp{vunpack(67)}
20507 returns @samp{[[0 .. 1], 6]}. If the input is negative, the set
20508 it represents is semi-infinite: @samp{vunpack(-4) = [2 .. inf)}.
20509 Use @kbd{V E} afterwards to expand intervals to individual
20510 values if you wish. Note that this command uses the @kbd{b}
20511 (binary) prefix key.
20513 The @kbd{b p} (@code{calc-pack-bits}) [@code{vpack}] command
20514 converts the other way, from a vector or interval representing
20515 a set of nonnegative integers into a binary integer describing
20516 the same set. The set may include positive infinity, but must
20517 not include any negative numbers. The input is interpreted as a
20518 set of integers in the sense of @kbd{V F} (@code{vfloor}). Beware
20519 that a simple input like @samp{[100]} can result in a huge integer
20521 @texline (@math{2^{100}}, a 31-digit integer, in this case).
20522 @infoline (@expr{2^100}, a 31-digit integer, in this case).
20524 @node Statistical Operations, Reducing and Mapping, Set Operations, Matrix Functions
20525 @section Statistical Operations on Vectors
20528 @cindex Statistical functions
20529 The commands in this section take vectors as arguments and compute
20530 various statistical measures on the data stored in the vectors. The
20531 references used in the definitions of these functions are Bevington's
20532 @emph{Data Reduction and Error Analysis for the Physical Sciences},
20533 and @emph{Numerical Recipes} by Press, Flannery, Teukolsky and
20536 The statistical commands use the @kbd{u} prefix key followed by
20537 a shifted letter or other character.
20539 @xref{Manipulating Vectors}, for a description of @kbd{V H}
20540 (@code{calc-histogram}).
20542 @xref{Curve Fitting}, for the @kbd{a F} command for doing
20543 least-squares fits to statistical data.
20545 @xref{Probability Distribution Functions}, for several common
20546 probability distribution functions.
20549 * Single-Variable Statistics::
20550 * Paired-Sample Statistics::
20553 @node Single-Variable Statistics, Paired-Sample Statistics, Statistical Operations, Statistical Operations
20554 @subsection Single-Variable Statistics
20557 These functions do various statistical computations on single
20558 vectors. Given a numeric prefix argument, they actually pop
20559 @var{n} objects from the stack and combine them into a data
20560 vector. Each object may be either a number or a vector; if a
20561 vector, any sub-vectors inside it are ``flattened'' as if by
20562 @kbd{v a 0}; @pxref{Manipulating Vectors}. By default one object
20563 is popped, which (in order to be useful) is usually a vector.
20565 If an argument is a variable name, and the value stored in that
20566 variable is a vector, then the stored vector is used. This method
20567 has the advantage that if your data vector is large, you can avoid
20568 the slow process of manipulating it directly on the stack.
20570 These functions are left in symbolic form if any of their arguments
20571 are not numbers or vectors, e.g., if an argument is a formula, or
20572 a non-vector variable. However, formulas embedded within vector
20573 arguments are accepted; the result is a symbolic representation
20574 of the computation, based on the assumption that the formula does
20575 not itself represent a vector. All varieties of numbers such as
20576 error forms and interval forms are acceptable.
20578 Some of the functions in this section also accept a single error form
20579 or interval as an argument. They then describe a property of the
20580 normal or uniform (respectively) statistical distribution described
20581 by the argument. The arguments are interpreted in the same way as
20582 the @var{M} argument of the random number function @kbd{k r}. In
20583 particular, an interval with integer limits is considered an integer
20584 distribution, so that @samp{[2 .. 6)} is the same as @samp{[2 .. 5]}.
20585 An interval with at least one floating-point limit is a continuous
20586 distribution: @samp{[2.0 .. 6.0)} is @emph{not} the same as
20587 @samp{[2.0 .. 5.0]}!
20590 @pindex calc-vector-count
20592 The @kbd{u #} (@code{calc-vector-count}) [@code{vcount}] command
20593 computes the number of data values represented by the inputs.
20594 For example, @samp{vcount(1, [2, 3], [[4, 5], [], x, y])} returns 7.
20595 If the argument is a single vector with no sub-vectors, this
20596 simply computes the length of the vector.
20600 @pindex calc-vector-sum
20601 @pindex calc-vector-prod
20604 @cindex Summations (statistical)
20605 The @kbd{u +} (@code{calc-vector-sum}) [@code{vsum}] command
20606 computes the sum of the data values. The @kbd{u *}
20607 (@code{calc-vector-prod}) [@code{vprod}] command computes the
20608 product of the data values. If the input is a single flat vector,
20609 these are the same as @kbd{V R +} and @kbd{V R *}
20610 (@pxref{Reducing and Mapping}).
20614 @pindex calc-vector-max
20615 @pindex calc-vector-min
20618 The @kbd{u X} (@code{calc-vector-max}) [@code{vmax}] command
20619 computes the maximum of the data values, and the @kbd{u N}
20620 (@code{calc-vector-min}) [@code{vmin}] command computes the minimum.
20621 If the argument is an interval, this finds the minimum or maximum
20622 value in the interval. (Note that @samp{vmax([2..6)) = 5} as
20623 described above.) If the argument is an error form, this returns
20624 plus or minus infinity.
20627 @pindex calc-vector-mean
20629 @cindex Mean of data values
20630 The @kbd{u M} (@code{calc-vector-mean}) [@code{vmean}] command
20631 computes the average (arithmetic mean) of the data values.
20632 If the inputs are error forms
20633 @texline @math{x \pm \sigma},
20634 @infoline @samp{x +/- s},
20635 this is the weighted mean of the @expr{x} values with weights
20636 @texline @math{1 /\sigma^2}.
20637 @infoline @expr{1 / s^2}.
20639 $$ \mu = { \displaystyle \sum { x_i \over \sigma_i^2 } \over
20640 \displaystyle \sum { 1 \over \sigma_i^2 } } $$
20642 If the inputs are not error forms, this is simply the sum of the
20643 values divided by the count of the values.
20645 Note that a plain number can be considered an error form with
20647 @texline @math{\sigma = 0}.
20648 @infoline @expr{s = 0}.
20649 If the input to @kbd{u M} is a mixture of
20650 plain numbers and error forms, the result is the mean of the
20651 plain numbers, ignoring all values with non-zero errors. (By the
20652 above definitions it's clear that a plain number effectively
20653 has an infinite weight, next to which an error form with a finite
20654 weight is completely negligible.)
20656 This function also works for distributions (error forms or
20657 intervals). The mean of an error form `@var{a} @tfn{+/-} @var{b}' is simply
20658 @expr{a}. The mean of an interval is the mean of the minimum
20659 and maximum values of the interval.
20662 @pindex calc-vector-mean-error
20664 The @kbd{I u M} (@code{calc-vector-mean-error}) [@code{vmeane}]
20665 command computes the mean of the data points expressed as an
20666 error form. This includes the estimated error associated with
20667 the mean. If the inputs are error forms, the error is the square
20668 root of the reciprocal of the sum of the reciprocals of the squares
20669 of the input errors. (I.e., the variance is the reciprocal of the
20670 sum of the reciprocals of the variances.)
20672 $$ \sigma_\mu^2 = {1 \over \displaystyle \sum {1 \over \sigma_i^2}} $$
20674 If the inputs are plain
20675 numbers, the error is equal to the standard deviation of the values
20676 divided by the square root of the number of values. (This works
20677 out to be equivalent to calculating the standard deviation and
20678 then assuming each value's error is equal to this standard
20681 $$ \sigma_\mu^2 = {\sigma^2 \over N} $$
20685 @pindex calc-vector-median
20687 @cindex Median of data values
20688 The @kbd{H u M} (@code{calc-vector-median}) [@code{vmedian}]
20689 command computes the median of the data values. The values are
20690 first sorted into numerical order; the median is the middle
20691 value after sorting. (If the number of data values is even,
20692 the median is taken to be the average of the two middle values.)
20693 The median function is different from the other functions in
20694 this section in that the arguments must all be real numbers;
20695 variables are not accepted even when nested inside vectors.
20696 (Otherwise it is not possible to sort the data values.) If
20697 any of the input values are error forms, their error parts are
20700 The median function also accepts distributions. For both normal
20701 (error form) and uniform (interval) distributions, the median is
20702 the same as the mean.
20705 @pindex calc-vector-harmonic-mean
20707 @cindex Harmonic mean
20708 The @kbd{H I u M} (@code{calc-vector-harmonic-mean}) [@code{vhmean}]
20709 command computes the harmonic mean of the data values. This is
20710 defined as the reciprocal of the arithmetic mean of the reciprocals
20713 $$ { N \over \displaystyle \sum {1 \over x_i} } $$
20717 @pindex calc-vector-geometric-mean
20719 @cindex Geometric mean
20720 The @kbd{u G} (@code{calc-vector-geometric-mean}) [@code{vgmean}]
20721 command computes the geometric mean of the data values. This
20722 is the @var{n}th root of the product of the values. This is also
20723 equal to the @code{exp} of the arithmetic mean of the logarithms
20724 of the data values.
20726 $$ \exp \left ( \sum { \ln x_i } \right ) =
20727 \left ( \prod { x_i } \right)^{1 / N} $$
20732 The @kbd{H u G} [@code{agmean}] command computes the ``arithmetic-geometric
20733 mean'' of two numbers taken from the stack. This is computed by
20734 replacing the two numbers with their arithmetic mean and geometric
20735 mean, then repeating until the two values converge.
20737 $$ a_{i+1} = { a_i + b_i \over 2 } , \qquad b_{i+1} = \sqrt{a_i b_i} $$
20740 @cindex Root-mean-square
20741 Another commonly used mean, the RMS (root-mean-square), can be computed
20742 for a vector of numbers simply by using the @kbd{A} command.
20745 @pindex calc-vector-sdev
20747 @cindex Standard deviation
20748 @cindex Sample statistics
20749 The @kbd{u S} (@code{calc-vector-sdev}) [@code{vsdev}] command
20750 computes the standard
20751 @texline deviation@tie{}@math{\sigma}
20752 @infoline deviation
20753 of the data values. If the values are error forms, the errors are used
20754 as weights just as for @kbd{u M}. This is the @emph{sample} standard
20755 deviation, whose value is the square root of the sum of the squares of
20756 the differences between the values and the mean of the @expr{N} values,
20757 divided by @expr{N-1}.
20759 $$ \sigma^2 = {1 \over N - 1} \sum (x_i - \mu)^2 $$
20762 This function also applies to distributions. The standard deviation
20763 of a single error form is simply the error part. The standard deviation
20764 of a continuous interval happens to equal the difference between the
20766 @texline @math{\sqrt{12}}.
20767 @infoline @expr{sqrt(12)}.
20768 The standard deviation of an integer interval is the same as the
20769 standard deviation of a vector of those integers.
20772 @pindex calc-vector-pop-sdev
20774 @cindex Population statistics
20775 The @kbd{I u S} (@code{calc-vector-pop-sdev}) [@code{vpsdev}]
20776 command computes the @emph{population} standard deviation.
20777 It is defined by the same formula as above but dividing
20778 by @expr{N} instead of by @expr{N-1}. The population standard
20779 deviation is used when the input represents the entire set of
20780 data values in the distribution; the sample standard deviation
20781 is used when the input represents a sample of the set of all
20782 data values, so that the mean computed from the input is itself
20783 only an estimate of the true mean.
20785 $$ \sigma^2 = {1 \over N} \sum (x_i - \mu)^2 $$
20788 For error forms and continuous intervals, @code{vpsdev} works
20789 exactly like @code{vsdev}. For integer intervals, it computes the
20790 population standard deviation of the equivalent vector of integers.
20794 @pindex calc-vector-variance
20795 @pindex calc-vector-pop-variance
20798 @cindex Variance of data values
20799 The @kbd{H u S} (@code{calc-vector-variance}) [@code{vvar}] and
20800 @kbd{H I u S} (@code{calc-vector-pop-variance}) [@code{vpvar}]
20801 commands compute the variance of the data values. The variance
20803 @texline square@tie{}@math{\sigma^2}
20805 of the standard deviation, i.e., the sum of the
20806 squares of the deviations of the data values from the mean.
20807 (This definition also applies when the argument is a distribution.)
20813 The @code{vflat} algebraic function returns a vector of its
20814 arguments, interpreted in the same way as the other functions
20815 in this section. For example, @samp{vflat(1, [2, [3, 4]], 5)}
20816 returns @samp{[1, 2, 3, 4, 5]}.
20818 @node Paired-Sample Statistics, , Single-Variable Statistics, Statistical Operations
20819 @subsection Paired-Sample Statistics
20822 The functions in this section take two arguments, which must be
20823 vectors of equal size. The vectors are each flattened in the same
20824 way as by the single-variable statistical functions. Given a numeric
20825 prefix argument of 1, these functions instead take one object from
20826 the stack, which must be an
20827 @texline @math{N\times2}
20829 matrix of data values. Once again, variable names can be used in place
20830 of actual vectors and matrices.
20833 @pindex calc-vector-covariance
20836 The @kbd{u C} (@code{calc-vector-covariance}) [@code{vcov}] command
20837 computes the sample covariance of two vectors. The covariance
20838 of vectors @var{x} and @var{y} is the sum of the products of the
20839 differences between the elements of @var{x} and the mean of @var{x}
20840 times the differences between the corresponding elements of @var{y}
20841 and the mean of @var{y}, all divided by @expr{N-1}. Note that
20842 the variance of a vector is just the covariance of the vector
20843 with itself. Once again, if the inputs are error forms the
20844 errors are used as weight factors. If both @var{x} and @var{y}
20845 are composed of error forms, the error for a given data point
20846 is taken as the square root of the sum of the squares of the two
20849 $$ \sigma_{x\!y}^2 = {1 \over N-1} \sum (x_i - \mu_x) (y_i - \mu_y) $$
20850 $$ \sigma_{x\!y}^2 =
20851 {\displaystyle {1 \over N-1}
20852 \sum {(x_i - \mu_x) (y_i - \mu_y) \over \sigma_i^2}
20853 \over \displaystyle {1 \over N} \sum {1 \over \sigma_i^2}}
20858 @pindex calc-vector-pop-covariance
20860 The @kbd{I u C} (@code{calc-vector-pop-covariance}) [@code{vpcov}]
20861 command computes the population covariance, which is the same as the
20862 sample covariance computed by @kbd{u C} except dividing by @expr{N}
20863 instead of @expr{N-1}.
20866 @pindex calc-vector-correlation
20868 @cindex Correlation coefficient
20869 @cindex Linear correlation
20870 The @kbd{H u C} (@code{calc-vector-correlation}) [@code{vcorr}]
20871 command computes the linear correlation coefficient of two vectors.
20872 This is defined by the covariance of the vectors divided by the
20873 product of their standard deviations. (There is no difference
20874 between sample or population statistics here.)
20876 $$ r_{x\!y} = { \sigma_{x\!y}^2 \over \sigma_x^2 \sigma_y^2 } $$
20879 @node Reducing and Mapping, Vector and Matrix Formats, Statistical Operations, Matrix Functions
20880 @section Reducing and Mapping Vectors
20883 The commands in this section allow for more general operations on the
20884 elements of vectors.
20890 The simplest of these operations is @kbd{V A} (@code{calc-apply})
20891 [@code{apply}], which applies a given operator to the elements of a vector.
20892 For example, applying the hypothetical function @code{f} to the vector
20893 @w{@samp{[1, 2, 3]}} would produce the function call @samp{f(1, 2, 3)}.
20894 Applying the @code{+} function to the vector @samp{[a, b]} gives
20895 @samp{a + b}. Applying @code{+} to the vector @samp{[a, b, c]} is an
20896 error, since the @code{+} function expects exactly two arguments.
20898 While @kbd{V A} is useful in some cases, you will usually find that either
20899 @kbd{V R} or @kbd{V M}, described below, is closer to what you want.
20902 * Specifying Operators::
20905 * Nesting and Fixed Points::
20906 * Generalized Products::
20909 @node Specifying Operators, Mapping, Reducing and Mapping, Reducing and Mapping
20910 @subsection Specifying Operators
20913 Commands in this section (like @kbd{V A}) prompt you to press the key
20914 corresponding to the desired operator. Press @kbd{?} for a partial
20915 list of the available operators. Generally, an operator is any key or
20916 sequence of keys that would normally take one or more arguments from
20917 the stack and replace them with a result. For example, @kbd{V A H C}
20918 uses the hyperbolic cosine operator, @code{cosh}. (Since @code{cosh}
20919 expects one argument, @kbd{V A H C} requires a vector with a single
20920 element as its argument.)
20922 You can press @kbd{x} at the operator prompt to select any algebraic
20923 function by name to use as the operator. This includes functions you
20924 have defined yourself using the @kbd{Z F} command. (@xref{Algebraic
20925 Definitions}.) If you give a name for which no function has been
20926 defined, the result is left in symbolic form, as in @samp{f(1, 2, 3)}.
20927 Calc will prompt for the number of arguments the function takes if it
20928 can't figure it out on its own (say, because you named a function that
20929 is currently undefined). It is also possible to type a digit key before
20930 the function name to specify the number of arguments, e.g.,
20931 @kbd{V M 3 x f @key{RET}} calls @code{f} with three arguments even if it
20932 looks like it ought to have only two. This technique may be necessary
20933 if the function allows a variable number of arguments. For example,
20934 the @kbd{v e} [@code{vexp}] function accepts two or three arguments;
20935 if you want to map with the three-argument version, you will have to
20936 type @kbd{V M 3 v e}.
20938 It is also possible to apply any formula to a vector by treating that
20939 formula as a function. When prompted for the operator to use, press
20940 @kbd{'} (the apostrophe) and type your formula as an algebraic entry.
20941 You will then be prompted for the argument list, which defaults to a
20942 list of all variables that appear in the formula, sorted into alphabetic
20943 order. For example, suppose you enter the formula @w{@samp{x + 2y^x}}.
20944 The default argument list would be @samp{(x y)}, which means that if
20945 this function is applied to the arguments @samp{[3, 10]} the result will
20946 be @samp{3 + 2*10^3}. (If you plan to use a certain formula in this
20947 way often, you might consider defining it as a function with @kbd{Z F}.)
20949 Another way to specify the arguments to the formula you enter is with
20950 @kbd{$}, @kbd{$$}, and so on. For example, @kbd{V A ' $$ + 2$^$$}
20951 has the same effect as the previous example. The argument list is
20952 automatically taken to be @samp{($$ $)}. (The order of the arguments
20953 may seem backwards, but it is analogous to the way normal algebraic
20954 entry interacts with the stack.)
20956 If you press @kbd{$} at the operator prompt, the effect is similar to
20957 the apostrophe except that the relevant formula is taken from top-of-stack
20958 instead. The actual vector arguments of the @kbd{V A $} or related command
20959 then start at the second-to-top stack position. You will still be
20960 prompted for an argument list.
20962 @cindex Nameless functions
20963 @cindex Generic functions
20964 A function can be written without a name using the notation @samp{<#1 - #2>},
20965 which means ``a function of two arguments that computes the first
20966 argument minus the second argument.'' The symbols @samp{#1} and @samp{#2}
20967 are placeholders for the arguments. You can use any names for these
20968 placeholders if you wish, by including an argument list followed by a
20969 colon: @samp{<x, y : x - y>}. When you type @kbd{V A ' $$ + 2$^$$ @key{RET}},
20970 Calc builds the nameless function @samp{<#1 + 2 #2^#1>} as the function
20971 to map across the vectors. When you type @kbd{V A ' x + 2y^x @key{RET} @key{RET}},
20972 Calc builds the nameless function @w{@samp{<x, y : x + 2 y^x>}}. In both
20973 cases, Calc also writes the nameless function to the Trail so that you
20974 can get it back later if you wish.
20976 If there is only one argument, you can write @samp{#} in place of @samp{#1}.
20977 (Note that @samp{< >} notation is also used for date forms. Calc tells
20978 that @samp{<@var{stuff}>} is a nameless function by the presence of
20979 @samp{#} signs inside @var{stuff}, or by the fact that @var{stuff}
20980 begins with a list of variables followed by a colon.)
20982 You can type a nameless function directly to @kbd{V A '}, or put one on
20983 the stack and use it with @w{@kbd{V A $}}. Calc will not prompt for an
20984 argument list in this case, since the nameless function specifies the
20985 argument list as well as the function itself. In @kbd{V A '}, you can
20986 omit the @samp{< >} marks if you use @samp{#} notation for the arguments,
20987 so that @kbd{V A ' #1+#2 @key{RET}} is the same as @kbd{V A ' <#1+#2> @key{RET}},
20988 which in turn is the same as @kbd{V A ' $$+$ @key{RET}}.
20990 @cindex Lambda expressions
20995 The internal format for @samp{<x, y : x + y>} is @samp{lambda(x, y, x + y)}.
20996 (The word @code{lambda} derives from Lisp notation and the theory of
20997 functions.) The internal format for @samp{<#1 + #2>} is @samp{lambda(ArgA,
20998 ArgB, ArgA + ArgB)}. Note that there is no actual Calc function called
20999 @code{lambda}; the whole point is that the @code{lambda} expression is
21000 used in its symbolic form, not evaluated for an answer until it is applied
21001 to specific arguments by a command like @kbd{V A} or @kbd{V M}.
21003 (Actually, @code{lambda} does have one special property: Its arguments
21004 are never evaluated; for example, putting @samp{<(2/3) #>} on the stack
21005 will not simplify the @samp{2/3} until the nameless function is actually
21034 As usual, commands like @kbd{V A} have algebraic function name equivalents.
21035 For example, @kbd{V A k g} with an argument of @samp{v} is equivalent to
21036 @samp{apply(gcd, v)}. The first argument specifies the operator name,
21037 and is either a variable whose name is the same as the function name,
21038 or a nameless function like @samp{<#^3+1>}. Operators that are normally
21039 written as algebraic symbols have the names @code{add}, @code{sub},
21040 @code{mul}, @code{div}, @code{pow}, @code{neg}, @code{mod}, and
21047 The @code{call} function builds a function call out of several arguments:
21048 @samp{call(gcd, x, y)} is the same as @samp{apply(gcd, [x, y])}, which
21049 in turn is the same as @samp{gcd(x, y)}. The first argument of @code{call},
21050 like the other functions described here, may be either a variable naming a
21051 function, or a nameless function (@samp{call(<#1+2#2>, x, y)} is the same
21054 (Experts will notice that it's not quite proper to use a variable to name
21055 a function, since the name @code{gcd} corresponds to the Lisp variable
21056 @code{var-gcd} but to the Lisp function @code{calcFunc-gcd}. Calc
21057 automatically makes this translation, so you don't have to worry
21060 @node Mapping, Reducing, Specifying Operators, Reducing and Mapping
21061 @subsection Mapping
21068 The @kbd{V M} (@code{calc-map}) [@code{map}] command applies a given
21069 operator elementwise to one or more vectors. For example, mapping
21070 @code{A} [@code{abs}] produces a vector of the absolute values of the
21071 elements in the input vector. Mapping @code{+} pops two vectors from
21072 the stack, which must be of equal length, and produces a vector of the
21073 pairwise sums of the elements. If either argument is a non-vector, it
21074 is duplicated for each element of the other vector. For example,
21075 @kbd{[1,2,3] 2 V M ^} squares the elements of the specified vector.
21076 With the 2 listed first, it would have computed a vector of powers of
21077 two. Mapping a user-defined function pops as many arguments from the
21078 stack as the function requires. If you give an undefined name, you will
21079 be prompted for the number of arguments to use.
21081 If any argument to @kbd{V M} is a matrix, the operator is normally mapped
21082 across all elements of the matrix. For example, given the matrix
21083 @expr{[[1, -2, 3], [-4, 5, -6]]}, @kbd{V M A} takes six absolute values to
21085 @texline @math{3\times2}
21087 matrix, @expr{[[1, 2, 3], [4, 5, 6]]}.
21090 The command @kbd{V M _} [@code{mapr}] (i.e., type an underscore at the
21091 operator prompt) maps by rows instead. For example, @kbd{V M _ A} views
21092 the above matrix as a vector of two 3-element row vectors. It produces
21093 a new vector which contains the absolute values of those row vectors,
21094 namely @expr{[3.74, 8.77]}. (Recall, the absolute value of a vector is
21095 defined as the square root of the sum of the squares of the elements.)
21096 Some operators accept vectors and return new vectors; for example,
21097 @kbd{v v} reverses a vector, so @kbd{V M _ v v} would reverse each row
21098 of the matrix to get a new matrix, @expr{[[3, -2, 1], [-6, 5, -4]]}.
21100 Sometimes a vector of vectors (representing, say, strings, sets, or lists)
21101 happens to look like a matrix. If so, remember to use @kbd{V M _} if you
21102 want to map a function across the whole strings or sets rather than across
21103 their individual elements.
21106 The command @kbd{V M :} [@code{mapc}] maps by columns. Basically, it
21107 transposes the input matrix, maps by rows, and then, if the result is a
21108 matrix, transposes again. For example, @kbd{V M : A} takes the absolute
21109 values of the three columns of the matrix, treating each as a 2-vector,
21110 and @kbd{V M : v v} reverses the columns to get the matrix
21111 @expr{[[-4, 5, -6], [1, -2, 3]]}.
21113 (The symbols @kbd{_} and @kbd{:} were chosen because they had row-like
21114 and column-like appearances, and were not already taken by useful
21115 operators. Also, they appear shifted on most keyboards so they are easy
21116 to type after @kbd{V M}.)
21118 The @kbd{_} and @kbd{:} modifiers have no effect on arguments that are
21119 not matrices (so if none of the arguments are matrices, they have no
21120 effect at all). If some of the arguments are matrices and others are
21121 plain numbers, the plain numbers are held constant for all rows of the
21122 matrix (so that @kbd{2 V M _ ^} squares every row of a matrix; squaring
21123 a vector takes a dot product of the vector with itself).
21125 If some of the arguments are vectors with the same lengths as the
21126 rows (for @kbd{V M _}) or columns (for @kbd{V M :}) of the matrix
21127 arguments, those vectors are also held constant for every row or
21130 Sometimes it is useful to specify another mapping command as the operator
21131 to use with @kbd{V M}. For example, @kbd{V M _ V A +} applies @kbd{V A +}
21132 to each row of the input matrix, which in turn adds the two values on that
21133 row. If you give another vector-operator command as the operator for
21134 @kbd{V M}, it automatically uses map-by-rows mode if you don't specify
21135 otherwise; thus @kbd{V M V A +} is equivalent to @kbd{V M _ V A +}. (If
21136 you really want to map-by-elements another mapping command, you can use
21137 a triple-nested mapping command: @kbd{V M V M V A +} means to map
21138 @kbd{V M V A +} over the rows of the matrix; in turn, @kbd{V A +} is
21139 mapped over the elements of each row.)
21143 Previous versions of Calc had ``map across'' and ``map down'' modes
21144 that are now considered obsolete; the old ``map across'' is now simply
21145 @kbd{V M V A}, and ``map down'' is now @kbd{V M : V A}. The algebraic
21146 functions @code{mapa} and @code{mapd} are still supported, though.
21147 Note also that, while the old mapping modes were persistent (once you
21148 set the mode, it would apply to later mapping commands until you reset
21149 it), the new @kbd{:} and @kbd{_} modifiers apply only to the current
21150 mapping command. The default @kbd{V M} always means map-by-elements.
21152 @xref{Algebraic Manipulation}, for the @kbd{a M} command, which is like
21153 @kbd{V M} but for equations and inequalities instead of vectors.
21154 @xref{Storing Variables}, for the @kbd{s m} command which modifies a
21155 variable's stored value using a @kbd{V M}-like operator.
21157 @node Reducing, Nesting and Fixed Points, Mapping, Reducing and Mapping
21158 @subsection Reducing
21163 @pindex calc-reduce
21165 The @kbd{V R} (@code{calc-reduce}) [@code{reduce}] command applies a given
21166 binary operator across all the elements of a vector. A binary operator is
21167 a function such as @code{+} or @code{max} which takes two arguments. For
21168 example, reducing @code{+} over a vector computes the sum of the elements
21169 of the vector. Reducing @code{-} computes the first element minus each of
21170 the remaining elements. Reducing @code{max} computes the maximum element
21171 and so on. In general, reducing @code{f} over the vector @samp{[a, b, c, d]}
21172 produces @samp{f(f(f(a, b), c), d)}.
21177 The @kbd{I V R} [@code{rreduce}] command is similar to @kbd{V R} except
21178 that works from right to left through the vector. For example, plain
21179 @kbd{V R -} on the vector @samp{[a, b, c, d]} produces @samp{a - b - c - d}
21180 but @kbd{I V R -} on the same vector produces @samp{a - (b - (c - d))},
21181 or @samp{a - b + c - d}. This ``alternating sum'' occurs frequently
21182 in power series expansions.
21187 The @kbd{V U} (@code{calc-accumulate}) [@code{accum}] command does an
21188 accumulation operation. Here Calc does the corresponding reduction
21189 operation, but instead of producing only the final result, it produces
21190 a vector of all the intermediate results. Accumulating @code{+} over
21191 the vector @samp{[a, b, c, d]} produces the vector
21192 @samp{[a, a + b, a + b + c, a + b + c + d]}.
21197 The @kbd{I V U} [@code{raccum}] command does a right-to-left accumulation.
21198 For example, @kbd{I V U -} on the vector @samp{[a, b, c, d]} produces the
21199 vector @samp{[a - b + c - d, b - c + d, c - d, d]}.
21205 As for @kbd{V M}, @kbd{V R} normally reduces a matrix elementwise. For
21206 example, given the matrix @expr{[[a, b, c], [d, e, f]]}, @kbd{V R +} will
21207 compute @expr{a + b + c + d + e + f}. You can type @kbd{V R _} or
21208 @kbd{V R :} to modify this behavior. The @kbd{V R _} [@code{reducea}]
21209 command reduces ``across'' the matrix; it reduces each row of the matrix
21210 as a vector, then collects the results. Thus @kbd{V R _ +} of this
21211 matrix would produce @expr{[a + b + c, d + e + f]}. Similarly, @kbd{V R :}
21212 [@code{reduced}] reduces down; @kbd{V R : +} would produce @expr{[a + d,
21217 There is a third ``by rows'' mode for reduction that is occasionally
21218 useful; @kbd{V R =} [@code{reducer}] simply reduces the operator over
21219 the rows of the matrix themselves. Thus @kbd{V R = +} on the above
21220 matrix would get the same result as @kbd{V R : +}, since adding two
21221 row vectors is equivalent to adding their elements. But @kbd{V R = *}
21222 would multiply the two rows (to get a single number, their dot product),
21223 while @kbd{V R : *} would produce a vector of the products of the columns.
21225 These three matrix reduction modes work with @kbd{V R} and @kbd{I V R},
21226 but they are not currently supported with @kbd{V U} or @kbd{I V U}.
21230 The obsolete reduce-by-columns function, @code{reducec}, is still
21231 supported but there is no way to get it through the @kbd{V R} command.
21233 The commands @kbd{C-x * :} and @kbd{C-x * _} are equivalent to typing
21234 @kbd{C-x * r} to grab a rectangle of data into Calc, and then typing
21235 @kbd{V R : +} or @kbd{V R _ +}, respectively, to sum the columns or
21236 rows of the matrix. @xref{Grabbing From Buffers}.
21238 @node Nesting and Fixed Points, Generalized Products, Reducing, Reducing and Mapping
21239 @subsection Nesting and Fixed Points
21245 The @kbd{H V R} [@code{nest}] command applies a function to a given
21246 argument repeatedly. It takes two values, @samp{a} and @samp{n}, from
21247 the stack, where @samp{n} must be an integer. It then applies the
21248 function nested @samp{n} times; if the function is @samp{f} and @samp{n}
21249 is 3, the result is @samp{f(f(f(a)))}. The number @samp{n} may be
21250 negative if Calc knows an inverse for the function @samp{f}; for
21251 example, @samp{nest(sin, a, -2)} returns @samp{arcsin(arcsin(a))}.
21256 The @kbd{H V U} [@code{anest}] command is an accumulating version of
21257 @code{nest}: It returns a vector of @samp{n+1} values, e.g.,
21258 @samp{[a, f(a), f(f(a)), f(f(f(a)))]}. If @samp{n} is negative and
21259 @samp{F} is the inverse of @samp{f}, then the result is of the
21260 form @samp{[a, F(a), F(F(a)), F(F(F(a)))]}.
21265 @cindex Fixed points
21266 The @kbd{H I V R} [@code{fixp}] command is like @kbd{H V R}, except
21267 that it takes only an @samp{a} value from the stack; the function is
21268 applied until it reaches a ``fixed point,'' i.e., until the result
21274 The @kbd{H I V U} [@code{afixp}] command is an accumulating @code{fixp}.
21275 The first element of the return vector will be the initial value @samp{a};
21276 the last element will be the final result that would have been returned
21279 For example, 0.739085 is a fixed point of the cosine function (in radians):
21280 @samp{cos(0.739085) = 0.739085}. You can find this value by putting, say,
21281 1.0 on the stack and typing @kbd{H I V U C}. (We use the accumulating
21282 version so we can see the intermediate results: @samp{[1, 0.540302, 0.857553,
21283 0.65329, ...]}. With a precision of six, this command will take 36 steps
21284 to converge to 0.739085.)
21286 Newton's method for finding roots is a classic example of iteration
21287 to a fixed point. To find the square root of five starting with an
21288 initial guess, Newton's method would look for a fixed point of the
21289 function @samp{(x + 5/x) / 2}. Putting a guess of 1 on the stack
21290 and typing @kbd{H I V R ' ($ + 5/$)/2 @key{RET}} quickly yields the result
21291 2.23607. This is equivalent to using the @kbd{a R} (@code{calc-find-root})
21292 command to find a root of the equation @samp{x^2 = 5}.
21294 These examples used numbers for @samp{a} values. Calc keeps applying
21295 the function until two successive results are equal to within the
21296 current precision. For complex numbers, both the real parts and the
21297 imaginary parts must be equal to within the current precision. If
21298 @samp{a} is a formula (say, a variable name), then the function is
21299 applied until two successive results are exactly the same formula.
21300 It is up to you to ensure that the function will eventually converge;
21301 if it doesn't, you may have to press @kbd{C-g} to stop the Calculator.
21303 The algebraic @code{fixp} function takes two optional arguments, @samp{n}
21304 and @samp{tol}. The first is the maximum number of steps to be allowed,
21305 and must be either an integer or the symbol @samp{inf} (infinity, the
21306 default). The second is a convergence tolerance. If a tolerance is
21307 specified, all results during the calculation must be numbers, not
21308 formulas, and the iteration stops when the magnitude of the difference
21309 between two successive results is less than or equal to the tolerance.
21310 (This implies that a tolerance of zero iterates until the results are
21313 Putting it all together, @samp{fixp(<(# + A/#)/2>, B, 20, 1e-10)}
21314 computes the square root of @samp{A} given the initial guess @samp{B},
21315 stopping when the result is correct within the specified tolerance, or
21316 when 20 steps have been taken, whichever is sooner.
21318 @node Generalized Products, , Nesting and Fixed Points, Reducing and Mapping
21319 @subsection Generalized Products
21323 @pindex calc-outer-product
21325 The @kbd{V O} (@code{calc-outer-product}) [@code{outer}] command applies
21326 a given binary operator to all possible pairs of elements from two
21327 vectors, to produce a matrix. For example, @kbd{V O *} with @samp{[a, b]}
21328 and @samp{[x, y, z]} on the stack produces a multiplication table:
21329 @samp{[[a x, a y, a z], [b x, b y, b z]]}. Element @var{r},@var{c} of
21330 the result matrix is obtained by applying the operator to element @var{r}
21331 of the lefthand vector and element @var{c} of the righthand vector.
21335 @pindex calc-inner-product
21337 The @kbd{V I} (@code{calc-inner-product}) [@code{inner}] command computes
21338 the generalized inner product of two vectors or matrices, given a
21339 ``multiplicative'' operator and an ``additive'' operator. These can each
21340 actually be any binary operators; if they are @samp{*} and @samp{+},
21341 respectively, the result is a standard matrix multiplication. Element
21342 @var{r},@var{c} of the result matrix is obtained by mapping the
21343 multiplicative operator across row @var{r} of the lefthand matrix and
21344 column @var{c} of the righthand matrix, and then reducing with the additive
21345 operator. Just as for the standard @kbd{*} command, this can also do a
21346 vector-matrix or matrix-vector inner product, or a vector-vector
21347 generalized dot product.
21349 Since @kbd{V I} requires two operators, it prompts twice. In each case,
21350 you can use any of the usual methods for entering the operator. If you
21351 use @kbd{$} twice to take both operator formulas from the stack, the
21352 first (multiplicative) operator is taken from the top of the stack
21353 and the second (additive) operator is taken from second-to-top.
21355 @node Vector and Matrix Formats, , Reducing and Mapping, Matrix Functions
21356 @section Vector and Matrix Display Formats
21359 Commands for controlling vector and matrix display use the @kbd{v} prefix
21360 instead of the usual @kbd{d} prefix. But they are display modes; in
21361 particular, they are influenced by the @kbd{I} and @kbd{H} prefix keys
21362 in the same way (@pxref{Display Modes}). Matrix display is also
21363 influenced by the @kbd{d O} (@code{calc-flat-language}) mode;
21364 @pxref{Normal Language Modes}.
21368 @pindex calc-matrix-left-justify
21371 @pindex calc-matrix-center-justify
21374 @pindex calc-matrix-right-justify
21375 The commands @kbd{v <} (@code{calc-matrix-left-justify}), @kbd{v >}
21376 (@code{calc-matrix-right-justify}), and @w{@kbd{v =}}
21377 (@code{calc-matrix-center-justify}) control whether matrix elements
21378 are justified to the left, right, or center of their columns.
21382 @pindex calc-vector-brackets
21385 @pindex calc-vector-braces
21388 @pindex calc-vector-parens
21389 The @kbd{v [} (@code{calc-vector-brackets}) command turns the square
21390 brackets that surround vectors and matrices displayed in the stack on
21391 and off. The @kbd{v @{} (@code{calc-vector-braces}) and @kbd{v (}
21392 (@code{calc-vector-parens}) commands use curly braces or parentheses,
21393 respectively, instead of square brackets. For example, @kbd{v @{} might
21394 be used in preparation for yanking a matrix into a buffer running
21395 Mathematica. (In fact, the Mathematica language mode uses this mode;
21396 @pxref{Mathematica Language Mode}.) Note that, regardless of the
21397 display mode, either brackets or braces may be used to enter vectors,
21398 and parentheses may never be used for this purpose.
21406 @pindex calc-matrix-brackets
21407 The @kbd{v ]} (@code{calc-matrix-brackets}) command controls the
21408 ``big'' style display of matrices, for matrices which have more than
21409 one row. It prompts for a string of code letters; currently
21410 implemented letters are @code{R}, which enables brackets on each row
21411 of the matrix; @code{O}, which enables outer brackets in opposite
21412 corners of the matrix; and @code{C}, which enables commas or
21413 semicolons at the ends of all rows but the last. The default format
21414 is @samp{RO}. (Before Calc 2.00, the format was fixed at @samp{ROC}.)
21415 Here are some example matrices:
21419 [ [ 123, 0, 0 ] [ [ 123, 0, 0 ],
21420 [ 0, 123, 0 ] [ 0, 123, 0 ],
21421 [ 0, 0, 123 ] ] [ 0, 0, 123 ] ]
21430 [ 123, 0, 0 [ 123, 0, 0 ;
21431 0, 123, 0 0, 123, 0 ;
21432 0, 0, 123 ] 0, 0, 123 ]
21441 [ 123, 0, 0 ] 123, 0, 0
21442 [ 0, 123, 0 ] 0, 123, 0
21443 [ 0, 0, 123 ] 0, 0, 123
21450 Note that of the formats shown here, @samp{RO}, @samp{ROC}, and
21451 @samp{OC} are all recognized as matrices during reading, while
21452 the others are useful for display only.
21456 @pindex calc-vector-commas
21457 The @kbd{v ,} (@code{calc-vector-commas}) command turns commas on and
21458 off in vector and matrix display.
21460 In vectors of length one, and in all vectors when commas have been
21461 turned off, Calc adds extra parentheses around formulas that might
21462 otherwise be ambiguous. For example, @samp{[a b]} could be a vector
21463 of the one formula @samp{a b}, or it could be a vector of two
21464 variables with commas turned off. Calc will display the former
21465 case as @samp{[(a b)]}. You can disable these extra parentheses
21466 (to make the output less cluttered at the expense of allowing some
21467 ambiguity) by adding the letter @code{P} to the control string you
21468 give to @kbd{v ]} (as described above).
21472 @pindex calc-full-vectors
21473 The @kbd{v .} (@code{calc-full-vectors}) command turns abbreviated
21474 display of long vectors on and off. In this mode, vectors of six
21475 or more elements, or matrices of six or more rows or columns, will
21476 be displayed in an abbreviated form that displays only the first
21477 three elements and the last element: @samp{[a, b, c, ..., z]}.
21478 When very large vectors are involved this will substantially
21479 improve Calc's display speed.
21482 @pindex calc-full-trail-vectors
21483 The @kbd{t .} (@code{calc-full-trail-vectors}) command controls a
21484 similar mode for recording vectors in the Trail. If you turn on
21485 this mode, vectors of six or more elements and matrices of six or
21486 more rows or columns will be abbreviated when they are put in the
21487 Trail. The @kbd{t y} (@code{calc-trail-yank}) command will be
21488 unable to recover those vectors. If you are working with very
21489 large vectors, this mode will improve the speed of all operations
21490 that involve the trail.
21494 @pindex calc-break-vectors
21495 The @kbd{v /} (@code{calc-break-vectors}) command turns multi-line
21496 vector display on and off. Normally, matrices are displayed with one
21497 row per line but all other types of vectors are displayed in a single
21498 line. This mode causes all vectors, whether matrices or not, to be
21499 displayed with a single element per line. Sub-vectors within the
21500 vectors will still use the normal linear form.
21502 @node Algebra, Units, Matrix Functions, Top
21506 This section covers the Calc features that help you work with
21507 algebraic formulas. First, the general sub-formula selection
21508 mechanism is described; this works in conjunction with any Calc
21509 commands. Then, commands for specific algebraic operations are
21510 described. Finally, the flexible @dfn{rewrite rule} mechanism
21513 The algebraic commands use the @kbd{a} key prefix; selection
21514 commands use the @kbd{j} (for ``just a letter that wasn't used
21515 for anything else'') prefix.
21517 @xref{Editing Stack Entries}, to see how to manipulate formulas
21518 using regular Emacs editing commands.
21520 When doing algebraic work, you may find several of the Calculator's
21521 modes to be helpful, including Algebraic Simplification mode (@kbd{m A})
21522 or No-Simplification mode (@kbd{m O}),
21523 Algebraic entry mode (@kbd{m a}), Fraction mode (@kbd{m f}), and
21524 Symbolic mode (@kbd{m s}). @xref{Mode Settings}, for discussions
21525 of these modes. You may also wish to select Big display mode (@kbd{d B}).
21526 @xref{Normal Language Modes}.
21529 * Selecting Subformulas::
21530 * Algebraic Manipulation::
21531 * Simplifying Formulas::
21534 * Solving Equations::
21535 * Numerical Solutions::
21538 * Logical Operations::
21542 @node Selecting Subformulas, Algebraic Manipulation, Algebra, Algebra
21543 @section Selecting Sub-Formulas
21547 @cindex Sub-formulas
21548 @cindex Parts of formulas
21549 When working with an algebraic formula it is often necessary to
21550 manipulate a portion of the formula rather than the formula as a
21551 whole. Calc allows you to ``select'' a portion of any formula on
21552 the stack. Commands which would normally operate on that stack
21553 entry will now operate only on the sub-formula, leaving the
21554 surrounding part of the stack entry alone.
21556 One common non-algebraic use for selection involves vectors. To work
21557 on one element of a vector in-place, simply select that element as a
21558 ``sub-formula'' of the vector.
21561 * Making Selections::
21562 * Changing Selections::
21563 * Displaying Selections::
21564 * Operating on Selections::
21565 * Rearranging with Selections::
21568 @node Making Selections, Changing Selections, Selecting Subformulas, Selecting Subformulas
21569 @subsection Making Selections
21573 @pindex calc-select-here
21574 To select a sub-formula, move the Emacs cursor to any character in that
21575 sub-formula, and press @w{@kbd{j s}} (@code{calc-select-here}). Calc will
21576 highlight the smallest portion of the formula that contains that
21577 character. By default the sub-formula is highlighted by blanking out
21578 all of the rest of the formula with dots. Selection works in any
21579 display mode but is perhaps easiest in Big mode (@kbd{d B}).
21580 Suppose you enter the following formula:
21592 (by typing @kbd{' ((a+b)^3 + sqrt(c)) / (2x+1)}). If you move the
21593 cursor to the letter @samp{b} and press @w{@kbd{j s}}, the display changes
21606 Every character not part of the sub-formula @samp{b} has been changed
21607 to a dot. (If the customizable variable
21608 @code{calc-highlight-selections-with-faces} is non-nil, then the characters
21609 not part of the sub-formula are de-emphasized by using a less
21610 noticeable face instead of using dots. @pxref{Displaying Selections}.)
21611 The @samp{*} next to the line number is to remind you that
21612 the formula has a portion of it selected. (In this case, it's very
21613 obvious, but it might not always be. If Embedded mode is enabled,
21614 the word @samp{Sel} also appears in the mode line because the stack
21615 may not be visible. @pxref{Embedded Mode}.)
21617 If you had instead placed the cursor on the parenthesis immediately to
21618 the right of the @samp{b}, the selection would have been:
21630 The portion selected is always large enough to be considered a complete
21631 formula all by itself, so selecting the parenthesis selects the whole
21632 formula that it encloses. Putting the cursor on the @samp{+} sign
21633 would have had the same effect.
21635 (Strictly speaking, the Emacs cursor is really the manifestation of
21636 the Emacs ``point,'' which is a position @emph{between} two characters
21637 in the buffer. So purists would say that Calc selects the smallest
21638 sub-formula which contains the character to the right of ``point.'')
21640 If you supply a numeric prefix argument @var{n}, the selection is
21641 expanded to the @var{n}th enclosing sub-formula. Thus, positioning
21642 the cursor on the @samp{b} and typing @kbd{C-u 1 j s} will select
21643 @samp{a + b}; typing @kbd{C-u 2 j s} will select @samp{(a + b)^3},
21646 If the cursor is not on any part of the formula, or if you give a
21647 numeric prefix that is too large, the entire formula is selected.
21649 If the cursor is on the @samp{.} line that marks the top of the stack
21650 (i.e., its normal ``rest position''), this command selects the entire
21651 formula at stack level 1. Most selection commands similarly operate
21652 on the formula at the top of the stack if you haven't positioned the
21653 cursor on any stack entry.
21656 @pindex calc-select-additional
21657 The @kbd{j a} (@code{calc-select-additional}) command enlarges the
21658 current selection to encompass the cursor. To select the smallest
21659 sub-formula defined by two different points, move to the first and
21660 press @kbd{j s}, then move to the other and press @kbd{j a}. This
21661 is roughly analogous to using @kbd{C-@@} (@code{set-mark-command}) to
21662 select the two ends of a region of text during normal Emacs editing.
21665 @pindex calc-select-once
21666 The @kbd{j o} (@code{calc-select-once}) command selects a formula in
21667 exactly the same way as @kbd{j s}, except that the selection will
21668 last only as long as the next command that uses it. For example,
21669 @kbd{j o 1 +} is a handy way to add one to the sub-formula indicated
21672 (A somewhat more precise definition: The @kbd{j o} command sets a flag
21673 such that the next command involving selected stack entries will clear
21674 the selections on those stack entries afterwards. All other selection
21675 commands except @kbd{j a} and @kbd{j O} clear this flag.)
21679 @pindex calc-select-here-maybe
21680 @pindex calc-select-once-maybe
21681 The @kbd{j S} (@code{calc-select-here-maybe}) and @kbd{j O}
21682 (@code{calc-select-once-maybe}) commands are equivalent to @kbd{j s}
21683 and @kbd{j o}, respectively, except that if the formula already
21684 has a selection they have no effect. This is analogous to the
21685 behavior of some commands such as @kbd{j r} (@code{calc-rewrite-selection};
21686 @pxref{Selections with Rewrite Rules}) and is mainly intended to be
21687 used in keyboard macros that implement your own selection-oriented
21690 Selection of sub-formulas normally treats associative terms like
21691 @samp{a + b - c + d} and @samp{x * y * z} as single levels of the formula.
21692 If you place the cursor anywhere inside @samp{a + b - c + d} except
21693 on one of the variable names and use @kbd{j s}, you will select the
21694 entire four-term sum.
21697 @pindex calc-break-selections
21698 The @kbd{j b} (@code{calc-break-selections}) command controls a mode
21699 in which the ``deep structure'' of these associative formulas shows
21700 through. Calc actually stores the above formulas as
21701 @samp{((a + b) - c) + d} and @samp{x * (y * z)}. (Note that for certain
21702 obscure reasons, by default Calc treats multiplication as
21703 right-associative.) Once you have enabled @kbd{j b} mode, selecting
21704 with the cursor on the @samp{-} sign would only select the @samp{a + b -
21705 c} portion, which makes sense when the deep structure of the sum is
21706 considered. There is no way to select the @samp{b - c + d} portion;
21707 although this might initially look like just as legitimate a sub-formula
21708 as @samp{a + b - c}, the deep structure shows that it isn't. The @kbd{d
21709 U} command can be used to view the deep structure of any formula
21710 (@pxref{Normal Language Modes}).
21712 When @kbd{j b} mode has not been enabled, the deep structure is
21713 generally hidden by the selection commands---what you see is what
21717 @pindex calc-unselect
21718 The @kbd{j u} (@code{calc-unselect}) command unselects the formula
21719 that the cursor is on. If there was no selection in the formula,
21720 this command has no effect. With a numeric prefix argument, it
21721 unselects the @var{n}th stack element rather than using the cursor
21725 @pindex calc-clear-selections
21726 The @kbd{j c} (@code{calc-clear-selections}) command unselects all
21729 @node Changing Selections, Displaying Selections, Making Selections, Selecting Subformulas
21730 @subsection Changing Selections
21734 @pindex calc-select-more
21735 Once you have selected a sub-formula, you can expand it using the
21736 @w{@kbd{j m}} (@code{calc-select-more}) command. If @samp{a + b} is
21737 selected, pressing @w{@kbd{j m}} repeatedly works as follows:
21742 (a + b) . . . (a + b) + V c (a + b) + V c
21743 1* ............... 1* ............... 1* ---------------
21744 . . . . . . . . 2 x + 1
21749 In the last example, the entire formula is selected. This is roughly
21750 the same as having no selection at all, but because there are subtle
21751 differences the @samp{*} character is still there on the line number.
21753 With a numeric prefix argument @var{n}, @kbd{j m} expands @var{n}
21754 times (or until the entire formula is selected). Note that @kbd{j s}
21755 with argument @var{n} is equivalent to plain @kbd{j s} followed by
21756 @kbd{j m} with argument @var{n}. If @w{@kbd{j m}} is used when there
21757 is no current selection, it is equivalent to @w{@kbd{j s}}.
21759 Even though @kbd{j m} does not explicitly use the location of the
21760 cursor within the formula, it nevertheless uses the cursor to determine
21761 which stack element to operate on. As usual, @kbd{j m} when the cursor
21762 is not on any stack element operates on the top stack element.
21765 @pindex calc-select-less
21766 The @kbd{j l} (@code{calc-select-less}) command reduces the current
21767 selection around the cursor position. That is, it selects the
21768 immediate sub-formula of the current selection which contains the
21769 cursor, the opposite of @kbd{j m}. If the cursor is not inside the
21770 current selection, the command de-selects the formula.
21773 @pindex calc-select-part
21774 The @kbd{j 1} through @kbd{j 9} (@code{calc-select-part}) commands
21775 select the @var{n}th sub-formula of the current selection. They are
21776 like @kbd{j l} (@code{calc-select-less}) except they use counting
21777 rather than the cursor position to decide which sub-formula to select.
21778 For example, if the current selection is @kbd{a + b + c} or
21779 @kbd{f(a, b, c)} or @kbd{[a, b, c]}, then @kbd{j 1} selects @samp{a},
21780 @kbd{j 2} selects @samp{b}, and @kbd{j 3} selects @samp{c}; in each of
21781 these cases, @kbd{j 4} through @kbd{j 9} would be errors.
21783 If there is no current selection, @kbd{j 1} through @kbd{j 9} select
21784 the @var{n}th top-level sub-formula. (In other words, they act as if
21785 the entire stack entry were selected first.) To select the @var{n}th
21786 sub-formula where @var{n} is greater than nine, you must instead invoke
21787 @w{@kbd{j 1}} with @var{n} as a numeric prefix argument.
21791 @pindex calc-select-next
21792 @pindex calc-select-previous
21793 The @kbd{j n} (@code{calc-select-next}) and @kbd{j p}
21794 (@code{calc-select-previous}) commands change the current selection
21795 to the next or previous sub-formula at the same level. For example,
21796 if @samp{b} is selected in @w{@samp{2 + a*b*c + x}}, then @kbd{j n}
21797 selects @samp{c}. Further @kbd{j n} commands would be in error because,
21798 even though there is something to the right of @samp{c} (namely, @samp{x}),
21799 it is not at the same level; in this case, it is not a term of the
21800 same product as @samp{b} and @samp{c}. However, @kbd{j m} (to select
21801 the whole product @samp{a*b*c} as a term of the sum) followed by
21802 @w{@kbd{j n}} would successfully select the @samp{x}.
21804 Similarly, @kbd{j p} moves the selection from the @samp{b} in this
21805 sample formula to the @samp{a}. Both commands accept numeric prefix
21806 arguments to move several steps at a time.
21808 It is interesting to compare Calc's selection commands with the
21809 Emacs Info system's commands for navigating through hierarchically
21810 organized documentation. Calc's @kbd{j n} command is completely
21811 analogous to Info's @kbd{n} command. Likewise, @kbd{j p} maps to
21812 @kbd{p}, @kbd{j 2} maps to @kbd{2}, and Info's @kbd{u} is like @kbd{j m}.
21813 (Note that @kbd{j u} stands for @code{calc-unselect}, not ``up''.)
21814 The Info @kbd{m} command is somewhat similar to Calc's @kbd{j s} and
21815 @kbd{j l}; in each case, you can jump directly to a sub-component
21816 of the hierarchy simply by pointing to it with the cursor.
21818 @node Displaying Selections, Operating on Selections, Changing Selections, Selecting Subformulas
21819 @subsection Displaying Selections
21823 @pindex calc-show-selections
21824 @vindex calc-highlight-selections-with-faces
21825 @vindex calc-selected-face
21826 @vindex calc-nonselected-face
21827 The @kbd{j d} (@code{calc-show-selections}) command controls how
21828 selected sub-formulas are displayed. One of the alternatives is
21829 illustrated in the above examples; if we press @kbd{j d} we switch
21830 to the other style in which the selected portion itself is obscured
21836 (a + b) . . . ## # ## + V c
21837 1* ............... 1* ---------------
21841 If the customizable variable
21842 @code{calc-highlight-selections-with-faces} is non-nil, then the
21843 non-selected portion of the formula will be de-emphasized by using a
21844 less noticeable face (@code{calc-nonselected-face}) instead of dots
21845 and the selected sub-formula will be highlighted by using a more
21846 noticeable face (@code{calc-selected-face}) instead of @samp{#}
21847 signs. (@pxref{Customizing Calc}.)
21849 @node Operating on Selections, Rearranging with Selections, Displaying Selections, Selecting Subformulas
21850 @subsection Operating on Selections
21853 Once a selection is made, all Calc commands that manipulate items
21854 on the stack will operate on the selected portions of the items
21855 instead. (Note that several stack elements may have selections
21856 at once, though there can be only one selection at a time in any
21857 given stack element.)
21860 @pindex calc-enable-selections
21861 The @kbd{j e} (@code{calc-enable-selections}) command disables the
21862 effect that selections have on Calc commands. The current selections
21863 still exist, but Calc commands operate on whole stack elements anyway.
21864 This mode can be identified by the fact that the @samp{*} markers on
21865 the line numbers are gone, even though selections are visible. To
21866 reactivate the selections, press @kbd{j e} again.
21868 To extract a sub-formula as a new formula, simply select the
21869 sub-formula and press @key{RET}. This normally duplicates the top
21870 stack element; here it duplicates only the selected portion of that
21873 To replace a sub-formula with something different, you can enter the
21874 new value onto the stack and press @key{TAB}. This normally exchanges
21875 the top two stack elements; here it swaps the value you entered into
21876 the selected portion of the formula, returning the old selected
21877 portion to the top of the stack.
21882 (a + b) . . . 17 x y . . . 17 x y + V c
21883 2* ............... 2* ............. 2: -------------
21884 . . . . . . . . 2 x + 1
21887 1: 17 x y 1: (a + b) 1: (a + b)
21891 In this example we select a sub-formula of our original example,
21892 enter a new formula, @key{TAB} it into place, then deselect to see
21893 the complete, edited formula.
21895 If you want to swap whole formulas around even though they contain
21896 selections, just use @kbd{j e} before and after.
21899 @pindex calc-enter-selection
21900 The @kbd{j '} (@code{calc-enter-selection}) command is another way
21901 to replace a selected sub-formula. This command does an algebraic
21902 entry just like the regular @kbd{'} key. When you press @key{RET},
21903 the formula you type replaces the original selection. You can use
21904 the @samp{$} symbol in the formula to refer to the original
21905 selection. If there is no selection in the formula under the cursor,
21906 the cursor is used to make a temporary selection for the purposes of
21907 the command. Thus, to change a term of a formula, all you have to
21908 do is move the Emacs cursor to that term and press @kbd{j '}.
21911 @pindex calc-edit-selection
21912 The @kbd{j `} (@code{calc-edit-selection}) command is a similar
21913 analogue of the @kbd{`} (@code{calc-edit}) command. It edits the
21914 selected sub-formula in a separate buffer. If there is no
21915 selection, it edits the sub-formula indicated by the cursor.
21917 To delete a sub-formula, press @key{DEL}. This generally replaces
21918 the sub-formula with the constant zero, but in a few suitable contexts
21919 it uses the constant one instead. The @key{DEL} key automatically
21920 deselects and re-simplifies the entire formula afterwards. Thus:
21925 17 x y + # # 17 x y 17 # y 17 y
21926 1* ------------- 1: ------- 1* ------- 1: -------
21927 2 x + 1 2 x + 1 2 x + 1 2 x + 1
21931 In this example, we first delete the @samp{sqrt(c)} term; Calc
21932 accomplishes this by replacing @samp{sqrt(c)} with zero and
21933 resimplifying. We then delete the @kbd{x} in the numerator;
21934 since this is part of a product, Calc replaces it with @samp{1}
21937 If you select an element of a vector and press @key{DEL}, that
21938 element is deleted from the vector. If you delete one side of
21939 an equation or inequality, only the opposite side remains.
21941 @kindex j @key{DEL}
21942 @pindex calc-del-selection
21943 The @kbd{j @key{DEL}} (@code{calc-del-selection}) command is like
21944 @key{DEL} but with the auto-selecting behavior of @kbd{j '} and
21945 @kbd{j `}. It deletes the selected portion of the formula
21946 indicated by the cursor, or, in the absence of a selection, it
21947 deletes the sub-formula indicated by the cursor position.
21949 @kindex j @key{RET}
21950 @pindex calc-grab-selection
21951 (There is also an auto-selecting @kbd{j @key{RET}} (@code{calc-copy-selection})
21954 Normal arithmetic operations also apply to sub-formulas. Here we
21955 select the denominator, press @kbd{5 -} to subtract five from the
21956 denominator, press @kbd{n} to negate the denominator, then
21957 press @kbd{Q} to take the square root.
21961 .. . .. . .. . .. .
21962 1* ....... 1* ....... 1* ....... 1* ..........
21963 2 x + 1 2 x - 4 4 - 2 x _________
21968 Certain types of operations on selections are not allowed. For
21969 example, for an arithmetic function like @kbd{-} no more than one of
21970 the arguments may be a selected sub-formula. (As the above example
21971 shows, the result of the subtraction is spliced back into the argument
21972 which had the selection; if there were more than one selection involved,
21973 this would not be well-defined.) If you try to subtract two selections,
21974 the command will abort with an error message.
21976 Operations on sub-formulas sometimes leave the formula as a whole
21977 in an ``un-natural'' state. Consider negating the @samp{2 x} term
21978 of our sample formula by selecting it and pressing @kbd{n}
21979 (@code{calc-change-sign}).
21984 1* .......... 1* ...........
21985 ......... ..........
21986 . . . 2 x . . . -2 x
21990 Unselecting the sub-formula reveals that the minus sign, which would
21991 normally have canceled out with the subtraction automatically, has
21992 not been able to do so because the subtraction was not part of the
21993 selected portion. Pressing @kbd{=} (@code{calc-evaluate}) or doing
21994 any other mathematical operation on the whole formula will cause it
22000 1: ----------- 1: ----------
22001 __________ _________
22002 V 4 - -2 x V 4 + 2 x
22006 @node Rearranging with Selections, , Operating on Selections, Selecting Subformulas
22007 @subsection Rearranging Formulas using Selections
22011 @pindex calc-commute-right
22012 The @kbd{j R} (@code{calc-commute-right}) command moves the selected
22013 sub-formula to the right in its surrounding formula. Generally the
22014 selection is one term of a sum or product; the sum or product is
22015 rearranged according to the commutative laws of algebra.
22017 As with @kbd{j '} and @kbd{j @key{DEL}}, the term under the cursor is used
22018 if there is no selection in the current formula. All commands described
22019 in this section share this property. In this example, we place the
22020 cursor on the @samp{a} and type @kbd{j R}, then repeat.
22023 1: a + b - c 1: b + a - c 1: b - c + a
22027 Note that in the final step above, the @samp{a} is switched with
22028 the @samp{c} but the signs are adjusted accordingly. When moving
22029 terms of sums and products, @kbd{j R} will never change the
22030 mathematical meaning of the formula.
22032 The selected term may also be an element of a vector or an argument
22033 of a function. The term is exchanged with the one to its right.
22034 In this case, the ``meaning'' of the vector or function may of
22035 course be drastically changed.
22038 1: [a, b, c] 1: [b, a, c] 1: [b, c, a]
22040 1: f(a, b, c) 1: f(b, a, c) 1: f(b, c, a)
22044 @pindex calc-commute-left
22045 The @kbd{j L} (@code{calc-commute-left}) command is like @kbd{j R}
22046 except that it swaps the selected term with the one to its left.
22048 With numeric prefix arguments, these commands move the selected
22049 term several steps at a time. It is an error to try to move a
22050 term left or right past the end of its enclosing formula.
22051 With numeric prefix arguments of zero, these commands move the
22052 selected term as far as possible in the given direction.
22055 @pindex calc-sel-distribute
22056 The @kbd{j D} (@code{calc-sel-distribute}) command mixes the selected
22057 sum or product into the surrounding formula using the distributive
22058 law. For example, in @samp{a * (b - c)} with the @samp{b - c}
22059 selected, the result is @samp{a b - a c}. This also distributes
22060 products or quotients into surrounding powers, and can also do
22061 transformations like @samp{exp(a + b)} to @samp{exp(a) exp(b)},
22062 where @samp{a + b} is the selected term, and @samp{ln(a ^ b)}
22063 to @samp{ln(a) b}, where @samp{a ^ b} is the selected term.
22065 For multiple-term sums or products, @kbd{j D} takes off one term
22066 at a time: @samp{a * (b + c - d)} goes to @samp{a * (c - d) + a b}
22067 with the @samp{c - d} selected so that you can type @kbd{j D}
22068 repeatedly to expand completely. The @kbd{j D} command allows a
22069 numeric prefix argument which specifies the maximum number of
22070 times to expand at once; the default is one time only.
22072 @vindex DistribRules
22073 The @kbd{j D} command is implemented using rewrite rules.
22074 @xref{Selections with Rewrite Rules}. The rules are stored in
22075 the Calc variable @code{DistribRules}. A convenient way to view
22076 these rules is to use @kbd{s e} (@code{calc-edit-variable}) which
22077 displays and edits the stored value of a variable. Press @kbd{C-c C-c}
22078 to return from editing mode; be careful not to make any actual changes
22079 or else you will affect the behavior of future @kbd{j D} commands!
22081 To extend @kbd{j D} to handle new cases, just edit @code{DistribRules}
22082 as described above. You can then use the @kbd{s p} command to save
22083 this variable's value permanently for future Calc sessions.
22084 @xref{Operations on Variables}.
22087 @pindex calc-sel-merge
22089 The @kbd{j M} (@code{calc-sel-merge}) command is the complement
22090 of @kbd{j D}; given @samp{a b - a c} with either @samp{a b} or
22091 @samp{a c} selected, the result is @samp{a * (b - c)}. Once
22092 again, @kbd{j M} can also merge calls to functions like @code{exp}
22093 and @code{ln}; examine the variable @code{MergeRules} to see all
22094 the relevant rules.
22097 @pindex calc-sel-commute
22098 @vindex CommuteRules
22099 The @kbd{j C} (@code{calc-sel-commute}) command swaps the arguments
22100 of the selected sum, product, or equation. It always behaves as
22101 if @kbd{j b} mode were in effect, i.e., the sum @samp{a + b + c} is
22102 treated as the nested sums @samp{(a + b) + c} by this command.
22103 If you put the cursor on the first @samp{+}, the result is
22104 @samp{(b + a) + c}; if you put the cursor on the second @samp{+}, the
22105 result is @samp{c + (a + b)} (which the default simplifications
22106 will rearrange to @samp{(c + a) + b}). The relevant rules are stored
22107 in the variable @code{CommuteRules}.
22109 You may need to turn default simplifications off (with the @kbd{m O}
22110 command) in order to get the full benefit of @kbd{j C}. For example,
22111 commuting @samp{a - b} produces @samp{-b + a}, but the default
22112 simplifications will ``simplify'' this right back to @samp{a - b} if
22113 you don't turn them off. The same is true of some of the other
22114 manipulations described in this section.
22117 @pindex calc-sel-negate
22118 @vindex NegateRules
22119 The @kbd{j N} (@code{calc-sel-negate}) command replaces the selected
22120 term with the negative of that term, then adjusts the surrounding
22121 formula in order to preserve the meaning. For example, given
22122 @samp{exp(a - b)} where @samp{a - b} is selected, the result is
22123 @samp{1 / exp(b - a)}. By contrast, selecting a term and using the
22124 regular @kbd{n} (@code{calc-change-sign}) command negates the
22125 term without adjusting the surroundings, thus changing the meaning
22126 of the formula as a whole. The rules variable is @code{NegateRules}.
22129 @pindex calc-sel-invert
22130 @vindex InvertRules
22131 The @kbd{j &} (@code{calc-sel-invert}) command is similar to @kbd{j N}
22132 except it takes the reciprocal of the selected term. For example,
22133 given @samp{a - ln(b)} with @samp{b} selected, the result is
22134 @samp{a + ln(1/b)}. The rules variable is @code{InvertRules}.
22137 @pindex calc-sel-jump-equals
22139 The @kbd{j E} (@code{calc-sel-jump-equals}) command moves the
22140 selected term from one side of an equation to the other. Given
22141 @samp{a + b = c + d} with @samp{c} selected, the result is
22142 @samp{a + b - c = d}. This command also works if the selected
22143 term is part of a @samp{*}, @samp{/}, or @samp{^} formula. The
22144 relevant rules variable is @code{JumpRules}.
22148 @pindex calc-sel-isolate
22149 The @kbd{j I} (@code{calc-sel-isolate}) command isolates the
22150 selected term on its side of an equation. It uses the @kbd{a S}
22151 (@code{calc-solve-for}) command to solve the equation, and the
22152 Hyperbolic flag affects it in the same way. @xref{Solving Equations}.
22153 When it applies, @kbd{j I} is often easier to use than @kbd{j E}.
22154 It understands more rules of algebra, and works for inequalities
22155 as well as equations.
22159 @pindex calc-sel-mult-both-sides
22160 @pindex calc-sel-div-both-sides
22161 The @kbd{j *} (@code{calc-sel-mult-both-sides}) command prompts for a
22162 formula using algebraic entry, then multiplies both sides of the
22163 selected quotient or equation by that formula. It performs the
22164 default algebraic simplifications before re-forming the
22165 quotient or equation. You can suppress this simplification by
22166 providing a prefix argument: @kbd{C-u j *}. There is also a @kbd{j /}
22167 (@code{calc-sel-div-both-sides}) which is similar to @kbd{j *} but
22168 dividing instead of multiplying by the factor you enter.
22170 If the selection is a quotient with numerator 1, then Calc's default
22171 simplifications would normally cancel the new factors. To prevent
22172 this, when the @kbd{j *} command is used on a selection whose numerator is
22173 1 or -1, the denominator is expanded at the top level using the
22174 distributive law (as if using the @kbd{C-u 1 a x} command). Suppose the
22175 formula on the stack is @samp{1 / (a + 1)} and you wish to multiplying the
22176 top and bottom by @samp{a - 1}. Calc's default simplifications would
22177 normally change the result @samp{(a - 1) /(a + 1) (a - 1)} back
22178 to the original form by cancellation; when @kbd{j *} is used, Calc
22179 expands the denominator to @samp{a (a - 1) + a - 1} to prevent this.
22181 If you wish the @kbd{j *} command to completely expand the denominator
22182 of a quotient you can call it with a zero prefix: @kbd{C-u 0 j *}. For
22183 example, if the formula on the stack is @samp{1 / (sqrt(a) + 1)}, you may
22184 wish to eliminate the square root in the denominator by multiplying
22185 the top and bottom by @samp{sqrt(a) - 1}. If you did this simply by using
22186 a simple @kbd{j *} command, you would get
22187 @samp{(sqrt(a)-1)/ (sqrt(a) (sqrt(a) - 1) + sqrt(a) - 1)}. Instead,
22188 you would probably want to use @kbd{C-u 0 j *}, which would expand the
22189 bottom and give you the desired result @samp{(sqrt(a)-1)/(a-1)}. More
22190 generally, if @kbd{j *} is called with an argument of a positive
22191 integer @var{n}, then the denominator of the expression will be
22192 expanded @var{n} times (as if with the @kbd{C-u @var{n} a x} command).
22194 If the selection is an inequality, @kbd{j *} and @kbd{j /} will
22195 accept any factor, but will warn unless they can prove the factor
22196 is either positive or negative. (In the latter case the direction
22197 of the inequality will be switched appropriately.) @xref{Declarations},
22198 for ways to inform Calc that a given variable is positive or
22199 negative. If Calc can't tell for sure what the sign of the factor
22200 will be, it will assume it is positive and display a warning
22203 For selections that are not quotients, equations, or inequalities,
22204 these commands pull out a multiplicative factor: They divide (or
22205 multiply) by the entered formula, simplify, then multiply (or divide)
22206 back by the formula.
22210 @pindex calc-sel-add-both-sides
22211 @pindex calc-sel-sub-both-sides
22212 The @kbd{j +} (@code{calc-sel-add-both-sides}) and @kbd{j -}
22213 (@code{calc-sel-sub-both-sides}) commands analogously add to or
22214 subtract from both sides of an equation or inequality. For other
22215 types of selections, they extract an additive factor. A numeric
22216 prefix argument suppresses simplification of the intermediate
22220 @pindex calc-sel-unpack
22221 The @kbd{j U} (@code{calc-sel-unpack}) command replaces the
22222 selected function call with its argument. For example, given
22223 @samp{a + sin(x^2)} with @samp{sin(x^2)} selected, the result
22224 is @samp{a + x^2}. (The @samp{x^2} will remain selected; if you
22225 wanted to change the @code{sin} to @code{cos}, just press @kbd{C}
22226 now to take the cosine of the selected part.)
22229 @pindex calc-sel-evaluate
22230 The @kbd{j v} (@code{calc-sel-evaluate}) command performs the
22231 basic simplifications on the selected sub-formula.
22232 These simplifications would normally be done automatically
22233 on all results, but may have been partially inhibited by
22234 previous selection-related operations, or turned off altogether
22235 by the @kbd{m O} command. This command is just an auto-selecting
22236 version of the @w{@kbd{a v}} command (@pxref{Algebraic Manipulation}).
22238 With a numeric prefix argument of 2, @kbd{C-u 2 j v} applies
22239 the default algebraic simplifications to the selected
22240 sub-formula. With a prefix argument of 3 or more, e.g., @kbd{C-u j v}
22241 applies the @kbd{a e} (@code{calc-simplify-extended}) command.
22242 @xref{Simplifying Formulas}. With a negative prefix argument
22243 it simplifies at the top level only, just as with @kbd{a v}.
22244 Here the ``top'' level refers to the top level of the selected
22248 @pindex calc-sel-expand-formula
22249 The @kbd{j "} (@code{calc-sel-expand-formula}) command is to @kbd{a "}
22250 (@pxref{Algebraic Manipulation}) what @kbd{j v} is to @kbd{a v}.
22252 You can use the @kbd{j r} (@code{calc-rewrite-selection}) command
22253 to define other algebraic operations on sub-formulas. @xref{Rewrite Rules}.
22255 @node Algebraic Manipulation, Simplifying Formulas, Selecting Subformulas, Algebra
22256 @section Algebraic Manipulation
22259 The commands in this section perform general-purpose algebraic
22260 manipulations. They work on the whole formula at the top of the
22261 stack (unless, of course, you have made a selection in that
22264 Many algebra commands prompt for a variable name or formula. If you
22265 answer the prompt with a blank line, the variable or formula is taken
22266 from top-of-stack, and the normal argument for the command is taken
22267 from the second-to-top stack level.
22270 @pindex calc-alg-evaluate
22271 The @kbd{a v} (@code{calc-alg-evaluate}) command performs the normal
22272 default simplifications on a formula; for example, @samp{a - -b} is
22273 changed to @samp{a + b}. These simplifications are normally done
22274 automatically on all Calc results, so this command is useful only if
22275 you have turned default simplifications off with an @kbd{m O}
22276 command. @xref{Simplification Modes}.
22278 It is often more convenient to type @kbd{=}, which is like @kbd{a v}
22279 but which also substitutes stored values for variables in the formula.
22280 Use @kbd{a v} if you want the variables to ignore their stored values.
22282 If you give a numeric prefix argument of 2 to @kbd{a v}, it simplifies
22283 using Calc's algebraic simplifications; @pxref{Simplifying Formulas}.
22284 If you give a numeric prefix of 3 or more, it uses Extended
22285 Simplification mode (@kbd{a e}).
22287 If you give a negative prefix argument @mathit{-1}, @mathit{-2}, or @mathit{-3},
22288 it simplifies in the corresponding mode but only works on the top-level
22289 function call of the formula. For example, @samp{(2 + 3) * (2 + 3)} will
22290 simplify to @samp{(2 + 3)^2}, without simplifying the sub-formulas
22291 @samp{2 + 3}. As another example, typing @kbd{V R +} to sum the vector
22292 @samp{[1, 2, 3, 4]} produces the formula @samp{reduce(add, [1, 2, 3, 4])}
22293 in No-Simplify mode. Using @kbd{a v} will evaluate this all the way to
22294 10; using @kbd{C-u - a v} will evaluate it only to @samp{1 + 2 + 3 + 4}.
22295 (@xref{Reducing and Mapping}.)
22299 The @kbd{=} command corresponds to the @code{evalv} function, and
22300 the related @kbd{N} command, which is like @kbd{=} but temporarily
22301 disables Symbolic mode (@kbd{m s}) during the evaluation, corresponds
22302 to the @code{evalvn} function. (These commands interpret their prefix
22303 arguments differently than @kbd{a v}; @kbd{=} treats the prefix as
22304 the number of stack elements to evaluate at once, and @kbd{N} treats
22305 it as a temporary different working precision.)
22307 The @code{evalvn} function can take an alternate working precision
22308 as an optional second argument. This argument can be either an
22309 integer, to set the precision absolutely, or a vector containing
22310 a single integer, to adjust the precision relative to the current
22311 precision. Note that @code{evalvn} with a larger than current
22312 precision will do the calculation at this higher precision, but the
22313 result will as usual be rounded back down to the current precision
22314 afterward. For example, @samp{evalvn(pi - 3.1415)} at a precision
22315 of 12 will return @samp{9.265359e-5}; @samp{evalvn(pi - 3.1415, 30)}
22316 will return @samp{9.26535897932e-5} (computing a 25-digit result which
22317 is then rounded down to 12); and @samp{evalvn(pi - 3.1415, [-2])}
22318 will return @samp{9.2654e-5}.
22321 @pindex calc-expand-formula
22322 The @kbd{a "} (@code{calc-expand-formula}) command expands functions
22323 into their defining formulas wherever possible. For example,
22324 @samp{deg(x^2)} is changed to @samp{180 x^2 / pi}. Most functions,
22325 like @code{sin} and @code{gcd}, are not defined by simple formulas
22326 and so are unaffected by this command. One important class of
22327 functions which @emph{can} be expanded is the user-defined functions
22328 created by the @kbd{Z F} command. @xref{Algebraic Definitions}.
22329 Other functions which @kbd{a "} can expand include the probability
22330 distribution functions, most of the financial functions, and the
22331 hyperbolic and inverse hyperbolic functions. A numeric prefix argument
22332 affects @kbd{a "} in the same way as it does @kbd{a v}: A positive
22333 argument expands all functions in the formula and then simplifies in
22334 various ways; a negative argument expands and simplifies only the
22335 top-level function call.
22338 @pindex calc-map-equation
22340 The @kbd{a M} (@code{calc-map-equation}) [@code{mapeq}] command applies
22341 a given function or operator to one or more equations. It is analogous
22342 to @kbd{V M}, which operates on vectors instead of equations.
22343 @pxref{Reducing and Mapping}. For example, @kbd{a M S} changes
22344 @samp{x = y+1} to @samp{sin(x) = sin(y+1)}, and @kbd{a M +} with
22345 @samp{x = y+1} and @expr{6} on the stack produces @samp{x+6 = y+7}.
22346 With two equations on the stack, @kbd{a M +} would add the lefthand
22347 sides together and the righthand sides together to get the two
22348 respective sides of a new equation.
22350 Mapping also works on inequalities. Mapping two similar inequalities
22351 produces another inequality of the same type. Mapping an inequality
22352 with an equation produces an inequality of the same type. Mapping a
22353 @samp{<=} with a @samp{<} or @samp{!=} (not-equal) produces a @samp{<}.
22354 If inequalities with opposite direction (e.g., @samp{<} and @samp{>})
22355 are mapped, the direction of the second inequality is reversed to
22356 match the first: Using @kbd{a M +} on @samp{a < b} and @samp{a > 2}
22357 reverses the latter to get @samp{2 < a}, which then allows the
22358 combination @samp{a + 2 < b + a}, which the algebraic simplifications
22359 can reduce to @samp{2 < b}.
22361 Using @kbd{a M *}, @kbd{a M /}, @kbd{a M n}, or @kbd{a M &} to negate
22362 or invert an inequality will reverse the direction of the inequality.
22363 Other adjustments to inequalities are @emph{not} done automatically;
22364 @kbd{a M S} will change @w{@samp{x < y}} to @samp{sin(x) < sin(y)} even
22365 though this is not true for all values of the variables.
22369 With the Hyperbolic flag, @kbd{H a M} [@code{mapeqp}] does a plain
22370 mapping operation without reversing the direction of any inequalities.
22371 Thus, @kbd{H a M &} would change @kbd{x > 2} to @kbd{1/x > 0.5}.
22372 (This change is mathematically incorrect, but perhaps you were
22373 fixing an inequality which was already incorrect.)
22377 With the Inverse flag, @kbd{I a M} [@code{mapeqr}] always reverses
22378 the direction of the inequality. You might use @kbd{I a M C} to
22379 change @samp{x < y} to @samp{cos(x) > cos(y)} if you know you are
22380 working with small positive angles.
22383 @pindex calc-substitute
22385 The @kbd{a b} (@code{calc-substitute}) [@code{subst}] command substitutes
22387 of some variable or sub-expression of an expression with a new
22388 sub-expression. For example, substituting @samp{sin(x)} with @samp{cos(y)}
22389 in @samp{2 sin(x)^2 + x sin(x) + sin(2 x)} produces
22390 @samp{2 cos(y)^2 + x cos(y) + @w{sin(2 x)}}.
22391 Note that this is a purely structural substitution; the lone @samp{x} and
22392 the @samp{sin(2 x)} stayed the same because they did not look like
22393 @samp{sin(x)}. @xref{Rewrite Rules}, for a more general method for
22394 doing substitutions.
22396 The @kbd{a b} command normally prompts for two formulas, the old
22397 one and the new one. If you enter a blank line for the first
22398 prompt, all three arguments are taken from the stack (new, then old,
22399 then target expression). If you type an old formula but then enter a
22400 blank line for the new one, the new formula is taken from top-of-stack
22401 and the target from second-to-top. If you answer both prompts, the
22402 target is taken from top-of-stack as usual.
22404 Note that @kbd{a b} has no understanding of commutativity or
22405 associativity. The pattern @samp{x+y} will not match the formula
22406 @samp{y+x}. Also, @samp{y+z} will not match inside the formula @samp{x+y+z}
22407 because the @samp{+} operator is left-associative, so the ``deep
22408 structure'' of that formula is @samp{(x+y) + z}. Use @kbd{d U}
22409 (@code{calc-unformatted-language}) mode to see the true structure of
22410 a formula. The rewrite rule mechanism, discussed later, does not have
22413 As an algebraic function, @code{subst} takes three arguments:
22414 Target expression, old, new. Note that @code{subst} is always
22415 evaluated immediately, even if its arguments are variables, so if
22416 you wish to put a call to @code{subst} onto the stack you must
22417 turn the default simplifications off first (with @kbd{m O}).
22419 @node Simplifying Formulas, Polynomials, Algebraic Manipulation, Algebra
22420 @section Simplifying Formulas
22426 @pindex calc-simplify
22429 The sections below describe all the various kinds of
22430 simplifications Calc provides in full detail. None of Calc's
22431 simplification commands are designed to pull rabbits out of hats;
22432 they simply apply certain specific rules to put formulas into
22433 less redundant or more pleasing forms. Serious algebra in Calc
22434 must be done manually, usually with a combination of selections
22435 and rewrite rules. @xref{Rearranging with Selections}.
22436 @xref{Rewrite Rules}.
22438 @xref{Simplification Modes}, for commands to control what level of
22439 simplification occurs automatically. Normally the algebraic
22440 simplifications described below occur. If you have turned on a
22441 simplification mode which does not do these algebraic simplifications,
22442 you can still apply them to a formula with the @kbd{a s}
22443 (@code{calc-simplify}) [@code{simplify}] command.
22445 There are some simplifications that, while sometimes useful, are never
22446 done automatically. For example, the @kbd{I} prefix can be given to
22447 @kbd{a s}; the @kbd{I a s} command will change any trigonometric
22448 function to the appropriate combination of @samp{sin}s and @samp{cos}s
22449 before simplifying. This can be useful in simplifying even mildly
22450 complicated trigonometric expressions. For example, while the algebraic
22451 simplifications can reduce @samp{sin(x) csc(x)} to @samp{1}, they will not
22452 simplify @samp{sin(x)^2 csc(x)}. The command @kbd{I a s} can be used to
22453 simplify this latter expression; it will transform @samp{sin(x)^2
22454 csc(x)} into @samp{sin(x)}. However, @kbd{I a s} will also perform
22455 some ``simplifications'' which may not be desired; for example, it
22456 will transform @samp{tan(x)^2} into @samp{sin(x)^2 / cos(x)^2}. The
22457 Hyperbolic prefix @kbd{H} can be used similarly; the @kbd{H a s} will
22458 replace any hyperbolic functions in the formula with the appropriate
22459 combinations of @samp{sinh}s and @samp{cosh}s before simplifying.
22463 * Basic Simplifications::
22464 * Algebraic Simplifications::
22465 * Unsafe Simplifications::
22466 * Simplification of Units::
22469 @node Basic Simplifications, Algebraic Simplifications, Simplifying Formulas, Simplifying Formulas
22470 @subsection Basic Simplifications
22473 @cindex Basic simplifications
22474 This section describes basic simplifications which Calc performs in many
22475 situations. For example, both binary simplifications and algebraic
22476 simplifications begin by performing these basic simplifications. You
22477 can type @kbd{m I} to restrict the simplifications done on the stack to
22478 these simplifications.
22480 The most basic simplification is the evaluation of functions.
22481 For example, @expr{2 + 3} is evaluated to @expr{5}, and @expr{@tfn{sqrt}(9)}
22482 is evaluated to @expr{3}. Evaluation does not occur if the arguments
22483 to a function are somehow of the wrong type @expr{@tfn{tan}([2,3,4])}),
22484 range (@expr{@tfn{tan}(90)}), or number (@expr{@tfn{tan}(3,5)}),
22485 or if the function name is not recognized (@expr{@tfn{f}(5)}), or if
22486 Symbolic mode (@pxref{Symbolic Mode}) prevents evaluation
22487 (@expr{@tfn{sqrt}(2)}).
22489 Calc simplifies (evaluates) the arguments to a function before it
22490 simplifies the function itself. Thus @expr{@tfn{sqrt}(5+4)} is
22491 simplified to @expr{@tfn{sqrt}(9)} before the @code{sqrt} function
22492 itself is applied. There are very few exceptions to this rule:
22493 @code{quote}, @code{lambda}, and @code{condition} (the @code{::}
22494 operator) do not evaluate their arguments, @code{if} (the @code{? :}
22495 operator) does not evaluate all of its arguments, and @code{evalto}
22496 does not evaluate its lefthand argument.
22498 Most commands apply at least these basic simplifications to all
22499 arguments they take from the stack, perform a particular operation,
22500 then simplify the result before pushing it back on the stack. In the
22501 common special case of regular arithmetic commands like @kbd{+} and
22502 @kbd{Q} [@code{sqrt}], the arguments are simply popped from the stack
22503 and collected into a suitable function call, which is then simplified
22504 (the arguments being simplified first as part of the process, as
22507 Even the basic set of simplifications are too numerous to describe
22508 completely here, but this section will describe the ones that apply to the
22509 major arithmetic operators. This list will be rather technical in
22510 nature, and will probably be interesting to you only if you are
22511 a serious user of Calc's algebra facilities.
22517 As well as the simplifications described here, if you have stored
22518 any rewrite rules in the variable @code{EvalRules} then these rules
22519 will also be applied before any of the basic simplifications.
22520 @xref{Automatic Rewrites}, for details.
22526 And now, on with the basic simplifications:
22528 Arithmetic operators like @kbd{+} and @kbd{*} always take two
22529 arguments in Calc's internal form. Sums and products of three or
22530 more terms are arranged by the associative law of algebra into
22531 a left-associative form for sums, @expr{((a + b) + c) + d}, and
22532 (by default) a right-associative form for products,
22533 @expr{a * (b * (c * d))}. Formulas like @expr{(a + b) + (c + d)} are
22534 rearranged to left-associative form, though this rarely matters since
22535 Calc's algebra commands are designed to hide the inner structure of sums
22536 and products as much as possible. Sums and products in their proper
22537 associative form will be written without parentheses in the examples
22540 Sums and products are @emph{not} rearranged according to the
22541 commutative law (@expr{a + b} to @expr{b + a}) except in a few
22542 special cases described below. Some algebra programs always
22543 rearrange terms into a canonical order, which enables them to
22544 see that @expr{a b + b a} can be simplified to @expr{2 a b}.
22545 If you are using Basic Simplification mode, Calc assumes you have put
22546 the terms into the order you want and generally leaves that order alone,
22547 with the consequence that formulas like the above will only be
22548 simplified if you explicitly give the @kbd{a s} command.
22549 @xref{Algebraic Simplifications}.
22551 Differences @expr{a - b} are treated like sums @expr{a + (-b)}
22552 for purposes of simplification; one of the default simplifications
22553 is to rewrite @expr{a + (-b)} or @expr{(-b) + a}, where @expr{-b}
22554 represents a ``negative-looking'' term, into @expr{a - b} form.
22555 ``Negative-looking'' means negative numbers, negated formulas like
22556 @expr{-x}, and products or quotients in which either term is
22559 Other simplifications involving negation are @expr{-(-x)} to @expr{x};
22560 @expr{-(a b)} or @expr{-(a/b)} where either @expr{a} or @expr{b} is
22561 negative-looking, simplified by negating that term, or else where
22562 @expr{a} or @expr{b} is any number, by negating that number;
22563 @expr{-(a + b)} to @expr{-a - b}, and @expr{-(b - a)} to @expr{a - b}.
22564 (This, and rewriting @expr{(-b) + a} to @expr{a - b}, are the only
22565 cases where the order of terms in a sum is changed by the default
22568 The distributive law is used to simplify sums in some cases:
22569 @expr{a x + b x} to @expr{(a + b) x}, where @expr{a} represents
22570 a number or an implicit 1 or @mathit{-1} (as in @expr{x} or @expr{-x})
22571 and similarly for @expr{b}. Use the @kbd{a c}, @w{@kbd{a f}}, or
22572 @kbd{j M} commands to merge sums with non-numeric coefficients
22573 using the distributive law.
22575 The distributive law is only used for sums of two terms, or
22576 for adjacent terms in a larger sum. Thus @expr{a + b + b + c}
22577 is simplified to @expr{a + 2 b + c}, but @expr{a + b + c + b}
22578 is not simplified. The reason is that comparing all terms of a
22579 sum with one another would require time proportional to the
22580 square of the number of terms; Calc omits potentially slow
22581 operations like this in basic simplification mode.
22583 Finally, @expr{a + 0} and @expr{0 + a} are simplified to @expr{a}.
22584 A consequence of the above rules is that @expr{0 - a} is simplified
22591 The products @expr{1 a} and @expr{a 1} are simplified to @expr{a};
22592 @expr{(-1) a} and @expr{a (-1)} are simplified to @expr{-a};
22593 @expr{0 a} and @expr{a 0} are simplified to @expr{0}, except that
22594 in Matrix mode where @expr{a} is not provably scalar the result
22595 is the generic zero matrix @samp{idn(0)}, and that if @expr{a} is
22596 infinite the result is @samp{nan}.
22598 Also, @expr{(-a) b} and @expr{a (-b)} are simplified to @expr{-(a b)},
22599 where this occurs for negated formulas but not for regular negative
22602 Products are commuted only to move numbers to the front:
22603 @expr{a b 2} is commuted to @expr{2 a b}.
22605 The product @expr{a (b + c)} is distributed over the sum only if
22606 @expr{a} and at least one of @expr{b} and @expr{c} are numbers:
22607 @expr{2 (x + 3)} goes to @expr{2 x + 6}. The formula
22608 @expr{(-a) (b - c)}, where @expr{-a} is a negative number, is
22609 rewritten to @expr{a (c - b)}.
