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3132f345 CW |
1 | ;;; calcalg2.el --- more algebraic functions for Calc |
2 | ||
58ba2f8f | 3 | ;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004, |
8b72699e | 4 | ;; 2005, 2006, 2007, 2008 Free Software Foundation, Inc. |
3132f345 CW |
5 | |
6 | ;; Author: David Gillespie <daveg@synaptics.com> | |
e8fff8ed | 7 | ;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com> |
136211a9 EZ |
8 | |
9 | ;; This file is part of GNU Emacs. | |
10 | ||
7c671b23 GM |
11 | ;; GNU Emacs is free software; you can redistribute it and/or modify |
12 | ;; it under the terms of the GNU General Public License as published by | |
075969b4 | 13 | ;; the Free Software Foundation; either version 3, or (at your option) |
7c671b23 GM |
14 | ;; any later version. |
15 | ||
136211a9 | 16 | ;; GNU Emacs is distributed in the hope that it will be useful, |
7c671b23 GM |
17 | ;; but WITHOUT ANY WARRANTY; without even the implied warranty of |
18 | ;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
19 | ;; GNU General Public License for more details. | |
20 | ||
21 | ;; You should have received a copy of the GNU General Public License | |
22 | ;; along with GNU Emacs; see the file COPYING. If not, write to the | |
23 | ;; Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, | |
24 | ;; Boston, MA 02110-1301, USA. | |
136211a9 | 25 | |
3132f345 | 26 | ;;; Commentary: |
136211a9 | 27 | |
3132f345 | 28 | ;;; Code: |
136211a9 EZ |
29 | |
30 | ;; This file is autoloaded from calc-ext.el. | |
136211a9 | 31 | |
59f2c6c0 | 32 | (require 'calc-ext) |
136211a9 EZ |
33 | (require 'calc-macs) |
34 | ||
136211a9 EZ |
35 | (defun calc-derivative (var num) |
36 | (interactive "sDifferentiate with respect to: \np") | |
37 | (calc-slow-wrapper | |
3132f345 CW |
38 | (when (< num 0) |
39 | (error "Order of derivative must be positive")) | |
136211a9 EZ |
40 | (let ((func (if (calc-is-hyperbolic) 'calcFunc-tderiv 'calcFunc-deriv)) |
41 | n expr) | |
42 | (if (or (equal var "") (equal var "$")) | |
43 | (setq n 2 | |
44 | expr (calc-top-n 2) | |
45 | var (calc-top-n 1)) | |
46 | (setq var (math-read-expr var)) | |
3132f345 CW |
47 | (when (eq (car-safe var) 'error) |
48 | (error "Bad format in expression: %s" (nth 1 var))) | |
136211a9 EZ |
49 | (setq n 1 |
50 | expr (calc-top-n 1))) | |
51 | (while (>= (setq num (1- num)) 0) | |
52 | (setq expr (list func expr var))) | |
bf77c646 | 53 | (calc-enter-result n "derv" expr)))) |
136211a9 | 54 | |
b682966a JB |
55 | (defun calc-integral (var &optional arg) |
56 | (interactive "sIntegration variable: \nP") | |
57 | (if arg | |
58 | (calc-tabular-command 'calcFunc-integ "Integration" "intg" nil var nil nil) | |
59 | (calc-slow-wrapper | |
60 | (if (or (equal var "") (equal var "$")) | |
61 | (calc-enter-result 2 "intg" (list 'calcFunc-integ | |
62 | (calc-top-n 2) | |
63 | (calc-top-n 1))) | |
64 | (let ((var (math-read-expr var))) | |
65 | (if (eq (car-safe var) 'error) | |
66 | (error "Bad format in expression: %s" (nth 1 var))) | |
67 | (calc-enter-result 1 "intg" (list 'calcFunc-integ | |
68 | (calc-top-n 1) | |
69 | var))))))) | |
136211a9 EZ |
70 | |
71 | (defun calc-num-integral (&optional varname lowname highname) | |