22611 The distributive law of products and powers is used for adjacent
22612 terms of the product: @expr{x^a x^b} goes to
22613 @texline @math{x^{a+b}}
22614 @infoline @expr{x^(a+b)}
22615 where @expr{a} is a number, or an implicit 1 (as in @expr{x}),
22616 or the implicit one-half of @expr{@tfn{sqrt}(x)}, and similarly for
22617 @expr{b}. The result is written using @samp{sqrt} or @samp{1/sqrt}
22618 if the sum of the powers is @expr{1/2} or @expr{-1/2}, respectively.
22619 If the sum of the powers is zero, the product is simplified to
22620 @expr{1} or to @samp{idn(1)} if Matrix mode is enabled.
22622 The product of a negative power times anything but another negative
22623 power is changed to use division:
22624 @texline @math{x^{-2} y}
22625 @infoline @expr{x^(-2) y}
22626 goes to @expr{y / x^2} unless Matrix mode is
22627 in effect and neither @expr{x} nor @expr{y} are scalar (in which
22628 case it is considered unsafe to rearrange the order of the terms).
22630 Finally, @expr{a (b/c)} is rewritten to @expr{(a b)/c}, and also
22631 @expr{(a/b) c} is changed to @expr{(a c)/b} unless in Matrix mode.
22637 Simplifications for quotients are analogous to those for products.
22638 The quotient @expr{0 / x} is simplified to @expr{0}, with the same
22639 exceptions that were noted for @expr{0 x}. Likewise, @expr{x / 1}
22640 and @expr{x / (-1)} are simplified to @expr{x} and @expr{-x},
22643 The quotient @expr{x / 0} is left unsimplified or changed to an
22644 infinite quantity, as directed by the current infinite mode.
22645 @xref{Infinite Mode}.
22648 @texline @math{a / b^{-c}}
22649 @infoline @expr{a / b^(-c)}
22650 is changed to @expr{a b^c}, where @expr{-c} is any negative-looking
22651 power. Also, @expr{1 / b^c} is changed to
22652 @texline @math{b^{-c}}
22653 @infoline @expr{b^(-c)}
22654 for any power @expr{c}.
22656 Also, @expr{(-a) / b} and @expr{a / (-b)} go to @expr{-(a/b)};
22657 @expr{(a/b) / c} goes to @expr{a / (b c)}; and @expr{a / (b/c)}
22658 goes to @expr{(a c) / b} unless Matrix mode prevents this
22659 rearrangement. Similarly, @expr{a / (b:c)} is simplified to
22660 @expr{(c:b) a} for any fraction @expr{b:c}.
22662 The distributive law is applied to @expr{(a + b) / c} only if
22663 @expr{c} and at least one of @expr{a} and @expr{b} are numbers.
22664 Quotients of powers and square roots are distributed just as
22665 described for multiplication.
22667 Quotients of products cancel only in the leading terms of the
22668 numerator and denominator. In other words, @expr{a x b / a y b}
22669 is canceled to @expr{x b / y b} but not to @expr{x / y}. Once
22670 again this is because full cancellation can be slow; use @kbd{a s}
22671 to cancel all terms of the quotient.
22673 Quotients of negative-looking values are simplified according
22674 to @expr{(-a) / (-b)} to @expr{a / b}, @expr{(-a) / (b - c)}
22675 to @expr{a / (c - b)}, and @expr{(a - b) / (-c)} to @expr{(b - a) / c}.
22681 The formula @expr{x^0} is simplified to @expr{1}, or to @samp{idn(1)}
22682 in Matrix mode. The formula @expr{0^x} is simplified to @expr{0}
22683 unless @expr{x} is a negative number, complex number or zero.
22684 If @expr{x} is negative, complex or @expr{0.0}, @expr{0^x} is an
22685 infinity or an unsimplified formula according to the current infinite
22686 mode. The expression @expr{0^0} is simplified to @expr{1}.
22688 Powers of products or quotients @expr{(a b)^c}, @expr{(a/b)^c}
22689 are distributed to @expr{a^c b^c}, @expr{a^c / b^c} only if @expr{c}
22690 is an integer, or if either @expr{a} or @expr{b} are nonnegative
22691 real numbers. Powers of powers @expr{(a^b)^c} are simplified to
22692 @texline @math{a^{b c}}
22693 @infoline @expr{a^(b c)}
22694 only when @expr{c} is an integer and @expr{b c} also
22695 evaluates to an integer. Without these restrictions these simplifications
22696 would not be safe because of problems with principal values.
22698 @texline @math{((-3)^{1/2})^2}
22699 @infoline @expr{((-3)^1:2)^2}
22700 is safe to simplify, but
22701 @texline @math{((-3)^2)^{1/2}}
22702 @infoline @expr{((-3)^2)^1:2}
22703 is not.) @xref{Declarations}, for ways to inform Calc that your
22704 variables satisfy these requirements.
22706 As a special case of this rule, @expr{@tfn{sqrt}(x)^n} is simplified to
22707 @texline @math{x^{n/2}}
22708 @infoline @expr{x^(n/2)}
22709 only for even integers @expr{n}.
22711 If @expr{a} is known to be real, @expr{b} is an even integer, and
22712 @expr{c} is a half- or quarter-integer, then @expr{(a^b)^c} is
22713 simplified to @expr{@tfn{abs}(a^(b c))}.
22715 Also, @expr{(-a)^b} is simplified to @expr{a^b} if @expr{b} is an
22716 even integer, or to @expr{-(a^b)} if @expr{b} is an odd integer,
22717 for any negative-looking expression @expr{-a}.
22719 Square roots @expr{@tfn{sqrt}(x)} generally act like one-half powers
22720 @texline @math{x^{1:2}}
22721 @infoline @expr{x^1:2}
22722 for the purposes of the above-listed simplifications.
22725 @texline @math{1 / x^{1:2}}
22726 @infoline @expr{1 / x^1:2}
22728 @texline @math{x^{-1:2}},
22729 @infoline @expr{x^(-1:2)},
22730 but @expr{1 / @tfn{sqrt}(x)} is left alone.
22736 Generic identity matrices (@pxref{Matrix Mode}) are simplified by the
22737 following rules: @expr{@tfn{idn}(a) + b} to @expr{a + b} if @expr{b}
22738 is provably scalar, or expanded out if @expr{b} is a matrix;
22739 @expr{@tfn{idn}(a) + @tfn{idn}(b)} to @expr{@tfn{idn}(a + b)};
22740 @expr{-@tfn{idn}(a)} to @expr{@tfn{idn}(-a)}; @expr{a @tfn{idn}(b)} to
22741 @expr{@tfn{idn}(a b)} if @expr{a} is provably scalar, or to @expr{a b}
22742 if @expr{a} is provably non-scalar; @expr{@tfn{idn}(a) @tfn{idn}(b)} to
22743 @expr{@tfn{idn}(a b)}; analogous simplifications for quotients involving
22744 @code{idn}; and @expr{@tfn{idn}(a)^n} to @expr{@tfn{idn}(a^n)} where
22745 @expr{n} is an integer.
22751 The @code{floor} function and other integer truncation functions
22752 vanish if the argument is provably integer-valued, so that
22753 @expr{@tfn{floor}(@tfn{round}(x))} simplifies to @expr{@tfn{round}(x)}.
22754 Also, combinations of @code{float}, @code{floor} and its friends,
22755 and @code{ffloor} and its friends, are simplified in appropriate
22756 ways. @xref{Integer Truncation}.
22758 The expression @expr{@tfn{abs}(-x)} changes to @expr{@tfn{abs}(x)}.
22759 The expression @expr{@tfn{abs}(@tfn{abs}(x))} changes to
22760 @expr{@tfn{abs}(x)}; in fact, @expr{@tfn{abs}(x)} changes to @expr{x} or
22761 @expr{-x} if @expr{x} is provably nonnegative or nonpositive
22762 (@pxref{Declarations}).
22764 While most functions do not recognize the variable @code{i} as an
22765 imaginary number, the @code{arg} function does handle the two cases
22766 @expr{@tfn{arg}(@tfn{i})} and @expr{@tfn{arg}(-@tfn{i})} just for convenience.
22768 The expression @expr{@tfn{conj}(@tfn{conj}(x))} simplifies to @expr{x}.
22769 Various other expressions involving @code{conj}, @code{re}, and
22770 @code{im} are simplified, especially if some of the arguments are
22771 provably real or involve the constant @code{i}. For example,
22772 @expr{@tfn{conj}(a + b i)} is changed to
22773 @expr{@tfn{conj}(a) - @tfn{conj}(b) i}, or to @expr{a - b i} if @expr{a}
22774 and @expr{b} are known to be real.
22776 Functions like @code{sin} and @code{arctan} generally don't have
22777 any default simplifications beyond simply evaluating the functions
22778 for suitable numeric arguments and infinity. The algebraic
22779 simplifications described in the next section do provide some
22780 simplifications for these functions, though.
22782 One important simplification that does occur is that
22783 @expr{@tfn{ln}(@tfn{e})} is simplified to 1, and @expr{@tfn{ln}(@tfn{e}^x)} is
22784 simplified to @expr{x} for any @expr{x}. This occurs even if you have
22785 stored a different value in the Calc variable @samp{e}; but this would
22786 be a bad idea in any case if you were also using natural logarithms!
22788 Among the logical functions, @tfn{!(@var{a} <= @var{b})} changes to
22789 @tfn{@var{a} > @var{b}} and so on. Equations and inequalities where both sides
22790 are either negative-looking or zero are simplified by negating both sides
22791 and reversing the inequality. While it might seem reasonable to simplify
22792 @expr{!!x} to @expr{x}, this would not be valid in general because
22793 @expr{!!2} is 1, not 2.
22795 Most other Calc functions have few if any basic simplifications
22796 defined, aside of course from evaluation when the arguments are
22799 @node Algebraic Simplifications, Unsafe Simplifications, Basic Simplifications, Simplifying Formulas
22800 @subsection Algebraic Simplifications
22803 @cindex Algebraic simplifications
22806 This section describes all simplifications that are performed by
22807 the algebraic simplification mode, which is the default simplification
22808 mode. If you have switched to a different simplification mode, you can
22809 switch back with the @kbd{m A} command. Even in other simplification
22810 modes, the @kbd{a s} command will use these algebraic simplifications to
22811 simplify the formula.
22813 There is a variable, @code{AlgSimpRules}, in which you can put rewrites
22814 to be applied. Its use is analogous to @code{EvalRules},
22815 but without the special restrictions. Basically, the simplifier does
22816 @samp{@w{a r} AlgSimpRules} with an infinite repeat count on the whole
22817 expression being simplified, then it traverses the expression applying
22818 the built-in rules described below. If the result is different from
22819 the original expression, the process repeats with the basic
22820 simplifications (including @code{EvalRules}), then @code{AlgSimpRules},
22821 then the built-in simplifications, and so on.
22827 Sums are simplified in two ways. Constant terms are commuted to the
22828 end of the sum, so that @expr{a + 2 + b} changes to @expr{a + b + 2}.
22829 The only exception is that a constant will not be commuted away
22830 from the first position of a difference, i.e., @expr{2 - x} is not
22831 commuted to @expr{-x + 2}.
22833 Also, terms of sums are combined by the distributive law, as in
22834 @expr{x + y + 2 x} to @expr{y + 3 x}. This always occurs for
22835 adjacent terms, but Calc's algebraic simplifications compare all pairs
22836 of terms including non-adjacent ones.
22842 Products are sorted into a canonical order using the commutative
22843 law. For example, @expr{b c a} is commuted to @expr{a b c}.
22844 This allows easier comparison of products; for example, the basic
22845 simplifications will not change @expr{x y + y x} to @expr{2 x y},
22846 but the algebraic simplifications; it first rewrites the sum to
22847 @expr{x y + x y} which can then be recognized as a sum of identical
22850 The canonical ordering used to sort terms of products has the
22851 property that real-valued numbers, interval forms and infinities
22852 come first, and are sorted into increasing order. The @kbd{V S}
22853 command uses the same ordering when sorting a vector.
22855 Sorting of terms of products is inhibited when Matrix mode is
22856 turned on; in this case, Calc will never exchange the order of
22857 two terms unless it knows at least one of the terms is a scalar.
22859 Products of powers are distributed by comparing all pairs of
22860 terms, using the same method that the default simplifications
22861 use for adjacent terms of products.
22863 Even though sums are not sorted, the commutative law is still
22864 taken into account when terms of a product are being compared.
22865 Thus @expr{(x + y) (y + x)} will be simplified to @expr{(x + y)^2}.
22866 A subtle point is that @expr{(x - y) (y - x)} will @emph{not}
22867 be simplified to @expr{-(x - y)^2}; Calc does not notice that
22868 one term can be written as a constant times the other, even if
22869 that constant is @mathit{-1}.
22871 A fraction times any expression, @expr{(a:b) x}, is changed to
22872 a quotient involving integers: @expr{a x / b}. This is not
22873 done for floating-point numbers like @expr{0.5}, however. This
22874 is one reason why you may find it convenient to turn Fraction mode
22875 on while doing algebra; @pxref{Fraction Mode}.
22881 Quotients are simplified by comparing all terms in the numerator
22882 with all terms in the denominator for possible cancellation using
22883 the distributive law. For example, @expr{a x^2 b / c x^3 d} will
22884 cancel @expr{x^2} from the top and bottom to get @expr{a b / c x d}.
22885 (The terms in the denominator will then be rearranged to @expr{c d x}
22886 as described above.) If there is any common integer or fractional
22887 factor in the numerator and denominator, it is canceled out;
22888 for example, @expr{(4 x + 6) / 8 x} simplifies to @expr{(2 x + 3) / 4 x}.
22890 Non-constant common factors are not found even by algebraic
22891 simplifications. To cancel the factor @expr{a} in
22892 @expr{(a x + a) / a^2} you could first use @kbd{j M} on the product
22893 @expr{a x} to Merge the numerator to @expr{a (1+x)}, which can then be
22894 simplified successfully.
22900 Integer powers of the variable @code{i} are simplified according
22901 to the identity @expr{i^2 = -1}. If you store a new value other
22902 than the complex number @expr{(0,1)} in @code{i}, this simplification
22903 will no longer occur. This is not done by the basic
22904 simplifications; in case someone (unwisely) wants to use the name
22905 @code{i} for a variable unrelated to complex numbers, they can use
22906 basic simplification mode.
22908 Square roots of integer or rational arguments are simplified in
22909 several ways. (Note that these will be left unevaluated only in
22910 Symbolic mode.) First, square integer or rational factors are
22911 pulled out so that @expr{@tfn{sqrt}(8)} is rewritten as
22912 @texline @math{2\,@tfn{sqrt}(2)}.
22913 @infoline @expr{2 sqrt(2)}.
22914 Conceptually speaking this implies factoring the argument into primes
22915 and moving pairs of primes out of the square root, but for reasons of
22916 efficiency Calc only looks for primes up to 29.
22918 Square roots in the denominator of a quotient are moved to the
22919 numerator: @expr{1 / @tfn{sqrt}(3)} changes to @expr{@tfn{sqrt}(3) / 3}.
22920 The same effect occurs for the square root of a fraction:
22921 @expr{@tfn{sqrt}(2:3)} changes to @expr{@tfn{sqrt}(6) / 3}.
22927 The @code{%} (modulo) operator is simplified in several ways
22928 when the modulus @expr{M} is a positive real number. First, if
22929 the argument is of the form @expr{x + n} for some real number
22930 @expr{n}, then @expr{n} is itself reduced modulo @expr{M}. For
22931 example, @samp{(x - 23) % 10} is simplified to @samp{(x + 7) % 10}.
22933 If the argument is multiplied by a constant, and this constant
22934 has a common integer divisor with the modulus, then this factor is
22935 canceled out. For example, @samp{12 x % 15} is changed to
22936 @samp{3 (4 x % 5)} by factoring out 3. Also, @samp{(12 x + 1) % 15}
22937 is changed to @samp{3 ((4 x + 1:3) % 5)}. While these forms may
22938 not seem ``simpler,'' they allow Calc to discover useful information
22939 about modulo forms in the presence of declarations.
22941 If the modulus is 1, then Calc can use @code{int} declarations to
22942 evaluate the expression. For example, the idiom @samp{x % 2} is
22943 often used to check whether a number is odd or even. As described
22944 above, @w{@samp{2 n % 2}} and @samp{(2 n + 1) % 2} are simplified to
22945 @samp{2 (n % 1)} and @samp{2 ((n + 1:2) % 1)}, respectively; Calc
22946 can simplify these to 0 and 1 (respectively) if @code{n} has been
22947 declared to be an integer.
22953 Trigonometric functions are simplified in several ways. Whenever a
22954 products of two trigonometric functions can be replaced by a single
22955 function, the replacement is made; for example,
22956 @expr{@tfn{tan}(x) @tfn{cos}(x)} is simplified to @expr{@tfn{sin}(x)}.
22957 Reciprocals of trigonometric functions are replaced by their reciprocal
22958 function; for example, @expr{1/@tfn{sec}(x)} is simplified to
22959 @expr{@tfn{cos}(x)}. The corresponding simplifications for the
22960 hyperbolic functions are also handled.
22962 Trigonometric functions of their inverse functions are
22963 simplified. The expression @expr{@tfn{sin}(@tfn{arcsin}(x))} is
22964 simplified to @expr{x}, and similarly for @code{cos} and @code{tan}.
22965 Trigonometric functions of inverses of different trigonometric
22966 functions can also be simplified, as in @expr{@tfn{sin}(@tfn{arccos}(x))}
22967 to @expr{@tfn{sqrt}(1 - x^2)}.
22969 If the argument to @code{sin} is negative-looking, it is simplified to
22970 @expr{-@tfn{sin}(x)}, and similarly for @code{cos} and @code{tan}.
22971 Finally, certain special values of the argument are recognized;
22972 @pxref{Trigonometric and Hyperbolic Functions}.
22974 Hyperbolic functions of their inverses and of negative-looking
22975 arguments are also handled, as are exponentials of inverse
22976 hyperbolic functions.
22978 No simplifications for inverse trigonometric and hyperbolic
22979 functions are known, except for negative arguments of @code{arcsin},
22980 @code{arctan}, @code{arcsinh}, and @code{arctanh}. Note that
22981 @expr{@tfn{arcsin}(@tfn{sin}(x))} can @emph{not} safely change to
22982 @expr{x}, since this only correct within an integer multiple of
22983 @texline @math{2 \pi}
22984 @infoline @expr{2 pi}
22985 radians or 360 degrees. However, @expr{@tfn{arcsinh}(@tfn{sinh}(x))} is
22986 simplified to @expr{x} if @expr{x} is known to be real.
22988 Several simplifications that apply to logarithms and exponentials
22989 are that @expr{@tfn{exp}(@tfn{ln}(x))},
22990 @texline @tfn{e}@math{^{\ln(x)}},
22991 @infoline @expr{e^@tfn{ln}(x)},
22993 @texline @math{10^{{\rm log10}(x)}}
22994 @infoline @expr{10^@tfn{log10}(x)}
22995 all reduce to @expr{x}. Also, @expr{@tfn{ln}(@tfn{exp}(x))}, etc., can
22996 reduce to @expr{x} if @expr{x} is provably real. The form
22997 @expr{@tfn{exp}(x)^y} is simplified to @expr{@tfn{exp}(x y)}. If @expr{x}
22998 is a suitable multiple of
22999 @texline @math{\pi i}
23000 @infoline @expr{pi i}
23001 (as described above for the trigonometric functions), then
23002 @expr{@tfn{exp}(x)} or @expr{e^x} will be expanded. Finally,
23003 @expr{@tfn{ln}(x)} is simplified to a form involving @code{pi} and
23004 @code{i} where @expr{x} is provably negative, positive imaginary, or
23005 negative imaginary.
23007 The error functions @code{erf} and @code{erfc} are simplified when
23008 their arguments are negative-looking or are calls to the @code{conj}
23015 Equations and inequalities are simplified by canceling factors
23016 of products, quotients, or sums on both sides. Inequalities
23017 change sign if a negative multiplicative factor is canceled.
23018 Non-constant multiplicative factors as in @expr{a b = a c} are
23019 canceled from equations only if they are provably nonzero (generally
23020 because they were declared so; @pxref{Declarations}). Factors
23021 are canceled from inequalities only if they are nonzero and their
23024 Simplification also replaces an equation or inequality with
23025 1 or 0 (``true'' or ``false'') if it can through the use of
23026 declarations. If @expr{x} is declared to be an integer greater
23027 than 5, then @expr{x < 3}, @expr{x = 3}, and @expr{x = 7.5} are
23028 all simplified to 0, but @expr{x > 3} is simplified to 1.
23029 By a similar analysis, @expr{abs(x) >= 0} is simplified to 1,
23030 as is @expr{x^2 >= 0} if @expr{x} is known to be real.
23032 @node Unsafe Simplifications, Simplification of Units, Algebraic Simplifications, Simplifying Formulas
23033 @subsection ``Unsafe'' Simplifications
23036 @cindex Unsafe simplifications
23037 @cindex Extended simplification
23040 @pindex calc-simplify-extended
23042 @mindex esimpl@idots
23045 Calc is capable of performing some simplifications which may sometimes
23046 be desired but which are not ``safe'' in all cases. The @kbd{a e}
23047 (@code{calc-simplify-extended}) [@code{esimplify}] command
23048 applies the algebraic simplifications as well as these extended, or
23049 ``unsafe'', simplifications. Use this only if you know the values in
23050 your formula lie in the restricted ranges for which these
23051 simplifications are valid. You can use Extended Simplification mode
23052 (@kbd{m E}) to have these simplifications done automatically.
23054 The symbolic integrator uses these extended simplifications; one effect
23055 of this is that the integrator's results must be used with caution.
23056 Where an integral table will often attach conditions like ``for positive
23057 @expr{a} only,'' Calc (like most other symbolic integration programs)
23058 will simply produce an unqualified result.
23060 Because @kbd{a e}'s simplifications are unsafe, it is sometimes better
23061 to type @kbd{C-u -3 a v}, which does extended simplification only
23062 on the top level of the formula without affecting the sub-formulas.
23063 In fact, @kbd{C-u -3 j v} allows you to target extended simplification
23064 to any specific part of a formula.
23066 The variable @code{ExtSimpRules} contains rewrites to be applied when
23067 the extended simplifications are used. These are applied in addition to
23068 @code{EvalRules} and @code{AlgSimpRules}. (The @kbd{a r AlgSimpRules}
23069 step described above is simply followed by an @kbd{a r ExtSimpRules} step.)
23071 Following is a complete list of the ``unsafe'' simplifications.
23077 Inverse trigonometric or hyperbolic functions, called with their
23078 corresponding non-inverse functions as arguments, are simplified.
23079 For example, @expr{@tfn{arcsin}(@tfn{sin}(x))} changes
23080 to @expr{x}. Also, @expr{@tfn{arcsin}(@tfn{cos}(x))} and
23081 @expr{@tfn{arccos}(@tfn{sin}(x))} both change to @expr{@tfn{pi}/2 - x}.
23082 These simplifications are unsafe because they are valid only for
23083 values of @expr{x} in a certain range; outside that range, values
23084 are folded down to the 360-degree range that the inverse trigonometric
23085 functions always produce.
23087 Powers of powers @expr{(x^a)^b} are simplified to
23088 @texline @math{x^{a b}}
23089 @infoline @expr{x^(a b)}
23090 for all @expr{a} and @expr{b}. These results will be valid only
23091 in a restricted range of @expr{x}; for example, in
23092 @texline @math{(x^2)^{1:2}}
23093 @infoline @expr{(x^2)^1:2}
23094 the powers cancel to get @expr{x}, which is valid for positive values
23095 of @expr{x} but not for negative or complex values.
23097 Similarly, @expr{@tfn{sqrt}(x^a)} and @expr{@tfn{sqrt}(x)^a} are both
23098 simplified (possibly unsafely) to
23099 @texline @math{x^{a/2}}.
23100 @infoline @expr{x^(a/2)}.
23102 Forms like @expr{@tfn{sqrt}(1 - sin(x)^2)} are simplified to, e.g.,
23103 @expr{@tfn{cos}(x)}. Calc has identities of this sort for @code{sin},
23104 @code{cos}, @code{tan}, @code{sinh}, and @code{cosh}.
23106 Arguments of square roots are partially factored to look for
23107 squared terms that can be extracted. For example,
23108 @expr{@tfn{sqrt}(a^2 b^3 + a^3 b^2)} simplifies to
23109 @expr{a b @tfn{sqrt}(a+b)}.
23111 The simplifications of @expr{@tfn{ln}(@tfn{exp}(x))},
23112 @expr{@tfn{ln}(@tfn{e}^x)}, and @expr{@tfn{log10}(10^x)} to @expr{x} are also
23113 unsafe because of problems with principal values (although these
23114 simplifications are safe if @expr{x} is known to be real).
23116 Common factors are canceled from products on both sides of an
23117 equation, even if those factors may be zero: @expr{a x / b x}
23118 to @expr{a / b}. Such factors are never canceled from
23119 inequalities: Even the extended simplifications are not bold enough to
23120 reduce @expr{a x < b x} to @expr{a < b} (or @expr{a > b}, depending
23121 on whether you believe @expr{x} is positive or negative).
23122 The @kbd{a M /} command can be used to divide a factor out of
23123 both sides of an inequality.
23125 @node Simplification of Units, , Unsafe Simplifications, Simplifying Formulas
23126 @subsection Simplification of Units
23129 The simplifications described in this section (as well as the algebraic
23130 simplifications) are applied when units need to be simplified. They can
23131 be applied using the @kbd{u s} (@code{calc-simplify-units}) command, or
23132 will be done automatically in Units Simplification mode (@kbd{m U}).
23133 @xref{Basic Operations on Units}.
23135 The variable @code{UnitSimpRules} contains rewrites to be applied by
23136 units simplifications. These are applied in addition to @code{EvalRules}
23137 and @code{AlgSimpRules}.
23139 Scalar mode is automatically put into effect when simplifying units.
23140 @xref{Matrix Mode}.
23142 Sums @expr{a + b} involving units are simplified by extracting the
23143 units of @expr{a} as if by the @kbd{u x} command (call the result
23144 @expr{u_a}), then simplifying the expression @expr{b / u_a}
23145 using @kbd{u b} and @kbd{u s}. If the result has units then the sum
23146 is inconsistent and is left alone. Otherwise, it is rewritten
23147 in terms of the units @expr{u_a}.
23149 If units auto-ranging mode is enabled, products or quotients in
23150 which the first argument is a number which is out of range for the
23151 leading unit are modified accordingly.
23153 When canceling and combining units in products and quotients,
23154 Calc accounts for unit names that differ only in the prefix letter.
23155 For example, @samp{2 km m} is simplified to @samp{2000 m^2}.
23156 However, compatible but different units like @code{ft} and @code{in}
23157 are not combined in this way.
23159 Quotients @expr{a / b} are simplified in three additional ways. First,
23160 if @expr{b} is a number or a product beginning with a number, Calc
23161 computes the reciprocal of this number and moves it to the numerator.
23163 Second, for each pair of unit names from the numerator and denominator
23164 of a quotient, if the units are compatible (e.g., they are both
23165 units of area) then they are replaced by the ratio between those
23166 units. For example, in @samp{3 s in N / kg cm} the units
23167 @samp{in / cm} will be replaced by @expr{2.54}.
23169 Third, if the units in the quotient exactly cancel out, so that
23170 a @kbd{u b} command on the quotient would produce a dimensionless
23171 number for an answer, then the quotient simplifies to that number.
23173 For powers and square roots, the ``unsafe'' simplifications
23174 @expr{(a b)^c} to @expr{a^c b^c}, @expr{(a/b)^c} to @expr{a^c / b^c},
23175 and @expr{(a^b)^c} to
23176 @texline @math{a^{b c}}
23177 @infoline @expr{a^(b c)}
23178 are done if the powers are real numbers. (These are safe in the context
23179 of units because all numbers involved can reasonably be assumed to be
23182 Also, if a unit name is raised to a fractional power, and the
23183 base units in that unit name all occur to powers which are a
23184 multiple of the denominator of the power, then the unit name
23185 is expanded out into its base units, which can then be simplified
23186 according to the previous paragraph. For example, @samp{acre^1.5}
23187 is simplified by noting that @expr{1.5 = 3:2}, that @samp{acre}
23188 is defined in terms of @samp{m^2}, and that the 2 in the power of
23189 @code{m} is a multiple of 2 in @expr{3:2}. Thus, @code{acre^1.5} is
23190 replaced by approximately
23191 @texline @math{(4046 m^2)^{1.5}}
23192 @infoline @expr{(4046 m^2)^1.5},
23193 which is then changed to
23194 @texline @math{4046^{1.5} \, (m^2)^{1.5}},
23195 @infoline @expr{4046^1.5 (m^2)^1.5},
23196 then to @expr{257440 m^3}.
23198 The functions @code{float}, @code{frac}, @code{clean}, @code{abs},
23199 as well as @code{floor} and the other integer truncation functions,
23200 applied to unit names or products or quotients involving units, are
23201 simplified. For example, @samp{round(1.6 in)} is changed to
23202 @samp{round(1.6) round(in)}; the lefthand term evaluates to 2,
23203 and the righthand term simplifies to @code{in}.
23205 The functions @code{sin}, @code{cos}, and @code{tan} with arguments
23206 that have angular units like @code{rad} or @code{arcmin} are
23207 simplified by converting to base units (radians), then evaluating
23208 with the angular mode temporarily set to radians.
23210 @node Polynomials, Calculus, Simplifying Formulas, Algebra
23211 @section Polynomials
23213 A @dfn{polynomial} is a sum of terms which are coefficients times
23214 various powers of a ``base'' variable. For example, @expr{2 x^2 + 3 x - 4}
23215 is a polynomial in @expr{x}. Some formulas can be considered
23216 polynomials in several different variables: @expr{1 + 2 x + 3 y + 4 x y^2}
23217 is a polynomial in both @expr{x} and @expr{y}. Polynomial coefficients
23218 are often numbers, but they may in general be any formulas not
23219 involving the base variable.
23222 @pindex calc-factor
23224 The @kbd{a f} (@code{calc-factor}) [@code{factor}] command factors a
23225 polynomial into a product of terms. For example, the polynomial
23226 @expr{x^3 + 2 x^2 + x} is factored into @samp{x*(x+1)^2}. As another
23227 example, @expr{a c + b d + b c + a d} is factored into the product
23228 @expr{(a + b) (c + d)}.
23230 Calc currently has three algorithms for factoring. Formulas which are
23231 linear in several variables, such as the second example above, are
23232 merged according to the distributive law. Formulas which are
23233 polynomials in a single variable, with constant integer or fractional
23234 coefficients, are factored into irreducible linear and/or quadratic
23235 terms. The first example above factors into three linear terms
23236 (@expr{x}, @expr{x+1}, and @expr{x+1} again). Finally, formulas
23237 which do not fit the above criteria are handled by the algebraic
23240 Calc's polynomial factorization algorithm works by using the general
23241 root-finding command (@w{@kbd{a P}}) to solve for the roots of the
23242 polynomial. It then looks for roots which are rational numbers
23243 or complex-conjugate pairs, and converts these into linear and
23244 quadratic terms, respectively. Because it uses floating-point
23245 arithmetic, it may be unable to find terms that involve large
23246 integers (whose number of digits approaches the current precision).
23247 Also, irreducible factors of degree higher than quadratic are not
23248 found, and polynomials in more than one variable are not treated.
23249 (A more robust factorization algorithm may be included in a future
23252 @vindex FactorRules
23264 The rewrite-based factorization method uses rules stored in the variable
23265 @code{FactorRules}. @xref{Rewrite Rules}, for a discussion of the
23266 operation of rewrite rules. The default @code{FactorRules} are able
23267 to factor quadratic forms symbolically into two linear terms,
23268 @expr{(a x + b) (c x + d)}. You can edit these rules to include other
23269 cases if you wish. To use the rules, Calc builds the formula
23270 @samp{thecoefs(x, [a, b, c, ...])} where @code{x} is the polynomial
23271 base variable and @code{a}, @code{b}, etc., are polynomial coefficients
23272 (which may be numbers or formulas). The constant term is written first,
23273 i.e., in the @code{a} position. When the rules complete, they should have
23274 changed the formula into the form @samp{thefactors(x, [f1, f2, f3, ...])}
23275 where each @code{fi} should be a factored term, e.g., @samp{x - ai}.
23276 Calc then multiplies these terms together to get the complete
23277 factored form of the polynomial. If the rules do not change the
23278 @code{thecoefs} call to a @code{thefactors} call, @kbd{a f} leaves the
23279 polynomial alone on the assumption that it is unfactorable. (Note that
23280 the function names @code{thecoefs} and @code{thefactors} are used only
23281 as placeholders; there are no actual Calc functions by those names.)
23285 The @kbd{H a f} [@code{factors}] command also factors a polynomial,
23286 but it returns a list of factors instead of an expression which is the
23287 product of the factors. Each factor is represented by a sub-vector
23288 of the factor, and the power with which it appears. For example,
23289 @expr{x^5 + x^4 - 33 x^3 + 63 x^2} factors to @expr{(x + 7) x^2 (x - 3)^2}
23290 in @kbd{a f}, or to @expr{[ [x, 2], [x+7, 1], [x-3, 2] ]} in @kbd{H a f}.
23291 If there is an overall numeric factor, it always comes first in the list.
23292 The functions @code{factor} and @code{factors} allow a second argument
23293 when written in algebraic form; @samp{factor(x,v)} factors @expr{x} with
23294 respect to the specific variable @expr{v}. The default is to factor with
23295 respect to all the variables that appear in @expr{x}.
23298 @pindex calc-collect
23300 The @kbd{a c} (@code{calc-collect}) [@code{collect}] command rearranges a
23302 polynomial in a given variable, ordered in decreasing powers of that
23303 variable. For example, given @expr{1 + 2 x + 3 y + 4 x y^2} on
23304 the stack, @kbd{a c x} would produce @expr{(2 + 4 y^2) x + (1 + 3 y)},
23305 and @kbd{a c y} would produce @expr{(4 x) y^2 + 3 y + (1 + 2 x)}.
23306 The polynomial will be expanded out using the distributive law as
23307 necessary: Collecting @expr{x} in @expr{(x - 1)^3} produces
23308 @expr{x^3 - 3 x^2 + 3 x - 1}. Terms not involving @expr{x} will
23311 The ``variable'' you specify at the prompt can actually be any
23312 expression: @kbd{a c ln(x+1)} will collect together all terms multiplied
23313 by @samp{ln(x+1)} or integer powers thereof. If @samp{x} also appears
23314 in the formula in a context other than @samp{ln(x+1)}, @kbd{a c} will
23315 treat those occurrences as unrelated to @samp{ln(x+1)}, i.e., as constants.
23318 @pindex calc-expand
23320 The @kbd{a x} (@code{calc-expand}) [@code{expand}] command expands an
23321 expression by applying the distributive law everywhere. It applies to
23322 products, quotients, and powers involving sums. By default, it fully
23323 distributes all parts of the expression. With a numeric prefix argument,
23324 the distributive law is applied only the specified number of times, then
23325 the partially expanded expression is left on the stack.
23327 The @kbd{a x} and @kbd{j D} commands are somewhat redundant. Use
23328 @kbd{a x} if you want to expand all products of sums in your formula.
23329 Use @kbd{j D} if you want to expand a particular specified term of
23330 the formula. There is an exactly analogous correspondence between
23331 @kbd{a f} and @kbd{j M}. (The @kbd{j D} and @kbd{j M} commands
23332 also know many other kinds of expansions, such as
23333 @samp{exp(a + b) = exp(a) exp(b)}, which @kbd{a x} and @kbd{a f}
23336 Calc's automatic simplifications will sometimes reverse a partial
23337 expansion. For example, the first step in expanding @expr{(x+1)^3} is
23338 to write @expr{(x+1) (x+1)^2}. If @kbd{a x} stops there and tries
23339 to put this formula onto the stack, though, Calc will automatically
23340 simplify it back to @expr{(x+1)^3} form. The solution is to turn
23341 simplification off first (@pxref{Simplification Modes}), or to run
23342 @kbd{a x} without a numeric prefix argument so that it expands all
23343 the way in one step.
23348 The @kbd{a a} (@code{calc-apart}) [@code{apart}] command expands a
23349 rational function by partial fractions. A rational function is the
23350 quotient of two polynomials; @code{apart} pulls this apart into a
23351 sum of rational functions with simple denominators. In algebraic
23352 notation, the @code{apart} function allows a second argument that
23353 specifies which variable to use as the ``base''; by default, Calc
23354 chooses the base variable automatically.
23357 @pindex calc-normalize-rat
23359 The @kbd{a n} (@code{calc-normalize-rat}) [@code{nrat}] command
23360 attempts to arrange a formula into a quotient of two polynomials.
23361 For example, given @expr{1 + (a + b/c) / d}, the result would be
23362 @expr{(b + a c + c d) / c d}. The quotient is reduced, so that
23363 @kbd{a n} will simplify @expr{(x^2 + 2x + 1) / (x^2 - 1)} by dividing
23364 out the common factor @expr{x + 1}, yielding @expr{(x + 1) / (x - 1)}.
23367 @pindex calc-poly-div
23369 The @kbd{a \} (@code{calc-poly-div}) [@code{pdiv}] command divides
23370 two polynomials @expr{u} and @expr{v}, yielding a new polynomial
23371 @expr{q}. If several variables occur in the inputs, the inputs are
23372 considered multivariate polynomials. (Calc divides by the variable
23373 with the largest power in @expr{u} first, or, in the case of equal
23374 powers, chooses the variables in alphabetical order.) For example,
23375 dividing @expr{x^2 + 3 x + 2} by @expr{x + 2} yields @expr{x + 1}.
23376 The remainder from the division, if any, is reported at the bottom
23377 of the screen and is also placed in the Trail along with the quotient.
23379 Using @code{pdiv} in algebraic notation, you can specify the particular
23380 variable to be used as the base: @code{pdiv(@var{a},@var{b},@var{x})}.
23381 If @code{pdiv} is given only two arguments (as is always the case with
23382 the @kbd{a \} command), then it does a multivariate division as outlined
23386 @pindex calc-poly-rem
23388 The @kbd{a %} (@code{calc-poly-rem}) [@code{prem}] command divides
23389 two polynomials and keeps the remainder @expr{r}. The quotient
23390 @expr{q} is discarded. For any formulas @expr{a} and @expr{b}, the
23391 results of @kbd{a \} and @kbd{a %} satisfy @expr{a = q b + r}.
23392 (This is analogous to plain @kbd{\} and @kbd{%}, which compute the
23393 integer quotient and remainder from dividing two numbers.)
23397 @pindex calc-poly-div-rem
23400 The @kbd{a /} (@code{calc-poly-div-rem}) [@code{pdivrem}] command
23401 divides two polynomials and reports both the quotient and the
23402 remainder as a vector @expr{[q, r]}. The @kbd{H a /} [@code{pdivide}]
23403 command divides two polynomials and constructs the formula
23404 @expr{q + r/b} on the stack. (Naturally if the remainder is zero,
23405 this will immediately simplify to @expr{q}.)
23408 @pindex calc-poly-gcd
23410 The @kbd{a g} (@code{calc-poly-gcd}) [@code{pgcd}] command computes
23411 the greatest common divisor of two polynomials. (The GCD actually
23412 is unique only to within a constant multiplier; Calc attempts to
23413 choose a GCD which will be unsurprising.) For example, the @kbd{a n}
23414 command uses @kbd{a g} to take the GCD of the numerator and denominator
23415 of a quotient, then divides each by the result using @kbd{a \}. (The
23416 definition of GCD ensures that this division can take place without
23417 leaving a remainder.)
23419 While the polynomials used in operations like @kbd{a /} and @kbd{a g}
23420 often have integer coefficients, this is not required. Calc can also
23421 deal with polynomials over the rationals or floating-point reals.
23422 Polynomials with modulo-form coefficients are also useful in many
23423 applications; if you enter @samp{(x^2 + 3 x - 1) mod 5}, Calc
23424 automatically transforms this into a polynomial over the field of
23425 integers mod 5: @samp{(1 mod 5) x^2 + (3 mod 5) x + (4 mod 5)}.
23427 Congratulations and thanks go to Ove Ewerlid
23428 (@code{ewerlid@@mizar.DoCS.UU.SE}), who contributed many of the
23429 polynomial routines used in the above commands.
23431 @xref{Decomposing Polynomials}, for several useful functions for
23432 extracting the individual coefficients of a polynomial.
23434 @node Calculus, Solving Equations, Polynomials, Algebra
23438 The following calculus commands do not automatically simplify their
23439 inputs or outputs using @code{calc-simplify}. You may find it helps
23440 to do this by hand by typing @kbd{a s} or @kbd{a e}. It may also help
23441 to use @kbd{a x} and/or @kbd{a c} to arrange a result in the most
23445 * Differentiation::
23447 * Customizing the Integrator::
23448 * Numerical Integration::
23452 @node Differentiation, Integration, Calculus, Calculus
23453 @subsection Differentiation
23458 @pindex calc-derivative
23461 The @kbd{a d} (@code{calc-derivative}) [@code{deriv}] command computes
23462 the derivative of the expression on the top of the stack with respect to
23463 some variable, which it will prompt you to enter. Normally, variables
23464 in the formula other than the specified differentiation variable are
23465 considered constant, i.e., @samp{deriv(y,x)} is reduced to zero. With
23466 the Hyperbolic flag, the @code{tderiv} (total derivative) operation is used
23467 instead, in which derivatives of variables are not reduced to zero
23468 unless those variables are known to be ``constant,'' i.e., independent
23469 of any other variables. (The built-in special variables like @code{pi}
23470 are considered constant, as are variables that have been declared
23471 @code{const}; @pxref{Declarations}.)
23473 With a numeric prefix argument @var{n}, this command computes the
23474 @var{n}th derivative.
23476 When working with trigonometric functions, it is best to switch to
23477 Radians mode first (with @w{@kbd{m r}}). The derivative of @samp{sin(x)}
23478 in degrees is @samp{(pi/180) cos(x)}, probably not the expected
23481 If you use the @code{deriv} function directly in an algebraic formula,
23482 you can write @samp{deriv(f,x,x0)} which represents the derivative
23483 of @expr{f} with respect to @expr{x}, evaluated at the point
23484 @texline @math{x=x_0}.
23485 @infoline @expr{x=x0}.
23487 If the formula being differentiated contains functions which Calc does
23488 not know, the derivatives of those functions are produced by adding
23489 primes (apostrophe characters). For example, @samp{deriv(f(2x), x)}
23490 produces @samp{2 f'(2 x)}, where the function @code{f'} represents the
23491 derivative of @code{f}.
23493 For functions you have defined with the @kbd{Z F} command, Calc expands
23494 the functions according to their defining formulas unless you have
23495 also defined @code{f'} suitably. For example, suppose we define
23496 @samp{sinc(x) = sin(x)/x} using @kbd{Z F}. If we then differentiate
23497 the formula @samp{sinc(2 x)}, the formula will be expanded to
23498 @samp{sin(2 x) / (2 x)} and differentiated. However, if we also
23499 define @samp{sinc'(x) = dsinc(x)}, say, then Calc will write the
23500 result as @samp{2 dsinc(2 x)}. @xref{Algebraic Definitions}.
23502 For multi-argument functions @samp{f(x,y,z)}, the derivative with respect
23503 to the first argument is written @samp{f'(x,y,z)}; derivatives with
23504 respect to the other arguments are @samp{f'2(x,y,z)} and @samp{f'3(x,y,z)}.
23505 Various higher-order derivatives can be formed in the obvious way, e.g.,
23506 @samp{f'@var{}'(x)} (the second derivative of @code{f}) or
23507 @samp{f'@var{}'2'3(x,y,z)} (@code{f} differentiated with respect to each
23510 @node Integration, Customizing the Integrator, Differentiation, Calculus
23511 @subsection Integration
23515 @pindex calc-integral
23517 The @kbd{a i} (@code{calc-integral}) [@code{integ}] command computes the
23518 indefinite integral of the expression on the top of the stack with
23519 respect to a prompted-for variable. The integrator is not guaranteed to
23520 work for all integrable functions, but it is able to integrate several
23521 large classes of formulas. In particular, any polynomial or rational
23522 function (a polynomial divided by a polynomial) is acceptable.
23523 (Rational functions don't have to be in explicit quotient form, however;
23524 @texline @math{x/(1+x^{-2})}
23525 @infoline @expr{x/(1+x^-2)}
23526 is not strictly a quotient of polynomials, but it is equivalent to
23527 @expr{x^3/(x^2+1)}, which is.) Also, square roots of terms involving
23528 @expr{x} and @expr{x^2} may appear in rational functions being
23529 integrated. Finally, rational functions involving trigonometric or
23530 hyperbolic functions can be integrated.
23532 With an argument (@kbd{C-u a i}), this command will compute the definite
23533 integral of the expression on top of the stack. In this case, the
23534 command will again prompt for an integration variable, then prompt for a
23535 lower limit and an upper limit.
23538 If you use the @code{integ} function directly in an algebraic formula,
23539 you can also write @samp{integ(f,x,v)} which expresses the resulting
23540 indefinite integral in terms of variable @code{v} instead of @code{x}.
23541 With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
23542 integral from @code{a} to @code{b}.
23545 If you use the @code{integ} function directly in an algebraic formula,
23546 you can also write @samp{integ(f,x,v)} which expresses the resulting
23547 indefinite integral in terms of variable @code{v} instead of @code{x}.
23548 With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
23549 integral $\int_a^b f(x) \, dx$.
23552 Please note that the current implementation of Calc's integrator sometimes
23553 produces results that are significantly more complex than they need to
23554 be. For example, the integral Calc finds for
23555 @texline @math{1/(x+\sqrt{x^2+1})}
23556 @infoline @expr{1/(x+sqrt(x^2+1))}
23557 is several times more complicated than the answer Mathematica
23558 returns for the same input, although the two forms are numerically
23559 equivalent. Also, any indefinite integral should be considered to have
23560 an arbitrary constant of integration added to it, although Calc does not
23561 write an explicit constant of integration in its result. For example,
23562 Calc's solution for
23563 @texline @math{1/(1+\tan x)}
23564 @infoline @expr{1/(1+tan(x))}
23565 differs from the solution given in the @emph{CRC Math Tables} by a
23567 @texline @math{\pi i / 2}
23568 @infoline @expr{pi i / 2},
23569 due to a different choice of constant of integration.
23571 The Calculator remembers all the integrals it has done. If conditions
23572 change in a way that would invalidate the old integrals, say, a switch
23573 from Degrees to Radians mode, then they will be thrown out. If you
23574 suspect this is not happening when it should, use the
23575 @code{calc-flush-caches} command; @pxref{Caches}.
23578 Calc normally will pursue integration by substitution or integration by
23579 parts up to 3 nested times before abandoning an approach as fruitless.
23580 If the integrator is taking too long, you can lower this limit by storing
23581 a number (like 2) in the variable @code{IntegLimit}. (The @kbd{s I}
23582 command is a convenient way to edit @code{IntegLimit}.) If this variable
23583 has no stored value or does not contain a nonnegative integer, a limit
23584 of 3 is used. The lower this limit is, the greater the chance that Calc
23585 will be unable to integrate a function it could otherwise handle. Raising
23586 this limit allows the Calculator to solve more integrals, though the time
23587 it takes may grow exponentially. You can monitor the integrator's actions
23588 by creating an Emacs buffer called @code{*Trace*}. If such a buffer
23589 exists, the @kbd{a i} command will write a log of its actions there.
23591 If you want to manipulate integrals in a purely symbolic way, you can
23592 set the integration nesting limit to 0 to prevent all but fast
23593 table-lookup solutions of integrals. You might then wish to define
23594 rewrite rules for integration by parts, various kinds of substitutions,
23595 and so on. @xref{Rewrite Rules}.
23597 @node Customizing the Integrator, Numerical Integration, Integration, Calculus
23598 @subsection Customizing the Integrator
23602 Calc has two built-in rewrite rules called @code{IntegRules} and
23603 @code{IntegAfterRules} which you can edit to define new integration
23604 methods. @xref{Rewrite Rules}. At each step of the integration process,
23605 Calc wraps the current integrand in a call to the fictitious function
23606 @samp{integtry(@var{expr},@var{var})}, where @var{expr} is the
23607 integrand and @var{var} is the integration variable. If your rules
23608 rewrite this to be a plain formula (not a call to @code{integtry}), then
23609 Calc will use this formula as the integral of @var{expr}. For example,
23610 the rule @samp{integtry(mysin(x),x) := -mycos(x)} would define a rule to
23611 integrate a function @code{mysin} that acts like the sine function.
23612 Then, putting @samp{4 mysin(2y+1)} on the stack and typing @kbd{a i y}
23613 will produce the integral @samp{-2 mycos(2y+1)}. Note that Calc has
23614 automatically made various transformations on the integral to allow it
23615 to use your rule; integral tables generally give rules for
23616 @samp{mysin(a x + b)}, but you don't need to use this much generality
23617 in your @code{IntegRules}.
23619 @cindex Exponential integral Ei(x)
23624 As a more serious example, the expression @samp{exp(x)/x} cannot be
23625 integrated in terms of the standard functions, so the ``exponential
23626 integral'' function
23627 @texline @math{{\rm Ei}(x)}
23628 @infoline @expr{Ei(x)}
23629 was invented to describe it.
23630 We can get Calc to do this integral in terms of a made-up @code{Ei}
23631 function by adding the rule @samp{[integtry(exp(x)/x, x) := Ei(x)]}
23632 to @code{IntegRules}. Now entering @samp{exp(2x)/x} on the stack
23633 and typing @kbd{a i x} yields @samp{Ei(2 x)}. This new rule will
23634 work with Calc's various built-in integration methods (such as
23635 integration by substitution) to solve a variety of other problems
23636 involving @code{Ei}: For example, now Calc will also be able to
23637 integrate @samp{exp(exp(x))} and @samp{ln(ln(x))} (to get @samp{Ei(exp(x))}
23638 and @samp{x ln(ln(x)) - Ei(ln(x))}, respectively).
23640 Your rule may do further integration by calling @code{integ}. For
23641 example, @samp{integtry(twice(u),x) := twice(integ(u))} allows Calc
23642 to integrate @samp{twice(sin(x))} to get @samp{twice(-cos(x))}.
23643 Note that @code{integ} was called with only one argument. This notation
23644 is allowed only within @code{IntegRules}; it means ``integrate this
23645 with respect to the same integration variable.'' If Calc is unable
23646 to integrate @code{u}, the integration that invoked @code{IntegRules}
23647 also fails. Thus integrating @samp{twice(f(x))} fails, returning the
23648 unevaluated integral @samp{integ(twice(f(x)), x)}. It is still valid
23649 to call @code{integ} with two or more arguments, however; in this case,
23650 if @code{u} is not integrable, @code{twice} itself will still be
23651 integrated: If the above rule is changed to @samp{... := twice(integ(u,x))},
23652 then integrating @samp{twice(f(x))} will yield @samp{twice(integ(f(x),x))}.
23654 If a rule instead produces the formula @samp{integsubst(@var{sexpr},
23655 @var{svar})}, either replacing the top-level @code{integtry} call or
23656 nested anywhere inside the expression, then Calc will apply the
23657 substitution @samp{@var{u} = @var{sexpr}(@var{svar})} to try to
23658 integrate the original @var{expr}. For example, the rule
23659 @samp{sqrt(a) := integsubst(sqrt(x),x)} says that if Calc ever finds
23660 a square root in the integrand, it should attempt the substitution
23661 @samp{u = sqrt(x)}. (This particular rule is unnecessary because
23662 Calc always tries ``obvious'' substitutions where @var{sexpr} actually
23663 appears in the integrand.) The variable @var{svar} may be the same
23664 as the @var{var} that appeared in the call to @code{integtry}, but
23667 When integrating according to an @code{integsubst}, Calc uses the
23668 equation solver to find the inverse of @var{sexpr} (if the integrand
23669 refers to @var{var} anywhere except in subexpressions that exactly
23670 match @var{sexpr}). It uses the differentiator to find the derivative
23671 of @var{sexpr} and/or its inverse (it has two methods that use one
23672 derivative or the other). You can also specify these items by adding
23673 extra arguments to the @code{integsubst} your rules construct; the
23674 general form is @samp{integsubst(@var{sexpr}, @var{svar}, @var{sinv},
23675 @var{sprime})}, where @var{sinv} is the inverse of @var{sexpr} (still
23676 written as a function of @var{svar}), and @var{sprime} is the
23677 derivative of @var{sexpr} with respect to @var{svar}. If you don't
23678 specify these things, and Calc is not able to work them out on its
23679 own with the information it knows, then your substitution rule will
23680 work only in very specific, simple cases.
23682 Calc applies @code{IntegRules} as if by @kbd{C-u 1 a r IntegRules};
23683 in other words, Calc stops rewriting as soon as any rule in your rule
23684 set succeeds. (If it weren't for this, the @samp{integsubst(sqrt(x),x)}
23685 example above would keep on adding layers of @code{integsubst} calls
23688 @vindex IntegSimpRules
23689 Another set of rules, stored in @code{IntegSimpRules}, are applied
23690 every time the integrator uses algebraic simplifications to simplify an
23691 intermediate result. For example, putting the rule
23692 @samp{twice(x) := 2 x} into @code{IntegSimpRules} would tell Calc to
23693 convert the @code{twice} function into a form it knows whenever
23694 integration is attempted.
23696 One more way to influence the integrator is to define a function with
23697 the @kbd{Z F} command (@pxref{Algebraic Definitions}). Calc's
23698 integrator automatically expands such functions according to their
23699 defining formulas, even if you originally asked for the function to
23700 be left unevaluated for symbolic arguments. (Certain other Calc
23701 systems, such as the differentiator and the equation solver, also
23704 @vindex IntegAfterRules
23705 Sometimes Calc is able to find a solution to your integral, but it
23706 expresses the result in a way that is unnecessarily complicated. If
23707 this happens, you can either use @code{integsubst} as described
23708 above to try to hint at a more direct path to the desired result, or
23709 you can use @code{IntegAfterRules}. This is an extra rule set that
23710 runs after the main integrator returns its result; basically, Calc does
23711 an @kbd{a r IntegAfterRules} on the result before showing it to you.
23712 (It also does algebraic simplifications, without @code{IntegSimpRules},
23713 after that to further simplify the result.) For example, Calc's integrator
23714 sometimes produces expressions of the form @samp{ln(1+x) - ln(1-x)};
23715 the default @code{IntegAfterRules} rewrite this into the more readable
23716 form @samp{2 arctanh(x)}. Note that, unlike @code{IntegRules},
23717 @code{IntegSimpRules} and @code{IntegAfterRules} are applied any number
23718 of times until no further changes are possible. Rewriting by
23719 @code{IntegAfterRules} occurs only after the main integrator has
23720 finished, not at every step as for @code{IntegRules} and
23721 @code{IntegSimpRules}.
23723 @node Numerical Integration, Taylor Series, Customizing the Integrator, Calculus
23724 @subsection Numerical Integration
23728 @pindex calc-num-integral
23730 If you want a purely numerical answer to an integration problem, you can
23731 use the @kbd{a I} (@code{calc-num-integral}) [@code{ninteg}] command. This
23732 command prompts for an integration variable, a lower limit, and an
23733 upper limit. Except for the integration variable, all other variables
23734 that appear in the integrand formula must have stored values. (A stored
23735 value, if any, for the integration variable itself is ignored.)
23737 Numerical integration works by evaluating your formula at many points in
23738 the specified interval. Calc uses an ``open Romberg'' method; this means
23739 that it does not evaluate the formula actually at the endpoints (so that
23740 it is safe to integrate @samp{sin(x)/x} from zero, for example). Also,
23741 the Romberg method works especially well when the function being
23742 integrated is fairly smooth. If the function is not smooth, Calc will
23743 have to evaluate it at quite a few points before it can accurately
23744 determine the value of the integral.
23746 Integration is much faster when the current precision is small. It is
23747 best to set the precision to the smallest acceptable number of digits
23748 before you use @kbd{a I}. If Calc appears to be taking too long, press
23749 @kbd{C-g} to halt it and try a lower precision. If Calc still appears
23750 to need hundreds of evaluations, check to make sure your function is
23751 well-behaved in the specified interval.
23753 It is possible for the lower integration limit to be @samp{-inf} (minus
23754 infinity). Likewise, the upper limit may be plus infinity. Calc
23755 internally transforms the integral into an equivalent one with finite
23756 limits. However, integration to or across singularities is not supported:
23757 The integral of @samp{1/sqrt(x)} from 0 to 1 exists (it can be found
23758 by Calc's symbolic integrator, for example), but @kbd{a I} will fail
23759 because the integrand goes to infinity at one of the endpoints.
23761 @node Taylor Series, , Numerical Integration, Calculus
23762 @subsection Taylor Series
23766 @pindex calc-taylor
23768 The @kbd{a t} (@code{calc-taylor}) [@code{taylor}] command computes a
23769 power series expansion or Taylor series of a function. You specify the
23770 variable and the desired number of terms. You may give an expression of
23771 the form @samp{@var{var} = @var{a}} or @samp{@var{var} - @var{a}} instead
23772 of just a variable to produce a Taylor expansion about the point @var{a}.
23773 You may specify the number of terms with a numeric prefix argument;
23774 otherwise the command will prompt you for the number of terms. Note that
23775 many series expansions have coefficients of zero for some terms, so you
23776 may appear to get fewer terms than you asked for.
23778 If the @kbd{a i} command is unable to find a symbolic integral for a
23779 function, you can get an approximation by integrating the function's
23782 @node Solving Equations, Numerical Solutions, Calculus, Algebra
23783 @section Solving Equations
23787 @pindex calc-solve-for
23789 @cindex Equations, solving
23790 @cindex Solving equations
23791 The @kbd{a S} (@code{calc-solve-for}) [@code{solve}] command rearranges
23792 an equation to solve for a specific variable. An equation is an
23793 expression of the form @expr{L = R}. For example, the command @kbd{a S x}
23794 will rearrange @expr{y = 3x + 6} to the form, @expr{x = y/3 - 2}. If the
23795 input is not an equation, it is treated like an equation of the
23798 This command also works for inequalities, as in @expr{y < 3x + 6}.
23799 Some inequalities cannot be solved where the analogous equation could
23800 be; for example, solving
23801 @texline @math{a < b \, c}
23802 @infoline @expr{a < b c}
23803 for @expr{b} is impossible
23804 without knowing the sign of @expr{c}. In this case, @kbd{a S} will
23806 @texline @math{b \mathbin{\hbox{\code{!=}}} a/c}
23807 @infoline @expr{b != a/c}
23808 (using the not-equal-to operator) to signify that the direction of the
23809 inequality is now unknown. The inequality
23810 @texline @math{a \le b \, c}
23811 @infoline @expr{a <= b c}
23812 is not even partially solved. @xref{Declarations}, for a way to tell
23813 Calc that the signs of the variables in a formula are in fact known.
23815 Two useful commands for working with the result of @kbd{a S} are
23816 @kbd{a .} (@pxref{Logical Operations}), which converts @expr{x = y/3 - 2}
23817 to @expr{y/3 - 2}, and @kbd{s l} (@pxref{Let Command}) which evaluates
23818 another formula with @expr{x} set equal to @expr{y/3 - 2}.
23821 * Multiple Solutions::
23822 * Solving Systems of Equations::
23823 * Decomposing Polynomials::
23826 @node Multiple Solutions, Solving Systems of Equations, Solving Equations, Solving Equations
23827 @subsection Multiple Solutions
23832 Some equations have more than one solution. The Hyperbolic flag
23833 (@code{H a S}) [@code{fsolve}] tells the solver to report the fully
23834 general family of solutions. It will invent variables @code{n1},
23835 @code{n2}, @dots{}, which represent independent arbitrary integers, and
23836 @code{s1}, @code{s2}, @dots{}, which represent independent arbitrary
23837 signs (either @mathit{+1} or @mathit{-1}). If you don't use the Hyperbolic
23838 flag, Calc will use zero in place of all arbitrary integers, and plus
23839 one in place of all arbitrary signs. Note that variables like @code{n1}
23840 and @code{s1} are not given any special interpretation in Calc except by
23841 the equation solver itself. As usual, you can use the @w{@kbd{s l}}
23842 (@code{calc-let}) command to obtain solutions for various actual values
23843 of these variables.
23845 For example, @kbd{' x^2 = y @key{RET} H a S x @key{RET}} solves to
23846 get @samp{x = s1 sqrt(y)}, indicating that the two solutions to the
23847 equation are @samp{sqrt(y)} and @samp{-sqrt(y)}. Another way to
23848 think about it is that the square-root operation is really a
23849 two-valued function; since every Calc function must return a
23850 single result, @code{sqrt} chooses to return the positive result.
23851 Then @kbd{H a S} doctors this result using @code{s1} to indicate
23852 the full set of possible values of the mathematical square-root.
23854 There is a similar phenomenon going the other direction: Suppose
23855 we solve @samp{sqrt(y) = x} for @code{y}. Calc squares both sides
23856 to get @samp{y = x^2}. This is correct, except that it introduces
23857 some dubious solutions. Consider solving @samp{sqrt(y) = -3}:
23858 Calc will report @expr{y = 9} as a valid solution, which is true
23859 in the mathematical sense of square-root, but false (there is no
23860 solution) for the actual Calc positive-valued @code{sqrt}. This
23861 happens for both @kbd{a S} and @kbd{H a S}.
23863 @cindex @code{GenCount} variable
23873 If you store a positive integer in the Calc variable @code{GenCount},
23874 then Calc will generate formulas of the form @samp{as(@var{n})} for
23875 arbitrary signs, and @samp{an(@var{n})} for arbitrary integers,
23876 where @var{n} represents successive values taken by incrementing
23877 @code{GenCount} by one. While the normal arbitrary sign and
23878 integer symbols start over at @code{s1} and @code{n1} with each
23879 new Calc command, the @code{GenCount} approach will give each
23880 arbitrary value a name that is unique throughout the entire Calc
23881 session. Also, the arbitrary values are function calls instead
23882 of variables, which is advantageous in some cases. For example,
23883 you can make a rewrite rule that recognizes all arbitrary signs
23884 using a pattern like @samp{as(n)}. The @kbd{s l} command only works
23885 on variables, but you can use the @kbd{a b} (@code{calc-substitute})
23886 command to substitute actual values for function calls like @samp{as(3)}.
23888 The @kbd{s G} (@code{calc-edit-GenCount}) command is a convenient
23889 way to create or edit this variable. Press @kbd{C-c C-c} to finish.
23891 If you have not stored a value in @code{GenCount}, or if the value
23892 in that variable is not a positive integer, the regular
23893 @code{s1}/@code{n1} notation is used.
23899 With the Inverse flag, @kbd{I a S} [@code{finv}] treats the expression
23900 on top of the stack as a function of the specified variable and solves
23901 to find the inverse function, written in terms of the same variable.
23902 For example, @kbd{I a S x} inverts @expr{2x + 6} to @expr{x/2 - 3}.
23903 You can use both Inverse and Hyperbolic [@code{ffinv}] to obtain a
23904 fully general inverse, as described above.
23907 @pindex calc-poly-roots
23909 Some equations, specifically polynomials, have a known, finite number
23910 of solutions. The @kbd{a P} (@code{calc-poly-roots}) [@code{roots}]
23911 command uses @kbd{H a S} to solve an equation in general form, then, for
23912 all arbitrary-sign variables like @code{s1}, and all arbitrary-integer
23913 variables like @code{n1} for which @code{n1} only usefully varies over
23914 a finite range, it expands these variables out to all their possible
23915 values. The results are collected into a vector, which is returned.
23916 For example, @samp{roots(x^4 = 1, x)} returns the four solutions
23917 @samp{[1, -1, (0, 1), (0, -1)]}. Generally an @var{n}th degree
23918 polynomial will always have @var{n} roots on the complex plane.
23919 (If you have given a @code{real} declaration for the solution
23920 variable, then only the real-valued solutions, if any, will be
23921 reported; @pxref{Declarations}.)
23923 Note that because @kbd{a P} uses @kbd{H a S}, it is able to deliver
23924 symbolic solutions if the polynomial has symbolic coefficients. Also
23925 note that Calc's solver is not able to get exact symbolic solutions
23926 to all polynomials. Polynomials containing powers up to @expr{x^4}
23927 can always be solved exactly; polynomials of higher degree sometimes
23928 can be: @expr{x^6 + x^3 + 1} is converted to @expr{(x^3)^2 + (x^3) + 1},
23929 which can be solved for @expr{x^3} using the quadratic equation, and then
23930 for @expr{x} by taking cube roots. But in many cases, like
23931 @expr{x^6 + x + 1}, Calc does not know how to rewrite the polynomial
23932 into a form it can solve. The @kbd{a P} command can still deliver a
23933 list of numerical roots, however, provided that Symbolic mode (@kbd{m s})
23934 is not turned on. (If you work with Symbolic mode on, recall that the
23935 @kbd{N} (@code{calc-eval-num}) key is a handy way to reevaluate the
23936 formula on the stack with Symbolic mode temporarily off.) Naturally,
23937 @kbd{a P} can only provide numerical roots if the polynomial coefficients
23938 are all numbers (real or complex).
23940 @node Solving Systems of Equations, Decomposing Polynomials, Multiple Solutions, Solving Equations
23941 @subsection Solving Systems of Equations
23944 @cindex Systems of equations, symbolic
23945 You can also use the commands described above to solve systems of
23946 simultaneous equations. Just create a vector of equations, then
23947 specify a vector of variables for which to solve. (You can omit
23948 the surrounding brackets when entering the vector of variables
23951 For example, putting @samp{[x + y = a, x - y = b]} on the stack
23952 and typing @kbd{a S x,y @key{RET}} produces the vector of solutions
23953 @samp{[x = a - (a-b)/2, y = (a-b)/2]}. The result vector will
23954 have the same length as the variables vector, and the variables
23955 will be listed in the same order there. Note that the solutions
23956 are not always simplified as far as possible; the solution for
23957 @expr{x} here could be improved by an application of the @kbd{a n}
23960 Calc's algorithm works by trying to eliminate one variable at a
23961 time by solving one of the equations for that variable and then
23962 substituting into the other equations. Calc will try all the
23963 possibilities, but you can speed things up by noting that Calc
23964 first tries to eliminate the first variable with the first
23965 equation, then the second variable with the second equation,
23966 and so on. It also helps to put the simpler (e.g., more linear)
23967 equations toward the front of the list. Calc's algorithm will
23968 solve any system of linear equations, and also many kinds of
23975 Normally there will be as many variables as equations. If you
23976 give fewer variables than equations (an ``over-determined'' system
23977 of equations), Calc will find a partial solution. For example,
23978 typing @kbd{a S y @key{RET}} with the above system of equations
23979 would produce @samp{[y = a - x]}. There are now several ways to
23980 express this solution in terms of the original variables; Calc uses
23981 the first one that it finds. You can control the choice by adding
23982 variable specifiers of the form @samp{elim(@var{v})} to the
23983 variables list. This says that @var{v} should be eliminated from
23984 the equations; the variable will not appear at all in the solution.
23985 For example, typing @kbd{a S y,elim(x)} would yield
23986 @samp{[y = a - (b+a)/2]}.
23988 If the variables list contains only @code{elim} specifiers,
23989 Calc simply eliminates those variables from the equations
23990 and then returns the resulting set of equations. For example,
23991 @kbd{a S elim(x)} produces @samp{[a - 2 y = b]}. Every variable
23992 eliminated will reduce the number of equations in the system
23995 Again, @kbd{a S} gives you one solution to the system of
23996 equations. If there are several solutions, you can use @kbd{H a S}
23997 to get a general family of solutions, or, if there is a finite
23998 number of solutions, you can use @kbd{a P} to get a list. (In
23999 the latter case, the result will take the form of a matrix where
24000 the rows are different solutions and the columns correspond to the
24001 variables you requested.)
24003 Another way to deal with certain kinds of overdetermined systems of
24004 equations is the @kbd{a F} command, which does least-squares fitting
24005 to satisfy the equations. @xref{Curve Fitting}.
24007 @node Decomposing Polynomials, , Solving Systems of Equations, Solving Equations
24008 @subsection Decomposing Polynomials
24015 The @code{poly} function takes a polynomial and a variable as
24016 arguments, and returns a vector of polynomial coefficients (constant
24017 coefficient first). For example, @samp{poly(x^3 + 2 x, x)} returns
24018 @expr{[0, 2, 0, 1]}. If the input is not a polynomial in @expr{x},
24019 the call to @code{poly} is left in symbolic form. If the input does
24020 not involve the variable @expr{x}, the input is returned in a list
24021 of length one, representing a polynomial with only a constant
24022 coefficient. The call @samp{poly(x, x)} returns the vector @expr{[0, 1]}.
24023 The last element of the returned vector is guaranteed to be nonzero;
24024 note that @samp{poly(0, x)} returns the empty vector @expr{[]}.
24025 Note also that @expr{x} may actually be any formula; for example,
24026 @samp{poly(sin(x)^2 - sin(x) + 3, sin(x))} returns @expr{[3, -1, 1]}.
24028 @cindex Coefficients of polynomial
24029 @cindex Degree of polynomial
24030 To get the @expr{x^k} coefficient of polynomial @expr{p}, use
24031 @samp{poly(p, x)_(k+1)}. To get the degree of polynomial @expr{p},
24032 use @samp{vlen(poly(p, x)) - 1}. For example, @samp{poly((x+1)^4, x)}
24033 returns @samp{[1, 4, 6, 4, 1]}, so @samp{poly((x+1)^4, x)_(2+1)}
24034 gives the @expr{x^2} coefficient of this polynomial, 6.
24040 One important feature of the solver is its ability to recognize
24041 formulas which are ``essentially'' polynomials. This ability is
24042 made available to the user through the @code{gpoly} function, which
24043 is used just like @code{poly}: @samp{gpoly(@var{expr}, @var{var})}.
24044 If @var{expr} is a polynomial in some term which includes @var{var}, then
24045 this function will return a vector @samp{[@var{x}, @var{c}, @var{a}]}
24046 where @var{x} is the term that depends on @var{var}, @var{c} is a
24047 vector of polynomial coefficients (like the one returned by @code{poly}),
24048 and @var{a} is a multiplier which is usually 1. Basically,
24049 @samp{@var{expr} = @var{a}*(@var{c}_1 + @var{c}_2 @var{x} +
24050 @var{c}_3 @var{x}^2 + ...)}. The last element of @var{c} is
24051 guaranteed to be non-zero, and @var{c} will not equal @samp{[1]}
24052 (i.e., the trivial decomposition @var{expr} = @var{x} is not
24053 considered a polynomial). One side effect is that @samp{gpoly(x, x)}
24054 and @samp{gpoly(6, x)}, both of which might be expected to recognize
24055 their arguments as polynomials, will not because the decomposition
24056 is considered trivial.
24058 For example, @samp{gpoly((x-2)^2, x)} returns @samp{[x, [4, -4, 1], 1]},
24059 since the expanded form of this polynomial is @expr{4 - 4 x + x^2}.
24061 The term @var{x} may itself be a polynomial in @var{var}. This is
24062 done to reduce the size of the @var{c} vector. For example,
24063 @samp{gpoly(x^4 + x^2 - 1, x)} returns @samp{[x^2, [-1, 1, 1], 1]},
24064 since a quadratic polynomial in @expr{x^2} is easier to solve than
24065 a quartic polynomial in @expr{x}.
24067 A few more examples of the kinds of polynomials @code{gpoly} can
24071 sin(x) - 1 [sin(x), [-1, 1], 1]
24072 x + 1/x - 1 [x, [1, -1, 1], 1/x]
24073 x + 1/x [x^2, [1, 1], 1/x]
24074 x^3 + 2 x [x^2, [2, 1], x]
24075 x + x^2:3 + sqrt(x) [x^1:6, [1, 1, 0, 1], x^1:2]
24076 x^(2a) + 2 x^a + 5 [x^a, [5, 2, 1], 1]
24077 (exp(-x) + exp(x)) / 2 [e^(2 x), [0.5, 0.5], e^-x]
24080 The @code{poly} and @code{gpoly} functions accept a third integer argument
24081 which specifies the largest degree of polynomial that is acceptable.
24082 If this is @expr{n}, then only @var{c} vectors of length @expr{n+1}
24083 or less will be returned. Otherwise, the @code{poly} or @code{gpoly}
24084 call will remain in symbolic form. For example, the equation solver
24085 can handle quartics and smaller polynomials, so it calls
24086 @samp{gpoly(@var{expr}, @var{var}, 4)} to discover whether @var{expr}
24087 can be treated by its linear, quadratic, cubic, or quartic formulas.
24093 The @code{pdeg} function computes the degree of a polynomial;
24094 @samp{pdeg(p,x)} is the highest power of @code{x} that appears in
24095 @code{p}. This is the same as @samp{vlen(poly(p,x))-1}, but is
24096 much more efficient. If @code{p} is constant with respect to @code{x},
24097 then @samp{pdeg(p,x) = 0}. If @code{p} is not a polynomial in @code{x}
24098 (e.g., @samp{pdeg(2 cos(x), x)}, the function remains unevaluated.
24099 It is possible to omit the second argument @code{x}, in which case
24100 @samp{pdeg(p)} returns the highest total degree of any term of the
24101 polynomial, counting all variables that appear in @code{p}. Note
24102 that @code{pdeg(c) = pdeg(c,x) = 0} for any nonzero constant @code{c};
24103 the degree of the constant zero is considered to be @code{-inf}
24110 The @code{plead} function finds the leading term of a polynomial.
24111 Thus @samp{plead(p,x)} is equivalent to @samp{poly(p,x)_vlen(poly(p,x))},
24112 though again more efficient. In particular, @samp{plead((2x+1)^10, x)}
24113 returns 1024 without expanding out the list of coefficients. The
24114 value of @code{plead(p,x)} will be zero only if @expr{p = 0}.
24120 The @code{pcont} function finds the @dfn{content} of a polynomial. This
24121 is the greatest common divisor of all the coefficients of the polynomial.
24122 With two arguments, @code{pcont(p,x)} effectively uses @samp{poly(p,x)}
24123 to get a list of coefficients, then uses @code{pgcd} (the polynomial
24124 GCD function) to combine these into an answer. For example,
24125 @samp{pcont(4 x y^2 + 6 x^2 y, x)} is @samp{2 y}. The content is
24126 basically the ``biggest'' polynomial that can be divided into @code{p}
24127 exactly. The sign of the content is the same as the sign of the leading
24130 With only one argument, @samp{pcont(p)} computes the numerical
24131 content of the polynomial, i.e., the @code{gcd} of the numerical
24132 coefficients of all the terms in the formula. Note that @code{gcd}
24133 is defined on rational numbers as well as integers; it computes
24134 the @code{gcd} of the numerators and the @code{lcm} of the
24135 denominators. Thus @samp{pcont(4:3 x y^2 + 6 x^2 y)} returns 2:3.
24136 Dividing the polynomial by this number will clear all the
24137 denominators, as well as dividing by any common content in the
24138 numerators. The numerical content of a polynomial is negative only
24139 if all the coefficients in the polynomial are negative.
24145 The @code{pprim} function finds the @dfn{primitive part} of a
24146 polynomial, which is simply the polynomial divided (using @code{pdiv}
24147 if necessary) by its content. If the input polynomial has rational
24148 coefficients, the result will have integer coefficients in simplest
24151 @node Numerical Solutions, Curve Fitting, Solving Equations, Algebra
24152 @section Numerical Solutions
24155 Not all equations can be solved symbolically. The commands in this
24156 section use numerical algorithms that can find a solution to a specific
24157 instance of an equation to any desired accuracy. Note that the
24158 numerical commands are slower than their algebraic cousins; it is a
24159 good idea to try @kbd{a S} before resorting to these commands.
24161 (@xref{Curve Fitting}, for some other, more specialized, operations
24162 on numerical data.)
24167 * Numerical Systems of Equations::
24170 @node Root Finding, Minimization, Numerical Solutions, Numerical Solutions
24171 @subsection Root Finding
24175 @pindex calc-find-root
24177 @cindex Newton's method
24178 @cindex Roots of equations
24179 @cindex Numerical root-finding
24180 The @kbd{a R} (@code{calc-find-root}) [@code{root}] command finds a
24181 numerical solution (or @dfn{root}) of an equation. (This command treats
24182 inequalities the same as equations. If the input is any other kind
24183 of formula, it is interpreted as an equation of the form @expr{X = 0}.)
24185 The @kbd{a R} command requires an initial guess on the top of the
24186 stack, and a formula in the second-to-top position. It prompts for a
24187 solution variable, which must appear in the formula. All other variables
24188 that appear in the formula must have assigned values, i.e., when
24189 a value is assigned to the solution variable and the formula is
24190 evaluated with @kbd{=}, it should evaluate to a number. Any assigned
24191 value for the solution variable itself is ignored and unaffected by
24194 When the command completes, the initial guess is replaced on the stack
24195 by a vector of two numbers: The value of the solution variable that
24196 solves the equation, and the difference between the lefthand and
24197 righthand sides of the equation at that value. Ordinarily, the second
24198 number will be zero or very nearly zero. (Note that Calc uses a
24199 slightly higher precision while finding the root, and thus the second
24200 number may be slightly different from the value you would compute from
24201 the equation yourself.)
24203 The @kbd{v h} (@code{calc-head}) command is a handy way to extract
24204 the first element of the result vector, discarding the error term.
24206 The initial guess can be a real number, in which case Calc searches
24207 for a real solution near that number, or a complex number, in which
24208 case Calc searches the whole complex plane near that number for a
24209 solution, or it can be an interval form which restricts the search
24210 to real numbers inside that interval.
24212 Calc tries to use @kbd{a d} to take the derivative of the equation.
24213 If this succeeds, it uses Newton's method. If the equation is not
24214 differentiable Calc uses a bisection method. (If Newton's method
24215 appears to be going astray, Calc switches over to bisection if it
24216 can, or otherwise gives up. In this case it may help to try again
24217 with a slightly different initial guess.) If the initial guess is a
24218 complex number, the function must be differentiable.
24220 If the formula (or the difference between the sides of an equation)
24221 is negative at one end of the interval you specify and positive at
24222 the other end, the root finder is guaranteed to find a root.
24223 Otherwise, Calc subdivides the interval into small parts looking for
24224 positive and negative values to bracket the root. When your guess is
24225 an interval, Calc will not look outside that interval for a root.
24229 The @kbd{H a R} [@code{wroot}] command is similar to @kbd{a R}, except
24230 that if the initial guess is an interval for which the function has
24231 the same sign at both ends, then rather than subdividing the interval
24232 Calc attempts to widen it to enclose a root. Use this mode if
24233 you are not sure if the function has a root in your interval.
24235 If the function is not differentiable, and you give a simple number
24236 instead of an interval as your initial guess, Calc uses this widening
24237 process even if you did not type the Hyperbolic flag. (If the function
24238 @emph{is} differentiable, Calc uses Newton's method which does not
24239 require a bounding interval in order to work.)
24241 If Calc leaves the @code{root} or @code{wroot} function in symbolic
24242 form on the stack, it will normally display an explanation for why
24243 no root was found. If you miss this explanation, press @kbd{w}
24244 (@code{calc-why}) to get it back.
24246 @node Minimization, Numerical Systems of Equations, Root Finding, Numerical Solutions
24247 @subsection Minimization
24254 @pindex calc-find-minimum
24255 @pindex calc-find-maximum
24258 @cindex Minimization, numerical
24259 The @kbd{a N} (@code{calc-find-minimum}) [@code{minimize}] command
24260 finds a minimum value for a formula. It is very similar in operation
24261 to @kbd{a R} (@code{calc-find-root}): You give the formula and an initial
24262 guess on the stack, and are prompted for the name of a variable. The guess
24263 may be either a number near the desired minimum, or an interval enclosing
24264 the desired minimum. The function returns a vector containing the
24265 value of the variable which minimizes the formula's value, along
24266 with the minimum value itself.
24268 Note that this command looks for a @emph{local} minimum. Many functions
24269 have more than one minimum; some, like
24270 @texline @math{x \sin x},
24271 @infoline @expr{x sin(x)},
24272 have infinitely many. In fact, there is no easy way to define the
24273 ``global'' minimum of
24274 @texline @math{x \sin x}
24275 @infoline @expr{x sin(x)}
24276 but Calc can still locate any particular local minimum
24277 for you. Calc basically goes downhill from the initial guess until it
24278 finds a point at which the function's value is greater both to the left
24279 and to the right. Calc does not use derivatives when minimizing a function.
24281 If your initial guess is an interval and it looks like the minimum
24282 occurs at one or the other endpoint of the interval, Calc will return
24283 that endpoint only if that endpoint is closed; thus, minimizing @expr{17 x}
24284 over @expr{[2..3]} will return @expr{[2, 38]}, but minimizing over
24285 @expr{(2..3]} would report no minimum found. In general, you should
24286 use closed intervals to find literally the minimum value in that
24287 range of @expr{x}, or open intervals to find the local minimum, if
24288 any, that happens to lie in that range.
24290 Most functions are smooth and flat near their minimum values. Because
24291 of this flatness, if the current precision is, say, 12 digits, the
24292 variable can only be determined meaningfully to about six digits. Thus
24293 you should set the precision to twice as many digits as you need in your
24304 The @kbd{H a N} [@code{wminimize}] command, analogously to @kbd{H a R},
24305 expands the guess interval to enclose a minimum rather than requiring
24306 that the minimum lie inside the interval you supply.
24308 The @kbd{a X} (@code{calc-find-maximum}) [@code{maximize}] and
24309 @kbd{H a X} [@code{wmaximize}] commands effectively minimize the
24310 negative of the formula you supply.
24312 The formula must evaluate to a real number at all points inside the
24313 interval (or near the initial guess if the guess is a number). If
24314 the initial guess is a complex number the variable will be minimized
24315 over the complex numbers; if it is real or an interval it will
24316 be minimized over the reals.
24318 @node Numerical Systems of Equations, , Minimization, Numerical Solutions
24319 @subsection Systems of Equations
24322 @cindex Systems of equations, numerical
24323 The @kbd{a R} command can also solve systems of equations. In this
24324 case, the equation should instead be a vector of equations, the
24325 guess should instead be a vector of numbers (intervals are not
24326 supported), and the variable should be a vector of variables. You
24327 can omit the brackets while entering the list of variables. Each
24328 equation must be differentiable by each variable for this mode to
24329 work. The result will be a vector of two vectors: The variable
24330 values that solved the system of equations, and the differences
24331 between the sides of the equations with those variable values.
24332 There must be the same number of equations as variables. Since
24333 only plain numbers are allowed as guesses, the Hyperbolic flag has
24334 no effect when solving a system of equations.
24336 It is also possible to minimize over many variables with @kbd{a N}
24337 (or maximize with @kbd{a X}). Once again the variable name should
24338 be replaced by a vector of variables, and the initial guess should
24339 be an equal-sized vector of initial guesses. But, unlike the case of
24340 multidimensional @kbd{a R}, the formula being minimized should
24341 still be a single formula, @emph{not} a vector. Beware that
24342 multidimensional minimization is currently @emph{very} slow.
24344 @node Curve Fitting, Summations, Numerical Solutions, Algebra
24345 @section Curve Fitting
24348 The @kbd{a F} command fits a set of data to a @dfn{model formula},
24349 such as @expr{y = m x + b} where @expr{m} and @expr{b} are parameters
24350 to be determined. For a typical set of measured data there will be
24351 no single @expr{m} and @expr{b} that exactly fit the data; in this
24352 case, Calc chooses values of the parameters that provide the closest
24353 possible fit. The model formula can be entered in various ways after
24354 the key sequence @kbd{a F} is pressed.
24356 If the letter @kbd{P} is pressed after @kbd{a F} but before the model
24357 description is entered, the data as well as the model formula will be
24358 plotted after the formula is determined. This will be indicated by a
24359 ``P'' in the minibuffer after the help message.
24363 * Polynomial and Multilinear Fits::
24364 * Error Estimates for Fits::
24365 * Standard Nonlinear Models::
24366 * Curve Fitting Details::
24370 @node Linear Fits, Polynomial and Multilinear Fits, Curve Fitting, Curve Fitting
24371 @subsection Linear Fits
24375 @pindex calc-curve-fit
24377 @cindex Linear regression
24378 @cindex Least-squares fits
24379 The @kbd{a F} (@code{calc-curve-fit}) [@code{fit}] command attempts
24380 to fit a set of data (@expr{x} and @expr{y} vectors of numbers) to a
24381 straight line, polynomial, or other function of @expr{x}. For the
24382 moment we will consider only the case of fitting to a line, and we
24383 will ignore the issue of whether or not the model was in fact a good
24386 In a standard linear least-squares fit, we have a set of @expr{(x,y)}
24387 data points that we wish to fit to the model @expr{y = m x + b}
24388 by adjusting the parameters @expr{m} and @expr{b} to make the @expr{y}
24389 values calculated from the formula be as close as possible to the actual
24390 @expr{y} values in the data set. (In a polynomial fit, the model is
24391 instead, say, @expr{y = a x^3 + b x^2 + c x + d}. In a multilinear fit,
24392 we have data points of the form @expr{(x_1,x_2,x_3,y)} and our model is
24393 @expr{y = a x_1 + b x_2 + c x_3 + d}. These will be discussed later.)
24395 In the model formula, variables like @expr{x} and @expr{x_2} are called
24396 the @dfn{independent variables}, and @expr{y} is the @dfn{dependent
24397 variable}. Variables like @expr{m}, @expr{a}, and @expr{b} are called
24398 the @dfn{parameters} of the model.
24400 The @kbd{a F} command takes the data set to be fitted from the stack.
24401 By default, it expects the data in the form of a matrix. For example,
24402 for a linear or polynomial fit, this would be a
24403 @texline @math{2\times N}
24405 matrix where the first row is a list of @expr{x} values and the second
24406 row has the corresponding @expr{y} values. For the multilinear fit
24407 shown above, the matrix would have four rows (@expr{x_1}, @expr{x_2},
24408 @expr{x_3}, and @expr{y}, respectively).
24410 If you happen to have an
24411 @texline @math{N\times2}
24413 matrix instead of a
24414 @texline @math{2\times N}
24416 matrix, just press @kbd{v t} first to transpose the matrix.
24418 After you type @kbd{a F}, Calc prompts you to select a model. For a
24419 linear fit, press the digit @kbd{1}.
24421 Calc then prompts for you to name the variables. By default it chooses
24422 high letters like @expr{x} and @expr{y} for independent variables and
24423 low letters like @expr{a} and @expr{b} for parameters. (The dependent
24424 variable doesn't need a name.) The two kinds of variables are separated
24425 by a semicolon. Since you generally care more about the names of the
24426 independent variables than of the parameters, Calc also allows you to
24427 name only those and let the parameters use default names.
24429 For example, suppose the data matrix
24434 [ [ 1, 2, 3, 4, 5 ]
24435 [ 5, 7, 9, 11, 13 ] ]
24441 $$ \pmatrix{ 1 & 2 & 3 & 4 & 5 \cr
24442 5 & 7 & 9 & 11 & 13 }
24448 is on the stack and we wish to do a simple linear fit. Type
24449 @kbd{a F}, then @kbd{1} for the model, then @key{RET} to use
24450 the default names. The result will be the formula @expr{3. + 2. x}
24451 on the stack. Calc has created the model expression @kbd{a + b x},
24452 then found the optimal values of @expr{a} and @expr{b} to fit the
24453 data. (In this case, it was able to find an exact fit.) Calc then
24454 substituted those values for @expr{a} and @expr{b} in the model
24457 The @kbd{a F} command puts two entries in the trail. One is, as
24458 always, a copy of the result that went to the stack; the other is
24459 a vector of the actual parameter values, written as equations:
24460 @expr{[a = 3, b = 2]}, in case you'd rather read them in a list
24461 than pick them out of the formula. (You can type @kbd{t y}
24462 to move this vector to the stack; see @ref{Trail Commands}.
24464 Specifying a different independent variable name will affect the
24465 resulting formula: @kbd{a F 1 k @key{RET}} produces @kbd{3 + 2 k}.
24466 Changing the parameter names (say, @kbd{a F 1 k;b,m @key{RET}}) will affect
24467 the equations that go into the trail.
24473 To see what happens when the fit is not exact, we could change
24474 the number 13 in the data matrix to 14 and try the fit again.
24481 Evaluating this formula, say with @kbd{v x 5 @key{RET} @key{TAB} V M $ @key{RET}}, shows
24482 a reasonably close match to the y-values in the data.
24485 [4.8, 7., 9.2, 11.4, 13.6]
24488 Since there is no line which passes through all the @var{n} data points,
24489 Calc has chosen a line that best approximates the data points using
24490 the method of least squares. The idea is to define the @dfn{chi-square}
24495 chi^2 = sum((y_i - (a + b x_i))^2, i, 1, N)
24500 $$ \chi^2 = \sum_{i=1}^N (y_i - (a + b x_i))^2 $$
24505 which is clearly zero if @expr{a + b x} exactly fits all data points,
24506 and increases as various @expr{a + b x_i} values fail to match the
24507 corresponding @expr{y_i} values. There are several reasons why the
24508 summand is squared, one of them being to ensure that
24509 @texline @math{\chi^2 \ge 0}.
24510 @infoline @expr{chi^2 >= 0}.
24511 Least-squares fitting simply chooses the values of @expr{a} and @expr{b}
24512 for which the error
24513 @texline @math{\chi^2}
24514 @infoline @expr{chi^2}
24515 is as small as possible.
24517 Other kinds of models do the same thing but with a different model
24518 formula in place of @expr{a + b x_i}.
24524 A numeric prefix argument causes the @kbd{a F} command to take the
24525 data in some other form than one big matrix. A positive argument @var{n}
24526 will take @var{N} items from the stack, corresponding to the @var{n} rows
24527 of a data matrix. In the linear case, @var{n} must be 2 since there
24528 is always one independent variable and one dependent variable.
24530 A prefix of zero or plain @kbd{C-u} is a compromise; Calc takes two
24531 items from the stack, an @var{n}-row matrix of @expr{x} values, and a
24532 vector of @expr{y} values. If there is only one independent variable,
24533 the @expr{x} values can be either a one-row matrix or a plain vector,
24534 in which case the @kbd{C-u} prefix is the same as a @w{@kbd{C-u 2}} prefix.
24536 @node Polynomial and Multilinear Fits, Error Estimates for Fits, Linear Fits, Curve Fitting
24537 @subsection Polynomial and Multilinear Fits
24540 To fit the data to higher-order polynomials, just type one of the
24541 digits @kbd{2} through @kbd{9} when prompted for a model. For example,
24542 we could fit the original data matrix from the previous section
24543 (with 13, not 14) to a parabola instead of a line by typing
24544 @kbd{a F 2 @key{RET}}.