72 | (interactive "sIntegration variable: ") | |
73 | (calc-tabular-command 'calcFunc-ninteg "Integration" "nint" | |
bf77c646 | 74 | nil varname lowname highname)) |
136211a9 EZ |
75 | |
76 | (defun calc-summation (arg &optional varname lowname highname) | |
77 | (interactive "P\nsSummation variable: ") | |
78 | (calc-tabular-command 'calcFunc-sum "Summation" "sum" | |
bf77c646 | 79 | arg varname lowname highname)) |
136211a9 EZ |
80 | |
81 | (defun calc-alt-summation (arg &optional varname lowname highname) | |
82 | (interactive "P\nsSummation variable: ") | |
83 | (calc-tabular-command 'calcFunc-asum "Summation" "asum" | |
bf77c646 | 84 | arg varname lowname highname)) |
136211a9 EZ |
85 | |
86 | (defun calc-product (arg &optional varname lowname highname) | |
87 | (interactive "P\nsIndex variable: ") | |
88 | (calc-tabular-command 'calcFunc-prod "Index" "prod" | |
bf77c646 | 89 | arg varname lowname highname)) |
136211a9 EZ |
90 | |
91 | (defun calc-tabulate (arg &optional varname lowname highname) | |
92 | (interactive "P\nsIndex variable: ") | |
93 | (calc-tabular-command 'calcFunc-table "Index" "tabl" | |
bf77c646 | 94 | arg varname lowname highname)) |
136211a9 EZ |
95 | |
96 | (defun calc-tabular-command (func prompt prefix arg varname lowname highname) | |
97 | (calc-slow-wrapper | |
98 | (let (var (low nil) (high nil) (step nil) stepname stepnum (num 1) expr) | |
99 | (if (consp arg) | |
100 | (setq stepnum 1) | |
101 | (setq stepnum 0)) | |
102 | (if (or (equal varname "") (equal varname "$") (null varname)) | |
103 | (setq high (calc-top-n (+ stepnum 1)) | |
104 | low (calc-top-n (+ stepnum 2)) | |
105 | var (calc-top-n (+ stepnum 3)) | |
106 | num (+ stepnum 4)) | |
107 | (setq var (if (stringp varname) (math-read-expr varname) varname)) | |
108 | (if (eq (car-safe var) 'error) | |
109 | (error "Bad format in expression: %s" (nth 1 var))) | |
110 | (or lowname | |
111 | (setq lowname (read-string (concat prompt " variable: " varname | |
112 | ", from: ")))) | |
113 | (if (or (equal lowname "") (equal lowname "$")) | |
114 | (setq high (calc-top-n (+ stepnum 1)) | |
115 | low (calc-top-n (+ stepnum 2)) | |
116 | num (+ stepnum 3)) | |
117 | (setq low (if (stringp lowname) (math-read-expr lowname) lowname)) | |
118 | (if (eq (car-safe low) 'error) | |
119 | (error "Bad format in expression: %s" (nth 1 low))) | |
120 | (or highname | |
121 | (setq highname (read-string (concat prompt " variable: " varname | |
122 | ", from: " lowname | |
123 | ", to: ")))) | |
124 | (if (or (equal highname "") (equal highname "$")) | |
125 | (setq high (calc-top-n (+ stepnum 1)) | |
126 | num (+ stepnum 2)) | |
127 | (setq high (if (stringp highname) (math-read-expr highname) | |
128 | highname)) | |
129 | (if (eq (car-safe high) 'error) | |
130 | (error "Bad format in expression: %s" (nth 1 high))) | |
131 | (if (consp arg) | |
132 | (progn | |
133 | (setq stepname (read-string (concat prompt " variable: " | |
134 | varname | |
135 | ", from: " lowname | |
136 | ", to: " highname | |
137 | ", step: "))) | |
138 | (if (or (equal stepname "") (equal stepname "$")) | |
139 | (setq step (calc-top-n 1) | |
140 | num 2) | |
141 | (setq step (math-read-expr stepname)) | |