24547 2.00000000001 x - 1.5e-12 x^2 + 2.99999999999
24550 Note that since the constant and linear terms are enough to fit the
24551 data exactly, it's no surprise that Calc chose a tiny contribution
24552 for @expr{x^2}. (The fact that it's not exactly zero is due only
24553 to roundoff error. Since our data are exact integers, we could get
24554 an exact answer by typing @kbd{m f} first to get Fraction mode.
24555 Then the @expr{x^2} term would vanish altogether. Usually, though,
24556 the data being fitted will be approximate floats so Fraction mode
24559 Doing the @kbd{a F 2} fit on the data set with 14 instead of 13
24560 gives a much larger @expr{x^2} contribution, as Calc bends the
24561 line slightly to improve the fit.
24564 0.142857142855 x^2 + 1.34285714287 x + 3.59999999998
24567 An important result from the theory of polynomial fitting is that it
24568 is always possible to fit @var{n} data points exactly using a polynomial
24569 of degree @mathit{@var{n}-1}, sometimes called an @dfn{interpolating polynomial}.
24570 Using the modified (14) data matrix, a model number of 4 gives
24571 a polynomial that exactly matches all five data points:
24574 0.04167 x^4 - 0.4167 x^3 + 1.458 x^2 - 0.08333 x + 4.
24577 The actual coefficients we get with a precision of 12, like
24578 @expr{0.0416666663588}, clearly suffer from loss of precision.
24579 It is a good idea to increase the working precision to several
24580 digits beyond what you need when you do a fitting operation.
24581 Or, if your data are exact, use Fraction mode to get exact
24584 You can type @kbd{i} instead of a digit at the model prompt to fit
24585 the data exactly to a polynomial. This just counts the number of
24586 columns of the data matrix to choose the degree of the polynomial
24589 Fitting data ``exactly'' to high-degree polynomials is not always
24590 a good idea, though. High-degree polynomials have a tendency to
24591 wiggle uncontrollably in between the fitting data points. Also,
24592 if the exact-fit polynomial is going to be used to interpolate or
24593 extrapolate the data, it is numerically better to use the @kbd{a p}
24594 command described below. @xref{Interpolation}.
24600 Another generalization of the linear model is to assume the
24601 @expr{y} values are a sum of linear contributions from several
24602 @expr{x} values. This is a @dfn{multilinear} fit, and it is also
24603 selected by the @kbd{1} digit key. (Calc decides whether the fit
24604 is linear or multilinear by counting the rows in the data matrix.)
24606 Given the data matrix,
24610 [ [ 1, 2, 3, 4, 5 ]
24612 [ 14.5, 15, 18.5, 22.5, 24 ] ]
24617 the command @kbd{a F 1 @key{RET}} will call the first row @expr{x} and the
24618 second row @expr{y}, and will fit the values in the third row to the
24619 model @expr{a + b x + c y}.
24625 Calc can do multilinear fits with any number of independent variables
24626 (i.e., with any number of data rows).
24632 Yet another variation is @dfn{homogeneous} linear models, in which
24633 the constant term is known to be zero. In the linear case, this
24634 means the model formula is simply @expr{a x}; in the multilinear
24635 case, the model might be @expr{a x + b y + c z}; and in the polynomial
24636 case, the model could be @expr{a x + b x^2 + c x^3}. You can get
24637 a homogeneous linear or multilinear model by pressing the letter
24638 @kbd{h} followed by a regular model key, like @kbd{1} or @kbd{2}.
24639 This will be indicated by an ``h'' in the minibuffer after the help
24642 It is certainly possible to have other constrained linear models,
24643 like @expr{2.3 + a x} or @expr{a - 4 x}. While there is no single
24644 key to select models like these, a later section shows how to enter
24645 any desired model by hand. In the first case, for example, you
24646 would enter @kbd{a F ' 2.3 + a x}.
24648 Another class of models that will work but must be entered by hand
24649 are multinomial fits, e.g., @expr{a + b x + c y + d x^2 + e y^2 + f x y}.
24651 @node Error Estimates for Fits, Standard Nonlinear Models, Polynomial and Multilinear Fits, Curve Fitting
24652 @subsection Error Estimates for Fits
24657 With the Hyperbolic flag, @kbd{H a F} [@code{efit}] performs the same
24658 fitting operation as @kbd{a F}, but reports the coefficients as error
24659 forms instead of plain numbers. Fitting our two data matrices (first
24660 with 13, then with 14) to a line with @kbd{H a F} gives the results,
24664 2.6 +/- 0.382970843103 + 2.2 +/- 0.115470053838 x
24667 In the first case the estimated errors are zero because the linear
24668 fit is perfect. In the second case, the errors are nonzero but
24669 moderately small, because the data are still very close to linear.
24671 It is also possible for the @emph{input} to a fitting operation to
24672 contain error forms. The data values must either all include errors
24673 or all be plain numbers. Error forms can go anywhere but generally
24674 go on the numbers in the last row of the data matrix. If the last
24675 row contains error forms
24676 @texline `@var{y_i}@w{ @tfn{+/-} }@math{\sigma_i}',
24677 @infoline `@var{y_i}@w{ @tfn{+/-} }@var{sigma_i}',
24679 @texline @math{\chi^2}
24680 @infoline @expr{chi^2}
24685 chi^2 = sum(((y_i - (a + b x_i)) / sigma_i)^2, i, 1, N)
24690 $$ \chi^2 = \sum_{i=1}^N \left(y_i - (a + b x_i) \over \sigma_i\right)^2 $$
24695 so that data points with larger error estimates contribute less to
24696 the fitting operation.
24698 If there are error forms on other rows of the data matrix, all the
24699 errors for a given data point are combined; the square root of the
24700 sum of the squares of the errors forms the
24701 @texline @math{\sigma_i}
24702 @infoline @expr{sigma_i}
24703 used for the data point.
24705 Both @kbd{a F} and @kbd{H a F} can accept error forms in the input
24706 matrix, although if you are concerned about error analysis you will
24707 probably use @kbd{H a F} so that the output also contains error
24710 If the input contains error forms but all the
24711 @texline @math{\sigma_i}
24712 @infoline @expr{sigma_i}
24713 values are the same, it is easy to see that the resulting fitted model
24714 will be the same as if the input did not have error forms at all
24715 @texline (@math{\chi^2}
24716 @infoline (@expr{chi^2}
24717 is simply scaled uniformly by
24718 @texline @math{1 / \sigma^2},
24719 @infoline @expr{1 / sigma^2},
24720 which doesn't affect where it has a minimum). But there @emph{will} be
24721 a difference in the estimated errors of the coefficients reported by
24724 Consult any text on statistical modeling of data for a discussion
24725 of where these error estimates come from and how they should be
24734 With the Inverse flag, @kbd{I a F} [@code{xfit}] produces even more
24735 information. The result is a vector of six items:
24739 The model formula with error forms for its coefficients or
24740 parameters. This is the result that @kbd{H a F} would have
24744 A vector of ``raw'' parameter values for the model. These are the
24745 polynomial coefficients or other parameters as plain numbers, in the
24746 same order as the parameters appeared in the final prompt of the
24747 @kbd{I a F} command. For polynomials of degree @expr{d}, this vector
24748 will have length @expr{M = d+1} with the constant term first.
24751 The covariance matrix @expr{C} computed from the fit. This is
24752 an @var{m}x@var{m} symmetric matrix; the diagonal elements
24753 @texline @math{C_{jj}}
24754 @infoline @expr{C_j_j}
24756 @texline @math{\sigma_j^2}
24757 @infoline @expr{sigma_j^2}
24758 of the parameters. The other elements are covariances
24759 @texline @math{\sigma_{ij}^2}
24760 @infoline @expr{sigma_i_j^2}
24761 that describe the correlation between pairs of parameters. (A related
24762 set of numbers, the @dfn{linear correlation coefficients}
24763 @texline @math{r_{ij}},
24764 @infoline @expr{r_i_j},
24766 @texline @math{\sigma_{ij}^2 / \sigma_i \, \sigma_j}.)
24767 @infoline @expr{sigma_i_j^2 / sigma_i sigma_j}.)
24770 A vector of @expr{M} ``parameter filter'' functions whose
24771 meanings are described below. If no filters are necessary this
24772 will instead be an empty vector; this is always the case for the
24773 polynomial and multilinear fits described so far.
24777 @texline @math{\chi^2}
24778 @infoline @expr{chi^2}
24779 for the fit, calculated by the formulas shown above. This gives a
24780 measure of the quality of the fit; statisticians consider
24781 @texline @math{\chi^2 \approx N - M}
24782 @infoline @expr{chi^2 = N - M}
24783 to indicate a moderately good fit (where again @expr{N} is the number of
24784 data points and @expr{M} is the number of parameters).
24787 A measure of goodness of fit expressed as a probability @expr{Q}.
24788 This is computed from the @code{utpc} probability distribution
24790 @texline @math{\chi^2}
24791 @infoline @expr{chi^2}
24792 with @expr{N - M} degrees of freedom. A
24793 value of 0.5 implies a good fit; some texts recommend that often
24794 @expr{Q = 0.1} or even 0.001 can signify an acceptable fit. In
24796 @texline @math{\chi^2}
24797 @infoline @expr{chi^2}
24798 statistics assume the errors in your inputs
24799 follow a normal (Gaussian) distribution; if they don't, you may
24800 have to accept smaller values of @expr{Q}.
24802 The @expr{Q} value is computed only if the input included error
24803 estimates. Otherwise, Calc will report the symbol @code{nan}
24804 for @expr{Q}. The reason is that in this case the
24805 @texline @math{\chi^2}
24806 @infoline @expr{chi^2}
24807 value has effectively been used to estimate the original errors
24808 in the input, and thus there is no redundant information left
24809 over to use for a confidence test.
24812 @node Standard Nonlinear Models, Curve Fitting Details, Error Estimates for Fits, Curve Fitting
24813 @subsection Standard Nonlinear Models
24816 The @kbd{a F} command also accepts other kinds of models besides
24817 lines and polynomials. Some common models have quick single-key
24818 abbreviations; others must be entered by hand as algebraic formulas.
24820 Here is a complete list of the standard models recognized by @kbd{a F}:
24824 Linear or multilinear. @mathit{a + b x + c y + d z}.
24826 Polynomials. @mathit{a + b x + c x^2 + d x^3}.
24828 Exponential. @mathit{a} @tfn{exp}@mathit{(b x)} @tfn{exp}@mathit{(c y)}.
24830 Base-10 exponential. @mathit{a} @tfn{10^}@mathit{(b x)} @tfn{10^}@mathit{(c y)}.
24832 Exponential (alternate notation). @tfn{exp}@mathit{(a + b x + c y)}.
24834 Base-10 exponential (alternate). @tfn{10^}@mathit{(a + b x + c y)}.
24836 Logarithmic. @mathit{a + b} @tfn{ln}@mathit{(x) + c} @tfn{ln}@mathit{(y)}.
24838 Base-10 logarithmic. @mathit{a + b} @tfn{log10}@mathit{(x) + c} @tfn{log10}@mathit{(y)}.
24840 General exponential. @mathit{a b^x c^y}.
24842 Power law. @mathit{a x^b y^c}.
24844 Quadratic. @mathit{a + b (x-c)^2 + d (x-e)^2}.
24847 @texline @math{{a \over b \sqrt{2 \pi}} \exp\left( -{1 \over 2} \left( x - c \over b \right)^2 \right)}.
24848 @infoline @mathit{(a / b sqrt(2 pi)) exp(-0.5*((x-c)/b)^2)}.
24850 Logistic @emph{s} curve.
24851 @texline @math{a/(1+e^{b(x-c)})}.
24852 @infoline @mathit{a/(1 + exp(b (x - c)))}.
24854 Logistic bell curve.
24855 @texline @math{ae^{b(x-c)}/(1+e^{b(x-c)})^2}.
24856 @infoline @mathit{a exp(b (x - c))/(1 + exp(b (x - c)))^2}.
24858 Hubbert linearization.
24859 @texline @math{{y \over x} = a(1-x/b)}.
24860 @infoline @mathit{(y/x) = a (1 - x/b)}.
24863 All of these models are used in the usual way; just press the appropriate
24864 letter at the model prompt, and choose variable names if you wish. The
24865 result will be a formula as shown in the above table, with the best-fit
24866 values of the parameters substituted. (You may find it easier to read
24867 the parameter values from the vector that is placed in the trail.)
24869 All models except Gaussian, logistics, Hubbert and polynomials can
24870 generalize as shown to any number of independent variables. Also, all
24871 the built-in models except for the logistic and Hubbert curves have an
24872 additive or multiplicative parameter shown as @expr{a} in the above table
24873 which can be replaced by zero or one, as appropriate, by typing @kbd{h}
24874 before the model key.
24876 Note that many of these models are essentially equivalent, but express
24877 the parameters slightly differently. For example, @expr{a b^x} and
24878 the other two exponential models are all algebraic rearrangements of
24879 each other. Also, the ``quadratic'' model is just a degree-2 polynomial
24880 with the parameters expressed differently. Use whichever form best
24881 matches the problem.
24883 The HP-28/48 calculators support four different models for curve
24884 fitting, called @code{LIN}, @code{LOG}, @code{EXP}, and @code{PWR}.
24885 These correspond to Calc models @samp{a + b x}, @samp{a + b ln(x)},
24886 @samp{a exp(b x)}, and @samp{a x^b}, respectively. In each case,
24887 @expr{a} is what the HP-48 identifies as the ``intercept,'' and
24888 @expr{b} is what it calls the ``slope.''
24894 If the model you want doesn't appear on this list, press @kbd{'}
24895 (the apostrophe key) at the model prompt to enter any algebraic
24896 formula, such as @kbd{m x - b}, as the model. (Not all models
24897 will work, though---see the next section for details.)
24899 The model can also be an equation like @expr{y = m x + b}.
24900 In this case, Calc thinks of all the rows of the data matrix on
24901 equal terms; this model effectively has two parameters
24902 (@expr{m} and @expr{b}) and two independent variables (@expr{x}
24903 and @expr{y}), with no ``dependent'' variables. Model equations
24904 do not need to take this @expr{y =} form. For example, the
24905 implicit line equation @expr{a x + b y = 1} works fine as a
24908 When you enter a model, Calc makes an alphabetical list of all
24909 the variables that appear in the model. These are used for the
24910 default parameters, independent variables, and dependent variable
24911 (in that order). If you enter a plain formula (not an equation),
24912 Calc assumes the dependent variable does not appear in the formula
24913 and thus does not need a name.
24915 For example, if the model formula has the variables @expr{a,mu,sigma,t,x},
24916 and the data matrix has three rows (meaning two independent variables),
24917 Calc will use @expr{a,mu,sigma} as the default parameters, and the
24918 data rows will be named @expr{t} and @expr{x}, respectively. If you
24919 enter an equation instead of a plain formula, Calc will use @expr{a,mu}
24920 as the parameters, and @expr{sigma,t,x} as the three independent
24923 You can, of course, override these choices by entering something
24924 different at the prompt. If you leave some variables out of the list,
24925 those variables must have stored values and those stored values will
24926 be used as constants in the model. (Stored values for the parameters
24927 and independent variables are ignored by the @kbd{a F} command.)
24928 If you list only independent variables, all the remaining variables
24929 in the model formula will become parameters.
24931 If there are @kbd{$} signs in the model you type, they will stand
24932 for parameters and all other variables (in alphabetical order)
24933 will be independent. Use @kbd{$} for one parameter, @kbd{$$} for
24934 another, and so on. Thus @kbd{$ x + $$} is another way to describe
24937 If you type a @kbd{$} instead of @kbd{'} at the model prompt itself,
24938 Calc will take the model formula from the stack. (The data must then
24939 appear at the second stack level.) The same conventions are used to
24940 choose which variables in the formula are independent by default and
24941 which are parameters.
24943 Models taken from the stack can also be expressed as vectors of
24944 two or three elements, @expr{[@var{model}, @var{vars}]} or
24945 @expr{[@var{model}, @var{vars}, @var{params}]}. Each of @var{vars}
24946 and @var{params} may be either a variable or a vector of variables.
24947 (If @var{params} is omitted, all variables in @var{model} except
24948 those listed as @var{vars} are parameters.)
24950 When you enter a model manually with @kbd{'}, Calc puts a 3-vector
24951 describing the model in the trail so you can get it back if you wish.
24959 Finally, you can store a model in one of the Calc variables
24960 @code{Model1} or @code{Model2}, then use this model by typing
24961 @kbd{a F u} or @kbd{a F U} (respectively). The value stored in
24962 the variable can be any of the formats that @kbd{a F $} would
24963 accept for a model on the stack.
24969 Calc uses the principal values of inverse functions like @code{ln}
24970 and @code{arcsin} when doing fits. For example, when you enter
24971 the model @samp{y = sin(a t + b)} Calc actually uses the easier
24972 form @samp{arcsin(y) = a t + b}. The @code{arcsin} function always
24973 returns results in the range from @mathit{-90} to 90 degrees (or the
24974 equivalent range in radians). Suppose you had data that you
24975 believed to represent roughly three oscillations of a sine wave,
24976 so that the argument of the sine might go from zero to
24977 @texline @math{3\times360}
24978 @infoline @mathit{3*360}
24980 The above model would appear to be a good way to determine the
24981 true frequency and phase of the sine wave, but in practice it
24982 would fail utterly. The righthand side of the actual model
24983 @samp{arcsin(y) = a t + b} will grow smoothly with @expr{t}, but
24984 the lefthand side will bounce back and forth between @mathit{-90} and 90.
24985 No values of @expr{a} and @expr{b} can make the two sides match,
24986 even approximately.
24988 There is no good solution to this problem at present. You could
24989 restrict your data to small enough ranges so that the above problem
24990 doesn't occur (i.e., not straddling any peaks in the sine wave).
24991 Or, in this case, you could use a totally different method such as
24992 Fourier analysis, which is beyond the scope of the @kbd{a F} command.
24993 (Unfortunately, Calc does not currently have any facilities for
24994 taking Fourier and related transforms.)
24996 @node Curve Fitting Details, Interpolation, Standard Nonlinear Models, Curve Fitting
24997 @subsection Curve Fitting Details
25000 Calc's internal least-squares fitter can only handle multilinear
25001 models. More precisely, it can handle any model of the form
25002 @expr{a f(x,y,z) + b g(x,y,z) + c h(x,y,z)}, where @expr{a,b,c}
25003 are the parameters and @expr{x,y,z} are the independent variables
25004 (of course there can be any number of each, not just three).
25006 In a simple multilinear or polynomial fit, it is easy to see how
25007 to convert the model into this form. For example, if the model
25008 is @expr{a + b x + c x^2}, then @expr{f(x) = 1}, @expr{g(x) = x},
25009 and @expr{h(x) = x^2} are suitable functions.
25011 For most other models, Calc uses a variety of algebraic manipulations
25012 to try to put the problem into the form
25015 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)
25019 where @expr{Y,A,B,C,F,G,H} are arbitrary functions. It computes
25020 @expr{Y}, @expr{F}, @expr{G}, and @expr{H} for all the data points,
25021 does a standard linear fit to find the values of @expr{A}, @expr{B},
25022 and @expr{C}, then uses the equation solver to solve for @expr{a,b,c}
25023 in terms of @expr{A,B,C}.
25025 A remarkable number of models can be cast into this general form.
25026 We'll look at two examples here to see how it works. The power-law
25027 model @expr{y = a x^b} with two independent variables and two parameters
25028 can be rewritten as follows:
25033 y = exp(ln(a) + b ln(x))
25034 ln(y) = ln(a) + b ln(x)
25038 which matches the desired form with
25039 @texline @math{Y = \ln(y)},
25040 @infoline @expr{Y = ln(y)},
25041 @texline @math{A = \ln(a)},
25042 @infoline @expr{A = ln(a)},
25043 @expr{F = 1}, @expr{B = b}, and
25044 @texline @math{G = \ln(x)}.
25045 @infoline @expr{G = ln(x)}.
25046 Calc thus computes the logarithms of your @expr{y} and @expr{x} values,
25047 does a linear fit for @expr{A} and @expr{B}, then solves to get
25048 @texline @math{a = \exp(A)}
25049 @infoline @expr{a = exp(A)}
25052 Another interesting example is the ``quadratic'' model, which can
25053 be handled by expanding according to the distributive law.
25056 y = a + b*(x - c)^2
25057 y = a + b c^2 - 2 b c x + b x^2
25061 which matches with @expr{Y = y}, @expr{A = a + b c^2}, @expr{F = 1},
25062 @expr{B = -2 b c}, @expr{G = x} (the @mathit{-2} factor could just as easily
25063 have been put into @expr{G} instead of @expr{B}), @expr{C = b}, and
25066 The Gaussian model looks quite complicated, but a closer examination
25067 shows that it's actually similar to the quadratic model but with an
25068 exponential that can be brought to the top and moved into @expr{Y}.
25070 The logistic models cannot be put into general linear form. For these
25071 models, and the Hubbert linearization, Calc computes a rough
25072 approximation for the parameters, then uses the Levenberg-Marquardt
25073 iterative method to refine the approximations.
25075 Another model that cannot be put into general linear
25076 form is a Gaussian with a constant background added on, i.e.,
25077 @expr{d} + the regular Gaussian formula. If you have a model like
25078 this, your best bet is to replace enough of your parameters with
25079 constants to make the model linearizable, then adjust the constants
25080 manually by doing a series of fits. You can compare the fits by
25081 graphing them, by examining the goodness-of-fit measures returned by
25082 @kbd{I a F}, or by some other method suitable to your application.
25083 Note that some models can be linearized in several ways. The
25084 Gaussian-plus-@var{d} model can be linearized by setting @expr{d}
25085 (the background) to a constant, or by setting @expr{b} (the standard
25086 deviation) and @expr{c} (the mean) to constants.
25088 To fit a model with constants substituted for some parameters, just
25089 store suitable values in those parameter variables, then omit them
25090 from the list of parameters when you answer the variables prompt.
25096 A last desperate step would be to use the general-purpose
25097 @code{minimize} function rather than @code{fit}. After all, both
25098 functions solve the problem of minimizing an expression (the
25099 @texline @math{\chi^2}
25100 @infoline @expr{chi^2}
25101 sum) by adjusting certain parameters in the expression. The @kbd{a F}
25102 command is able to use a vastly more efficient algorithm due to its
25103 special knowledge about linear chi-square sums, but the @kbd{a N}
25104 command can do the same thing by brute force.
25106 A compromise would be to pick out a few parameters without which the
25107 fit is linearizable, and use @code{minimize} on a call to @code{fit}
25108 which efficiently takes care of the rest of the parameters. The thing
25109 to be minimized would be the value of
25110 @texline @math{\chi^2}
25111 @infoline @expr{chi^2}
25112 returned as the fifth result of the @code{xfit} function:
25115 minimize(xfit(gaus(a,b,c,d,x), x, [a,b,c], data)_5, d, guess)
25119 where @code{gaus} represents the Gaussian model with background,
25120 @code{data} represents the data matrix, and @code{guess} represents
25121 the initial guess for @expr{d} that @code{minimize} requires.
25122 This operation will only be, shall we say, extraordinarily slow
25123 rather than astronomically slow (as would be the case if @code{minimize}
25124 were used by itself to solve the problem).
25130 The @kbd{I a F} [@code{xfit}] command is somewhat trickier when
25131 nonlinear models are used. The second item in the result is the
25132 vector of ``raw'' parameters @expr{A}, @expr{B}, @expr{C}. The
25133 covariance matrix is written in terms of those raw parameters.
25134 The fifth item is a vector of @dfn{filter} expressions. This
25135 is the empty vector @samp{[]} if the raw parameters were the same
25136 as the requested parameters, i.e., if @expr{A = a}, @expr{B = b},
25137 and so on (which is always true if the model is already linear
25138 in the parameters as written, e.g., for polynomial fits). If the
25139 parameters had to be rearranged, the fifth item is instead a vector
25140 of one formula per parameter in the original model. The raw
25141 parameters are expressed in these ``filter'' formulas as
25142 @samp{fitdummy(1)} for @expr{A}, @samp{fitdummy(2)} for @expr{B},
25145 When Calc needs to modify the model to return the result, it replaces
25146 @samp{fitdummy(1)} in all the filters with the first item in the raw
25147 parameters list, and so on for the other raw parameters, then
25148 evaluates the resulting filter formulas to get the actual parameter
25149 values to be substituted into the original model. In the case of
25150 @kbd{H a F} and @kbd{I a F} where the parameters must be error forms,
25151 Calc uses the square roots of the diagonal entries of the covariance
25152 matrix as error values for the raw parameters, then lets Calc's
25153 standard error-form arithmetic take it from there.
25155 If you use @kbd{I a F} with a nonlinear model, be sure to remember
25156 that the covariance matrix is in terms of the raw parameters,
25157 @emph{not} the actual requested parameters. It's up to you to
25158 figure out how to interpret the covariances in the presence of
25159 nontrivial filter functions.
25161 Things are also complicated when the input contains error forms.
25162 Suppose there are three independent and dependent variables, @expr{x},
25163 @expr{y}, and @expr{z}, one or more of which are error forms in the
25164 data. Calc combines all the error values by taking the square root
25165 of the sum of the squares of the errors. It then changes @expr{x}
25166 and @expr{y} to be plain numbers, and makes @expr{z} into an error
25167 form with this combined error. The @expr{Y(x,y,z)} part of the
25168 linearized model is evaluated, and the result should be an error
25169 form. The error part of that result is used for
25170 @texline @math{\sigma_i}
25171 @infoline @expr{sigma_i}
25172 for the data point. If for some reason @expr{Y(x,y,z)} does not return
25173 an error form, the combined error from @expr{z} is used directly for
25174 @texline @math{\sigma_i}.
25175 @infoline @expr{sigma_i}.
25176 Finally, @expr{z} is also stripped of its error
25177 for use in computing @expr{F(x,y,z)}, @expr{G(x,y,z)} and so on;
25178 the righthand side of the linearized model is computed in regular
25179 arithmetic with no error forms.
25181 (While these rules may seem complicated, they are designed to do
25182 the most reasonable thing in the typical case that @expr{Y(x,y,z)}
25183 depends only on the dependent variable @expr{z}, and in fact is
25184 often simply equal to @expr{z}. For common cases like polynomials
25185 and multilinear models, the combined error is simply used as the
25186 @texline @math{\sigma}
25187 @infoline @expr{sigma}
25188 for the data point with no further ado.)
25195 It may be the case that the model you wish to use is linearizable,
25196 but Calc's built-in rules are unable to figure it out. Calc uses
25197 its algebraic rewrite mechanism to linearize a model. The rewrite
25198 rules are kept in the variable @code{FitRules}. You can edit this
25199 variable using the @kbd{s e FitRules} command; in fact, there is
25200 a special @kbd{s F} command just for editing @code{FitRules}.
25201 @xref{Operations on Variables}.
25203 @xref{Rewrite Rules}, for a discussion of rewrite rules.
25237 Calc uses @code{FitRules} as follows. First, it converts the model
25238 to an equation if necessary and encloses the model equation in a
25239 call to the function @code{fitmodel} (which is not actually a defined
25240 function in Calc; it is only used as a placeholder by the rewrite rules).
25241 Parameter variables are renamed to function calls @samp{fitparam(1)},
25242 @samp{fitparam(2)}, and so on, and independent variables are renamed
25243 to @samp{fitvar(1)}, @samp{fitvar(2)}, etc. The dependent variable
25244 is the highest-numbered @code{fitvar}. For example, the power law
25245 model @expr{a x^b} is converted to @expr{y = a x^b}, then to
25249 fitmodel(fitvar(2) = fitparam(1) fitvar(1)^fitparam(2))
25253 Calc then applies the rewrites as if by @samp{C-u 0 a r FitRules}.
25254 (The zero prefix means that rewriting should continue until no further
25255 changes are possible.)
25257 When rewriting is complete, the @code{fitmodel} call should have
25258 been replaced by a @code{fitsystem} call that looks like this:
25261 fitsystem(@var{Y}, @var{FGH}, @var{abc})
25265 where @var{Y} is a formula that describes the function @expr{Y(x,y,z)},
25266 @var{FGH} is the vector of formulas @expr{[F(x,y,z), G(x,y,z), H(x,y,z)]},
25267 and @var{abc} is the vector of parameter filters which refer to the
25268 raw parameters as @samp{fitdummy(1)} for @expr{A}, @samp{fitdummy(2)}
25269 for @expr{B}, etc. While the number of raw parameters (the length of
25270 the @var{FGH} vector) is usually the same as the number of original
25271 parameters (the length of the @var{abc} vector), this is not required.
25273 The power law model eventually boils down to
25277 fitsystem(ln(fitvar(2)),
25278 [1, ln(fitvar(1))],
25279 [exp(fitdummy(1)), fitdummy(2)])
25283 The actual implementation of @code{FitRules} is complicated; it
25284 proceeds in four phases. First, common rearrangements are done
25285 to try to bring linear terms together and to isolate functions like
25286 @code{exp} and @code{ln} either all the way ``out'' (so that they
25287 can be put into @var{Y}) or all the way ``in'' (so that they can
25288 be put into @var{abc} or @var{FGH}). In particular, all
25289 non-constant powers are converted to logs-and-exponentials form,
25290 and the distributive law is used to expand products of sums.
25291 Quotients are rewritten to use the @samp{fitinv} function, where
25292 @samp{fitinv(x)} represents @expr{1/x} while the @code{FitRules}
25293 are operating. (The use of @code{fitinv} makes recognition of
25294 linear-looking forms easier.) If you modify @code{FitRules}, you
25295 will probably only need to modify the rules for this phase.
25297 Phase two, whose rules can actually also apply during phases one
25298 and three, first rewrites @code{fitmodel} to a two-argument
25299 form @samp{fitmodel(@var{Y}, @var{model})}, where @var{Y} is
25300 initially zero and @var{model} has been changed from @expr{a=b}
25301 to @expr{a-b} form. It then tries to peel off invertible functions
25302 from the outside of @var{model} and put them into @var{Y} instead,
25303 calling the equation solver to invert the functions. Finally, when
25304 this is no longer possible, the @code{fitmodel} is changed to a
25305 four-argument @code{fitsystem}, where the fourth argument is
25306 @var{model} and the @var{FGH} and @var{abc} vectors are initially
25307 empty. (The last vector is really @var{ABC}, corresponding to
25308 raw parameters, for now.)
25310 Phase three converts a sum of items in the @var{model} to a sum
25311 of @samp{fitpart(@var{a}, @var{b}, @var{c})} terms which represent
25312 terms @samp{@var{a}*@var{b}*@var{c}} of the sum, where @var{a}
25313 is all factors that do not involve any variables, @var{b} is all
25314 factors that involve only parameters, and @var{c} is the factors
25315 that involve only independent variables. (If this decomposition
25316 is not possible, the rule set will not complete and Calc will
25317 complain that the model is too complex.) Then @code{fitpart}s
25318 with equal @var{b} or @var{c} components are merged back together
25319 using the distributive law in order to minimize the number of
25320 raw parameters needed.
25322 Phase four moves the @code{fitpart} terms into the @var{FGH} and
25323 @var{ABC} vectors. Also, some of the algebraic expansions that
25324 were done in phase 1 are undone now to make the formulas more
25325 computationally efficient. Finally, it calls the solver one more
25326 time to convert the @var{ABC} vector to an @var{abc} vector, and
25327 removes the fourth @var{model} argument (which by now will be zero)
25328 to obtain the three-argument @code{fitsystem} that the linear
25329 least-squares solver wants to see.
25335 @mindex hasfit@idots
25337 @tindex hasfitparams
25345 Two functions which are useful in connection with @code{FitRules}
25346 are @samp{hasfitparams(x)} and @samp{hasfitvars(x)}, which check
25347 whether @expr{x} refers to any parameters or independent variables,
25348 respectively. Specifically, these functions return ``true'' if the
25349 argument contains any @code{fitparam} (or @code{fitvar}) function
25350 calls, and ``false'' otherwise. (Recall that ``true'' means a
25351 nonzero number, and ``false'' means zero. The actual nonzero number
25352 returned is the largest @var{n} from all the @samp{fitparam(@var{n})}s
25353 or @samp{fitvar(@var{n})}s, respectively, that appear in the formula.)
25359 The @code{fit} function in algebraic notation normally takes four
25360 arguments, @samp{fit(@var{model}, @var{vars}, @var{params}, @var{data})},
25361 where @var{model} is the model formula as it would be typed after
25362 @kbd{a F '}, @var{vars} is the independent variable or a vector of
25363 independent variables, @var{params} likewise gives the parameter(s),
25364 and @var{data} is the data matrix. Note that the length of @var{vars}
25365 must be equal to the number of rows in @var{data} if @var{model} is
25366 an equation, or one less than the number of rows if @var{model} is
25367 a plain formula. (Actually, a name for the dependent variable is
25368 allowed but will be ignored in the plain-formula case.)
25370 If @var{params} is omitted, the parameters are all variables in
25371 @var{model} except those that appear in @var{vars}. If @var{vars}
25372 is also omitted, Calc sorts all the variables that appear in
25373 @var{model} alphabetically and uses the higher ones for @var{vars}
25374 and the lower ones for @var{params}.
25376 Alternatively, @samp{fit(@var{modelvec}, @var{data})} is allowed
25377 where @var{modelvec} is a 2- or 3-vector describing the model
25378 and variables, as discussed previously.
25380 If Calc is unable to do the fit, the @code{fit} function is left
25381 in symbolic form, ordinarily with an explanatory message. The
25382 message will be ``Model expression is too complex'' if the
25383 linearizer was unable to put the model into the required form.
25385 The @code{efit} (corresponding to @kbd{H a F}) and @code{xfit}
25386 (for @kbd{I a F}) functions are completely analogous.
25388 @node Interpolation, , Curve Fitting Details, Curve Fitting
25389 @subsection Polynomial Interpolation
25392 @pindex calc-poly-interp
25394 The @kbd{a p} (@code{calc-poly-interp}) [@code{polint}] command does
25395 a polynomial interpolation at a particular @expr{x} value. It takes
25396 two arguments from the stack: A data matrix of the sort used by
25397 @kbd{a F}, and a single number which represents the desired @expr{x}
25398 value. Calc effectively does an exact polynomial fit as if by @kbd{a F i},
25399 then substitutes the @expr{x} value into the result in order to get an
25400 approximate @expr{y} value based on the fit. (Calc does not actually
25401 use @kbd{a F i}, however; it uses a direct method which is both more
25402 efficient and more numerically stable.)
25404 The result of @kbd{a p} is actually a vector of two values: The @expr{y}
25405 value approximation, and an error measure @expr{dy} that reflects Calc's
25406 estimation of the probable error of the approximation at that value of
25407 @expr{x}. If the input @expr{x} is equal to any of the @expr{x} values
25408 in the data matrix, the output @expr{y} will be the corresponding @expr{y}
25409 value from the matrix, and the output @expr{dy} will be exactly zero.
25411 A prefix argument of 2 causes @kbd{a p} to take separate x- and
25412 y-vectors from the stack instead of one data matrix.
25414 If @expr{x} is a vector of numbers, @kbd{a p} will return a matrix of
25415 interpolated results for each of those @expr{x} values. (The matrix will
25416 have two columns, the @expr{y} values and the @expr{dy} values.)
25417 If @expr{x} is a formula instead of a number, the @code{polint} function
25418 remains in symbolic form; use the @kbd{a "} command to expand it out to
25419 a formula that describes the fit in symbolic terms.
25421 In all cases, the @kbd{a p} command leaves the data vectors or matrix
25422 on the stack. Only the @expr{x} value is replaced by the result.
25426 The @kbd{H a p} [@code{ratint}] command does a rational function
25427 interpolation. It is used exactly like @kbd{a p}, except that it
25428 uses as its model the quotient of two polynomials. If there are
25429 @expr{N} data points, the numerator and denominator polynomials will
25430 each have degree @expr{N/2} (if @expr{N} is odd, the denominator will
25431 have degree one higher than the numerator).
25433 Rational approximations have the advantage that they can accurately
25434 describe functions that have poles (points at which the function's value
25435 goes to infinity, so that the denominator polynomial of the approximation
25436 goes to zero). If @expr{x} corresponds to a pole of the fitted rational
25437 function, then the result will be a division by zero. If Infinite mode
25438 is enabled, the result will be @samp{[uinf, uinf]}.
25440 There is no way to get the actual coefficients of the rational function
25441 used by @kbd{H a p}. (The algorithm never generates these coefficients
25442 explicitly, and quotients of polynomials are beyond @w{@kbd{a F}}'s
25443 capabilities to fit.)
25445 @node Summations, Logical Operations, Curve Fitting, Algebra
25446 @section Summations
25449 @cindex Summation of a series
25451 @pindex calc-summation
25453 The @kbd{a +} (@code{calc-summation}) [@code{sum}] command computes
25454 the sum of a formula over a certain range of index values. The formula
25455 is taken from the top of the stack; the command prompts for the
25456 name of the summation index variable, the lower limit of the
25457 sum (any formula), and the upper limit of the sum. If you
25458 enter a blank line at any of these prompts, that prompt and
25459 any later ones are answered by reading additional elements from
25460 the stack. Thus, @kbd{' k^2 @key{RET} ' k @key{RET} 1 @key{RET} 5 @key{RET} a + @key{RET}}
25461 produces the result 55.
25463 $$ \sum_{k=1}^5 k^2 = 55 $$
25466 The choice of index variable is arbitrary, but it's best not to
25467 use a variable with a stored value. In particular, while
25468 @code{i} is often a favorite index variable, it should be avoided
25469 in Calc because @code{i} has the imaginary constant @expr{(0, 1)}
25470 as a value. If you pressed @kbd{=} on a sum over @code{i}, it would
25471 be changed to a nonsensical sum over the ``variable'' @expr{(0, 1)}!
25472 If you really want to use @code{i} as an index variable, use
25473 @w{@kbd{s u i @key{RET}}} first to ``unstore'' this variable.
25474 (@xref{Storing Variables}.)
25476 A numeric prefix argument steps the index by that amount rather
25477 than by one. Thus @kbd{' a_k @key{RET} C-u -2 a + k @key{RET} 10 @key{RET} 0 @key{RET}}
25478 yields @samp{a_10 + a_8 + a_6 + a_4 + a_2 + a_0}. A prefix
25479 argument of plain @kbd{C-u} causes @kbd{a +} to prompt for the
25480 step value, in which case you can enter any formula or enter
25481 a blank line to take the step value from the stack. With the
25482 @kbd{C-u} prefix, @kbd{a +} can take up to five arguments from
25483 the stack: The formula, the variable, the lower limit, the
25484 upper limit, and (at the top of the stack), the step value.
25486 Calc knows how to do certain sums in closed form. For example,
25487 @samp{sum(6 k^2, k, 1, n) = @w{2 n^3} + 3 n^2 + n}. In particular,
25488 this is possible if the formula being summed is polynomial or
25489 exponential in the index variable. Sums of logarithms are
25490 transformed into logarithms of products. Sums of trigonometric
25491 and hyperbolic functions are transformed to sums of exponentials
25492 and then done in closed form. Also, of course, sums in which the
25493 lower and upper limits are both numbers can always be evaluated
25494 just by grinding them out, although Calc will use closed forms
25495 whenever it can for the sake of efficiency.
25497 The notation for sums in algebraic formulas is
25498 @samp{sum(@var{expr}, @var{var}, @var{low}, @var{high}, @var{step})}.
25499 If @var{step} is omitted, it defaults to one. If @var{high} is
25500 omitted, @var{low} is actually the upper limit and the lower limit
25501 is one. If @var{low} is also omitted, the limits are @samp{-inf}
25502 and @samp{inf}, respectively.
25504 Infinite sums can sometimes be evaluated: @samp{sum(.5^k, k, 1, inf)}
25505 returns @expr{1}. This is done by evaluating the sum in closed
25506 form (to @samp{1. - 0.5^n} in this case), then evaluating this
25507 formula with @code{n} set to @code{inf}. Calc's usual rules
25508 for ``infinite'' arithmetic can find the answer from there. If
25509 infinite arithmetic yields a @samp{nan}, or if the sum cannot be
25510 solved in closed form, Calc leaves the @code{sum} function in
25511 symbolic form. @xref{Infinities}.
25513 As a special feature, if the limits are infinite (or omitted, as
25514 described above) but the formula includes vectors subscripted by
25515 expressions that involve the iteration variable, Calc narrows
25516 the limits to include only the range of integers which result in
25517 valid subscripts for the vector. For example, the sum
25518 @samp{sum(k [a,b,c,d,e,f,g]_(2k),k)} evaluates to @samp{b + 2 d + 3 f}.
25520 The limits of a sum do not need to be integers. For example,
25521 @samp{sum(a_k, k, 0, 2 n, n)} produces @samp{a_0 + a_n + a_(2 n)}.
25522 Calc computes the number of iterations using the formula
25523 @samp{1 + (@var{high} - @var{low}) / @var{step}}, which must,
25524 after algebraic simplification, evaluate to an integer.
25526 If the number of iterations according to the above formula does
25527 not come out to an integer, the sum is invalid and will be left
25528 in symbolic form. However, closed forms are still supplied, and
25529 you are on your honor not to misuse the resulting formulas by
25530 substituting mismatched bounds into them. For example,
25531 @samp{sum(k, k, 1, 10, 2)} is invalid, but Calc will go ahead and
25532 evaluate the closed form solution for the limits 1 and 10 to get
25533 the rather dubious answer, 29.25.
25535 If the lower limit is greater than the upper limit (assuming a
25536 positive step size), the result is generally zero. However,
25537 Calc only guarantees a zero result when the upper limit is
25538 exactly one step less than the lower limit, i.e., if the number
25539 of iterations is @mathit{-1}. Thus @samp{sum(f(k), k, n, n-1)} is zero
25540 but the sum from @samp{n} to @samp{n-2} may report a nonzero value
25541 if Calc used a closed form solution.
25543 Calc's logical predicates like @expr{a < b} return 1 for ``true''
25544 and 0 for ``false.'' @xref{Logical Operations}. This can be
25545 used to advantage for building conditional sums. For example,
25546 @samp{sum(prime(k)*k^2, k, 1, 20)} is the sum of the squares of all
25547 prime numbers from 1 to 20; the @code{prime} predicate returns 1 if
25548 its argument is prime and 0 otherwise. You can read this expression
25549 as ``the sum of @expr{k^2}, where @expr{k} is prime.'' Indeed,
25550 @samp{sum(prime(k)*k^2, k)} would represent the sum of @emph{all} primes
25551 squared, since the limits default to plus and minus infinity, but
25552 there are no such sums that Calc's built-in rules can do in
25555 As another example, @samp{sum((k != k_0) * f(k), k, 1, n)} is the
25556 sum of @expr{f(k)} for all @expr{k} from 1 to @expr{n}, excluding
25557 one value @expr{k_0}. Slightly more tricky is the summand
25558 @samp{(k != k_0) / (k - k_0)}, which is an attempt to describe
25559 the sum of all @expr{1/(k-k_0)} except at @expr{k = k_0}, where
25560 this would be a division by zero. But at @expr{k = k_0}, this
25561 formula works out to the indeterminate form @expr{0 / 0}, which
25562 Calc will not assume is zero. Better would be to use
25563 @samp{(k != k_0) ? 1/(k-k_0) : 0}; the @samp{? :} operator does
25564 an ``if-then-else'' test: This expression says, ``if
25565 @texline @math{k \ne k_0},
25566 @infoline @expr{k != k_0},
25567 then @expr{1/(k-k_0)}, else zero.'' Now the formula @expr{1/(k-k_0)}
25568 will not even be evaluated by Calc when @expr{k = k_0}.
25570 @cindex Alternating sums
25572 @pindex calc-alt-summation
25574 The @kbd{a -} (@code{calc-alt-summation}) [@code{asum}] command
25575 computes an alternating sum. Successive terms of the sequence
25576 are given alternating signs, with the first term (corresponding
25577 to the lower index value) being positive. Alternating sums
25578 are converted to normal sums with an extra term of the form
25579 @samp{(-1)^(k-@var{low})}. This formula is adjusted appropriately
25580 if the step value is other than one. For example, the Taylor
25581 series for the sine function is @samp{asum(x^k / k!, k, 1, inf, 2)}.
25582 (Calc cannot evaluate this infinite series, but it can approximate
25583 it if you replace @code{inf} with any particular odd number.)
25584 Calc converts this series to a regular sum with a step of one,
25585 namely @samp{sum((-1)^k x^(2k+1) / (2k+1)!, k, 0, inf)}.
25587 @cindex Product of a sequence
25589 @pindex calc-product
25591 The @kbd{a *} (@code{calc-product}) [@code{prod}] command is
25592 the analogous way to take a product of many terms. Calc also knows
25593 some closed forms for products, such as @samp{prod(k, k, 1, n) = n!}.
25594 Conditional products can be written @samp{prod(k^prime(k), k, 1, n)}
25595 or @samp{prod(prime(k) ? k : 1, k, 1, n)}.
25598 @pindex calc-tabulate
25600 The @kbd{a T} (@code{calc-tabulate}) [@code{table}] command
25601 evaluates a formula at a series of iterated index values, just
25602 like @code{sum} and @code{prod}, but its result is simply a
25603 vector of the results. For example, @samp{table(a_i, i, 1, 7, 2)}
25604 produces @samp{[a_1, a_3, a_5, a_7]}.
25606 @node Logical Operations, Rewrite Rules, Summations, Algebra
25607 @section Logical Operations
25610 The following commands and algebraic functions return true/false values,
25611 where 1 represents ``true'' and 0 represents ``false.'' In cases where
25612 a truth value is required (such as for the condition part of a rewrite
25613 rule, or as the condition for a @w{@kbd{Z [ Z ]}} control structure), any
25614 nonzero value is accepted to mean ``true.'' (Specifically, anything
25615 for which @code{dnonzero} returns 1 is ``true,'' and anything for
25616 which @code{dnonzero} returns 0 or cannot decide is assumed ``false.''
25617 Note that this means that @w{@kbd{Z [ Z ]}} will execute the ``then''
25618 portion if its condition is provably true, but it will execute the
25619 ``else'' portion for any condition like @expr{a = b} that is not
25620 provably true, even if it might be true. Algebraic functions that
25621 have conditions as arguments, like @code{? :} and @code{&&}, remain
25622 unevaluated if the condition is neither provably true nor provably
25623 false. @xref{Declarations}.)
25626 @pindex calc-equal-to
25630 The @kbd{a =} (@code{calc-equal-to}) command, or @samp{eq(a,b)} function
25631 (which can also be written @samp{a = b} or @samp{a == b} in an algebraic
25632 formula) is true if @expr{a} and @expr{b} are equal, either because they
25633 are identical expressions, or because they are numbers which are
25634 numerically equal. (Thus the integer 1 is considered equal to the float
25635 1.0.) If the equality of @expr{a} and @expr{b} cannot be determined,
25636 the comparison is left in symbolic form. Note that as a command, this
25637 operation pops two values from the stack and pushes back either a 1 or
25638 a 0, or a formula @samp{a = b} if the values' equality cannot be determined.
25640 Many Calc commands use @samp{=} formulas to represent @dfn{equations}.
25641 For example, the @kbd{a S} (@code{calc-solve-for}) command rearranges
25642 an equation to solve for a given variable. The @kbd{a M}
25643 (@code{calc-map-equation}) command can be used to apply any
25644 function to both sides of an equation; for example, @kbd{2 a M *}
25645 multiplies both sides of the equation by two. Note that just
25646 @kbd{2 *} would not do the same thing; it would produce the formula
25647 @samp{2 (a = b)} which represents 2 if the equality is true or
25650 The @code{eq} function with more than two arguments (e.g., @kbd{C-u 3 a =}
25651 or @samp{a = b = c}) tests if all of its arguments are equal. In
25652 algebraic notation, the @samp{=} operator is unusual in that it is
25653 neither left- nor right-associative: @samp{a = b = c} is not the
25654 same as @samp{(a = b) = c} or @samp{a = (b = c)} (which each compare
25655 one variable with the 1 or 0 that results from comparing two other
25659 @pindex calc-not-equal-to
25662 The @kbd{a #} (@code{calc-not-equal-to}) command, or @samp{neq(a,b)} or
25663 @samp{a != b} function, is true if @expr{a} and @expr{b} are not equal.
25664 This also works with more than two arguments; @samp{a != b != c != d}
25665 tests that all four of @expr{a}, @expr{b}, @expr{c}, and @expr{d} are
25682 @pindex calc-less-than
25683 @pindex calc-greater-than
25684 @pindex calc-less-equal
25685 @pindex calc-greater-equal
25714 The @kbd{a <} (@code{calc-less-than}) [@samp{lt(a,b)} or @samp{a < b}]
25715 operation is true if @expr{a} is less than @expr{b}. Similar functions
25716 are @kbd{a >} (@code{calc-greater-than}) [@samp{gt(a,b)} or @samp{a > b}],
25717 @kbd{a [} (@code{calc-less-equal}) [@samp{leq(a,b)} or @samp{a <= b}], and
25718 @kbd{a ]} (@code{calc-greater-equal}) [@samp{geq(a,b)} or @samp{a >= b}].
25720 While the inequality functions like @code{lt} do not accept more
25721 than two arguments, the syntax @w{@samp{a <= b < c}} is translated to an
25722 equivalent expression involving intervals: @samp{b in [a .. c)}.
25723 (See the description of @code{in} below.) All four combinations
25724 of @samp{<} and @samp{<=} are allowed, or any of the four combinations
25725 of @samp{>} and @samp{>=}. Four-argument constructions like
25726 @samp{a < b < c < d}, and mixtures like @w{@samp{a < b = c}} that
25727 involve both equations and inequalities, are not allowed.
25730 @pindex calc-remove-equal
25732 The @kbd{a .} (@code{calc-remove-equal}) [@code{rmeq}] command extracts
25733 the righthand side of the equation or inequality on the top of the
25734 stack. It also works elementwise on vectors. For example, if
25735 @samp{[x = 2.34, y = z / 2]} is on the stack, then @kbd{a .} produces
25736 @samp{[2.34, z / 2]}. As a special case, if the righthand side is a
25737 variable and the lefthand side is a number (as in @samp{2.34 = x}), then
25738 Calc keeps the lefthand side instead. Finally, this command works with
25739 assignments @samp{x := 2.34} as well as equations, always taking the
25740 righthand side, and for @samp{=>} (evaluates-to) operators, always
25741 taking the lefthand side.
25744 @pindex calc-logical-and
25747 The @kbd{a &} (@code{calc-logical-and}) [@samp{land(a,b)} or @samp{a && b}]
25748 function is true if both of its arguments are true, i.e., are
25749 non-zero numbers. In this case, the result will be either @expr{a} or
25750 @expr{b}, chosen arbitrarily. If either argument is zero, the result is
25751 zero. Otherwise, the formula is left in symbolic form.
25754 @pindex calc-logical-or
25757 The @kbd{a |} (@code{calc-logical-or}) [@samp{lor(a,b)} or @samp{a || b}]
25758 function is true if either or both of its arguments are true (nonzero).
25759 The result is whichever argument was nonzero, choosing arbitrarily if both
25760 are nonzero. If both @expr{a} and @expr{b} are zero, the result is
25764 @pindex calc-logical-not
25767 The @kbd{a !} (@code{calc-logical-not}) [@samp{lnot(a)} or @samp{!@: a}]
25768 function is true if @expr{a} is false (zero), or false if @expr{a} is
25769 true (nonzero). It is left in symbolic form if @expr{a} is not a
25773 @pindex calc-logical-if
25783 @cindex Arguments, not evaluated
25784 The @kbd{a :} (@code{calc-logical-if}) [@samp{if(a,b,c)} or @samp{a ? b :@: c}]
25785 function is equal to either @expr{b} or @expr{c} if @expr{a} is a nonzero
25786 number or zero, respectively. If @expr{a} is not a number, the test is
25787 left in symbolic form and neither @expr{b} nor @expr{c} is evaluated in
25788 any way. In algebraic formulas, this is one of the few Calc functions
25789 whose arguments are not automatically evaluated when the function itself
25790 is evaluated. The others are @code{lambda}, @code{quote}, and
25793 One minor surprise to watch out for is that the formula @samp{a?3:4}
25794 will not work because the @samp{3:4} is parsed as a fraction instead of
25795 as three separate symbols. Type something like @samp{a ? 3 : 4} or
25796 @samp{a?(3):4} instead.
25798 As a special case, if @expr{a} evaluates to a vector, then both @expr{b}
25799 and @expr{c} are evaluated; the result is a vector of the same length
25800 as @expr{a} whose elements are chosen from corresponding elements of
25801 @expr{b} and @expr{c} according to whether each element of @expr{a}
25802 is zero or nonzero. Each of @expr{b} and @expr{c} must be either a
25803 vector of the same length as @expr{a}, or a non-vector which is matched
25804 with all elements of @expr{a}.
25807 @pindex calc-in-set
25809 The @kbd{a @{} (@code{calc-in-set}) [@samp{in(a,b)}] function is true if
25810 the number @expr{a} is in the set of numbers represented by @expr{b}.
25811 If @expr{b} is an interval form, @expr{a} must be one of the values
25812 encompassed by the interval. If @expr{b} is a vector, @expr{a} must be
25813 equal to one of the elements of the vector. (If any vector elements are
25814 intervals, @expr{a} must be in any of the intervals.) If @expr{b} is a
25815 plain number, @expr{a} must be numerically equal to @expr{b}.
25816 @xref{Set Operations}, for a group of commands that manipulate sets
25823 The @samp{typeof(a)} function produces an integer or variable which
25824 characterizes @expr{a}. If @expr{a} is a number, vector, or variable,
25825 the result will be one of the following numbers:
25830 3 Floating-point number
25832 5 Rectangular complex number
25833 6 Polar complex number
25839 12 Infinity (inf, uinf, or nan)
25841 101 Vector (but not a matrix)
25845 Otherwise, @expr{a} is a formula, and the result is a variable which
25846 represents the name of the top-level function call.
25860 The @samp{integer(a)} function returns true if @expr{a} is an integer.
25861 The @samp{real(a)} function
25862 is true if @expr{a} is a real number, either integer, fraction, or
25863 float. The @samp{constant(a)} function returns true if @expr{a} is
25864 any of the objects for which @code{typeof} would produce an integer
25865 code result except for variables, and provided that the components of
25866 an object like a vector or error form are themselves constant.
25867 Note that infinities do not satisfy any of these tests, nor do
25868 special constants like @code{pi} and @code{e}.
25870 @xref{Declarations}, for a set of similar functions that recognize
25871 formulas as well as actual numbers. For example, @samp{dint(floor(x))}
25872 is true because @samp{floor(x)} is provably integer-valued, but
25873 @samp{integer(floor(x))} does not because @samp{floor(x)} is not
25874 literally an integer constant.
25880 The @samp{refers(a,b)} function is true if the variable (or sub-expression)
25881 @expr{b} appears in @expr{a}, or false otherwise. Unlike the other
25882 tests described here, this function returns a definite ``no'' answer
25883 even if its arguments are still in symbolic form. The only case where
25884 @code{refers} will be left unevaluated is if @expr{a} is a plain
25885 variable (different from @expr{b}).
25891 The @samp{negative(a)} function returns true if @expr{a} ``looks'' negative,
25892 because it is a negative number, because it is of the form @expr{-x},
25893 or because it is a product or quotient with a term that looks negative.
25894 This is most useful in rewrite rules. Beware that @samp{negative(a)}
25895 evaluates to 1 or 0 for @emph{any} argument @expr{a}, so it can only
25896 be stored in a formula if the default simplifications are turned off
25897 first with @kbd{m O} (or if it appears in an unevaluated context such
25898 as a rewrite rule condition).
25904 The @samp{variable(a)} function is true if @expr{a} is a variable,
25905 or false if not. If @expr{a} is a function call, this test is left
25906 in symbolic form. Built-in variables like @code{pi} and @code{inf}
25907 are considered variables like any others by this test.
25913 The @samp{nonvar(a)} function is true if @expr{a} is a non-variable.
25914 If its argument is a variable it is left unsimplified; it never
25915 actually returns zero. However, since Calc's condition-testing
25916 commands consider ``false'' anything not provably true, this is
25935 @cindex Linearity testing
25936 The functions @code{lin}, @code{linnt}, @code{islin}, and @code{islinnt}
25937 check if an expression is ``linear,'' i.e., can be written in the form
25938 @expr{a + b x} for some constants @expr{a} and @expr{b}, and some
25939 variable or subformula @expr{x}. The function @samp{islin(f,x)} checks
25940 if formula @expr{f} is linear in @expr{x}, returning 1 if so. For
25941 example, @samp{islin(x,x)}, @samp{islin(-x,x)}, @samp{islin(3,x)}, and
25942 @samp{islin(x y / 3 - 2, x)} all return 1. The @samp{lin(f,x)} function
25943 is similar, except that instead of returning 1 it returns the vector
25944 @expr{[a, b, x]}. For the above examples, this vector would be
25945 @expr{[0, 1, x]}, @expr{[0, -1, x]}, @expr{[3, 0, x]}, and
25946 @expr{[-2, y/3, x]}, respectively. Both @code{lin} and @code{islin}
25947 generally remain unevaluated for expressions which are not linear,
25948 e.g., @samp{lin(2 x^2, x)} and @samp{lin(sin(x), x)}. The second
25949 argument can also be a formula; @samp{islin(2 + 3 sin(x), sin(x))}
25952 The @code{linnt} and @code{islinnt} functions perform a similar check,
25953 but require a ``non-trivial'' linear form, which means that the
25954 @expr{b} coefficient must be non-zero. For example, @samp{lin(2,x)}
25955 returns @expr{[2, 0, x]} and @samp{lin(y,x)} returns @expr{[y, 0, x]},
25956 but @samp{linnt(2,x)} and @samp{linnt(y,x)} are left unevaluated
25957 (in other words, these formulas are considered to be only ``trivially''
25958 linear in @expr{x}).
25960 All four linearity-testing functions allow you to omit the second
25961 argument, in which case the input may be linear in any non-constant
25962 formula. Here, the @expr{a=0}, @expr{b=1} case is also considered
25963 trivial, and only constant values for @expr{a} and @expr{b} are
25964 recognized. Thus, @samp{lin(2 x y)} returns @expr{[0, 2, x y]},
25965 @samp{lin(2 - x y)} returns @expr{[2, -1, x y]}, and @samp{lin(x y)}
25966 returns @expr{[0, 1, x y]}. The @code{linnt} function would allow the
25967 first two cases but not the third. Also, neither @code{lin} nor
25968 @code{linnt} accept plain constants as linear in the one-argument
25969 case: @samp{islin(2,x)} is true, but @samp{islin(2)} is false.
25975 The @samp{istrue(a)} function returns 1 if @expr{a} is a nonzero
25976 number or provably nonzero formula, or 0 if @expr{a} is anything else.
25977 Calls to @code{istrue} can only be manipulated if @kbd{m O} mode is
25978 used to make sure they are not evaluated prematurely. (Note that
25979 declarations are used when deciding whether a formula is true;
25980 @code{istrue} returns 1 when @code{dnonzero} would return 1, and
25981 it returns 0 when @code{dnonzero} would return 0 or leave itself
25984 @node Rewrite Rules, , Logical Operations, Algebra
25985 @section Rewrite Rules
25988 @cindex Rewrite rules
25989 @cindex Transformations
25990 @cindex Pattern matching
25992 @pindex calc-rewrite
25994 The @kbd{a r} (@code{calc-rewrite}) [@code{rewrite}] command makes
25995 substitutions in a formula according to a specified pattern or patterns
25996 known as @dfn{rewrite rules}. Whereas @kbd{a b} (@code{calc-substitute})
25997 matches literally, so that substituting @samp{sin(x)} with @samp{cos(x)}
25998 matches only the @code{sin} function applied to the variable @code{x},
25999 rewrite rules match general kinds of formulas; rewriting using the rule
26000 @samp{sin(x) := cos(x)} matches @code{sin} of any argument and replaces
26001 it with @code{cos} of that same argument. The only significance of the
26002 name @code{x} is that the same name is used on both sides of the rule.
26004 Rewrite rules rearrange formulas already in Calc's memory.
26005 @xref{Syntax Tables}, to read about @dfn{syntax rules}, which are
26006 similar to algebraic rewrite rules but operate when new algebraic
26007 entries are being parsed, converting strings of characters into
26011 * Entering Rewrite Rules::
26012 * Basic Rewrite Rules::
26013 * Conditional Rewrite Rules::
26014 * Algebraic Properties of Rewrite Rules::
26015 * Other Features of Rewrite Rules::
26016 * Composing Patterns in Rewrite Rules::
26017 * Nested Formulas with Rewrite Rules::
26018 * Multi-Phase Rewrite Rules::
26019 * Selections with Rewrite Rules::
26020 * Matching Commands::
26021 * Automatic Rewrites::
26022 * Debugging Rewrites::
26023 * Examples of Rewrite Rules::
26026 @node Entering Rewrite Rules, Basic Rewrite Rules, Rewrite Rules, Rewrite Rules
26027 @subsection Entering Rewrite Rules
26030 Rewrite rules normally use the ``assignment'' operator
26031 @samp{@var{old} := @var{new}}.
26032 This operator is equivalent to the function call @samp{assign(old, new)}.
26033 The @code{assign} function is undefined by itself in Calc, so an
26034 assignment formula such as a rewrite rule will be left alone by ordinary
26035 Calc commands. But certain commands, like the rewrite system, interpret
26036 assignments in special ways.
26038 For example, the rule @samp{sin(x)^2 := 1-cos(x)^2} says to replace
26039 every occurrence of the sine of something, squared, with one minus the
26040 square of the cosine of that same thing. All by itself as a formula
26041 on the stack it does nothing, but when given to the @kbd{a r} command
26042 it turns that command into a sine-squared-to-cosine-squared converter.
26044 To specify a set of rules to be applied all at once, make a vector of
26047 When @kbd{a r} prompts you to enter the rewrite rules, you can answer
26052 With a rule: @kbd{f(x) := g(x) @key{RET}}.
26054 With a vector of rules: @kbd{[f1(x) := g1(x), f2(x) := g2(x)] @key{RET}}.
26055 (You can omit the enclosing square brackets if you wish.)
26057 With the name of a variable that contains the rule or rules vector:
26058 @kbd{myrules @key{RET}}.
26060 With any formula except a rule, a vector, or a variable name; this
26061 will be interpreted as the @var{old} half of a rewrite rule,
26062 and you will be prompted a second time for the @var{new} half:
26063 @kbd{f(x) @key{RET} g(x) @key{RET}}.
26065 With a blank line, in which case the rule, rules vector, or variable
26066 will be taken from the top of the stack (and the formula to be
26067 rewritten will come from the second-to-top position).
26070 If you enter the rules directly (as opposed to using rules stored
26071 in a variable), those rules will be put into the Trail so that you
26072 can retrieve them later. @xref{Trail Commands}.
26074 It is most convenient to store rules you use often in a variable and
26075 invoke them by giving the variable name. The @kbd{s e}
26076 (@code{calc-edit-variable}) command is an easy way to create or edit a
26077 rule set stored in a variable. You may also wish to use @kbd{s p}
26078 (@code{calc-permanent-variable}) to save your rules permanently;
26079 @pxref{Operations on Variables}.
26081 Rewrite rules are compiled into a special internal form for faster
26082 matching. If you enter a rule set directly it must be recompiled
26083 every time. If you store the rules in a variable and refer to them
26084 through that variable, they will be compiled once and saved away
26085 along with the variable for later reference. This is another good
26086 reason to store your rules in a variable.
26088 Calc also accepts an obsolete notation for rules, as vectors
26089 @samp{[@var{old}, @var{new}]}. But because it is easily confused with a
26090 vector of two rules, the use of this notation is no longer recommended.
26092 @node Basic Rewrite Rules, Conditional Rewrite Rules, Entering Rewrite Rules, Rewrite Rules
26093 @subsection Basic Rewrite Rules
26096 To match a particular formula @expr{x} with a particular rewrite rule
26097 @samp{@var{old} := @var{new}}, Calc compares the structure of @expr{x} with
26098 the structure of @var{old}. Variables that appear in @var{old} are
26099 treated as @dfn{meta-variables}; the corresponding positions in @expr{x}
26100 may contain any sub-formulas. For example, the pattern @samp{f(x,y)}
26101 would match the expression @samp{f(12, a+1)} with the meta-variable
26102 @samp{x} corresponding to 12 and with @samp{y} corresponding to
26103 @samp{a+1}. However, this pattern would not match @samp{f(12)} or
26104 @samp{g(12, a+1)}, since there is no assignment of the meta-variables
26105 that will make the pattern match these expressions. Notice that if
26106 the pattern is a single meta-variable, it will match any expression.
26108 If a given meta-variable appears more than once in @var{old}, the
26109 corresponding sub-formulas of @expr{x} must be identical. Thus
26110 the pattern @samp{f(x,x)} would match @samp{f(12, 12)} and
26111 @samp{f(a+1, a+1)} but not @samp{f(12, a+1)} or @samp{f(a+b, b+a)}.
26112 (@xref{Conditional Rewrite Rules}, for a way to match the latter.)
26114 Things other than variables must match exactly between the pattern
26115 and the target formula. To match a particular variable exactly, use
26116 the pseudo-function @samp{quote(v)} in the pattern. For example, the
26117 pattern @samp{x+quote(y)} matches @samp{x+y}, @samp{2+y}, or
26120 The special variable names @samp{e}, @samp{pi}, @samp{i}, @samp{phi},
26121 @samp{gamma}, @samp{inf}, @samp{uinf}, and @samp{nan} always match
26122 literally. Thus the pattern @samp{sin(d + e + f)} acts exactly like
26123 @samp{sin(d + quote(e) + f)}.
26125 If the @var{old} pattern is found to match a given formula, that
26126 formula is replaced by @var{new}, where any occurrences in @var{new}
26127 of meta-variables from the pattern are replaced with the sub-formulas
26128 that they matched. Thus, applying the rule @samp{f(x,y) := g(y+x,x)}
26129 to @samp{f(12, a+1)} would produce @samp{g(a+13, 12)}.
26131 The normal @kbd{a r} command applies rewrite rules over and over
26132 throughout the target formula until no further changes are possible
26133 (up to a limit of 100 times). Use @kbd{C-u 1 a r} to make only one
26136 @node Conditional Rewrite Rules, Algebraic Properties of Rewrite Rules, Basic Rewrite Rules, Rewrite Rules
26137 @subsection Conditional Rewrite Rules
26140 A rewrite rule can also be @dfn{conditional}, written in the form
26141 @samp{@var{old} := @var{new} :: @var{cond}}. (There is also the obsolete
26142 form @samp{[@var{old}, @var{new}, @var{cond}]}.) If a @var{cond} part
26144 rule, this is an additional condition that must be satisfied before
26145 the rule is accepted. Once @var{old} has been successfully matched
26146 to the target expression, @var{cond} is evaluated (with all the
26147 meta-variables substituted for the values they matched) and simplified
26148 with Calc's algebraic simplifications. If the result is a nonzero
26149 number or any other object known to be nonzero (@pxref{Declarations}),
26150 the rule is accepted. If the result is zero or if it is a symbolic
26151 formula that is not known to be nonzero, the rule is rejected.
26152 @xref{Logical Operations}, for a number of functions that return
26153 1 or 0 according to the results of various tests.
26155 For example, the formula @samp{n > 0} simplifies to 1 or 0 if @expr{n}
26156 is replaced by a positive or nonpositive number, respectively (or if
26157 @expr{n} has been declared to be positive or nonpositive). Thus,
26158 the rule @samp{f(x,y) := g(y+x,x) :: x+y > 0} would apply to
26159 @samp{f(0, 4)} but not to @samp{f(-3, 2)} or @samp{f(12, a+1)}
26160 (assuming no outstanding declarations for @expr{a}). In the case of
26161 @samp{f(-3, 2)}, the condition can be shown not to be satisfied; in
26162 the case of @samp{f(12, a+1)}, the condition merely cannot be shown
26163 to be satisfied, but that is enough to reject the rule.
26165 While Calc will use declarations to reason about variables in the
26166 formula being rewritten, declarations do not apply to meta-variables.
26167 For example, the rule @samp{f(a) := g(a+1)} will match for any values
26168 of @samp{a}, such as complex numbers, vectors, or formulas, even if
26169 @samp{a} has been declared to be real or scalar. If you want the
26170 meta-variable @samp{a} to match only literal real numbers, use
26171 @samp{f(a) := g(a+1) :: real(a)}. If you want @samp{a} to match only
26172 reals and formulas which are provably real, use @samp{dreal(a)} as
26175 The @samp{::} operator is a shorthand for the @code{condition}
26176 function; @samp{@var{old} := @var{new} :: @var{cond}} is equivalent to
26177 the formula @samp{condition(assign(@var{old}, @var{new}), @var{cond})}.
26179 If you have several conditions, you can use @samp{... :: c1 :: c2 :: c3}
26180 or @samp{... :: c1 && c2 && c3}. The two are entirely equivalent.
26182 It is also possible to embed conditions inside the pattern:
26183 @samp{f(x :: x>0, y) := g(y+x, x)}. This is purely a notational
26184 convenience, though; where a condition appears in a rule has no
26185 effect on when it is tested. The rewrite-rule compiler automatically
26186 decides when it is best to test each condition while a rule is being
26189 Certain conditions are handled as special cases by the rewrite rule
26190 system and are tested very efficiently: Where @expr{x} is any
26191 meta-variable, these conditions are @samp{integer(x)}, @samp{real(x)},
26192 @samp{constant(x)}, @samp{negative(x)}, @samp{x >= y} where @expr{y}
26193 is either a constant or another meta-variable and @samp{>=} may be
26194 replaced by any of the six relational operators, and @samp{x % a = b}
26195 where @expr{a} and @expr{b} are constants. Other conditions, like
26196 @samp{x >= y+1} or @samp{dreal(x)}, will be less efficient to check
26197 since Calc must bring the whole evaluator and simplifier into play.
26199 An interesting property of @samp{::} is that neither of its arguments
26200 will be touched by Calc's default simplifications. This is important
26201 because conditions often are expressions that cannot safely be
26202 evaluated early. For example, the @code{typeof} function never
26203 remains in symbolic form; entering @samp{typeof(a)} will put the
26204 number 100 (the type code for variables like @samp{a}) on the stack.
26205 But putting the condition @samp{... :: typeof(a) = 6} on the stack
26206 is safe since @samp{::} prevents the @code{typeof} from being
26207 evaluated until the condition is actually used by the rewrite system.
26209 Since @samp{::} protects its lefthand side, too, you can use a dummy
26210 condition to protect a rule that must itself not evaluate early.
26211 For example, it's not safe to put @samp{a(f,x) := apply(f, [x])} on
26212 the stack because it will immediately evaluate to @samp{a(f,x) := f(x)},
26213 where the meta-variable-ness of @code{f} on the righthand side has been
26214 lost. But @samp{a(f,x) := apply(f, [x]) :: 1} is safe, and of course
26215 the condition @samp{1} is always true (nonzero) so it has no effect on
26216 the functioning of the rule. (The rewrite compiler will ensure that
26217 it doesn't even impact the speed of matching the rule.)
26219 @node Algebraic Properties of Rewrite Rules, Other Features of Rewrite Rules, Conditional Rewrite Rules, Rewrite Rules
26220 @subsection Algebraic Properties of Rewrite Rules
26223 The rewrite mechanism understands the algebraic properties of functions
26224 like @samp{+} and @samp{*}. In particular, pattern matching takes
26225 the associativity and commutativity of the following functions into
26229 + - * = != && || and or xor vint vunion vxor gcd lcm max min beta
26232 For example, the rewrite rule:
26235 a x + b x := (a + b) x
26239 will match formulas of the form,
26242 a x + b x, x a + x b, a x + x b, x a + b x
26245 Rewrites also understand the relationship between the @samp{+} and @samp{-}
26246 operators. The above rewrite rule will also match the formulas,
26249 a x - b x, x a - x b, a x - x b, x a - b x
26253 by matching @samp{b} in the pattern to @samp{-b} from the formula.
26255 Applied to a sum of many terms like @samp{r + a x + s + b x + t}, this
26256 pattern will check all pairs of terms for possible matches. The rewrite
26257 will take whichever suitable pair it discovers first.
26259 In general, a pattern using an associative operator like @samp{a + b}
26260 will try @var{2 n} different ways to match a sum of @var{n} terms
26261 like @samp{x + y + z - w}. First, @samp{a} is matched against each
26262 of @samp{x}, @samp{y}, @samp{z}, and @samp{-w} in turn, with @samp{b}
26263 being matched to the remainders @samp{y + z - w}, @samp{x + z - w}, etc.
26264 If none of these succeed, then @samp{b} is matched against each of the
26265 four terms with @samp{a} matching the remainder. Half-and-half matches,
26266 like @samp{(x + y) + (z - w)}, are not tried.
26268 Note that @samp{*} is not commutative when applied to matrices, but
26269 rewrite rules pretend that it is. If you type @kbd{m v} to enable
26270 Matrix mode (@pxref{Matrix Mode}), rewrite rules will match @samp{*}
26271 literally, ignoring its usual commutativity property. (In the
26272 current implementation, the associativity also vanishes---it is as
26273 if the pattern had been enclosed in a @code{plain} marker; see below.)
26274 If you are applying rewrites to formulas with matrices, it's best to
26275 enable Matrix mode first to prevent algebraically incorrect rewrites
26278 The pattern @samp{-x} will actually match any expression. For example,
26286 will rewrite @samp{f(a)} to @samp{-f(-a)}. To avoid this, either use
26287 a @code{plain} marker as described below, or add a @samp{negative(x)}
26288 condition. The @code{negative} function is true if its argument
26289 ``looks'' negative, for example, because it is a negative number or
26290 because it is a formula like @samp{-x}. The new rule using this
26294 f(x) := -f(-x) :: negative(x) @r{or, equivalently,}
26295 f(-x) := -f(x) :: negative(-x)
26298 In the same way, the pattern @samp{x - y} will match the sum @samp{a + b}
26299 by matching @samp{y} to @samp{-b}.
26301 The pattern @samp{a b} will also match the formula @samp{x/y} if
26302 @samp{y} is a number. Thus the rule @samp{a x + @w{b x} := (a+b) x}
26303 will also convert @samp{a x + x / 2} to @samp{(a + 0.5) x} (or
26304 @samp{(a + 1:2) x}, depending on the current fraction mode).
26306 Calc will @emph{not} take other liberties with @samp{*}, @samp{/}, and
26307 @samp{^}. For example, the pattern @samp{f(a b)} will not match
26308 @samp{f(x^2)}, and @samp{f(a + b)} will not match @samp{f(2 x)}, even
26309 though conceivably these patterns could match with @samp{a = b = x}.
26310 Nor will @samp{f(a b)} match @samp{f(x / y)} if @samp{y} is not a
26311 constant, even though it could be considered to match with @samp{a = x}
26312 and @samp{b = 1/y}. The reasons are partly for efficiency, and partly
26313 because while few mathematical operations are substantively different
26314 for addition and subtraction, often it is preferable to treat the cases
26315 of multiplication, division, and integer powers separately.
26317 Even more subtle is the rule set
26320 [ f(a) + f(b) := f(a + b), -f(a) := f(-a) ]
26324 attempting to match @samp{f(x) - f(y)}. You might think that Calc
26325 will view this subtraction as @samp{f(x) + (-f(y))} and then apply
26326 the above two rules in turn, but actually this will not work because
26327 Calc only does this when considering rules for @samp{+} (like the
26328 first rule in this set). So it will see first that @samp{f(x) + (-f(y))}
26329 does not match @samp{f(a) + f(b)} for any assignments of the
26330 meta-variables, and then it will see that @samp{f(x) - f(y)} does
26331 not match @samp{-f(a)} for any assignment of @samp{a}. Because Calc
26332 tries only one rule at a time, it will not be able to rewrite
26333 @samp{f(x) - f(y)} with this rule set. An explicit @samp{f(a) - f(b)}
26334 rule will have to be added.
26336 Another thing patterns will @emph{not} do is break up complex numbers.
26337 The pattern @samp{myconj(a + @w{b i)} := a - b i} will work for formulas
26338 involving the special constant @samp{i} (such as @samp{3 - 4 i}), but
26339 it will not match actual complex numbers like @samp{(3, -4)}. A version
26340 of the above rule for complex numbers would be
26343 myconj(a) := re(a) - im(a) (0,1) :: im(a) != 0
26347 (Because the @code{re} and @code{im} functions understand the properties
26348 of the special constant @samp{i}, this rule will also work for
26349 @samp{3 - 4 i}. In fact, this particular rule would probably be better
26350 without the @samp{im(a) != 0} condition, since if @samp{im(a) = 0} the
26351 righthand side of the rule will still give the correct answer for the
26352 conjugate of a real number.)
26354 It is also possible to specify optional arguments in patterns. The rule
26357 opt(a) x + opt(b) (x^opt(c) + opt(d)) := f(a, b, c, d)
26361 will match the formula
26368 in a fairly straightforward manner, but it will also match reduced
26372 x + x^2, 2(x + 1) - x, x + x
26376 producing, respectively,
26379 f(1, 1, 2, 0), f(-1, 2, 1, 1), f(1, 1, 1, 0)
26382 (The latter two formulas can be entered only if default simplifications
26383 have been turned off with @kbd{m O}.)
26385 The default value for a term of a sum is zero. The default value
26386 for a part of a product, for a power, or for the denominator of a
26387 quotient, is one. Also, @samp{-x} matches the pattern @samp{opt(a) b}
26388 with @samp{a = -1}.
26390 In particular, the distributive-law rule can be refined to
26393 opt(a) x + opt(b) x := (a + b) x
26397 so that it will convert, e.g., @samp{a x - x}, to @samp{(a - 1) x}.
26399 The pattern @samp{opt(a) + opt(b) x} matches almost any formulas which
26400 are linear in @samp{x}. You can also use the @code{lin} and @code{islin}
26401 functions with rewrite conditions to test for this; @pxref{Logical
26402 Operations}. These functions are not as convenient to use in rewrite
26403 rules, but they recognize more kinds of formulas as linear:
26404 @samp{x/z} is considered linear with @expr{b = 1/z} by @code{lin},
26405 but it will not match the above pattern because that pattern calls
26406 for a multiplication, not a division.
26408 As another example, the obvious rule to replace @samp{sin(x)^2 + cos(x)^2}
26412 sin(x)^2 + cos(x)^2 := 1
26416 misses many cases because the sine and cosine may both be multiplied by
26417 an equal factor. Here's a more successful rule:
26420 opt(a) sin(x)^2 + opt(a) cos(x)^2 := a
26423 Note that this rule will @emph{not} match @samp{sin(x)^2 + 6 cos(x)^2}
26424 because one @expr{a} would have ``matched'' 1 while the other matched 6.
26426 Calc automatically converts a rule like
26436 f(temp, x) := g(x) :: temp = x-1
26440 (where @code{temp} stands for a new, invented meta-variable that
26441 doesn't actually have a name). This modified rule will successfully
26442 match @samp{f(6, 7)}, binding @samp{temp} and @samp{x} to 6 and 7,
26443 respectively, then verifying that they differ by one even though
26444 @samp{6} does not superficially look like @samp{x-1}.
26446 However, Calc does not solve equations to interpret a rule. The
26450 f(x-1, x+1) := g(x)
26454 will not work. That is, it will match @samp{f(a - 1 + b, a + 1 + b)}
26455 but not @samp{f(6, 8)}. Calc always interprets at least one occurrence
26456 of a variable by literal matching. If the variable appears ``isolated''
26457 then Calc is smart enough to use it for literal matching. But in this
26458 last example, Calc is forced to rewrite the rule to @samp{f(x-1, temp)
26459 := g(x) :: temp = x+1} where the @samp{x-1} term must correspond to an
26460 actual ``something-minus-one'' in the target formula.
26462 A successful way to write this would be @samp{f(x, x+2) := g(x+1)}.
26463 You could make this resemble the original form more closely by using
26464 @code{let} notation, which is described in the next section:
26467 f(xm1, x+1) := g(x) :: let(x := xm1+1)
26470 Calc does this rewriting or ``conditionalizing'' for any sub-pattern
26471 which involves only the functions in the following list, operating
26472 only on constants and meta-variables which have already been matched
26473 elsewhere in the pattern. When matching a function call, Calc is
26474 careful to match arguments which are plain variables before arguments
26475 which are calls to any of the functions below, so that a pattern like
26476 @samp{f(x-1, x)} can be conditionalized even though the isolated
26477 @samp{x} comes after the @samp{x-1}.
26480 + - * / \ % ^ abs sign round rounde roundu trunc floor ceil
26481 max min re im conj arg
26484 You can suppress all of the special treatments described in this
26485 section by surrounding a function call with a @code{plain} marker.
26486 This marker causes the function call which is its argument to be
26487 matched literally, without regard to commutativity, associativity,
26488 negation, or conditionalization. When you use @code{plain}, the
26489 ``deep structure'' of the formula being matched can show through.
26493 plain(a - a b) := f(a, b)
26497 will match only literal subtractions. However, the @code{plain}
26498 marker does not affect its arguments' arguments. In this case,
26499 commutativity and associativity is still considered while matching
26500 the @w{@samp{a b}} sub-pattern, so the whole pattern will match
26501 @samp{x - y x} as well as @samp{x - x y}. We could go still
26505 plain(a - plain(a b)) := f(a, b)
26509 which would do a completely strict match for the pattern.
26511 By contrast, the @code{quote} marker means that not only the
26512 function name but also the arguments must be literally the same.
26513 The above pattern will match @samp{x - x y} but
26516 quote(a - a b) := f(a, b)
26520 will match only the single formula @samp{a - a b}. Also,
26523 quote(a - quote(a b)) := f(a, b)
26527 will match only @samp{a - quote(a b)}---probably not the desired
26530 A certain amount of algebra is also done when substituting the
26531 meta-variables on the righthand side of a rule. For example,
26535 a + f(b) := f(a + b)
26539 matching @samp{f(x) - y} would produce @samp{f((-y) + x)} if
26540 taken literally, but the rewrite mechanism will simplify the
26541 righthand side to @samp{f(x - y)} automatically. (Of course,
26542 the default simplifications would do this anyway, so this
26543 special simplification is only noticeable if you have turned the
26544 default simplifications off.) This rewriting is done only when
26545 a meta-variable expands to a ``negative-looking'' expression.
26546 If this simplification is not desirable, you can use a @code{plain}
26547 marker on the righthand side:
26550 a + f(b) := f(plain(a + b))
26554 In this example, we are still allowing the pattern-matcher to
26555 use all the algebra it can muster, but the righthand side will
26556 always simplify to a literal addition like @samp{f((-y) + x)}.
26558 @node Other Features of Rewrite Rules, Composing Patterns in Rewrite Rules, Algebraic Properties of Rewrite Rules, Rewrite Rules
26559 @subsection Other Features of Rewrite Rules
26562 Certain ``function names'' serve as markers in rewrite rules.
26563 Here is a complete list of these markers. First are listed the
26564 markers that work inside a pattern; then come the markers that
26565 work in the righthand side of a rule.
26571 One kind of marker, @samp{import(x)}, takes the place of a whole
26572 rule. Here @expr{x} is the name of a variable containing another
26573 rule set; those rules are ``spliced into'' the rule set that
26574 imports them. For example, if @samp{[f(a+b) := f(a) + f(b),
26575 f(a b) := a f(b) :: real(a)]} is stored in variable @samp{linearF},
26576 then the rule set @samp{[f(0) := 0, import(linearF)]} will apply
26577 all three rules. It is possible to modify the imported rules
26578 slightly: @samp{import(x, v1, x1, v2, x2, @dots{})} imports
26579 the rule set @expr{x} with all occurrences of
26580 @texline @math{v_1},
26581 @infoline @expr{v1},
26582 as either a variable name or a function name, replaced with
26583 @texline @math{x_1}
26584 @infoline @expr{x1}
26586 @texline @math{v_1}
26587 @infoline @expr{v1}
26588 is used as a function name, then
26589 @texline @math{x_1}
26590 @infoline @expr{x1}
26591 must be either a function name itself or a @w{@samp{< >}} nameless
26592 function; @pxref{Specifying Operators}.) For example, @samp{[g(0) := 0,
26593 import(linearF, f, g)]} applies the linearity rules to the function
26594 @samp{g} instead of @samp{f}. Imports can be nested, but the
26595 import-with-renaming feature may fail to rename sub-imports properly.
26597 The special functions allowed in patterns are:
26605 This pattern matches exactly @expr{x}; variable names in @expr{x} are
26606 not interpreted as meta-variables. The only flexibility is that
26607 numbers are compared for numeric equality, so that the pattern
26608 @samp{f(quote(12))} will match both @samp{f(12)} and @samp{f(12.0)}.
26609 (Numbers are always treated this way by the rewrite mechanism:
26610 The rule @samp{f(x,x) := g(x)} will match @samp{f(12, 12.0)}.
26611 The rewrite may produce either @samp{g(12)} or @samp{g(12.0)}
26612 as a result in this case.)
26619 Here @expr{x} must be a function call @samp{f(x1,x2,@dots{})}. This
26620 pattern matches a call to function @expr{f} with the specified
26621 argument patterns. No special knowledge of the properties of the
26622 function @expr{f} is used in this case; @samp{+} is not commutative or
26623 associative. Unlike @code{quote}, the arguments @samp{x1,x2,@dots{}}
26624 are treated as patterns. If you wish them to be treated ``plainly''
26625 as well, you must enclose them with more @code{plain} markers:
26626 @samp{plain(plain(@w{-a}) + plain(b c))}.
26633 Here @expr{x} must be a variable name. This must appear as an
26634 argument to a function or an element of a vector; it specifies that
26635 the argument or element is optional.
26636 As an argument to @samp{+}, @samp{-}, @samp{*}, @samp{&&}, or @samp{||},
26637 or as the second argument to @samp{/} or @samp{^}, the value @var{def}
26638 may be omitted. The pattern @samp{x + opt(y)} matches a sum by
26639 binding one summand to @expr{x} and the other to @expr{y}, and it
26640 matches anything else by binding the whole expression to @expr{x} and
26641 zero to @expr{y}. The other operators above work similarly.
26643 For general miscellaneous functions, the default value @code{def}
26644 must be specified. Optional arguments are dropped starting with
26645 the rightmost one during matching. For example, the pattern
26646 @samp{f(opt(a,0), b, opt(c,b))} will match @samp{f(b)}, @samp{f(a,b)},
26647 or @samp{f(a,b,c)}. Default values of zero and @expr{b} are
26648 supplied in this example for the omitted arguments. Note that
26649 the literal variable @expr{b} will be the default in the latter
26650 case, @emph{not} the value that matched the meta-variable @expr{b}.
26651 In other words, the default @var{def} is effectively quoted.
26653 @item condition(x,c)
26659 This matches the pattern @expr{x}, with the attached condition
26660 @expr{c}. It is the same as @samp{x :: c}.
26668 This matches anything that matches both pattern @expr{x} and
26669 pattern @expr{y}. It is the same as @samp{x &&& y}.
26670 @pxref{Composing Patterns in Rewrite Rules}.
26678 This matches anything that matches either pattern @expr{x} or
26679 pattern @expr{y}. It is the same as @w{@samp{x ||| y}}.
26687 This matches anything that does not match pattern @expr{x}.
26688 It is the same as @samp{!!! x}.
26694 @tindex cons (rewrites)
26695 This matches any vector of one or more elements. The first
26696 element is matched to @expr{h}; a vector of the remaining
26697 elements is matched to @expr{t}. Note that vectors of fixed
26698 length can also be matched as actual vectors: The rule
26699 @samp{cons(a,cons(b,[])) := cons(a+b,[])} is equivalent
26700 to the rule @samp{[a,b] := [a+b]}.
26706 @tindex rcons (rewrites)
26707 This is like @code{cons}, except that the @emph{last} element
26708 is matched to @expr{h}, with the remaining elements matched
26711 @item apply(f,args)
26715 @tindex apply (rewrites)
26716 This matches any function call. The name of the function, in
26717 the form of a variable, is matched to @expr{f}. The arguments
26718 of the function, as a vector of zero or more objects, are
26719 matched to @samp{args}. Constants, variables, and vectors
26720 do @emph{not} match an @code{apply} pattern. For example,
26721 @samp{apply(f,x)} matches any function call, @samp{apply(quote(f),x)}
26722 matches any call to the function @samp{f}, @samp{apply(f,[a,b])}
26723 matches any function call with exactly two arguments, and
26724 @samp{apply(quote(f), cons(a,cons(b,x)))} matches any call
26725 to the function @samp{f} with two or more arguments. Another
26726 way to implement the latter, if the rest of the rule does not
26727 need to refer to the first two arguments of @samp{f} by name,
26728 would be @samp{apply(quote(f), x :: vlen(x) >= 2)}.
26729 Here's a more interesting sample use of @code{apply}:
26732 apply(f,[x+n]) := n + apply(f,[x])
26733 :: in(f, [floor,ceil,round,trunc]) :: integer(n)
26736 Note, however, that this will be slower to match than a rule
26737 set with four separate rules. The reason is that Calc sorts
26738 the rules of a rule set according to top-level function name;
26739 if the top-level function is @code{apply}, Calc must try the
26740 rule for every single formula and sub-formula. If the top-level
26741 function in the pattern is, say, @code{floor}, then Calc invokes
26742 the rule only for sub-formulas which are calls to @code{floor}.
26744 Formulas normally written with operators like @code{+} are still
26745 considered function calls: @code{apply(f,x)} matches @samp{a+b}
26746 with @samp{f = add}, @samp{x = [a,b]}.
26748 You must use @code{apply} for meta-variables with function names
26749 on both sides of a rewrite rule: @samp{apply(f, [x]) := f(x+1)}
26750 is @emph{not} correct, because it rewrites @samp{spam(6)} into
26751 @samp{f(7)}. The righthand side should be @samp{apply(f, [x+1])}.
26752 Also note that you will have to use No-Simplify mode (@kbd{m O})
26753 when entering this rule so that the @code{apply} isn't
26754 evaluated immediately to get the new rule @samp{f(x) := f(x+1)}.
26755 Or, use @kbd{s e} to enter the rule without going through the stack,
26756 or enter the rule as @samp{apply(f, [x]) := apply(f, [x+1]) @w{:: 1}}.
26757 @xref{Conditional Rewrite Rules}.
26764 This is used for applying rules to formulas with selections;
26765 @pxref{Selections with Rewrite Rules}.
26768 Special functions for the righthand sides of rules are:
26772 The notation @samp{quote(x)} is changed to @samp{x} when the
26773 righthand side is used. As far as the rewrite rule is concerned,
26774 @code{quote} is invisible. However, @code{quote} has the special
26775 property in Calc that its argument is not evaluated. Thus,
26776 while it will not work to put the rule @samp{t(a) := typeof(a)}
26777 on the stack because @samp{typeof(a)} is evaluated immediately
26778 to produce @samp{t(a) := 100}, you can use @code{quote} to
26779 protect the righthand side: @samp{t(a) := quote(typeof(a))}.
26780 (@xref{Conditional Rewrite Rules}, for another trick for
26781 protecting rules from evaluation.)
26784 Special properties of and simplifications for the function call
26785 @expr{x} are not used. One interesting case where @code{plain}
26786 is useful is the rule, @samp{q(x) := quote(x)}, trying to expand a
26787 shorthand notation for the @code{quote} function. This rule will
26788 not work as shown; instead of replacing @samp{q(foo)} with
26789 @samp{quote(foo)}, it will replace it with @samp{foo}! The correct
26790 rule would be @samp{q(x) := plain(quote(x))}.
26793 Where @expr{t} is a vector, this is converted into an expanded
26794 vector during rewrite processing. Note that @code{cons} is a regular
26795 Calc function which normally does this anyway; the only way @code{cons}
26796 is treated specially by rewrites is that @code{cons} on the righthand
26797 side of a rule will be evaluated even if default simplifications
26798 have been turned off.
26801 Analogous to @code{cons} except putting @expr{h} at the @emph{end} of
26802 the vector @expr{t}.
26804 @item apply(f,args)
26805 Where @expr{f} is a variable and @var{args} is a vector, this
26806 is converted to a function call. Once again, note that @code{apply}
26807 is also a regular Calc function.
26814 The formula @expr{x} is handled in the usual way, then the
26815 default simplifications are applied to it even if they have
26816 been turned off normally. This allows you to treat any function
26817 similarly to the way @code{cons} and @code{apply} are always
26818 treated. However, there is a slight difference: @samp{cons(2+3, [])}
26819 with default simplifications off will be converted to @samp{[2+3]},
26820 whereas @samp{eval(cons(2+3, []))} will be converted to @samp{[5]}.
26827 The formula @expr{x} has meta-variables substituted in the usual
26828 way, then algebraically simplified.
26830 @item evalextsimp(x)
26834 @tindex evalextsimp
26835 The formula @expr{x} has meta-variables substituted in the normal
26836 way, then ``extendedly'' simplified as if by the @kbd{a e} command.
26839 @xref{Selections with Rewrite Rules}.
26842 There are also some special functions you can use in conditions.
26850 The expression @expr{x} is evaluated with meta-variables substituted.
26851 The algebraic simplifications are @emph{not} applied by
26852 default, but @expr{x} can include calls to @code{evalsimp} or
26853 @code{evalextsimp} as described above to invoke higher levels
26854 of simplification. The result of @expr{x} is then bound to the
26855 meta-variable @expr{v}. As usual, if this meta-variable has already
26856 been matched to something else the two values must be equal; if the
26857 meta-variable is new then it is bound to the result of the expression.
26858 This variable can then appear in later conditions, and on the righthand
26860 In fact, @expr{v} may be any pattern in which case the result of
26861 evaluating @expr{x} is matched to that pattern, binding any
26862 meta-variables that appear in that pattern. Note that @code{let}
26863 can only appear by itself as a condition, or as one term of an
26864 @samp{&&} which is a whole condition: It cannot be inside
26865 an @samp{||} term or otherwise buried.
26867 The alternate, equivalent form @samp{let(v, x)} is also recognized.
26868 Note that the use of @samp{:=} by @code{let}, while still being
26869 assignment-like in character, is unrelated to the use of @samp{:=}
26870 in the main part of a rewrite rule.
26872 As an example, @samp{f(a) := g(ia) :: let(ia := 1/a) :: constant(ia)}
26873 replaces @samp{f(a)} with @samp{g} of the inverse of @samp{a}, if
26874 that inverse exists and is constant. For example, if @samp{a} is a
26875 singular matrix the operation @samp{1/a} is left unsimplified and
26876 @samp{constant(ia)} fails, but if @samp{a} is an invertible matrix
26877 then the rule succeeds. Without @code{let} there would be no way
26878 to express this rule that didn't have to invert the matrix twice.
26879 Note that, because the meta-variable @samp{ia} is otherwise unbound
26880 in this rule, the @code{let} condition itself always ``succeeds''
26881 because no matter what @samp{1/a} evaluates to, it can successfully
26882 be bound to @code{ia}.
26884 Here's another example, for integrating cosines of linear
26885 terms: @samp{myint(cos(y),x) := sin(y)/b :: let([a,b,x] := lin(y,x))}.
26886 The @code{lin} function returns a 3-vector if its argument is linear,
26887 or leaves itself unevaluated if not. But an unevaluated @code{lin}
26888 call will not match the 3-vector on the lefthand side of the @code{let},
26889 so this @code{let} both verifies that @code{y} is linear, and binds
26890 the coefficients @code{a} and @code{b} for use elsewhere in the rule.
26891 (It would have been possible to use @samp{sin(a x + b)/b} for the
26892 righthand side instead, but using @samp{sin(y)/b} avoids gratuitous
26893 rearrangement of the argument of the sine.)
26899 Similarly, here is a rule that implements an inverse-@code{erf}
26900 function. It uses @code{root} to search for a solution. If
26901 @code{root} succeeds, it will return a vector of two numbers
26902 where the first number is the desired solution. If no solution
26903 is found, @code{root} remains in symbolic form. So we use
26904 @code{let} to check that the result was indeed a vector.
26907 ierf(x) := y :: let([y,z] := root(erf(a) = x, a, .5))
26911 The meta-variable @var{v}, which must already have been matched
26912 to something elsewhere in the rule, is compared against pattern
26913 @var{p}. Since @code{matches} is a standard Calc function, it
26914 can appear anywhere in a condition. But if it appears alone or
26915 as a term of a top-level @samp{&&}, then you get the special
26916 extra feature that meta-variables which are bound to things
26917 inside @var{p} can be used elsewhere in the surrounding rewrite
26920 The only real difference between @samp{let(p := v)} and
26921 @samp{matches(v, p)} is that the former evaluates @samp{v} using
26922 the default simplifications, while the latter does not.
26926 This is actually a variable, not a function. If @code{remember}
26927 appears as a condition in a rule, then when that rule succeeds
26928 the original expression and rewritten expression are added to the
26929 front of the rule set that contained the rule. If the rule set
26930 was not stored in a variable, @code{remember} is ignored. The
26931 lefthand side is enclosed in @code{quote} in the added rule if it
26932 contains any variables.
26934 For example, the rule @samp{f(n) := n f(n-1) :: remember} applied
26935 to @samp{f(7)} will add the rule @samp{f(7) := 7 f(6)} to the front
26936 of the rule set. The rule set @code{EvalRules} works slightly
26937 differently: There, the evaluation of @samp{f(6)} will complete before
26938 the result is added to the rule set, in this case as @samp{f(7) := 5040}.
26939 Thus @code{remember} is most useful inside @code{EvalRules}.
26941 It is up to you to ensure that the optimization performed by
26942 @code{remember} is safe. For example, the rule @samp{foo(n) := n
26943 :: evalv(eatfoo) > 0 :: remember} is a bad idea (@code{evalv} is
26944 the function equivalent of the @kbd{=} command); if the variable
26945 @code{eatfoo} ever contains 1, rules like @samp{foo(7) := 7} will
26946 be added to the rule set and will continue to operate even if
26947 @code{eatfoo} is later changed to 0.
26954 Remember the match as described above, but only if condition @expr{c}
26955 is true. For example, @samp{remember(n % 4 = 0)} in the above factorial
26956 rule remembers only every fourth result. Note that @samp{remember(1)}
26957 is equivalent to @samp{remember}, and @samp{remember(0)} has no effect.
26960 @node Composing Patterns in Rewrite Rules, Nested Formulas with Rewrite Rules, Other Features of Rewrite Rules, Rewrite Rules
26961 @subsection Composing Patterns in Rewrite Rules
26964 There are three operators, @samp{&&&}, @samp{|||}, and @samp{!!!},
26965 that combine rewrite patterns to make larger patterns. The
26966 combinations are ``and,'' ``or,'' and ``not,'' respectively, and
26967 these operators are the pattern equivalents of @samp{&&}, @samp{||}
26968 and @samp{!} (which operate on zero-or-nonzero logical values).
26970 Note that @samp{&&&}, @samp{|||}, and @samp{!!!} are left in symbolic
26971 form by all regular Calc features; they have special meaning only in
26972 the context of rewrite rule patterns.
26974 The pattern @samp{@var{p1} &&& @var{p2}} matches anything that
26975 matches both @var{p1} and @var{p2}. One especially useful case is
26976 when one of @var{p1} or @var{p2} is a meta-variable. For example,
26977 here is a rule that operates on error forms:
26980 f(x &&& a +/- b, x) := g(x)
26983 This does the same thing, but is arguably simpler than, the rule
26986 f(a +/- b, a +/- b) := g(a +/- b)
26993 Here's another interesting example:
26996 ends(cons(a, x) &&& rcons(y, b)) := [a, b]
27000 which effectively clips out the middle of a vector leaving just
27001 the first and last elements. This rule will change a one-element
27002 vector @samp{[a]} to @samp{[a, a]}. The similar rule
27005 ends(cons(a, rcons(y, b))) := [a, b]
27009 would do the same thing except that it would fail to match a
27010 one-element vector.
27016 The pattern @samp{@var{p1} ||| @var{p2}} matches anything that
27017 matches either @var{p1} or @var{p2}. Calc first tries matching
27018 against @var{p1}; if that fails, it goes on to try @var{p2}.
27024 A simple example of @samp{|||} is
27027 curve(inf ||| -inf) := 0
27031 which converts both @samp{curve(inf)} and @samp{curve(-inf)} to zero.
27033 Here is a larger example:
27036 log(a, b) ||| (ln(a) :: let(b := e)) := mylog(a, b)
27039 This matches both generalized and natural logarithms in a single rule.
27040 Note that the @samp{::} term must be enclosed in parentheses because
27041 that operator has lower precedence than @samp{|||} or @samp{:=}.
27043 (In practice this rule would probably include a third alternative,
27044 omitted here for brevity, to take care of @code{log10}.)
27046 While Calc generally treats interior conditions exactly the same as
27047 conditions on the outside of a rule, it does guarantee that if all the
27048 variables in the condition are special names like @code{e}, or already
27049 bound in the pattern to which the condition is attached (say, if
27050 @samp{a} had appeared in this condition), then Calc will process this
27051 condition right after matching the pattern to the left of the @samp{::}.
27052 Thus, we know that @samp{b} will be bound to @samp{e} only if the
27053 @code{ln} branch of the @samp{|||} was taken.
27055 Note that this rule was careful to bind the same set of meta-variables
27056 on both sides of the @samp{|||}. Calc does not check this, but if
27057 you bind a certain meta-variable only in one branch and then use that
27058 meta-variable elsewhere in the rule, results are unpredictable:
27061 f(a,b) ||| g(b) := h(a,b)
27064 Here if the pattern matches @samp{g(17)}, Calc makes no promises about
27065 the value that will be substituted for @samp{a} on the righthand side.
27071 The pattern @samp{!!! @var{pat}} matches anything that does not
27072 match @var{pat}. Any meta-variables that are bound while matching
27073 @var{pat} remain unbound outside of @var{pat}.
27078 f(x &&& !!! a +/- b, !!![]) := g(x)
27082 converts @code{f} whose first argument is anything @emph{except} an
27083 error form, and whose second argument is not the empty vector, into
27084 a similar call to @code{g} (but without the second argument).
27086 If we know that the second argument will be a vector (empty or not),
27087 then an equivalent rule would be:
27090 f(x, y) := g(x) :: typeof(x) != 7 :: vlen(y) > 0
27094 where of course 7 is the @code{typeof} code for error forms.
27095 Another final condition, that works for any kind of @samp{y},
27096 would be @samp{!istrue(y == [])}. (The @code{istrue} function
27097 returns an explicit 0 if its argument was left in symbolic form;
27098 plain @samp{!(y == [])} or @samp{y != []} would not work to replace
27099 @samp{!!![]} since these would be left unsimplified, and thus cause
27100 the rule to fail, if @samp{y} was something like a variable name.)
27102 It is possible for a @samp{!!!} to refer to meta-variables bound
27103 elsewhere in the pattern. For example,
27110 matches any call to @code{f} with different arguments, changing
27111 this to @code{g} with only the first argument.
27113 If a function call is to be matched and one of the argument patterns
27114 contains a @samp{!!!} somewhere inside it, that argument will be
27122 will be careful to bind @samp{a} to the second argument of @code{f}
27123 before testing the first argument. If Calc had tried to match the
27124 first argument of @code{f} first, the results would have been
27125 disastrous: since @code{a} was unbound so far, the pattern @samp{a}
27126 would have matched anything at all, and the pattern @samp{!!!a}
27127 therefore would @emph{not} have matched anything at all!
27129 @node Nested Formulas with Rewrite Rules, Multi-Phase Rewrite Rules, Composing Patterns in Rewrite Rules, Rewrite Rules
27130 @subsection Nested Formulas with Rewrite Rules
27133 When @kbd{a r} (@code{calc-rewrite}) is used, it takes an expression from
27134 the top of the stack and attempts to match any of the specified rules
27135 to any part of the expression, starting with the whole expression
27136 and then, if that fails, trying deeper and deeper sub-expressions.
27137 For each part of the expression, the rules are tried in the order
27138 they appear in the rules vector. The first rule to match the first
27139 sub-expression wins; it replaces the matched sub-expression according
27140 to the @var{new} part of the rule.
27142 Often, the rule set will match and change the formula several times.
27143 The top-level formula is first matched and substituted repeatedly until
27144 it no longer matches the pattern; then, sub-formulas are tried, and
27145 so on. Once every part of the formula has gotten its chance, the
27146 rewrite mechanism starts over again with the top-level formula
27147 (in case a substitution of one of its arguments has caused it again
27148 to match). This continues until no further matches can be made
27149 anywhere in the formula.
27151 It is possible for a rule set to get into an infinite loop. The
27152 most obvious case, replacing a formula with itself, is not a problem
27153 because a rule is not considered to ``succeed'' unless the righthand
27154 side actually comes out to something different than the original
27155 formula or sub-formula that was matched. But if you accidentally
27156 had both @samp{ln(a b) := ln(a) + ln(b)} and the reverse
27157 @samp{ln(a) + ln(b) := ln(a b)} in your rule set, Calc would
27158 run forever switching a formula back and forth between the two
27161 To avoid disaster, Calc normally stops after 100 changes have been
27162 made to the formula. This will be enough for most multiple rewrites,
27163 but it will keep an endless loop of rewrites from locking up the
27164 computer forever. (On most systems, you can also type @kbd{C-g} to
27165 halt any Emacs command prematurely.)
27167 To change this limit, give a positive numeric prefix argument.
27168 In particular, @kbd{M-1 a r} applies only one rewrite at a time,
27169 useful when you are first testing your rule (or just if repeated
27170 rewriting is not what is called for by your application).
27179 You can also put a ``function call'' @samp{iterations(@var{n})}
27180 in place of a rule anywhere in your rules vector (but usually at
27181 the top). Then, @var{n} will be used instead of 100 as the default
27182 number of iterations for this rule set. You can use
27183 @samp{iterations(inf)} if you want no iteration limit by default.
27184 A prefix argument will override the @code{iterations} limit in the
27192 More precisely, the limit controls the number of ``iterations,''
27193 where each iteration is a successful matching of a rule pattern whose
27194 righthand side, after substituting meta-variables and applying the
27195 default simplifications, is different from the original sub-formula
27198 A prefix argument of zero sets the limit to infinity. Use with caution!
27200 Given a negative numeric prefix argument, @kbd{a r} will match and
27201 substitute the top-level expression up to that many times, but
27202 will not attempt to match the rules to any sub-expressions.
27204 In a formula, @code{rewrite(@var{expr}, @var{rules}, @var{n})}
27205 does a rewriting operation. Here @var{expr} is the expression
27206 being rewritten, @var{rules} is the rule, vector of rules, or
27207 variable containing the rules, and @var{n} is the optional
27208 iteration limit, which may be a positive integer, a negative
27209 integer, or @samp{inf} or @samp{-inf}. If @var{n} is omitted
27210 the @code{iterations} value from the rule set is used; if both
27211 are omitted, 100 is used.
27213 @node Multi-Phase Rewrite Rules, Selections with Rewrite Rules, Nested Formulas with Rewrite Rules, Rewrite Rules
27214 @subsection Multi-Phase Rewrite Rules
27217 It is possible to separate a rewrite rule set into several @dfn{phases}.
27218 During each phase, certain rules will be enabled while certain others
27219 will be disabled. A @dfn{phase schedule} controls the order in which
27220 phases occur during the rewriting process.
27227 If a call to the marker function @code{phase} appears in the rules
27228 vector in place of a rule, all rules following that point will be
27229 members of the phase(s) identified in the arguments to @code{phase}.
27230 Phases are given integer numbers. The markers @samp{phase()} and
27231 @samp{phase(all)} both mean the following rules belong to all phases;
27232 this is the default at the start of the rule set.
27234 If you do not explicitly schedule the phases, Calc sorts all phase
27235 numbers that appear in the rule set and executes the phases in
27236 ascending order. For example, the rule set
27253 has three phases, 1 through 3. Phase 1 consists of the @code{f0},
27254 @code{f1}, and @code{f4} rules (in that order). Phase 2 consists of
27255 @code{f0}, @code{f2}, and @code{f4}. Phase 3 consists of @code{f0}
27258 When Calc rewrites a formula using this rule set, it first rewrites
27259 the formula using only the phase 1 rules until no further changes are
27260 possible. Then it switches to the phase 2 rule set and continues
27261 until no further changes occur, then finally rewrites with phase 3.
27262 When no more phase 3 rules apply, rewriting finishes. (This is
27263 assuming @kbd{a r} with a large enough prefix argument to allow the
27264 rewriting to run to completion; the sequence just described stops
27265 early if the number of iterations specified in the prefix argument,
27266 100 by default, is reached.)
27268 During each phase, Calc descends through the nested levels of the
27269 formula as described previously. (@xref{Nested Formulas with Rewrite
27270 Rules}.) Rewriting starts at the top of the formula, then works its
27271 way down to the parts, then goes back to the top and works down again.
27272 The phase 2 rules do not begin until no phase 1 rules apply anywhere
27279 A @code{schedule} marker appearing in the rule set (anywhere, but
27280 conventionally at the top) changes the default schedule of phases.
27281 In the simplest case, @code{schedule} has a sequence of phase numbers
27282 for arguments; each phase number is invoked in turn until the
27283 arguments to @code{schedule} are exhausted. Thus adding
27284 @samp{schedule(3,2,1)} at the top of the above rule set would
27285 reverse the order of the phases; @samp{schedule(1,2,3)} would have
27286 no effect since this is the default schedule; and @samp{schedule(1,2,1,3)}
27287 would give phase 1 a second chance after phase 2 has completed, before
27288 moving on to phase 3.
27290 Any argument to @code{schedule} can instead be a vector of phase
27291 numbers (or even of sub-vectors). Then the sub-sequence of phases
27292 described by the vector are tried repeatedly until no change occurs
27293 in any phase in the sequence. For example, @samp{schedule([1, 2], 3)}
27294 tries phase 1, then phase 2, then, if either phase made any changes
27295 to the formula, repeats these two phases until they can make no
27296 further progress. Finally, it goes on to phase 3 for finishing
27299 Also, items in @code{schedule} can be variable names as well as
27300 numbers. A variable name is interpreted as the name of a function
27301 to call on the whole formula. For example, @samp{schedule(1, simplify)}
27302 says to apply the phase-1 rules (presumably, all of them), then to
27303 call @code{simplify} which is the function name equivalent of @kbd{a s}.
27304 Likewise, @samp{schedule([1, simplify])} says to alternate between
27305 phase 1 and @kbd{a s} until no further changes occur.
27307 Phases can be used purely to improve efficiency; if it is known that
27308 a certain group of rules will apply only at the beginning of rewriting,
27309 and a certain other group will apply only at the end, then rewriting
27310 will be faster if these groups are identified as separate phases.
27311 Once the phase 1 rules are done, Calc can put them aside and no longer
27312 spend any time on them while it works on phase 2.
27314 There are also some problems that can only be solved with several
27315 rewrite phases. For a real-world example of a multi-phase rule set,
27316 examine the set @code{FitRules}, which is used by the curve-fitting
27317 command to convert a model expression to linear form.
27318 @xref{Curve Fitting Details}. This set is divided into four phases.
27319 The first phase rewrites certain kinds of expressions to be more
27320 easily linearizable, but less computationally efficient. After the
27321 linear components have been picked out, the final phase includes the
27322 opposite rewrites to put each component back into an efficient form.
27323 If both sets of rules were included in one big phase, Calc could get
27324 into an infinite loop going back and forth between the two forms.
27326 Elsewhere in @code{FitRules}, the components are first isolated,
27327 then recombined where possible to reduce the complexity of the linear
27328 fit, then finally packaged one component at a time into vectors.
27329 If the packaging rules were allowed to begin before the recombining
27330 rules were finished, some components might be put away into vectors
27331 before they had a chance to recombine. By putting these rules in
27332 two separate phases, this problem is neatly avoided.
27334 @node Selections with Rewrite Rules, Matching Commands, Multi-Phase Rewrite Rules, Rewrite Rules
27335 @subsection Selections with Rewrite Rules
27338 If a sub-formula of the current formula is selected (as by @kbd{j s};
27339 @pxref{Selecting Subformulas}), the @kbd{a r} (@code{calc-rewrite})
27340 command applies only to that sub-formula. Together with a negative
27341 prefix argument, you can use this fact to apply a rewrite to one
27342 specific part of a formula without affecting any other parts.
27345 @pindex calc-rewrite-selection
27346 The @kbd{j r} (@code{calc-rewrite-selection}) command allows more
27347 sophisticated operations on selections. This command prompts for
27348 the rules in the same way as @kbd{a r}, but it then applies those
27349 rules to the whole formula in question even though a sub-formula
27350 of it has been selected. However, the selected sub-formula will
27351 first have been surrounded by a @samp{select( )} function call.
27352 (Calc's evaluator does not understand the function name @code{select};
27353 this is only a tag used by the @kbd{j r} command.)
27355 For example, suppose the formula on the stack is @samp{2 (a + b)^2}
27356 and the sub-formula @samp{a + b} is selected. This formula will
27357 be rewritten to @samp{2 select(a + b)^2} and then the rewrite
27358 rules will be applied in the usual way. The rewrite rules can
27359 include references to @code{select} to tell where in the pattern
27360 the selected sub-formula should appear.
27362 If there is still exactly one @samp{select( )} function call in
27363 the formula after rewriting is done, it indicates which part of
27364 the formula should be selected afterwards. Otherwise, the
27365 formula will be unselected.
27367 You can make @kbd{j r} act much like @kbd{a r} by enclosing both parts
27368 of the rewrite rule with @samp{select()}. However, @kbd{j r}
27369 allows you to use the current selection in more flexible ways.
27370 Suppose you wished to make a rule which removed the exponent from
27371 the selected term; the rule @samp{select(a)^x := select(a)} would
27372 work. In the above example, it would rewrite @samp{2 select(a + b)^2}
27373 to @samp{2 select(a + b)}. This would then be returned to the
27374 stack as @samp{2 (a + b)} with the @samp{a + b} selected.
27376 The @kbd{j r} command uses one iteration by default, unlike
27377 @kbd{a r} which defaults to 100 iterations. A numeric prefix
27378 argument affects @kbd{j r} in the same way as @kbd{a r}.
27379 @xref{Nested Formulas with Rewrite Rules}.
27381 As with other selection commands, @kbd{j r} operates on the stack
27382 entry that contains the cursor. (If the cursor is on the top-of-stack
27383 @samp{.} marker, it works as if the cursor were on the formula
27386 If you don't specify a set of rules, the rules are taken from the
27387 top of the stack, just as with @kbd{a r}. In this case, the
27388 cursor must indicate stack entry 2 or above as the formula to be
27389 rewritten (otherwise the same formula would be used as both the
27390 target and the rewrite rules).
27392 If the indicated formula has no selection, the cursor position within
27393 the formula temporarily selects a sub-formula for the purposes of this
27394 command. If the cursor is not on any sub-formula (e.g., it is in
27395 the line-number area to the left of the formula), the @samp{select( )}
27396 markers are ignored by the rewrite mechanism and the rules are allowed
27397 to apply anywhere in the formula.
27399 As a special feature, the normal @kbd{a r} command also ignores
27400 @samp{select( )} calls in rewrite rules. For example, if you used the
27401 above rule @samp{select(a)^x := select(a)} with @kbd{a r}, it would apply
27402 the rule as if it were @samp{a^x := a}. Thus, you can write general
27403 purpose rules with @samp{select( )} hints inside them so that they
27404 will ``do the right thing'' in both @kbd{a r} and @kbd{j r},
27405 both with and without selections.
27407 @node Matching Commands, Automatic Rewrites, Selections with Rewrite Rules, Rewrite Rules
27408 @subsection Matching Commands
27414 The @kbd{a m} (@code{calc-match}) [@code{match}] function takes a
27415 vector of formulas and a rewrite-rule-style pattern, and produces
27416 a vector of all formulas which match the pattern. The command
27417 prompts you to enter the pattern; as for @kbd{a r}, you can enter
27418 a single pattern (i.e., a formula with meta-variables), or a
27419 vector of patterns, or a variable which contains patterns, or
27420 you can give a blank response in which case the patterns are taken
27421 from the top of the stack. The pattern set will be compiled once
27422 and saved if it is stored in a variable. If there are several
27423 patterns in the set, vector elements are kept if they match any
27426 For example, @samp{match(a+b, [x, x+y, x-y, 7, x+y+z])}
27427 will return @samp{[x+y, x-y, x+y+z]}.
27429 The @code{import} mechanism is not available for pattern sets.
27431 The @kbd{a m} command can also be used to extract all vector elements
27432 which satisfy any condition: The pattern @samp{x :: x>0} will select
27433 all the positive vector elements.
27437 With the Inverse flag [@code{matchnot}], this command extracts all
27438 vector elements which do @emph{not} match the given pattern.
27444 There is also a function @samp{matches(@var{x}, @var{p})} which
27445 evaluates to 1 if expression @var{x} matches pattern @var{p}, or
27446 to 0 otherwise. This is sometimes useful for including into the
27447 conditional clauses of other rewrite rules.
27453 The function @code{vmatches} is just like @code{matches}, except
27454 that if the match succeeds it returns a vector of assignments to
27455 the meta-variables instead of the number 1. For example,
27456 @samp{vmatches(f(1,2), f(a,b))} returns @samp{[a := 1, b := 2]}.
27457 If the match fails, the function returns the number 0.
27459 @node Automatic Rewrites, Debugging Rewrites, Matching Commands, Rewrite Rules
27460 @subsection Automatic Rewrites
27463 @cindex @code{EvalRules} variable
27465 It is possible to get Calc to apply a set of rewrite rules on all
27466 results, effectively adding to the built-in set of default
27467 simplifications. To do this, simply store your rule set in the
27468 variable @code{EvalRules}. There is a convenient @kbd{s E} command
27469 for editing @code{EvalRules}; @pxref{Operations on Variables}.
27471 For example, suppose you want @samp{sin(a + b)} to be expanded out
27472 to @samp{sin(b) cos(a) + cos(b) sin(a)} wherever it appears, and
27473 similarly for @samp{cos(a + b)}. The corresponding rewrite rule
27478 [ sin(a + b) := cos(a) sin(b) + sin(a) cos(b),
27479 cos(a + b) := cos(a) cos(b) - sin(a) sin(b) ]
27483 To apply these manually, you could put them in a variable called
27484 @code{trigexp} and then use @kbd{a r trigexp} every time you wanted
27485 to expand trig functions. But if instead you store them in the
27486 variable @code{EvalRules}, they will automatically be applied to all
27487 sines and cosines of sums. Then, with @samp{2 x} and @samp{45} on
27488 the stack, typing @kbd{+ S} will (assuming Degrees mode) result in
27489 @samp{0.7071 sin(2 x) + 0.7071 cos(2 x)} automatically.
27491 As each level of a formula is evaluated, the rules from
27492 @code{EvalRules} are applied before the default simplifications.
27493 Rewriting continues until no further @code{EvalRules} apply.
27494 Note that this is different from the usual order of application of
27495 rewrite rules: @code{EvalRules} works from the bottom up, simplifying
27496 the arguments to a function before the function itself, while @kbd{a r}
27497 applies rules from the top down.
27499 Because the @code{EvalRules} are tried first, you can use them to
27500 override the normal behavior of any built-in Calc function.
27502 It is important not to write a rule that will get into an infinite
27503 loop. For example, the rule set @samp{[f(0) := 1, f(n) := n f(n-1)]}
27504 appears to be a good definition of a factorial function, but it is
27505 unsafe. Imagine what happens if @samp{f(2.5)} is simplified. Calc
27506 will continue to subtract 1 from this argument forever without reaching
27507 zero. A safer second rule would be @samp{f(n) := n f(n-1) :: n>0}.
27508 Another dangerous rule is @samp{g(x, y) := g(y, x)}. Rewriting
27509 @samp{g(2, 4)}, this would bounce back and forth between that and
27510 @samp{g(4, 2)} forever. If an infinite loop in @code{EvalRules}
27511 occurs, Emacs will eventually stop with a ``Computation got stuck
27512 or ran too long'' message.
27514 Another subtle difference between @code{EvalRules} and regular rewrites
27515 concerns rules that rewrite a formula into an identical formula. For
27516 example, @samp{f(n) := f(floor(n))} ``fails to match'' when @expr{n} is
27517 already an integer. But in @code{EvalRules} this case is detected only
27518 if the righthand side literally becomes the original formula before any
27519 further simplification. This means that @samp{f(n) := f(floor(n))} will
27520 get into an infinite loop if it occurs in @code{EvalRules}. Calc will
27521 replace @samp{f(6)} with @samp{f(floor(6))}, which is different from
27522 @samp{f(6)}, so it will consider the rule to have matched and will
27523 continue simplifying that formula; first the argument is simplified
27524 to get @samp{f(6)}, then the rule matches again to get @samp{f(floor(6))}
27525 again, ad infinitum. A much safer rule would check its argument first,
27526 say, with @samp{f(n) := f(floor(n)) :: !dint(n)}.
27528 (What really happens is that the rewrite mechanism substitutes the
27529 meta-variables in the righthand side of a rule, compares to see if the
27530 result is the same as the original formula and fails if so, then uses
27531 the default simplifications to simplify the result and compares again
27532 (and again fails if the formula has simplified back to its original
27533 form). The only special wrinkle for the @code{EvalRules} is that the
27534 same rules will come back into play when the default simplifications
27535 are used. What Calc wants to do is build @samp{f(floor(6))}, see that
27536 this is different from the original formula, simplify to @samp{f(6)},
27537 see that this is the same as the original formula, and thus halt the
27538 rewriting. But while simplifying, @samp{f(6)} will again trigger
27539 the same @code{EvalRules} rule and Calc will get into a loop inside
27540 the rewrite mechanism itself.)
27542 The @code{phase}, @code{schedule}, and @code{iterations} markers do
27543 not work in @code{EvalRules}. If the rule set is divided into phases,
27544 only the phase 1 rules are applied, and the schedule is ignored.
27545 The rules are always repeated as many times as possible.
27547 The @code{EvalRules} are applied to all function calls in a formula,
27548 but not to numbers (and other number-like objects like error forms),
27549 nor to vectors or individual variable names. (Though they will apply
27550 to @emph{components} of vectors and error forms when appropriate.) You
27551 might try to make a variable @code{phihat} which automatically expands
27552 to its definition without the need to press @kbd{=} by writing the
27553 rule @samp{quote(phihat) := (1-sqrt(5))/2}, but unfortunately this rule
27554 will not work as part of @code{EvalRules}.
27556 Finally, another limitation is that Calc sometimes calls its built-in
27557 functions directly rather than going through the default simplifications.
27558 When it does this, @code{EvalRules} will not be able to override those
27559 functions. For example, when you take the absolute value of the complex
27560 number @expr{(2, 3)}, Calc computes @samp{sqrt(2*2 + 3*3)} by calling
27561 the multiplication, addition, and square root functions directly rather
27562 than applying the default simplifications to this formula. So an
27563 @code{EvalRules} rule that (perversely) rewrites @samp{sqrt(13) := 6}
27564 would not apply. (However, if you put Calc into Symbolic mode so that
27565 @samp{sqrt(13)} will be left in symbolic form by the built-in square
27566 root function, your rule will be able to apply. But if the complex
27567 number were @expr{(3,4)}, so that @samp{sqrt(25)} must be calculated,
27568 then Symbolic mode will not help because @samp{sqrt(25)} can be
27569 evaluated exactly to 5.)
27571 One subtle restriction that normally only manifests itself with
27572 @code{EvalRules} is that while a given rewrite rule is in the process
27573 of being checked, that same rule cannot be recursively applied. Calc
27574 effectively removes the rule from its rule set while checking the rule,
27575 then puts it back once the match succeeds or fails. (The technical
27576 reason for this is that compiled pattern programs are not reentrant.)
27577 For example, consider the rule @samp{foo(x) := x :: foo(x/2) > 0}
27578 attempting to match @samp{foo(8)}. This rule will be inactive while
27579 the condition @samp{foo(4) > 0} is checked, even though it might be
27580 an integral part of evaluating that condition. Note that this is not
27581 a problem for the more usual recursive type of rule, such as
27582 @samp{foo(x) := foo(x/2)}, because there the rule has succeeded and
27583 been reactivated by the time the righthand side is evaluated.
27585 If @code{EvalRules} has no stored value (its default state), or if
27586 anything but a vector is stored in it, then it is ignored.
27588 Even though Calc's rewrite mechanism is designed to compare rewrite
27589 rules to formulas as quickly as possible, storing rules in
27590 @code{EvalRules} may make Calc run substantially slower. This is
27591 particularly true of rules where the top-level call is a commonly used
27592 function, or is not fixed. The rule @samp{f(n) := n f(n-1) :: n>0} will
27593 only activate the rewrite mechanism for calls to the function @code{f},
27594 but @samp{lg(n) + lg(m) := lg(n m)} will check every @samp{+} operator.
27597 apply(f, [a*b]) := apply(f, [a]) + apply(f, [b]) :: in(f, [ln, log10])
27601 may seem more ``efficient'' than two separate rules for @code{ln} and
27602 @code{log10}, but actually it is vastly less efficient because rules
27603 with @code{apply} as the top-level pattern must be tested against
27604 @emph{every} function call that is simplified.
27606 @cindex @code{AlgSimpRules} variable
27607 @vindex AlgSimpRules
27608 Suppose you want @samp{sin(a + b)} to be expanded out not all the time,
27609 but only when algebraic simplifications are used to simplify the
27610 formula. The variable @code{AlgSimpRules} holds rules for this purpose.
27611 The @kbd{a s} command will apply @code{EvalRules} and
27612 @code{AlgSimpRules} to the formula, as well as all of its built-in
27615 Most of the special limitations for @code{EvalRules} don't apply to
27616 @code{AlgSimpRules}. Calc simply does an @kbd{a r AlgSimpRules}
27617 command with an infinite repeat count as the first step of algebraic
27618 simplifications. It then applies its own built-in simplifications
27619 throughout the formula, and then repeats these two steps (along with
27620 applying the default simplifications) until no further changes are
27623 @cindex @code{ExtSimpRules} variable
27624 @cindex @code{UnitSimpRules} variable
27625 @vindex ExtSimpRules
27626 @vindex UnitSimpRules
27627 There are also @code{ExtSimpRules} and @code{UnitSimpRules} variables
27628 that are used by @kbd{a e} and @kbd{u s}, respectively; these commands
27629 also apply @code{EvalRules} and @code{AlgSimpRules}. The variable
27630 @code{IntegSimpRules} contains simplification rules that are used
27631 only during integration by @kbd{a i}.
27633 @node Debugging Rewrites, Examples of Rewrite Rules, Automatic Rewrites, Rewrite Rules
27634 @subsection Debugging Rewrites
27637 If a buffer named @samp{*Trace*} exists, the rewrite mechanism will
27638 record some useful information there as it operates. The original
27639 formula is written there, as is the result of each successful rewrite,
27640 and the final result of the rewriting. All phase changes are also
27643 Calc always appends to @samp{*Trace*}. You must empty this buffer
27644 yourself periodically if it is in danger of growing unwieldy.
27646 Note that the rewriting mechanism is substantially slower when the
27647 @samp{*Trace*} buffer exists, even if the buffer is not visible on
27648 the screen. Once you are done, you will probably want to kill this
27649 buffer (with @kbd{C-x k *Trace* @key{RET}}). If you leave it in
27650 existence and forget about it, all your future rewrite commands will
27651 be needlessly slow.
27653 @node Examples of Rewrite Rules, , Debugging Rewrites, Rewrite Rules
27654 @subsection Examples of Rewrite Rules
27657 Returning to the example of substituting the pattern
27658 @samp{sin(x)^2 + cos(x)^2} with 1, we saw that the rule
27659 @samp{opt(a) sin(x)^2 + opt(a) cos(x)^2 := a} does a good job of
27660 finding suitable cases. Another solution would be to use the rule
27661 @samp{cos(x)^2 := 1 - sin(x)^2}, followed by algebraic simplification
27662 if necessary. This rule will be the most effective way to do the job,
27663 but at the expense of making some changes that you might not desire.
27665 Another algebraic rewrite rule is @samp{exp(x+y) := exp(x) exp(y)}.
27666 To make this work with the @w{@kbd{j r}} command so that it can be
27667 easily targeted to a particular exponential in a large formula,
27668 you might wish to write the rule as @samp{select(exp(x+y)) :=
27669 select(exp(x) exp(y))}. The @samp{select} markers will be
27670 ignored by the regular @kbd{a r} command
27671 (@pxref{Selections with Rewrite Rules}).
27673 A surprisingly useful rewrite rule is @samp{a/(b-c) := a*(b+c)/(b^2-c^2)}.
27674 This will simplify the formula whenever @expr{b} and/or @expr{c} can
27675 be made simpler by squaring. For example, applying this rule to
27676 @samp{2 / (sqrt(2) + 3)} yields @samp{6:7 - 2:7 sqrt(2)} (assuming
27677 Symbolic mode has been enabled to keep the square root from being
27678 evaluated to a floating-point approximation). This rule is also
27679 useful when working with symbolic complex numbers, e.g.,
27680 @samp{(a + b i) / (c + d i)}.
27682 As another example, we could define our own ``triangular numbers'' function
27683 with the rules @samp{[tri(0) := 0, tri(n) := n + tri(n-1) :: n>0]}. Enter
27684 this vector and store it in a variable: @kbd{@w{s t} trirules}. Now, given
27685 a suitable formula like @samp{tri(5)} on the stack, type @samp{a r trirules}
27686 to apply these rules repeatedly. After six applications, @kbd{a r} will
27687 stop with 15 on the stack. Once these rules are debugged, it would probably
27688 be most useful to add them to @code{EvalRules} so that Calc will evaluate
27689 the new @code{tri} function automatically. We could then use @kbd{Z K} on
27690 the keyboard macro @kbd{' tri($) @key{RET}} to make a command that applies
27691 @code{tri} to the value on the top of the stack. @xref{Programming}.
27693 @cindex Quaternions
27694 The following rule set, contributed by
27695 @texline Fran\c cois
27697 Pinard, implements @dfn{quaternions}, a generalization of the concept of
27698 complex numbers. Quaternions have four components, and are here
27699 represented by function calls @samp{quat(@var{w}, [@var{x}, @var{y},
27700 @var{z}])} with ``real part'' @var{w} and the three ``imaginary'' parts
27701 collected into a vector. Various arithmetical operations on quaternions
27702 are supported. To use these rules, either add them to @code{EvalRules},
27703 or create a command based on @kbd{a r} for simplifying quaternion
27704 formulas. A convenient way to enter quaternions would be a command
27705 defined by a keyboard macro containing: @kbd{' quat($$$$, [$$$, $$, $])
27709 [ quat(w, x, y, z) := quat(w, [x, y, z]),
27710 quat(w, [0, 0, 0]) := w,
27711 abs(quat(w, v)) := hypot(w, v),
27712 -quat(w, v) := quat(-w, -v),
27713 r + quat(w, v) := quat(r + w, v) :: real(r),
27714 r - quat(w, v) := quat(r - w, -v) :: real(r),
27715 quat(w1, v1) + quat(w2, v2) := quat(w1 + w2, v1 + v2),
27716 r * quat(w, v) := quat(r * w, r * v) :: real(r),
27717 plain(quat(w1, v1) * quat(w2, v2))
27718 := quat(w1 * w2 - v1 * v2, w1 * v2 + w2 * v1 + cross(v1, v2)),
27719 quat(w1, v1) / r := quat(w1 / r, v1 / r) :: real(r),
27720 z / quat(w, v) := z * quatinv(quat(w, v)),
27721 quatinv(quat(w, v)) := quat(w, -v) / (w^2 + v^2),
27722 quatsqr(quat(w, v)) := quat(w^2 - v^2, 2 * w * v),
27723 quat(w, v)^k := quatsqr(quat(w, v)^(k / 2))
27724 :: integer(k) :: k > 0 :: k % 2 = 0,
27725 quat(w, v)^k := quatsqr(quat(w, v)^((k - 1) / 2)) * quat(w, v)
27726 :: integer(k) :: k > 2,
27727 quat(w, v)^-k := quatinv(quat(w, v)^k) :: integer(k) :: k > 0 ]
27730 Quaternions, like matrices, have non-commutative multiplication.
27731 In other words, @expr{q1 * q2 = q2 * q1} is not necessarily true if
27732 @expr{q1} and @expr{q2} are @code{quat} forms. The @samp{quat*quat}
27733 rule above uses @code{plain} to prevent Calc from rearranging the
27734 product. It may also be wise to add the line @samp{[quat(), matrix]}
27735 to the @code{Decls} matrix, to ensure that Calc's other algebraic
27736 operations will not rearrange a quaternion product. @xref{Declarations}.
27738 These rules also accept a four-argument @code{quat} form, converting
27739 it to the preferred form in the first rule. If you would rather see
27740 results in the four-argument form, just append the two items
27741 @samp{phase(2), quat(w, [x, y, z]) := quat(w, x, y, z)} to the end
27742 of the rule set. (But remember that multi-phase rule sets don't work
27743 in @code{EvalRules}.)
27745 @node Units, Store and Recall, Algebra, Top
27746 @chapter Operating on Units
27749 One special interpretation of algebraic formulas is as numbers with units.
27750 For example, the formula @samp{5 m / s^2} can be read ``five meters
27751 per second squared.'' The commands in this chapter help you
27752 manipulate units expressions in this form. Units-related commands
27753 begin with the @kbd{u} prefix key.
27756 * Basic Operations on Units::
27757 * The Units Table::
27758 * Predefined Units::
27759 * User-Defined Units::
27760 * Logarithmic Units::
27764 @node Basic Operations on Units, The Units Table, Units, Units
27765 @section Basic Operations on Units
27768 A @dfn{units expression} is a formula which is basically a number
27769 multiplied and/or divided by one or more @dfn{unit names}, which may
27770 optionally be raised to integer powers. Actually, the value part need not
27771 be a number; any product or quotient involving unit names is a units
27772 expression. Many of the units commands will also accept any formula,
27773 where the command applies to all units expressions which appear in the
27776 A unit name is a variable whose name appears in the @dfn{unit table},
27777 or a variable whose name is a prefix character like @samp{k} (for ``kilo'')
27778 or @samp{u} (for ``micro'') followed by a name in the unit table.
27779 A substantial table of built-in units is provided with Calc;
27780 @pxref{Predefined Units}. You can also define your own unit names;
27781 @pxref{User-Defined Units}.
27783 Note that if the value part of a units expression is exactly @samp{1},
27784 it will be removed by the Calculator's automatic algebra routines: The
27785 formula @samp{1 mm} is ``simplified'' to @samp{mm}. This is only a
27786 display anomaly, however; @samp{mm} will work just fine as a
27787 representation of one millimeter.
27789 You may find that Algebraic mode (@pxref{Algebraic Entry}) makes working
27790 with units expressions easier. Otherwise, you will have to remember
27791 to hit the apostrophe key every time you wish to enter units.
27794 @pindex calc-simplify-units
27796 @mindex usimpl@idots
27799 The @kbd{u s} (@code{calc-simplify-units}) [@code{usimplify}] command
27801 expression. It uses Calc's algebraic simplifications to simplify the
27802 expression first as a regular algebraic formula; it then looks for
27803 features that can be further simplified by converting one object's units
27804 to be compatible with another's. For example, @samp{5 m + 23 mm} will
27805 simplify to @samp{5.023 m}. When different but compatible units are
27806 added, the righthand term's units are converted to match those of the
27807 lefthand term. @xref{Simplification Modes}, for a way to have this done
27808 automatically at all times.
27810 Units simplification also handles quotients of two units with the same
27811 dimensionality, as in @w{@samp{2 in s/L cm}} to @samp{5.08 s/L}; fractional
27812 powers of unit expressions, as in @samp{sqrt(9 mm^2)} to @samp{3 mm} and
27813 @samp{sqrt(9 acre)} to a quantity in meters; and @code{floor},
27814 @code{ceil}, @code{round}, @code{rounde}, @code{roundu}, @code{trunc},
27815 @code{float}, @code{frac}, @code{abs}, and @code{clean}
27816 applied to units expressions, in which case
27817 the operation in question is applied only to the numeric part of the
27818 expression. Finally, trigonometric functions of quantities with units
27819 of angle are evaluated, regardless of the current angular mode.
27822 @pindex calc-convert-units
27823 The @kbd{u c} (@code{calc-convert-units}) command converts a units
27824 expression to new, compatible units. For example, given the units
27825 expression @samp{55 mph}, typing @kbd{u c m/s @key{RET}} produces
27826 @samp{24.5872 m/s}. If you have previously converted a units expression
27827 with the same type of units (in this case, distance over time), you will
27828 be offered the previous choice of new units as a default. Continuing
27829 the above example, entering the units expression @samp{100 km/hr} and
27830 typing @kbd{u c @key{RET}} (without specifying new units) produces
27831 @samp{27.7777777778 m/s}.
27834 @pindex calc-convert-temperature
27835 @cindex Temperature conversion
27836 The @kbd{u c} command treats temperature units (like @samp{degC} and
27837 @samp{K}) as relative temperatures. For example, @kbd{u c} converts
27838 @samp{10 degC} to @samp{18 degF}: A change of 10 degrees Celsius
27839 corresponds to a change of 18 degrees Fahrenheit. To convert absolute
27840 temperatures, you can use the @kbd{u t}
27841 (@code{calc-convert-temperature}) command. The value on the stack
27842 must be a simple units expression with units of temperature only.
27843 This command would convert @samp{10 degC} to @samp{50 degF}, the
27844 equivalent temperature on the Fahrenheit scale.
27846 While many of Calc's conversion factors are exact, some are necessarily
27847 approximate. If Calc is in fraction mode (@pxref{Fraction Mode}), then
27848 unit conversions will try to give exact, rational conversions, but it
27849 isn't always possible. Given @samp{55 mph} in fraction mode, typing
27850 @kbd{u c m/s @key{RET}} produces @samp{15367:625 m/s}, for example,
27851 while typing @kbd{u c au/yr @key{RET}} produces
27852 @samp{5.18665819999e-3 au/yr}.
27854 If the units you request are inconsistent with the original units, the
27855 number will be converted into your units times whatever ``remainder''
27856 units are left over. For example, converting @samp{55 mph} into acres
27857 produces @samp{6.08e-3 acre / m s}. (Recall that multiplication binds
27858 more strongly than division in Calc formulas, so the units here are
27859 acres per meter-second.) Remainder units are expressed in terms of
27860 ``fundamental'' units like @samp{m} and @samp{s}, regardless of the
27863 If you want to disallow using inconsistent units, you can set the customizable variable
27864 @code{calc-ensure-consistent-units} to @code{t} (@pxref{Customizing Calc}). In this case,
27865 if you request units which are inconsistent with the original units, you will be warned about
27866 it and no conversion will occur.
27868 One special exception is that if you specify a single unit name, and
27869 a compatible unit appears somewhere in the units expression, then
27870 that compatible unit will be converted to the new unit and the
27871 remaining units in the expression will be left alone. For example,
27872 given the input @samp{980 cm/s^2}, the command @kbd{u c ms} will
27873 change the @samp{s} to @samp{ms} to get @samp{9.8e-4 cm/ms^2}.
27874 The ``remainder unit'' @samp{cm} is left alone rather than being
27875 changed to the base unit @samp{m}.
27877 You can use explicit unit conversion instead of the @kbd{u s} command
27878 to gain more control over the units of the result of an expression.
27879 For example, given @samp{5 m + 23 mm}, you can type @kbd{u c m} or
27880 @kbd{u c mm} to express the result in either meters or millimeters.
27881 (For that matter, you could type @kbd{u c fath} to express the result
27882 in fathoms, if you preferred!)
27884 In place of a specific set of units, you can also enter one of the
27885 units system names @code{si}, @code{mks} (equivalent), or @code{cgs}.
27886 For example, @kbd{u c si @key{RET}} converts the expression into
27887 International System of Units (SI) base units. Also, @kbd{u c base}
27888 converts to Calc's base units, which are the same as @code{si} units
27889 except that @code{base} uses @samp{g} as the fundamental unit of mass
27890 whereas @code{si} uses @samp{kg}.
27892 @cindex Composite units
27893 The @kbd{u c} command also accepts @dfn{composite units}, which
27894 are expressed as the sum of several compatible unit names. For
27895 example, converting @samp{30.5 in} to units @samp{mi+ft+in} (miles,
27896 feet, and inches) produces @samp{2 ft + 6.5 in}. Calc first
27897 sorts the unit names into order of decreasing relative size.
27898 It then accounts for as much of the input quantity as it can
27899 using an integer number times the largest unit, then moves on
27900 to the next smaller unit, and so on. Only the smallest unit
27901 may have a non-integer amount attached in the result. A few
27902 standard unit names exist for common combinations, such as
27903 @code{mfi} for @samp{mi+ft+in}, and @code{tpo} for @samp{ton+lb+oz}.
27904 Composite units are expanded as if by @kbd{a x}, so that
27905 @samp{(ft+in)/hr} is first converted to @samp{ft/hr+in/hr}.
27907 If the value on the stack does not contain any units, @kbd{u c} will
27908 prompt first for the old units which this value should be considered
27909 to have, then for the new units. Assuming the old and new units you
27910 give are consistent with each other, the result also will not contain
27911 any units. For example, @kbd{@w{u c} cm @key{RET} in @key{RET}}
27912 converts the number 2 on the stack to 5.08.
27915 @pindex calc-base-units
27916 The @kbd{u b} (@code{calc-base-units}) command is shorthand for
27917 @kbd{u c base}; it converts the units expression on the top of the
27918 stack into @code{base} units. If @kbd{u s} does not simplify a
27919 units expression as far as you would like, try @kbd{u b}.
27921 Like the @kbd{u c} command, the @kbd{u b} command treats temperature
27922 units as relative temperatures.
27925 @pindex calc-remove-units
27927 @pindex calc-extract-units
27928 The @kbd{u r} (@code{calc-remove-units}) command removes units from the
27929 formula at the top of the stack. The @kbd{u x}
27930 (@code{calc-extract-units}) command extracts only the units portion of a
27931 formula. These commands essentially replace every term of the formula
27932 that does or doesn't (respectively) look like a unit name by the
27933 constant 1, then resimplify the formula.
27936 @pindex calc-autorange-units
27937 The @kbd{u a} (@code{calc-autorange-units}) command turns on and off a
27938 mode in which unit prefixes like @code{k} (``kilo'') are automatically
27939 applied to keep the numeric part of a units expression in a reasonable
27940 range. This mode affects @kbd{u s} and all units conversion commands
27941 except @kbd{u b}. For example, with autoranging on, @samp{12345 Hz}
27942 will be simplified to @samp{12.345 kHz}. Autoranging is useful for
27943 some kinds of units (like @code{Hz} and @code{m}), but is probably
27944 undesirable for non-metric units like @code{ft} and @code{tbsp}.
27945 (Composite units are more appropriate for those; see above.)
27947 Autoranging always applies the prefix to the leftmost unit name.
27948 Calc chooses the largest prefix that causes the number to be greater
27949 than or equal to 1.0. Thus an increasing sequence of adjusted times
27950 would be @samp{1 ms, 10 ms, 100 ms, 1 s, 10 s, 100 s, 1 ks}.
27951 Generally the rule of thumb is that the number will be adjusted
27952 to be in the interval @samp{[1 .. 1000)}, although there are several
27953 exceptions to this rule. First, if the unit has a power then this
27954 is not possible; @samp{0.1 s^2} simplifies to @samp{100000 ms^2}.
27955 Second, the ``centi-'' prefix is allowed to form @code{cm} (centimeters),
27956 but will not apply to other units. The ``deci-,'' ``deka-,'' and
27957 ``hecto-'' prefixes are never used. Thus the allowable interval is
27958 @samp{[1 .. 10)} for millimeters and @samp{[1 .. 100)} for centimeters.
27959 Finally, a prefix will not be added to a unit if the resulting name
27960 is also the actual name of another unit; @samp{1e-15 t} would normally
27961 be considered a ``femto-ton,'' but it is written as @samp{1000 at}
27962 (1000 atto-tons) instead because @code{ft} would be confused with feet.
27964 @node The Units Table, Predefined Units, Basic Operations on Units, Units
27965 @section The Units Table
27969 @pindex calc-enter-units-table
27970 The @kbd{u v} (@code{calc-enter-units-table}) command displays the units table
27971 in another buffer called @code{*Units Table*}. Each entry in this table
27972 gives the unit name as it would appear in an expression, the definition
27973 of the unit in terms of simpler units, and a full name or description of
27974 the unit. Fundamental units are defined as themselves; these are the
27975 units produced by the @kbd{u b} command. The fundamental units are
27976 meters, seconds, grams, kelvins, amperes, candelas, moles, radians,
27979 The Units Table buffer also displays the Unit Prefix Table. Note that
27980 two prefixes, ``kilo'' and ``hecto,'' accept either upper- or lower-case
27981 prefix letters. @samp{Meg} is also accepted as a synonym for the @samp{M}
27982 prefix. Whenever a unit name can be interpreted as either a built-in name
27983 or a prefix followed by another built-in name, the former interpretation
27984 wins. For example, @samp{2 pt} means two pints, not two pico-tons.
27986 The Units Table buffer, once created, is not rebuilt unless you define
27987 new units. To force the buffer to be rebuilt, give any numeric prefix
27988 argument to @kbd{u v}.
27991 @pindex calc-view-units-table
27992 The @kbd{u V} (@code{calc-view-units-table}) command is like @kbd{u v} except
27993 that the cursor is not moved into the Units Table buffer. You can
27994 type @kbd{u V} again to remove the Units Table from the display. To
27995 return from the Units Table buffer after a @kbd{u v}, type @kbd{C-x * c}
27996 again or use the regular Emacs @w{@kbd{C-x o}} (@code{other-window})
27997 command. You can also kill the buffer with @kbd{C-x k} if you wish;
27998 the actual units table is safely stored inside the Calculator.
28001 @pindex calc-get-unit-definition
28002 The @kbd{u g} (@code{calc-get-unit-definition}) command retrieves a unit's
28003 defining expression and pushes it onto the Calculator stack. For example,
28004 @kbd{u g in} will produce the expression @samp{2.54 cm}. This is the
28005 same definition for the unit that would appear in the Units Table buffer.
28006 Note that this command works only for actual unit names; @kbd{u g km}
28007 will report that no such unit exists, for example, because @code{km} is
28008 really the unit @code{m} with a @code{k} (``kilo'') prefix. To see a
28009 definition of a unit in terms of base units, it is easier to push the
28010 unit name on the stack and then reduce it to base units with @kbd{u b}.
28013 @pindex calc-explain-units
28014 The @kbd{u e} (@code{calc-explain-units}) command displays an English
28015 description of the units of the expression on the stack. For example,
28016 for the expression @samp{62 km^2 g / s^2 mol K}, the description is
28017 ``Square-Kilometer Gram per (Second-squared Mole Degree-Kelvin).'' This
28018 command uses the English descriptions that appear in the righthand
28019 column of the Units Table.
28021 @node Predefined Units, User-Defined Units, The Units Table, Units
28022 @section Predefined Units
28025 The definitions of many units have changed over the years. For example,
28026 the meter was originally defined in 1791 as one ten-millionth of the
28027 distance from the equator to the north pole. In order to be more
28028 precise, the definition was adjusted several times, and now a meter is
28029 defined as the distance that light will travel in a vacuum in
28030 1/299792458 of a second; consequently, the speed of light in a
28031 vacuum is exactly 299792458 m/s. Many other units have been
28032 redefined in terms of fundamental physical processes; a second, for
28033 example, is currently defined as 9192631770 periods of a certain
28034 radiation related to the cesium-133 atom. The only SI unit that is not
28035 based on a fundamental physical process (although there are efforts to
28036 change this) is the kilogram, which was originally defined as the mass
28037 of one liter of water, but is now defined as the mass of the
28038 International Prototype Kilogram (IPK), a cylinder of platinum-iridium
28039 kept at the Bureau International des Poids et Mesures in S@`evres,
28040 France. (There are several copies of the IPK throughout the world.)
28041 The British imperial units, once defined in terms of physical objects,
28042 were redefined in 1963 in terms of SI units. The US customary units,
28043 which were the same as British units until the British imperial system
28044 was created in 1824, were also defined in terms of the SI units in 1893.
28045 Because of these redefinitions, conversions between metric, British
28046 Imperial, and US customary units can often be done precisely.
28048 Since the exact definitions of many kinds of units have evolved over the
28049 years, and since certain countries sometimes have local differences in
28050 their definitions, it is a good idea to examine Calc's definition of a
28051 unit before depending on its exact value. For example, there are three
28052 different units for gallons, corresponding to the US (@code{gal}),
28053 Canadian (@code{galC}), and British (@code{galUK}) definitions. Also,
28054 note that @code{oz} is a standard ounce of mass, @code{ozt} is a Troy
28055 ounce, and @code{ozfl} is a fluid ounce.
28057 The temperature units corresponding to degrees Kelvin and Centigrade
28058 (Celsius) are the same in this table, since most units commands treat
28059 temperatures as being relative. The @code{calc-convert-temperature}
28060 command has special rules for handling the different absolute magnitudes
28061 of the various temperature scales.
28063 The unit of volume ``liters'' can be referred to by either the lower-case
28064 @code{l} or the upper-case @code{L}.
28066 The unit @code{A} stands for Amperes; the name @code{Ang} is used
28074 The unit @code{pt} stands for pints; the name @code{point} stands for
28075 a typographical point, defined by @samp{72 point = 1 in}. This is
28076 slightly different than the point defined by the American Typefounder's
28077 Association in 1886, but the point used by Calc has become standard
28078 largely due to its use by the PostScript page description language.
28079 There is also @code{texpt}, which stands for a printer's point as
28080 defined by the @TeX{} typesetting system: @samp{72.27 texpt = 1 in}.
28081 Other units used by @TeX{} are available; they are @code{texpc} (a pica),
28082 @code{texbp} (a ``big point'', equal to a standard point which is larger
28083 than the point used by @TeX{}), @code{texdd} (a Didot point),
28084 @code{texcc} (a Cicero) and @code{texsp} (a scaled @TeX{} point,
28085 all dimensions representable in @TeX{} are multiples of this value).
28087 When Calc is using the @TeX{} or @LaTeX{} language mode (@pxref{TeX
28088 and LaTeX Language Modes}), the @TeX{} specific unit names will not
28089 use the @samp{tex} prefix; the unit name for a @TeX{} point will be
28090 @samp{pt} instead of @samp{texpt}, for example. To avoid conflicts,
28091 the unit names for pint and parsec will simply be @samp{pint} and
28092 @samp{parsec} instead of @samp{pt} and @samp{pc}.
28095 The unit @code{e} stands for the elementary (electron) unit of charge;
28096 because algebra command could mistake this for the special constant
28097 @expr{e}, Calc provides the alternate unit name @code{ech} which is
28098 preferable to @code{e}.
28100 The name @code{g} stands for one gram of mass; there is also @code{gf},
28101 one gram of force. (Likewise for @kbd{lb}, pounds, and @kbd{lbf}.)
28102 Meanwhile, one ``@expr{g}'' of acceleration is denoted @code{ga}.
28104 The unit @code{ton} is a U.S. ton of @samp{2000 lb}, and @code{t} is
28105 a metric ton of @samp{1000 kg}.
28107 The names @code{s} (or @code{sec}) and @code{min} refer to units of
28108 time; @code{arcsec} and @code{arcmin} are units of angle.
28110 Some ``units'' are really physical constants; for example, @code{c}
28111 represents the speed of light, and @code{h} represents Planck's
28112 constant. You can use these just like other units: converting
28113 @samp{.5 c} to @samp{m/s} expresses one-half the speed of light in
28114 meters per second. You can also use this merely as a handy reference;
28115 the @kbd{u g} command gets the definition of one of these constants
28116 in its normal terms, and @kbd{u b} expresses the definition in base
28119 Two units, @code{pi} and @code{alpha} (the fine structure constant,
28120 approximately @mathit{1/137}) are dimensionless. The units simplification
28121 commands simply treat these names as equivalent to their corresponding
28122 values. However you can, for example, use @kbd{u c} to convert a pure
28123 number into multiples of the fine structure constant, or @kbd{u b} to
28124 convert this back into a pure number. (When @kbd{u c} prompts for the
28125 ``old units,'' just enter a blank line to signify that the value
28126 really is unitless.)
28128 @c Describe angular units, luminosity vs. steradians problem.
28130 @node User-Defined Units, Logarithmic Units, Predefined Units, Units
28131 @section User-Defined Units
28134 Calc provides ways to get quick access to your selected ``favorite''
28135 units, as well as ways to define your own new units.
28138 @pindex calc-quick-units
28140 @cindex @code{Units} variable
28141 @cindex Quick units
28142 To select your favorite units, store a vector of unit names or
28143 expressions in the Calc variable @code{Units}. The @kbd{u 1}
28144 through @kbd{u 9} commands (@code{calc-quick-units}) provide access
28145 to these units. If the value on the top of the stack is a plain
28146 number (with no units attached), then @kbd{u 1} gives it the
28147 specified units. (Basically, it multiplies the number by the
28148 first item in the @code{Units} vector.) If the number on the
28149 stack @emph{does} have units, then @kbd{u 1} converts that number
28150 to the new units. For example, suppose the vector @samp{[in, ft]}
28151 is stored in @code{Units}. Then @kbd{30 u 1} will create the
28152 expression @samp{30 in}, and @kbd{u 2} will convert that expression
28155 The @kbd{u 0} command accesses the tenth element of @code{Units}.
28156 Only ten quick units may be defined at a time. If the @code{Units}
28157 variable has no stored value (the default), or if its value is not
28158 a vector, then the quick-units commands will not function. The
28159 @kbd{s U} command is a convenient way to edit the @code{Units}
28160 variable; @pxref{Operations on Variables}.
28163 @pindex calc-define-unit
28164 @cindex User-defined units
28165 The @kbd{u d} (@code{calc-define-unit}) command records the units
28166 expression on the top of the stack as the definition for a new,
28167 user-defined unit. For example, putting @samp{16.5 ft} on the stack and
28168 typing @kbd{u d rod} defines the new unit @samp{rod} to be equivalent to
28169 16.5 feet. The unit conversion and simplification commands will now
28170 treat @code{rod} just like any other unit of length. You will also be
28171 prompted for an optional English description of the unit, which will
28172 appear in the Units Table. If you wish the definition of this unit to
28173 be displayed in a special way in the Units Table buffer (such as with an
28174 asterisk to indicate an approximate value), then you can call this
28175 command with an argument, @kbd{C-u u d}; you will then also be prompted
28176 for a string that will be used to display the definition.
28179 @pindex calc-undefine-unit
28180 The @kbd{u u} (@code{calc-undefine-unit}) command removes a user-defined
28181 unit. It is not possible to remove one of the predefined units,
28184 If you define a unit with an existing unit name, your new definition
28185 will replace the original definition of that unit. If the unit was a
28186 predefined unit, the old definition will not be replaced, only
28187 ``shadowed.'' The built-in definition will reappear if you later use
28188 @kbd{u u} to remove the shadowing definition.
28190 To create a new fundamental unit, use either 1 or the unit name itself
28191 as the defining expression. Otherwise the expression can involve any
28192 other units that you like (except for composite units like @samp{mfi}).
28193 You can create a new composite unit with a sum of other units as the
28194 defining expression. The next unit operation like @kbd{u c} or @kbd{u v}
28195 will rebuild the internal unit table incorporating your modifications.
28196 Note that erroneous definitions (such as two units defined in terms of
28197 each other) will not be detected until the unit table is next rebuilt;
28198 @kbd{u v} is a convenient way to force this to happen.
28200 Temperature units are treated specially inside the Calculator; it is not
28201 possible to create user-defined temperature units.
28204 @pindex calc-permanent-units
28205 @cindex Calc init file, user-defined units
28206 The @kbd{u p} (@code{calc-permanent-units}) command stores the user-defined
28207 units in your Calc init file (the file given by the variable
28208 @code{calc-settings-file}, typically @file{~/.emacs.d/calc.el}), so that the
28209 units will still be available in subsequent Emacs sessions. If there
28210 was already a set of user-defined units in your Calc init file, it
28211 is replaced by the new set. (@xref{General Mode Commands}, for a way to
28212 tell Calc to use a different file for the Calc init file.)
28214 @node Logarithmic Units, Musical Notes, User-Defined Units, Units
28215 @section Logarithmic Units
28217 The units @code{dB} (decibels) and @code{Np} (nepers) are logarithmic
28218 units which are manipulated differently than standard units. Calc
28219 provides commands to work with these logarithmic units.
28221 Decibels and nepers are used to measure power quantities as well as
28222 field quantities (quantities whose squares are proportional to power);
28223 these two types of quantities are handled slightly different from each
28224 other. By default the Calc commands work as if power quantities are
28225 being used; with the @kbd{H} prefix the Calc commands work as if field
28226 quantities are being used.
28228 The decibel level of a power
28229 @infoline @math{P1},
28230 @texline @math{P_1},
28231 relative to a reference power
28232 @infoline @math{P0},
28233 @texline @math{P_0},
28235 @infoline @math{10 log10(P1/P0) dB}.
28236 @texline @math{10 \log_{10}(P_{1}/P_{0}) {\rm dB}}.
28237 (The factor of 10 is because a decibel, as its name implies, is
28238 one-tenth of a bel. The bel, named after Alexander Graham Bell, was
28239 considered to be too large of a unit and was effectively replaced by
28240 the decibel.) If @math{F} is a field quantity with power
28241 @math{P=k F^2}, then a reference quantity of
28242 @infoline @math{F0}
28243 @texline @math{F_0}
28244 would correspond to a power of
28245 @infoline @math{P0=k F0^2}.
28246 @texline @math{P_{0}=kF_{0}^2}.
28248 @infoline @math{P1=k F1^2},
28249 @texline @math{P_{1}=kF_{1}^2},
28254 10 log10(P1/P0) = 10 log10(F1^2/F0^2) = 20 log10(F1/F0).
28258 $$ 10 \log_{10}(P_1/P_0) = 10 \log_{10}(F_1^2/F_0^2) = 20
28259 \log_{10}(F_1/F_0)$$
28263 In order to get the same decibel level regardless of whether a field
28264 quantity or the corresponding power quantity is used, the decibel
28265 level of a field quantity
28266 @infoline @math{F1},
28267 @texline @math{F_1},
28268 relative to a reference
28269 @infoline @math{F0},
28270 @texline @math{F_0},
28272 @infoline @math{20 log10(F1/F0) dB}.
28273 @texline @math{20 \log_{10}(F_{1}/F_{0}) {\rm dB}}.
28274 For example, the decibel value of a sound pressure level of
28275 @infoline @math{60 uPa}
28276 @texline @math{60 \mu{\rm Pa}}
28278 @infoline @math{20 uPa}
28279 @texline @math{20 \mu{\rm Pa}}
28280 (the threshold of human hearing) is
28281 @infoline @math{20 log10(60 uPa/ 20 uPa) dB = 20 log10(3) dB},
28282 @texline @math{20 \log_{10}(60 \mu{\rm Pa}/20 \mu{\rm Pa}) {\rm dB} = 20 \log_{10}(3) {\rm dB}},
28284 @infoline @math{9.54 dB}.
28285 @texline @math{9.54 {\rm dB}}.
28286 Note that in taking the ratio, the original units cancel and so these
28287 logarithmic units are dimensionless.
28289 Nepers (named after John Napier, who is credited with inventing the
28290 logarithm) are similar to bels except they use natural logarithms instead
28291 of common logarithms. The neper level of a power
28292 @infoline @math{P1},
28293 @texline @math{P_1},
28294 relative to a reference power
28295 @infoline @math{P0},
28296 @texline @math{P_0},
28298 @infoline @math{(1/2) ln(P1/P0) Np}.
28299 @texline @math{(1/2) \ln(P_1/P_0) {\rm Np}}.
28300 The neper level of a field
28301 @infoline @math{F1},
28302 @texline @math{F_1},
28303 relative to a reference field
28304 @infoline @math{F0},
28305 @texline @math{F_0},
28307 @infoline @math{ln(F1/F0) Np}.
28308 @texline @math{\ln(F_1/F_0) {\rm Np}}.
28310 @vindex calc-lu-power-reference
28311 @vindex calc-lu-field-reference
28312 For power quantities, Calc uses
28313 @infoline @math{1 mW}
28314 @texline @math{1 {\rm mW}}
28315 as the default reference quantity; this default can be changed by changing
28316 the value of the customizable variable
28317 @code{calc-lu-power-reference} (@pxref{Customizing Calc}).
28318 For field quantities, Calc uses
28319 @infoline @math{20 uPa}
28320 @texline @math{20 \mu{\rm Pa}}
28321 as the default reference quantity; this is the value used in acoustics
28322 which is where decibels are commonly encountered. This default can be
28323 changed by changing the value of the customizable variable
28324 @code{calc-lu-field-reference} (@pxref{Customizing Calc}). A
28325 non-default reference quantity will be read from the stack if the
28326 capital @kbd{O} prefix is used.
28329 @pindex calc-lu-quant
28332 The @kbd{l q} (@code{calc-lu-quant}) [@code{lupquant}]
28333 command computes the power quantity corresponding to a given number of
28334 logarithmic units. With the capital @kbd{O} prefix, @kbd{O l q}, the
28335 reference level will be read from the top of the stack. (In an
28336 algebraic formula, @code{lupquant} can be given an optional second
28337 argument which will be used for the reference level.) For example,
28338 @code{20 dB @key{RET} l q} will return @code{100 mW};
28339 @code{20 dB @key{RET} 4 W @key{RET} O l q} will return @code{400 W}.
28340 The @kbd{H l q} [@code{lufquant}] command behaves like @kbd{l q} but
28341 computes field quantities instead of power quantities.
28351 The @kbd{l d} (@code{calc-db}) [@code{dbpower}] command will compute
28352 the decibel level of a power quantity using the default reference
28353 level; @kbd{H l d} [@code{dbfield}] will compute the decibel level of
28354 a field quantity. The commands @kbd{l n} (@code{calc-np})
28355 [@code{nppower}] and @kbd{H l n} [@code{npfield}] will similarly
28356 compute neper levels. With the capital @kbd{O} prefix these commands
28357 will read a reference level from the stack; in an algebraic formula
28358 the reference level can be given as an optional second argument.
28361 @pindex calc-lu-plus
28365 @pindex calc-lu-minus
28369 @pindex calc-lu-times
28373 @pindex calc-lu-divide
28376 The sum of two power or field quantities doesn't correspond to the sum
28377 of the corresponding decibel or neper levels. If the powers
28378 corresponding to decibel levels
28379 @infoline @math{D1}
28380 @texline @math{D_1}
28382 @infoline @math{D2}
28383 @texline @math{D_2}
28384 are added, the corresponding decibel level ``sum'' will be
28388 10 log10(10^(D1/10) + 10^(D2/10)) dB.
28392 $$ 10 \log_{10}(10^{D_1/10} + 10^{D_2/10}) {\rm dB}.$$
28396 When field quantities are combined, it often means the corresponding
28397 powers are added and so the above formula might be used. In
28398 acoustics, for example, the sound pressure level is a field quantity
28399 and so the decibels are often defined using the field formula, but the
28400 sound pressure levels are combined as the sound power levels, and so
28401 the above formula should be used. If two field quantities themselves
28402 are added, the new decibel level will be
28406 20 log10(10^(D1/20) + 10^(D2/20)) dB.
28410 $$ 20 \log_{10}(10^{D_1/20} + 10^{D_2/20}) {\rm dB}.$$
28414 If the power corresponding to @math{D} dB is multiplied by a number @math{N},
28415 then the corresponding decibel level will be
28419 D + 10 log10(N) dB,
28423 $$ D + 10 \log_{10}(N) {\rm dB},$$
28427 if a field quantity is multiplied by @math{N} the corresponding decibel level
28432 D + 20 log10(N) dB.
28436 $$ D + 20 \log_{10}(N) {\rm dB}.$$
28440 There are similar formulas for combining nepers. The @kbd{l +}
28441 (@code{calc-lu-plus}) [@code{lupadd}] command will ``add'' two
28442 logarithmic unit power levels this way; with the @kbd{H} prefix,
28443 @kbd{H l +} [@code{lufadd}] will add logarithmic unit field levels.
28444 Similarly, logarithmic units can be ``subtracted'' with @kbd{l -}
28445 (@code{calc-lu-minus}) [@code{lupsub}] or @kbd{H l -} [@code{lufsub}].
28446 The @kbd{l *} (@code{calc-lu-times}) [@code{lupmul}] and @kbd{H l *}
28447 [@code{lufmul}] commands will ``multiply'' a logarithmic unit by a
28448 number; the @kbd{l /} (@code{calc-lu-divide}) [@code{lupdiv}] and
28449 @kbd{H l /} [@code{lufdiv}] commands will ``divide'' a logarithmic
28450 unit by a number. Note that the reference quantities don't play a role
28451 in this arithmetic.
28453 @node Musical Notes, , Logarithmic Units, Units
28454 @section Musical Notes
28456 Calc can convert between musical notes and their associated
28457 frequencies. Notes can be given using either scientific pitch
28458 notation or midi numbers. Since these note systems are basically
28459 logarithmic scales, Calc uses the @kbd{l} prefix for functions
28460 operating on notes.
28462 Scientific pitch notation refers to a note by giving a letter
28463 A through G, possibly followed by a flat or sharp) with a subscript
28464 indicating an octave number. Each octave starts with C and ends with
28466 @c increasing each note by a semitone will result
28467 @c in the sequence @expr{C}, @expr{C} sharp, @expr{D}, @expr{E} flat, @expr{E},
28468 @c @expr{F}, @expr{F} sharp, @expr{G}, @expr{A} flat, @expr{A}, @expr{B}
28469 @c flat and @expr{B}.
28470 the octave numbered 0 was chosen to correspond to the lowest
28471 audible frequency. Using this system, middle C (about 261.625 Hz)
28472 corresponds to the note @expr{C} in octave 4 and is denoted
28473 @expr{C_4}. Any frequency can be described by giving a note plus an
28474 offset in cents (where a cent is a ratio of frequencies so that a
28475 semitone consists of 100 cents).
28477 The midi note number system assigns numbers to notes so that
28478 @expr{C_(-1)} corresponds to the midi note number 0 and @expr{G_9}
28479 corresponds to the midi note number 127. A midi controller can have
28480 up to 128 keys and each midi note number from 0 to 127 corresponds to
28486 The @kbd{l s} (@code{calc-spn}) [@code{spn}] command converts either
28487 a frequency or a midi number to scientific pitch notation. For
28488 example, @code{500 Hz} gets converted to
28489 @code{B_4 + 21.3094853649 cents} and @code{84} to @code{C_6}.
28495 The @kbd{l m} (@code{calc-midi}) [@code{midi}] command converts either
28496 a frequency or a note given in scientific pitch notation to the
28497 corresponding midi number. For example, @code{C_6} gets converted to 84
28498 and @code{440 Hz} to 69.
28503 The @kbd{l f} (@code{calc-freq}) [@code{freq}] command converts either
28504 either a midi number or a note given in scientific pitch notation to
28505 the corresponding frequency. For example, @code{Asharp_2 + 30 cents}
28506 gets converted to @code{118.578040134 Hz} and @code{55} to
28507 @code{195.99771799 Hz}.
28509 Since the frequencies of notes are not usually given exactly (and are
28510 typically irrational), the customizable variable
28511 @code{calc-note-threshold} determines how close (in cents) a frequency
28512 needs to be to a note to be recognized as that note
28513 (@pxref{Customizing Calc}). This variable has a default value of
28514 @code{1}. For example, middle @var{C} is approximately
28515 @expr{261.625565302 Hz}; this frequency is often shortened to
28516 @expr{261.625 Hz}. Without @code{calc-note-threshold} (or a value of
28517 @expr{0}), Calc would convert @code{261.625 Hz} to scientific pitch
28518 notation @code{B_3 + 99.9962592773 cents}; with the default value of
28519 @code{1}, Calc converts @code{261.625 Hz} to @code{C_4}.
28523 @node Store and Recall, Graphics, Units, Top
28524 @chapter Storing and Recalling
28527 Calculator variables are really just Lisp variables that contain numbers
28528 or formulas in a form that Calc can understand. The commands in this
28529 section allow you to manipulate variables conveniently. Commands related
28530 to variables use the @kbd{s} prefix key.
28533 * Storing Variables::
28534 * Recalling Variables::
28535 * Operations on Variables::
28537 * Evaluates-To Operator::
28540 @node Storing Variables, Recalling Variables, Store and Recall, Store and Recall
28541 @section Storing Variables
28546 @cindex Storing variables
28547 @cindex Quick variables
28550 The @kbd{s s} (@code{calc-store}) command stores the value at the top of
28551 the stack into a specified variable. It prompts you to enter the
28552 name of the variable. If you press a single digit, the value is stored
28553 immediately in one of the ``quick'' variables @code{q0} through
28554 @code{q9}. Or you can enter any variable name.
28557 @pindex calc-store-into
28558 The @kbd{s s} command leaves the stored value on the stack. There is
28559 also an @kbd{s t} (@code{calc-store-into}) command, which removes a
28560 value from the stack and stores it in a variable.
28562 If the top of stack value is an equation @samp{a = 7} or assignment
28563 @samp{a := 7} with a variable on the lefthand side, then Calc will
28564 assign that variable with that value by default, i.e., if you type
28565 @kbd{s s @key{RET}} or @kbd{s t @key{RET}}. In this example, the
28566 value 7 would be stored in the variable @samp{a}. (If you do type
28567 a variable name at the prompt, the top-of-stack value is stored in
28568 its entirety, even if it is an equation: @samp{s s b @key{RET}}
28569 with @samp{a := 7} on the stack stores @samp{a := 7} in @code{b}.)
28571 In fact, the top of stack value can be a vector of equations or
28572 assignments with different variables on their lefthand sides; the
28573 default will be to store all the variables with their corresponding
28574 righthand sides simultaneously.
28576 It is also possible to type an equation or assignment directly at
28577 the prompt for the @kbd{s s} or @kbd{s t} command: @kbd{s s foo = 7}.
28578 In this case the expression to the right of the @kbd{=} or @kbd{:=}
28579 symbol is evaluated as if by the @kbd{=} command, and that value is
28580 stored in the variable. No value is taken from the stack; @kbd{s s}
28581 and @kbd{s t} are equivalent when used in this way.
28585 The prefix keys @kbd{s} and @kbd{t} may be followed immediately by a
28586 digit; @kbd{s 9} is equivalent to @kbd{s s 9}, and @kbd{t 9} is
28587 equivalent to @kbd{s t 9}. (The @kbd{t} prefix is otherwise used
28588 for trail and time/date commands.)
28624 @pindex calc-store-plus
28625 @pindex calc-store-minus
28626 @pindex calc-store-times
28627 @pindex calc-store-div
28628 @pindex calc-store-power
28629 @pindex calc-store-concat
28630 @pindex calc-store-neg
28631 @pindex calc-store-inv
28632 @pindex calc-store-decr
28633 @pindex calc-store-incr
28634 There are also several ``arithmetic store'' commands. For example,
28635 @kbd{s +} removes a value from the stack and adds it to the specified
28636 variable. The other arithmetic stores are @kbd{s -}, @kbd{s *}, @kbd{s /},
28637 @kbd{s ^}, and @w{@kbd{s |}} (vector concatenation), plus @kbd{s n} and
28638 @kbd{s &} which negate or invert the value in a variable, and @w{@kbd{s [}}
28639 and @kbd{s ]} which decrease or increase a variable by one.
28641 All the arithmetic stores accept the Inverse prefix to reverse the
28642 order of the operands. If @expr{v} represents the contents of the
28643 variable, and @expr{a} is the value drawn from the stack, then regular
28644 @w{@kbd{s -}} assigns
28645 @texline @math{v \coloneq v - a},
28646 @infoline @expr{v := v - a},
28647 but @kbd{I s -} assigns
28648 @texline @math{v \coloneq a - v}.
28649 @infoline @expr{v := a - v}.
28650 While @kbd{I s *} might seem pointless, it is
28651 useful if matrix multiplication is involved. Actually, all the
28652 arithmetic stores use formulas designed to behave usefully both
28653 forwards and backwards:
28657 s + v := v + a v := a + v
28658 s - v := v - a v := a - v
28659 s * v := v * a v := a * v
28660 s / v := v / a v := a / v
28661 s ^ v := v ^ a v := a ^ v
28662 s | v := v | a v := a | v
28663 s n v := v / (-1) v := (-1) / v
28664 s & v := v ^ (-1) v := (-1) ^ v
28665 s [ v := v - 1 v := 1 - v
28666 s ] v := v - (-1) v := (-1) - v
28670 In the last four cases, a numeric prefix argument will be used in
28671 place of the number one. (For example, @kbd{M-2 s ]} increases
28672 a variable by 2, and @kbd{M-2 I s ]} replaces a variable by
28673 minus-two minus the variable.
28675 The first six arithmetic stores can also be typed @kbd{s t +}, @kbd{s t -},
28676 etc. The commands @kbd{s s +}, @kbd{s s -}, and so on are analogous
28677 arithmetic stores that don't remove the value @expr{a} from the stack.
28679 All arithmetic stores report the new value of the variable in the
28680 Trail for your information. They signal an error if the variable
28681 previously had no stored value. If default simplifications have been
28682 turned off, the arithmetic stores temporarily turn them on for numeric
28683 arguments only (i.e., they temporarily do an @kbd{m N} command).
28684 @xref{Simplification Modes}. Large vectors put in the trail by
28685 these commands always use abbreviated (@kbd{t .}) mode.
28688 @pindex calc-store-map
28689 The @kbd{s m} command is a general way to adjust a variable's value
28690 using any Calc function. It is a ``mapping'' command analogous to
28691 @kbd{V M}, @kbd{V R}, etc. @xref{Reducing and Mapping}, to see
28692 how to specify a function for a mapping command. Basically,
28693 all you do is type the Calc command key that would invoke that
28694 function normally. For example, @kbd{s m n} applies the @kbd{n}
28695 key to negate the contents of the variable, so @kbd{s m n} is
28696 equivalent to @kbd{s n}. Also, @kbd{s m Q} takes the square root
28697 of the value stored in a variable, @kbd{s m v v} uses @kbd{v v} to
28698 reverse the vector stored in the variable, and @kbd{s m H I S}
28699 takes the hyperbolic arcsine of the variable contents.
28701 If the mapping function takes two or more arguments, the additional
28702 arguments are taken from the stack; the old value of the variable
28703 is provided as the first argument. Thus @kbd{s m -} with @expr{a}
28704 on the stack computes @expr{v - a}, just like @kbd{s -}. With the
28705 Inverse prefix, the variable's original value becomes the @emph{last}
28706 argument instead of the first. Thus @kbd{I s m -} is also
28707 equivalent to @kbd{I s -}.
28710 @pindex calc-store-exchange
28711 The @kbd{s x} (@code{calc-store-exchange}) command exchanges the value
28712 of a variable with the value on the top of the stack. Naturally, the
28713 variable must already have a stored value for this to work.
28715 You can type an equation or assignment at the @kbd{s x} prompt. The
28716 command @kbd{s x a=6} takes no values from the stack; instead, it
28717 pushes the old value of @samp{a} on the stack and stores @samp{a = 6}.
28720 @pindex calc-unstore
28721 @cindex Void variables
28722 @cindex Un-storing variables
28723 Until you store something in them, most variables are ``void,'' that is,
28724 they contain no value at all. If they appear in an algebraic formula
28725 they will be left alone even if you press @kbd{=} (@code{calc-evaluate}).
28726 The @kbd{s u} (@code{calc-unstore}) command returns a variable to the
28730 @pindex calc-copy-variable
28731 The @kbd{s c} (@code{calc-copy-variable}) command copies the stored
28732 value of one variable to another. One way it differs from a simple
28733 @kbd{s r} followed by an @kbd{s t} (aside from saving keystrokes) is
28734 that the value never goes on the stack and thus is never rounded,
28735 evaluated, or simplified in any way; it is not even rounded down to the
28738 The only variables with predefined values are the ``special constants''
28739 @code{pi}, @code{e}, @code{i}, @code{phi}, and @code{gamma}. You are free
28740 to unstore these variables or to store new values into them if you like,
28741 although some of the algebraic-manipulation functions may assume these
28742 variables represent their standard values. Calc displays a warning if
28743 you change the value of one of these variables, or of one of the other
28744 special variables @code{inf}, @code{uinf}, and @code{nan} (which are
28747 Note that @code{pi} doesn't actually have 3.14159265359 stored in it,
28748 but rather a special magic value that evaluates to @cpi{} at the current
28749 precision. Likewise @code{e}, @code{i}, and @code{phi} evaluate
28750 according to the current precision or polar mode. If you recall a value
28751 from @code{pi} and store it back, this magic property will be lost. The
28752 magic property is preserved, however, when a variable is copied with
28756 @pindex calc-copy-special-constant
28757 If one of the ``special constants'' is redefined (or undefined) so that
28758 it no longer has its magic property, the property can be restored with
28759 @kbd{s k} (@code{calc-copy-special-constant}). This command will prompt
28760 for a special constant and a variable to store it in, and so a special
28761 constant can be stored in any variable. Here, the special constant that
28762 you enter doesn't depend on the value of the corresponding variable;
28763 @code{pi} will represent 3.14159@dots{} regardless of what is currently
28764 stored in the Calc variable @code{pi}. If one of the other special
28765 variables, @code{inf}, @code{uinf} or @code{nan}, is given a value, its
28766 original behavior can be restored by voiding it with @kbd{s u}.
28768 @node Recalling Variables, Operations on Variables, Storing Variables, Store and Recall
28769 @section Recalling Variables
28773 @pindex calc-recall
28774 @cindex Recalling variables
28775 The most straightforward way to extract the stored value from a variable
28776 is to use the @kbd{s r} (@code{calc-recall}) command. This command prompts
28777 for a variable name (similarly to @code{calc-store}), looks up the value
28778 of the specified variable, and pushes that value onto the stack. It is
28779 an error to try to recall a void variable.
28781 It is also possible to recall the value from a variable by evaluating a
28782 formula containing that variable. For example, @kbd{' a @key{RET} =} is
28783 the same as @kbd{s r a @key{RET}} except that if the variable is void, the
28784 former will simply leave the formula @samp{a} on the stack whereas the
28785 latter will produce an error message.
28788 The @kbd{r} prefix may be followed by a digit, so that @kbd{r 9} is
28789 equivalent to @kbd{s r 9}.
28791 @node Operations on Variables, Let Command, Recalling Variables, Store and Recall
28792 @section Other Operations on Variables
28796 @pindex calc-edit-variable
28797 The @kbd{s e} (@code{calc-edit-variable}) command edits the stored
28798 value of a variable without ever putting that value on the stack
28799 or simplifying or evaluating the value. It prompts for the name of
28800 the variable to edit. If the variable has no stored value, the
28801 editing buffer will start out empty. If the editing buffer is
28802 empty when you press @kbd{C-c C-c} to finish, the variable will
28803 be made void. @xref{Editing Stack Entries}, for a general
28804 description of editing.
28806 The @kbd{s e} command is especially useful for creating and editing
28807 rewrite rules which are stored in variables. Sometimes these rules
28808 contain formulas which must not be evaluated until the rules are
28809 actually used. (For example, they may refer to @samp{deriv(x,y)},
28810 where @code{x} will someday become some expression involving @code{y};
28811 if you let Calc evaluate the rule while you are defining it, Calc will
28812 replace @samp{deriv(x,y)} with 0 because the formula @code{x} does
28813 not itself refer to @code{y}.) By contrast, recalling the variable,
28814 editing with @kbd{`}, and storing will evaluate the variable's value
28815 as a side effect of putting the value on the stack.
28863 @pindex calc-store-AlgSimpRules
28864 @pindex calc-store-Decls
28865 @pindex calc-store-EvalRules
28866 @pindex calc-store-FitRules
28867 @pindex calc-store-GenCount
28868 @pindex calc-store-Holidays
28869 @pindex calc-store-IntegLimit
28870 @pindex calc-store-LineStyles
28871 @pindex calc-store-PointStyles
28872 @pindex calc-store-PlotRejects
28873 @pindex calc-store-TimeZone
28874 @pindex calc-store-Units
28875 @pindex calc-store-ExtSimpRules
28876 There are several special-purpose variable-editing commands that
28877 use the @kbd{s} prefix followed by a shifted letter:
28881 Edit @code{AlgSimpRules}. @xref{Algebraic Simplifications}.
28883 Edit @code{Decls}. @xref{Declarations}.
28885 Edit @code{EvalRules}. @xref{Basic Simplifications}.
28887 Edit @code{FitRules}. @xref{Curve Fitting}.
28889 Edit @code{GenCount}. @xref{Solving Equations}.
28891 Edit @code{Holidays}. @xref{Business Days}.
28893 Edit @code{IntegLimit}. @xref{Calculus}.
28895 Edit @code{LineStyles}. @xref{Graphics}.
28897 Edit @code{PointStyles}. @xref{Graphics}.
28899 Edit @code{PlotRejects}. @xref{Graphics}.
28901 Edit @code{TimeZone}. @xref{Time Zones}.
28903 Edit @code{Units}. @xref{User-Defined Units}.
28905 Edit @code{ExtSimpRules}. @xref{Unsafe Simplifications}.
28908 These commands are just versions of @kbd{s e} that use fixed variable
28909 names rather than prompting for the variable name.
28912 @pindex calc-permanent-variable
28913 @cindex Storing variables
28914 @cindex Permanent variables
28915 @cindex Calc init file, variables
28916 The @kbd{s p} (@code{calc-permanent-variable}) command saves a
28917 variable's value permanently in your Calc init file (the file given by
28918 the variable @code{calc-settings-file}, typically @file{~/.emacs.d/calc.el}), so
28919 that its value will still be available in future Emacs sessions. You
28920 can re-execute @w{@kbd{s p}} later on to update the saved value, but the
28921 only way to remove a saved variable is to edit your calc init file
28922 by hand. (@xref{General Mode Commands}, for a way to tell Calc to
28923 use a different file for the Calc init file.)
28925 If you do not specify the name of a variable to save (i.e.,
28926 @kbd{s p @key{RET}}), all Calc variables with defined values
28927 are saved except for the special constants @code{pi}, @code{e},
28928 @code{i}, @code{phi}, and @code{gamma}; the variables @code{TimeZone}
28929 and @code{PlotRejects};
28930 @code{FitRules}, @code{DistribRules}, and other built-in rewrite
28931 rules; and @code{PlotData@var{n}} variables generated
28932 by the graphics commands. (You can still save these variables by
28933 explicitly naming them in an @kbd{s p} command.)
28936 @pindex calc-insert-variables
28937 The @kbd{s i} (@code{calc-insert-variables}) command writes
28938 the values of all Calc variables into a specified buffer.
28939 The variables are written with the prefix @code{var-} in the form of
28940 Lisp @code{setq} commands
28941 which store the values in string form. You can place these commands
28942 in your Calc init file (or @file{.emacs}) if you wish, though in this case it
28943 would be easier to use @kbd{s p @key{RET}}. (Note that @kbd{s i}
28944 omits the same set of variables as @w{@kbd{s p @key{RET}}}; the difference
28945 is that @kbd{s i} will store the variables in any buffer, and it also
28946 stores in a more human-readable format.)
28948 @node Let Command, Evaluates-To Operator, Operations on Variables, Store and Recall
28949 @section The Let Command
28954 @cindex Variables, temporary assignment
28955 @cindex Temporary assignment to variables
28956 If you have an expression like @samp{a+b^2} on the stack and you wish to
28957 compute its value where @expr{b=3}, you can simply store 3 in @expr{b} and
28958 then press @kbd{=} to reevaluate the formula. This has the side-effect
28959 of leaving the stored value of 3 in @expr{b} for future operations.
28961 The @kbd{s l} (@code{calc-let}) command evaluates a formula under a
28962 @emph{temporary} assignment of a variable. It stores the value on the
28963 top of the stack into the specified variable, then evaluates the
28964 second-to-top stack entry, then restores the original value (or lack of one)
28965 in the variable. Thus after @kbd{'@w{ }a+b^2 @key{RET} 3 s l b @key{RET}},
28966 the stack will contain the formula @samp{a + 9}. The subsequent command
28967 @kbd{@w{5 s l a} @key{RET}} will replace this formula with the number 14.
28968 The variables @samp{a} and @samp{b} are not permanently affected in any way
28971 The value on the top of the stack may be an equation or assignment, or
28972 a vector of equations or assignments, in which case the default will be
28973 analogous to the case of @kbd{s t @key{RET}}. @xref{Storing Variables}.
28975 Also, you can answer the variable-name prompt with an equation or
28976 assignment: @kbd{s l b=3 @key{RET}} is the same as storing 3 on the stack
28977 and typing @kbd{s l b @key{RET}}.
28979 The @kbd{a b} (@code{calc-substitute}) command is another way to substitute
28980 a variable with a value in a formula. It does an actual substitution
28981 rather than temporarily assigning the variable and evaluating. For
28982 example, letting @expr{n=2} in @samp{f(n pi)} with @kbd{a b} will
28983 produce @samp{f(2 pi)}, whereas @kbd{s l} would give @samp{f(6.28)}
28984 since the evaluation step will also evaluate @code{pi}.
28986 @node Evaluates-To Operator, , Let Command, Store and Recall
28987 @section The Evaluates-To Operator
28992 @cindex Evaluates-to operator
28993 @cindex @samp{=>} operator
28994 The special algebraic symbol @samp{=>} is known as the @dfn{evaluates-to
28995 operator}. (It will show up as an @code{evalto} function call in
28996 other language modes like Pascal and @LaTeX{}.) This is a binary
28997 operator, that is, it has a lefthand and a righthand argument,
28998 although it can be entered with the righthand argument omitted.
29000 A formula like @samp{@var{a} => @var{b}} is evaluated by Calc as
29001 follows: First, @var{a} is not simplified or modified in any
29002 way. The previous value of argument @var{b} is thrown away; the
29003 formula @var{a} is then copied and evaluated as if by the @kbd{=}
29004 command according to all current modes and stored variable values,
29005 and the result is installed as the new value of @var{b}.
29007 For example, suppose you enter the algebraic formula @samp{2 + 3 => 17}.
29008 The number 17 is ignored, and the lefthand argument is left in its
29009 unevaluated form; the result is the formula @samp{2 + 3 => 5}.
29012 @pindex calc-evalto
29013 You can enter an @samp{=>} formula either directly using algebraic
29014 entry (in which case the righthand side may be omitted since it is
29015 going to be replaced right away anyhow), or by using the @kbd{s =}
29016 (@code{calc-evalto}) command, which takes @var{a} from the stack
29017 and replaces it with @samp{@var{a} => @var{b}}.
29019 Calc keeps track of all @samp{=>} operators on the stack, and
29020 recomputes them whenever anything changes that might affect their
29021 values, i.e., a mode setting or variable value. This occurs only
29022 if the @samp{=>} operator is at the top level of the formula, or
29023 if it is part of a top-level vector. In other words, pushing
29024 @samp{2 + (a => 17)} will change the 17 to the actual value of
29025 @samp{a} when you enter the formula, but the result will not be
29026 dynamically updated when @samp{a} is changed later because the
29027 @samp{=>} operator is buried inside a sum. However, a vector
29028 of @samp{=>} operators will be recomputed, since it is convenient
29029 to push a vector like @samp{[a =>, b =>, c =>]} on the stack to
29030 make a concise display of all the variables in your problem.
29031 (Another way to do this would be to use @samp{[a, b, c] =>},
29032 which provides a slightly different format of display. You
29033 can use whichever you find easiest to read.)
29036 @pindex calc-auto-recompute
29037 The @kbd{m C} (@code{calc-auto-recompute}) command allows you to
29038 turn this automatic recomputation on or off. If you turn
29039 recomputation off, you must explicitly recompute an @samp{=>}
29040 operator on the stack in one of the usual ways, such as by
29041 pressing @kbd{=}. Turning recomputation off temporarily can save
29042 a lot of time if you will be changing several modes or variables
29043 before you look at the @samp{=>} entries again.
29045 Most commands are not especially useful with @samp{=>} operators
29046 as arguments. For example, given @samp{x + 2 => 17}, it won't
29047 work to type @kbd{1 +} to get @samp{x + 3 => 18}. If you want
29048 to operate on the lefthand side of the @samp{=>} operator on
29049 the top of the stack, type @kbd{j 1} (that's the digit ``one'')
29050 to select the lefthand side, execute your commands, then type
29051 @kbd{j u} to unselect.
29053 All current modes apply when an @samp{=>} operator is computed,
29054 including the current simplification mode. Recall that the
29055 formula @samp{arcsin(sin(x))} will not be handled by Calc's algebraic
29056 simplifications, but Calc's unsafe simplifications will reduce it to
29057 @samp{x}. If you enter @samp{arcsin(sin(x)) =>} normally, the result
29058 will be @samp{arcsin(sin(x)) => arcsin(sin(x))}. If you change to
29059 Extended Simplification mode, the result will be
29060 @samp{arcsin(sin(x)) => x}. However, just pressing @kbd{a e}
29061 once will have no effect on @samp{arcsin(sin(x)) => arcsin(sin(x))},
29062 because the righthand side depends only on the lefthand side
29063 and the current mode settings, and the lefthand side is not
29064 affected by commands like @kbd{a e}.
29066 The ``let'' command (@kbd{s l}) has an interesting interaction
29067 with the @samp{=>} operator. The @kbd{s l} command evaluates the
29068 second-to-top stack entry with the top stack entry supplying
29069 a temporary value for a given variable. As you might expect,
29070 if that stack entry is an @samp{=>} operator its righthand
29071 side will temporarily show this value for the variable. In
29072 fact, all @samp{=>}s on the stack will be updated if they refer
29073 to that variable. But this change is temporary in the sense
29074 that the next command that causes Calc to look at those stack
29075 entries will make them revert to the old variable value.
29079 2: a => a 2: a => 17 2: a => a
29080 1: a + 1 => a + 1 1: a + 1 => 18 1: a + 1 => a + 1
29083 17 s l a @key{RET} p 8 @key{RET}
29087 Here the @kbd{p 8} command changes the current precision,
29088 thus causing the @samp{=>} forms to be recomputed after the
29089 influence of the ``let'' is gone. The @kbd{d @key{SPC}} command
29090 (@code{calc-refresh}) is a handy way to force the @samp{=>}
29091 operators on the stack to be recomputed without any other
29095 @pindex calc-assign
29098 Embedded mode also uses @samp{=>} operators. In Embedded mode,
29099 the lefthand side of an @samp{=>} operator can refer to variables
29100 assigned elsewhere in the file by @samp{:=} operators. The
29101 assignment operator @samp{a := 17} does not actually do anything
29102 by itself. But Embedded mode recognizes it and marks it as a sort
29103 of file-local definition of the variable. You can enter @samp{:=}
29104 operators in Algebraic mode, or by using the @kbd{s :}
29105 (@code{calc-assign}) [@code{assign}] command which takes a variable
29106 and value from the stack and replaces them with an assignment.
29108 @xref{TeX and LaTeX Language Modes}, for the way @samp{=>} appears in
29109 @TeX{} language output. The @dfn{eqn} mode gives similar
29110 treatment to @samp{=>}.
29112 @node Graphics, Kill and Yank, Store and Recall, Top
29116 The commands for graphing data begin with the @kbd{g} prefix key. Calc
29117 uses GNUPLOT 2.0 or later to do graphics. These commands will only work
29118 if GNUPLOT is available on your system. (While GNUPLOT sounds like
29119 a relative of GNU Emacs, it is actually completely unrelated.
29120 However, it is free software. It can be obtained from
29121 @samp{http://www.gnuplot.info}.)
29123 @vindex calc-gnuplot-name
29124 If you have GNUPLOT installed on your system but Calc is unable to
29125 find it, you may need to set the @code{calc-gnuplot-name} variable in
29126 your Calc init file or @file{.emacs}. You may also need to set some
29127 Lisp variables to show Calc how to run GNUPLOT on your system; these
29128 are described under @kbd{g D} and @kbd{g O} below. If you are using
29129 the X window system or MS-Windows, Calc will configure GNUPLOT for you
29130 automatically. If you have GNUPLOT 3.0 or later and you are using a
29131 Unix or GNU system without X, Calc will configure GNUPLOT to display
29132 graphs using simple character graphics that will work on any
29133 Posix-compatible terminal.
29137 * Three Dimensional Graphics::
29138 * Managing Curves::
29139 * Graphics Options::
29143 @node Basic Graphics, Three Dimensional Graphics, Graphics, Graphics
29144 @section Basic Graphics
29148 @pindex calc-graph-fast
29149 The easiest graphics command is @kbd{g f} (@code{calc-graph-fast}).
29150 This command takes two vectors of equal length from the stack.
29151 The vector at the top of the stack represents the ``y'' values of
29152 the various data points. The vector in the second-to-top position
29153 represents the corresponding ``x'' values. This command runs
29154 GNUPLOT (if it has not already been started by previous graphing
29155 commands) and displays the set of data points. The points will
29156 be connected by lines, and there will also be some kind of symbol
29157 to indicate the points themselves.
29159 The ``x'' entry may instead be an interval form, in which case suitable
29160 ``x'' values are interpolated between the minimum and maximum values of
29161 the interval (whether the interval is open or closed is ignored).
29163 The ``x'' entry may also be a number, in which case Calc uses the
29164 sequence of ``x'' values @expr{x}, @expr{x+1}, @expr{x+2}, etc.
29165 (Generally the number 0 or 1 would be used for @expr{x} in this case.)
29167 The ``y'' entry may be any formula instead of a vector. Calc effectively
29168 uses @kbd{N} (@code{calc-eval-num}) to evaluate variables in the formula;
29169 the result of this must be a formula in a single (unassigned) variable.
29170 The formula is plotted with this variable taking on the various ``x''
29171 values. Graphs of formulas by default use lines without symbols at the
29172 computed data points. Note that if neither ``x'' nor ``y'' is a vector,
29173 Calc guesses at a reasonable number of data points to use. See the
29174 @kbd{g N} command below. (The ``x'' values must be either a vector
29175 or an interval if ``y'' is a formula.)
29181 If ``y'' is (or evaluates to) a formula of the form
29182 @samp{xy(@var{x}, @var{y})} then the result is a
29183 parametric plot. The two arguments of the fictitious @code{xy} function
29184 are used as the ``x'' and ``y'' coordinates of the curve, respectively.
29185 In this case the ``x'' vector or interval you specified is not directly
29186 visible in the graph. For example, if ``x'' is the interval @samp{[0..360]}
29187 and ``y'' is the formula @samp{xy(sin(t), cos(t))}, the resulting graph
29190 Also, ``x'' and ``y'' may each be variable names, in which case Calc
29191 looks for suitable vectors, intervals, or formulas stored in those
29194 The ``x'' and ``y'' values for the data points (as pulled from the vectors,
29195 calculated from the formulas, or interpolated from the intervals) should
29196 be real numbers (integers, fractions, or floats). One exception to this
29197 is that the ``y'' entry can consist of a vector of numbers combined with
29198 error forms, in which case the points will be plotted with the
29199 appropriate error bars. Other than this, if either the ``x''
29200 value or the ``y'' value of a given data point is not a real number, that
29201 data point will be omitted from the graph. The points on either side
29202 of the invalid point will @emph{not} be connected by a line.
29204 See the documentation for @kbd{g a} below for a description of the way
29205 numeric prefix arguments affect @kbd{g f}.
29207 @cindex @code{PlotRejects} variable
29208 @vindex PlotRejects
29209 If you store an empty vector in the variable @code{PlotRejects}
29210 (i.e., @kbd{[ ] s t PlotRejects}), Calc will append information to
29211 this vector for every data point which was rejected because its
29212 ``x'' or ``y'' values were not real numbers. The result will be
29213 a matrix where each row holds the curve number, data point number,
29214 ``x'' value, and ``y'' value for a rejected data point.
29215 @xref{Evaluates-To Operator}, for a handy way to keep tabs on the
29216 current value of @code{PlotRejects}. @xref{Operations on Variables},
29217 for the @kbd{s R} command which is another easy way to examine
29218 @code{PlotRejects}.
29221 @pindex calc-graph-clear
29222 To clear the graphics display, type @kbd{g c} (@code{calc-graph-clear}).
29223 If the GNUPLOT output device is an X window, the window will go away.
29224 Effects on other kinds of output devices will vary. You don't need
29225 to use @kbd{g c} if you don't want to---if you give another @kbd{g f}
29226 or @kbd{g p} command later on, it will reuse the existing graphics
29227 window if there is one.
29229 @node Three Dimensional Graphics, Managing Curves, Basic Graphics, Graphics
29230 @section Three-Dimensional Graphics
29233 @pindex calc-graph-fast-3d
29234 The @kbd{g F} (@code{calc-graph-fast-3d}) command makes a three-dimensional
29235 graph. It works only if you have GNUPLOT 3.0 or later; with GNUPLOT 2.0,
29236 you will see a GNUPLOT error message if you try this command.
29238 The @kbd{g F} command takes three values from the stack, called ``x'',
29239 ``y'', and ``z'', respectively. As was the case for 2D graphs, there
29240 are several options for these values.
29242 In the first case, ``x'' and ``y'' are each vectors (not necessarily of
29243 the same length); either or both may instead be interval forms. The
29244 ``z'' value must be a matrix with the same number of rows as elements
29245 in ``x'', and the same number of columns as elements in ``y''. The
29246 result is a surface plot where
29247 @texline @math{z_{ij}}
29248 @infoline @expr{z_ij}
29249 is the height of the point
29250 at coordinate @expr{(x_i, y_j)} on the surface. The 3D graph will
29251 be displayed from a certain default viewpoint; you can change this
29252 viewpoint by adding a @samp{set view} to the @samp{*Gnuplot Commands*}
29253 buffer as described later. See the GNUPLOT documentation for a
29254 description of the @samp{set view} command.
29256 Each point in the matrix will be displayed as a dot in the graph,
29257 and these points will be connected by a grid of lines (@dfn{isolines}).
29259 In the second case, ``x'', ``y'', and ``z'' are all vectors of equal
29260 length. The resulting graph displays a 3D line instead of a surface,
29261 where the coordinates of points along the line are successive triplets
29262 of values from the input vectors.
29264 In the third case, ``x'' and ``y'' are vectors or interval forms, and
29265 ``z'' is any formula involving two variables (not counting variables
29266 with assigned values). These variables are sorted into alphabetical
29267 order; the first takes on values from ``x'' and the second takes on
29268 values from ``y'' to form a matrix of results that are graphed as a
29275 If the ``z'' formula evaluates to a call to the fictitious function
29276 @samp{xyz(@var{x}, @var{y}, @var{z})}, then the result is a
29277 ``parametric surface.'' In this case, the axes of the graph are
29278 taken from the @var{x} and @var{y} values in these calls, and the
29279 ``x'' and ``y'' values from the input vectors or intervals are used only
29280 to specify the range of inputs to the formula. For example, plotting
29281 @samp{[0..360], [0..180], xyz(sin(x)*sin(y), cos(x)*sin(y), cos(y))}
29282 will draw a sphere. (Since the default resolution for 3D plots is
29283 5 steps in each of ``x'' and ``y'', this will draw a very crude
29284 sphere. You could use the @kbd{g N} command, described below, to
29285 increase this resolution, or specify the ``x'' and ``y'' values as
29286 vectors with more than 5 elements.
29288 It is also possible to have a function in a regular @kbd{g f} plot
29289 evaluate to an @code{xyz} call. Since @kbd{g f} plots a line, not
29290 a surface, the result will be a 3D parametric line. For example,
29291 @samp{[[0..720], xyz(sin(x), cos(x), x)]} will plot two turns of a
29292 helix (a three-dimensional spiral).
29294 As for @kbd{g f}, each of ``x'', ``y'', and ``z'' may instead be
29295 variables containing the relevant data.
29297 @node Managing Curves, Graphics Options, Three Dimensional Graphics, Graphics
29298 @section Managing Curves
29301 The @kbd{g f} command is really shorthand for the following commands:
29302 @kbd{C-u g d g a g p}. Likewise, @w{@kbd{g F}} is shorthand for
29303 @kbd{C-u g d g A g p}. You can gain more control over your graph
29304 by using these commands directly.
29307 @pindex calc-graph-add
29308 The @kbd{g a} (@code{calc-graph-add}) command adds the ``curve''
29309 represented by the two values on the top of the stack to the current
29310 graph. You can have any number of curves in the same graph. When
29311 you give the @kbd{g p} command, all the curves will be drawn superimposed
29314 The @kbd{g a} command (and many others that affect the current graph)
29315 will cause a special buffer, @samp{*Gnuplot Commands*}, to be displayed
29316 in another window. This buffer is a template of the commands that will
29317 be sent to GNUPLOT when it is time to draw the graph. The first
29318 @kbd{g a} command adds a @code{plot} command to this buffer. Succeeding
29319 @kbd{g a} commands add extra curves onto that @code{plot} command.
29320 Other graph-related commands put other GNUPLOT commands into this
29321 buffer. In normal usage you never need to work with this buffer
29322 directly, but you can if you wish. The only constraint is that there
29323 must be only one @code{plot} command, and it must be the last command
29324 in the buffer. If you want to save and later restore a complete graph
29325 configuration, you can use regular Emacs commands to save and restore
29326 the contents of the @samp{*Gnuplot Commands*} buffer.
29330 If the values on the stack are not variable names, @kbd{g a} will invent
29331 variable names for them (of the form @samp{PlotData@var{n}}) and store
29332 the values in those variables. The ``x'' and ``y'' variables are what
29333 go into the @code{plot} command in the template. If you add a curve
29334 that uses a certain variable and then later change that variable, you
29335 can replot the graph without having to delete and re-add the curve.
29336 That's because the variable name, not the vector, interval or formula
29337 itself, is what was added by @kbd{g a}.
29339 A numeric prefix argument on @kbd{g a} or @kbd{g f} changes the way
29340 stack entries are interpreted as curves. With a positive prefix
29341 argument @expr{n}, the top @expr{n} stack entries are ``y'' values
29342 for @expr{n} different curves which share a common ``x'' value in
29343 the @expr{n+1}st stack entry. (Thus @kbd{g a} with no prefix
29344 argument is equivalent to @kbd{C-u 1 g a}.)
29346 A prefix of zero or plain @kbd{C-u} means to take two stack entries,
29347 ``x'' and ``y'' as usual, but to interpret ``y'' as a vector of
29348 ``y'' values for several curves that share a common ``x''.
29350 A negative prefix argument tells Calc to read @expr{n} vectors from
29351 the stack; each vector @expr{[x, y]} describes an independent curve.
29352 This is the only form of @kbd{g a} that creates several curves at once
29353 that don't have common ``x'' values. (Of course, the range of ``x''
29354 values covered by all the curves ought to be roughly the same if
29355 they are to look nice on the same graph.)
29357 For example, to plot
29358 @texline @math{\sin n x}
29359 @infoline @expr{sin(n x)}
29360 for integers @expr{n}
29361 from 1 to 5, you could use @kbd{v x} to create a vector of integers
29362 (@expr{n}), then @kbd{V M '} or @kbd{V M $} to map @samp{sin(n x)}
29363 across this vector. The resulting vector of formulas is suitable
29364 for use as the ``y'' argument to a @kbd{C-u g a} or @kbd{C-u g f}
29368 @pindex calc-graph-add-3d
29369 The @kbd{g A} (@code{calc-graph-add-3d}) command adds a 3D curve
29370 to the graph. It is not valid to intermix 2D and 3D curves in a
29371 single graph. This command takes three arguments, ``x'', ``y'',
29372 and ``z'', from the stack. With a positive prefix @expr{n}, it
29373 takes @expr{n+2} arguments (common ``x'' and ``y'', plus @expr{n}
29374 separate ``z''s). With a zero prefix, it takes three stack entries
29375 but the ``z'' entry is a vector of curve values. With a negative
29376 prefix @expr{-n}, it takes @expr{n} vectors of the form @expr{[x, y, z]}.
29377 The @kbd{g A} command works by adding a @code{splot} (surface-plot)
29378 command to the @samp{*Gnuplot Commands*} buffer.
29380 (Although @kbd{g a} adds a 2D @code{plot} command to the
29381 @samp{*Gnuplot Commands*} buffer, Calc changes this to @code{splot}
29382 before sending it to GNUPLOT if it notices that the data points are
29383 evaluating to @code{xyz} calls. It will not work to mix 2D and 3D
29384 @kbd{g a} curves in a single graph, although Calc does not currently
29388 @pindex calc-graph-delete
29389 The @kbd{g d} (@code{calc-graph-delete}) command deletes the most
29390 recently added curve from the graph. It has no effect if there are
29391 no curves in the graph. With a numeric prefix argument of any kind,
29392 it deletes all of the curves from the graph.
29395 @pindex calc-graph-hide
29396 The @kbd{g H} (@code{calc-graph-hide}) command ``hides'' or ``unhides''
29397 the most recently added curve. A hidden curve will not appear in
29398 the actual plot, but information about it such as its name and line and
29399 point styles will be retained.
29402 @pindex calc-graph-juggle
29403 The @kbd{g j} (@code{calc-graph-juggle}) command moves the curve
29404 at the end of the list (the ``most recently added curve'') to the
29405 front of the list. The next-most-recent curve is thus exposed for
29406 @w{@kbd{g d}} or similar commands to use. With @kbd{g j} you can work
29407 with any curve in the graph even though curve-related commands only
29408 affect the last curve in the list.
29411 @pindex calc-graph-plot
29412 The @kbd{g p} (@code{calc-graph-plot}) command uses GNUPLOT to draw
29413 the graph described in the @samp{*Gnuplot Commands*} buffer. Any
29414 GNUPLOT parameters which are not defined by commands in this buffer
29415 are reset to their default values. The variables named in the @code{plot}
29416 command are written to a temporary data file and the variable names
29417 are then replaced by the file name in the template. The resulting
29418 plotting commands are fed to the GNUPLOT program. See the documentation
29419 for the GNUPLOT program for more specific information. All temporary
29420 files are removed when Emacs or GNUPLOT exits.
29422 If you give a formula for ``y'', Calc will remember all the values that
29423 it calculates for the formula so that later plots can reuse these values.
29424 Calc throws out these saved values when you change any circumstances
29425 that may affect the data, such as switching from Degrees to Radians
29426 mode, or changing the value of a parameter in the formula. You can
29427 force Calc to recompute the data from scratch by giving a negative
29428 numeric prefix argument to @kbd{g p}.
29430 Calc uses a fairly rough step size when graphing formulas over intervals.
29431 This is to ensure quick response. You can ``refine'' a plot by giving
29432 a positive numeric prefix argument to @kbd{g p}. Calc goes through
29433 the data points it has computed and saved from previous plots of the
29434 function, and computes and inserts a new data point midway between
29435 each of the existing points. You can refine a plot any number of times,
29436 but beware that the amount of calculation involved doubles each time.
29438 Calc does not remember computed values for 3D graphs. This means the
29439 numerix prefix argument, if any, to @kbd{g p} is effectively ignored if
29440 the current graph is three-dimensional.
29443 @pindex calc-graph-print
29444 The @kbd{g P} (@code{calc-graph-print}) command is like @kbd{g p},
29445 except that it sends the output to a printer instead of to the
29446 screen. More precisely, @kbd{g p} looks for @samp{set terminal}
29447 or @samp{set output} commands in the @samp{*Gnuplot Commands*} buffer;
29448 lacking these it uses the default settings. However, @kbd{g P}
29449 ignores @samp{set terminal} and @samp{set output} commands and
29450 uses a different set of default values. All of these values are
29451 controlled by the @kbd{g D} and @kbd{g O} commands discussed below.
29452 Provided everything is set up properly, @kbd{g p} will plot to
29453 the screen unless you have specified otherwise and @kbd{g P} will
29454 always plot to the printer.
29456 @node Graphics Options, Devices, Managing Curves, Graphics
29457 @section Graphics Options
29461 @pindex calc-graph-grid
29462 The @kbd{g g} (@code{calc-graph-grid}) command turns the ``grid''
29463 on and off. It is off by default; tick marks appear only at the
29464 edges of the graph. With the grid turned on, dotted lines appear
29465 across the graph at each tick mark. Note that this command only
29466 changes the setting in @samp{*Gnuplot Commands*}; to see the effects
29467 of the change you must give another @kbd{g p} command.
29470 @pindex calc-graph-border
29471 The @kbd{g b} (@code{calc-graph-border}) command turns the border
29472 (the box that surrounds the graph) on and off. It is on by default.
29473 This command will only work with GNUPLOT 3.0 and later versions.
29476 @pindex calc-graph-key
29477 The @kbd{g k} (@code{calc-graph-key}) command turns the ``key''
29478 on and off. The key is a chart in the corner of the graph that
29479 shows the correspondence between curves and line styles. It is
29480 off by default, and is only really useful if you have several
29481 curves on the same graph.
29484 @pindex calc-graph-num-points
29485 The @kbd{g N} (@code{calc-graph-num-points}) command allows you
29486 to select the number of data points in the graph. This only affects
29487 curves where neither ``x'' nor ``y'' is specified as a vector.
29488 Enter a blank line to revert to the default value (initially 15).
29489 With no prefix argument, this command affects only the current graph.
29490 With a positive prefix argument this command changes or, if you enter
29491 a blank line, displays the default number of points used for all
29492 graphs created by @kbd{g a} that don't specify the resolution explicitly.
29493 With a negative prefix argument, this command changes or displays
29494 the default value (initially 5) used for 3D graphs created by @kbd{g A}.
29495 Note that a 3D setting of 5 means that a total of @expr{5^2 = 25} points
29496 will be computed for the surface.
29498 Data values in the graph of a function are normally computed to a
29499 precision of five digits, regardless of the current precision at the
29500 time. This is usually more than adequate, but there are cases where
29501 it will not be. For example, plotting @expr{1 + x} with @expr{x} in the
29502 interval @samp{[0 ..@: 1e-6]} will round all the data points down
29503 to 1.0! Putting the command @samp{set precision @var{n}} in the
29504 @samp{*Gnuplot Commands*} buffer will cause the data to be computed
29505 at precision @var{n} instead of 5. Since this is such a rare case,
29506 there is no keystroke-based command to set the precision.
29509 @pindex calc-graph-header
29510 The @kbd{g h} (@code{calc-graph-header}) command sets the title
29511 for the graph. This will show up centered above the graph.
29512 The default title is blank (no title).
29515 @pindex calc-graph-name
29516 The @kbd{g n} (@code{calc-graph-name}) command sets the title of an
29517 individual curve. Like the other curve-manipulating commands, it
29518 affects the most recently added curve, i.e., the last curve on the
29519 list in the @samp{*Gnuplot Commands*} buffer. To set the title of
29520 the other curves you must first juggle them to the end of the list
29521 with @kbd{g j}, or edit the @samp{*Gnuplot Commands*} buffer by hand.
29522 Curve titles appear in the key; if the key is turned off they are
29527 @pindex calc-graph-title-x
29528 @pindex calc-graph-title-y
29529 The @kbd{g t} (@code{calc-graph-title-x}) and @kbd{g T}
29530 (@code{calc-graph-title-y}) commands set the titles on the ``x''
29531 and ``y'' axes, respectively. These titles appear next to the
29532 tick marks on the left and bottom edges of the graph, respectively.
29533 Calc does not have commands to control the tick marks themselves,
29534 but you can edit them into the @samp{*Gnuplot Commands*} buffer if
29535 you wish. See the GNUPLOT documentation for details.
29539 @pindex calc-graph-range-x
29540 @pindex calc-graph-range-y
29541 The @kbd{g r} (@code{calc-graph-range-x}) and @kbd{g R}
29542 (@code{calc-graph-range-y}) commands set the range of values on the
29543 ``x'' and ``y'' axes, respectively. You are prompted to enter a
29544 suitable range. This should be either a pair of numbers of the
29545 form, @samp{@var{min}:@var{max}}, or a blank line to revert to the
29546 default behavior of setting the range based on the range of values
29547 in the data, or @samp{$} to take the range from the top of the stack.
29548 Ranges on the stack can be represented as either interval forms or
29549 vectors: @samp{[@var{min} ..@: @var{max}]} or @samp{[@var{min}, @var{max}]}.
29553 @pindex calc-graph-log-x
29554 @pindex calc-graph-log-y
29555 The @kbd{g l} (@code{calc-graph-log-x}) and @kbd{g L} (@code{calc-graph-log-y})
29556 commands allow you to set either or both of the axes of the graph to
29557 be logarithmic instead of linear.
29562 @pindex calc-graph-log-z
29563 @pindex calc-graph-range-z
29564 @pindex calc-graph-title-z
29565 For 3D plots, @kbd{g C-t}, @kbd{g C-r}, and @kbd{g C-l} (those are
29566 letters with the Control key held down) are the corresponding commands
29567 for the ``z'' axis.
29571 @pindex calc-graph-zero-x
29572 @pindex calc-graph-zero-y
29573 The @kbd{g z} (@code{calc-graph-zero-x}) and @kbd{g Z}
29574 (@code{calc-graph-zero-y}) commands control whether a dotted line is
29575 drawn to indicate the ``x'' and/or ``y'' zero axes. (These are the same
29576 dotted lines that would be drawn there anyway if you used @kbd{g g} to
29577 turn the ``grid'' feature on.) Zero-axis lines are on by default, and
29578 may be turned off only in GNUPLOT 3.0 and later versions. They are
29579 not available for 3D plots.
29582 @pindex calc-graph-line-style
29583 The @kbd{g s} (@code{calc-graph-line-style}) command turns the connecting
29584 lines on or off for the most recently added curve, and optionally selects
29585 the style of lines to be used for that curve. Plain @kbd{g s} simply
29586 toggles the lines on and off. With a numeric prefix argument, @kbd{g s}
29587 turns lines on and sets a particular line style. Line style numbers
29588 start at one and their meanings vary depending on the output device.
29589 GNUPLOT guarantees that there will be at least six different line styles
29590 available for any device.
29593 @pindex calc-graph-point-style
29594 The @kbd{g S} (@code{calc-graph-point-style}) command similarly turns
29595 the symbols at the data points on or off, or sets the point style.
29596 If you turn both lines and points off, the data points will show as
29597 tiny dots. If the ``y'' values being plotted contain error forms and
29598 the connecting lines are turned off, then this command will also turn
29599 the error bars on or off.
29601 @cindex @code{LineStyles} variable
29602 @cindex @code{PointStyles} variable
29604 @vindex PointStyles
29605 Another way to specify curve styles is with the @code{LineStyles} and
29606 @code{PointStyles} variables. These variables initially have no stored
29607 values, but if you store a vector of integers in one of these variables,
29608 the @kbd{g a} and @kbd{g f} commands will use those style numbers
29609 instead of the defaults for new curves that are added to the graph.
29610 An entry should be a positive integer for a specific style, or 0 to let
29611 the style be chosen automatically, or @mathit{-1} to turn off lines or points
29612 altogether. If there are more curves than elements in the vector, the
29613 last few curves will continue to have the default styles. Of course,
29614 you can later use @kbd{g s} and @kbd{g S} to change any of these styles.
29616 For example, @kbd{'[2 -1 3] @key{RET} s t LineStyles} causes the first curve
29617 to have lines in style number 2, the second curve to have no connecting
29618 lines, and the third curve to have lines in style 3. Point styles will
29619 still be assigned automatically, but you could store another vector in
29620 @code{PointStyles} to define them, too.
29622 @node Devices, , Graphics Options, Graphics
29623 @section Graphical Devices
29627 @pindex calc-graph-device
29628 The @kbd{g D} (@code{calc-graph-device}) command sets the device name
29629 (or ``terminal name'' in GNUPLOT lingo) to be used by @kbd{g p} commands
29630 on this graph. It does not affect the permanent default device name.
29631 If you enter a blank name, the device name reverts to the default.
29632 Enter @samp{?} to see a list of supported devices.
29634 With a positive numeric prefix argument, @kbd{g D} instead sets
29635 the default device name, used by all plots in the future which do
29636 not override it with a plain @kbd{g D} command. If you enter a
29637 blank line this command shows you the current default. The special
29638 name @code{default} signifies that Calc should choose @code{x11} if
29639 the X window system is in use (as indicated by the presence of a
29640 @code{DISPLAY} environment variable), @code{windows} on MS-Windows, or
29641 otherwise @code{dumb} under GNUPLOT 3.0 and later, or
29642 @code{postscript} under GNUPLOT 2.0. This is the initial default
29645 The @code{dumb} device is an interface to ``dumb terminals,'' i.e.,
29646 terminals with no special graphics facilities. It writes a crude
29647 picture of the graph composed of characters like @code{-} and @code{|}
29648 to a buffer called @samp{*Gnuplot Trail*}, which Calc then displays.
29649 The graph is made the same size as the Emacs screen, which on most
29650 dumb terminals will be
29651 @texline @math{80\times24}
29653 characters. The graph is displayed in
29654 an Emacs ``recursive edit''; type @kbd{q} or @kbd{C-c C-c} to exit
29655 the recursive edit and return to Calc. Note that the @code{dumb}
29656 device is present only in GNUPLOT 3.0 and later versions.
29658 The word @code{dumb} may be followed by two numbers separated by
29659 spaces. These are the desired width and height of the graph in
29660 characters. Also, the device name @code{big} is like @code{dumb}
29661 but creates a graph four times the width and height of the Emacs
29662 screen. You will then have to scroll around to view the entire
29663 graph. In the @samp{*Gnuplot Trail*} buffer, @key{SPC}, @key{DEL},
29664 @kbd{<}, and @kbd{>} are defined to scroll by one screenful in each
29665 of the four directions.
29667 With a negative numeric prefix argument, @kbd{g D} sets or displays
29668 the device name used by @kbd{g P} (@code{calc-graph-print}). This
29669 is initially @code{postscript}. If you don't have a PostScript
29670 printer, you may decide once again to use @code{dumb} to create a
29671 plot on any text-only printer.
29674 @pindex calc-graph-output
29675 The @kbd{g O} (@code{calc-graph-output}) command sets the name of the
29676 output file used by GNUPLOT@. For some devices, notably @code{x11} and
29677 @code{windows}, there is no output file and this information is not
29678 used. Many other ``devices'' are really file formats like
29679 @code{postscript}; in these cases the output in the desired format
29680 goes into the file you name with @kbd{g O}. Type @kbd{g O stdout
29681 @key{RET}} to set GNUPLOT to write to its standard output stream,
29682 i.e., to @samp{*Gnuplot Trail*}. This is the default setting.
29684 Another special output name is @code{tty}, which means that GNUPLOT
29685 is going to write graphics commands directly to its standard output,
29686 which you wish Emacs to pass through to your terminal. Tektronix
29687 graphics terminals, among other devices, operate this way. Calc does
29688 this by telling GNUPLOT to write to a temporary file, then running a
29689 sub-shell executing the command @samp{cat tempfile >/dev/tty}. On
29690 typical Unix systems, this will copy the temporary file directly to
29691 the terminal, bypassing Emacs entirely. You will have to type @kbd{C-l}
29692 to Emacs afterwards to refresh the screen.
29694 Once again, @kbd{g O} with a positive or negative prefix argument
29695 sets the default or printer output file names, respectively. In each
29696 case you can specify @code{auto}, which causes Calc to invent a temporary
29697 file name for each @kbd{g p} (or @kbd{g P}) command. This temporary file
29698 will be deleted once it has been displayed or printed. If the output file
29699 name is not @code{auto}, the file is not automatically deleted.
29701 The default and printer devices and output files can be saved
29702 permanently by the @kbd{m m} (@code{calc-save-modes}) command. The
29703 default number of data points (see @kbd{g N}) and the X geometry
29704 (see @kbd{g X}) are also saved. Other graph information is @emph{not}
29705 saved; you can save a graph's configuration simply by saving the contents
29706 of the @samp{*Gnuplot Commands*} buffer.
29708 @vindex calc-gnuplot-plot-command
29709 @vindex calc-gnuplot-default-device
29710 @vindex calc-gnuplot-default-output
29711 @vindex calc-gnuplot-print-command
29712 @vindex calc-gnuplot-print-device
29713 @vindex calc-gnuplot-print-output
29714 You may wish to configure the default and
29715 printer devices and output files for the whole system. The relevant
29716 Lisp variables are @code{calc-gnuplot-default-device} and @code{-output},
29717 and @code{calc-gnuplot-print-device} and @code{-output}. The output
29718 file names must be either strings as described above, or Lisp
29719 expressions which are evaluated on the fly to get the output file names.
29721 Other important Lisp variables are @code{calc-gnuplot-plot-command} and
29722 @code{calc-gnuplot-print-command}, which give the system commands to
29723 display or print the output of GNUPLOT, respectively. These may be
29724 @code{nil} if no command is necessary, or strings which can include
29725 @samp{%s} to signify the name of the file to be displayed or printed.
29726 Or, these variables may contain Lisp expressions which are evaluated
29727 to display or print the output. These variables are customizable
29728 (@pxref{Customizing Calc}).
29731 @pindex calc-graph-display
29732 The @kbd{g x} (@code{calc-graph-display}) command lets you specify
29733 on which X window system display your graphs should be drawn. Enter
29734 a blank line to see the current display name. This command has no
29735 effect unless the current device is @code{x11}.
29738 @pindex calc-graph-geometry
29739 The @kbd{g X} (@code{calc-graph-geometry}) command is a similar
29740 command for specifying the position and size of the X window.
29741 The normal value is @code{default}, which generally means your
29742 window manager will let you place the window interactively.
29743 Entering @samp{800x500+0+0} would create an 800-by-500 pixel
29744 window in the upper-left corner of the screen. This command has no
29745 effect if the current device is @code{windows}.
29747 The buffer called @samp{*Gnuplot Trail*} holds a transcript of the
29748 session with GNUPLOT@. This shows the commands Calc has ``typed'' to
29749 GNUPLOT and the responses it has received. Calc tries to notice when an
29750 error message has appeared here and display the buffer for you when
29751 this happens. You can check this buffer yourself if you suspect
29752 something has gone wrong@footnote{
29753 On MS-Windows, due to the peculiarities of how the Windows version of
29754 GNUPLOT (called @command{wgnuplot}) works, the GNUPLOT responses are
29755 not communicated back to Calc. Instead, you need to look them up in
29756 the GNUPLOT command window that is displayed as in normal interactive
29761 @pindex calc-graph-command
29762 The @kbd{g C} (@code{calc-graph-command}) command prompts you to
29763 enter any line of text, then simply sends that line to the current
29764 GNUPLOT process. The @samp{*Gnuplot Trail*} buffer looks deceptively
29765 like a Shell buffer but you can't type commands in it yourself.
29766 Instead, you must use @kbd{g C} for this purpose.
29770 @pindex calc-graph-view-commands
29771 @pindex calc-graph-view-trail
29772 The @kbd{g v} (@code{calc-graph-view-commands}) and @kbd{g V}
29773 (@code{calc-graph-view-trail}) commands display the @samp{*Gnuplot Commands*}
29774 and @samp{*Gnuplot Trail*} buffers, respectively, in another window.
29775 This happens automatically when Calc thinks there is something you
29776 will want to see in either of these buffers. If you type @kbd{g v}
29777 or @kbd{g V} when the relevant buffer is already displayed, the
29778 buffer is hidden again. (Note that on MS-Windows, the @samp{*Gnuplot
29779 Trail*} buffer will usually show nothing of interest, because
29780 GNUPLOT's responses are not communicated back to Calc.)
29782 One reason to use @kbd{g v} is to add your own commands to the
29783 @samp{*Gnuplot Commands*} buffer. Press @kbd{g v}, then use
29784 @kbd{C-x o} to switch into that window. For example, GNUPLOT has
29785 @samp{set label} and @samp{set arrow} commands that allow you to
29786 annotate your plots. Since Calc doesn't understand these commands,
29787 you have to add them to the @samp{*Gnuplot Commands*} buffer
29788 yourself, then use @w{@kbd{g p}} to replot using these new commands. Note
29789 that your commands must appear @emph{before} the @code{plot} command.
29790 To get help on any GNUPLOT feature, type, e.g., @kbd{g C help set label}.
29791 You may have to type @kbd{g C @key{RET}} a few times to clear the
29792 ``press return for more'' or ``subtopic of @dots{}'' requests.
29793 Note that Calc always sends commands (like @samp{set nolabel}) to
29794 reset all plotting parameters to the defaults before each plot, so
29795 to delete a label all you need to do is delete the @samp{set label}
29796 line you added (or comment it out with @samp{#}) and then replot
29800 @pindex calc-graph-quit
29801 You can use @kbd{g q} (@code{calc-graph-quit}) to kill the GNUPLOT
29802 process that is running. The next graphing command you give will
29803 start a fresh GNUPLOT process. The word @samp{Graph} appears in
29804 the Calc window's mode line whenever a GNUPLOT process is currently
29805 running. The GNUPLOT process is automatically killed when you
29806 exit Emacs if you haven't killed it manually by then.
29809 @pindex calc-graph-kill
29810 The @kbd{g K} (@code{calc-graph-kill}) command is like @kbd{g q}
29811 except that it also views the @samp{*Gnuplot Trail*} buffer so that
29812 you can see the process being killed. This is better if you are
29813 killing GNUPLOT because you think it has gotten stuck.
29815 @node Kill and Yank, Keypad Mode, Graphics, Top
29816 @chapter Kill and Yank Functions
29819 The commands in this chapter move information between the Calculator and
29820 other Emacs editing buffers.
29822 In many cases Embedded mode is an easier and more natural way to
29823 work with Calc from a regular editing buffer. @xref{Embedded Mode}.
29826 * Killing From Stack::
29827 * Yanking Into Stack::
29828 * Saving Into Registers::
29829 * Inserting From Registers::
29830 * Grabbing From Buffers::
29831 * Yanking Into Buffers::
29832 * X Cut and Paste::
29835 @node Killing From Stack, Yanking Into Stack, Kill and Yank, Kill and Yank
29836 @section Killing from the Stack
29842 @pindex calc-copy-as-kill
29844 @pindex calc-kill-region
29846 @pindex calc-copy-region-as-kill
29849 @dfn{Kill} commands are Emacs commands that insert text into the ``kill
29850 ring,'' from which it can later be ``yanked'' by a @kbd{C-y} command.
29851 Three common kill commands in normal Emacs are @kbd{C-k}, which kills
29852 one line, @kbd{C-w}, which kills the region between mark and point, and
29853 @kbd{M-w}, which puts the region into the kill ring without actually
29854 deleting it. All of these commands work in the Calculator, too,
29855 although in the Calculator they operate on whole stack entries, so they
29856 ``round up'' the specified region to encompass full lines. (To copy
29857 only parts of lines, the @kbd{M-C-w} command in the Calculator will copy
29858 the region to the kill ring without any ``rounding up'', just like the
29859 @kbd{M-w} command in normal Emacs.) Also, @kbd{M-k} has been provided
29860 to complete the set; it puts the current line into the kill ring without
29863 The kill commands are unusual in that they pay attention to the location
29864 of the cursor in the Calculator buffer. If the cursor is on or below
29865 the bottom line, the kill commands operate on the top of the stack.
29866 Otherwise, they operate on whatever stack element the cursor is on. The
29867 text is copied into the kill ring exactly as it appears on the screen,
29868 including line numbers if they are enabled.
29870 A numeric prefix argument to @kbd{C-k} or @kbd{M-k} affects the number
29871 of lines killed. A positive argument kills the current line and @expr{n-1}
29872 lines below it. A negative argument kills the @expr{-n} lines above the
29873 current line. Again this mirrors the behavior of the standard Emacs
29874 @kbd{C-k} command. Although a whole line is always deleted, @kbd{C-k}
29875 with no argument copies only the number itself into the kill ring, whereas
29876 @kbd{C-k} with a prefix argument of 1 copies the number with its trailing
29879 @node Yanking Into Stack, Saving Into Registers, Killing From Stack, Kill and Yank
29880 @section Yanking into the Stack
29885 The @kbd{C-y} command yanks the most recently killed text back into the
29886 Calculator. It pushes this value onto the top of the stack regardless of
29887 the cursor position. In general it re-parses the killed text as a number
29888 or formula (or a list of these separated by commas or newlines). However if
29889 the thing being yanked is something that was just killed from the Calculator
29890 itself, its full internal structure is yanked. For example, if you have
29891 set the floating-point display mode to show only four significant digits,
29892 then killing and re-yanking 3.14159 (which displays as 3.142) will yank the
29893 full 3.14159, even though yanking it into any other buffer would yank the
29894 number in its displayed form, 3.142. (Since the default display modes
29895 show all objects to their full precision, this feature normally makes no
29898 @node Saving Into Registers, Inserting From Registers, Yanking Into Stack, Kill and Yank
29899 @section Saving into Registers
29903 @pindex calc-copy-to-register
29904 @pindex calc-prepend-to-register
29905 @pindex calc-append-to-register
29907 An alternative to killing and yanking stack entries is using
29908 registers in Calc. Saving stack entries in registers is like
29909 saving text in normal Emacs registers; although, like Calc's kill
29910 commands, register commands always operate on whole stack
29913 Registers in Calc are places to store stack entries for later use;
29914 each register is indexed by a single character. To store the current
29915 region (rounded up, of course, to include full stack entries) into a
29916 register, use the command @kbd{r s} (@code{calc-copy-to-register}).
29917 You will then be prompted for a register to use, the next character
29918 you type will be the index for the register. To store the region in
29919 register @var{r}, the full command will be @kbd{r s @var{r}}. With an
29920 argument, @kbd{C-u r s @var{r}}, the region being copied to the
29921 register will be deleted from the Calc buffer.
29923 It is possible to add additional stack entries to a register. The
29924 command @kbd{M-x calc-append-to-register} will prompt for a register,
29925 then add the stack entries in the region to the end of the register
29926 contents. The command @kbd{M-x calc-prepend-to-register} will
29927 similarly prompt for a register and add the stack entries in the
29928 region to the beginning of the register contents. Both commands take
29929 @kbd{C-u} arguments, which will cause the region to be deleted after being
29930 added to the register.
29932 @node Inserting From Registers, Grabbing From Buffers, Saving Into Registers, Kill and Yank
29933 @section Inserting from Registers
29936 @pindex calc-insert-register
29937 The command @kbd{r i} (@code{calc-insert-register}) will prompt for a
29938 register, then insert the contents of that register into the
29939 Calculator. If the contents of the register were placed there from
29940 within Calc, then the full internal structure of the contents will be
29941 inserted into the Calculator, otherwise whatever text is in the
29942 register is reparsed and then inserted into the Calculator.
29944 @node Grabbing From Buffers, Yanking Into Buffers, Inserting From Registers, Kill and Yank
29945 @section Grabbing from Other Buffers
29949 @pindex calc-grab-region
29950 The @kbd{C-x * g} (@code{calc-grab-region}) command takes the text between
29951 point and mark in the current buffer and attempts to parse it as a
29952 vector of values. Basically, it wraps the text in vector brackets
29953 @samp{[ ]} unless the text already is enclosed in vector brackets,
29954 then reads the text as if it were an algebraic entry. The contents
29955 of the vector may be numbers, formulas, or any other Calc objects.
29956 If the @kbd{C-x * g} command works successfully, it does an automatic
29957 @kbd{C-x * c} to enter the Calculator buffer.
29959 A numeric prefix argument grabs the specified number of lines around
29960 point, ignoring the mark. A positive prefix grabs from point to the
29961 @expr{n}th following newline (so that @kbd{M-1 C-x * g} grabs from point
29962 to the end of the current line); a negative prefix grabs from point
29963 back to the @expr{n+1}st preceding newline. In these cases the text
29964 that is grabbed is exactly the same as the text that @kbd{C-k} would
29965 delete given that prefix argument.
29967 A prefix of zero grabs the current line; point may be anywhere on the
29970 A plain @kbd{C-u} prefix interprets the region between point and mark
29971 as a single number or formula rather than a vector. For example,
29972 @kbd{C-x * g} on the text @samp{2 a b} produces the vector of three
29973 values @samp{[2, a, b]}, but @kbd{C-u C-x * g} on the same region
29974 reads a formula which is a product of three things: @samp{2 a b}.
29975 (The text @samp{a + b}, on the other hand, will be grabbed as a
29976 vector of one element by plain @kbd{C-x * g} because the interpretation
29977 @samp{[a, +, b]} would be a syntax error.)
29979 If a different language has been specified (@pxref{Language Modes}),
29980 the grabbed text will be interpreted according to that language.
29983 @pindex calc-grab-rectangle
29984 The @kbd{C-x * r} (@code{calc-grab-rectangle}) command takes the text between
29985 point and mark and attempts to parse it as a matrix. If point and mark
29986 are both in the leftmost column, the lines in between are parsed in their
29987 entirety. Otherwise, point and mark define the corners of a rectangle
29988 whose contents are parsed.
29990 Each line of the grabbed area becomes a row of the matrix. The result
29991 will actually be a vector of vectors, which Calc will treat as a matrix
29992 only if every row contains the same number of values.
29994 If a line contains a portion surrounded by square brackets (or curly
29995 braces), that portion is interpreted as a vector which becomes a row
29996 of the matrix. Any text surrounding the bracketed portion on the line
29999 Otherwise, the entire line is interpreted as a row vector as if it
30000 were surrounded by square brackets. Leading line numbers (in the
30001 format used in the Calc stack buffer) are ignored. If you wish to
30002 force this interpretation (even if the line contains bracketed
30003 portions), give a negative numeric prefix argument to the
30004 @kbd{C-x * r} command.
30006 If you give a numeric prefix argument of zero or plain @kbd{C-u}, each
30007 line is instead interpreted as a single formula which is converted into
30008 a one-element vector. Thus the result of @kbd{C-u C-x * r} will be a
30009 one-column matrix. For example, suppose one line of the data is the
30010 expression @samp{2 a}. A plain @w{@kbd{C-x * r}} will interpret this as
30011 @samp{[2 a]}, which in turn is read as a two-element vector that forms
30012 one row of the matrix. But a @kbd{C-u C-x * r} will interpret this row
30015 If you give a positive numeric prefix argument @var{n}, then each line
30016 will be split up into columns of width @var{n}; each column is parsed
30017 separately as a matrix element. If a line contained
30018 @w{@samp{2 +/- 3 4 +/- 5}}, then grabbing with a prefix argument of 8
30019 would correctly split the line into two error forms.
30021 @xref{Matrix Functions}, to see how to pull the matrix apart into its
30022 constituent rows and columns. (If it is a
30023 @texline @math{1\times1}
30025 matrix, just hit @kbd{v u} (@code{calc-unpack}) twice.)
30029 @pindex calc-grab-sum-across
30030 @pindex calc-grab-sum-down
30031 @cindex Summing rows and columns of data
30032 The @kbd{C-x * :} (@code{calc-grab-sum-down}) command is a handy way to
30033 grab a rectangle of data and sum its columns. It is equivalent to
30034 typing @kbd{C-x * r}, followed by @kbd{V R : +} (the vector reduction
30035 command that sums the columns of a matrix; @pxref{Reducing}). The
30036 result of the command will be a vector of numbers, one for each column
30037 in the input data. The @kbd{C-x * _} (@code{calc-grab-sum-across}) command
30038 similarly grabs a rectangle and sums its rows by executing @w{@kbd{V R _ +}}.
30040 As well as being more convenient, @kbd{C-x * :} and @kbd{C-x * _} are also
30041 much faster because they don't actually place the grabbed vector on
30042 the stack. In a @kbd{C-x * r V R : +} sequence, formatting the vector
30043 for display on the stack takes a large fraction of the total time
30044 (unless you have planned ahead and used @kbd{v .} and @kbd{t .} modes).
30046 For example, suppose we have a column of numbers in a file which we
30047 wish to sum. Go to one corner of the column and press @kbd{C-@@} to
30048 set the mark; go to the other corner and type @kbd{C-x * :}. Since there
30049 is only one column, the result will be a vector of one number, the sum.
30050 (You can type @kbd{v u} to unpack this vector into a plain number if
30051 you want to do further arithmetic with it.)
30053 To compute the product of the column of numbers, we would have to do
30054 it ``by hand'' since there's no special grab-and-multiply command.
30055 Use @kbd{C-x * r} to grab the column of numbers into the calculator in
30056 the form of a column matrix. The statistics command @kbd{u *} is a
30057 handy way to find the product of a vector or matrix of numbers.
30058 @xref{Statistical Operations}. Another approach would be to use
30059 an explicit column reduction command, @kbd{V R : *}.
30061 @node Yanking Into Buffers, X Cut and Paste, Grabbing From Buffers, Kill and Yank
30062 @section Yanking into Other Buffers
30066 @pindex calc-copy-to-buffer
30067 The plain @kbd{y} (@code{calc-copy-to-buffer}) command inserts the number
30068 at the top of the stack into the most recently used normal editing buffer.
30069 (More specifically, this is the most recently used buffer which is displayed
30070 in a window and whose name does not begin with @samp{*}. If there is no
30071 such buffer, this is the most recently used buffer except for Calculator
30072 and Calc Trail buffers.) The number is inserted exactly as it appears and
30073 without a newline. (If line-numbering is enabled, the line number is
30074 normally not included.) The number is @emph{not} removed from the stack.
30076 With a prefix argument, @kbd{y} inserts several numbers, one per line.
30077 A positive argument inserts the specified number of values from the top
30078 of the stack. A negative argument inserts the @expr{n}th value from the
30079 top of the stack. An argument of zero inserts the entire stack. Note
30080 that @kbd{y} with an argument of 1 is slightly different from @kbd{y}
30081 with no argument; the former always copies full lines, whereas the
30082 latter strips off the trailing newline.
30084 With a lone @kbd{C-u} as a prefix argument, @kbd{y} @emph{replaces} the
30085 region in the other buffer with the yanked text, then quits the
30086 Calculator, leaving you in that buffer. A typical use would be to use
30087 @kbd{C-x * g} to read a region of data into the Calculator, operate on the
30088 data to produce a new matrix, then type @kbd{C-u y} to replace the
30089 original data with the new data. One might wish to alter the matrix
30090 display style (@pxref{Vector and Matrix Formats}) or change the current
30091 display language (@pxref{Language Modes}) before doing this. Also, note
30092 that this command replaces a linear region of text (as grabbed by
30093 @kbd{C-x * g}), not a rectangle (as grabbed by @kbd{C-x * r}).
30095 If the editing buffer is in overwrite (as opposed to insert) mode,
30096 and the @kbd{C-u} prefix was not used, then the yanked number will
30097 overwrite the characters following point rather than being inserted
30098 before those characters. The usual conventions of overwrite mode
30099 are observed; for example, characters will be inserted at the end of
30100 a line rather than overflowing onto the next line. Yanking a multi-line
30101 object such as a matrix in overwrite mode overwrites the next @var{n}
30102 lines in the buffer, lengthening or shortening each line as necessary.
30103 Finally, if the thing being yanked is a simple integer or floating-point
30104 number (like @samp{-1.2345e-3}) and the characters following point also
30105 make up such a number, then Calc will replace that number with the new
30106 number, lengthening or shortening as necessary. The concept of
30107 ``overwrite mode'' has thus been generalized from overwriting characters
30108 to overwriting one complete number with another.
30111 The @kbd{C-x * y} key sequence is equivalent to @kbd{y} except that
30112 it can be typed anywhere, not just in Calc. This provides an easy
30113 way to guarantee that Calc knows which editing buffer you want to use!
30115 @node X Cut and Paste, , Yanking Into Buffers, Kill and Yank
30116 @section X Cut and Paste
30119 If you are using Emacs with the X window system, there is an easier
30120 way to move small amounts of data into and out of the calculator:
30121 Use the mouse-oriented cut and paste facilities of X.
30123 The default bindings for a three-button mouse cause the left button
30124 to move the Emacs cursor to the given place, the right button to
30125 select the text between the cursor and the clicked location, and
30126 the middle button to yank the selection into the buffer at the
30127 clicked location. So, if you have a Calc window and an editing
30128 window on your Emacs screen, you can use left-click/right-click
30129 to select a number, vector, or formula from one window, then
30130 middle-click to paste that value into the other window. When you
30131 paste text into the Calc window, Calc interprets it as an algebraic
30132 entry. It doesn't matter where you click in the Calc window; the
30133 new value is always pushed onto the top of the stack.
30135 The @code{xterm} program that is typically used for general-purpose
30136 shell windows in X interprets the mouse buttons in the same way.
30137 So you can use the mouse to move data between Calc and any other
30138 Unix program. One nice feature of @code{xterm} is that a double
30139 left-click selects one word, and a triple left-click selects a
30140 whole line. So you can usually transfer a single number into Calc
30141 just by double-clicking on it in the shell, then middle-clicking
30142 in the Calc window.
30144 @node Keypad Mode, Embedded Mode, Kill and Yank, Top
30145 @chapter Keypad Mode
30149 @pindex calc-keypad
30150 The @kbd{C-x * k} (@code{calc-keypad}) command starts the Calculator
30151 and displays a picture of a calculator-style keypad. If you are using
30152 the X window system, you can click on any of the ``keys'' in the
30153 keypad using the left mouse button to operate the calculator.
30154 The original window remains the selected window; in Keypad mode
30155 you can type in your file while simultaneously performing
30156 calculations with the mouse.
30158 @pindex full-calc-keypad
30159 If you have used @kbd{C-x * b} first, @kbd{C-x * k} instead invokes
30160 the @code{full-calc-keypad} command, which takes over the whole
30161 Emacs screen and displays the keypad, the Calc stack, and the Calc
30162 trail all at once. This mode would normally be used when running
30163 Calc standalone (@pxref{Standalone Operation}).
30165 If you aren't using the X window system, you must switch into
30166 the @samp{*Calc Keypad*} window, place the cursor on the desired
30167 ``key,'' and type @key{SPC} or @key{RET}. If you think this
30168 is easier than using Calc normally, go right ahead.
30170 Calc commands are more or less the same in Keypad mode. Certain
30171 keypad keys differ slightly from the corresponding normal Calc
30172 keystrokes; all such deviations are described below.
30174 Keypad mode includes many more commands than will fit on the keypad
30175 at once. Click the right mouse button [@code{calc-keypad-menu}]
30176 to switch to the next menu. The bottom five rows of the keypad
30177 stay the same; the top three rows change to a new set of commands.
30178 To return to earlier menus, click the middle mouse button
30179 [@code{calc-keypad-menu-back}] or simply advance through the menus
30180 until you wrap around. Typing @key{TAB} inside the keypad window
30181 is equivalent to clicking the right mouse button there.
30183 You can always click the @key{EXEC} button and type any normal
30184 Calc key sequence. This is equivalent to switching into the
30185 Calc buffer, typing the keys, then switching back to your
30189 * Keypad Main Menu::
30190 * Keypad Functions Menu::
30191 * Keypad Binary Menu::
30192 * Keypad Vectors Menu::
30193 * Keypad Modes Menu::
30196 @node Keypad Main Menu, Keypad Functions Menu, Keypad Mode, Keypad Mode
30201 |----+----+--Calc---+----+----1
30202 |FLR |CEIL|RND |TRNC|CLN2|FLT |
30203 |----+----+----+----+----+----|
30204 | LN |EXP | |ABS |IDIV|MOD |
30205 |----+----+----+----+----+----|
30206 |SIN |COS |TAN |SQRT|y^x |1/x |
30207 |----+----+----+----+----+----|
30208 | ENTER |+/- |EEX |UNDO| <- |
30209 |-----+---+-+--+--+-+---++----|
30210 | INV | 7 | 8 | 9 | / |
30211 |-----+-----+-----+-----+-----|
30212 | HYP | 4 | 5 | 6 | * |
30213 |-----+-----+-----+-----+-----|
30214 |EXEC | 1 | 2 | 3 | - |
30215 |-----+-----+-----+-----+-----|
30216 | OFF | 0 | . | PI | + |
30217 |-----+-----+-----+-----+-----+
30222 This is the menu that appears the first time you start Keypad mode.
30223 It will show up in a vertical window on the right side of your screen.
30224 Above this menu is the traditional Calc stack display. On a 24-line
30225 screen you will be able to see the top three stack entries.
30227 The ten digit keys, decimal point, and @key{EEX} key are used for
30228 entering numbers in the obvious way. @key{EEX} begins entry of an
30229 exponent in scientific notation. Just as with regular Calc, the
30230 number is pushed onto the stack as soon as you press @key{ENTER}
30231 or any other function key.
30233 The @key{+/-} key corresponds to normal Calc's @kbd{n} key. During
30234 numeric entry it changes the sign of the number or of the exponent.
30235 At other times it changes the sign of the number on the top of the
30238 The @key{INV} and @key{HYP} keys modify other keys. As well as
30239 having the effects described elsewhere in this manual, Keypad mode
30240 defines several other ``inverse'' operations. These are described
30241 below and in the following sections.
30243 The @key{ENTER} key finishes the current numeric entry, or otherwise
30244 duplicates the top entry on the stack.
30246 The @key{UNDO} key undoes the most recent Calc operation.
30247 @kbd{INV UNDO} is the ``redo'' command, and @kbd{HYP UNDO} is
30248 ``last arguments'' (@kbd{M-@key{RET}}).
30250 The @key{<-} key acts as a ``backspace'' during numeric entry.
30251 At other times it removes the top stack entry. @kbd{INV <-}
30252 clears the entire stack. @kbd{HYP <-} takes an integer from
30253 the stack, then removes that many additional stack elements.
30255 The @key{EXEC} key prompts you to enter any keystroke sequence
30256 that would normally work in Calc mode. This can include a
30257 numeric prefix if you wish. It is also possible simply to
30258 switch into the Calc window and type commands in it; there is
30259 nothing ``magic'' about this window when Keypad mode is active.
30261 The other keys in this display perform their obvious calculator
30262 functions. @key{CLN2} rounds the top-of-stack by temporarily
30263 reducing the precision by 2 digits. @key{FLT} converts an
30264 integer or fraction on the top of the stack to floating-point.
30266 The @key{INV} and @key{HYP} keys combined with several of these keys
30267 give you access to some common functions even if the appropriate menu
30268 is not displayed. Obviously you don't need to learn these keys
30269 unless you find yourself wasting time switching among the menus.
30273 is the same as @key{1/x}.
30275 is the same as @key{SQRT}.
30277 is the same as @key{CONJ}.
30279 is the same as @key{y^x}.
30281 is the same as @key{INV y^x} (the @expr{x}th root of @expr{y}).
30283 are the same as @key{SIN} / @kbd{INV SIN}.
30285 are the same as @key{COS} / @kbd{INV COS}.
30287 are the same as @key{TAN} / @kbd{INV TAN}.
30289 are the same as @key{LN} / @kbd{HYP LN}.
30291 are the same as @key{EXP} / @kbd{HYP EXP}.
30293 is the same as @key{ABS}.
30295 is the same as @key{RND} (@code{calc-round}).
30297 is the same as @key{CLN2}.
30299 is the same as @key{FLT} (@code{calc-float}).
30301 is the same as @key{IMAG}.
30303 is the same as @key{PREC}.
30305 is the same as @key{SWAP}.
30307 is the same as @key{RLL3}.
30308 @item INV HYP ENTER
30309 is the same as @key{OVER}.
30311 packs the top two stack entries as an error form.
30313 packs the top two stack entries as a modulo form.
30315 creates an interval form; this removes an integer which is one
30316 of 0 @samp{[]}, 1 @samp{[)}, 2 @samp{(]} or 3 @samp{()}, followed
30317 by the two limits of the interval.
30320 The @kbd{OFF} key turns Calc off; typing @kbd{C-x * k} or @kbd{C-x * *}
30321 again has the same effect. This is analogous to typing @kbd{q} or
30322 hitting @kbd{C-x * c} again in the normal calculator. If Calc is
30323 running standalone (the @code{full-calc-keypad} command appeared in the
30324 command line that started Emacs), then @kbd{OFF} is replaced with
30325 @kbd{EXIT}; clicking on this actually exits Emacs itself.
30327 @node Keypad Functions Menu, Keypad Binary Menu, Keypad Main Menu, Keypad Mode
30328 @section Functions Menu
30332 |----+----+----+----+----+----2
30333 |IGAM|BETA|IBET|ERF |BESJ|BESY|
30334 |----+----+----+----+----+----|
30335 |IMAG|CONJ| RE |ATN2|RAND|RAGN|
30336 |----+----+----+----+----+----|
30337 |GCD |FACT|DFCT|BNOM|PERM|NXTP|
30338 |----+----+----+----+----+----|
30343 This menu provides various operations from the @kbd{f} and @kbd{k}
30346 @key{IMAG} multiplies the number on the stack by the imaginary
30347 number @expr{i = (0, 1)}.
30349 @key{RE} extracts the real part a complex number. @kbd{INV RE}
30350 extracts the imaginary part.
30352 @key{RAND} takes a number from the top of the stack and computes
30353 a random number greater than or equal to zero but less than that
30354 number. (@xref{Random Numbers}.) @key{RAGN} is the ``random
30355 again'' command; it computes another random number using the
30356 same limit as last time.
30358 @key{INV GCD} computes the LCM (least common multiple) function.
30360 @key{INV FACT} is the gamma function.
30361 @texline @math{\Gamma(x) = (x-1)!}.
30362 @infoline @expr{gamma(x) = (x-1)!}.
30364 @key{PERM} is the number-of-permutations function, which is on the
30365 @kbd{H k c} key in normal Calc.
30367 @key{NXTP} finds the next prime after a number. @kbd{INV NXTP}
30368 finds the previous prime.
30370 @node Keypad Binary Menu, Keypad Vectors Menu, Keypad Functions Menu, Keypad Mode
30371 @section Binary Menu
30375 |----+----+----+----+----+----3
30376 |AND | OR |XOR |NOT |LSH |RSH |
30377 |----+----+----+----+----+----|
30378 |DEC |HEX |OCT |BIN |WSIZ|ARSH|
30379 |----+----+----+----+----+----|
30380 | A | B | C | D | E | F |
30381 |----+----+----+----+----+----|
30386 The keys in this menu perform operations on binary integers.
30387 Note that both logical and arithmetic right-shifts are provided.
30388 @key{INV LSH} rotates one bit to the left.
30390 The ``difference'' function (normally on @kbd{b d}) is on @key{INV AND}.
30391 The ``clip'' function (normally on @w{@kbd{b c}}) is on @key{INV NOT}.
30393 The @key{DEC}, @key{HEX}, @key{OCT}, and @key{BIN} keys select the
30394 current radix for display and entry of numbers: Decimal, hexadecimal,
30395 octal, or binary. The six letter keys @key{A} through @key{F} are used
30396 for entering hexadecimal numbers.
30398 The @key{WSIZ} key displays the current word size for binary operations
30399 and allows you to enter a new word size. You can respond to the prompt
30400 using either the keyboard or the digits and @key{ENTER} from the keypad.
30401 The initial word size is 32 bits.
30403 @node Keypad Vectors Menu, Keypad Modes Menu, Keypad Binary Menu, Keypad Mode
30404 @section Vectors Menu
30408 |----+----+----+----+----+----4
30409 |SUM |PROD|MAX |MAP*|MAP^|MAP$|
30410 |----+----+----+----+----+----|
30411 |MINV|MDET|MTRN|IDNT|CRSS|"x" |
30412 |----+----+----+----+----+----|
30413 |PACK|UNPK|INDX|BLD |LEN |... |
30414 |----+----+----+----+----+----|
30419 The keys in this menu operate on vectors and matrices.
30421 @key{PACK} removes an integer @var{n} from the top of the stack;
30422 the next @var{n} stack elements are removed and packed into a vector,
30423 which is replaced onto the stack. Thus the sequence
30424 @kbd{1 ENTER 3 ENTER 5 ENTER 3 PACK} enters the vector
30425 @samp{[1, 3, 5]} onto the stack. To enter a matrix, build each row
30426 on the stack as a vector, then use a final @key{PACK} to collect the
30427 rows into a matrix.
30429 @key{UNPK} unpacks the vector on the stack, pushing each of its
30430 components separately.
30432 @key{INDX} removes an integer @var{n}, then builds a vector of
30433 integers from 1 to @var{n}. @kbd{INV INDX} takes three numbers
30434 from the stack: The vector size @var{n}, the starting number,
30435 and the increment. @kbd{BLD} takes an integer @var{n} and any
30436 value @var{x} and builds a vector of @var{n} copies of @var{x}.
30438 @key{IDNT} removes an integer @var{n}, then builds an @var{n}-by-@var{n}
30441 @key{LEN} replaces a vector by its length, an integer.
30443 @key{...} turns on or off ``abbreviated'' display mode for large vectors.
30445 @key{MINV}, @key{MDET}, @key{MTRN}, and @key{CROSS} are the matrix
30446 inverse, determinant, and transpose, and vector cross product.
30448 @key{SUM} replaces a vector by the sum of its elements. It is
30449 equivalent to @kbd{u +} in normal Calc (@pxref{Statistical Operations}).
30450 @key{PROD} computes the product of the elements of a vector, and
30451 @key{MAX} computes the maximum of all the elements of a vector.
30453 @key{INV SUM} computes the alternating sum of the first element
30454 minus the second, plus the third, minus the fourth, and so on.
30455 @key{INV MAX} computes the minimum of the vector elements.
30457 @key{HYP SUM} computes the mean of the vector elements.
30458 @key{HYP PROD} computes the sample standard deviation.
30459 @key{HYP MAX} computes the median.
30461 @key{MAP*} multiplies two vectors elementwise. It is equivalent
30462 to the @kbd{V M *} command. @key{MAP^} computes powers elementwise.
30463 The arguments must be vectors of equal length, or one must be a vector
30464 and the other must be a plain number. For example, @kbd{2 MAP^} squares
30465 all the elements of a vector.
30467 @key{MAP$} maps the formula on the top of the stack across the
30468 vector in the second-to-top position. If the formula contains
30469 several variables, Calc takes that many vectors starting at the
30470 second-to-top position and matches them to the variables in
30471 alphabetical order. The result is a vector of the same size as
30472 the input vectors, whose elements are the formula evaluated with
30473 the variables set to the various sets of numbers in those vectors.
30474 For example, you could simulate @key{MAP^} using @key{MAP$} with
30475 the formula @samp{x^y}.
30477 The @kbd{"x"} key pushes the variable name @expr{x} onto the
30478 stack. To build the formula @expr{x^2 + 6}, you would use the
30479 key sequence @kbd{"x" 2 y^x 6 +}. This formula would then be
30480 suitable for use with the @key{MAP$} key described above.
30481 With @key{INV}, @key{HYP}, or @key{INV} and @key{HYP}, the
30482 @kbd{"x"} key pushes the variable names @expr{y}, @expr{z}, and
30483 @expr{t}, respectively.
30485 @node Keypad Modes Menu, , Keypad Vectors Menu, Keypad Mode
30486 @section Modes Menu
30490 |----+----+----+----+----+----5
30491 |FLT |FIX |SCI |ENG |GRP | |
30492 |----+----+----+----+----+----|
30493 |RAD |DEG |FRAC|POLR|SYMB|PREC|
30494 |----+----+----+----+----+----|
30495 |SWAP|RLL3|RLL4|OVER|STO |RCL |
30496 |----+----+----+----+----+----|
30501 The keys in this menu manipulate modes, variables, and the stack.
30503 The @key{FLT}, @key{FIX}, @key{SCI}, and @key{ENG} keys select
30504 floating-point, fixed-point, scientific, or engineering notation.
30505 @key{FIX} displays two digits after the decimal by default; the
30506 others display full precision. With the @key{INV} prefix, these
30507 keys pop a number-of-digits argument from the stack.
30509 The @key{GRP} key turns grouping of digits with commas on or off.
30510 @kbd{INV GRP} enables grouping to the right of the decimal point as
30511 well as to the left.
30513 The @key{RAD} and @key{DEG} keys switch between radians and degrees
30514 for trigonometric functions.
30516 The @key{FRAC} key turns Fraction mode on or off. This affects
30517 whether commands like @kbd{/} with integer arguments produce
30518 fractional or floating-point results.
30520 The @key{POLR} key turns Polar mode on or off, determining whether
30521 polar or rectangular complex numbers are used by default.
30523 The @key{SYMB} key turns Symbolic mode on or off, in which
30524 operations that would produce inexact floating-point results
30525 are left unevaluated as algebraic formulas.
30527 The @key{PREC} key selects the current precision. Answer with
30528 the keyboard or with the keypad digit and @key{ENTER} keys.
30530 The @key{SWAP} key exchanges the top two stack elements.
30531 The @key{RLL3} key rotates the top three stack elements upwards.
30532 The @key{RLL4} key rotates the top four stack elements upwards.
30533 The @key{OVER} key duplicates the second-to-top stack element.
30535 The @key{STO} and @key{RCL} keys are analogous to @kbd{s t} and
30536 @kbd{s r} in regular Calc. @xref{Store and Recall}. Click the
30537 @key{STO} or @key{RCL} key, then one of the ten digits. (Named
30538 variables are not available in Keypad mode.) You can also use,
30539 for example, @kbd{STO + 3} to add to register 3.
30541 @node Embedded Mode, Programming, Keypad Mode, Top
30542 @chapter Embedded Mode
30545 Embedded mode in Calc provides an alternative to copying numbers
30546 and formulas back and forth between editing buffers and the Calc
30547 stack. In Embedded mode, your editing buffer becomes temporarily
30548 linked to the stack and this copying is taken care of automatically.
30551 * Basic Embedded Mode::
30552 * More About Embedded Mode::
30553 * Assignments in Embedded Mode::
30554 * Mode Settings in Embedded Mode::
30555 * Customizing Embedded Mode::
30558 @node Basic Embedded Mode, More About Embedded Mode, Embedded Mode, Embedded Mode
30559 @section Basic Embedded Mode
30563 @pindex calc-embedded
30564 To enter Embedded mode, position the Emacs point (cursor) on a
30565 formula in any buffer and press @kbd{C-x * e} (@code{calc-embedded}).
30566 Note that @kbd{C-x * e} is not to be used in the Calc stack buffer
30567 like most Calc commands, but rather in regular editing buffers that
30568 are visiting your own files.
30570 Calc will try to guess an appropriate language based on the major mode
30571 of the editing buffer. (@xref{Language Modes}.) If the current buffer is
30572 in @code{latex-mode}, for example, Calc will set its language to @LaTeX{}.
30573 Similarly, Calc will use @TeX{} language for @code{tex-mode},
30574 @code{plain-tex-mode} and @code{context-mode}, C language for
30575 @code{c-mode} and @code{c++-mode}, FORTRAN language for
30576 @code{fortran-mode} and @code{f90-mode}, Pascal for @code{pascal-mode},
30577 and eqn for @code{nroff-mode} (@pxref{Customizing Calc}).
30578 These can be overridden with Calc's mode
30579 changing commands (@pxref{Mode Settings in Embedded Mode}). If no
30580 suitable language is available, Calc will continue with its current language.
30582 Calc normally scans backward and forward in the buffer for the
30583 nearest opening and closing @dfn{formula delimiters}. The simplest
30584 delimiters are blank lines. Other delimiters that Embedded mode
30589 The @TeX{} and @LaTeX{} math delimiters @samp{$ $}, @samp{$$ $$},
30590 @samp{\[ \]}, and @samp{\( \)};
30592 Lines beginning with @samp{\begin} and @samp{\end} (except matrix delimiters);
30594 Lines beginning with @samp{@@} (Texinfo delimiters).
30596 Lines beginning with @samp{.EQ} and @samp{.EN} (@dfn{eqn} delimiters);
30598 Lines containing a single @samp{%} or @samp{.\"} symbol and nothing else.
30601 @xref{Customizing Embedded Mode}, to see how to make Calc recognize
30602 your own favorite delimiters. Delimiters like @samp{$ $} can appear
30603 on their own separate lines or in-line with the formula.
30605 If you give a positive or negative numeric prefix argument, Calc
30606 instead uses the current point as one end of the formula, and includes
30607 that many lines forward or backward (respectively, including the current
30608 line). Explicit delimiters are not necessary in this case.
30610 With a prefix argument of zero, Calc uses the current region (delimited
30611 by point and mark) instead of formula delimiters. With a prefix
30612 argument of @kbd{C-u} only, Calc uses the current line as the formula.
30615 @pindex calc-embedded-word
30616 The @kbd{C-x * w} (@code{calc-embedded-word}) command will start Embedded
30617 mode on the current ``word''; in this case Calc will scan for the first
30618 non-numeric character (i.e., the first character that is not a digit,
30619 sign, decimal point, or upper- or lower-case @samp{e}) forward and
30620 backward to delimit the formula.
30622 When you enable Embedded mode for a formula, Calc reads the text
30623 between the delimiters and tries to interpret it as a Calc formula.
30624 Calc can generally identify @TeX{} formulas and
30625 Big-style formulas even if the language mode is wrong. If Calc
30626 can't make sense of the formula, it beeps and refuses to enter
30627 Embedded mode. But if the current language is wrong, Calc can
30628 sometimes parse the formula successfully (but incorrectly);
30629 for example, the C expression @samp{atan(a[1])} can be parsed
30630 in Normal language mode, but the @code{atan} won't correspond to
30631 the built-in @code{arctan} function, and the @samp{a[1]} will be
30632 interpreted as @samp{a} times the vector @samp{[1]}!
30634 If you press @kbd{C-x * e} or @kbd{C-x * w} to activate an embedded
30635 formula which is blank, say with the cursor on the space between
30636 the two delimiters @samp{$ $}, Calc will immediately prompt for
30637 an algebraic entry.
30639 Only one formula in one buffer can be enabled at a time. If you
30640 move to another area of the current buffer and give Calc commands,
30641 Calc turns Embedded mode off for the old formula and then tries
30642 to restart Embedded mode at the new position. Other buffers are
30643 not affected by Embedded mode.
30645 When Embedded mode begins, Calc pushes the current formula onto
30646 the stack. No Calc stack window is created; however, Calc copies
30647 the top-of-stack position into the original buffer at all times.
30648 You can create a Calc window by hand with @kbd{C-x * o} if you
30649 find you need to see the entire stack.
30651 For example, typing @kbd{C-x * e} while somewhere in the formula
30652 @samp{n>2} in the following line enables Embedded mode on that
30656 We define $F_n = F_(n-1)+F_(n-2)$ for all $n>2$.
30660 The formula @expr{n>2} will be pushed onto the Calc stack, and
30661 the top of stack will be copied back into the editing buffer.
30662 This means that spaces will appear around the @samp{>} symbol
30663 to match Calc's usual display style:
30666 We define $F_n = F_(n-1)+F_(n-2)$ for all $n > 2$.
30670 No spaces have appeared around the @samp{+} sign because it's
30671 in a different formula, one which we have not yet touched with
30674 Now that Embedded mode is enabled, keys you type in this buffer
30675 are interpreted as Calc commands. At this point we might use
30676 the ``commute'' command @kbd{j C} to reverse the inequality.
30677 This is a selection-based command for which we first need to
30678 move the cursor onto the operator (@samp{>} in this case) that
30679 needs to be commuted.
30682 We define $F_n = F_(n-1)+F_(n-2)$ for all $2 < n$.
30685 The @kbd{C-x * o} command is a useful way to open a Calc window
30686 without actually selecting that window. Giving this command
30687 verifies that @samp{2 < n} is also on the Calc stack. Typing
30688 @kbd{17 @key{RET}} would produce:
30691 We define $F_n = F_(n-1)+F_(n-2)$ for all $17$.
30695 with @samp{2 < n} and @samp{17} on the stack; typing @key{TAB}
30696 at this point will exchange the two stack values and restore
30697 @samp{2 < n} to the embedded formula. Even though you can't
30698 normally see the stack in Embedded mode, it is still there and
30699 it still operates in the same way. But, as with old-fashioned
30700 RPN calculators, you can only see the value at the top of the
30701 stack at any given time (unless you use @kbd{C-x * o}).
30703 Typing @kbd{C-x * e} again turns Embedded mode off. The Calc
30704 window reveals that the formula @w{@samp{2 < n}} is automatically
30705 removed from the stack, but the @samp{17} is not. Entering
30706 Embedded mode always pushes one thing onto the stack, and
30707 leaving Embedded mode always removes one thing. Anything else
30708 that happens on the stack is entirely your business as far as
30709 Embedded mode is concerned.
30711 If you press @kbd{C-x * e} in the wrong place by accident, it is
30712 possible that Calc will be able to parse the nearby text as a
30713 formula and will mangle that text in an attempt to redisplay it
30714 ``properly'' in the current language mode. If this happens,
30715 press @kbd{C-x * e} again to exit Embedded mode, then give the
30716 regular Emacs ``undo'' command (@kbd{C-_} or @kbd{C-x u}) to put
30717 the text back the way it was before Calc edited it. Note that Calc's
30718 own Undo command (typed before you turn Embedded mode back off)
30719 will not do you any good, because as far as Calc is concerned
30720 you haven't done anything with this formula yet.
30722 @node More About Embedded Mode, Assignments in Embedded Mode, Basic Embedded Mode, Embedded Mode
30723 @section More About Embedded Mode
30726 When Embedded mode ``activates'' a formula, i.e., when it examines
30727 the formula for the first time since the buffer was created or
30728 loaded, Calc tries to sense the language in which the formula was
30729 written. If the formula contains any @LaTeX{}-like @samp{\} sequences,
30730 it is parsed (i.e., read) in @LaTeX{} mode. If the formula appears to
30731 be written in multi-line Big mode, it is parsed in Big mode. Otherwise,
30732 it is parsed according to the current language mode.
30734 Note that Calc does not change the current language mode according
30735 the formula it reads in. Even though it can read a @LaTeX{} formula when
30736 not in @LaTeX{} mode, it will immediately rewrite this formula using
30737 whatever language mode is in effect.
30744 @pindex calc-show-plain
30745 Calc's parser is unable to read certain kinds of formulas. For
30746 example, with @kbd{v ]} (@code{calc-matrix-brackets}) you can
30747 specify matrix display styles which the parser is unable to
30748 recognize as matrices. The @kbd{d p} (@code{calc-show-plain})
30749 command turns on a mode in which a ``plain'' version of a
30750 formula is placed in front of the fully-formatted version.
30751 When Calc reads a formula that has such a plain version in
30752 front, it reads the plain version and ignores the formatted
30755 Plain formulas are preceded and followed by @samp{%%%} signs
30756 by default. This notation has the advantage that the @samp{%}
30757 character begins a comment in @TeX{} and @LaTeX{}, so if your formula is
30758 embedded in a @TeX{} or @LaTeX{} document its plain version will be
30759 invisible in the final printed copy. Certain major modes have different
30760 delimiters to ensure that the ``plain'' version will be
30761 in a comment for those modes, also.
30762 See @ref{Customizing Embedded Mode} to see how to change the ``plain''
30763 formula delimiters.
30765 There are several notations which Calc's parser for ``big''
30766 formatted formulas can't yet recognize. In particular, it can't
30767 read the large symbols for @code{sum}, @code{prod}, and @code{integ},
30768 and it can't handle @samp{=>} with the righthand argument omitted.
30769 Also, Calc won't recognize special formats you have defined with
30770 the @kbd{Z C} command (@pxref{User-Defined Compositions}). In
30771 these cases it is important to use ``plain'' mode to make sure
30772 Calc will be able to read your formula later.
30774 Another example where ``plain'' mode is important is if you have
30775 specified a float mode with few digits of precision. Normally
30776 any digits that are computed but not displayed will simply be
30777 lost when you save and re-load your embedded buffer, but ``plain''
30778 mode allows you to make sure that the complete number is present
30779 in the file as well as the rounded-down number.
30785 Embedded buffers remember active formulas for as long as they
30786 exist in Emacs memory. Suppose you have an embedded formula
30787 which is @cpi{} to the normal 12 decimal places, and then
30788 type @w{@kbd{C-u 5 d n}} to display only five decimal places.
30789 If you then type @kbd{d n}, all 12 places reappear because the
30790 full number is still there on the Calc stack. More surprisingly,
30791 even if you exit Embedded mode and later re-enter it for that
30792 formula, typing @kbd{d n} will restore all 12 places because
30793 each buffer remembers all its active formulas. However, if you
30794 save the buffer in a file and reload it in a new Emacs session,
30795 all non-displayed digits will have been lost unless you used
30802 In some applications of Embedded mode, you will want to have a
30803 sequence of copies of a formula that show its evolution as you
30804 work on it. For example, you might want to have a sequence
30805 like this in your file (elaborating here on the example from
30806 the ``Getting Started'' chapter):
30815 @r{(the derivative of }ln(ln(x))@r{)}
30817 whose value at x = 2 is
30827 @pindex calc-embedded-duplicate
30828 The @kbd{C-x * d} (@code{calc-embedded-duplicate}) command is a
30829 handy way to make sequences like this. If you type @kbd{C-x * d},
30830 the formula under the cursor (which may or may not have Embedded
30831 mode enabled for it at the time) is copied immediately below and
30832 Embedded mode is then enabled for that copy.
30834 For this example, you would start with just
30843 and press @kbd{C-x * d} with the cursor on this formula. The result
30856 with the second copy of the formula enabled in Embedded mode.
30857 You can now press @kbd{a d x @key{RET}} to take the derivative, and
30858 @kbd{C-x * d C-x * d} to make two more copies of the derivative.
30859 To complete the computations, type @kbd{3 s l x @key{RET}} to evaluate
30860 the last formula, then move up to the second-to-last formula
30861 and type @kbd{2 s l x @key{RET}}.
30863 Finally, you would want to press @kbd{C-x * e} to exit Embedded
30864 mode, then go up and insert the necessary text in between the
30865 various formulas and numbers.
30873 @pindex calc-embedded-new-formula
30874 The @kbd{C-x * f} (@code{calc-embedded-new-formula}) command
30875 creates a new embedded formula at the current point. It inserts
30876 some default delimiters, which are usually just blank lines,
30877 and then does an algebraic entry to get the formula (which is
30878 then enabled for Embedded mode). This is just shorthand for
30879 typing the delimiters yourself, positioning the cursor between
30880 the new delimiters, and pressing @kbd{C-x * e}. The key sequence
30881 @kbd{C-x * '} is equivalent to @kbd{C-x * f}.
30885 @pindex calc-embedded-next
30886 @pindex calc-embedded-previous
30887 The @kbd{C-x * n} (@code{calc-embedded-next}) and @kbd{C-x * p}
30888 (@code{calc-embedded-previous}) commands move the cursor to the
30889 next or previous active embedded formula in the buffer. They
30890 can take positive or negative prefix arguments to move by several
30891 formulas. Note that these commands do not actually examine the
30892 text of the buffer looking for formulas; they only see formulas
30893 which have previously been activated in Embedded mode. In fact,
30894 @kbd{C-x * n} and @kbd{C-x * p} are a useful way to tell which
30895 embedded formulas are currently active. Also, note that these
30896 commands do not enable Embedded mode on the next or previous
30897 formula, they just move the cursor.
30900 @pindex calc-embedded-edit
30901 The @kbd{C-x * `} (@code{calc-embedded-edit}) command edits the
30902 embedded formula at the current point as if by @kbd{`} (@code{calc-edit}).
30903 Embedded mode does not have to be enabled for this to work. Press
30904 @kbd{C-c C-c} to finish the edit, or @kbd{C-x k} to cancel.
30906 @node Assignments in Embedded Mode, Mode Settings in Embedded Mode, More About Embedded Mode, Embedded Mode
30907 @section Assignments in Embedded Mode
30910 The @samp{:=} (assignment) and @samp{=>} (``evaluates-to'') operators
30911 are especially useful in Embedded mode. They allow you to make
30912 a definition in one formula, then refer to that definition in
30913 other formulas embedded in the same buffer.
30915 An embedded formula which is an assignment to a variable, as in
30922 records @expr{5} as the stored value of @code{foo} for the
30923 purposes of Embedded mode operations in the current buffer. It
30924 does @emph{not} actually store @expr{5} as the ``global'' value
30925 of @code{foo}, however. Regular Calc operations, and Embedded
30926 formulas in other buffers, will not see this assignment.
30928 One way to use this assigned value is simply to create an
30929 Embedded formula elsewhere that refers to @code{foo}, and to press
30930 @kbd{=} in that formula. However, this permanently replaces the
30931 @code{foo} in the formula with its current value. More interesting
30932 is to use @samp{=>} elsewhere:
30938 @xref{Evaluates-To Operator}, for a general discussion of @samp{=>}.
30940 If you move back and change the assignment to @code{foo}, any
30941 @samp{=>} formulas which refer to it are automatically updated.
30949 The obvious question then is, @emph{how} can one easily change the
30950 assignment to @code{foo}? If you simply select the formula in
30951 Embedded mode and type 17, the assignment itself will be replaced
30952 by the 17. The effect on the other formula will be that the
30953 variable @code{foo} becomes unassigned:
30961 The right thing to do is first to use a selection command (@kbd{j 2}
30962 will do the trick) to select the righthand side of the assignment.
30963 Then, @kbd{17 @key{TAB} @key{DEL}} will swap the 17 into place (@pxref{Selecting
30964 Subformulas}, to see how this works).
30967 @pindex calc-embedded-select
30968 The @kbd{C-x * j} (@code{calc-embedded-select}) command provides an
30969 easy way to operate on assignments. It is just like @kbd{C-x * e},
30970 except that if the enabled formula is an assignment, it uses
30971 @kbd{j 2} to select the righthand side. If the enabled formula
30972 is an evaluates-to, it uses @kbd{j 1} to select the lefthand side.
30973 A formula can also be a combination of both:
30976 bar := foo + 3 => 20
30980 in which case @kbd{C-x * j} will select the middle part (@samp{foo + 3}).
30982 The formula is automatically deselected when you leave Embedded
30986 @pindex calc-embedded-update-formula
30987 Another way to change the assignment to @code{foo} would simply be
30988 to edit the number using regular Emacs editing rather than Embedded
30989 mode. Then, we have to find a way to get Embedded mode to notice
30990 the change. The @kbd{C-x * u} (@code{calc-embedded-update-formula})
30991 command is a convenient way to do this.
30999 Pressing @kbd{C-x * u} is much like pressing @kbd{C-x * e = C-x * e}, that
31000 is, temporarily enabling Embedded mode for the formula under the
31001 cursor and then evaluating it with @kbd{=}. But @kbd{C-x * u} does
31002 not actually use @kbd{C-x * e}, and in fact another formula somewhere
31003 else can be enabled in Embedded mode while you use @kbd{C-x * u} and
31004 that formula will not be disturbed.
31006 With a numeric prefix argument, @kbd{C-x * u} updates all active
31007 @samp{=>} formulas in the buffer. Formulas which have not yet
31008 been activated in Embedded mode, and formulas which do not have
31009 @samp{=>} as their top-level operator, are not affected by this.
31010 (This is useful only if you have used @kbd{m C}; see below.)
31012 With a plain @kbd{C-u} prefix, @kbd{C-u C-x * u} updates only in the
31013 region between mark and point rather than in the whole buffer.
31015 @kbd{C-x * u} is also a handy way to activate a formula, such as an
31016 @samp{=>} formula that has freshly been typed in or loaded from a
31020 @pindex calc-embedded-activate
31021 The @kbd{C-x * a} (@code{calc-embedded-activate}) command scans
31022 through the current buffer and activates all embedded formulas
31023 that contain @samp{:=} or @samp{=>} symbols. This does not mean
31024 that Embedded mode is actually turned on, but only that the
31025 formulas' positions are registered with Embedded mode so that
31026 the @samp{=>} values can be properly updated as assignments are
31029 It is a good idea to type @kbd{C-x * a} right after loading a file
31030 that uses embedded @samp{=>} operators. Emacs includes a nifty
31031 ``buffer-local variables'' feature that you can use to do this
31032 automatically. The idea is to place near the end of your file
31033 a few lines that look like this:
31036 --- Local Variables: ---
31037 --- eval:(calc-embedded-activate) ---
31042 where the leading and trailing @samp{---} can be replaced by
31043 any suitable strings (which must be the same on all three lines)
31044 or omitted altogether; in a @TeX{} or @LaTeX{} file, @samp{%} would be a good
31045 leading string and no trailing string would be necessary. In a
31046 C program, @samp{/*} and @samp{*/} would be good leading and
31049 When Emacs loads a file into memory, it checks for a Local Variables
31050 section like this one at the end of the file. If it finds this
31051 section, it does the specified things (in this case, running
31052 @kbd{C-x * a} automatically) before editing of the file begins.
31053 The Local Variables section must be within 3000 characters of the
31054 end of the file for Emacs to find it, and it must be in the last
31055 page of the file if the file has any page separators.
31056 @xref{File Variables, , Local Variables in Files, emacs, the
31059 Note that @kbd{C-x * a} does not update the formulas it finds.
31060 To do this, type, say, @kbd{M-1 C-x * u} after @w{@kbd{C-x * a}}.
31061 Generally this should not be a problem, though, because the
31062 formulas will have been up-to-date already when the file was
31065 Normally, @kbd{C-x * a} activates all the formulas it finds, but
31066 any previous active formulas remain active as well. With a
31067 positive numeric prefix argument, @kbd{C-x * a} first deactivates
31068 all current active formulas, then actives the ones it finds in
31069 its scan of the buffer. With a negative prefix argument,
31070 @kbd{C-x * a} simply deactivates all formulas.
31072 Embedded mode has two symbols, @samp{Active} and @samp{~Active},
31073 which it puts next to the major mode name in a buffer's mode line.
31074 It puts @samp{Active} if it has reason to believe that all
31075 formulas in the buffer are active, because you have typed @kbd{C-x * a}
31076 and Calc has not since had to deactivate any formulas (which can
31077 happen if Calc goes to update an @samp{=>} formula somewhere because
31078 a variable changed, and finds that the formula is no longer there
31079 due to some kind of editing outside of Embedded mode). Calc puts
31080 @samp{~Active} in the mode line if some, but probably not all,
31081 formulas in the buffer are active. This happens if you activate
31082 a few formulas one at a time but never use @kbd{C-x * a}, or if you
31083 used @kbd{C-x * a} but then Calc had to deactivate a formula
31084 because it lost track of it. If neither of these symbols appears
31085 in the mode line, no embedded formulas are active in the buffer
31086 (e.g., before Embedded mode has been used, or after a @kbd{M-- C-x * a}).
31088 Embedded formulas can refer to assignments both before and after them
31089 in the buffer. If there are several assignments to a variable, the
31090 nearest preceding assignment is used if there is one, otherwise the
31091 following assignment is used.
31105 As well as simple variables, you can also assign to subscript
31106 expressions of the form @samp{@var{var}_@var{number}} (as in
31107 @code{x_0}), or @samp{@var{var}_@var{var}} (as in @code{x_max}).
31108 Assignments to other kinds of objects can be represented by Calc,
31109 but the automatic linkage between assignments and references works
31110 only for plain variables and these two kinds of subscript expressions.
31112 If there are no assignments to a given variable, the global
31113 stored value for the variable is used (@pxref{Storing Variables}),
31114 or, if no value is stored, the variable is left in symbolic form.
31115 Note that global stored values will be lost when the file is saved
31116 and loaded in a later Emacs session, unless you have used the
31117 @kbd{s p} (@code{calc-permanent-variable}) command to save them;
31118 @pxref{Operations on Variables}.
31120 The @kbd{m C} (@code{calc-auto-recompute}) command turns automatic
31121 recomputation of @samp{=>} forms on and off. If you turn automatic
31122 recomputation off, you will have to use @kbd{C-x * u} to update these
31123 formulas manually after an assignment has been changed. If you
31124 plan to change several assignments at once, it may be more efficient
31125 to type @kbd{m C}, change all the assignments, then use @kbd{M-1 C-x * u}
31126 to update the entire buffer afterwards. The @kbd{m C} command also
31127 controls @samp{=>} formulas on the stack; @pxref{Evaluates-To
31128 Operator}. When you turn automatic recomputation back on, the
31129 stack will be updated but the Embedded buffer will not; you must
31130 use @kbd{C-x * u} to update the buffer by hand.
31132 @node Mode Settings in Embedded Mode, Customizing Embedded Mode, Assignments in Embedded Mode, Embedded Mode
31133 @section Mode Settings in Embedded Mode
31136 @pindex calc-embedded-preserve-modes
31138 The mode settings can be changed while Calc is in embedded mode, but
31139 by default they will revert to their original values when embedded mode
31140 is ended. However, the modes saved when the mode-recording mode is
31141 @code{Save} (see below) and the modes in effect when the @kbd{m e}
31142 (@code{calc-embedded-preserve-modes}) command is given
31143 will be preserved when embedded mode is ended.
31145 Embedded mode has a rather complicated mechanism for handling mode
31146 settings in Embedded formulas. It is possible to put annotations
31147 in the file that specify mode settings either global to the entire
31148 file or local to a particular formula or formulas. In the latter
31149 case, different modes can be specified for use when a formula
31150 is the enabled Embedded mode formula.
31152 When you give any mode-setting command, like @kbd{m f} (for Fraction
31153 mode) or @kbd{d s} (for scientific notation), Embedded mode adds
31154 a line like the following one to the file just before the opening
31155 delimiter of the formula.
31158 % [calc-mode: fractions: t]
31159 % [calc-mode: float-format: (sci 0)]
31162 When Calc interprets an embedded formula, it scans the text before
31163 the formula for mode-setting annotations like these and sets the
31164 Calc buffer to match these modes. Modes not explicitly described
31165 in the file are not changed. Calc scans all the way to the top of
31166 the file, or up to a line of the form
31173 which you can insert at strategic places in the file if this backward
31174 scan is getting too slow, or just to provide a barrier between one
31175 ``zone'' of mode settings and another.
31177 If the file contains several annotations for the same mode, the
31178 closest one before the formula is used. Annotations after the
31179 formula are never used (except for global annotations, described
31182 The scan does not look for the leading @samp{% }, only for the
31183 square brackets and the text they enclose. In fact, the leading
31184 characters are different for different major modes. You can edit the
31185 mode annotations to a style that works better in context if you wish.
31186 @xref{Customizing Embedded Mode}, to see how to change the style
31187 that Calc uses when it generates the annotations. You can write
31188 mode annotations into the file yourself if you know the syntax;
31189 the easiest way to find the syntax for a given mode is to let
31190 Calc write the annotation for it once and see what it does.
31192 If you give a mode-changing command for a mode that already has
31193 a suitable annotation just above the current formula, Calc will
31194 modify that annotation rather than generating a new, conflicting
31197 Mode annotations have three parts, separated by colons. (Spaces
31198 after the colons are optional.) The first identifies the kind
31199 of mode setting, the second is a name for the mode itself, and
31200 the third is the value in the form of a Lisp symbol, number,
31201 or list. Annotations with unrecognizable text in the first or
31202 second parts are ignored. The third part is not checked to make
31203 sure the value is of a valid type or range; if you write an
31204 annotation by hand, be sure to give a proper value or results
31205 will be unpredictable. Mode-setting annotations are case-sensitive.
31207 While Embedded mode is enabled, the word @code{Local} appears in
31208 the mode line. This is to show that mode setting commands generate
31209 annotations that are ``local'' to the current formula or set of
31210 formulas. The @kbd{m R} (@code{calc-mode-record-mode}) command
31211 causes Calc to generate different kinds of annotations. Pressing
31212 @kbd{m R} repeatedly cycles through the possible modes.
31214 @code{LocEdit} and @code{LocPerm} modes generate annotations
31215 that look like this, respectively:
31218 % [calc-edit-mode: float-format: (sci 0)]
31219 % [calc-perm-mode: float-format: (sci 5)]
31222 The first kind of annotation will be used only while a formula
31223 is enabled in Embedded mode. The second kind will be used only
31224 when the formula is @emph{not} enabled. (Whether the formula
31225 is ``active'' or not, i.e., whether Calc has seen this formula
31226 yet, is not relevant here.)
31228 @code{Global} mode generates an annotation like this at the end
31232 % [calc-global-mode: fractions t]
31235 Global mode annotations affect all formulas throughout the file,
31236 and may appear anywhere in the file. This allows you to tuck your
31237 mode annotations somewhere out of the way, say, on a new page of
31238 the file, as long as those mode settings are suitable for all
31239 formulas in the file.
31241 Enabling a formula with @kbd{C-x * e} causes a fresh scan for local
31242 mode annotations; you will have to use this after adding annotations
31243 above a formula by hand to get the formula to notice them. Updating
31244 a formula with @kbd{C-x * u} will also re-scan the local modes, but
31245 global modes are only re-scanned by @kbd{C-x * a}.
31247 Another way that modes can get out of date is if you add a local
31248 mode annotation to a formula that has another formula after it.
31249 In this example, we have used the @kbd{d s} command while the
31250 first of the two embedded formulas is active. But the second
31251 formula has not changed its style to match, even though by the
31252 rules of reading annotations the @samp{(sci 0)} applies to it, too.
31255 % [calc-mode: float-format: (sci 0)]
31261 We would have to go down to the other formula and press @kbd{C-x * u}
31262 on it in order to get it to notice the new annotation.
31264 Two more mode-recording modes selectable by @kbd{m R} are available
31265 which are also available outside of Embedded mode.
31266 (@pxref{General Mode Commands}.) They are @code{Save}, in which mode
31267 settings are recorded permanently in your Calc init file (the file given
31268 by the variable @code{calc-settings-file}, typically @file{~/.emacs.d/calc.el})
31269 rather than by annotating the current document, and no-recording
31270 mode (where there is no symbol like @code{Save} or @code{Local} in
31271 the mode line), in which mode-changing commands do not leave any
31272 annotations at all.
31274 When Embedded mode is not enabled, mode-recording modes except
31275 for @code{Save} have no effect.
31277 @node Customizing Embedded Mode, , Mode Settings in Embedded Mode, Embedded Mode
31278 @section Customizing Embedded Mode
31281 You can modify Embedded mode's behavior by setting various Lisp
31282 variables described here. These variables are customizable
31283 (@pxref{Customizing Calc}), or you can use @kbd{M-x set-variable}
31284 or @kbd{M-x edit-options} to adjust a variable on the fly.
31285 (Another possibility would be to use a file-local variable annotation at
31286 the end of the file;
31287 @pxref{File Variables, , Local Variables in Files, emacs, the Emacs manual}.)
31288 Many of the variables given mentioned here can be set to depend on the
31289 major mode of the editing buffer (@pxref{Customizing Calc}).
31291 @vindex calc-embedded-open-formula
31292 The @code{calc-embedded-open-formula} variable holds a regular
31293 expression for the opening delimiter of a formula. @xref{Regexp Search,
31294 , Regular Expression Search, emacs, the Emacs manual}, to see
31295 how regular expressions work. Basically, a regular expression is a
31296 pattern that Calc can search for. A regular expression that considers
31297 blank lines, @samp{$}, and @samp{$$} to be opening delimiters is
31298 @code{"\\`\\|^\n\\|\\$\\$?"}. Just in case the meaning of this
31299 regular expression is not completely plain, let's go through it
31302 The surrounding @samp{" "} marks quote the text between them as a
31303 Lisp string. If you left them off, @code{set-variable} or
31304 @code{edit-options} would try to read the regular expression as a
31307 The most obvious property of this regular expression is that it
31308 contains indecently many backslashes. There are actually two levels
31309 of backslash usage going on here. First, when Lisp reads a quoted
31310 string, all pairs of characters beginning with a backslash are
31311 interpreted as special characters. Here, @code{\n} changes to a
31312 new-line character, and @code{\\} changes to a single backslash.
31313 So the actual regular expression seen by Calc is
31314 @samp{\`\|^ @r{(newline)} \|\$\$?}.
31316 Regular expressions also consider pairs beginning with backslash
31317 to have special meanings. Sometimes the backslash is used to quote
31318 a character that otherwise would have a special meaning in a regular
31319 expression, like @samp{$}, which normally means ``end-of-line,''
31320 or @samp{?}, which means that the preceding item is optional. So
31321 @samp{\$\$?} matches either one or two dollar signs.
31323 The other codes in this regular expression are @samp{^}, which matches
31324 ``beginning-of-line,'' @samp{\|}, which means ``or,'' and @samp{\`},
31325 which matches ``beginning-of-buffer.'' So the whole pattern means
31326 that a formula begins at the beginning of the buffer, or on a newline
31327 that occurs at the beginning of a line (i.e., a blank line), or at
31328 one or two dollar signs.
31330 The default value of @code{calc-embedded-open-formula} looks just
31331 like this example, with several more alternatives added on to
31332 recognize various other common kinds of delimiters.
31334 By the way, the reason to use @samp{^\n} rather than @samp{^$}
31335 or @samp{\n\n}, which also would appear to match blank lines,
31336 is that the former expression actually ``consumes'' only one
31337 newline character as @emph{part of} the delimiter, whereas the
31338 latter expressions consume zero or two newlines, respectively.
31339 The former choice gives the most natural behavior when Calc
31340 must operate on a whole formula including its delimiters.
31342 See the Emacs manual for complete details on regular expressions.
31343 But just for your convenience, here is a list of all characters
31344 which must be quoted with backslash (like @samp{\$}) to avoid
31345 some special interpretation: @samp{. * + ? [ ] ^ $ \}. (Note
31346 the backslash in this list; for example, to match @samp{\[} you
31347 must use @code{"\\\\\\["}. An exercise for the reader is to
31348 account for each of these six backslashes!)
31350 @vindex calc-embedded-close-formula
31351 The @code{calc-embedded-close-formula} variable holds a regular
31352 expression for the closing delimiter of a formula. A closing
31353 regular expression to match the above example would be
31354 @code{"\\'\\|\n$\\|\\$\\$?"}. This is almost the same as the
31355 other one, except it now uses @samp{\'} (``end-of-buffer'') and
31356 @samp{\n$} (newline occurring at end of line, yet another way
31357 of describing a blank line that is more appropriate for this
31360 @vindex calc-embedded-word-regexp
31361 The @code{calc-embedded-word-regexp} variable holds a regular expression
31362 used to define an expression to look for (a ``word'') when you type
31363 @kbd{C-x * w} to enable Embedded mode.
31365 @vindex calc-embedded-open-plain
31366 The @code{calc-embedded-open-plain} variable is a string which
31367 begins a ``plain'' formula written in front of the formatted
31368 formula when @kbd{d p} mode is turned on. Note that this is an
31369 actual string, not a regular expression, because Calc must be able
31370 to write this string into a buffer as well as to recognize it.
31371 The default string is @code{"%%% "} (note the trailing space), but may
31372 be different for certain major modes.
31374 @vindex calc-embedded-close-plain
31375 The @code{calc-embedded-close-plain} variable is a string which
31376 ends a ``plain'' formula. The default is @code{" %%%\n"}, but may be
31377 different for different major modes. Without
31378 the trailing newline here, the first line of a Big mode formula
31379 that followed might be shifted over with respect to the other lines.
31381 @vindex calc-embedded-open-new-formula
31382 The @code{calc-embedded-open-new-formula} variable is a string
31383 which is inserted at the front of a new formula when you type
31384 @kbd{C-x * f}. Its default value is @code{"\n\n"}. If this
31385 string begins with a newline character and the @kbd{C-x * f} is
31386 typed at the beginning of a line, @kbd{C-x * f} will skip this
31387 first newline to avoid introducing unnecessary blank lines in
31390 @vindex calc-embedded-close-new-formula
31391 The @code{calc-embedded-close-new-formula} variable is the corresponding
31392 string which is inserted at the end of a new formula. Its default
31393 value is also @code{"\n\n"}. The final newline is omitted by
31394 @w{@kbd{C-x * f}} if typed at the end of a line. (It follows that if
31395 @kbd{C-x * f} is typed on a blank line, both a leading opening
31396 newline and a trailing closing newline are omitted.)
31398 @vindex calc-embedded-announce-formula
31399 The @code{calc-embedded-announce-formula} variable is a regular
31400 expression which is sure to be followed by an embedded formula.
31401 The @kbd{C-x * a} command searches for this pattern as well as for
31402 @samp{=>} and @samp{:=} operators. Note that @kbd{C-x * a} will
31403 not activate just anything surrounded by formula delimiters; after
31404 all, blank lines are considered formula delimiters by default!
31405 But if your language includes a delimiter which can only occur
31406 actually in front of a formula, you can take advantage of it here.
31407 The default pattern is @code{"%Embed\n\\(% .*\n\\)*"}, but may be
31408 different for different major modes.
31409 This pattern will check for @samp{%Embed} followed by any number of
31410 lines beginning with @samp{%} and a space. This last is important to
31411 make Calc consider mode annotations part of the pattern, so that the
31412 formula's opening delimiter really is sure to follow the pattern.
31414 @vindex calc-embedded-open-mode
31415 The @code{calc-embedded-open-mode} variable is a string (not a
31416 regular expression) which should precede a mode annotation.
31417 Calc never scans for this string; Calc always looks for the
31418 annotation itself. But this is the string that is inserted before
31419 the opening bracket when Calc adds an annotation on its own.
31420 The default is @code{"% "}, but may be different for different major
31423 @vindex calc-embedded-close-mode
31424 The @code{calc-embedded-close-mode} variable is a string which
31425 follows a mode annotation written by Calc. Its default value
31426 is simply a newline, @code{"\n"}, but may be different for different
31427 major modes. If you change this, it is a good idea still to end with a
31428 newline so that mode annotations will appear on lines by themselves.
31430 @node Programming, Copying, Embedded Mode, Top
31431 @chapter Programming
31434 There are several ways to ``program'' the Emacs Calculator, depending
31435 on the nature of the problem you need to solve.
31439 @dfn{Keyboard macros} allow you to record a sequence of keystrokes
31440 and play them back at a later time. This is just the standard Emacs
31441 keyboard macro mechanism, dressed up with a few more features such
31442 as loops and conditionals.
31445 @dfn{Algebraic definitions} allow you to use any formula to define a
31446 new function. This function can then be used in algebraic formulas or
31447 as an interactive command.
31450 @dfn{Rewrite rules} are discussed in the section on algebra commands.
31451 @xref{Rewrite Rules}. If you put your rewrite rules in the variable
31452 @code{EvalRules}, they will be applied automatically to all Calc
31453 results in just the same way as an internal ``rule'' is applied to
31454 evaluate @samp{sqrt(9)} to 3 and so on. @xref{Automatic Rewrites}.
31457 @dfn{Lisp} is the programming language that Calc (and most of Emacs)
31458 is written in. If the above techniques aren't powerful enough, you
31459 can write Lisp functions to do anything that built-in Calc commands
31460 can do. Lisp code is also somewhat faster than keyboard macros or
31465 Programming features are available through the @kbd{z} and @kbd{Z}
31466 prefix keys. New commands that you define are two-key sequences
31467 beginning with @kbd{z}. Commands for managing these definitions
31468 use the shift-@kbd{Z} prefix. (The @kbd{Z T} (@code{calc-timing})
31469 command is described elsewhere; @pxref{Troubleshooting Commands}.
31470 The @kbd{Z C} (@code{calc-user-define-composition}) command is also
31471 described elsewhere; @pxref{User-Defined Compositions}.)
31474 * Creating User Keys::
31475 * Keyboard Macros::
31476 * Invocation Macros::
31477 * Algebraic Definitions::
31478 * Lisp Definitions::
31481 @node Creating User Keys, Keyboard Macros, Programming, Programming
31482 @section Creating User Keys
31486 @pindex calc-user-define
31487 Any Calculator command may be bound to a key using the @kbd{Z D}
31488 (@code{calc-user-define}) command. Actually, it is bound to a two-key
31489 sequence beginning with the lower-case @kbd{z} prefix.
31491 The @kbd{Z D} command first prompts for the key to define. For example,
31492 press @kbd{Z D a} to define the new key sequence @kbd{z a}. You are then
31493 prompted for the name of the Calculator command that this key should
31494 run. For example, the @code{calc-sincos} command is not normally
31495 available on a key. Typing @kbd{Z D s sincos @key{RET}} programs the
31496 @kbd{z s} key sequence to run @code{calc-sincos}. This definition will remain
31497 in effect for the rest of this Emacs session, or until you redefine
31498 @kbd{z s} to be something else.
31500 You can actually bind any Emacs command to a @kbd{z} key sequence by
31501 backspacing over the @samp{calc-} when you are prompted for the command name.
31503 As with any other prefix key, you can type @kbd{z ?} to see a list of
31504 all the two-key sequences you have defined that start with @kbd{z}.
31505 Initially, no @kbd{z} sequences (except @kbd{z ?} itself) are defined.
31507 User keys are typically letters, but may in fact be any key.
31508 (@key{META}-keys are not permitted, nor are a terminal's special
31509 function keys which generate multi-character sequences when pressed.)
31510 You can define different commands on the shifted and unshifted versions
31511 of a letter if you wish.
31514 @pindex calc-user-undefine
31515 The @kbd{Z U} (@code{calc-user-undefine}) command unbinds a user key.
31516 For example, the key sequence @kbd{Z U s} will undefine the @code{sincos}
31517 key we defined above.
31520 @pindex calc-user-define-permanent
31521 @cindex Storing user definitions
31522 @cindex Permanent user definitions
31523 @cindex Calc init file, user-defined commands
31524 The @kbd{Z P} (@code{calc-user-define-permanent}) command makes a key
31525 binding permanent so that it will remain in effect even in future Emacs
31526 sessions. (It does this by adding a suitable bit of Lisp code into
31527 your Calc init file; that is, the file given by the variable
31528 @code{calc-settings-file}, typically @file{~/.emacs.d/calc.el}.) For example,
31529 @kbd{Z P s} would register our @code{sincos} command permanently. If
31530 you later wish to unregister this command you must edit your Calc init
31531 file by hand. (@xref{General Mode Commands}, for a way to tell Calc to
31532 use a different file for the Calc init file.)
31534 The @kbd{Z P} command also saves the user definition, if any, for the
31535 command bound to the key. After @kbd{Z F} and @kbd{Z C}, a given user
31536 key could invoke a command, which in turn calls an algebraic function,
31537 which might have one or more special display formats. A single @kbd{Z P}
31538 command will save all of these definitions.
31539 To save an algebraic function, type @kbd{'} (the apostrophe)
31540 when prompted for a key, and type the function name. To save a command
31541 without its key binding, type @kbd{M-x} and enter a function name. (The
31542 @samp{calc-} prefix will automatically be inserted for you.)
31543 (If the command you give implies a function, the function will be saved,
31544 and if the function has any display formats, those will be saved, but
31545 not the other way around: Saving a function will not save any commands
31546 or key bindings associated with the function.)
31549 @pindex calc-user-define-edit
31550 @cindex Editing user definitions
31551 The @kbd{Z E} (@code{calc-user-define-edit}) command edits the definition
31552 of a user key. This works for keys that have been defined by either
31553 keyboard macros or formulas; further details are contained in the relevant
31554 following sections.
31556 @node Keyboard Macros, Invocation Macros, Creating User Keys, Programming
31557 @section Programming with Keyboard Macros
31561 @cindex Programming with keyboard macros
31562 @cindex Keyboard macros
31563 The easiest way to ``program'' the Emacs Calculator is to use standard
31564 keyboard macros. Press @w{@kbd{C-x (}} to begin recording a macro. From
31565 this point on, keystrokes you type will be saved away as well as
31566 performing their usual functions. Press @kbd{C-x )} to end recording.
31567 Press shift-@kbd{X} (or the standard Emacs key sequence @kbd{C-x e}) to
31568 execute your keyboard macro by replaying the recorded keystrokes.
31569 @xref{Keyboard Macros, , , emacs, the Emacs Manual}, for further
31572 When you use @kbd{X} to invoke a keyboard macro, the entire macro is
31573 treated as a single command by the undo and trail features. The stack
31574 display buffer is not updated during macro execution, but is instead
31575 fixed up once the macro completes. Thus, commands defined with keyboard
31576 macros are convenient and efficient. The @kbd{C-x e} command, on the
31577 other hand, invokes the keyboard macro with no special treatment: Each
31578 command in the macro will record its own undo information and trail entry,
31579 and update the stack buffer accordingly. If your macro uses features
31580 outside of Calc's control to operate on the contents of the Calc stack
31581 buffer, or if it includes Undo, Redo, or last-arguments commands, you
31582 must use @kbd{C-x e} to make sure the buffer and undo list are up-to-date
31583 at all times. You could also consider using @kbd{K} (@code{calc-keep-args})
31584 instead of @kbd{M-@key{RET}} (@code{calc-last-args}).
31586 Calc extends the standard Emacs keyboard macros in several ways.
31587 Keyboard macros can be used to create user-defined commands. Keyboard
31588 macros can include conditional and iteration structures, somewhat
31589 analogous to those provided by a traditional programmable calculator.
31592 * Naming Keyboard Macros::
31593 * Conditionals in Macros::
31594 * Loops in Macros::
31595 * Local Values in Macros::
31596 * Queries in Macros::
31599 @node Naming Keyboard Macros, Conditionals in Macros, Keyboard Macros, Keyboard Macros
31600 @subsection Naming Keyboard Macros
31604 @pindex calc-user-define-kbd-macro
31605 Once you have defined a keyboard macro, you can bind it to a @kbd{z}
31606 key sequence with the @kbd{Z K} (@code{calc-user-define-kbd-macro}) command.
31607 This command prompts first for a key, then for a command name. For
31608 example, if you type @kbd{C-x ( n @key{TAB} n @key{TAB} C-x )} you will
31609 define a keyboard macro which negates the top two numbers on the stack
31610 (@key{TAB} swaps the top two stack elements). Now you can type
31611 @kbd{Z K n @key{RET}} to define this keyboard macro onto the @kbd{z n} key
31612 sequence. The default command name (if you answer the second prompt with
31613 just the @key{RET} key as in this example) will be something like
31614 @samp{calc-User-n}. The keyboard macro will now be available as both
31615 @kbd{z n} and @kbd{M-x calc-User-n}. You can backspace and enter a more
31616 descriptive command name if you wish.
31618 Macros defined by @kbd{Z K} act like single commands; they are executed
31619 in the same way as by the @kbd{X} key. If you wish to define the macro
31620 as a standard no-frills Emacs macro (to be executed as if by @kbd{C-x e}),
31621 give a negative prefix argument to @kbd{Z K}.
31623 Once you have bound your keyboard macro to a key, you can use
31624 @kbd{Z P} to register it permanently with Emacs. @xref{Creating User Keys}.
31626 @cindex Keyboard macros, editing
31627 The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has
31628 been defined by a keyboard macro tries to use the @code{edmacro} package
31629 edit the macro. Type @kbd{C-c C-c} to finish editing and update
31630 the definition stored on the key, or, to cancel the edit, kill the
31631 buffer with @kbd{C-x k}.
31632 The special characters @code{RET}, @code{LFD}, @code{TAB}, @code{SPC},
31633 @code{DEL}, and @code{NUL} must be entered as these three character
31634 sequences, written in all uppercase, as must the prefixes @code{C-} and
31635 @code{M-}. Spaces and line breaks are ignored. Other characters are
31636 copied verbatim into the keyboard macro. Basically, the notation is the
31637 same as is used in all of this manual's examples, except that the manual
31638 takes some liberties with spaces: When we say @kbd{' [1 2 3] @key{RET}},
31639 we take it for granted that it is clear we really mean
31640 @kbd{' [1 @key{SPC} 2 @key{SPC} 3] @key{RET}}.
31643 @pindex read-kbd-macro
31644 The @kbd{C-x * m} (@code{read-kbd-macro}) command reads an Emacs ``region''
31645 of spelled-out keystrokes and defines it as the current keyboard macro.
31646 It is a convenient way to define a keyboard macro that has been stored
31647 in a file, or to define a macro without executing it at the same time.
31649 @node Conditionals in Macros, Loops in Macros, Naming Keyboard Macros, Keyboard Macros
31650 @subsection Conditionals in Keyboard Macros
31655 @pindex calc-kbd-if
31656 @pindex calc-kbd-else
31657 @pindex calc-kbd-else-if
31658 @pindex calc-kbd-end-if
31659 @cindex Conditional structures
31660 The @kbd{Z [} (@code{calc-kbd-if}) and @kbd{Z ]} (@code{calc-kbd-end-if})
31661 commands allow you to put simple tests in a keyboard macro. When Calc
31662 sees the @kbd{Z [}, it pops an object from the stack and, if the object is
31663 a non-zero value, continues executing keystrokes. But if the object is
31664 zero, or if it is not provably nonzero, Calc skips ahead to the matching
31665 @kbd{Z ]} keystroke. @xref{Logical Operations}, for a set of commands for
31666 performing tests which conveniently produce 1 for true and 0 for false.
31668 For example, @kbd{@key{RET} 0 a < Z [ n Z ]} implements an absolute-value
31669 function in the form of a keyboard macro. This macro duplicates the
31670 number on the top of the stack, pushes zero and compares using @kbd{a <}
31671 (@code{calc-less-than}), then, if the number was less than zero,
31672 executes @kbd{n} (@code{calc-change-sign}). Otherwise, the change-sign
31673 command is skipped.
31675 To program this macro, type @kbd{C-x (}, type the above sequence of
31676 keystrokes, then type @kbd{C-x )}. Note that the keystrokes will be
31677 executed while you are making the definition as well as when you later
31678 re-execute the macro by typing @kbd{X}. Thus you should make sure a
31679 suitable number is on the stack before defining the macro so that you
31680 don't get a stack-underflow error during the definition process.
31682 Conditionals can be nested arbitrarily. However, there should be exactly
31683 one @kbd{Z ]} for each @kbd{Z [} in a keyboard macro.
31686 The @kbd{Z :} (@code{calc-kbd-else}) command allows you to choose between
31687 two keystroke sequences. The general format is @kbd{@var{cond} Z [
31688 @var{then-part} Z : @var{else-part} Z ]}. If @var{cond} is true
31689 (i.e., if the top of stack contains a non-zero number after @var{cond}
31690 has been executed), the @var{then-part} will be executed and the
31691 @var{else-part} will be skipped. Otherwise, the @var{then-part} will
31692 be skipped and the @var{else-part} will be executed.
31695 The @kbd{Z |} (@code{calc-kbd-else-if}) command allows you to choose
31696 between any number of alternatives. For example,
31697 @kbd{@var{cond1} Z [ @var{part1} Z : @var{cond2} Z | @var{part2} Z :
31698 @var{part3} Z ]} will execute @var{part1} if @var{cond1} is true,
31699 otherwise it will execute @var{part2} if @var{cond2} is true, otherwise
31700 it will execute @var{part3}.
31702 More precisely, @kbd{Z [} pops a number and conditionally skips to the
31703 next matching @kbd{Z :} or @kbd{Z ]} key. @w{@kbd{Z ]}} has no effect when
31704 actually executed. @kbd{Z :} skips to the next matching @kbd{Z ]}.
31705 @kbd{Z |} pops a number and conditionally skips to the next matching
31706 @kbd{Z :} or @kbd{Z ]}; thus, @kbd{Z [} and @kbd{Z |} are functionally
31707 equivalent except that @kbd{Z [} participates in nesting but @kbd{Z |}
31710 Calc's conditional and looping constructs work by scanning the
31711 keyboard macro for occurrences of character sequences like @samp{Z:}
31712 and @samp{Z]}. One side-effect of this is that if you use these
31713 constructs you must be careful that these character pairs do not
31714 occur by accident in other parts of the macros. Since Calc rarely
31715 uses shift-@kbd{Z} for any purpose except as a prefix character, this
31716 is not likely to be a problem. Another side-effect is that it will
31717 not work to define your own custom key bindings for these commands.
31718 Only the standard shift-@kbd{Z} bindings will work correctly.
31721 If Calc gets stuck while skipping characters during the definition of a
31722 macro, type @kbd{Z C-g} to cancel the definition. (Typing plain @kbd{C-g}
31723 actually adds a @kbd{C-g} keystroke to the macro.)
31725 @node Loops in Macros, Local Values in Macros, Conditionals in Macros, Keyboard Macros
31726 @subsection Loops in Keyboard Macros
31731 @pindex calc-kbd-repeat
31732 @pindex calc-kbd-end-repeat
31733 @cindex Looping structures
31734 @cindex Iterative structures
31735 The @kbd{Z <} (@code{calc-kbd-repeat}) and @kbd{Z >}
31736 (@code{calc-kbd-end-repeat}) commands pop a number from the stack,
31737 which must be an integer, then repeat the keystrokes between the brackets
31738 the specified number of times. If the integer is zero or negative, the
31739 body is skipped altogether. For example, @kbd{1 @key{TAB} Z < 2 * Z >}
31740 computes two to a nonnegative integer power. First, we push 1 on the
31741 stack and then swap the integer argument back to the top. The @kbd{Z <}
31742 pops that argument leaving the 1 back on top of the stack. Then, we
31743 repeat a multiply-by-two step however many times.
31745 Once again, the keyboard macro is executed as it is being entered.
31746 In this case it is especially important to set up reasonable initial
31747 conditions before making the definition: Suppose the integer 1000 just
31748 happened to be sitting on the stack before we typed the above definition!
31749 Another approach is to enter a harmless dummy definition for the macro,
31750 then go back and edit in the real one with a @kbd{Z E} command. Yet
31751 another approach is to type the macro as written-out keystroke names
31752 in a buffer, then use @kbd{C-x * m} (@code{read-kbd-macro}) to read the
31757 The @kbd{Z /} (@code{calc-kbd-break}) command allows you to break out
31758 of a keyboard macro loop prematurely. It pops an object from the stack;
31759 if that object is true (a non-zero number), control jumps out of the
31760 innermost enclosing @kbd{Z <} @dots{} @kbd{Z >} loop and continues
31761 after the @kbd{Z >}. If the object is false, the @kbd{Z /} has no
31762 effect. Thus @kbd{@var{cond} Z /} is similar to @samp{if (@var{cond}) break;}
31767 @pindex calc-kbd-for
31768 @pindex calc-kbd-end-for
31769 The @kbd{Z (} (@code{calc-kbd-for}) and @kbd{Z )} (@code{calc-kbd-end-for})
31770 commands are similar to @kbd{Z <} and @kbd{Z >}, except that they make the
31771 value of the counter available inside the loop. The general layout is
31772 @kbd{@var{init} @var{final} Z ( @var{body} @var{step} Z )}. The @kbd{Z (}
31773 command pops initial and final values from the stack. It then creates
31774 a temporary internal counter and initializes it with the value @var{init}.
31775 The @kbd{Z (} command then repeatedly pushes the counter value onto the
31776 stack and executes @var{body} and @var{step}, adding @var{step} to the
31777 counter each time until the loop finishes.
31779 @cindex Summations (by keyboard macros)
31780 By default, the loop finishes when the counter becomes greater than (or
31781 less than) @var{final}, assuming @var{initial} is less than (greater
31782 than) @var{final}. If @var{initial} is equal to @var{final}, the body
31783 executes exactly once. The body of the loop always executes at least
31784 once. For example, @kbd{0 1 10 Z ( 2 ^ + 1 Z )} computes the sum of the
31785 squares of the integers from 1 to 10, in steps of 1.
31787 If you give a numeric prefix argument of 1 to @kbd{Z (}, the loop is
31788 forced to use upward-counting conventions. In this case, if @var{initial}
31789 is greater than @var{final} the body will not be executed at all.
31790 Note that @var{step} may still be negative in this loop; the prefix
31791 argument merely constrains the loop-finished test. Likewise, a prefix
31792 argument of @mathit{-1} forces downward-counting conventions.
31796 @pindex calc-kbd-loop
31797 @pindex calc-kbd-end-loop
31798 The @kbd{Z @{} (@code{calc-kbd-loop}) and @kbd{Z @}}
31799 (@code{calc-kbd-end-loop}) commands are similar to @kbd{Z <} and
31800 @kbd{Z >}, except that they do not pop a count from the stack---they
31801 effectively create an infinite loop. Every @kbd{Z @{} @dots{} @kbd{Z @}}
31802 loop ought to include at least one @kbd{Z /} to make sure the loop
31803 doesn't run forever. (If any error message occurs which causes Emacs
31804 to beep, the keyboard macro will also be halted; this is a standard
31805 feature of Emacs. You can also generally press @kbd{C-g} to halt a
31806 running keyboard macro, although not all versions of Unix support
31809 The conditional and looping constructs are not actually tied to
31810 keyboard macros, but they are most often used in that context.
31811 For example, the keystrokes @kbd{10 Z < 23 @key{RET} Z >} push
31812 ten copies of 23 onto the stack. This can be typed ``live'' just
31813 as easily as in a macro definition.
31815 @xref{Conditionals in Macros}, for some additional notes about
31816 conditional and looping commands.
31818 @node Local Values in Macros, Queries in Macros, Loops in Macros, Keyboard Macros
31819 @subsection Local Values in Macros
31822 @cindex Local variables
31823 @cindex Restoring saved modes
31824 Keyboard macros sometimes want to operate under known conditions
31825 without affecting surrounding conditions. For example, a keyboard
31826 macro may wish to turn on Fraction mode, or set a particular
31827 precision, independent of the user's normal setting for those
31832 @pindex calc-kbd-push
31833 @pindex calc-kbd-pop
31834 Macros also sometimes need to use local variables. Assignments to
31835 local variables inside the macro should not affect any variables
31836 outside the macro. The @kbd{Z `} (@code{calc-kbd-push}) and @kbd{Z '}
31837 (@code{calc-kbd-pop}) commands give you both of these capabilities.
31839 When you type @kbd{Z `} (with a backquote or accent grave character),
31840 the values of various mode settings are saved away. The ten ``quick''
31841 variables @code{q0} through @code{q9} are also saved. When
31842 you type @w{@kbd{Z '}} (with an apostrophe), these values are restored.
31843 Pairs of @kbd{Z `} and @kbd{Z '} commands may be nested.
31845 If a keyboard macro halts due to an error in between a @kbd{Z `} and
31846 a @kbd{Z '}, the saved values will be restored correctly even though
31847 the macro never reaches the @kbd{Z '} command. Thus you can use
31848 @kbd{Z `} and @kbd{Z '} without having to worry about what happens
31849 in exceptional conditions.
31851 If you type @kbd{Z `} ``live'' (not in a keyboard macro), Calc puts
31852 you into a ``recursive edit.'' You can tell you are in a recursive
31853 edit because there will be extra square brackets in the mode line,
31854 as in @samp{[(Calculator)]}. These brackets will go away when you
31855 type the matching @kbd{Z '} command. The modes and quick variables
31856 will be saved and restored in just the same way as if actual keyboard
31857 macros were involved.
31859 The modes saved by @kbd{Z `} and @kbd{Z '} are the current precision
31860 and binary word size, the angular mode (Deg, Rad, or HMS), the
31861 simplification mode, Algebraic mode, Symbolic mode, Infinite mode,
31862 Matrix or Scalar mode, Fraction mode, and the current complex mode
31863 (Polar or Rectangular). The ten ``quick'' variables' values (or lack
31864 thereof) are also saved.
31866 Most mode-setting commands act as toggles, but with a numeric prefix
31867 they force the mode either on (positive prefix) or off (negative
31868 or zero prefix). Since you don't know what the environment might
31869 be when you invoke your macro, it's best to use prefix arguments
31870 for all mode-setting commands inside the macro.
31872 In fact, @kbd{C-u Z `} is like @kbd{Z `} except that it sets the modes
31873 listed above to their default values. As usual, the matching @kbd{Z '}
31874 will restore the modes to their settings from before the @kbd{C-u Z `}.
31875 Also, @w{@kbd{Z `}} with a negative prefix argument resets the algebraic mode
31876 to its default (off) but leaves the other modes the same as they were
31877 outside the construct.
31879 The contents of the stack and trail, values of non-quick variables, and
31880 other settings such as the language mode and the various display modes,
31881 are @emph{not} affected by @kbd{Z `} and @kbd{Z '}.
31883 @node Queries in Macros, , Local Values in Macros, Keyboard Macros
31884 @subsection Queries in Keyboard Macros
31888 @c @pindex calc-kbd-report
31889 @c The @kbd{Z =} (@code{calc-kbd-report}) command displays an informative
31890 @c message including the value on the top of the stack. You are prompted
31891 @c to enter a string. That string, along with the top-of-stack value,
31892 @c is displayed unless @kbd{m w} (@code{calc-working}) has been used
31893 @c to turn such messages off.
31897 @pindex calc-kbd-query
31898 The @kbd{Z #} (@code{calc-kbd-query}) command prompts for an algebraic
31899 entry which takes its input from the keyboard, even during macro
31900 execution. All the normal conventions of algebraic input, including the
31901 use of @kbd{$} characters, are supported. The prompt message itself is
31902 taken from the top of the stack, and so must be entered (as a string)
31903 before the @kbd{Z #} command. (Recall, as a string it can be entered by
31904 pressing the @kbd{"} key and will appear as a vector when it is put on
31905 the stack. The prompt message is only put on the stack to provide a
31906 prompt for the @kbd{Z #} command; it will not play any role in any
31907 subsequent calculations.) This command allows your keyboard macros to
31908 accept numbers or formulas as interactive input.
31911 @kbd{2 @key{RET} "Power: " @key{RET} Z # 3 @key{RET} ^} will prompt for
31912 input with ``Power: '' in the minibuffer, then return 2 to the provided
31913 power. (The response to the prompt that's given, 3 in this example,
31914 will not be part of the macro.)
31916 @xref{Keyboard Macro Query, , , emacs, the Emacs Manual}, for a description of
31917 @kbd{C-x q} (@code{kbd-macro-query}), the standard Emacs way to accept
31918 keyboard input during a keyboard macro. In particular, you can use
31919 @kbd{C-x q} to enter a recursive edit, which allows the user to perform
31920 any Calculator operations interactively before pressing @kbd{C-M-c} to
31921 return control to the keyboard macro.
31923 @node Invocation Macros, Algebraic Definitions, Keyboard Macros, Programming
31924 @section Invocation Macros
31928 @pindex calc-user-invocation
31929 @pindex calc-user-define-invocation
31930 Calc provides one special keyboard macro, called up by @kbd{C-x * z}
31931 (@code{calc-user-invocation}), that is intended to allow you to define
31932 your own special way of starting Calc. To define this ``invocation
31933 macro,'' create the macro in the usual way with @kbd{C-x (} and
31934 @kbd{C-x )}, then type @kbd{Z I} (@code{calc-user-define-invocation}).
31935 There is only one invocation macro, so you don't need to type any
31936 additional letters after @kbd{Z I}. From now on, you can type
31937 @kbd{C-x * z} at any time to execute your invocation macro.
31939 For example, suppose you find yourself often grabbing rectangles of
31940 numbers into Calc and multiplying their columns. You can do this
31941 by typing @kbd{C-x * r} to grab, and @kbd{V R : *} to multiply columns.
31942 To make this into an invocation macro, just type @kbd{C-x ( C-x * r
31943 V R : * C-x )}, then @kbd{Z I}. Then, to multiply a rectangle of data,
31944 just mark the data in its buffer in the usual way and type @kbd{C-x * z}.
31946 Invocation macros are treated like regular Emacs keyboard macros;
31947 all the special features described above for @kbd{Z K}-style macros
31948 do not apply. @kbd{C-x * z} is just like @kbd{C-x e}, except that it
31949 uses the macro that was last stored by @kbd{Z I}. (In fact, the
31950 macro does not even have to have anything to do with Calc!)
31952 The @kbd{m m} command saves the last invocation macro defined by
31953 @kbd{Z I} along with all the other Calc mode settings.
31954 @xref{General Mode Commands}.
31956 @node Algebraic Definitions, Lisp Definitions, Invocation Macros, Programming
31957 @section Programming with Formulas
31961 @pindex calc-user-define-formula
31962 @cindex Programming with algebraic formulas
31963 Another way to create a new Calculator command uses algebraic formulas.
31964 The @kbd{Z F} (@code{calc-user-define-formula}) command stores the
31965 formula at the top of the stack as the definition for a key. This
31966 command prompts for five things: The key, the command name, the function
31967 name, the argument list, and the behavior of the command when given
31968 non-numeric arguments.
31970 For example, suppose we type @kbd{' a+2b @key{RET}} to push the formula
31971 @samp{a + 2*b} onto the stack. We now type @kbd{Z F m} to define this
31972 formula on the @kbd{z m} key sequence. The next prompt is for a command
31973 name, beginning with @samp{calc-}, which should be the long (@kbd{M-x}) form
31974 for the new command. If you simply press @key{RET}, a default name like
31975 @code{calc-User-m} will be constructed. In our example, suppose we enter
31976 @kbd{spam @key{RET}} to define the new command as @code{calc-spam}.
31978 If you want to give the formula a long-style name only, you can press
31979 @key{SPC} or @key{RET} when asked which single key to use. For example
31980 @kbd{Z F @key{RET} spam @key{RET}} defines the new command as
31981 @kbd{M-x calc-spam}, with no keyboard equivalent.
31983 The third prompt is for an algebraic function name. The default is to
31984 use the same name as the command name but without the @samp{calc-}
31985 prefix. (If this is of the form @samp{User-m}, the hyphen is removed so
31986 it won't be taken for a minus sign in algebraic formulas.)
31987 This is the name you will use if you want to enter your
31988 new function in an algebraic formula. Suppose we enter @kbd{yow @key{RET}}.
31989 Then the new function can be invoked by pushing two numbers on the
31990 stack and typing @kbd{z m} or @kbd{x spam}, or by entering the algebraic
31991 formula @samp{yow(x,y)}.
31993 The fourth prompt is for the function's argument list. This is used to
31994 associate values on the stack with the variables that appear in the formula.
31995 The default is a list of all variables which appear in the formula, sorted
31996 into alphabetical order. In our case, the default would be @samp{(a b)}.
31997 This means that, when the user types @kbd{z m}, the Calculator will remove
31998 two numbers from the stack, substitute these numbers for @samp{a} and
31999 @samp{b} (respectively) in the formula, then simplify the formula and
32000 push the result on the stack. In other words, @kbd{10 @key{RET} 100 z m}
32001 would replace the 10 and 100 on the stack with the number 210, which is
32002 @expr{a + 2 b} with @expr{a=10} and @expr{b=100}. Likewise, the formula
32003 @samp{yow(10, 100)} will be evaluated by substituting @expr{a=10} and
32004 @expr{b=100} in the definition.
32006 You can rearrange the order of the names before pressing @key{RET} to
32007 control which stack positions go to which variables in the formula. If
32008 you remove a variable from the argument list, that variable will be left
32009 in symbolic form by the command. Thus using an argument list of @samp{(b)}
32010 for our function would cause @kbd{10 z m} to replace the 10 on the stack
32011 with the formula @samp{a + 20}. If we had used an argument list of
32012 @samp{(b a)}, the result with inputs 10 and 100 would have been 120.
32014 You can also put a nameless function on the stack instead of just a
32015 formula, as in @samp{<a, b : a + 2 b>}. @xref{Specifying Operators}.
32016 In this example, the command will be defined by the formula @samp{a + 2 b}
32017 using the argument list @samp{(a b)}.
32019 The final prompt is a y-or-n question concerning what to do if symbolic
32020 arguments are given to your function. If you answer @kbd{y}, then
32021 executing @kbd{z m} (using the original argument list @samp{(a b)}) with
32022 arguments @expr{10} and @expr{x} will leave the function in symbolic
32023 form, i.e., @samp{yow(10,x)}. On the other hand, if you answer @kbd{n},
32024 then the formula will always be expanded, even for non-constant
32025 arguments: @samp{10 + 2 x}. If you never plan to feed algebraic
32026 formulas to your new function, it doesn't matter how you answer this
32029 If you answered @kbd{y} to this question you can still cause a function
32030 call to be expanded by typing @kbd{a "} (@code{calc-expand-formula}).
32031 Also, Calc will expand the function if necessary when you take a
32032 derivative or integral or solve an equation involving the function.
32035 @pindex calc-get-user-defn
32036 Once you have defined a formula on a key, you can retrieve this formula
32037 with the @kbd{Z G} (@code{calc-user-define-get-defn}) command. Press a
32038 key, and this command pushes the formula that was used to define that
32039 key onto the stack. Actually, it pushes a nameless function that
32040 specifies both the argument list and the defining formula. You will get
32041 an error message if the key is undefined, or if the key was not defined
32042 by a @kbd{Z F} command.
32044 The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has
32045 been defined by a formula uses a variant of the @code{calc-edit} command
32046 to edit the defining formula. Press @kbd{C-c C-c} to finish editing and
32047 store the new formula back in the definition, or kill the buffer with
32049 cancel the edit. (The argument list and other properties of the
32050 definition are unchanged; to adjust the argument list, you can use
32051 @kbd{Z G} to grab the function onto the stack, edit with @kbd{`}, and
32052 then re-execute the @kbd{Z F} command.)
32054 As usual, the @kbd{Z P} command records your definition permanently.
32055 In this case it will permanently record all three of the relevant
32056 definitions: the key, the command, and the function.
32058 You may find it useful to turn off the default simplifications with
32059 @kbd{m O} (@code{calc-no-simplify-mode}) when entering a formula to be
32060 used as a function definition. For example, the formula @samp{deriv(a^2,v)}
32061 which might be used to define a new function @samp{dsqr(a,v)} will be
32062 ``simplified'' to 0 immediately upon entry since @code{deriv} considers
32063 @expr{a} to be constant with respect to @expr{v}. Turning off
32064 default simplifications cures this problem: The definition will be stored
32065 in symbolic form without ever activating the @code{deriv} function. Press
32066 @kbd{m D} to turn the default simplifications back on afterwards.
32068 @node Lisp Definitions, , Algebraic Definitions, Programming
32069 @section Programming with Lisp
32072 The Calculator can be programmed quite extensively in Lisp. All you
32073 do is write a normal Lisp function definition, but with @code{defmath}
32074 in place of @code{defun}. This has the same form as @code{defun}, but it
32075 automagically replaces calls to standard Lisp functions like @code{+} and
32076 @code{zerop} with calls to the corresponding functions in Calc's own library.
32077 Thus you can write natural-looking Lisp code which operates on all of the
32078 standard Calculator data types. You can then use @kbd{Z D} if you wish to
32079 bind your new command to a @kbd{z}-prefix key sequence. The @kbd{Z E} command
32080 will not edit a Lisp-based definition.
32082 Emacs Lisp is described in the GNU Emacs Lisp Reference Manual. This section
32083 assumes a familiarity with Lisp programming concepts; if you do not know
32084 Lisp, you may find keyboard macros or rewrite rules to be an easier way
32085 to program the Calculator.
32087 This section first discusses ways to write commands, functions, or
32088 small programs to be executed inside of Calc. Then it discusses how
32089 your own separate programs are able to call Calc from the outside.
32090 Finally, there is a list of internal Calc functions and data structures
32091 for the true Lisp enthusiast.
32094 * Defining Functions::
32095 * Defining Simple Commands::
32096 * Defining Stack Commands::
32097 * Argument Qualifiers::
32098 * Example Definitions::
32100 * Calling Calc from Your Programs::
32104 @node Defining Functions, Defining Simple Commands, Lisp Definitions, Lisp Definitions
32105 @subsection Defining New Functions
32109 The @code{defmath} function (actually a Lisp macro) is like @code{defun}
32110 except that code in the body of the definition can make use of the full
32111 range of Calculator data types. The prefix @samp{calcFunc-} is added
32112 to the specified name to get the actual Lisp function name. As a simple
32116 (defmath myfact (n)
32118 (* n (myfact (1- n)))
32123 This actually expands to the code,
32126 (defun calcFunc-myfact (n)
32128 (math-mul n (calcFunc-myfact (math-add n -1)))
32133 This function can be used in algebraic expressions, e.g., @samp{myfact(5)}.
32135 The @samp{myfact} function as it is defined above has the bug that an
32136 expression @samp{myfact(a+b)} will be simplified to 1 because the
32137 formula @samp{a+b} is not considered to be @code{posp}. A robust
32138 factorial function would be written along the following lines:
32141 (defmath myfact (n)
32143 (* n (myfact (1- n)))
32146 nil))) ; this could be simplified as: (and (= n 0) 1)
32149 If a function returns @code{nil}, it is left unsimplified by the Calculator
32150 (except that its arguments will be simplified). Thus, @samp{myfact(a+1+2)}
32151 will be simplified to @samp{myfact(a+3)} but no further. Beware that every
32152 time the Calculator reexamines this formula it will attempt to resimplify
32153 it, so your function ought to detect the returning-@code{nil} case as
32154 efficiently as possible.
32156 The following standard Lisp functions are treated by @code{defmath}:
32157 @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^} or
32158 @code{expt}, @code{=}, @code{<}, @code{>}, @code{<=}, @code{>=},
32159 @code{/=}, @code{1+}, @code{1-}, @code{logand}, @code{logior}, @code{logxor},
32160 @code{logandc2}, @code{lognot}. Also, @code{~=} is an abbreviation for
32161 @code{math-nearly-equal}, which is useful in implementing Taylor series.
32163 For other functions @var{func}, if a function by the name
32164 @samp{calcFunc-@var{func}} exists it is used, otherwise if a function by the
32165 name @samp{math-@var{func}} exists it is used, otherwise if @var{func} itself
32166 is defined as a function it is used, otherwise @samp{calcFunc-@var{func}} is
32167 used on the assumption that this is a to-be-defined math function. Also, if
32168 the function name is quoted as in @samp{('integerp a)} the function name is
32169 always used exactly as written (but not quoted).
32171 Variable names have @samp{var-} prepended to them unless they appear in
32172 the function's argument list or in an enclosing @code{let}, @code{let*},
32173 @code{for}, or @code{foreach} form,
32174 or their names already contain a @samp{-} character. Thus a reference to
32175 @samp{foo} is the same as a reference to @samp{var-foo}.
32177 A few other Lisp extensions are available in @code{defmath} definitions:
32181 The @code{elt} function accepts any number of index variables.
32182 Note that Calc vectors are stored as Lisp lists whose first
32183 element is the symbol @code{vec}; thus, @samp{(elt v 2)} yields
32184 the second element of vector @code{v}, and @samp{(elt m i j)}
32185 yields one element of a Calc matrix.
32188 The @code{setq} function has been extended to act like the Common
32189 Lisp @code{setf} function. (The name @code{setf} is recognized as
32190 a synonym of @code{setq}.) Specifically, the first argument of
32191 @code{setq} can be an @code{nth}, @code{elt}, @code{car}, or @code{cdr} form,
32192 in which case the effect is to store into the specified
32193 element of a list. Thus, @samp{(setq (elt m i j) x)} stores @expr{x}
32194 into one element of a matrix.
32197 A @code{for} looping construct is available. For example,
32198 @samp{(for ((i 0 10)) body)} executes @code{body} once for each
32199 binding of @expr{i} from zero to 10. This is like a @code{let}
32200 form in that @expr{i} is temporarily bound to the loop count
32201 without disturbing its value outside the @code{for} construct.
32202 Nested loops, as in @samp{(for ((i 0 10) (j 0 (1- i) 2)) body)},
32203 are also available. For each value of @expr{i} from zero to 10,
32204 @expr{j} counts from 0 to @expr{i-1} in steps of two. Note that
32205 @code{for} has the same general outline as @code{let*}, except
32206 that each element of the header is a list of three or four
32207 things, not just two.
32210 The @code{foreach} construct loops over elements of a list.
32211 For example, @samp{(foreach ((x (cdr v))) body)} executes
32212 @code{body} with @expr{x} bound to each element of Calc vector
32213 @expr{v} in turn. The purpose of @code{cdr} here is to skip over
32214 the initial @code{vec} symbol in the vector.
32217 The @code{break} function breaks out of the innermost enclosing
32218 @code{while}, @code{for}, or @code{foreach} loop. If given a
32219 value, as in @samp{(break x)}, this value is returned by the
32220 loop. (Lisp loops otherwise always return @code{nil}.)
32223 The @code{return} function prematurely returns from the enclosing
32224 function. For example, @samp{(return (+ x y))} returns @expr{x+y}
32225 as the value of a function. You can use @code{return} anywhere
32226 inside the body of the function.
32229 Non-integer numbers (and extremely large integers) cannot be included
32230 directly into a @code{defmath} definition. This is because the Lisp
32231 reader will fail to parse them long before @code{defmath} ever gets control.
32232 Instead, use the notation, @samp{:"3.1415"}. In fact, any algebraic
32233 formula can go between the quotes. For example,
32236 (defmath sqexp (x) ; sqexp(x) == sqrt(exp(x)) == exp(x*0.5)
32244 (defun calcFunc-sqexp (x)
32245 (and (math-numberp x)
32246 (calcFunc-exp (math-mul x '(float 5 -1)))))
32249 Note the use of @code{numberp} as a guard to ensure that the argument is
32250 a number first, returning @code{nil} if not. The exponential function
32251 could itself have been included in the expression, if we had preferred:
32252 @samp{:"exp(x * 0.5)"}. As another example, the multiplication-and-recursion
32253 step of @code{myfact} could have been written
32259 A good place to put your @code{defmath} commands is your Calc init file
32260 (the file given by @code{calc-settings-file}, typically
32261 @file{~/.emacs.d/calc.el}), which will not be loaded until Calc starts.
32262 If a file named @file{.emacs} exists in your home directory, Emacs reads
32263 and executes the Lisp forms in this file as it starts up. While it may
32264 seem reasonable to put your favorite @code{defmath} commands there,
32265 this has the unfortunate side-effect that parts of the Calculator must be
32266 loaded in to process the @code{defmath} commands whether or not you will
32267 actually use the Calculator! If you want to put the @code{defmath}
32268 commands there (for example, if you redefine @code{calc-settings-file}
32269 to be @file{.emacs}), a better effect can be had by writing
32272 (put 'calc-define 'thing '(progn
32279 @vindex calc-define
32280 The @code{put} function adds a @dfn{property} to a symbol. Each Lisp
32281 symbol has a list of properties associated with it. Here we add a
32282 property with a name of @code{thing} and a @samp{(progn ...)} form as
32283 its value. When Calc starts up, and at the start of every Calc command,
32284 the property list for the symbol @code{calc-define} is checked and the
32285 values of any properties found are evaluated as Lisp forms. The
32286 properties are removed as they are evaluated. The property names
32287 (like @code{thing}) are not used; you should choose something like the
32288 name of your project so as not to conflict with other properties.
32290 The net effect is that you can put the above code in your @file{.emacs}
32291 file and it will not be executed until Calc is loaded. Or, you can put
32292 that same code in another file which you load by hand either before or
32293 after Calc itself is loaded.
32295 The properties of @code{calc-define} are evaluated in the same order
32296 that they were added. They can assume that the Calc modules @file{calc.el},
32297 @file{calc-ext.el}, and @file{calc-macs.el} have been fully loaded, and
32298 that the @samp{*Calculator*} buffer will be the current buffer.
32300 If your @code{calc-define} property only defines algebraic functions,
32301 you can be sure that it will have been evaluated before Calc tries to
32302 call your function, even if the file defining the property is loaded
32303 after Calc is loaded. But if the property defines commands or key
32304 sequences, it may not be evaluated soon enough. (Suppose it defines the
32305 new command @code{tweak-calc}; the user can load your file, then type
32306 @kbd{M-x tweak-calc} before Calc has had chance to do anything.) To
32307 protect against this situation, you can put
32310 (run-hooks 'calc-check-defines)
32313 @findex calc-check-defines
32315 at the end of your file. The @code{calc-check-defines} function is what
32316 looks for and evaluates properties on @code{calc-define}; @code{run-hooks}
32317 has the advantage that it is quietly ignored if @code{calc-check-defines}
32318 is not yet defined because Calc has not yet been loaded.
32320 Examples of things that ought to be enclosed in a @code{calc-define}
32321 property are @code{defmath} calls, @code{define-key} calls that modify
32322 the Calc key map, and any calls that redefine things defined inside Calc.
32323 Ordinary @code{defun}s need not be enclosed with @code{calc-define}.
32325 @node Defining Simple Commands, Defining Stack Commands, Defining Functions, Lisp Definitions
32326 @subsection Defining New Simple Commands
32329 @findex interactive
32330 If a @code{defmath} form contains an @code{interactive} clause, it defines
32331 a Calculator command. Actually such a @code{defmath} results in @emph{two}
32332 function definitions: One, a @samp{calcFunc-} function as was just described,
32333 with the @code{interactive} clause removed. Two, a @samp{calc-} function
32334 with a suitable @code{interactive} clause and some sort of wrapper to make
32335 the command work in the Calc environment.
32337 In the simple case, the @code{interactive} clause has the same form as
32338 for normal Emacs Lisp commands:
32341 (defmath increase-precision (delta)
32342 "Increase precision by DELTA." ; This is the "documentation string"
32343 (interactive "p") ; Register this as a M-x-able command
32344 (setq calc-internal-prec (+ calc-internal-prec delta)))
32347 This expands to the pair of definitions,
32350 (defun calc-increase-precision (delta)
32351 "Increase precision by DELTA."
32354 (setq calc-internal-prec (math-add calc-internal-prec delta))))
32356 (defun calcFunc-increase-precision (delta)
32357 "Increase precision by DELTA."
32358 (setq calc-internal-prec (math-add calc-internal-prec delta)))
32362 where in this case the latter function would never really be used! Note
32363 that since the Calculator stores small integers as plain Lisp integers,
32364 the @code{math-add} function will work just as well as the native
32365 @code{+} even when the intent is to operate on native Lisp integers.
32367 @findex calc-wrapper
32368 The @samp{calc-wrapper} call invokes a macro which surrounds the body of
32369 the function with code that looks roughly like this:
32372 (let ((calc-command-flags nil))
32374 (save-current-buffer
32375 (calc-select-buffer)
32376 @emph{body of function}
32377 @emph{renumber stack}
32378 @emph{clear} Working @emph{message})
32379 @emph{realign cursor and window}
32380 @emph{clear Inverse, Hyperbolic, and Keep Args flags}
32381 @emph{update Emacs mode line}))
32384 @findex calc-select-buffer
32385 The @code{calc-select-buffer} function selects the @samp{*Calculator*}
32386 buffer if necessary, say, because the command was invoked from inside
32387 the @samp{*Calc Trail*} window.
32389 @findex calc-set-command-flag
32390 You can call, for example, @code{(calc-set-command-flag 'no-align)} to
32391 set the above-mentioned command flags. Calc routines recognize the
32392 following command flags:
32396 Stack line numbers @samp{1:}, @samp{2:}, and so on must be renumbered
32397 after this command completes. This is set by routines like
32400 @item clear-message
32401 Calc should call @samp{(message "")} if this command completes normally
32402 (to clear a ``Working@dots{}'' message out of the echo area).
32405 Do not move the cursor back to the @samp{.} top-of-stack marker.
32407 @item position-point
32408 Use the variables @code{calc-position-point-line} and
32409 @code{calc-position-point-column} to position the cursor after
32410 this command finishes.
32413 Do not clear @code{calc-inverse-flag}, @code{calc-hyperbolic-flag},
32414 and @code{calc-keep-args-flag} at the end of this command.
32417 Switch to buffer @samp{*Calc Edit*} after this command.
32420 Do not move trail pointer to end of trail when something is recorded
32426 @vindex calc-Y-help-msgs
32427 Calc reserves a special prefix key, shift-@kbd{Y}, for user-written
32428 extensions to Calc. There are no built-in commands that work with
32429 this prefix key; you must call @code{define-key} from Lisp (probably
32430 from inside a @code{calc-define} property) to add to it. Initially only
32431 @kbd{Y ?} is defined; it takes help messages from a list of strings
32432 (initially @code{nil}) in the variable @code{calc-Y-help-msgs}. All
32433 other undefined keys except for @kbd{Y} are reserved for use by
32434 future versions of Calc.
32436 If you are writing a Calc enhancement which you expect to give to
32437 others, it is best to minimize the number of @kbd{Y}-key sequences
32438 you use. In fact, if you have more than one key sequence you should
32439 consider defining three-key sequences with a @kbd{Y}, then a key that
32440 stands for your package, then a third key for the particular command
32441 within your package.
32443 Users may wish to install several Calc enhancements, and it is possible
32444 that several enhancements will choose to use the same key. In the
32445 example below, a variable @code{inc-prec-base-key} has been defined
32446 to contain the key that identifies the @code{inc-prec} package. Its
32447 value is initially @code{"P"}, but a user can change this variable
32448 if necessary without having to modify the file.
32450 Here is a complete file, @file{inc-prec.el}, which makes a @kbd{Y P I}
32451 command that increases the precision, and a @kbd{Y P D} command that
32452 decreases the precision.
32455 ;;; Increase and decrease Calc precision. Dave Gillespie, 5/31/91.
32456 ;; (Include copyright or copyleft stuff here.)
32458 (defvar inc-prec-base-key "P"
32459 "Base key for inc-prec.el commands.")
32461 (put 'calc-define 'inc-prec '(progn
32463 (define-key calc-mode-map (format "Y%sI" inc-prec-base-key)
32464 'increase-precision)
32465 (define-key calc-mode-map (format "Y%sD" inc-prec-base-key)
32466 'decrease-precision)
32468 (setq calc-Y-help-msgs
32469 (cons (format "%s + Inc-prec, Dec-prec" inc-prec-base-key)
32472 (defmath increase-precision (delta)
32473 "Increase precision by DELTA."
32475 (setq calc-internal-prec (+ calc-internal-prec delta)))
32477 (defmath decrease-precision (delta)
32478 "Decrease precision by DELTA."
32480 (setq calc-internal-prec (- calc-internal-prec delta)))
32482 )) ; end of calc-define property
32484 (run-hooks 'calc-check-defines)
32487 @node Defining Stack Commands, Argument Qualifiers, Defining Simple Commands, Lisp Definitions
32488 @subsection Defining New Stack-Based Commands
32491 To define a new computational command which takes and/or leaves arguments
32492 on the stack, a special form of @code{interactive} clause is used.
32495 (interactive @var{num} @var{tag})
32499 where @var{num} is an integer, and @var{tag} is a string. The effect is
32500 to pop @var{num} values off the stack, resimplify them by calling
32501 @code{calc-normalize}, and hand them to your function according to the
32502 function's argument list. Your function may include @code{&optional} and
32503 @code{&rest} parameters, so long as calling the function with @var{num}
32504 parameters is valid.
32506 Your function must return either a number or a formula in a form
32507 acceptable to Calc, or a list of such numbers or formulas. These value(s)
32508 are pushed onto the stack when the function completes. They are also
32509 recorded in the Calc Trail buffer on a line beginning with @var{tag},
32510 a string of (normally) four characters or less. If you omit @var{tag}
32511 or use @code{nil} as a tag, the result is not recorded in the trail.
32513 As an example, the definition
32516 (defmath myfact (n)
32517 "Compute the factorial of the integer at the top of the stack."
32518 (interactive 1 "fact")
32520 (* n (myfact (1- n)))
32525 is a version of the factorial function shown previously which can be used
32526 as a command as well as an algebraic function. It expands to
32529 (defun calc-myfact ()
32530 "Compute the factorial of the integer at the top of the stack."
32533 (calc-enter-result 1 "fact"
32534 (cons 'calcFunc-myfact (calc-top-list-n 1)))))
32536 (defun calcFunc-myfact (n)
32537 "Compute the factorial of the integer at the top of the stack."
32539 (math-mul n (calcFunc-myfact (math-add n -1)))
32540 (and (math-zerop n) 1)))
32543 @findex calc-slow-wrapper
32544 The @code{calc-slow-wrapper} function is a version of @code{calc-wrapper}
32545 that automatically puts up a @samp{Working...} message before the
32546 computation begins. (This message can be turned off by the user
32547 with an @kbd{m w} (@code{calc-working}) command.)
32549 @findex calc-top-list-n
32550 The @code{calc-top-list-n} function returns a list of the specified number
32551 of values from the top of the stack. It resimplifies each value by
32552 calling @code{calc-normalize}. If its argument is zero it returns an
32553 empty list. It does not actually remove these values from the stack.
32555 @findex calc-enter-result
32556 The @code{calc-enter-result} function takes an integer @var{num} and string
32557 @var{tag} as described above, plus a third argument which is either a
32558 Calculator data object or a list of such objects. These objects are
32559 resimplified and pushed onto the stack after popping the specified number
32560 of values from the stack. If @var{tag} is non-@code{nil}, the values
32561 being pushed are also recorded in the trail.
32563 Note that if @code{calcFunc-myfact} returns @code{nil} this represents
32564 ``leave the function in symbolic form.'' To return an actual empty list,
32565 in the sense that @code{calc-enter-result} will push zero elements back
32566 onto the stack, you should return the special value @samp{'(nil)}, a list
32567 containing the single symbol @code{nil}.
32569 The @code{interactive} declaration can actually contain a limited
32570 Emacs-style code string as well which comes just before @var{num} and
32571 @var{tag}. Currently the only Emacs code supported is @samp{"p"}, as in
32574 (defmath foo (a b &optional c)
32575 (interactive "p" 2 "foo")
32579 In this example, the command @code{calc-foo} will evaluate the expression
32580 @samp{foo(a,b)} if executed with no argument, or @samp{foo(a,b,n)} if
32581 executed with a numeric prefix argument of @expr{n}.
32583 The other code string allowed is @samp{"m"} (unrelated to the usual @samp{"m"}
32584 code as used with @code{defun}). It uses the numeric prefix argument as the
32585 number of objects to remove from the stack and pass to the function.
32586 In this case, the integer @var{num} serves as a default number of
32587 arguments to be used when no prefix is supplied.
32589 @node Argument Qualifiers, Example Definitions, Defining Stack Commands, Lisp Definitions
32590 @subsection Argument Qualifiers
32593 Anywhere a parameter name can appear in the parameter list you can also use
32594 an @dfn{argument qualifier}. Thus the general form of a definition is:
32597 (defmath @var{name} (@var{param} @var{param...}
32598 &optional @var{param} @var{param...}
32604 where each @var{param} is either a symbol or a list of the form
32607 (@var{qual} @var{param})
32610 The following qualifiers are recognized:
32615 The argument must not be an incomplete vector, interval, or complex number.
32616 (This is rarely needed since the Calculator itself will never call your
32617 function with an incomplete argument. But there is nothing stopping your
32618 own Lisp code from calling your function with an incomplete argument.)
32622 The argument must be an integer. If it is an integer-valued float
32623 it will be accepted but converted to integer form. Non-integers and
32624 formulas are rejected.
32628 Like @samp{integer}, but the argument must be non-negative.
32632 Like @samp{integer}, but the argument must fit into a native Lisp integer,
32633 which on most systems means less than 2^23 in absolute value. The
32634 argument is converted into Lisp-integer form if necessary.
32638 The argument is converted to floating-point format if it is a number or
32639 vector. If it is a formula it is left alone. (The argument is never
32640 actually rejected by this qualifier.)
32643 The argument must satisfy predicate @var{pred}, which is one of the
32644 standard Calculator predicates. @xref{Predicates}.
32646 @item not-@var{pred}
32647 The argument must @emph{not} satisfy predicate @var{pred}.
32653 (defmath foo (a (constp (not-matrixp b)) &optional (float c)
32662 (defun calcFunc-foo (a b &optional c &rest d)
32663 (and (math-matrixp b)
32664 (math-reject-arg b 'not-matrixp))
32665 (or (math-constp b)
32666 (math-reject-arg b 'constp))
32667 (and c (setq c (math-check-float c)))
32668 (setq d (mapcar 'math-check-integer d))
32673 which performs the necessary checks and conversions before executing the
32674 body of the function.
32676 @node Example Definitions, Calling Calc from Your Programs, Argument Qualifiers, Lisp Definitions
32677 @subsection Example Definitions
32680 This section includes some Lisp programming examples on a larger scale.
32681 These programs make use of some of the Calculator's internal functions;
32685 * Bit Counting Example::
32689 @node Bit Counting Example, Sine Example, Example Definitions, Example Definitions
32690 @subsubsection Bit-Counting
32697 Calc does not include a built-in function for counting the number of
32698 ``one'' bits in a binary integer. It's easy to invent one using @kbd{b u}
32699 to convert the integer to a set, and @kbd{V #} to count the elements of
32700 that set; let's write a function that counts the bits without having to
32701 create an intermediate set.
32704 (defmath bcount ((natnum n))
32705 (interactive 1 "bcnt")
32709 (setq count (1+ count)))
32710 (setq n (lsh n -1)))
32715 When this is expanded by @code{defmath}, it will become the following
32716 Emacs Lisp function:
32719 (defun calcFunc-bcount (n)
32720 (setq n (math-check-natnum n))
32722 (while (math-posp n)
32724 (setq count (math-add count 1)))
32725 (setq n (calcFunc-lsh n -1)))
32729 If the input numbers are large, this function involves a fair amount
32730 of arithmetic. A binary right shift is essentially a division by two;
32731 recall that Calc stores integers in decimal form so bit shifts must
32732 involve actual division.
32734 To gain a bit more efficiency, we could divide the integer into
32735 @var{n}-bit chunks, each of which can be handled quickly because
32736 they fit into Lisp integers. It turns out that Calc's arithmetic
32737 routines are especially fast when dividing by an integer less than
32738 1000, so we can set @var{n = 9} bits and use repeated division by 512:
32741 (defmath bcount ((natnum n))
32742 (interactive 1 "bcnt")
32744 (while (not (fixnump n))
32745 (let ((qr (idivmod n 512)))
32746 (setq count (+ count (bcount-fixnum (cdr qr)))
32748 (+ count (bcount-fixnum n))))
32750 (defun bcount-fixnum (n)
32753 (setq count (+ count (logand n 1))
32759 Note that the second function uses @code{defun}, not @code{defmath}.
32760 Because this function deals only with native Lisp integers (``fixnums''),
32761 it can use the actual Emacs @code{+} and related functions rather
32762 than the slower but more general Calc equivalents which @code{defmath}
32765 The @code{idivmod} function does an integer division, returning both
32766 the quotient and the remainder at once. Again, note that while it
32767 might seem that @samp{(logand n 511)} and @samp{(lsh n -9)} are
32768 more efficient ways to split off the bottom nine bits of @code{n},
32769 actually they are less efficient because each operation is really
32770 a division by 512 in disguise; @code{idivmod} allows us to do the
32771 same thing with a single division by 512.
32773 @node Sine Example, , Bit Counting Example, Example Definitions
32774 @subsubsection The Sine Function
32781 A somewhat limited sine function could be defined as follows, using the
32782 well-known Taylor series expansion for
32783 @texline @math{\sin x}:
32784 @infoline @samp{sin(x)}:
32787 (defmath mysin ((float (anglep x)))
32788 (interactive 1 "mysn")
32789 (setq x (to-radians x)) ; Convert from current angular mode.
32790 (let ((sum x) ; Initial term of Taylor expansion of sin.
32792 (nfact 1) ; "nfact" equals "n" factorial at all times.
32793 (xnegsqr :"-(x^2)")) ; "xnegsqr" equals -x^2.
32794 (for ((n 3 100 2)) ; Upper limit of 100 is a good precaution.
32795 (working "mysin" sum) ; Display "Working" message, if enabled.
32796 (setq nfact (* nfact (1- n) n)
32798 newsum (+ sum (/ x nfact)))
32799 (if (~= newsum sum) ; If newsum is "nearly equal to" sum,
32800 (break)) ; then we are done.
32805 The actual @code{sin} function in Calc works by first reducing the problem
32806 to a sine or cosine of a nonnegative number less than @cpiover{4}. This
32807 ensures that the Taylor series will converge quickly. Also, the calculation
32808 is carried out with two extra digits of precision to guard against cumulative
32809 round-off in @samp{sum}. Finally, complex arguments are allowed and handled
32810 by a separate algorithm.
32813 (defmath mysin ((float (scalarp x)))
32814 (interactive 1 "mysn")
32815 (setq x (to-radians x)) ; Convert from current angular mode.
32816 (with-extra-prec 2 ; Evaluate with extra precision.
32817 (cond ((complexp x)
32820 (- (mysin-raw (- x))) ; Always call mysin-raw with x >= 0.
32821 (t (mysin-raw x))))))
32823 (defmath mysin-raw (x)
32825 (mysin-raw (% x (two-pi)))) ; Now x < 7.
32827 (- (mysin-raw (- x (pi))))) ; Now -pi/2 <= x <= pi/2.
32829 (mycos-raw (- x (pi-over-2)))) ; Now -pi/2 <= x <= pi/4.
32830 ((< x (- (pi-over-4)))
32831 (- (mycos-raw (+ x (pi-over-2))))) ; Now -pi/4 <= x <= pi/4,
32832 (t (mysin-series x)))) ; so the series will be efficient.
32836 where @code{mysin-complex} is an appropriate function to handle complex
32837 numbers, @code{mysin-series} is the routine to compute the sine Taylor
32838 series as before, and @code{mycos-raw} is a function analogous to
32839 @code{mysin-raw} for cosines.
32841 The strategy is to ensure that @expr{x} is nonnegative before calling
32842 @code{mysin-raw}. This function then recursively reduces its argument
32843 to a suitable range, namely, plus-or-minus @cpiover{4}. Note that each
32844 test, and particularly the first comparison against 7, is designed so
32845 that small roundoff errors cannot produce an infinite loop. (Suppose
32846 we compared with @samp{(two-pi)} instead; if due to roundoff problems
32847 the modulo operator ever returned @samp{(two-pi)} exactly, an infinite
32848 recursion could result!) We use modulo only for arguments that will
32849 clearly get reduced, knowing that the next rule will catch any reductions
32850 that this rule misses.
32852 If a program is being written for general use, it is important to code
32853 it carefully as shown in this second example. For quick-and-dirty programs,
32854 when you know that your own use of the sine function will never encounter
32855 a large argument, a simpler program like the first one shown is fine.
32857 @node Calling Calc from Your Programs, Internals, Example Definitions, Lisp Definitions
32858 @subsection Calling Calc from Your Lisp Programs
32861 A later section (@pxref{Internals}) gives a full description of
32862 Calc's internal Lisp functions. It's not hard to call Calc from
32863 inside your programs, but the number of these functions can be daunting.
32864 So Calc provides one special ``programmer-friendly'' function called
32865 @code{calc-eval} that can be made to do just about everything you
32866 need. It's not as fast as the low-level Calc functions, but it's
32867 much simpler to use!
32869 It may seem that @code{calc-eval} itself has a daunting number of
32870 options, but they all stem from one simple operation.
32872 In its simplest manifestation, @samp{(calc-eval "1+2")} parses the
32873 string @code{"1+2"} as if it were a Calc algebraic entry and returns
32874 the result formatted as a string: @code{"3"}.
32876 Since @code{calc-eval} is on the list of recommended @code{autoload}
32877 functions, you don't need to make any special preparations to load
32878 Calc before calling @code{calc-eval} the first time. Calc will be
32879 loaded and initialized for you.
32881 All the Calc modes that are currently in effect will be used when
32882 evaluating the expression and formatting the result.
32889 @subsubsection Additional Arguments to @code{calc-eval}
32892 If the input string parses to a list of expressions, Calc returns
32893 the results separated by @code{", "}. You can specify a different
32894 separator by giving a second string argument to @code{calc-eval}:
32895 @samp{(calc-eval "1+2,3+4" ";")} returns @code{"3;7"}.
32897 The ``separator'' can also be any of several Lisp symbols which
32898 request other behaviors from @code{calc-eval}. These are discussed
32901 You can give additional arguments to be substituted for
32902 @samp{$}, @samp{$$}, and so on in the main expression. For
32903 example, @samp{(calc-eval "$/$$" nil "7" "1+1")} evaluates the
32904 expression @code{"7/(1+1)"} to yield the result @code{"3.5"}
32905 (assuming Fraction mode is not in effect). Note the @code{nil}
32906 used as a placeholder for the item-separator argument.
32913 @subsubsection Error Handling
32916 If @code{calc-eval} encounters an error, it returns a list containing
32917 the character position of the error, plus a suitable message as a
32918 string. Note that @samp{1 / 0} is @emph{not} an error by Calc's
32919 standards; it simply returns the string @code{"1 / 0"} which is the
32920 division left in symbolic form. But @samp{(calc-eval "1/")} will
32921 return the list @samp{(2 "Expected a number")}.
32923 If you bind the variable @code{calc-eval-error} to @code{t}
32924 using a @code{let} form surrounding the call to @code{calc-eval},
32925 errors instead call the Emacs @code{error} function which aborts
32926 to the Emacs command loop with a beep and an error message.
32928 If you bind this variable to the symbol @code{string}, error messages
32929 are returned as strings instead of lists. The character position is
32932 As a courtesy to other Lisp code which may be using Calc, be sure
32933 to bind @code{calc-eval-error} using @code{let} rather than changing
32934 it permanently with @code{setq}.
32941 @subsubsection Numbers Only
32944 Sometimes it is preferable to treat @samp{1 / 0} as an error
32945 rather than returning a symbolic result. If you pass the symbol
32946 @code{num} as the second argument to @code{calc-eval}, results
32947 that are not constants are treated as errors. The error message
32948 reported is the first @code{calc-why} message if there is one,
32949 or otherwise ``Number expected.''
32951 A result is ``constant'' if it is a number, vector, or other
32952 object that does not include variables or function calls. If it
32953 is a vector, the components must themselves be constants.
32960 @subsubsection Default Modes
32963 If the first argument to @code{calc-eval} is a list whose first
32964 element is a formula string, then @code{calc-eval} sets all the
32965 various Calc modes to their default values while the formula is
32966 evaluated and formatted. For example, the precision is set to 12
32967 digits, digit grouping is turned off, and the Normal language
32970 This same principle applies to the other options discussed below.
32971 If the first argument would normally be @var{x}, then it can also
32972 be the list @samp{(@var{x})} to use the default mode settings.
32974 If there are other elements in the list, they are taken as
32975 variable-name/value pairs which override the default mode
32976 settings. Look at the documentation at the front of the
32977 @file{calc.el} file to find the names of the Lisp variables for
32978 the various modes. The mode settings are restored to their
32979 original values when @code{calc-eval} is done.
32981 For example, @samp{(calc-eval '("$+$$" calc-internal-prec 8) 'num a b)}
32982 computes the sum of two numbers, requiring a numeric result, and
32983 using default mode settings except that the precision is 8 instead
32984 of the default of 12.
32986 It's usually best to use this form of @code{calc-eval} unless your
32987 program actually considers the interaction with Calc's mode settings
32988 to be a feature. This will avoid all sorts of potential ``gotchas'';
32989 consider what happens with @samp{(calc-eval "sqrt(2)" 'num)}
32990 when the user has left Calc in Symbolic mode or No-Simplify mode.
32992 As another example, @samp{(equal (calc-eval '("$<$$") nil a b) "1")}
32993 checks if the number in string @expr{a} is less than the one in
32994 string @expr{b}. Without using a list, the integer 1 might
32995 come out in a variety of formats which would be hard to test for
32996 conveniently: @code{"1"}, @code{"8#1"}, @code{"00001"}. (But
32997 see ``Predicates'' mode, below.)
33004 @subsubsection Raw Numbers
33007 Normally all input and output for @code{calc-eval} is done with strings.
33008 You can do arithmetic with, say, @samp{(calc-eval "$+$$" nil a b)}
33009 in place of @samp{(+ a b)}, but this is very inefficient since the
33010 numbers must be converted to and from string format as they are passed
33011 from one @code{calc-eval} to the next.
33013 If the separator is the symbol @code{raw}, the result will be returned
33014 as a raw Calc data structure rather than a string. You can read about
33015 how these objects look in the following sections, but usually you can
33016 treat them as ``black box'' objects with no important internal
33019 There is also a @code{rawnum} symbol, which is a combination of
33020 @code{raw} (returning a raw Calc object) and @code{num} (signaling
33021 an error if that object is not a constant).
33023 You can pass a raw Calc object to @code{calc-eval} in place of a
33024 string, either as the formula itself or as one of the @samp{$}
33025 arguments. Thus @samp{(calc-eval "$+$$" 'raw a b)} is an
33026 addition function that operates on raw Calc objects. Of course
33027 in this case it would be easier to call the low-level @code{math-add}
33028 function in Calc, if you can remember its name.
33030 In particular, note that a plain Lisp integer is acceptable to Calc
33031 as a raw object. (All Lisp integers are accepted on input, but
33032 integers of more than six decimal digits are converted to ``big-integer''
33033 form for output. @xref{Data Type Formats}.)
33035 When it comes time to display the object, just use @samp{(calc-eval a)}
33036 to format it as a string.
33038 It is an error if the input expression evaluates to a list of
33039 values. The separator symbol @code{list} is like @code{raw}
33040 except that it returns a list of one or more raw Calc objects.
33042 Note that a Lisp string is not a valid Calc object, nor is a list
33043 containing a string. Thus you can still safely distinguish all the
33044 various kinds of error returns discussed above.
33051 @subsubsection Predicates
33054 If the separator symbol is @code{pred}, the result of the formula is
33055 treated as a true/false value; @code{calc-eval} returns @code{t} or
33056 @code{nil}, respectively. A value is considered ``true'' if it is a
33057 non-zero number, or false if it is zero or if it is not a number.
33059 For example, @samp{(calc-eval "$<$$" 'pred a b)} tests whether
33060 one value is less than another.
33062 As usual, it is also possible for @code{calc-eval} to return one of
33063 the error indicators described above. Lisp will interpret such an
33064 indicator as ``true'' if you don't check for it explicitly. If you
33065 wish to have an error register as ``false'', use something like
33066 @samp{(eq (calc-eval ...) t)}.
33073 @subsubsection Variable Values
33076 Variables in the formula passed to @code{calc-eval} are not normally
33077 replaced by their values. If you wish this, you can use the
33078 @code{evalv} function (@pxref{Algebraic Manipulation}). For example,
33079 if 4 is stored in Calc variable @code{a} (i.e., in Lisp variable
33080 @code{var-a}), then @samp{(calc-eval "a+pi")} will return the
33081 formula @code{"a + pi"}, but @samp{(calc-eval "evalv(a+pi)")}
33082 will return @code{"7.14159265359"}.
33084 To store in a Calc variable, just use @code{setq} to store in the
33085 corresponding Lisp variable. (This is obtained by prepending
33086 @samp{var-} to the Calc variable name.) Calc routines will
33087 understand either string or raw form values stored in variables,
33088 although raw data objects are much more efficient. For example,
33089 to increment the Calc variable @code{a}:
33092 (setq var-a (calc-eval "evalv(a+1)" 'raw))
33100 @subsubsection Stack Access
33103 If the separator symbol is @code{push}, the formula argument is
33104 evaluated (with possible @samp{$} expansions, as usual). The
33105 result is pushed onto the Calc stack. The return value is @code{nil}
33106 (unless there is an error from evaluating the formula, in which
33107 case the return value depends on @code{calc-eval-error} in the
33110 If the separator symbol is @code{pop}, the first argument to
33111 @code{calc-eval} must be an integer instead of a string. That
33112 many values are popped from the stack and thrown away. A negative
33113 argument deletes the entry at that stack level. The return value
33114 is the number of elements remaining in the stack after popping;
33115 @samp{(calc-eval 0 'pop)} is a good way to measure the size of
33118 If the separator symbol is @code{top}, the first argument to
33119 @code{calc-eval} must again be an integer. The value at that
33120 stack level is formatted as a string and returned. Thus
33121 @samp{(calc-eval 1 'top)} returns the top-of-stack value. If the
33122 integer is out of range, @code{nil} is returned.
33124 The separator symbol @code{rawtop} is just like @code{top} except
33125 that the stack entry is returned as a raw Calc object instead of
33128 In all of these cases the first argument can be made a list in
33129 order to force the default mode settings, as described above.
33130 Thus @samp{(calc-eval '(2 calc-number-radix 16) 'top)} returns the
33131 second-to-top stack entry, formatted as a string using the default
33132 instead of current display modes, except that the radix is
33133 hexadecimal instead of decimal.
33135 It is, of course, polite to put the Calc stack back the way you
33136 found it when you are done, unless the user of your program is
33137 actually expecting it to affect the stack.
33139 Note that you do not actually have to switch into the @samp{*Calculator*}
33140 buffer in order to use @code{calc-eval}; it temporarily switches into
33141 the stack buffer if necessary.
33148 @subsubsection Keyboard Macros
33151 If the separator symbol is @code{macro}, the first argument must be a
33152 string of characters which Calc can execute as a sequence of keystrokes.
33153 This switches into the Calc buffer for the duration of the macro.
33154 For example, @samp{(calc-eval "vx5\rVR+" 'macro)} pushes the
33155 vector @samp{[1,2,3,4,5]} on the stack and then replaces it
33156 with the sum of those numbers. Note that @samp{\r} is the Lisp
33157 notation for the carriage-return, @key{RET}, character.
33159 If your keyboard macro wishes to pop the stack, @samp{\C-d} is
33160 safer than @samp{\177} (the @key{DEL} character) because some
33161 installations may have switched the meanings of @key{DEL} and
33162 @kbd{C-h}. Calc always interprets @kbd{C-d} as a synonym for
33163 ``pop-stack'' regardless of key mapping.
33165 If you provide a third argument to @code{calc-eval}, evaluation
33166 of the keyboard macro will leave a record in the Trail using
33167 that argument as a tag string. Normally the Trail is unaffected.
33169 The return value in this case is always @code{nil}.
33176 @subsubsection Lisp Evaluation
33179 Finally, if the separator symbol is @code{eval}, then the Lisp
33180 @code{eval} function is called on the first argument, which must
33181 be a Lisp expression rather than a Calc formula. Remember to
33182 quote the expression so that it is not evaluated until inside
33185 The difference from plain @code{eval} is that @code{calc-eval}
33186 switches to the Calc buffer before evaluating the expression.
33187 For example, @samp{(calc-eval '(setq calc-internal-prec 17) 'eval)}
33188 will correctly affect the buffer-local Calc precision variable.
33190 An alternative would be @samp{(calc-eval '(calc-precision 17) 'eval)}.
33191 This is evaluating a call to the function that is normally invoked
33192 by the @kbd{p} key, giving it 17 as its ``numeric prefix argument.''
33193 Note that this function will leave a message in the echo area as
33194 a side effect. Also, all Calc functions switch to the Calc buffer
33195 automatically if not invoked from there, so the above call is
33196 also equivalent to @samp{(calc-precision 17)} by itself.
33197 In all cases, Calc uses @code{save-excursion} to switch back to
33198 your original buffer when it is done.
33200 As usual the first argument can be a list that begins with a Lisp
33201 expression to use default instead of current mode settings.
33203 The result of @code{calc-eval} in this usage is just the result
33204 returned by the evaluated Lisp expression.
33211 @subsubsection Example
33214 @findex convert-temp
33215 Here is a sample Emacs command that uses @code{calc-eval}. Suppose
33216 you have a document with lots of references to temperatures on the
33217 Fahrenheit scale, say ``98.6 F'', and you wish to convert these
33218 references to Centigrade. The following command does this conversion.
33219 Place the Emacs cursor right after the letter ``F'' and invoke the
33220 command to change ``98.6 F'' to ``37 C''. Or, if the temperature is
33221 already in Centigrade form, the command changes it back to Fahrenheit.
33224 (defun convert-temp ()
33227 (re-search-backward "[^-.0-9]\\([-.0-9]+\\) *\\([FC]\\)")
33228 (let* ((top1 (match-beginning 1))
33229 (bot1 (match-end 1))
33230 (number (buffer-substring top1 bot1))
33231 (top2 (match-beginning 2))
33232 (bot2 (match-end 2))
33233 (type (buffer-substring top2 bot2)))
33234 (if (equal type "F")
33236 number (calc-eval "($ - 32)*5/9" nil number))
33238 number (calc-eval "$*9/5 + 32" nil number)))
33240 (delete-region top2 bot2)
33241 (insert-before-markers type)
33243 (delete-region top1 bot1)
33244 (if (string-match "\\.$" number) ; change "37." to "37"
33245 (setq number (substring number 0 -1)))
33249 Note the use of @code{insert-before-markers} when changing between
33250 ``F'' and ``C'', so that the character winds up before the cursor
33251 instead of after it.
33253 @node Internals, , Calling Calc from Your Programs, Lisp Definitions
33254 @subsection Calculator Internals
33257 This section describes the Lisp functions defined by the Calculator that
33258 may be of use to user-written Calculator programs (as described in the
33259 rest of this chapter). These functions are shown by their names as they
33260 conventionally appear in @code{defmath}. Their full Lisp names are
33261 generally gotten by prepending @samp{calcFunc-} or @samp{math-} to their
33262 apparent names. (Names that begin with @samp{calc-} are already in
33263 their full Lisp form.) You can use the actual full names instead if you
33264 prefer them, or if you are calling these functions from regular Lisp.
33266 The functions described here are scattered throughout the various
33267 Calc component files. Note that @file{calc.el} includes @code{autoload}s
33268 for only a few component files; when Calc wants to call an advanced
33269 function it calls @samp{(calc-extensions)} first; this function
33270 autoloads @file{calc-ext.el}, which in turn autoloads all the functions
33271 in the remaining component files.
33273 Because @code{defmath} itself uses the extensions, user-written code
33274 generally always executes with the extensions already loaded, so
33275 normally you can use any Calc function and be confident that it will
33276 be autoloaded for you when necessary. If you are doing something
33277 special, check carefully to make sure each function you are using is
33278 from @file{calc.el} or its components, and call @samp{(calc-extensions)}
33279 before using any function based in @file{calc-ext.el} if you can't
33280 prove this file will already be loaded.
33283 * Data Type Formats::
33284 * Interactive Lisp Functions::
33285 * Stack Lisp Functions::
33287 * Computational Lisp Functions::
33288 * Vector Lisp Functions::
33289 * Symbolic Lisp Functions::
33290 * Formatting Lisp Functions::
33294 @node Data Type Formats, Interactive Lisp Functions, Internals, Internals
33295 @subsubsection Data Type Formats
33298 Integers are stored in either of two ways, depending on their magnitude.
33299 Integers less than one million in absolute value are stored as standard
33300 Lisp integers. This is the only storage format for Calc data objects
33301 which is not a Lisp list.
33303 Large integers are stored as lists of the form @samp{(bigpos @var{d0}
33304 @var{d1} @var{d2} @dots{})} for positive integers 1000000 or more, or
33305 @samp{(bigneg @var{d0} @var{d1} @var{d2} @dots{})} for negative integers
33306 @mathit{-1000000} or less. Each @var{d} is a base-1000 ``digit,'' a Lisp integer
33307 from 0 to 999. The least significant digit is @var{d0}; the last digit,
33308 @var{dn}, which is always nonzero, is the most significant digit. For
33309 example, the integer @mathit{-12345678} is stored as @samp{(bigneg 678 345 12)}.
33311 The distinction between small and large integers is entirely hidden from
33312 the user. In @code{defmath} definitions, the Lisp predicate @code{integerp}
33313 returns true for either kind of integer, and in general both big and small
33314 integers are accepted anywhere the word ``integer'' is used in this manual.
33315 If the distinction must be made, native Lisp integers are called @dfn{fixnums}
33316 and large integers are called @dfn{bignums}.
33318 Fractions are stored as a list of the form, @samp{(frac @var{n} @var{d})}
33319 where @var{n} is an integer (big or small) numerator, @var{d} is an
33320 integer denominator greater than one, and @var{n} and @var{d} are relatively
33321 prime. Note that fractions where @var{d} is one are automatically converted
33322 to plain integers by all math routines; fractions where @var{d} is negative
33323 are normalized by negating the numerator and denominator.
33325 Floating-point numbers are stored in the form, @samp{(float @var{mant}
33326 @var{exp})}, where @var{mant} (the ``mantissa'') is an integer less than
33327 @samp{10^@var{p}} in absolute value (@var{p} represents the current
33328 precision), and @var{exp} (the ``exponent'') is a fixnum. The value of
33329 the float is @samp{@var{mant} * 10^@var{exp}}. For example, the number
33330 @mathit{-3.14} is stored as @samp{(float -314 -2) = -314*10^-2}. Other constraints
33331 are that the number 0.0 is always stored as @samp{(float 0 0)}, and,
33332 except for the 0.0 case, the rightmost base-10 digit of @var{mant} is
33333 always nonzero. (If the rightmost digit is zero, the number is
33334 rearranged by dividing @var{mant} by ten and incrementing @var{exp}.)
33336 Rectangular complex numbers are stored in the form @samp{(cplx @var{re}
33337 @var{im})}, where @var{re} and @var{im} are each real numbers, either
33338 integers, fractions, or floats. The value is @samp{@var{re} + @var{im}i}.
33339 The @var{im} part is nonzero; complex numbers with zero imaginary
33340 components are converted to real numbers automatically.
33342 Polar complex numbers are stored in the form @samp{(polar @var{r}
33343 @var{theta})}, where @var{r} is a positive real value and @var{theta}
33344 is a real value or HMS form representing an angle. This angle is
33345 usually normalized to lie in the interval @samp{(-180 ..@: 180)} degrees,
33346 or @samp{(-pi ..@: pi)} radians, according to the current angular mode.
33347 If the angle is 0 the value is converted to a real number automatically.
33348 (If the angle is 180 degrees, the value is usually also converted to a
33349 negative real number.)
33351 Hours-minutes-seconds forms are stored as @samp{(hms @var{h} @var{m}
33352 @var{s})}, where @var{h} is an integer or an integer-valued float (i.e.,
33353 a float with @samp{@var{exp} >= 0}), @var{m} is an integer or integer-valued
33354 float in the range @w{@samp{[0 ..@: 60)}}, and @var{s} is any real number
33355 in the range @samp{[0 ..@: 60)}.
33357 Date forms are stored as @samp{(date @var{n})}, where @var{n} is
33358 a real number that counts days since midnight on the morning of
33359 January 1, 1 AD@. If @var{n} is an integer, this is a pure date
33360 form. If @var{n} is a fraction or float, this is a date/time form.
33362 Modulo forms are stored as @samp{(mod @var{n} @var{m})}, where @var{m} is a
33363 positive real number or HMS form, and @var{n} is a real number or HMS
33364 form in the range @samp{[0 ..@: @var{m})}.
33366 Error forms are stored as @samp{(sdev @var{x} @var{sigma})}, where @var{x}
33367 is the mean value and @var{sigma} is the standard deviation. Each
33368 component is either a number, an HMS form, or a symbolic object
33369 (a variable or function call). If @var{sigma} is zero, the value is
33370 converted to a plain real number. If @var{sigma} is negative or
33371 complex, it is automatically normalized to be a positive real.
33373 Interval forms are stored as @samp{(intv @var{mask} @var{lo} @var{hi})},
33374 where @var{mask} is one of the integers 0, 1, 2, or 3, and @var{lo} and
33375 @var{hi} are real numbers, HMS forms, or symbolic objects. The @var{mask}
33376 is a binary integer where 1 represents the fact that the interval is
33377 closed on the high end, and 2 represents the fact that it is closed on
33378 the low end. (Thus 3 represents a fully closed interval.) The interval
33379 @w{@samp{(intv 3 @var{x} @var{x})}} is converted to the plain number @var{x};
33380 intervals @samp{(intv @var{mask} @var{x} @var{x})} for any other @var{mask}
33381 represent empty intervals. If @var{hi} is less than @var{lo}, the interval
33382 is converted to a standard empty interval by replacing @var{hi} with @var{lo}.
33384 Vectors are stored as @samp{(vec @var{v1} @var{v2} @dots{})}, where @var{v1}
33385 is the first element of the vector, @var{v2} is the second, and so on.
33386 An empty vector is stored as @samp{(vec)}. A matrix is simply a vector
33387 where all @var{v}'s are themselves vectors of equal lengths. Note that
33388 Calc vectors are unrelated to the Emacs Lisp ``vector'' type, which is
33389 generally unused by Calc data structures.
33391 Variables are stored as @samp{(var @var{name} @var{sym})}, where
33392 @var{name} is a Lisp symbol whose print name is used as the visible name
33393 of the variable, and @var{sym} is a Lisp symbol in which the variable's
33394 value is actually stored. Thus, @samp{(var pi var-pi)} represents the
33395 special constant @samp{pi}. Almost always, the form is @samp{(var
33396 @var{v} var-@var{v})}. If the variable name was entered with @code{#}
33397 signs (which are converted to hyphens internally), the form is
33398 @samp{(var @var{u} @var{v})}, where @var{u} is a symbol whose name
33399 contains @code{#} characters, and @var{v} is a symbol that contains
33400 @code{-} characters instead. The value of a variable is the Calc
33401 object stored in its @var{sym} symbol's value cell. If the symbol's
33402 value cell is void or if it contains @code{nil}, the variable has no
33403 value. Special constants have the form @samp{(special-const
33404 @var{value})} stored in their value cell, where @var{value} is a formula
33405 which is evaluated when the constant's value is requested. Variables
33406 which represent units are not stored in any special way; they are units
33407 only because their names appear in the units table. If the value
33408 cell contains a string, it is parsed to get the variable's value when
33409 the variable is used.
33411 A Lisp list with any other symbol as the first element is a function call.
33412 The symbols @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^},
33413 and @code{|} represent special binary operators; these lists are always
33414 of the form @samp{(@var{op} @var{lhs} @var{rhs})} where @var{lhs} is the
33415 sub-formula on the lefthand side and @var{rhs} is the sub-formula on the
33416 right. The symbol @code{neg} represents unary negation; this list is always
33417 of the form @samp{(neg @var{arg})}. Any other symbol @var{func} represents a
33418 function that would be displayed in function-call notation; the symbol
33419 @var{func} is in general always of the form @samp{calcFunc-@var{name}}.
33420 The function cell of the symbol @var{func} should contain a Lisp function
33421 for evaluating a call to @var{func}. This function is passed the remaining
33422 elements of the list (themselves already evaluated) as arguments; such
33423 functions should return @code{nil} or call @code{reject-arg} to signify
33424 that they should be left in symbolic form, or they should return a Calc
33425 object which represents their value, or a list of such objects if they
33426 wish to return multiple values. (The latter case is allowed only for
33427 functions which are the outer-level call in an expression whose value is
33428 about to be pushed on the stack; this feature is considered obsolete
33429 and is not used by any built-in Calc functions.)
33431 @node Interactive Lisp Functions, Stack Lisp Functions, Data Type Formats, Internals
33432 @subsubsection Interactive Functions
33435 The functions described here are used in implementing interactive Calc
33436 commands. Note that this list is not exhaustive! If there is an
33437 existing command that behaves similarly to the one you want to define,
33438 you may find helpful tricks by checking the source code for that command.
33440 @defun calc-set-command-flag flag
33441 Set the command flag @var{flag}. This is generally a Lisp symbol, but
33442 may in fact be anything. The effect is to add @var{flag} to the list
33443 stored in the variable @code{calc-command-flags}, unless it is already
33444 there. @xref{Defining Simple Commands}.
33447 @defun calc-clear-command-flag flag
33448 If @var{flag} appears among the list of currently-set command flags,
33449 remove it from that list.
33452 @defun calc-record-undo rec
33453 Add the ``undo record'' @var{rec} to the list of steps to take if the
33454 current operation should need to be undone. Stack push and pop functions
33455 automatically call @code{calc-record-undo}, so the kinds of undo records
33456 you might need to create take the form @samp{(set @var{sym} @var{value})},
33457 which says that the Lisp variable @var{sym} was changed and had previously
33458 contained @var{value}; @samp{(store @var{var} @var{value})} which says that
33459 the Calc variable @var{var} (a string which is the name of the symbol that
33460 contains the variable's value) was stored and its previous value was
33461 @var{value} (either a Calc data object, or @code{nil} if the variable was
33462 previously void); or @samp{(eval @var{undo} @var{redo} @var{args} @dots{})},
33463 which means that to undo requires calling the function @samp{(@var{undo}
33464 @var{args} @dots{})} and, if the undo is later redone, calling
33465 @samp{(@var{redo} @var{args} @dots{})}.
33468 @defun calc-record-why msg args
33469 Record the error or warning message @var{msg}, which is normally a string.
33470 This message will be replayed if the user types @kbd{w} (@code{calc-why});
33471 if the message string begins with a @samp{*}, it is considered important
33472 enough to display even if the user doesn't type @kbd{w}. If one or more
33473 @var{args} are present, the displayed message will be of the form,
33474 @samp{@var{msg}: @var{arg1}, @var{arg2}, @dots{}}, where the arguments are
33475 formatted on the assumption that they are either strings or Calc objects of
33476 some sort. If @var{msg} is a symbol, it is the name of a Calc predicate
33477 (such as @code{integerp} or @code{numvecp}) which the arguments did not
33478 satisfy; it is expanded to a suitable string such as ``Expected an
33479 integer.'' The @code{reject-arg} function calls @code{calc-record-why}
33480 automatically; @pxref{Predicates}.
33483 @defun calc-is-inverse
33484 This predicate returns true if the current command is inverse,
33485 i.e., if the Inverse (@kbd{I} key) flag was set.
33488 @defun calc-is-hyperbolic
33489 This predicate is the analogous function for the @kbd{H} key.
33492 @node Stack Lisp Functions, Predicates, Interactive Lisp Functions, Internals
33493 @subsubsection Stack-Oriented Functions
33496 The functions described here perform various operations on the Calc
33497 stack and trail. They are to be used in interactive Calc commands.
33499 @defun calc-push-list vals n
33500 Push the Calc objects in list @var{vals} onto the stack at stack level
33501 @var{n}. If @var{n} is omitted it defaults to 1, so that the elements
33502 are pushed at the top of the stack. If @var{n} is greater than 1, the
33503 elements will be inserted into the stack so that the last element will
33504 end up at level @var{n}, the next-to-last at level @var{n}+1, etc.
33505 The elements of @var{vals} are assumed to be valid Calc objects, and
33506 are not evaluated, rounded, or renormalized in any way. If @var{vals}
33507 is an empty list, nothing happens.
33509 The stack elements are pushed without any sub-formula selections.
33510 You can give an optional third argument to this function, which must
33511 be a list the same size as @var{vals} of selections. Each selection
33512 must be @code{eq} to some sub-formula of the corresponding formula
33513 in @var{vals}, or @code{nil} if that formula should have no selection.
33516 @defun calc-top-list n m
33517 Return a list of the @var{n} objects starting at level @var{m} of the
33518 stack. If @var{m} is omitted it defaults to 1, so that the elements are
33519 taken from the top of the stack. If @var{n} is omitted, it also
33520 defaults to 1, so that the top stack element (in the form of a
33521 one-element list) is returned. If @var{m} is greater than 1, the
33522 @var{m}th stack element will be at the end of the list, the @var{m}+1st
33523 element will be next-to-last, etc. If @var{n} or @var{m} are out of
33524 range, the command is aborted with a suitable error message. If @var{n}
33525 is zero, the function returns an empty list. The stack elements are not
33526 evaluated, rounded, or renormalized.
33528 If any stack elements contain selections, and selections have not
33529 been disabled by the @kbd{j e} (@code{calc-enable-selections}) command,
33530 this function returns the selected portions rather than the entire
33531 stack elements. It can be given a third ``selection-mode'' argument
33532 which selects other behaviors. If it is the symbol @code{t}, then
33533 a selection in any of the requested stack elements produces an
33534 ``invalid operation on selections'' error. If it is the symbol @code{full},
33535 the whole stack entry is always returned regardless of selections.
33536 If it is the symbol @code{sel}, the selected portion is always returned,
33537 or @code{nil} if there is no selection. (This mode ignores the @kbd{j e}
33538 command.) If the symbol is @code{entry}, the complete stack entry in
33539 list form is returned; the first element of this list will be the whole
33540 formula, and the third element will be the selection (or @code{nil}).
33543 @defun calc-pop-stack n m
33544 Remove the specified elements from the stack. The parameters @var{n}
33545 and @var{m} are defined the same as for @code{calc-top-list}. The return
33546 value of @code{calc-pop-stack} is uninteresting.
33548 If there are any selected sub-formulas among the popped elements, and
33549 @kbd{j e} has not been used to disable selections, this produces an
33550 error without changing the stack. If you supply an optional third
33551 argument of @code{t}, the stack elements are popped even if they
33552 contain selections.
33555 @defun calc-record-list vals tag
33556 This function records one or more results in the trail. The @var{vals}
33557 are a list of strings or Calc objects. The @var{tag} is the four-character
33558 tag string to identify the values. If @var{tag} is omitted, a blank tag
33562 @defun calc-normalize n
33563 This function takes a Calc object and ``normalizes'' it. At the very
33564 least this involves re-rounding floating-point values according to the
33565 current precision and other similar jobs. Also, unless the user has
33566 selected No-Simplify mode (@pxref{Simplification Modes}), this involves
33567 actually evaluating a formula object by executing the function calls
33568 it contains, and possibly also doing algebraic simplification, etc.
33571 @defun calc-top-list-n n m
33572 This function is identical to @code{calc-top-list}, except that it calls
33573 @code{calc-normalize} on the values that it takes from the stack. They
33574 are also passed through @code{check-complete}, so that incomplete
33575 objects will be rejected with an error message. All computational
33576 commands should use this in preference to @code{calc-top-list}; the only
33577 standard Calc commands that operate on the stack without normalizing
33578 are stack management commands like @code{calc-enter} and @code{calc-roll-up}.
33579 This function accepts the same optional selection-mode argument as
33580 @code{calc-top-list}.
33583 @defun calc-top-n m
33584 This function is a convenient form of @code{calc-top-list-n} in which only
33585 a single element of the stack is taken and returned, rather than a list
33586 of elements. This also accepts an optional selection-mode argument.
33589 @defun calc-enter-result n tag vals
33590 This function is a convenient interface to most of the above functions.
33591 The @var{vals} argument should be either a single Calc object, or a list
33592 of Calc objects; the object or objects are normalized, and the top @var{n}
33593 stack entries are replaced by the normalized objects. If @var{tag} is
33594 non-@code{nil}, the normalized objects are also recorded in the trail.
33595 A typical stack-based computational command would take the form,
33598 (calc-enter-result @var{n} @var{tag} (cons 'calcFunc-@var{func}
33599 (calc-top-list-n @var{n})))
33602 If any of the @var{n} stack elements replaced contain sub-formula
33603 selections, and selections have not been disabled by @kbd{j e},
33604 this function takes one of two courses of action. If @var{n} is
33605 equal to the number of elements in @var{vals}, then each element of
33606 @var{vals} is spliced into the corresponding selection; this is what
33607 happens when you use the @key{TAB} key, or when you use a unary
33608 arithmetic operation like @code{sqrt}. If @var{vals} has only one
33609 element but @var{n} is greater than one, there must be only one
33610 selection among the top @var{n} stack elements; the element from
33611 @var{vals} is spliced into that selection. This is what happens when
33612 you use a binary arithmetic operation like @kbd{+}. Any other
33613 combination of @var{n} and @var{vals} is an error when selections
33617 @defun calc-unary-op tag func arg
33618 This function implements a unary operator that allows a numeric prefix
33619 argument to apply the operator over many stack entries. If the prefix
33620 argument @var{arg} is @code{nil}, this uses @code{calc-enter-result}
33621 as outlined above. Otherwise, it maps the function over several stack
33622 elements; @pxref{Prefix Arguments}. For example,
33625 (defun calc-zeta (arg)
33627 (calc-unary-op "zeta" 'calcFunc-zeta arg))
33631 @defun calc-binary-op tag func arg ident unary
33632 This function implements a binary operator, analogously to
33633 @code{calc-unary-op}. The optional @var{ident} and @var{unary}
33634 arguments specify the behavior when the prefix argument is zero or
33635 one, respectively. If the prefix is zero, the value @var{ident}
33636 is pushed onto the stack, if specified, otherwise an error message
33637 is displayed. If the prefix is one, the unary function @var{unary}
33638 is applied to the top stack element, or, if @var{unary} is not
33639 specified, nothing happens. When the argument is two or more,
33640 the binary function @var{func} is reduced across the top @var{arg}
33641 stack elements; when the argument is negative, the function is
33642 mapped between the next-to-top @mathit{-@var{arg}} stack elements and the
33646 @defun calc-stack-size
33647 Return the number of elements on the stack as an integer. This count
33648 does not include elements that have been temporarily hidden by stack
33649 truncation; @pxref{Truncating the Stack}.
33652 @defun calc-cursor-stack-index n
33653 Move the point to the @var{n}th stack entry. If @var{n} is zero, this
33654 will be the @samp{.} line. If @var{n} is from 1 to the current stack size,
33655 this will be the beginning of the first line of that stack entry's display.
33656 If line numbers are enabled, this will move to the first character of the
33657 line number, not the stack entry itself.
33660 @defun calc-substack-height n
33661 Return the number of lines between the beginning of the @var{n}th stack
33662 entry and the bottom of the buffer. If @var{n} is zero, this
33663 will be one (assuming no stack truncation). If all stack entries are
33664 one line long (i.e., no matrices are displayed), the return value will
33665 be equal @var{n}+1 as long as @var{n} is in range. (Note that in Big
33666 mode, the return value includes the blank lines that separate stack
33670 @defun calc-refresh
33671 Erase the @code{*Calculator*} buffer and reformat its contents from memory.
33672 This must be called after changing any parameter, such as the current
33673 display radix, which might change the appearance of existing stack
33674 entries. (During a keyboard macro invoked by the @kbd{X} key, refreshing
33675 is suppressed, but a flag is set so that the entire stack will be refreshed
33676 rather than just the top few elements when the macro finishes.)
33679 @node Predicates, Computational Lisp Functions, Stack Lisp Functions, Internals
33680 @subsubsection Predicates
33683 The functions described here are predicates, that is, they return a
33684 true/false value where @code{nil} means false and anything else means
33685 true. These predicates are expanded by @code{defmath}, for example,
33686 from @code{zerop} to @code{math-zerop}. In many cases they correspond
33687 to native Lisp functions by the same name, but are extended to cover
33688 the full range of Calc data types.
33691 Returns true if @var{x} is numerically zero, in any of the Calc data
33692 types. (Note that for some types, such as error forms and intervals,
33693 it never makes sense to return true.) In @code{defmath}, the expression
33694 @samp{(= x 0)} will automatically be converted to @samp{(math-zerop x)},
33695 and @samp{(/= x 0)} will be converted to @samp{(not (math-zerop x))}.
33699 Returns true if @var{x} is negative. This accepts negative real numbers
33700 of various types, negative HMS and date forms, and intervals in which
33701 all included values are negative. In @code{defmath}, the expression
33702 @samp{(< x 0)} will automatically be converted to @samp{(math-negp x)},
33703 and @samp{(>= x 0)} will be converted to @samp{(not (math-negp x))}.
33707 Returns true if @var{x} is positive (and non-zero). For complex
33708 numbers, none of these three predicates will return true.
33711 @defun looks-negp x
33712 Returns true if @var{x} is ``negative-looking.'' This returns true if
33713 @var{x} is a negative number, or a formula with a leading minus sign
33714 such as @samp{-a/b}. In other words, this is an object which can be
33715 made simpler by calling @code{(- @var{x})}.
33719 Returns true if @var{x} is an integer of any size.
33723 Returns true if @var{x} is a native Lisp integer.
33727 Returns true if @var{x} is a nonnegative integer of any size.
33730 @defun fixnatnump x
33731 Returns true if @var{x} is a nonnegative Lisp integer.
33734 @defun num-integerp x
33735 Returns true if @var{x} is numerically an integer, i.e., either a
33736 true integer or a float with no significant digits to the right of
33740 @defun messy-integerp x
33741 Returns true if @var{x} is numerically, but not literally, an integer.
33742 A value is @code{num-integerp} if it is @code{integerp} or
33743 @code{messy-integerp} (but it is never both at once).
33746 @defun num-natnump x
33747 Returns true if @var{x} is numerically a nonnegative integer.
33751 Returns true if @var{x} is an even integer.
33754 @defun looks-evenp x
33755 Returns true if @var{x} is an even integer, or a formula with a leading
33756 multiplicative coefficient which is an even integer.
33760 Returns true if @var{x} is an odd integer.
33764 Returns true if @var{x} is a rational number, i.e., an integer or a
33769 Returns true if @var{x} is a real number, i.e., an integer, fraction,
33770 or floating-point number.
33774 Returns true if @var{x} is a real number or HMS form.
33778 Returns true if @var{x} is a float, or a complex number, error form,
33779 interval, date form, or modulo form in which at least one component
33784 Returns true if @var{x} is a rectangular or polar complex number
33785 (but not a real number).
33788 @defun rect-complexp x
33789 Returns true if @var{x} is a rectangular complex number.
33792 @defun polar-complexp x
33793 Returns true if @var{x} is a polar complex number.
33797 Returns true if @var{x} is a real number or a complex number.
33801 Returns true if @var{x} is a real or complex number or an HMS form.
33805 Returns true if @var{x} is a vector (this simply checks if its argument
33806 is a list whose first element is the symbol @code{vec}).
33810 Returns true if @var{x} is a number or vector.
33814 Returns true if @var{x} is a matrix, i.e., a vector of one or more vectors,
33815 all of the same size.
33818 @defun square-matrixp x
33819 Returns true if @var{x} is a square matrix.
33823 Returns true if @var{x} is any numeric Calc object, including real and
33824 complex numbers, HMS forms, date forms, error forms, intervals, and
33825 modulo forms. (Note that error forms and intervals may include formulas
33826 as their components; see @code{constp} below.)
33830 Returns true if @var{x} is an object or a vector. This also accepts
33831 incomplete objects, but it rejects variables and formulas (except as
33832 mentioned above for @code{objectp}).
33836 Returns true if @var{x} is a ``primitive'' or ``atomic'' Calc object,
33837 i.e., one whose components cannot be regarded as sub-formulas. This
33838 includes variables, and all @code{objectp} types except error forms
33843 Returns true if @var{x} is constant, i.e., a real or complex number,
33844 HMS form, date form, or error form, interval, or vector all of whose
33845 components are @code{constp}.
33849 Returns true if @var{x} is numerically less than @var{y}. Returns false
33850 if @var{x} is greater than or equal to @var{y}, or if the order is
33851 undefined or cannot be determined. Generally speaking, this works
33852 by checking whether @samp{@var{x} - @var{y}} is @code{negp}. In
33853 @code{defmath}, the expression @samp{(< x y)} will automatically be
33854 converted to @samp{(lessp x y)}; expressions involving @code{>}, @code{<=},
33855 and @code{>=} are similarly converted in terms of @code{lessp}.
33859 Returns true if @var{x} comes before @var{y} in a canonical ordering
33860 of Calc objects. If @var{x} and @var{y} are both real numbers, this
33861 will be the same as @code{lessp}. But whereas @code{lessp} considers
33862 other types of objects to be unordered, @code{beforep} puts any two
33863 objects into a definite, consistent order. The @code{beforep}
33864 function is used by the @kbd{V S} vector-sorting command, and also
33865 by Calc's algebraic simplifications to put the terms of a product into
33866 canonical order: This allows @samp{x y + y x} to be simplified easily to
33871 This is the standard Lisp @code{equal} predicate; it returns true if
33872 @var{x} and @var{y} are structurally identical. This is the usual way
33873 to compare numbers for equality, but note that @code{equal} will treat
33874 0 and 0.0 as different.
33877 @defun math-equal x y
33878 Returns true if @var{x} and @var{y} are numerically equal, either because
33879 they are @code{equal}, or because their difference is @code{zerop}. In
33880 @code{defmath}, the expression @samp{(= x y)} will automatically be
33881 converted to @samp{(math-equal x y)}.
33884 @defun equal-int x n
33885 Returns true if @var{x} and @var{n} are numerically equal, where @var{n}
33886 is a fixnum which is not a multiple of 10. This will automatically be
33887 used by @code{defmath} in place of the more general @code{math-equal}
33891 @defun nearly-equal x y
33892 Returns true if @var{x} and @var{y}, as floating-point numbers, are
33893 equal except possibly in the last decimal place. For example,
33894 314.159 and 314.166 are considered nearly equal if the current
33895 precision is 6 (since they differ by 7 units), but not if the current
33896 precision is 7 (since they differ by 70 units). Most functions which
33897 use series expansions use @code{with-extra-prec} to evaluate the
33898 series with 2 extra digits of precision, then use @code{nearly-equal}
33899 to decide when the series has converged; this guards against cumulative
33900 error in the series evaluation without doing extra work which would be
33901 lost when the result is rounded back down to the current precision.
33902 In @code{defmath}, this can be written @samp{(~= @var{x} @var{y})}.
33903 The @var{x} and @var{y} can be numbers of any kind, including complex.
33906 @defun nearly-zerop x y
33907 Returns true if @var{x} is nearly zero, compared to @var{y}. This
33908 checks whether @var{x} plus @var{y} would by be @code{nearly-equal}
33909 to @var{y} itself, to within the current precision, in other words,
33910 if adding @var{x} to @var{y} would have a negligible effect on @var{y}
33911 due to roundoff error. @var{X} may be a real or complex number, but
33912 @var{y} must be real.
33916 Return true if the formula @var{x} represents a true value in
33917 Calc, not Lisp, terms. It tests if @var{x} is a non-zero number
33918 or a provably non-zero formula.
33921 @defun reject-arg val pred
33922 Abort the current function evaluation due to unacceptable argument values.
33923 This calls @samp{(calc-record-why @var{pred} @var{val})}, then signals a
33924 Lisp error which @code{normalize} will trap. The net effect is that the
33925 function call which led here will be left in symbolic form.
33928 @defun inexact-value
33929 If Symbolic mode is enabled, this will signal an error that causes
33930 @code{normalize} to leave the formula in symbolic form, with the message
33931 ``Inexact result.'' (This function has no effect when not in Symbolic mode.)
33932 Note that if your function calls @samp{(sin 5)} in Symbolic mode, the
33933 @code{sin} function will call @code{inexact-value}, which will cause your
33934 function to be left unsimplified. You may instead wish to call
33935 @samp{(normalize (list 'calcFunc-sin 5))}, which in Symbolic mode will
33936 return the formula @samp{sin(5)} to your function.
33940 This signals an error that will be reported as a floating-point overflow.
33944 This signals a floating-point underflow.
33947 @node Computational Lisp Functions, Vector Lisp Functions, Predicates, Internals
33948 @subsubsection Computational Functions
33951 The functions described here do the actual computational work of the
33952 Calculator. In addition to these, note that any function described in
33953 the main body of this manual may be called from Lisp; for example, if
33954 the documentation refers to the @code{calc-sqrt} [@code{sqrt}] command,
33955 this means @code{calc-sqrt} is an interactive stack-based square-root
33956 command and @code{sqrt} (which @code{defmath} expands to @code{calcFunc-sqrt})
33957 is the actual Lisp function for taking square roots.
33959 The functions @code{math-add}, @code{math-sub}, @code{math-mul},
33960 @code{math-div}, @code{math-mod}, and @code{math-neg} are not included
33961 in this list, since @code{defmath} allows you to write native Lisp
33962 @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, and unary @code{-},
33963 respectively, instead.
33965 @defun normalize val
33966 (Full form: @code{math-normalize}.)
33967 Reduce the value @var{val} to standard form. For example, if @var{val}
33968 is a fixnum, it will be converted to a bignum if it is too large, and
33969 if @var{val} is a bignum it will be normalized by clipping off trailing
33970 (i.e., most-significant) zero digits and converting to a fixnum if it is
33971 small. All the various data types are similarly converted to their standard
33972 forms. Variables are left alone, but function calls are actually evaluated
33973 in formulas. For example, normalizing @samp{(+ 2 (calcFunc-abs -4))} will
33976 If a function call fails, because the function is void or has the wrong
33977 number of parameters, or because it returns @code{nil} or calls
33978 @code{reject-arg} or @code{inexact-result}, @code{normalize} returns
33979 the formula still in symbolic form.
33981 If the current simplification mode is ``none'' or ``numeric arguments
33982 only,'' @code{normalize} will act appropriately. However, the more
33983 powerful simplification modes (like Algebraic Simplification) are
33984 not handled by @code{normalize}. They are handled by @code{calc-normalize},
33985 which calls @code{normalize} and possibly some other routines, such
33986 as @code{simplify} or @code{simplify-units}. Programs generally will
33987 never call @code{calc-normalize} except when popping or pushing values
33991 @defun evaluate-expr expr
33992 Replace all variables in @var{expr} that have values with their values,
33993 then use @code{normalize} to simplify the result. This is what happens
33994 when you press the @kbd{=} key interactively.
33997 @defmac with-extra-prec n body
33998 Evaluate the Lisp forms in @var{body} with precision increased by @var{n}
33999 digits. This is a macro which expands to
34003 (let ((calc-internal-prec (+ calc-internal-prec @var{n})))
34007 The surrounding call to @code{math-normalize} causes a floating-point
34008 result to be rounded down to the original precision afterwards. This
34009 is important because some arithmetic operations assume a number's
34010 mantissa contains no more digits than the current precision allows.
34013 @defun make-frac n d
34014 Build a fraction @samp{@var{n}:@var{d}}. This is equivalent to calling
34015 @samp{(normalize (list 'frac @var{n} @var{d}))}, but more efficient.
34018 @defun make-float mant exp
34019 Build a floating-point value out of @var{mant} and @var{exp}, both
34020 of which are arbitrary integers. This function will return a
34021 properly normalized float value, or signal an overflow or underflow
34022 if @var{exp} is out of range.
34025 @defun make-sdev x sigma
34026 Build an error form out of @var{x} and the absolute value of @var{sigma}.
34027 If @var{sigma} is zero, the result is the number @var{x} directly.
34028 If @var{sigma} is negative or complex, its absolute value is used.
34029 If @var{x} or @var{sigma} is not a valid type of object for use in
34030 error forms, this calls @code{reject-arg}.
34033 @defun make-intv mask lo hi
34034 Build an interval form out of @var{mask} (which is assumed to be an
34035 integer from 0 to 3), and the limits @var{lo} and @var{hi}. If
34036 @var{lo} is greater than @var{hi}, an empty interval form is returned.
34037 This calls @code{reject-arg} if @var{lo} or @var{hi} is unsuitable.
34040 @defun sort-intv mask lo hi
34041 Build an interval form, similar to @code{make-intv}, except that if
34042 @var{lo} is less than @var{hi} they are simply exchanged, and the
34043 bits of @var{mask} are swapped accordingly.
34046 @defun make-mod n m
34047 Build a modulo form out of @var{n} and the modulus @var{m}. Since modulo
34048 forms do not allow formulas as their components, if @var{n} or @var{m}
34049 is not a real number or HMS form the result will be a formula which
34050 is a call to @code{makemod}, the algebraic version of this function.
34054 Convert @var{x} to floating-point form. Integers and fractions are
34055 converted to numerically equivalent floats; components of complex
34056 numbers, vectors, HMS forms, date forms, error forms, intervals, and
34057 modulo forms are recursively floated. If the argument is a variable
34058 or formula, this calls @code{reject-arg}.
34062 Compare the numbers @var{x} and @var{y}, and return @mathit{-1} if
34063 @samp{(lessp @var{x} @var{y})}, 1 if @samp{(lessp @var{y} @var{x})},
34064 0 if @samp{(math-equal @var{x} @var{y})}, or 2 if the order is
34065 undefined or cannot be determined.
34069 Return the number of digits of integer @var{n}, effectively
34070 @samp{ceil(log10(@var{n}))}, but much more efficient. Zero is
34071 considered to have zero digits.
34074 @defun scale-int x n
34075 Shift integer @var{x} left @var{n} decimal digits, or right @mathit{-@var{n}}
34076 digits with truncation toward zero.
34079 @defun scale-rounding x n
34080 Like @code{scale-int}, except that a right shift rounds to the nearest
34081 integer rather than truncating.
34085 Return the integer @var{n} as a fixnum, i.e., a native Lisp integer.
34086 If @var{n} is outside the permissible range for Lisp integers (usually
34087 24 binary bits) the result is undefined.
34091 Compute the square of @var{x}; short for @samp{(* @var{x} @var{x})}.
34094 @defun quotient x y
34095 Divide integer @var{x} by integer @var{y}; return an integer quotient
34096 and discard the remainder. If @var{x} or @var{y} is negative, the
34097 direction of rounding is undefined.
34101 Perform an integer division; if @var{x} and @var{y} are both nonnegative
34102 integers, this uses the @code{quotient} function, otherwise it computes
34103 @samp{floor(@var{x}/@var{y})}. Thus the result is well-defined but
34104 slower than for @code{quotient}.
34108 Divide integer @var{x} by integer @var{y}; return the integer remainder
34109 and discard the quotient. Like @code{quotient}, this works only for
34110 integer arguments and is not well-defined for negative arguments.
34111 For a more well-defined result, use @samp{(% @var{x} @var{y})}.
34115 Divide integer @var{x} by integer @var{y}; return a cons cell whose
34116 @code{car} is @samp{(quotient @var{x} @var{y})} and whose @code{cdr}
34117 is @samp{(imod @var{x} @var{y})}.
34121 Compute @var{x} to the power @var{y}. In @code{defmath} code, this can
34122 also be written @samp{(^ @var{x} @var{y})} or
34123 @w{@samp{(expt @var{x} @var{y})}}.
34126 @defun abs-approx x
34127 Compute a fast approximation to the absolute value of @var{x}. For
34128 example, for a rectangular complex number the result is the sum of
34129 the absolute values of the components.
34133 @findex gamma-const
34139 @findex pi-over-180
34140 @findex sqrt-two-pi
34144 The function @samp{(pi)} computes @samp{pi} to the current precision.
34145 Other related constant-generating functions are @code{two-pi},
34146 @code{pi-over-2}, @code{pi-over-4}, @code{pi-over-180}, @code{sqrt-two-pi},
34147 @code{e}, @code{sqrt-e}, @code{ln-2}, @code{ln-10}, @code{phi} and
34148 @code{gamma-const}. Each function returns a floating-point value in the
34149 current precision, and each uses caching so that all calls after the
34150 first are essentially free.
34153 @defmac math-defcache @var{func} @var{initial} @var{form}
34154 This macro, usually used as a top-level call like @code{defun} or
34155 @code{defvar}, defines a new cached constant analogous to @code{pi}, etc.
34156 It defines a function @code{func} which returns the requested value;
34157 if @var{initial} is non-@code{nil} it must be a @samp{(float @dots{})}
34158 form which serves as an initial value for the cache. If @var{func}
34159 is called when the cache is empty or does not have enough digits to
34160 satisfy the current precision, the Lisp expression @var{form} is evaluated
34161 with the current precision increased by four, and the result minus its
34162 two least significant digits is stored in the cache. For example,
34163 calling @samp{(pi)} with a precision of 30 computes @samp{pi} to 34
34164 digits, rounds it down to 32 digits for future use, then rounds it
34165 again to 30 digits for use in the present request.
34168 @findex half-circle
34169 @findex quarter-circle
34170 @defun full-circle symb
34171 If the current angular mode is Degrees or HMS, this function returns the
34172 integer 360. In Radians mode, this function returns either the
34173 corresponding value in radians to the current precision, or the formula
34174 @samp{2*pi}, depending on the Symbolic mode. There are also similar
34175 function @code{half-circle} and @code{quarter-circle}.
34178 @defun power-of-2 n
34179 Compute two to the integer power @var{n}, as a (potentially very large)
34180 integer. Powers of two are cached, so only the first call for a
34181 particular @var{n} is expensive.
34184 @defun integer-log2 n
34185 Compute the base-2 logarithm of @var{n}, which must be an integer which
34186 is a power of two. If @var{n} is not a power of two, this function will
34190 @defun div-mod a b m
34191 Divide @var{a} by @var{b}, modulo @var{m}. This returns @code{nil} if
34192 there is no solution, or if any of the arguments are not integers.
34195 @defun pow-mod a b m
34196 Compute @var{a} to the power @var{b}, modulo @var{m}. If @var{a},
34197 @var{b}, and @var{m} are integers, this uses an especially efficient
34198 algorithm. Otherwise, it simply computes @samp{(% (^ a b) m)}.
34202 Compute the integer square root of @var{n}. This is the square root
34203 of @var{n} rounded down toward zero, i.e., @samp{floor(sqrt(@var{n}))}.
34204 If @var{n} is itself an integer, the computation is especially efficient.
34207 @defun to-hms a ang
34208 Convert the argument @var{a} into an HMS form. If @var{ang} is specified,
34209 it is the angular mode in which to interpret @var{a}, either @code{deg}
34210 or @code{rad}. Otherwise, the current angular mode is used. If @var{a}
34211 is already an HMS form it is returned as-is.
34214 @defun from-hms a ang
34215 Convert the HMS form @var{a} into a real number. If @var{ang} is specified,
34216 it is the angular mode in which to express the result, otherwise the
34217 current angular mode is used. If @var{a} is already a real number, it
34221 @defun to-radians a
34222 Convert the number or HMS form @var{a} to radians from the current
34226 @defun from-radians a
34227 Convert the number @var{a} from radians to the current angular mode.
34228 If @var{a} is a formula, this returns the formula @samp{deg(@var{a})}.
34231 @defun to-radians-2 a
34232 Like @code{to-radians}, except that in Symbolic mode a degrees to
34233 radians conversion yields a formula like @samp{@var{a}*pi/180}.
34236 @defun from-radians-2 a
34237 Like @code{from-radians}, except that in Symbolic mode a radians to
34238 degrees conversion yields a formula like @samp{@var{a}*180/pi}.
34241 @defun random-digit
34242 Produce a random base-1000 digit in the range 0 to 999.
34245 @defun random-digits n
34246 Produce a random @var{n}-digit integer; this will be an integer
34247 in the interval @samp{[0, 10^@var{n})}.
34250 @defun random-float
34251 Produce a random float in the interval @samp{[0, 1)}.
34254 @defun prime-test n iters
34255 Determine whether the integer @var{n} is prime. Return a list which has
34256 one of these forms: @samp{(nil @var{f})} means the number is non-prime
34257 because it was found to be divisible by @var{f}; @samp{(nil)} means it
34258 was found to be non-prime by table look-up (so no factors are known);
34259 @samp{(nil unknown)} means it is definitely non-prime but no factors
34260 are known because @var{n} was large enough that Fermat's probabilistic
34261 test had to be used; @samp{(t)} means the number is definitely prime;
34262 and @samp{(maybe @var{i} @var{p})} means that Fermat's test, after @var{i}
34263 iterations, is @var{p} percent sure that the number is prime. The
34264 @var{iters} parameter is the number of Fermat iterations to use, in the
34265 case that this is necessary. If @code{prime-test} returns ``maybe,''
34266 you can call it again with the same @var{n} to get a greater certainty;
34267 @code{prime-test} remembers where it left off.
34270 @defun to-simple-fraction f
34271 If @var{f} is a floating-point number which can be represented exactly
34272 as a small rational number. return that number, else return @var{f}.
34273 For example, 0.75 would be converted to 3:4. This function is very
34277 @defun to-fraction f tol
34278 Find a rational approximation to floating-point number @var{f} to within
34279 a specified tolerance @var{tol}; this corresponds to the algebraic
34280 function @code{frac}, and can be rather slow.
34283 @defun quarter-integer n
34284 If @var{n} is an integer or integer-valued float, this function
34285 returns zero. If @var{n} is a half-integer (i.e., an integer plus
34286 @mathit{1:2} or 0.5), it returns 2. If @var{n} is a quarter-integer,
34287 it returns 1 or 3. If @var{n} is anything else, this function
34288 returns @code{nil}.
34291 @node Vector Lisp Functions, Symbolic Lisp Functions, Computational Lisp Functions, Internals
34292 @subsubsection Vector Functions
34295 The functions described here perform various operations on vectors and
34298 @defun math-concat x y
34299 Do a vector concatenation; this operation is written @samp{@var{x} | @var{y}}
34300 in a symbolic formula. @xref{Building Vectors}.
34303 @defun vec-length v
34304 Return the length of vector @var{v}. If @var{v} is not a vector, the
34305 result is zero. If @var{v} is a matrix, this returns the number of
34306 rows in the matrix.
34309 @defun mat-dimens m
34310 Determine the dimensions of vector or matrix @var{m}. If @var{m} is not
34311 a vector, the result is an empty list. If @var{m} is a plain vector
34312 but not a matrix, the result is a one-element list containing the length
34313 of the vector. If @var{m} is a matrix with @var{r} rows and @var{c} columns,
34314 the result is the list @samp{(@var{r} @var{c})}. Higher-order tensors
34315 produce lists of more than two dimensions. Note that the object
34316 @samp{[[1, 2, 3], [4, 5]]} is a vector of vectors not all the same size,
34317 and is treated by this and other Calc routines as a plain vector of two
34321 @defun dimension-error
34322 Abort the current function with a message of ``Dimension error.''
34323 The Calculator will leave the function being evaluated in symbolic
34324 form; this is really just a special case of @code{reject-arg}.
34327 @defun build-vector args
34328 Return a Calc vector with @var{args} as elements.
34329 For example, @samp{(build-vector 1 2 3)} returns the Calc vector
34330 @samp{[1, 2, 3]}, stored internally as the list @samp{(vec 1 2 3)}.
34333 @defun make-vec obj dims
34334 Return a Calc vector or matrix all of whose elements are equal to
34335 @var{obj}. For example, @samp{(make-vec 27 3 4)} returns a 3x4 matrix
34339 @defun row-matrix v
34340 If @var{v} is a plain vector, convert it into a row matrix, i.e.,
34341 a matrix whose single row is @var{v}. If @var{v} is already a matrix,
34345 @defun col-matrix v
34346 If @var{v} is a plain vector, convert it into a column matrix, i.e., a
34347 matrix with each element of @var{v} as a separate row. If @var{v} is
34348 already a matrix, leave it alone.
34352 Map the Lisp function @var{f} over the Calc vector @var{v}. For example,
34353 @samp{(map-vec 'math-floor v)} returns a vector of the floored components
34357 @defun map-vec-2 f a b
34358 Map the Lisp function @var{f} over the two vectors @var{a} and @var{b}.
34359 If @var{a} and @var{b} are vectors of equal length, the result is a
34360 vector of the results of calling @samp{(@var{f} @var{ai} @var{bi})}
34361 for each pair of elements @var{ai} and @var{bi}. If either @var{a} or
34362 @var{b} is a scalar, it is matched with each value of the other vector.
34363 For example, @samp{(map-vec-2 'math-add v 1)} returns the vector @var{v}
34364 with each element increased by one. Note that using @samp{'+} would not
34365 work here, since @code{defmath} does not expand function names everywhere,
34366 just where they are in the function position of a Lisp expression.
34369 @defun reduce-vec f v
34370 Reduce the function @var{f} over the vector @var{v}. For example, if
34371 @var{v} is @samp{[10, 20, 30, 40]}, this calls @samp{(f (f (f 10 20) 30) 40)}.
34372 If @var{v} is a matrix, this reduces over the rows of @var{v}.
34375 @defun reduce-cols f m
34376 Reduce the function @var{f} over the columns of matrix @var{m}. For
34377 example, if @var{m} is @samp{[[1, 2], [3, 4], [5, 6]]}, the result
34378 is a vector of the two elements @samp{(f (f 1 3) 5)} and @samp{(f (f 2 4) 6)}.
34382 Return the @var{n}th row of matrix @var{m}. This is equivalent to
34383 @samp{(elt m n)}. For a slower but safer version, use @code{mrow}.
34384 (@xref{Extracting Elements}.)
34388 Return the @var{n}th column of matrix @var{m}, in the form of a vector.
34389 The arguments are not checked for correctness.
34392 @defun mat-less-row m n
34393 Return a copy of matrix @var{m} with its @var{n}th row deleted. The
34394 number @var{n} must be in range from 1 to the number of rows in @var{m}.
34397 @defun mat-less-col m n
34398 Return a copy of matrix @var{m} with its @var{n}th column deleted.
34402 Return the transpose of matrix @var{m}.
34405 @defun flatten-vector v
34406 Flatten nested vector @var{v} into a vector of scalars. For example,
34407 if @var{v} is @samp{[[1, 2, 3], [4, 5]]} the result is @samp{[1, 2, 3, 4, 5]}.
34410 @defun copy-matrix m
34411 If @var{m} is a matrix, return a copy of @var{m}. This maps
34412 @code{copy-sequence} over the rows of @var{m}; in Lisp terms, each
34413 element of the result matrix will be @code{eq} to the corresponding
34414 element of @var{m}, but none of the @code{cons} cells that make up
34415 the structure of the matrix will be @code{eq}. If @var{m} is a plain
34416 vector, this is the same as @code{copy-sequence}.
34419 @defun swap-rows m r1 r2
34420 Exchange rows @var{r1} and @var{r2} of matrix @var{m} in-place. In
34421 other words, unlike most of the other functions described here, this
34422 function changes @var{m} itself rather than building up a new result
34423 matrix. The return value is @var{m}, i.e., @samp{(eq (swap-rows m 1 2) m)}
34424 is true, with the side effect of exchanging the first two rows of
34428 @node Symbolic Lisp Functions, Formatting Lisp Functions, Vector Lisp Functions, Internals
34429 @subsubsection Symbolic Functions
34432 The functions described here operate on symbolic formulas in the
34435 @defun calc-prepare-selection num
34436 Prepare a stack entry for selection operations. If @var{num} is
34437 omitted, the stack entry containing the cursor is used; otherwise,
34438 it is the number of the stack entry to use. This function stores
34439 useful information about the current stack entry into a set of
34440 variables. @code{calc-selection-cache-num} contains the number of
34441 the stack entry involved (equal to @var{num} if you specified it);
34442 @code{calc-selection-cache-entry} contains the stack entry as a
34443 list (such as @code{calc-top-list} would return with @code{entry}
34444 as the selection mode); and @code{calc-selection-cache-comp} contains
34445 a special ``tagged'' composition (@pxref{Formatting Lisp Functions})
34446 which allows Calc to relate cursor positions in the buffer with
34447 their corresponding sub-formulas.
34449 A slight complication arises in the selection mechanism because
34450 formulas may contain small integers. For example, in the vector
34451 @samp{[1, 2, 1]} the first and last elements are @code{eq} to each
34452 other; selections are recorded as the actual Lisp object that
34453 appears somewhere in the tree of the whole formula, but storing
34454 @code{1} would falsely select both @code{1}'s in the vector. So
34455 @code{calc-prepare-selection} also checks the stack entry and
34456 replaces any plain integers with ``complex number'' lists of the form
34457 @samp{(cplx @var{n} 0)}. This list will be displayed the same as a
34458 plain @var{n} and the change will be completely invisible to the
34459 user, but it will guarantee that no two sub-formulas of the stack
34460 entry will be @code{eq} to each other. Next time the stack entry
34461 is involved in a computation, @code{calc-normalize} will replace
34462 these lists with plain numbers again, again invisibly to the user.
34465 @defun calc-encase-atoms x
34466 This modifies the formula @var{x} to ensure that each part of the
34467 formula is a unique atom, using the @samp{(cplx @var{n} 0)} trick
34468 described above. This function may use @code{setcar} to modify
34469 the formula in-place.
34472 @defun calc-find-selected-part
34473 Find the smallest sub-formula of the current formula that contains
34474 the cursor. This assumes @code{calc-prepare-selection} has been
34475 called already. If the cursor is not actually on any part of the
34476 formula, this returns @code{nil}.
34479 @defun calc-change-current-selection selection
34480 Change the currently prepared stack element's selection to
34481 @var{selection}, which should be @code{eq} to some sub-formula
34482 of the stack element, or @code{nil} to unselect the formula.
34483 The stack element's appearance in the Calc buffer is adjusted
34484 to reflect the new selection.
34487 @defun calc-find-nth-part expr n
34488 Return the @var{n}th sub-formula of @var{expr}. This function is used
34489 by the selection commands, and (unless @kbd{j b} has been used) treats
34490 sums and products as flat many-element formulas. Thus if @var{expr}
34491 is @samp{((a + b) - c) + d}, calling @code{calc-find-nth-part} with
34492 @var{n} equal to four will return @samp{d}.
34495 @defun calc-find-parent-formula expr part
34496 Return the sub-formula of @var{expr} which immediately contains
34497 @var{part}. If @var{expr} is @samp{a*b + (c+1)*d} and @var{part}
34498 is @code{eq} to the @samp{c+1} term of @var{expr}, then this function
34499 will return @samp{(c+1)*d}. If @var{part} turns out not to be a
34500 sub-formula of @var{expr}, the function returns @code{nil}. If
34501 @var{part} is @code{eq} to @var{expr}, the function returns @code{t}.
34502 This function does not take associativity into account.
34505 @defun calc-find-assoc-parent-formula expr part
34506 This is the same as @code{calc-find-parent-formula}, except that
34507 (unless @kbd{j b} has been used) it continues widening the selection
34508 to contain a complete level of the formula. Given @samp{a} from
34509 @samp{((a + b) - c) + d}, @code{calc-find-parent-formula} will
34510 return @samp{a + b} but @code{calc-find-assoc-parent-formula} will
34511 return the whole expression.
34514 @defun calc-grow-assoc-formula expr part
34515 This expands sub-formula @var{part} of @var{expr} to encompass a
34516 complete level of the formula. If @var{part} and its immediate
34517 parent are not compatible associative operators, or if @kbd{j b}
34518 has been used, this simply returns @var{part}.
34521 @defun calc-find-sub-formula expr part
34522 This finds the immediate sub-formula of @var{expr} which contains
34523 @var{part}. It returns an index @var{n} such that
34524 @samp{(calc-find-nth-part @var{expr} @var{n})} would return @var{part}.
34525 If @var{part} is not a sub-formula of @var{expr}, it returns @code{nil}.
34526 If @var{part} is @code{eq} to @var{expr}, it returns @code{t}. This
34527 function does not take associativity into account.
34530 @defun calc-replace-sub-formula expr old new
34531 This function returns a copy of formula @var{expr}, with the
34532 sub-formula that is @code{eq} to @var{old} replaced by @var{new}.
34535 @defun simplify expr
34536 Simplify the expression @var{expr} by applying Calc's algebraic
34537 simplifications. This always returns a copy of the expression; the
34538 structure @var{expr} points to remains unchanged in memory.
34540 More precisely, here is what @code{simplify} does: The expression is
34541 first normalized and evaluated by calling @code{normalize}. If any
34542 @code{AlgSimpRules} have been defined, they are then applied. Then
34543 the expression is traversed in a depth-first, bottom-up fashion; at
34544 each level, any simplifications that can be made are made until no
34545 further changes are possible. Once the entire formula has been
34546 traversed in this way, it is compared with the original formula (from
34547 before the call to @code{normalize}) and, if it has changed,
34548 the entire procedure is repeated (starting with @code{normalize})
34549 until no further changes occur. Usually only two iterations are
34550 needed: one to simplify the formula, and another to verify that no
34551 further simplifications were possible.
34554 @defun simplify-extended expr
34555 Simplify the expression @var{expr}, with additional rules enabled that
34556 help do a more thorough job, while not being entirely ``safe'' in all
34557 circumstances. (For example, this mode will simplify @samp{sqrt(x^2)}
34558 to @samp{x}, which is only valid when @var{x} is positive.) This is
34559 implemented by temporarily binding the variable @code{math-living-dangerously}
34560 to @code{t} (using a @code{let} form) and calling @code{simplify}.
34561 Dangerous simplification rules are written to check this variable
34562 before taking any action.
34565 @defun simplify-units expr
34566 Simplify the expression @var{expr}, treating variable names as units
34567 whenever possible. This works by binding the variable
34568 @code{math-simplifying-units} to @code{t} while calling @code{simplify}.
34571 @defmac math-defsimplify funcs body
34572 Register a new simplification rule; this is normally called as a top-level
34573 form, like @code{defun} or @code{defmath}. If @var{funcs} is a symbol
34574 (like @code{+} or @code{calcFunc-sqrt}), this simplification rule is
34575 applied to the formulas which are calls to the specified function. Or,
34576 @var{funcs} can be a list of such symbols; the rule applies to all
34577 functions on the list. The @var{body} is written like the body of a
34578 function with a single argument called @code{expr}. The body will be
34579 executed with @code{expr} bound to a formula which is a call to one of
34580 the functions @var{funcs}. If the function body returns @code{nil}, or
34581 if it returns a result @code{equal} to the original @code{expr}, it is
34582 ignored and Calc goes on to try the next simplification rule that applies.
34583 If the function body returns something different, that new formula is
34584 substituted for @var{expr} in the original formula.
34586 At each point in the formula, rules are tried in the order of the
34587 original calls to @code{math-defsimplify}; the search stops after the
34588 first rule that makes a change. Thus later rules for that same
34589 function will not have a chance to trigger until the next iteration
34590 of the main @code{simplify} loop.
34592 Note that, since @code{defmath} is not being used here, @var{body} must
34593 be written in true Lisp code without the conveniences that @code{defmath}
34594 provides. If you prefer, you can have @var{body} simply call another
34595 function (defined with @code{defmath}) which does the real work.
34597 The arguments of a function call will already have been simplified
34598 before any rules for the call itself are invoked. Since a new argument
34599 list is consed up when this happens, this means that the rule's body is
34600 allowed to rearrange the function's arguments destructively if that is
34601 convenient. Here is a typical example of a simplification rule:
34604 (math-defsimplify calcFunc-arcsinh
34605 (or (and (math-looks-negp (nth 1 expr))
34606 (math-neg (list 'calcFunc-arcsinh
34607 (math-neg (nth 1 expr)))))
34608 (and (eq (car-safe (nth 1 expr)) 'calcFunc-sinh)
34609 (or math-living-dangerously
34610 (math-known-realp (nth 1 (nth 1 expr))))
34611 (nth 1 (nth 1 expr)))))
34614 This is really a pair of rules written with one @code{math-defsimplify}
34615 for convenience; the first replaces @samp{arcsinh(-x)} with
34616 @samp{-arcsinh(x)}, and the second, which is safe only for real @samp{x},
34617 replaces @samp{arcsinh(sinh(x))} with @samp{x}.
34620 @defun common-constant-factor expr
34621 Check @var{expr} to see if it is a sum of terms all multiplied by the
34622 same rational value. If so, return this value. If not, return @code{nil}.
34623 For example, if called on @samp{6x + 9y + 12z}, it would return 3, since
34624 3 is a common factor of all the terms.
34627 @defun cancel-common-factor expr factor
34628 Assuming @var{expr} is a sum with @var{factor} as a common factor,
34629 divide each term of the sum by @var{factor}. This is done by
34630 destructively modifying parts of @var{expr}, on the assumption that
34631 it is being used by a simplification rule (where such things are
34632 allowed; see above). For example, consider this built-in rule for
34636 (math-defsimplify calcFunc-sqrt
34637 (let ((fac (math-common-constant-factor (nth 1 expr))))
34638 (and fac (not (eq fac 1))
34639 (math-mul (math-normalize (list 'calcFunc-sqrt fac))
34641 (list 'calcFunc-sqrt
34642 (math-cancel-common-factor
34643 (nth 1 expr) fac)))))))
34647 @defun frac-gcd a b
34648 Compute a ``rational GCD'' of @var{a} and @var{b}, which must both be
34649 rational numbers. This is the fraction composed of the GCD of the
34650 numerators of @var{a} and @var{b}, over the GCD of the denominators.
34651 It is used by @code{common-constant-factor}. Note that the standard
34652 @code{gcd} function uses the LCM to combine the denominators.
34655 @defun map-tree func expr many
34656 Try applying Lisp function @var{func} to various sub-expressions of
34657 @var{expr}. Initially, call @var{func} with @var{expr} itself as an
34658 argument. If this returns an expression which is not @code{equal} to
34659 @var{expr}, apply @var{func} again until eventually it does return
34660 @var{expr} with no changes. Then, if @var{expr} is a function call,
34661 recursively apply @var{func} to each of the arguments. This keeps going
34662 until no changes occur anywhere in the expression; this final expression
34663 is returned by @code{map-tree}. Note that, unlike simplification rules,
34664 @var{func} functions may @emph{not} make destructive changes to
34665 @var{expr}. If a third argument @var{many} is provided, it is an
34666 integer which says how many times @var{func} may be applied; the
34667 default, as described above, is infinitely many times.
34670 @defun compile-rewrites rules
34671 Compile the rewrite rule set specified by @var{rules}, which should
34672 be a formula that is either a vector or a variable name. If the latter,
34673 the compiled rules are saved so that later @code{compile-rules} calls
34674 for that same variable can return immediately. If there are problems
34675 with the rules, this function calls @code{error} with a suitable
34679 @defun apply-rewrites expr crules heads
34680 Apply the compiled rewrite rule set @var{crules} to the expression
34681 @var{expr}. This will make only one rewrite and only checks at the
34682 top level of the expression. The result @code{nil} if no rules
34683 matched, or if the only rules that matched did not actually change
34684 the expression. The @var{heads} argument is optional; if is given,
34685 it should be a list of all function names that (may) appear in
34686 @var{expr}. The rewrite compiler tags each rule with the
34687 rarest-looking function name in the rule; if you specify @var{heads},
34688 @code{apply-rewrites} can use this information to narrow its search
34689 down to just a few rules in the rule set.
34692 @defun rewrite-heads expr
34693 Compute a @var{heads} list for @var{expr} suitable for use with
34694 @code{apply-rewrites}, as discussed above.
34697 @defun rewrite expr rules many
34698 This is an all-in-one rewrite function. It compiles the rule set
34699 specified by @var{rules}, then uses @code{map-tree} to apply the
34700 rules throughout @var{expr} up to @var{many} (default infinity)
34704 @defun match-patterns pat vec not-flag
34705 Given a Calc vector @var{vec} and an uncompiled pattern set or
34706 pattern set variable @var{pat}, this function returns a new vector
34707 of all elements of @var{vec} which do (or don't, if @var{not-flag} is
34708 non-@code{nil}) match any of the patterns in @var{pat}.
34711 @defun deriv expr var value symb
34712 Compute the derivative of @var{expr} with respect to variable @var{var}
34713 (which may actually be any sub-expression). If @var{value} is specified,
34714 the derivative is evaluated at the value of @var{var}; otherwise, the
34715 derivative is left in terms of @var{var}. If the expression contains
34716 functions for which no derivative formula is known, new derivative
34717 functions are invented by adding primes to the names; @pxref{Calculus}.
34718 However, if @var{symb} is non-@code{nil}, the presence of nondifferentiable
34719 functions in @var{expr} instead cancels the whole differentiation, and
34720 @code{deriv} returns @code{nil} instead.
34722 Derivatives of an @var{n}-argument function can be defined by
34723 adding a @code{math-derivative-@var{n}} property to the property list
34724 of the symbol for the function's derivative, which will be the
34725 function name followed by an apostrophe. The value of the property
34726 should be a Lisp function; it is called with the same arguments as the
34727 original function call that is being differentiated. It should return
34728 a formula for the derivative. For example, the derivative of @code{ln}
34732 (put 'calcFunc-ln\' 'math-derivative-1
34733 (function (lambda (u) (math-div 1 u))))
34736 The two-argument @code{log} function has two derivatives,
34738 (put 'calcFunc-log\' 'math-derivative-2 ; d(log(x,b)) / dx
34739 (function (lambda (x b) ... )))
34740 (put 'calcFunc-log\'2 'math-derivative-2 ; d(log(x,b)) / db
34741 (function (lambda (x b) ... )))
34745 @defun tderiv expr var value symb
34746 Compute the total derivative of @var{expr}. This is the same as
34747 @code{deriv}, except that variables other than @var{var} are not
34748 assumed to be constant with respect to @var{var}.
34751 @defun integ expr var low high
34752 Compute the integral of @var{expr} with respect to @var{var}.
34753 @xref{Calculus}, for further details.
34756 @defmac math-defintegral funcs body
34757 Define a rule for integrating a function or functions of one argument;
34758 this macro is very similar in format to @code{math-defsimplify}.
34759 The main difference is that here @var{body} is the body of a function
34760 with a single argument @code{u} which is bound to the argument to the
34761 function being integrated, not the function call itself. Also, the
34762 variable of integration is available as @code{math-integ-var}. If
34763 evaluation of the integral requires doing further integrals, the body
34764 should call @samp{(math-integral @var{x})} to find the integral of
34765 @var{x} with respect to @code{math-integ-var}; this function returns
34766 @code{nil} if the integral could not be done. Some examples:
34769 (math-defintegral calcFunc-conj
34770 (let ((int (math-integral u)))
34772 (list 'calcFunc-conj int))))
34774 (math-defintegral calcFunc-cos
34775 (and (equal u math-integ-var)
34776 (math-from-radians-2 (list 'calcFunc-sin u))))
34779 In the @code{cos} example, we define only the integral of @samp{cos(x) dx},
34780 relying on the general integration-by-substitution facility to handle
34781 cosines of more complicated arguments. An integration rule should return
34782 @code{nil} if it can't do the integral; if several rules are defined for
34783 the same function, they are tried in order until one returns a non-@code{nil}
34787 @defmac math-defintegral-2 funcs body
34788 Define a rule for integrating a function or functions of two arguments.
34789 This is exactly analogous to @code{math-defintegral}, except that @var{body}
34790 is written as the body of a function with two arguments, @var{u} and
34794 @defun solve-for lhs rhs var full
34795 Attempt to solve the equation @samp{@var{lhs} = @var{rhs}} by isolating
34796 the variable @var{var} on the lefthand side; return the resulting righthand
34797 side, or @code{nil} if the equation cannot be solved. The variable
34798 @var{var} must appear at least once in @var{lhs} or @var{rhs}. Note that
34799 the return value is a formula which does not contain @var{var}; this is
34800 different from the user-level @code{solve} and @code{finv} functions,
34801 which return a rearranged equation or a functional inverse, respectively.
34802 If @var{full} is non-@code{nil}, a full solution including dummy signs
34803 and dummy integers will be produced. User-defined inverses are provided
34804 as properties in a manner similar to derivatives:
34807 (put 'calcFunc-ln 'math-inverse
34808 (function (lambda (x) (list 'calcFunc-exp x))))
34811 This function can call @samp{(math-solve-get-sign @var{x})} to create
34812 a new arbitrary sign variable, returning @var{x} times that sign, and
34813 @samp{(math-solve-get-int @var{x})} to create a new arbitrary integer
34814 variable multiplied by @var{x}. These functions simply return @var{x}
34815 if the caller requested a non-``full'' solution.
34818 @defun solve-eqn expr var full
34819 This version of @code{solve-for} takes an expression which will
34820 typically be an equation or inequality. (If it is not, it will be
34821 interpreted as the equation @samp{@var{expr} = 0}.) It returns an
34822 equation or inequality, or @code{nil} if no solution could be found.
34825 @defun solve-system exprs vars full
34826 This function solves a system of equations. Generally, @var{exprs}
34827 and @var{vars} will be vectors of equal length.
34828 @xref{Solving Systems of Equations}, for other options.
34831 @defun expr-contains expr var
34832 Returns a non-@code{nil} value if @var{var} occurs as a subexpression
34835 This function might seem at first to be identical to
34836 @code{calc-find-sub-formula}. The key difference is that
34837 @code{expr-contains} uses @code{equal} to test for matches, whereas
34838 @code{calc-find-sub-formula} uses @code{eq}. In the formula
34839 @samp{f(a, a)}, the two @samp{a}s will be @code{equal} but not
34840 @code{eq} to each other.
34843 @defun expr-contains-count expr var
34844 Returns the number of occurrences of @var{var} as a subexpression
34845 of @var{expr}, or @code{nil} if there are no occurrences.
34848 @defun expr-depends expr var
34849 Returns true if @var{expr} refers to any variable the occurs in @var{var}.
34850 In other words, it checks if @var{expr} and @var{var} have any variables
34854 @defun expr-contains-vars expr
34855 Return true if @var{expr} contains any variables, or @code{nil} if @var{expr}
34856 contains only constants and functions with constant arguments.
34859 @defun expr-subst expr old new
34860 Returns a copy of @var{expr}, with all occurrences of @var{old} replaced
34861 by @var{new}. This treats @code{lambda} forms specially with respect
34862 to the dummy argument variables, so that the effect is always to return
34863 @var{expr} evaluated at @var{old} = @var{new}.
34866 @defun multi-subst expr old new
34867 This is like @code{expr-subst}, except that @var{old} and @var{new}
34868 are lists of expressions to be substituted simultaneously. If one
34869 list is shorter than the other, trailing elements of the longer list
34873 @defun expr-weight expr
34874 Returns the ``weight'' of @var{expr}, basically a count of the total
34875 number of objects and function calls that appear in @var{expr}. For
34876 ``primitive'' objects, this will be one.
34879 @defun expr-height expr
34880 Returns the ``height'' of @var{expr}, which is the deepest level to
34881 which function calls are nested. (Note that @samp{@var{a} + @var{b}}
34882 counts as a function call.) For primitive objects, this returns zero.
34885 @defun polynomial-p expr var
34886 Check if @var{expr} is a polynomial in variable (or sub-expression)
34887 @var{var}. If so, return the degree of the polynomial, that is, the
34888 highest power of @var{var} that appears in @var{expr}. For example,
34889 for @samp{(x^2 + 3)^3 + 4} this would return 6. This function returns
34890 @code{nil} unless @var{expr}, when expanded out by @kbd{a x}
34891 (@code{calc-expand}), would consist of a sum of terms in which @var{var}
34892 appears only raised to nonnegative integer powers. Note that if
34893 @var{var} does not occur in @var{expr}, then @var{expr} is considered
34894 a polynomial of degree 0.
34897 @defun is-polynomial expr var degree loose
34898 Check if @var{expr} is a polynomial in variable or sub-expression
34899 @var{var}, and, if so, return a list representation of the polynomial
34900 where the elements of the list are coefficients of successive powers of
34901 @var{var}: @samp{@var{a} + @var{b} x + @var{c} x^3} would produce the
34902 list @samp{(@var{a} @var{b} 0 @var{c})}, and @samp{(x + 1)^2} would
34903 produce the list @samp{(1 2 1)}. The highest element of the list will
34904 be non-zero, with the special exception that if @var{expr} is the
34905 constant zero, the returned value will be @samp{(0)}. Return @code{nil}
34906 if @var{expr} is not a polynomial in @var{var}. If @var{degree} is
34907 specified, this will not consider polynomials of degree higher than that
34908 value. This is a good precaution because otherwise an input of
34909 @samp{(x+1)^1000} will cause a huge coefficient list to be built. If
34910 @var{loose} is non-@code{nil}, then a looser definition of a polynomial
34911 is used in which coefficients are no longer required not to depend on
34912 @var{var}, but are only required not to take the form of polynomials
34913 themselves. For example, @samp{sin(x) x^2 + cos(x)} is a loose
34914 polynomial with coefficients @samp{((calcFunc-cos x) 0 (calcFunc-sin
34915 x))}. The result will never be @code{nil} in loose mode, since any
34916 expression can be interpreted as a ``constant'' loose polynomial.
34919 @defun polynomial-base expr pred
34920 Check if @var{expr} is a polynomial in any variable that occurs in it;
34921 if so, return that variable. (If @var{expr} is a multivariate polynomial,
34922 this chooses one variable arbitrarily.) If @var{pred} is specified, it should
34923 be a Lisp function which is called as @samp{(@var{pred} @var{subexpr})},
34924 and which should return true if @code{mpb-top-expr} (a global name for
34925 the original @var{expr}) is a suitable polynomial in @var{subexpr}.
34926 The default predicate uses @samp{(polynomial-p mpb-top-expr @var{subexpr})};
34927 you can use @var{pred} to specify additional conditions. Or, you could
34928 have @var{pred} build up a list of every suitable @var{subexpr} that
34932 @defun poly-simplify poly
34933 Simplify polynomial coefficient list @var{poly} by (destructively)
34934 clipping off trailing zeros.
34937 @defun poly-mix a ac b bc
34938 Mix two polynomial lists @var{a} and @var{b} (in the form returned by
34939 @code{is-polynomial}) in a linear combination with coefficient expressions
34940 @var{ac} and @var{bc}. The result is a (not necessarily simplified)
34941 polynomial list representing @samp{@var{ac} @var{a} + @var{bc} @var{b}}.
34944 @defun poly-mul a b
34945 Multiply two polynomial coefficient lists @var{a} and @var{b}. The
34946 result will be in simplified form if the inputs were simplified.
34949 @defun build-polynomial-expr poly var
34950 Construct a Calc formula which represents the polynomial coefficient
34951 list @var{poly} applied to variable @var{var}. The @kbd{a c}
34952 (@code{calc-collect}) command uses @code{is-polynomial} to turn an
34953 expression into a coefficient list, then @code{build-polynomial-expr}
34954 to turn the list back into an expression in regular form.
34957 @defun check-unit-name var
34958 Check if @var{var} is a variable which can be interpreted as a unit
34959 name. If so, return the units table entry for that unit. This
34960 will be a list whose first element is the unit name (not counting
34961 prefix characters) as a symbol and whose second element is the
34962 Calc expression which defines the unit. (Refer to the Calc sources
34963 for details on the remaining elements of this list.) If @var{var}
34964 is not a variable or is not a unit name, return @code{nil}.
34967 @defun units-in-expr-p expr sub-exprs
34968 Return true if @var{expr} contains any variables which can be
34969 interpreted as units. If @var{sub-exprs} is @code{t}, the entire
34970 expression is searched. If @var{sub-exprs} is @code{nil}, this
34971 checks whether @var{expr} is directly a units expression.
34974 @defun single-units-in-expr-p expr
34975 Check whether @var{expr} contains exactly one units variable. If so,
34976 return the units table entry for the variable. If @var{expr} does
34977 not contain any units, return @code{nil}. If @var{expr} contains
34978 two or more units, return the symbol @code{wrong}.
34981 @defun to-standard-units expr which
34982 Convert units expression @var{expr} to base units. If @var{which}
34983 is @code{nil}, use Calc's native base units. Otherwise, @var{which}
34984 can specify a units system, which is a list of two-element lists,
34985 where the first element is a Calc base symbol name and the second
34986 is an expression to substitute for it.
34989 @defun remove-units expr
34990 Return a copy of @var{expr} with all units variables replaced by ones.
34991 This expression is generally normalized before use.
34994 @defun extract-units expr
34995 Return a copy of @var{expr} with everything but units variables replaced
34999 @node Formatting Lisp Functions, Hooks, Symbolic Lisp Functions, Internals
35000 @subsubsection I/O and Formatting Functions
35003 The functions described here are responsible for parsing and formatting
35004 Calc numbers and formulas.
35006 @defun calc-eval str sep arg1 arg2 @dots{}
35007 This is the simplest interface to the Calculator from another Lisp program.
35008 @xref{Calling Calc from Your Programs}.
35011 @defun read-number str
35012 If string @var{str} contains a valid Calc number, either integer,
35013 fraction, float, or HMS form, this function parses and returns that
35014 number. Otherwise, it returns @code{nil}.
35017 @defun read-expr str
35018 Read an algebraic expression from string @var{str}. If @var{str} does
35019 not have the form of a valid expression, return a list of the form
35020 @samp{(error @var{pos} @var{msg})} where @var{pos} is an integer index
35021 into @var{str} of the general location of the error, and @var{msg} is
35022 a string describing the problem.
35025 @defun read-exprs str
35026 Read a list of expressions separated by commas, and return it as a
35027 Lisp list. If an error occurs in any expressions, an error list as
35028 shown above is returned instead.
35031 @defun calc-do-alg-entry initial prompt no-norm
35032 Read an algebraic formula or formulas using the minibuffer. All
35033 conventions of regular algebraic entry are observed. The return value
35034 is a list of Calc formulas; there will be more than one if the user
35035 entered a list of values separated by commas. The result is @code{nil}
35036 if the user presses Return with a blank line. If @var{initial} is
35037 given, it is a string which the minibuffer will initially contain.
35038 If @var{prompt} is given, it is the prompt string to use; the default
35039 is ``Algebraic:''. If @var{no-norm} is @code{t}, the formulas will
35040 be returned exactly as parsed; otherwise, they will be passed through
35041 @code{calc-normalize} first.
35043 To support the use of @kbd{$} characters in the algebraic entry, use
35044 @code{let} to bind @code{calc-dollar-values} to a list of the values
35045 to be substituted for @kbd{$}, @kbd{$$}, and so on, and bind
35046 @code{calc-dollar-used} to 0. Upon return, @code{calc-dollar-used}
35047 will have been changed to the highest number of consecutive @kbd{$}s
35048 that actually appeared in the input.
35051 @defun format-number a
35052 Convert the real or complex number or HMS form @var{a} to string form.
35055 @defun format-flat-expr a prec
35056 Convert the arbitrary Calc number or formula @var{a} to string form,
35057 in the style used by the trail buffer and the @code{calc-edit} command.
35058 This is a simple format designed
35059 mostly to guarantee the string is of a form that can be re-parsed by
35060 @code{read-expr}. Most formatting modes, such as digit grouping,
35061 complex number format, and point character, are ignored to ensure the
35062 result will be re-readable. The @var{prec} parameter is normally 0; if
35063 you pass a large integer like 1000 instead, the expression will be
35064 surrounded by parentheses unless it is a plain number or variable name.
35067 @defun format-nice-expr a width
35068 This is like @code{format-flat-expr} (with @var{prec} equal to 0),
35069 except that newlines will be inserted to keep lines down to the
35070 specified @var{width}, and vectors that look like matrices or rewrite
35071 rules are written in a pseudo-matrix format. The @code{calc-edit}
35072 command uses this when only one stack entry is being edited.
35075 @defun format-value a width
35076 Convert the Calc number or formula @var{a} to string form, using the
35077 format seen in the stack buffer. Beware the string returned may
35078 not be re-readable by @code{read-expr}, for example, because of digit
35079 grouping. Multi-line objects like matrices produce strings that
35080 contain newline characters to separate the lines. The @var{w}
35081 parameter, if given, is the target window size for which to format
35082 the expressions. If @var{w} is omitted, the width of the Calculator
35086 @defun compose-expr a prec
35087 Format the Calc number or formula @var{a} according to the current
35088 language mode, returning a ``composition.'' To learn about the
35089 structure of compositions, see the comments in the Calc source code.
35090 You can specify the format of a given type of function call by putting
35091 a @code{math-compose-@var{lang}} property on the function's symbol,
35092 whose value is a Lisp function that takes @var{a} and @var{prec} as
35093 arguments and returns a composition. Here @var{lang} is a language
35094 mode name, one of @code{normal}, @code{big}, @code{c}, @code{pascal},
35095 @code{fortran}, @code{tex}, @code{eqn}, @code{math}, or @code{maple}.
35096 In Big mode, Calc actually tries @code{math-compose-big} first, then
35097 tries @code{math-compose-normal}. If this property does not exist,
35098 or if the function returns @code{nil}, the function is written in the
35099 normal function-call notation for that language.
35102 @defun composition-to-string c w
35103 Convert a composition structure returned by @code{compose-expr} into
35104 a string. Multi-line compositions convert to strings containing
35105 newline characters. The target window size is given by @var{w}.
35106 The @code{format-value} function basically calls @code{compose-expr}
35107 followed by @code{composition-to-string}.
35110 @defun comp-width c
35111 Compute the width in characters of composition @var{c}.
35114 @defun comp-height c
35115 Compute the height in lines of composition @var{c}.
35118 @defun comp-ascent c
35119 Compute the portion of the height of composition @var{c} which is on or
35120 above the baseline. For a one-line composition, this will be one.
35123 @defun comp-descent c
35124 Compute the portion of the height of composition @var{c} which is below
35125 the baseline. For a one-line composition, this will be zero.
35128 @defun comp-first-char c
35129 If composition @var{c} is a ``flat'' composition, return the first
35130 (leftmost) character of the composition as an integer. Otherwise,
35134 @defun comp-last-char c
35135 If composition @var{c} is a ``flat'' composition, return the last
35136 (rightmost) character, otherwise return @code{nil}.
35139 @comment @node Lisp Variables, Hooks, Formatting Lisp Functions, Internals
35140 @comment @subsubsection Lisp Variables
35143 @comment (This section is currently unfinished.)
35145 @node Hooks, , Formatting Lisp Functions, Internals
35146 @subsubsection Hooks
35149 Hooks are variables which contain Lisp functions (or lists of functions)
35150 which are called at various times. Calc defines a number of hooks
35151 that help you to customize it in various ways. Calc uses the Lisp
35152 function @code{run-hooks} to invoke the hooks shown below. Several
35153 other customization-related variables are also described here.
35155 @defvar calc-load-hook
35156 This hook is called at the end of @file{calc.el}, after the file has
35157 been loaded, before any functions in it have been called, but after
35158 @code{calc-mode-map} and similar variables have been set up.
35161 @defvar calc-ext-load-hook
35162 This hook is called at the end of @file{calc-ext.el}.
35165 @defvar calc-start-hook
35166 This hook is called as the last step in a @kbd{M-x calc} command.
35167 At this point, the Calc buffer has been created and initialized if
35168 necessary, the Calc window and trail window have been created,
35169 and the ``Welcome to Calc'' message has been displayed.
35172 @defvar calc-mode-hook
35173 This hook is called when the Calc buffer is being created. Usually
35174 this will only happen once per Emacs session. The hook is called
35175 after Emacs has switched to the new buffer, the mode-settings file
35176 has been read if necessary, and all other buffer-local variables
35177 have been set up. After this hook returns, Calc will perform a
35178 @code{calc-refresh} operation, set up the mode line display, then
35179 evaluate any deferred @code{calc-define} properties that have not
35180 been evaluated yet.
35183 @defvar calc-trail-mode-hook
35184 This hook is called when the Calc Trail buffer is being created.
35185 It is called as the very last step of setting up the Trail buffer.
35186 Like @code{calc-mode-hook}, this will normally happen only once
35190 @defvar calc-end-hook
35191 This hook is called by @code{calc-quit}, generally because the user
35192 presses @kbd{q} or @kbd{C-x * c} while in Calc. The Calc buffer will
35193 be the current buffer. The hook is called as the very first
35194 step, before the Calc window is destroyed.
35197 @defvar calc-window-hook
35198 If this hook is non-@code{nil}, it is called to create the Calc window.
35199 Upon return, this new Calc window should be the current window.
35200 (The Calc buffer will already be the current buffer when the
35201 hook is called.) If the hook is not defined, Calc will
35202 generally use @code{split-window}, @code{set-window-buffer},
35203 and @code{select-window} to create the Calc window.
35206 @defvar calc-trail-window-hook
35207 If this hook is non-@code{nil}, it is called to create the Calc Trail
35208 window. The variable @code{calc-trail-buffer} will contain the buffer
35209 which the window should use. Unlike @code{calc-window-hook}, this hook
35210 must @emph{not} switch into the new window.
35213 @defvar calc-embedded-mode-hook
35214 This hook is called the first time that Embedded mode is entered.
35217 @defvar calc-embedded-new-buffer-hook
35218 This hook is called each time that Embedded mode is entered in a
35222 @defvar calc-embedded-new-formula-hook
35223 This hook is called each time that Embedded mode is enabled for a
35227 @defvar calc-edit-mode-hook
35228 This hook is called by @code{calc-edit} (and the other ``edit''
35229 commands) when the temporary editing buffer is being created.
35230 The buffer will have been selected and set up to be in
35231 @code{calc-edit-mode}, but will not yet have been filled with
35232 text. (In fact it may still have leftover text from a previous
35233 @code{calc-edit} command.)
35236 @defvar calc-mode-save-hook
35237 This hook is called by the @code{calc-save-modes} command,
35238 after Calc's own mode features have been inserted into the
35239 Calc init file and just before the ``End of mode settings''
35240 message is inserted.
35243 @defvar calc-reset-hook
35244 This hook is called after @kbd{C-x * 0} (@code{calc-reset}) has
35245 reset all modes. The Calc buffer will be the current buffer.
35248 @defvar calc-other-modes
35249 This variable contains a list of strings. The strings are
35250 concatenated at the end of the modes portion of the Calc
35251 mode line (after standard modes such as ``Deg'', ``Inv'' and
35252 ``Hyp''). Each string should be a short, single word followed
35253 by a space. The variable is @code{nil} by default.
35256 @defvar calc-mode-map
35257 This is the keymap that is used by Calc mode. The best time
35258 to adjust it is probably in a @code{calc-mode-hook}. If the
35259 Calc extensions package (@file{calc-ext.el}) has not yet been
35260 loaded, many of these keys will be bound to @code{calc-missing-key},
35261 which is a command that loads the extensions package and
35262 ``retypes'' the key. If your @code{calc-mode-hook} rebinds
35263 one of these keys, it will probably be overridden when the
35264 extensions are loaded.
35267 @defvar calc-digit-map
35268 This is the keymap that is used during numeric entry. Numeric
35269 entry uses the minibuffer, but this map binds every non-numeric
35270 key to @code{calcDigit-nondigit} which generally calls
35271 @code{exit-minibuffer} and ``retypes'' the key.
35274 @defvar calc-alg-ent-map
35275 This is the keymap that is used during algebraic entry. This is
35276 mostly a copy of @code{minibuffer-local-map}.
35279 @defvar calc-store-var-map
35280 This is the keymap that is used during entry of variable names for
35281 commands like @code{calc-store} and @code{calc-recall}. This is
35282 mostly a copy of @code{minibuffer-local-completion-map}.
35285 @defvar calc-edit-mode-map
35286 This is the (sparse) keymap used by @code{calc-edit} and other
35287 temporary editing commands. It binds @key{RET}, @key{LFD},
35288 and @kbd{C-c C-c} to @code{calc-edit-finish}.
35291 @defvar calc-mode-var-list
35292 This is a list of variables which are saved by @code{calc-save-modes}.
35293 Each entry is a list of two items, the variable (as a Lisp symbol)
35294 and its default value. When modes are being saved, each variable
35295 is compared with its default value (using @code{equal}) and any
35296 non-default variables are written out.
35299 @defvar calc-local-var-list
35300 This is a list of variables which should be buffer-local to the
35301 Calc buffer. Each entry is a variable name (as a Lisp symbol).
35302 These variables also have their default values manipulated by
35303 the @code{calc} and @code{calc-quit} commands; @pxref{Multiple Calculators}.
35304 Since @code{calc-mode-hook} is called after this list has been
35305 used the first time, your hook should add a variable to the
35306 list and also call @code{make-local-variable} itself.
35309 @node Copying, GNU Free Documentation License, Programming, Top
35310 @appendix GNU GENERAL PUBLIC LICENSE
35313 @node GNU Free Documentation License, Customizing Calc, Copying, Top
35314 @appendix GNU Free Documentation License
35315 @include doclicense.texi
35317 @node Customizing Calc, Reporting Bugs, GNU Free Documentation License, Top
35318 @appendix Customizing Calc
35320 The usual prefix for Calc is the key sequence @kbd{C-x *}. If you wish
35321 to use a different prefix, you can put
35324 (global-set-key "NEWPREFIX" 'calc-dispatch)
35328 in your .emacs file.
35329 (@xref{Key Bindings,,Customizing Key Bindings,emacs,
35330 The GNU Emacs Manual}, for more information on binding keys.)
35331 A convenient way to start Calc is with @kbd{C-x * *}; to make it equally
35332 convenient for users who use a different prefix, the prefix can be
35333 followed by @kbd{=}, @kbd{&}, @kbd{#}, @kbd{\}, @kbd{/}, @kbd{+} or
35334 @kbd{-} as well as @kbd{*} to start Calc, and so in many cases the last
35335 character of the prefix can simply be typed twice.
35337 Calc is controlled by many variables, most of which can be reset
35338 from within Calc. Some variables are less involved with actual
35339 calculation and can be set outside of Calc using Emacs's
35340 customization facilities. These variables are listed below.
35341 Typing @kbd{M-x customize-variable RET @var{variable-name} RET}
35342 will bring up a buffer in which the variable's value can be redefined.
35343 Typing @kbd{M-x customize-group RET calc RET} will bring up a buffer which
35344 contains all of Calc's customizable variables. (These variables can
35345 also be reset by putting the appropriate lines in your .emacs file;
35346 @xref{Init File, ,Init File, emacs, The GNU Emacs Manual}.)
35348 Some of the customizable variables are regular expressions. A regular
35349 expression is basically a pattern that Calc can search for.
35350 See @ref{Regexp Search,, Regular Expression Search, emacs, The GNU Emacs Manual}
35351 to see how regular expressions work.
35353 @defvar calc-settings-file
35354 The variable @code{calc-settings-file} holds the file name in
35355 which commands like @kbd{m m} and @kbd{Z P} store ``permanent''
35357 If @code{calc-settings-file} is not your user init file (typically
35358 @file{~/.emacs}) and if the variable @code{calc-loaded-settings-file} is
35359 @code{nil}, then Calc will automatically load your settings file (if it
35360 exists) the first time Calc is invoked.
35362 The default value for this variable is @code{"~/.emacs.d/calc.el"}
35363 unless the file @file{~/.calc.el} exists, in which case the default
35364 value will be @code{"~/.calc.el"}.
35367 @defvar calc-gnuplot-name
35368 See @ref{Graphics}.@*
35369 The variable @code{calc-gnuplot-name} should be the name of the
35370 GNUPLOT program (a string). If you have GNUPLOT installed on your
35371 system but Calc is unable to find it, you may need to set this
35372 variable. You may also need to set some Lisp variables to show Calc how
35373 to run GNUPLOT on your system, see @ref{Devices, ,Graphical Devices} .
35374 The default value of @code{calc-gnuplot-name} is @code{"gnuplot"}.
35377 @defvar calc-gnuplot-plot-command
35378 @defvarx calc-gnuplot-print-command
35379 See @ref{Devices, ,Graphical Devices}.@*
35380 The variables @code{calc-gnuplot-plot-command} and
35381 @code{calc-gnuplot-print-command} represent system commands to
35382 display and print the output of GNUPLOT, respectively. These may be
35383 @code{nil} if no command is necessary, or strings which can include
35384 @samp{%s} to signify the name of the file to be displayed or printed.
35385 Or, these variables may contain Lisp expressions which are evaluated
35386 to display or print the output.
35388 The default value of @code{calc-gnuplot-plot-command} is @code{nil},
35389 and the default value of @code{calc-gnuplot-print-command} is
35393 @defvar calc-language-alist
35394 See @ref{Basic Embedded Mode}.@*
35395 The variable @code{calc-language-alist} controls the languages that
35396 Calc will associate with major modes. When Calc embedded mode is
35397 enabled, it will try to use the current major mode to
35398 determine what language should be used. (This can be overridden using
35399 Calc's mode changing commands, @xref{Mode Settings in Embedded Mode}.)
35400 The variable @code{calc-language-alist} consists of a list of pairs of
35401 the form @code{(@var{MAJOR-MODE} . @var{LANGUAGE})}; for example,
35402 @code{(latex-mode . latex)} is one such pair. If Calc embedded is
35403 activated in a buffer whose major mode is @var{MAJOR-MODE}, it will set itself
35404 to use the language @var{LANGUAGE}.
35406 The default value of @code{calc-language-alist} is
35408 ((latex-mode . latex)
35410 (plain-tex-mode . tex)
35411 (context-mode . tex)
35413 (pascal-mode . pascal)
35416 (fortran-mode . fortran)
35417 (f90-mode . fortran))
35421 @defvar calc-embedded-announce-formula
35422 @defvarx calc-embedded-announce-formula-alist
35423 See @ref{Customizing Embedded Mode}.@*
35424 The variable @code{calc-embedded-announce-formula} helps determine
35425 what formulas @kbd{C-x * a} will activate in a buffer. It is a
35426 regular expression, and when activating embedded formulas with
35427 @kbd{C-x * a}, it will tell Calc that what follows is a formula to be
35428 activated. (Calc also uses other patterns to find formulas, such as
35429 @samp{=>} and @samp{:=}.)
35431 The default pattern is @code{"%Embed\n\\(% .*\n\\)*"}, which checks
35432 for @samp{%Embed} followed by any number of lines beginning with
35433 @samp{%} and a space.
35435 The variable @code{calc-embedded-announce-formula-alist} is used to
35436 set @code{calc-embedded-announce-formula} to different regular
35437 expressions depending on the major mode of the editing buffer.
35438 It consists of a list of pairs of the form @code{(@var{MAJOR-MODE} .
35439 @var{REGEXP})}, and its default value is
35441 ((c++-mode . "//Embed\n\\(// .*\n\\)*")
35442 (c-mode . "/\\*Embed\\*/\n\\(/\\* .*\\*/\n\\)*")
35443 (f90-mode . "!Embed\n\\(! .*\n\\)*")
35444 (fortran-mode . "C Embed\n\\(C .*\n\\)*")
35445 (html-helper-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
35446 (html-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
35447 (nroff-mode . "\\\\\"Embed\n\\(\\\\\" .*\n\\)*")
35448 (pascal-mode . "@{Embed@}\n\\(@{.*@}\n\\)*")
35449 (sgml-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
35450 (xml-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
35451 (texinfo-mode . "@@c Embed\n\\(@@c .*\n\\)*"))
35453 Any major modes added to @code{calc-embedded-announce-formula-alist}
35454 should also be added to @code{calc-embedded-open-close-plain-alist}
35455 and @code{calc-embedded-open-close-mode-alist}.
35458 @defvar calc-embedded-open-formula
35459 @defvarx calc-embedded-close-formula
35460 @defvarx calc-embedded-open-close-formula-alist
35461 See @ref{Customizing Embedded Mode}.@*
35462 The variables @code{calc-embedded-open-formula} and
35463 @code{calc-embedded-close-formula} control the region that Calc will
35464 activate as a formula when Embedded mode is entered with @kbd{C-x * e}.
35465 They are regular expressions;
35466 Calc normally scans backward and forward in the buffer for the
35467 nearest text matching these regular expressions to be the ``formula
35470 The simplest delimiters are blank lines. Other delimiters that
35471 Embedded mode understands by default are:
35474 The @TeX{} and @LaTeX{} math delimiters @samp{$ $}, @samp{$$ $$},
35475 @samp{\[ \]}, and @samp{\( \)};
35477 Lines beginning with @samp{\begin} and @samp{\end} (except matrix delimiters);
35479 Lines beginning with @samp{@@} (Texinfo delimiters).
35481 Lines beginning with @samp{.EQ} and @samp{.EN} (@dfn{eqn} delimiters);
35483 Lines containing a single @samp{%} or @samp{.\"} symbol and nothing else.
35486 The variable @code{calc-embedded-open-close-formula-alist} is used to
35487 set @code{calc-embedded-open-formula} and
35488 @code{calc-embedded-close-formula} to different regular
35489 expressions depending on the major mode of the editing buffer.
35490 It consists of a list of lists of the form
35491 @code{(@var{MAJOR-MODE} @var{OPEN-FORMULA-REGEXP}
35492 @var{CLOSE-FORMULA-REGEXP})}, and its default value is
35496 @defvar calc-embedded-word-regexp
35497 @defvarx calc-embedded-word-regexp-alist
35498 See @ref{Customizing Embedded Mode}.@*
35499 The variable @code{calc-embedded-word-regexp} determines the expression
35500 that Calc will activate when Embedded mode is entered with @kbd{C-x *
35501 w}. It is a regular expressions.
35503 The default value of @code{calc-embedded-word-regexp} is
35504 @code{"[-+]?[0-9]+\\(\\.[0-9]+\\)?\\([eE][-+]?[0-9]+\\)?"}.
35506 The variable @code{calc-embedded-word-regexp-alist} is used to
35507 set @code{calc-embedded-word-regexp} to a different regular
35508 expression depending on the major mode of the editing buffer.
35509 It consists of a list of lists of the form
35510 @code{(@var{MAJOR-MODE} @var{WORD-REGEXP})}, and its default value is
35514 @defvar calc-embedded-open-plain
35515 @defvarx calc-embedded-close-plain
35516 @defvarx calc-embedded-open-close-plain-alist
35517 See @ref{Customizing Embedded Mode}.@*
35518 The variables @code{calc-embedded-open-plain} and
35519 @code{calc-embedded-open-plain} are used to delimit ``plain''
35520 formulas. Note that these are actual strings, not regular
35521 expressions, because Calc must be able to write these string into a
35522 buffer as well as to recognize them.
35524 The default string for @code{calc-embedded-open-plain} is
35525 @code{"%%% "}, note the trailing space. The default string for
35526 @code{calc-embedded-close-plain} is @code{" %%%\n"}, without
35527 the trailing newline here, the first line of a Big mode formula
35528 that followed might be shifted over with respect to the other lines.
35530 The variable @code{calc-embedded-open-close-plain-alist} is used to
35531 set @code{calc-embedded-open-plain} and
35532 @code{calc-embedded-close-plain} to different strings
35533 depending on the major mode of the editing buffer.
35534 It consists of a list of lists of the form
35535 @code{(@var{MAJOR-MODE} @var{OPEN-PLAIN-STRING}
35536 @var{CLOSE-PLAIN-STRING})}, and its default value is
35538 ((c++-mode "// %% " " %%\n")
35539 (c-mode "/* %% " " %% */\n")
35540 (f90-mode "! %% " " %%\n")
35541 (fortran-mode "C %% " " %%\n")
35542 (html-helper-mode "<!-- %% " " %% -->\n")
35543 (html-mode "<!-- %% " " %% -->\n")
35544 (nroff-mode "\\\" %% " " %%\n")
35545 (pascal-mode "@{%% " " %%@}\n")
35546 (sgml-mode "<!-- %% " " %% -->\n")
35547 (xml-mode "<!-- %% " " %% -->\n")
35548 (texinfo-mode "@@c %% " " %%\n"))
35550 Any major modes added to @code{calc-embedded-open-close-plain-alist}
35551 should also be added to @code{calc-embedded-announce-formula-alist}
35552 and @code{calc-embedded-open-close-mode-alist}.
35555 @defvar calc-embedded-open-new-formula
35556 @defvarx calc-embedded-close-new-formula
35557 @defvarx calc-embedded-open-close-new-formula-alist
35558 See @ref{Customizing Embedded Mode}.@*
35559 The variables @code{calc-embedded-open-new-formula} and
35560 @code{calc-embedded-close-new-formula} are strings which are
35561 inserted before and after a new formula when you type @kbd{C-x * f}.
35563 The default value of @code{calc-embedded-open-new-formula} is
35564 @code{"\n\n"}. If this string begins with a newline character and the
35565 @kbd{C-x * f} is typed at the beginning of a line, @kbd{C-x * f} will skip
35566 this first newline to avoid introducing unnecessary blank lines in the
35567 file. The default value of @code{calc-embedded-close-new-formula} is
35568 also @code{"\n\n"}. The final newline is omitted by @w{@kbd{C-x * f}}
35569 if typed at the end of a line. (It follows that if @kbd{C-x * f} is
35570 typed on a blank line, both a leading opening newline and a trailing
35571 closing newline are omitted.)
35573 The variable @code{calc-embedded-open-close-new-formula-alist} is used to
35574 set @code{calc-embedded-open-new-formula} and
35575 @code{calc-embedded-close-new-formula} to different strings
35576 depending on the major mode of the editing buffer.
35577 It consists of a list of lists of the form
35578 @code{(@var{MAJOR-MODE} @var{OPEN-NEW-FORMULA-STRING}
35579 @var{CLOSE-NEW-FORMULA-STRING})}, and its default value is
35583 @defvar calc-embedded-open-mode
35584 @defvarx calc-embedded-close-mode
35585 @defvarx calc-embedded-open-close-mode-alist
35586 See @ref{Customizing Embedded Mode}.@*
35587 The variables @code{calc-embedded-open-mode} and
35588 @code{calc-embedded-close-mode} are strings which Calc will place before
35589 and after any mode annotations that it inserts. Calc never scans for
35590 these strings; Calc always looks for the annotation itself, so it is not
35591 necessary to add them to user-written annotations.
35593 The default value of @code{calc-embedded-open-mode} is @code{"% "}
35594 and the default value of @code{calc-embedded-close-mode} is
35596 If you change the value of @code{calc-embedded-close-mode}, it is a good
35597 idea still to end with a newline so that mode annotations will appear on
35598 lines by themselves.
35600 The variable @code{calc-embedded-open-close-mode-alist} is used to
35601 set @code{calc-embedded-open-mode} and
35602 @code{calc-embedded-close-mode} to different strings
35603 expressions depending on the major mode of the editing buffer.
35604 It consists of a list of lists of the form
35605 @code{(@var{MAJOR-MODE} @var{OPEN-MODE-STRING}
35606 @var{CLOSE-MODE-STRING})}, and its default value is
35608 ((c++-mode "// " "\n")
35609 (c-mode "/* " " */\n")
35610 (f90-mode "! " "\n")
35611 (fortran-mode "C " "\n")
35612 (html-helper-mode "<!-- " " -->\n")
35613 (html-mode "<!-- " " -->\n")
35614 (nroff-mode "\\\" " "\n")
35615 (pascal-mode "@{ " " @}\n")
35616 (sgml-mode "<!-- " " -->\n")
35617 (xml-mode "<!-- " " -->\n")
35618 (texinfo-mode "@@c " "\n"))
35620 Any major modes added to @code{calc-embedded-open-close-mode-alist}
35621 should also be added to @code{calc-embedded-announce-formula-alist}
35622 and @code{calc-embedded-open-close-plain-alist}.
35625 @defvar calc-lu-power-reference
35626 @defvarx calc-lu-field-reference
35627 See @ref{Logarithmic Units}.@*
35628 The variables @code{calc-lu-power-reference} and
35629 @code{calc-lu-field-reference} are unit expressions (written as
35630 strings) which Calc will use as reference quantities for logarithmic
35633 The default value of @code{calc-lu-power-reference} is @code{"mW"}
35634 and the default value of @code{calc-lu-field-reference} is
35638 @defvar calc-note-threshold
35639 See @ref{Musical Notes}.@*
35640 The variable @code{calc-note-threshold} is a number (written as a
35641 string) which determines how close (in cents) a frequency needs to be
35642 to a note to be recognized as that note.
35644 The default value of @code{calc-note-threshold} is 1.
35647 @defvar calc-highlight-selections-with-faces
35648 @defvarx calc-selected-face
35649 @defvarx calc-nonselected-face
35650 See @ref{Displaying Selections}.@*
35651 The variable @code{calc-highlight-selections-with-faces}
35652 determines how selected sub-formulas are distinguished.
35653 If @code{calc-highlight-selections-with-faces} is nil, then
35654 a selected sub-formula is distinguished either by changing every
35655 character not part of the sub-formula with a dot or by changing every
35656 character in the sub-formula with a @samp{#} sign.
35657 If @code{calc-highlight-selections-with-faces} is t,
35658 then a selected sub-formula is distinguished either by displaying the
35659 non-selected portion of the formula with @code{calc-nonselected-face}
35660 or by displaying the selected sub-formula with
35661 @code{calc-nonselected-face}.
35664 @defvar calc-multiplication-has-precedence
35665 The variable @code{calc-multiplication-has-precedence} determines
35666 whether multiplication has precedence over division in algebraic
35667 formulas in normal language modes. If
35668 @code{calc-multiplication-has-precedence} is non-@code{nil}, then
35669 multiplication has precedence (and, for certain obscure reasons, is
35670 right associative), and so for example @samp{a/b*c} will be interpreted
35671 as @samp{a/(b*c)}. If @code{calc-multiplication-has-precedence} is
35672 @code{nil}, then multiplication has the same precedence as division
35673 (and, like division, is left associative), and so for example
35674 @samp{a/b*c} will be interpreted as @samp{(a/b)*c}. The default value
35675 of @code{calc-multiplication-has-precedence} is @code{t}.
35678 @defvar calc-ensure-consistent-units
35679 When converting units, the variable @code{calc-ensure-consistent-units}
35680 determines whether or not the target units need to be consistent with the
35681 original units. If @code{calc-ensure-consistent-units} is @code{nil}, then
35682 the target units don't need to have the same dimensions as the original units;
35683 for example, converting @samp{100 ft/s} to @samp{m} will produce @samp{30.48 m/s}.
35684 If @code{calc-ensure-consistent-units} is non-@code{nil}, then the target units
35685 need to have the same dimensions as the original units; for example, converting
35686 @samp{100 ft/s} to @samp{m} will result in an error, since @samp{ft/s} and @samp{m}
35687 have different dimensions. The default value of @code{calc-ensure-consistent-units}
35691 @defvar calc-undo-length
35692 The variable @code{calc-undo-length} determines the number of undo
35693 steps that Calc will keep track of when @code{calc-quit} is called.
35694 If @code{calc-undo-length} is a non-negative integer, then this is the
35695 number of undo steps that will be preserved; if
35696 @code{calc-undo-length} has any other value, then all undo steps will
35697 be preserved. The default value of @code{calc-undo-length} is @expr{100}.
35700 @defvar calc-gregorian-switch
35701 See @ref{Date Forms}.@*
35702 The variable @code{calc-gregorian-switch} is either a list of integers
35703 @code{(@var{YEAR} @var{MONTH} @var{DAY})} or @code{nil}.
35704 If it is @code{nil}, then Calc's date forms always represent Gregorian dates.
35705 Otherwise, @code{calc-gregorian-switch} represents the date that the
35706 calendar switches from Julian dates to Gregorian dates;
35707 @code{(@var{YEAR} @var{MONTH} @var{DAY})} will be the first Gregorian
35708 date. The customization buffer will offer several standard dates to
35709 choose from, or the user can enter their own date.
35711 The default value of @code{calc-gregorian-switch} is @code{nil}.
35714 @node Reporting Bugs, Summary, Customizing Calc, Top
35715 @appendix Reporting Bugs
35718 If you find a bug in Calc, send e-mail to Jay Belanger,
35721 jay.p.belanger@@gmail.com
35725 There is an automatic command @kbd{M-x report-calc-bug} which helps
35726 you to report bugs. This command prompts you for a brief subject
35727 line, then leaves you in a mail editing buffer. Type @kbd{C-c C-c} to
35728 send your mail. Make sure your subject line indicates that you are
35729 reporting a Calc bug; this command sends mail to the maintainer's
35732 If you have suggestions for additional features for Calc, please send
35733 them. Some have dared to suggest that Calc is already top-heavy with
35734 features; this obviously cannot be the case, so if you have ideas, send
35737 At the front of the source file, @file{calc.el}, is a list of ideas for
35738 future work. If any enthusiastic souls wish to take it upon themselves
35739 to work on these, please send a message (using @kbd{M-x report-calc-bug})
35740 so any efforts can be coordinated.
35742 The latest version of Calc is available from Savannah, in the Emacs
35743 repository. See @uref{http://savannah.gnu.org/projects/emacs}.
35746 @node Summary, Key Index, Reporting Bugs, Top
35747 @appendix Calc Summary
35750 This section includes a complete list of Calc keystroke commands.
35751 Each line lists the stack entries used by the command (top-of-stack
35752 last), the keystrokes themselves, the prompts asked by the command,
35753 and the result of the command (also with top-of-stack last).
35754 The result is expressed using the equivalent algebraic function.
35755 Commands which put no results on the stack show the full @kbd{M-x}
35756 command name in that position. Numbers preceding the result or
35757 command name refer to notes at the end.
35759 Algebraic functions and @kbd{M-x} commands that don't have corresponding
35760 keystrokes are not listed in this summary.
35761 @xref{Command Index}. @xref{Function Index}.
35766 \vskip-2\baselineskip \null
35767 \gdef\sumrow#1{\sumrowx#1\relax}%
35768 \gdef\sumrowx#1\:#2\:#3\:#4\:#5\:#6\relax{%
35771 \hbox to5em{\sl\hss#1}%
35772 \hbox to5em{\tt#2\hss}%
35773 \hbox to4em{\sl#3\hss}%
35774 \hbox to5em{\rm\hss#4}%
35779 \gdef\sumlpar{{\rm(}}%
35780 \gdef\sumrpar{{\rm)}}%
35781 \gdef\sumcomma{{\rm,\thinspace}}%
35782 \gdef\sumexcl{{\rm!}}%
35783 \gdef\sumbreak{\vskip-2.5\baselineskip\goodbreak}%
35784 \gdef\minus#1{{\tt-}}%
35788 @catcode`@(=@active @let(=@sumlpar
35789 @catcode`@)=@active @let)=@sumrpar
35790 @catcode`@,=@active @let,=@sumcomma
35791 @catcode`@!=@active @let!=@sumexcl
35795 @advance@baselineskip-2.5pt
35798 @r{ @: C-x * a @: @: 33 @:calc-embedded-activate@:}
35799 @r{ @: C-x * b @: @: @:calc-big-or-small@:}
35800 @r{ @: C-x * c @: @: @:calc@:}
35801 @r{ @: C-x * d @: @: @:calc-embedded-duplicate@:}
35802 @r{ @: C-x * e @: @: 34 @:calc-embedded@:}
35803 @r{ @: C-x * f @:formula @: @:calc-embedded-new-formula@:}
35804 @r{ @: C-x * g @: @: 35 @:calc-grab-region@:}
35805 @r{ @: C-x * i @: @: @:calc-info@:}
35806 @r{ @: C-x * j @: @: @:calc-embedded-select@:}
35807 @r{ @: C-x * k @: @: @:calc-keypad@:}
35808 @r{ @: C-x * l @: @: @:calc-load-everything@:}
35809 @r{ @: C-x * m @: @: @:read-kbd-macro@:}
35810 @r{ @: C-x * n @: @: 4 @:calc-embedded-next@:}
35811 @r{ @: C-x * o @: @: @:calc-other-window@:}
35812 @r{ @: C-x * p @: @: 4 @:calc-embedded-previous@:}
35813 @r{ @: C-x * q @:formula @: @:quick-calc@:}
35814 @r{ @: C-x * r @: @: 36 @:calc-grab-rectangle@:}
35815 @r{ @: C-x * s @: @: @:calc-info-summary@:}
35816 @r{ @: C-x * t @: @: @:calc-tutorial@:}
35817 @r{ @: C-x * u @: @: @:calc-embedded-update-formula@:}
35818 @r{ @: C-x * w @: @: @:calc-embedded-word@:}
35819 @r{ @: C-x * x @: @: @:calc-quit@:}
35820 @r{ @: C-x * y @: @:1,28,49 @:calc-copy-to-buffer@:}
35821 @r{ @: C-x * z @: @: @:calc-user-invocation@:}
35822 @r{ @: C-x * : @: @: 36 @:calc-grab-sum-down@:}
35823 @r{ @: C-x * _ @: @: 36 @:calc-grab-sum-across@:}
35824 @r{ @: C-x * ` @:editing @: 30 @:calc-embedded-edit@:}
35825 @r{ @: C-x * 0 @:(zero) @: @:calc-reset@:}
35828 @r{ @: 0-9 @:number @: @:@:number}
35829 @r{ @: . @:number @: @:@:0.number}
35830 @r{ @: _ @:number @: @:-@:number}
35831 @r{ @: e @:number @: @:@:1e number}
35832 @r{ @: # @:number @: @:@:current-radix@tfn{#}number}
35833 @r{ @: P @:(in number) @: @:+/-@:}
35834 @r{ @: M @:(in number) @: @:mod@:}
35835 @r{ @: @@ ' " @: (in number)@: @:@:HMS form}
35836 @r{ @: h m s @: (in number)@: @:@:HMS form}
35839 @r{ @: ' @:formula @: 37,46 @:@:formula}
35840 @r{ @: $ @:formula @: 37,46 @:$@:formula}
35841 @r{ @: " @:string @: 37,46 @:@:string}
35844 @r{ a b@: + @: @: 2 @:add@:(a,b) a+b}
35845 @r{ a b@: - @: @: 2 @:sub@:(a,b) a@minus{}b}
35846 @r{ a b@: * @: @: 2 @:mul@:(a,b) a b, a*b}
35847 @r{ a b@: / @: @: 2 @:div@:(a,b) a/b}
35848 @r{ a b@: ^ @: @: 2 @:pow@:(a,b) a^b}
35849 @r{ a b@: I ^ @: @: 2 @:nroot@:(a,b) a^(1/b)}
35850 @r{ a b@: % @: @: 2 @:mod@:(a,b) a%b}
35851 @r{ a b@: \ @: @: 2 @:idiv@:(a,b) a\b}
35852 @r{ a b@: : @: @: 2 @:fdiv@:(a,b)}
35853 @r{ a b@: | @: @: 2 @:vconcat@:(a,b) a|b}
35854 @r{ a b@: I | @: @: @:vconcat@:(b,a) b|a}
35855 @r{ a b@: H | @: @: 2 @:append@:(a,b)}
35856 @r{ a b@: I H | @: @: @:append@:(b,a)}
35857 @r{ a@: & @: @: 1 @:inv@:(a) 1/a}
35858 @r{ a@: ! @: @: 1 @:fact@:(a) a!}
35859 @r{ a@: = @: @: 1 @:evalv@:(a)}
35860 @r{ a@: M-% @: @: @:percent@:(a) a%}
35863 @r{ ... a@: @summarykey{RET} @: @: 1 @:@:... a a}
35864 @r{ ... a@: @summarykey{SPC} @: @: 1 @:@:... a a}
35865 @r{... a b@: @summarykey{TAB} @: @: 3 @:@:... b a}
35866 @r{. a b c@: M-@summarykey{TAB} @: @: 3 @:@:... b c a}
35867 @r{... a b@: @summarykey{LFD} @: @: 1 @:@:... a b a}
35868 @r{ ... a@: @summarykey{DEL} @: @: 1 @:@:...}
35869 @r{... a b@: M-@summarykey{DEL} @: @: 1 @:@:... b}
35870 @r{ @: M-@summarykey{RET} @: @: 4 @:calc-last-args@:}
35871 @r{ a@: ` @:editing @: 1,30 @:calc-edit@:}
35874 @r{ ... a@: C-d @: @: 1 @:@:...}
35875 @r{ @: C-k @: @: 27 @:calc-kill@:}
35876 @r{ @: C-w @: @: 27 @:calc-kill-region@:}
35877 @r{ @: C-y @: @: @:calc-yank@:}
35878 @r{ @: C-_ @: @: 4 @:calc-undo@:}
35879 @r{ @: M-k @: @: 27 @:calc-copy-as-kill@:}
35880 @r{ @: M-w @: @: 27 @:calc-copy-region-as-kill@:}
35883 @r{ @: [ @: @: @:@:[...}
35884 @r{[.. a b@: ] @: @: @:@:[a,b]}
35885 @r{ @: ( @: @: @:@:(...}
35886 @r{(.. a b@: ) @: @: @:@:(a,b)}
35887 @r{ @: , @: @: @:@:vector or rect complex}
35888 @r{ @: ; @: @: @:@:matrix or polar complex}
35889 @r{ @: .. @: @: @:@:interval}
35892 @r{ @: ~ @: @: @:calc-num-prefix@:}
35893 @r{ @: < @: @: 4 @:calc-scroll-left@:}
35894 @r{ @: > @: @: 4 @:calc-scroll-right@:}
35895 @r{ @: @{ @: @: 4 @:calc-scroll-down@:}
35896 @r{ @: @} @: @: 4 @:calc-scroll-up@:}
35897 @r{ @: ? @: @: @:calc-help@:}
35900 @r{ a@: n @: @: 1 @:neg@:(a) @minus{}a}
35901 @r{ @: o @: @: 4 @:calc-realign@:}
35902 @r{ @: p @:precision @: 31 @:calc-precision@:}
35903 @r{ @: q @: @: @:calc-quit@:}
35904 @r{ @: w @: @: @:calc-why@:}
35905 @r{ @: x @:command @: @:M-x calc-@:command}
35906 @r{ a@: y @: @:1,28,49 @:calc-copy-to-buffer@:}
35909 @r{ a@: A @: @: 1 @:abs@:(a)}
35910 @r{ a b@: B @: @: 2 @:log@:(a,b)}
35911 @r{ a b@: I B @: @: 2 @:alog@:(a,b) b^a}
35912 @r{ a@: C @: @: 1 @:cos@:(a)}
35913 @r{ a@: I C @: @: 1 @:arccos@:(a)}
35914 @r{ a@: H C @: @: 1 @:cosh@:(a)}
35915 @r{ a@: I H C @: @: 1 @:arccosh@:(a)}
35916 @r{ @: D @: @: 4 @:calc-redo@:}
35917 @r{ a@: E @: @: 1 @:exp@:(a)}
35918 @r{ a@: H E @: @: 1 @:exp10@:(a) 10.^a}
35919 @r{ a@: F @: @: 1,11 @:floor@:(a,d)}
35920 @r{ a@: I F @: @: 1,11 @:ceil@:(a,d)}
35921 @r{ a@: H F @: @: 1,11 @:ffloor@:(a,d)}
35922 @r{ a@: I H F @: @: 1,11 @:fceil@:(a,d)}
35923 @r{ a@: G @: @: 1 @:arg@:(a)}
35924 @r{ @: H @:command @: 32 @:@:Hyperbolic}
35925 @r{ @: I @:command @: 32 @:@:Inverse}
35926 @r{ a@: J @: @: 1 @:conj@:(a)}
35927 @r{ @: K @:command @: 32 @:@:Keep-args}
35928 @r{ a@: L @: @: 1 @:ln@:(a)}
35929 @r{ a@: H L @: @: 1 @:log10@:(a)}
35930 @r{ @: M @: @: @:calc-more-recursion-depth@:}
35931 @r{ @: I M @: @: @:calc-less-recursion-depth@:}
35932 @r{ a@: N @: @: 5 @:evalvn@:(a)}
35933 @r{ @: O @:command @: 32 @:@:Option}
35934 @r{ @: P @: @: @:@:pi}
35935 @r{ @: I P @: @: @:@:gamma}
35936 @r{ @: H P @: @: @:@:e}
35937 @r{ @: I H P @: @: @:@:phi}
35938 @r{ a@: Q @: @: 1 @:sqrt@:(a)}
35939 @r{ a@: I Q @: @: 1 @:sqr@:(a) a^2}
35940 @r{ a@: R @: @: 1,11 @:round@:(a,d)}
35941 @r{ a@: I R @: @: 1,11 @:trunc@:(a,d)}
35942 @r{ a@: H R @: @: 1,11 @:fround@:(a,d)}
35943 @r{ a@: I H R @: @: 1,11 @:ftrunc@:(a,d)}
35944 @r{ a@: S @: @: 1 @:sin@:(a)}
35945 @r{ a@: I S @: @: 1 @:arcsin@:(a)}
35946 @r{ a@: H S @: @: 1 @:sinh@:(a)}
35947 @r{ a@: I H S @: @: 1 @:arcsinh@:(a)}
35948 @r{ a@: T @: @: 1 @:tan@:(a)}
35949 @r{ a@: I T @: @: 1 @:arctan@:(a)}
35950 @r{ a@: H T @: @: 1 @:tanh@:(a)}
35951 @r{ a@: I H T @: @: 1 @:arctanh@:(a)}
35952 @r{ @: U @: @: 4 @:calc-undo@:}
35953 @r{ @: X @: @: 4 @:calc-call-last-kbd-macro@:}
35956 @r{ a b@: a = @: @: 2 @:eq@:(a,b) a=b}
35957 @r{ a b@: a # @: @: 2 @:neq@:(a,b) a!=b}
35958 @r{ a b@: a < @: @: 2 @:lt@:(a,b) a<b}
35959 @r{ a b@: a > @: @: 2 @:gt@:(a,b) a>b}
35960 @r{ a b@: a [ @: @: 2 @:leq@:(a,b) a<=b}
35961 @r{ a b@: a ] @: @: 2 @:geq@:(a,b) a>=b}
35962 @r{ a b@: a @{ @: @: 2 @:in@:(a,b)}
35963 @r{ a b@: a & @: @: 2,45 @:land@:(a,b) a&&b}
35964 @r{ a b@: a | @: @: 2,45 @:lor@:(a,b) a||b}
35965 @r{ a@: a ! @: @: 1,45 @:lnot@:(a) !a}
35966 @r{ a b c@: a : @: @: 45 @:if@:(a,b,c) a?b:c}
35967 @r{ a@: a . @: @: 1 @:rmeq@:(a)}
35968 @r{ a@: a " @: @: 7,8 @:calc-expand-formula@:}
35971 @r{ a@: a + @:i, l, h @: 6,38 @:sum@:(a,i,l,h)}
35972 @r{ a@: a - @:i, l, h @: 6,38 @:asum@:(a,i,l,h)}
35973 @r{ a@: a * @:i, l, h @: 6,38 @:prod@:(a,i,l,h)}
35974 @r{ a b@: a _ @: @: 2 @:subscr@:(a,b) a_b}
35977 @r{ a b@: a \ @: @: 2 @:pdiv@:(a,b)}
35978 @r{ a b@: a % @: @: 2 @:prem@:(a,b)}
35979 @r{ a b@: a / @: @: 2 @:pdivrem@:(a,b) [q,r]}
35980 @r{ a b@: H a / @: @: 2 @:pdivide@:(a,b) q+r/b}
35983 @r{ a@: a a @: @: 1 @:apart@:(a)}
35984 @r{ a@: a b @:old, new @: 38 @:subst@:(a,old,new)}
35985 @r{ a@: a c @:v @: 38 @:collect@:(a,v)}
35986 @r{ a@: a d @:v @: 4,38 @:deriv@:(a,v)}
35987 @r{ a@: H a d @:v @: 4,38 @:tderiv@:(a,v)}
35988 @r{ a@: a e @: @: @:esimplify@:(a)}
35989 @r{ a@: a f @: @: 1 @:factor@:(a)}
35990 @r{ a@: H a f @: @: 1 @:factors@:(a)}
35991 @r{ a b@: a g @: @: 2 @:pgcd@:(a,b)}
35992 @r{ a@: a i @:v @: 38 @:integ@:(a,v)}
35993 @r{ a@: a m @:pats @: 38 @:match@:(a,pats)}
35994 @r{ a@: I a m @:pats @: 38 @:matchnot@:(a,pats)}
35995 @r{ data x@: a p @: @: 28 @:polint@:(data,x)}
35996 @r{ data x@: H a p @: @: 28 @:ratint@:(data,x)}
35997 @r{ a@: a n @: @: 1 @:nrat@:(a)}
35998 @r{ a@: a r @:rules @:4,8,38 @:rewrite@:(a,rules,n)}
35999 @r{ a@: a s @: @: @:simplify@:(a)}
36000 @r{ a@: a t @:v, n @: 31,39 @:taylor@:(a,v,n)}
36001 @r{ a@: a v @: @: 7,8 @:calc-alg-evaluate@:}
36002 @r{ a@: a x @: @: 4,8 @:expand@:(a)}
36005 @r{ data@: a F @:model, vars @: 48 @:fit@:(m,iv,pv,data)}
36006 @r{ data@: I a F @:model, vars @: 48 @:xfit@:(m,iv,pv,data)}
36007 @r{ data@: H a F @:model, vars @: 48 @:efit@:(m,iv,pv,data)}
36008 @r{ a@: a I @:v, l, h @: 38 @:ninteg@:(a,v,l,h)}
36009 @r{ a b@: a M @:op @: 22 @:mapeq@:(op,a,b)}
36010 @r{ a b@: I a M @:op @: 22 @:mapeqr@:(op,a,b)}
36011 @r{ a b@: H a M @:op @: 22 @:mapeqp@:(op,a,b)}
36012 @r{ a g@: a N @:v @: 38 @:minimize@:(a,v,g)}
36013 @r{ a g@: H a N @:v @: 38 @:wminimize@:(a,v,g)}
36014 @r{ a@: a P @:v @: 38 @:roots@:(a,v)}
36015 @r{ a g@: a R @:v @: 38 @:root@:(a,v,g)}
36016 @r{ a g@: H a R @:v @: 38 @:wroot@:(a,v,g)}
36017 @r{ a@: a S @:v @: 38 @:solve@:(a,v)}
36018 @r{ a@: I a S @:v @: 38 @:finv@:(a,v)}
36019 @r{ a@: H a S @:v @: 38 @:fsolve@:(a,v)}
36020 @r{ a@: I H a S @:v @: 38 @:ffinv@:(a,v)}
36021 @r{ a@: a T @:i, l, h @: 6,38 @:table@:(a,i,l,h)}
36022 @r{ a g@: a X @:v @: 38 @:maximize@:(a,v,g)}
36023 @r{ a g@: H a X @:v @: 38 @:wmaximize@:(a,v,g)}
36026 @r{ a b@: b a @: @: 9 @:and@:(a,b,w)}
36027 @r{ a@: b c @: @: 9 @:clip@:(a,w)}
36028 @r{ a b@: b d @: @: 9 @:diff@:(a,b,w)}
36029 @r{ a@: b l @: @: 10 @:lsh@:(a,n,w)}
36030 @r{ a n@: H b l @: @: 9 @:lsh@:(a,n,w)}
36031 @r{ a@: b n @: @: 9 @:not@:(a,w)}
36032 @r{ a b@: b o @: @: 9 @:or@:(a,b,w)}
36033 @r{ v@: b p @: @: 1 @:vpack@:(v)}
36034 @r{ a@: b r @: @: 10 @:rsh@:(a,n,w)}
36035 @r{ a n@: H b r @: @: 9 @:rsh@:(a,n,w)}
36036 @r{ a@: b t @: @: 10 @:rot@:(a,n,w)}
36037 @r{ a n@: H b t @: @: 9 @:rot@:(a,n,w)}
36038 @r{ a@: b u @: @: 1 @:vunpack@:(a)}
36039 @r{ @: b w @:w @: 9,50 @:calc-word-size@:}
36040 @r{ a b@: b x @: @: 9 @:xor@:(a,b,w)}
36043 @r{c s l p@: b D @: @: @:ddb@:(c,s,l,p)}
36044 @r{ r n p@: b F @: @: @:fv@:(r,n,p)}
36045 @r{ r n p@: I b F @: @: @:fvb@:(r,n,p)}
36046 @r{ r n p@: H b F @: @: @:fvl@:(r,n,p)}
36047 @r{ v@: b I @: @: 19 @:irr@:(v)}
36048 @r{ v@: I b I @: @: 19 @:irrb@:(v)}
36049 @r{ a@: b L @: @: 10 @:ash@:(a,n,w)}
36050 @r{ a n@: H b L @: @: 9 @:ash@:(a,n,w)}
36051 @r{ r n a@: b M @: @: @:pmt@:(r,n,a)}
36052 @r{ r n a@: I b M @: @: @:pmtb@:(r,n,a)}
36053 @r{ r n a@: H b M @: @: @:pmtl@:(r,n,a)}
36054 @r{ r v@: b N @: @: 19 @:npv@:(r,v)}
36055 @r{ r v@: I b N @: @: 19 @:npvb@:(r,v)}
36056 @r{ r n p@: b P @: @: @:pv@:(r,n,p)}
36057 @r{ r n p@: I b P @: @: @:pvb@:(r,n,p)}
36058 @r{ r n p@: H b P @: @: @:pvl@:(r,n,p)}
36059 @r{ a@: b R @: @: 10 @:rash@:(a,n,w)}
36060 @r{ a n@: H b R @: @: 9 @:rash@:(a,n,w)}
36061 @r{ c s l@: b S @: @: @:sln@:(c,s,l)}
36062 @r{ n p a@: b T @: @: @:rate@:(n,p,a)}
36063 @r{ n p a@: I b T @: @: @:rateb@:(n,p,a)}
36064 @r{ n p a@: H b T @: @: @:ratel@:(n,p,a)}
36065 @r{c s l p@: b Y @: @: @:syd@:(c,s,l,p)}
36067 @r{ r p a@: b # @: @: @:nper@:(r,p,a)}
36068 @r{ r p a@: I b # @: @: @:nperb@:(r,p,a)}
36069 @r{ r p a@: H b # @: @: @:nperl@:(r,p,a)}
36070 @r{ a b@: b % @: @: @:relch@:(a,b)}
36073 @r{ a@: c c @: @: 5 @:pclean@:(a,p)}
36074 @r{ a@: c 0-9 @: @: @:pclean@:(a,p)}
36075 @r{ a@: H c c @: @: 5 @:clean@:(a,p)}
36076 @r{ a@: H c 0-9 @: @: @:clean@:(a,p)}
36077 @r{ a@: c d @: @: 1 @:deg@:(a)}
36078 @r{ a@: c f @: @: 1 @:pfloat@:(a)}
36079 @r{ a@: H c f @: @: 1 @:float@:(a)}
36080 @r{ a@: c h @: @: 1 @:hms@:(a)}
36081 @r{ a@: c p @: @: @:polar@:(a)}
36082 @r{ a@: I c p @: @: @:rect@:(a)}
36083 @r{ a@: c r @: @: 1 @:rad@:(a)}
36086 @r{ a@: c F @: @: 5 @:pfrac@:(a,p)}
36087 @r{ a@: H c F @: @: 5 @:frac@:(a,p)}
36090 @r{ a@: c % @: @: @:percent@:(a*100)}
36093 @r{ @: d . @:char @: 50 @:calc-point-char@:}
36094 @r{ @: d , @:char @: 50 @:calc-group-char@:}
36095 @r{ @: d < @: @: 13,50 @:calc-left-justify@:}
36096 @r{ @: d = @: @: 13,50 @:calc-center-justify@:}
36097 @r{ @: d > @: @: 13,50 @:calc-right-justify@:}
36098 @r{ @: d @{ @:label @: 50 @:calc-left-label@:}
36099 @r{ @: d @} @:label @: 50 @:calc-right-label@:}
36100 @r{ @: d [ @: @: 4 @:calc-truncate-up@:}
36101 @r{ @: d ] @: @: 4 @:calc-truncate-down@:}
36102 @r{ @: d " @: @: 12,50 @:calc-display-strings@:}
36103 @r{ @: d @summarykey{SPC} @: @: @:calc-refresh@:}
36104 @r{ @: d @summarykey{RET} @: @: 1 @:calc-refresh-top@:}
36107 @r{ @: d 0 @: @: 50 @:calc-decimal-radix@:}
36108 @r{ @: d 2 @: @: 50 @:calc-binary-radix@:}
36109 @r{ @: d 6 @: @: 50 @:calc-hex-radix@:}
36110 @r{ @: d 8 @: @: 50 @:calc-octal-radix@:}
36113 @r{ @: d b @: @:12,13,50 @:calc-line-breaking@:}
36114 @r{ @: d c @: @: 50 @:calc-complex-notation@:}
36115 @r{ @: d d @:format @: 50 @:calc-date-notation@:}
36116 @r{ @: d e @: @: 5,50 @:calc-eng-notation@:}
36117 @r{ @: d f @:num @: 31,50 @:calc-fix-notation@:}
36118 @r{ @: d g @: @:12,13,50 @:calc-group-digits@:}
36119 @r{ @: d h @:format @: 50 @:calc-hms-notation@:}
36120 @r{ @: d i @: @: 50 @:calc-i-notation@:}
36121 @r{ @: d j @: @: 50 @:calc-j-notation@:}
36122 @r{ @: d l @: @: 12,50 @:calc-line-numbering@:}
36123 @r{ @: d n @: @: 5,50 @:calc-normal-notation@:}
36124 @r{ @: d o @:format @: 50 @:calc-over-notation@:}
36125 @r{ @: d p @: @: 12,50 @:calc-show-plain@:}
36126 @r{ @: d r @:radix @: 31,50 @:calc-radix@:}
36127 @r{ @: d s @: @: 5,50 @:calc-sci-notation@:}
36128 @r{ @: d t @: @: 27 @:calc-truncate-stack@:}
36129 @r{ @: d w @: @: 12,13 @:calc-auto-why@:}
36130 @r{ @: d z @: @: 12,50 @:calc-leading-zeros@:}
36133 @r{ @: d B @: @: 50 @:calc-big-language@:}
36134 @r{ @: d C @: @: 50 @:calc-c-language@:}
36135 @r{ @: d E @: @: 50 @:calc-eqn-language@:}
36136 @r{ @: d F @: @: 50 @:calc-fortran-language@:}
36137 @r{ @: d M @: @: 50 @:calc-mathematica-language@:}
36138 @r{ @: d N @: @: 50 @:calc-normal-language@:}
36139 @r{ @: d O @: @: 50 @:calc-flat-language@:}
36140 @r{ @: d P @: @: 50 @:calc-pascal-language@:}
36141 @r{ @: d T @: @: 50 @:calc-tex-language@:}
36142 @r{ @: d L @: @: 50 @:calc-latex-language@:}
36143 @r{ @: d U @: @: 50 @:calc-unformatted-language@:}
36144 @r{ @: d W @: @: 50 @:calc-maple-language@:}
36147 @r{ a@: f [ @: @: 4 @:decr@:(a,n)}
36148 @r{ a@: f ] @: @: 4 @:incr@:(a,n)}
36151 @r{ a b@: f b @: @: 2 @:beta@:(a,b)}
36152 @r{ a@: f e @: @: 1 @:erf@:(a)}
36153 @r{ a@: I f e @: @: 1 @:erfc@:(a)}
36154 @r{ a@: f g @: @: 1 @:gamma@:(a)}
36155 @r{ a b@: f h @: @: 2 @:hypot@:(a,b)}
36156 @r{ a@: f i @: @: 1 @:im@:(a)}
36157 @r{ n a@: f j @: @: 2 @:besJ@:(n,a)}
36158 @r{ a b@: f n @: @: 2 @:min@:(a,b)}
36159 @r{ a@: f r @: @: 1 @:re@:(a)}
36160 @r{ a@: f s @: @: 1 @:sign@:(a)}
36161 @r{ a b@: f x @: @: 2 @:max@:(a,b)}
36162 @r{ n a@: f y @: @: 2 @:besY@:(n,a)}
36165 @r{ a@: f A @: @: 1 @:abssqr@:(a)}
36166 @r{ x a b@: f B @: @: @:betaI@:(x,a,b)}
36167 @r{ x a b@: H f B @: @: @:betaB@:(x,a,b)}
36168 @r{ a@: f E @: @: 1 @:expm1@:(a)}
36169 @r{ a x@: f G @: @: 2 @:gammaP@:(a,x)}
36170 @r{ a x@: I f G @: @: 2 @:gammaQ@:(a,x)}
36171 @r{ a x@: H f G @: @: 2 @:gammag@:(a,x)}
36172 @r{ a x@: I H f G @: @: 2 @:gammaG@:(a,x)}
36173 @r{ a b@: f I @: @: 2 @:ilog@:(a,b)}
36174 @r{ a b@: I f I @: @: 2 @:alog@:(a,b) b^a}
36175 @r{ a@: f L @: @: 1 @:lnp1@:(a)}
36176 @r{ a@: f M @: @: 1 @:mant@:(a)}
36177 @r{ a@: f Q @: @: 1 @:isqrt@:(a)}
36178 @r{ a@: I f Q @: @: 1 @:sqr@:(a) a^2}
36179 @r{ a n@: f S @: @: 2 @:scf@:(a,n)}
36180 @r{ y x@: f T @: @: @:arctan2@:(y,x)}
36181 @r{ a@: f X @: @: 1 @:xpon@:(a)}
36184 @r{ x y@: g a @: @: 28,40 @:calc-graph-add@:}
36185 @r{ @: g b @: @: 12 @:calc-graph-border@:}
36186 @r{ @: g c @: @: @:calc-graph-clear@:}
36187 @r{ @: g d @: @: 41 @:calc-graph-delete@:}
36188 @r{ x y@: g f @: @: 28,40 @:calc-graph-fast@:}
36189 @r{ @: g g @: @: 12 @:calc-graph-grid@:}
36190 @r{ @: g h @:title @: @:calc-graph-header@:}
36191 @r{ @: g j @: @: 4 @:calc-graph-juggle@:}
36192 @r{ @: g k @: @: 12 @:calc-graph-key@:}
36193 @r{ @: g l @: @: 12 @:calc-graph-log-x@:}
36194 @r{ @: g n @:name @: @:calc-graph-name@:}
36195 @r{ @: g p @: @: 42 @:calc-graph-plot@:}
36196 @r{ @: g q @: @: @:calc-graph-quit@:}
36197 @r{ @: g r @:range @: @:calc-graph-range-x@:}
36198 @r{ @: g s @: @: 12,13 @:calc-graph-line-style@:}
36199 @r{ @: g t @:title @: @:calc-graph-title-x@:}
36200 @r{ @: g v @: @: @:calc-graph-view-commands@:}
36201 @r{ @: g x @:display @: @:calc-graph-display@:}
36202 @r{ @: g z @: @: 12 @:calc-graph-zero-x@:}
36205 @r{ x y z@: g A @: @: 28,40 @:calc-graph-add-3d@:}
36206 @r{ @: g C @:command @: @:calc-graph-command@:}
36207 @r{ @: g D @:device @: 43,44 @:calc-graph-device@:}
36208 @r{ x y z@: g F @: @: 28,40 @:calc-graph-fast-3d@:}
36209 @r{ @: g H @: @: 12 @:calc-graph-hide@:}
36210 @r{ @: g K @: @: @:calc-graph-kill@:}
36211 @r{ @: g L @: @: 12 @:calc-graph-log-y@:}
36212 @r{ @: g N @:number @: 43,51 @:calc-graph-num-points@:}
36213 @r{ @: g O @:filename @: 43,44 @:calc-graph-output@:}
36214 @r{ @: g P @: @: 42 @:calc-graph-print@:}
36215 @r{ @: g R @:range @: @:calc-graph-range-y@:}
36216 @r{ @: g S @: @: 12,13 @:calc-graph-point-style@:}
36217 @r{ @: g T @:title @: @:calc-graph-title-y@:}
36218 @r{ @: g V @: @: @:calc-graph-view-trail@:}
36219 @r{ @: g X @:format @: @:calc-graph-geometry@:}
36220 @r{ @: g Z @: @: 12 @:calc-graph-zero-y@:}
36223 @r{ @: g C-l @: @: 12 @:calc-graph-log-z@:}
36224 @r{ @: g C-r @:range @: @:calc-graph-range-z@:}
36225 @r{ @: g C-t @:title @: @:calc-graph-title-z@:}
36228 @r{ @: h b @: @: @:calc-describe-bindings@:}
36229 @r{ @: h c @:key @: @:calc-describe-key-briefly@:}
36230 @r{ @: h f @:function @: @:calc-describe-function@:}
36231 @r{ @: h h @: @: @:calc-full-help@:}
36232 @r{ @: h i @: @: @:calc-info@:}
36233 @r{ @: h k @:key @: @:calc-describe-key@:}
36234 @r{ @: h n @: @: @:calc-view-news@:}
36235 @r{ @: h s @: @: @:calc-info-summary@:}
36236 @r{ @: h t @: @: @:calc-tutorial@:}
36237 @r{ @: h v @:var @: @:calc-describe-variable@:}
36240 @r{ @: j 1-9 @: @: @:calc-select-part@:}
36241 @r{ @: j @summarykey{RET} @: @: 27 @:calc-copy-selection@:}
36242 @r{ @: j @summarykey{DEL} @: @: 27 @:calc-del-selection@:}
36243 @r{ @: j ' @:formula @: 27 @:calc-enter-selection@:}
36244 @r{ @: j ` @:editing @: 27,30 @:calc-edit-selection@:}
36245 @r{ @: j " @: @: 7,27 @:calc-sel-expand-formula@:}
36248 @r{ @: j + @:formula @: 27 @:calc-sel-add-both-sides@:}
36249 @r{ @: j - @:formula @: 27 @:calc-sel-sub-both-sides@:}
36250 @r{ @: j * @:formula @: 27 @:calc-sel-mul-both-sides@:}
36251 @r{ @: j / @:formula @: 27 @:calc-sel-div-both-sides@:}
36252 @r{ @: j & @: @: 27 @:calc-sel-invert@:}
36255 @r{ @: j a @: @: 27 @:calc-select-additional@:}
36256 @r{ @: j b @: @: 12 @:calc-break-selections@:}
36257 @r{ @: j c @: @: @:calc-clear-selections@:}
36258 @r{ @: j d @: @: 12,50 @:calc-show-selections@:}
36259 @r{ @: j e @: @: 12 @:calc-enable-selections@:}
36260 @r{ @: j l @: @: 4,27 @:calc-select-less@:}
36261 @r{ @: j m @: @: 4,27 @:calc-select-more@:}
36262 @r{ @: j n @: @: 4 @:calc-select-next@:}
36263 @r{ @: j o @: @: 4,27 @:calc-select-once@:}
36264 @r{ @: j p @: @: 4 @:calc-select-previous@:}
36265 @r{ @: j r @:rules @:4,8,27 @:calc-rewrite-selection@:}
36266 @r{ @: j s @: @: 4,27 @:calc-select-here@:}
36267 @r{ @: j u @: @: 27 @:calc-unselect@:}
36268 @r{ @: j v @: @: 7,27 @:calc-sel-evaluate@:}
36271 @r{ @: j C @: @: 27 @:calc-sel-commute@:}
36272 @r{ @: j D @: @: 4,27 @:calc-sel-distribute@:}
36273 @r{ @: j E @: @: 27 @:calc-sel-jump-equals@:}
36274 @r{ @: j I @: @: 27 @:calc-sel-isolate@:}
36275 @r{ @: H j I @: @: 27 @:calc-sel-isolate@: (full)}
36276 @r{ @: j L @: @: 4,27 @:calc-commute-left@:}
36277 @r{ @: j M @: @: 27 @:calc-sel-merge@:}
36278 @r{ @: j N @: @: 27 @:calc-sel-negate@:}
36279 @r{ @: j O @: @: 4,27 @:calc-select-once-maybe@:}
36280 @r{ @: j R @: @: 4,27 @:calc-commute-right@:}
36281 @r{ @: j S @: @: 4,27 @:calc-select-here-maybe@:}
36282 @r{ @: j U @: @: 27 @:calc-sel-unpack@:}
36285 @r{ @: k a @: @: @:calc-random-again@:}
36286 @r{ n@: k b @: @: 1 @:bern@:(n)}
36287 @r{ n x@: H k b @: @: 2 @:bern@:(n,x)}
36288 @r{ n m@: k c @: @: 2 @:choose@:(n,m)}
36289 @r{ n m@: H k c @: @: 2 @:perm@:(n,m)}
36290 @r{ n@: k d @: @: 1 @:dfact@:(n) n!!}
36291 @r{ n@: k e @: @: 1 @:euler@:(n)}
36292 @r{ n x@: H k e @: @: 2 @:euler@:(n,x)}
36293 @r{ n@: k f @: @: 4 @:prfac@:(n)}
36294 @r{ n m@: k g @: @: 2 @:gcd@:(n,m)}
36295 @r{ m n@: k h @: @: 14 @:shuffle@:(n,m)}
36296 @r{ n m@: k l @: @: 2 @:lcm@:(n,m)}
36297 @r{ n@: k m @: @: 1 @:moebius@:(n)}
36298 @r{ n@: k n @: @: 4 @:nextprime@:(n)}
36299 @r{ n@: I k n @: @: 4 @:prevprime@:(n)}
36300 @r{ n@: k p @: @: 4,28 @:calc-prime-test@:}
36301 @r{ m@: k r @: @: 14 @:random@:(m)}
36302 @r{ n m@: k s @: @: 2 @:stir1@:(n,m)}
36303 @r{ n m@: H k s @: @: 2 @:stir2@:(n,m)}
36304 @r{ n@: k t @: @: 1 @:totient@:(n)}
36307 @r{ n p x@: k B @: @: @:utpb@:(x,n,p)}
36308 @r{ n p x@: I k B @: @: @:ltpb@:(x,n,p)}
36309 @r{ v x@: k C @: @: @:utpc@:(x,v)}
36310 @r{ v x@: I k C @: @: @:ltpc@:(x,v)}
36311 @r{ n m@: k E @: @: @:egcd@:(n,m)}
36312 @r{v1 v2 x@: k F @: @: @:utpf@:(x,v1,v2)}
36313 @r{v1 v2 x@: I k F @: @: @:ltpf@:(x,v1,v2)}
36314 @r{ m s x@: k N @: @: @:utpn@:(x,m,s)}
36315 @r{ m s x@: I k N @: @: @:ltpn@:(x,m,s)}
36316 @r{ m x@: k P @: @: @:utpp@:(x,m)}
36317 @r{ m x@: I k P @: @: @:ltpp@:(x,m)}
36318 @r{ v x@: k T @: @: @:utpt@:(x,v)}
36319 @r{ v x@: I k T @: @: @:ltpt@:(x,v)}
36322 @r{ a b@: l + @: @: @:lupadd@:(a,b)}
36323 @r{ a b@: H l + @: @: @:lufadd@:(a,b)}
36324 @r{ a b@: l - @: @: @:lupsub@:(a,b)}
36325 @r{ a b@: H l - @: @: @:lufsub@:(a,b)}
36326 @r{ a b@: l * @: @: @:lupmul@:(a,b)}
36327 @r{ a b@: H l * @: @: @:lufmul@:(a,b)}
36328 @r{ a b@: l / @: @: @:lupdiv@:(a,b)}
36329 @r{ a b@: H l / @: @: @:lufdiv@:(a,b)}
36330 @r{ a@: l d @: @: @:dbpower@:(a)}
36331 @r{ a b@: O l d @: @: @:dbpower@:(a,b)}
36332 @r{ a@: H l d @: @: @:dbfield@:(a)}
36333 @r{ a b@: O H l d @: @: @:dbfield@:(a,b)}
36334 @r{ a@: l n @: @: @:nppower@:(a)}
36335 @r{ a b@: O l n @: @: @:nppower@:(a,b)}
36336 @r{ a@: H l n @: @: @:npfield@:(a)}
36337 @r{ a b@: O H l n @: @: @:npfield@:(a,b)}
36338 @r{ a@: l q @: @: @:lupquant@:(a)}
36339 @r{ a b@: O l q @: @: @:lupquant@:(a,b)}
36340 @r{ a@: H l q @: @: @:lufquant@:(a)}
36341 @r{ a b@: O H l q @: @: @:lufquant@:(a,b)}
36342 @r{ a@: l s @: @: @:spn@:(a)}
36343 @r{ a@: l m @: @: @:midi@:(a)}
36344 @r{ a@: l f @: @: @:freq@:(a)}
36347 @r{ @: m a @: @: 12,13 @:calc-algebraic-mode@:}
36348 @r{ @: m d @: @: @:calc-degrees-mode@:}
36349 @r{ @: m e @: @: @:calc-embedded-preserve-modes@:}
36350 @r{ @: m f @: @: 12 @:calc-frac-mode@:}
36351 @r{ @: m g @: @: 52 @:calc-get-modes@:}
36352 @r{ @: m h @: @: @:calc-hms-mode@:}
36353 @r{ @: m i @: @: 12,13 @:calc-infinite-mode@:}
36354 @r{ @: m m @: @: @:calc-save-modes@:}
36355 @r{ @: m p @: @: 12 @:calc-polar-mode@:}
36356 @r{ @: m r @: @: @:calc-radians-mode@:}
36357 @r{ @: m s @: @: 12 @:calc-symbolic-mode@:}
36358 @r{ @: m t @: @: 12 @:calc-total-algebraic-mode@:}
36359 @r{ @: m v @: @: 12,13 @:calc-matrix-mode@:}
36360 @r{ @: m w @: @: 13 @:calc-working@:}
36361 @r{ @: m x @: @: @:calc-always-load-extensions@:}
36364 @r{ @: m A @: @: 12 @:calc-alg-simplify-mode@:}
36365 @r{ @: m B @: @: 12 @:calc-bin-simplify-mode@:}
36366 @r{ @: m C @: @: 12 @:calc-auto-recompute@:}
36367 @r{ @: m D @: @: @:calc-default-simplify-mode@:}
36368 @r{ @: m E @: @: 12 @:calc-ext-simplify-mode@:}
36369 @r{ @: m F @:filename @: 13 @:calc-settings-file-name@:}
36370 @r{ @: m N @: @: 12 @:calc-num-simplify-mode@:}
36371 @r{ @: m O @: @: 12 @:calc-no-simplify-mode@:}
36372 @r{ @: m R @: @: 12,13 @:calc-mode-record-mode@:}
36373 @r{ @: m S @: @: 12 @:calc-shift-prefix@:}
36374 @r{ @: m U @: @: 12 @:calc-units-simplify-mode@:}
36377 @r{ @: r s @:register @: 27 @:calc-copy-to-register@:}
36378 @r{ @: r i @:register @: @:calc-insert-register@:}
36381 @r{ @: s c @:var1, var2 @: 29 @:calc-copy-variable@:}
36382 @r{ @: s d @:var, decl @: @:calc-declare-variable@:}
36383 @r{ @: s e @:var, editing @: 29,30 @:calc-edit-variable@:}
36384 @r{ @: s i @:buffer @: @:calc-insert-variables@:}
36385 @r{ @: s k @:const, var @: 29 @:calc-copy-special-constant@:}
36386 @r{ a b@: s l @:var @: 29 @:@:a (letting var=b)}
36387 @r{ a ...@: s m @:op, var @: 22,29 @:calc-store-map@:}
36388 @r{ @: s n @:var @: 29,47 @:calc-store-neg@: (v/-1)}
36389 @r{ @: s p @:var @: 29 @:calc-permanent-variable@:}
36390 @r{ @: s r @:var @: 29 @:@:v (recalled value)}
36391 @r{ @: r 0-9 @: @: @:calc-recall-quick@:}
36392 @r{ a@: s s @:var @: 28,29 @:calc-store@:}
36393 @r{ a@: s 0-9 @: @: @:calc-store-quick@:}
36394 @r{ a@: s t @:var @: 29 @:calc-store-into@:}
36395 @r{ a@: t 0-9 @: @: @:calc-store-into-quick@:}
36396 @r{ @: s u @:var @: 29 @:calc-unstore@:}
36397 @r{ a@: s x @:var @: 29 @:calc-store-exchange@:}
36400 @r{ @: s A @:editing @: 30 @:calc-edit-AlgSimpRules@:}
36401 @r{ @: s D @:editing @: 30 @:calc-edit-Decls@:}
36402 @r{ @: s E @:editing @: 30 @:calc-edit-EvalRules@:}
36403 @r{ @: s F @:editing @: 30 @:calc-edit-FitRules@:}
36404 @r{ @: s G @:editing @: 30 @:calc-edit-GenCount@:}
36405 @r{ @: s H @:editing @: 30 @:calc-edit-Holidays@:}
36406 @r{ @: s I @:editing @: 30 @:calc-edit-IntegLimit@:}
36407 @r{ @: s L @:editing @: 30 @:calc-edit-LineStyles@:}
36408 @r{ @: s P @:editing @: 30 @:calc-edit-PointStyles@:}
36409 @r{ @: s R @:editing @: 30 @:calc-edit-PlotRejects@:}
36410 @r{ @: s T @:editing @: 30 @:calc-edit-TimeZone@:}
36411 @r{ @: s U @:editing @: 30 @:calc-edit-Units@:}
36412 @r{ @: s X @:editing @: 30 @:calc-edit-ExtSimpRules@:}
36415 @r{ a@: s + @:var @: 29,47 @:calc-store-plus@: (v+a)}
36416 @r{ a@: s - @:var @: 29,47 @:calc-store-minus@: (v-a)}
36417 @r{ a@: s * @:var @: 29,47 @:calc-store-times@: (v*a)}
36418 @r{ a@: s / @:var @: 29,47 @:calc-store-div@: (v/a)}
36419 @r{ a@: s ^ @:var @: 29,47 @:calc-store-power@: (v^a)}
36420 @r{ a@: s | @:var @: 29,47 @:calc-store-concat@: (v|a)}
36421 @r{ @: s & @:var @: 29,47 @:calc-store-inv@: (v^-1)}
36422 @r{ @: s [ @:var @: 29,47 @:calc-store-decr@: (v-1)}
36423 @r{ @: s ] @:var @: 29,47 @:calc-store-incr@: (v-(-1))}
36424 @r{ a b@: s : @: @: 2 @:assign@:(a,b) a @tfn{:=} b}
36425 @r{ a@: s = @: @: 1 @:evalto@:(a,b) a @tfn{=>}}
36428 @r{ @: t [ @: @: 4 @:calc-trail-first@:}
36429 @r{ @: t ] @: @: 4 @:calc-trail-last@:}
36430 @r{ @: t < @: @: 4 @:calc-trail-scroll-left@:}
36431 @r{ @: t > @: @: 4 @:calc-trail-scroll-right@:}
36432 @r{ @: t . @: @: 12 @:calc-full-trail-vectors@:}
36435 @r{ @: t b @: @: 4 @:calc-trail-backward@:}
36436 @r{ @: t d @: @: 12,50 @:calc-trail-display@:}
36437 @r{ @: t f @: @: 4 @:calc-trail-forward@:}
36438 @r{ @: t h @: @: @:calc-trail-here@:}
36439 @r{ @: t i @: @: @:calc-trail-in@:}
36440 @r{ @: t k @: @: 4 @:calc-trail-kill@:}
36441 @r{ @: t m @:string @: @:calc-trail-marker@:}
36442 @r{ @: t n @: @: 4 @:calc-trail-next@:}
36443 @r{ @: t o @: @: @:calc-trail-out@:}
36444 @r{ @: t p @: @: 4 @:calc-trail-previous@:}
36445 @r{ @: t r @:string @: @:calc-trail-isearch-backward@:}
36446 @r{ @: t s @:string @: @:calc-trail-isearch-forward@:}
36447 @r{ @: t y @: @: 4 @:calc-trail-yank@:}
36450 @r{ d@: t C @:oz, nz @: @:tzconv@:(d,oz,nz)}
36451 @r{d oz nz@: t C @:$ @: @:tzconv@:(d,oz,nz)}
36452 @r{ d@: t D @: @: 15 @:date@:(d)}
36453 @r{ d@: t I @: @: 4 @:incmonth@:(d,n)}
36454 @r{ d@: t J @: @: 16 @:julian@:(d,z)}
36455 @r{ d@: t M @: @: 17 @:newmonth@:(d,n)}
36456 @r{ @: t N @: @: 16 @:now@:(z)}
36457 @r{ d@: t P @:1 @: 31 @:year@:(d)}
36458 @r{ d@: t P @:2 @: 31 @:month@:(d)}
36459 @r{ d@: t P @:3 @: 31 @:day@:(d)}
36460 @r{ d@: t P @:4 @: 31 @:hour@:(d)}
36461 @r{ d@: t P @:5 @: 31 @:minute@:(d)}
36462 @r{ d@: t P @:6 @: 31 @:second@:(d)}
36463 @r{ d@: t P @:7 @: 31 @:weekday@:(d)}
36464 @r{ d@: t P @:8 @: 31 @:yearday@:(d)}
36465 @r{ d@: t P @:9 @: 31 @:time@:(d)}
36466 @r{ d@: t U @: @: 16 @:unixtime@:(d,z)}
36467 @r{ d@: t W @: @: 17 @:newweek@:(d,w)}
36468 @r{ d@: t Y @: @: 17 @:newyear@:(d,n)}
36471 @r{ a b@: t + @: @: 2 @:badd@:(a,b)}
36472 @r{ a b@: t - @: @: 2 @:bsub@:(a,b)}
36475 @r{ @: u a @: @: 12 @:calc-autorange-units@:}
36476 @r{ a@: u b @: @: @:calc-base-units@:}
36477 @r{ a@: u c @:units @: 18 @:calc-convert-units@:}
36478 @r{ defn@: u d @:unit, descr @: @:calc-define-unit@:}
36479 @r{ @: u e @: @: @:calc-explain-units@:}
36480 @r{ @: u g @:unit @: @:calc-get-unit-definition@:}
36481 @r{ @: u p @: @: @:calc-permanent-units@:}
36482 @r{ a@: u r @: @: @:calc-remove-units@:}
36483 @r{ a@: u s @: @: @:usimplify@:(a)}
36484 @r{ a@: u t @:units @: 18 @:calc-convert-temperature@:}
36485 @r{ @: u u @:unit @: @:calc-undefine-unit@:}
36486 @r{ @: u v @: @: @:calc-enter-units-table@:}
36487 @r{ a@: u x @: @: @:calc-extract-units@:}
36488 @r{ a@: u 0-9 @: @: @:calc-quick-units@:}
36491 @r{ v1 v2@: u C @: @: 20 @:vcov@:(v1,v2)}
36492 @r{ v1 v2@: I u C @: @: 20 @:vpcov@:(v1,v2)}
36493 @r{ v1 v2@: H u C @: @: 20 @:vcorr@:(v1,v2)}
36494 @r{ v@: u G @: @: 19 @:vgmean@:(v)}
36495 @r{ a b@: H u G @: @: 2 @:agmean@:(a,b)}
36496 @r{ v@: u M @: @: 19 @:vmean@:(v)}
36497 @r{ v@: I u M @: @: 19 @:vmeane@:(v)}
36498 @r{ v@: H u M @: @: 19 @:vmedian@:(v)}
36499 @r{ v@: I H u M @: @: 19 @:vhmean@:(v)}
36500 @r{ v@: u N @: @: 19 @:vmin@:(v)}
36501 @r{ v@: u S @: @: 19 @:vsdev@:(v)}
36502 @r{ v@: I u S @: @: 19 @:vpsdev@:(v)}
36503 @r{ v@: H u S @: @: 19 @:vvar@:(v)}
36504 @r{ v@: I H u S @: @: 19 @:vpvar@:(v)}
36505 @r{ @: u V @: @: @:calc-view-units-table@:}
36506 @r{ v@: u X @: @: 19 @:vmax@:(v)}
36509 @r{ v@: u + @: @: 19 @:vsum@:(v)}
36510 @r{ v@: u * @: @: 19 @:vprod@:(v)}
36511 @r{ v@: u # @: @: 19 @:vcount@:(v)}
36514 @r{ @: V ( @: @: 50 @:calc-vector-parens@:}
36515 @r{ @: V @{ @: @: 50 @:calc-vector-braces@:}
36516 @r{ @: V [ @: @: 50 @:calc-vector-brackets@:}
36517 @r{ @: V ] @:ROCP @: 50 @:calc-matrix-brackets@:}
36518 @r{ @: V , @: @: 50 @:calc-vector-commas@:}
36519 @r{ @: V < @: @: 50 @:calc-matrix-left-justify@:}
36520 @r{ @: V = @: @: 50 @:calc-matrix-center-justify@:}
36521 @r{ @: V > @: @: 50 @:calc-matrix-right-justify@:}
36522 @r{ @: V / @: @: 12,50 @:calc-break-vectors@:}
36523 @r{ @: V . @: @: 12,50 @:calc-full-vectors@:}
36526 @r{ s t@: V ^ @: @: 2 @:vint@:(s,t)}
36527 @r{ s t@: V - @: @: 2 @:vdiff@:(s,t)}
36528 @r{ s@: V ~ @: @: 1 @:vcompl@:(s)}
36529 @r{ s@: V # @: @: 1 @:vcard@:(s)}
36530 @r{ s@: V : @: @: 1 @:vspan@:(s)}
36531 @r{ s@: V + @: @: 1 @:rdup@:(s)}
36534 @r{ m@: V & @: @: 1 @:inv@:(m) 1/m}
36537 @r{ v@: v a @:n @: @:arrange@:(v,n)}
36538 @r{ a@: v b @:n @: @:cvec@:(a,n)}
36539 @r{ v@: v c @:n >0 @: 21,31 @:mcol@:(v,n)}
36540 @r{ v@: v c @:n <0 @: 31 @:mrcol@:(v,-n)}
36541 @r{ m@: v c @:0 @: 31 @:getdiag@:(m)}
36542 @r{ v@: v d @: @: 25 @:diag@:(v,n)}
36543 @r{ v m@: v e @: @: 2 @:vexp@:(v,m)}
36544 @r{ v m f@: H v e @: @: 2 @:vexp@:(v,m,f)}
36545 @r{ v a@: v f @: @: 26 @:find@:(v,a,n)}
36546 @r{ v@: v h @: @: 1 @:head@:(v)}
36547 @r{ v@: I v h @: @: 1 @:tail@:(v)}
36548 @r{ v@: H v h @: @: 1 @:rhead@:(v)}
36549 @r{ v@: I H v h @: @: 1 @:rtail@:(v)}
36550 @r{ @: v i @:n @: 31 @:idn@:(1,n)}
36551 @r{ @: v i @:0 @: 31 @:idn@:(1)}
36552 @r{ h t@: v k @: @: 2 @:cons@:(h,t)}
36553 @r{ h t@: H v k @: @: 2 @:rcons@:(h,t)}
36554 @r{ v@: v l @: @: 1 @:vlen@:(v)}
36555 @r{ v@: H v l @: @: 1 @:mdims@:(v)}
36556 @r{ v m@: v m @: @: 2 @:vmask@:(v,m)}
36557 @r{ v@: v n @: @: 1 @:rnorm@:(v)}
36558 @r{ a b c@: v p @: @: 24 @:calc-pack@:}
36559 @r{ v@: v r @:n >0 @: 21,31 @:mrow@:(v,n)}
36560 @r{ v@: v r @:n <0 @: 31 @:mrrow@:(v,-n)}
36561 @r{ m@: v r @:0 @: 31 @:getdiag@:(m)}
36562 @r{ v i j@: v s @: @: @:subvec@:(v,i,j)}
36563 @r{ v i j@: I v s @: @: @:rsubvec@:(v,i,j)}
36564 @r{ m@: v t @: @: 1 @:trn@:(m)}
36565 @r{ v@: v u @: @: 24 @:calc-unpack@:}
36566 @r{ v@: v v @: @: 1 @:rev@:(v)}
36567 @r{ @: v x @:n @: 31 @:index@:(n)}
36568 @r{ n s i@: C-u v x @: @: @:index@:(n,s,i)}
36571 @r{ v@: V A @:op @: 22 @:apply@:(op,v)}
36572 @r{ v1 v2@: V C @: @: 2 @:cross@:(v1,v2)}
36573 @r{ m@: V D @: @: 1 @:det@:(m)}
36574 @r{ s@: V E @: @: 1 @:venum@:(s)}
36575 @r{ s@: V F @: @: 1 @:vfloor@:(s)}
36576 @r{ v@: V G @: @: @:grade@:(v)}
36577 @r{ v@: I V G @: @: @:rgrade@:(v)}
36578 @r{ v@: V H @:n @: 31 @:histogram@:(v,n)}
36579 @r{ v w@: H V H @:n @: 31 @:histogram@:(v,w,n)}
36580 @r{ v1 v2@: V I @:mop aop @: 22 @:inner@:(mop,aop,v1,v2)}
36581 @r{ m@: V J @: @: 1 @:ctrn@:(m)}
36582 @r{ m1 m2@: V K @: @: @:kron@:(m1,m2)}
36583 @r{ m@: V L @: @: 1 @:lud@:(m)}
36584 @r{ v@: V M @:op @: 22,23 @:map@:(op,v)}
36585 @r{ v@: V N @: @: 1 @:cnorm@:(v)}
36586 @r{ v1 v2@: V O @:op @: 22 @:outer@:(op,v1,v2)}
36587 @r{ v@: V R @:op @: 22,23 @:reduce@:(op,v)}
36588 @r{ v@: I V R @:op @: 22,23 @:rreduce@:(op,v)}
36589 @r{ a n@: H V R @:op @: 22 @:nest@:(op,a,n)}
36590 @r{ a@: I H V R @:op @: 22 @:fixp@:(op,a)}
36591 @r{ v@: V S @: @: @:sort@:(v)}
36592 @r{ v@: I V S @: @: @:rsort@:(v)}
36593 @r{ m@: V T @: @: 1 @:tr@:(m)}
36594 @r{ v@: V U @:op @: 22 @:accum@:(op,v)}
36595 @r{ v@: I V U @:op @: 22 @:raccum@:(op,v)}
36596 @r{ a n@: H V U @:op @: 22 @:anest@:(op,a,n)}
36597 @r{ a@: I H V U @:op @: 22 @:afixp@:(op,a)}
36598 @r{ s t@: V V @: @: 2 @:vunion@:(s,t)}
36599 @r{ s t@: V X @: @: 2 @:vxor@:(s,t)}
36602 @r{ @: Y @: @: @:@:user commands}
36605 @r{ @: z @: @: @:@:user commands}
36608 @r{ c@: Z [ @: @: 45 @:calc-kbd-if@:}
36609 @r{ c@: Z | @: @: 45 @:calc-kbd-else-if@:}
36610 @r{ @: Z : @: @: @:calc-kbd-else@:}
36611 @r{ @: Z ] @: @: @:calc-kbd-end-if@:}
36614 @r{ @: Z @{ @: @: 4 @:calc-kbd-loop@:}
36615 @r{ c@: Z / @: @: 45 @:calc-kbd-break@:}
36616 @r{ @: Z @} @: @: @:calc-kbd-end-loop@:}
36617 @r{ n@: Z < @: @: @:calc-kbd-repeat@:}
36618 @r{ @: Z > @: @: @:calc-kbd-end-repeat@:}
36619 @r{ n m@: Z ( @: @: @:calc-kbd-for@:}
36620 @r{ s@: Z ) @: @: @:calc-kbd-end-for@:}
36623 @r{ @: Z C-g @: @: @:@:cancel if/loop command}
36626 @r{ @: Z ` @: @: @:calc-kbd-push@:}
36627 @r{ @: Z ' @: @: @:calc-kbd-pop@:}
36628 @r{ @: Z # @: @: @:calc-kbd-query@:}
36631 @r{ comp@: Z C @:func, args @: 50 @:calc-user-define-composition@:}
36632 @r{ @: Z D @:key, command @: @:calc-user-define@:}
36633 @r{ @: Z E @:key, editing @: 30 @:calc-user-define-edit@:}
36634 @r{ defn@: Z F @:k, c, f, a, n@: 28 @:calc-user-define-formula@:}
36635 @r{ @: Z G @:key @: @:calc-get-user-defn@:}
36636 @r{ @: Z I @: @: @:calc-user-define-invocation@:}
36637 @r{ @: Z K @:key, command @: @:calc-user-define-kbd-macro@:}
36638 @r{ @: Z P @:key @: @:calc-user-define-permanent@:}
36639 @r{ @: Z S @: @: 30 @:calc-edit-user-syntax@:}
36640 @r{ @: Z T @: @: 12 @:calc-timing@:}
36641 @r{ @: Z U @:key @: @:calc-user-undefine@:}
36651 Positive prefix arguments apply to @expr{n} stack entries.
36652 Negative prefix arguments apply to the @expr{-n}th stack entry.
36653 A prefix of zero applies to the entire stack. (For @key{LFD} and
36654 @kbd{M-@key{DEL}}, the meaning of the sign is reversed.)
36658 Positive prefix arguments apply to @expr{n} stack entries.
36659 Negative prefix arguments apply to the top stack entry
36660 and the next @expr{-n} stack entries.
36664 Positive prefix arguments rotate top @expr{n} stack entries by one.
36665 Negative prefix arguments rotate the entire stack by @expr{-n}.
36666 A prefix of zero reverses the entire stack.
36670 Prefix argument specifies a repeat count or distance.
36674 Positive prefix arguments specify a precision @expr{p}.
36675 Negative prefix arguments reduce the current precision by @expr{-p}.
36679 A prefix argument is interpreted as an additional step-size parameter.
36680 A plain @kbd{C-u} prefix means to prompt for the step size.
36684 A prefix argument specifies simplification level and depth.
36685 1=Basic simplifications, 2=Algebraic simplifications, 3=Extended simplifications
36689 A negative prefix operates only on the top level of the input formula.
36693 Positive prefix arguments specify a word size of @expr{w} bits, unsigned.
36694 Negative prefix arguments specify a word size of @expr{w} bits, signed.
36698 Prefix arguments specify the shift amount @expr{n}. The @expr{w} argument
36699 cannot be specified in the keyboard version of this command.
36703 From the keyboard, @expr{d} is omitted and defaults to zero.
36707 Mode is toggled; a positive prefix always sets the mode, and a negative
36708 prefix always clears the mode.
36712 Some prefix argument values provide special variations of the mode.
36716 A prefix argument, if any, is used for @expr{m} instead of taking
36717 @expr{m} from the stack. @expr{M} may take any of these values:
36719 {@advance@tableindent10pt
36723 Random integer in the interval @expr{[0 .. m)}.
36725 Random floating-point number in the interval @expr{[0 .. m)}.
36727 Gaussian with mean 1 and standard deviation 0.
36729 Gaussian with specified mean and standard deviation.
36731 Random integer or floating-point number in that interval.
36733 Random element from the vector.
36741 A prefix argument from 1 to 6 specifies number of date components
36742 to remove from the stack. @xref{Date Conversions}.
36746 A prefix argument specifies a time zone; @kbd{C-u} says to take the
36747 time zone number or name from the top of the stack. @xref{Time Zones}.
36751 A prefix argument specifies a day number (0-6, 0-31, or 0-366).
36755 If the input has no units, you will be prompted for both the old and
36760 With a prefix argument, collect that many stack entries to form the
36761 input data set. Each entry may be a single value or a vector of values.
36765 With a prefix argument of 1, take a single
36766 @texline @var{n}@math{\times2}
36767 @infoline @mathit{@var{N}x2}
36768 matrix from the stack instead of two separate data vectors.
36772 The row or column number @expr{n} may be given as a numeric prefix
36773 argument instead. A plain @kbd{C-u} prefix says to take @expr{n}
36774 from the top of the stack. If @expr{n} is a vector or interval,
36775 a subvector/submatrix of the input is created.
36779 The @expr{op} prompt can be answered with the key sequence for the
36780 desired function, or with @kbd{x} or @kbd{z} followed by a function name,
36781 or with @kbd{$} to take a formula from the top of the stack, or with
36782 @kbd{'} and a typed formula. In the last two cases, the formula may
36783 be a nameless function like @samp{<#1+#2>} or @samp{<x, y : x+y>}, or it
36784 may include @kbd{$}, @kbd{$$}, etc. (where @kbd{$} will correspond to the
36785 last argument of the created function), or otherwise you will be
36786 prompted for an argument list. The number of vectors popped from the
36787 stack by @kbd{V M} depends on the number of arguments of the function.
36791 One of the mapping direction keys @kbd{_} (horizontal, i.e., map
36792 by rows or reduce across), @kbd{:} (vertical, i.e., map by columns or
36793 reduce down), or @kbd{=} (map or reduce by rows) may be used before
36794 entering @expr{op}; these modify the function name by adding the letter
36795 @code{r} for ``rows,'' @code{c} for ``columns,'' @code{a} for ``across,''
36796 or @code{d} for ``down.''
36800 The prefix argument specifies a packing mode. A nonnegative mode
36801 is the number of items (for @kbd{v p}) or the number of levels
36802 (for @kbd{v u}). A negative mode is as described below. With no
36803 prefix argument, the mode is taken from the top of the stack and
36804 may be an integer or a vector of integers.
36806 {@advance@tableindent-20pt
36810 (@var{2}) Rectangular complex number.
36812 (@var{2}) Polar complex number.
36814 (@var{3}) HMS form.
36816 (@var{2}) Error form.
36818 (@var{2}) Modulo form.
36820 (@var{2}) Closed interval.
36822 (@var{2}) Closed .. open interval.
36824 (@var{2}) Open .. closed interval.
36826 (@var{2}) Open interval.
36828 (@var{2}) Fraction.
36830 (@var{2}) Float with integer mantissa.
36832 (@var{2}) Float with mantissa in @expr{[1 .. 10)}.
36834 (@var{1}) Date form (using date numbers).
36836 (@var{3}) Date form (using year, month, day).
36838 (@var{6}) Date form (using year, month, day, hour, minute, second).
36846 A prefix argument specifies the size @expr{n} of the matrix. With no
36847 prefix argument, @expr{n} is omitted and the size is inferred from
36852 The prefix argument specifies the starting position @expr{n} (default 1).
36856 Cursor position within stack buffer affects this command.
36860 Arguments are not actually removed from the stack by this command.
36864 Variable name may be a single digit or a full name.
36868 Editing occurs in a separate buffer. Press @kbd{C-c C-c} (or
36869 @key{LFD}, or in some cases @key{RET}) to finish the edit, or kill the
36870 buffer with @kbd{C-x k} to cancel the edit. The @key{LFD} key prevents evaluation
36871 of the result of the edit.
36875 The number prompted for can also be provided as a prefix argument.
36879 Press this key a second time to cancel the prefix.
36883 With a negative prefix, deactivate all formulas. With a positive
36884 prefix, deactivate and then reactivate from scratch.
36888 Default is to scan for nearest formula delimiter symbols. With a
36889 prefix of zero, formula is delimited by mark and point. With a
36890 non-zero prefix, formula is delimited by scanning forward or
36891 backward by that many lines.
36895 Parse the region between point and mark as a vector. A nonzero prefix
36896 parses @var{n} lines before or after point as a vector. A zero prefix
36897 parses the current line as a vector. A @kbd{C-u} prefix parses the
36898 region between point and mark as a single formula.
36902 Parse the rectangle defined by point and mark as a matrix. A positive
36903 prefix @var{n} divides the rectangle into columns of width @var{n}.
36904 A zero or @kbd{C-u} prefix parses each line as one formula. A negative
36905 prefix suppresses special treatment of bracketed portions of a line.
36909 A numeric prefix causes the current language mode to be ignored.
36913 Responding to a prompt with a blank line answers that and all
36914 later prompts by popping additional stack entries.
36918 Answer for @expr{v} may also be of the form @expr{v = v_0} or
36923 With a positive prefix argument, stack contains many @expr{y}'s and one
36924 common @expr{x}. With a zero prefix, stack contains a vector of
36925 @expr{y}s and a common @expr{x}. With a negative prefix, stack
36926 contains many @expr{[x,y]} vectors. (For 3D plots, substitute
36927 @expr{z} for @expr{y} and @expr{x,y} for @expr{x}.)
36931 With any prefix argument, all curves in the graph are deleted.
36935 With a positive prefix, refines an existing plot with more data points.
36936 With a negative prefix, forces recomputation of the plot data.
36940 With any prefix argument, set the default value instead of the
36941 value for this graph.
36945 With a negative prefix argument, set the value for the printer.
36949 Condition is considered ``true'' if it is a nonzero real or complex
36950 number, or a formula whose value is known to be nonzero; it is ``false''
36955 Several formulas separated by commas are pushed as multiple stack
36956 entries. Trailing @kbd{)}, @kbd{]}, @kbd{@}}, @kbd{>}, and @kbd{"}
36957 delimiters may be omitted. The notation @kbd{$$$} refers to the value
36958 in stack level three, and causes the formula to replace the top three
36959 stack levels. The notation @kbd{$3} refers to stack level three without
36960 causing that value to be removed from the stack. Use @key{LFD} in place
36961 of @key{RET} to prevent evaluation; use @kbd{M-=} in place of @key{RET}
36962 to evaluate variables.
36966 The variable is replaced by the formula shown on the right. The
36967 Inverse flag reverses the order of the operands, e.g., @kbd{I s - x}
36969 @texline @math{x \coloneq a-x}.
36970 @infoline @expr{x := a-x}.
36974 Press @kbd{?} repeatedly to see how to choose a model. Answer the
36975 variables prompt with @expr{iv} or @expr{iv;pv} to specify
36976 independent and parameter variables. A positive prefix argument
36977 takes @mathit{@var{n}+1} vectors from the stack; a zero prefix takes a matrix
36978 and a vector from the stack.
36982 With a plain @kbd{C-u} prefix, replace the current region of the
36983 destination buffer with the yanked text instead of inserting.
36987 All stack entries are reformatted; the @kbd{H} prefix inhibits this.
36988 The @kbd{I} prefix sets the mode temporarily, redraws the top stack
36989 entry, then restores the original setting of the mode.
36993 A negative prefix sets the default 3D resolution instead of the
36994 default 2D resolution.
36998 This grabs a vector of the form [@var{prec}, @var{wsize}, @var{ssize},
36999 @var{radix}, @var{flfmt}, @var{ang}, @var{frac}, @var{symb}, @var{polar},
37000 @var{matrix}, @var{simp}, @var{inf}]. A prefix argument from 1 to 12
37001 grabs the @var{n}th mode value only.
37005 (Space is provided below for you to keep your own written notes.)
37013 @node Key Index, Command Index, Summary, Top
37014 @unnumbered Index of Key Sequences
37018 @node Command Index, Function Index, Key Index, Top
37019 @unnumbered Index of Calculator Commands
37021 Since all Calculator commands begin with the prefix @samp{calc-}, the
37022 @kbd{x} key has been provided as a variant of @kbd{M-x} which automatically
37023 types @samp{calc-} for you. Thus, @kbd{x last-args} is short for
37024 @kbd{M-x calc-last-args}.
37028 @node Function Index, Concept Index, Command Index, Top
37029 @unnumbered Index of Algebraic Functions
37031 This is a list of built-in functions and operators usable in algebraic
37032 expressions. Their full Lisp names are derived by adding the prefix
37033 @samp{calcFunc-}, as in @code{calcFunc-sqrt}.
37035 All functions except those noted with ``*'' have corresponding
37036 Calc keystrokes and can also be found in the Calc Summary.
37041 @node Concept Index, Variable Index, Function Index, Top
37042 @unnumbered Concept Index
37046 @node Variable Index, Lisp Function Index, Concept Index, Top
37047 @unnumbered Index of Variables
37049 The variables in this list that do not contain dashes are accessible
37050 as Calc variables. Add a @samp{var-} prefix to get the name of the
37051 corresponding Lisp variable.
37053 The remaining variables are Lisp variables suitable for @code{setq}ing
37054 in your Calc init file or @file{.emacs} file.
37058 @node Lisp Function Index, , Variable Index, Top
37059 @unnumbered Index of Lisp Math Functions
37061 The following functions are meant to be used with @code{defmath}, not
37062 @code{defun} definitions. For names that do not start with @samp{calc-},
37063 the corresponding full Lisp name is derived by adding a prefix of