142 | (if (eq (car-safe step) 'error) | |
143 | (error "Bad format in expression: %s" | |
144 | (nth 1 step))))))))) | |
145 | (or step | |
146 | (if (consp arg) | |
147 | (setq step (calc-top-n 1)) | |
148 | (if arg | |
149 | (setq step (prefix-numeric-value arg))))) | |
150 | (setq expr (calc-top-n num)) | |
151 | (calc-enter-result num prefix (append (list func expr var low high) | |
bf77c646 | 152 | (and step (list step))))))) |
136211a9 EZ |
153 | |
154 | (defun calc-solve-for (var) | |
bba984aa | 155 | (interactive "sVariable(s) to solve for: ") |
136211a9 EZ |
156 | (calc-slow-wrapper |
157 | (let ((func (if (calc-is-inverse) | |
158 | (if (calc-is-hyperbolic) 'calcFunc-ffinv 'calcFunc-finv) | |
159 | (if (calc-is-hyperbolic) 'calcFunc-fsolve 'calcFunc-solve)))) | |
160 | (if (or (equal var "") (equal var "$")) | |
161 | (calc-enter-result 2 "solv" (list func | |
162 | (calc-top-n 2) | |
163 | (calc-top-n 1))) | |
164 | (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var) | |
165 | (not (string-match "\\[" var))) | |
166 | (math-read-expr (concat "[" var "]")) | |
167 | (math-read-expr var)))) | |
168 | (if (eq (car-safe var) 'error) | |
169 | (error "Bad format in expression: %s" (nth 1 var))) | |
170 | (calc-enter-result 1 "solv" (list func | |
171 | (calc-top-n 1) | |
bf77c646 | 172 | var))))))) |
136211a9 EZ |
173 | |
174 | (defun calc-poly-roots (var) | |
175 | (interactive "sVariable to solve for: ") | |
176 | (calc-slow-wrapper | |
177 | (if (or (equal var "") (equal var "$")) | |
178 | (calc-enter-result 2 "prts" (list 'calcFunc-roots | |
179 | (calc-top-n 2) | |
180 | (calc-top-n 1))) | |
181 | (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var) | |
182 | (not (string-match "\\[" var))) | |
183 | (math-read-expr (concat "[" var "]")) | |
184 | (math-read-expr var)))) | |
185 | (if (eq (car-safe var) 'error) | |
186 | (error "Bad format in expression: %s" (nth 1 var))) | |
187 | (calc-enter-result 1 "prts" (list 'calcFunc-roots | |
188 | (calc-top-n 1) | |
bf77c646 | 189 | var)))))) |
136211a9 EZ |
190 | |
191 | (defun calc-taylor (var nterms) | |
192 | (interactive "sTaylor expansion variable: \nNNumber of terms: ") | |
193 | (calc-slow-wrapper | |
194 | (let ((var (math-read-expr var))) | |
195 | (if (eq (car-safe var) 'error) | |
196 | (error "Bad format in expression: %s" (nth 1 var))) | |
197 | (calc-enter-result 1 "tylr" (list 'calcFunc-taylor | |
198 | (calc-top-n 1) | |
199 | var | |
bf77c646 | 200 | (prefix-numeric-value nterms)))))) |
136211a9 EZ |
201 | |
202 | ||
f7adda54 JB |
203 | ;; The following are global variables used by math-derivative and some |
204 | ;; related functions | |
205 | (defvar math-deriv-var) | |
206 | (defvar math-deriv-total) | |
207 | (defvar math-deriv-symb) | |
9d871e6d JB |
208 | (defvar math-decls-cache) |
209 | (defvar math-decls-all) | |
f7adda54 JB |
210 | |
211 | (defun math-derivative (expr) | |
212 | (cond ((equal expr math-deriv-var) | |
136211a9 EZ |
213 | 1) |
214 | ((or (Math-scalarp expr) | |
215 | (eq (car expr) 'sdev) | |
216 | (and (eq (car expr) 'var) | |
f7adda54 | 217 | (or (not math-deriv-total) |
136211a9 EZ |
218 | (math-const-var expr) |
219 | (progn | |
220 | (math-setup-declarations) | |
221 | (memq 'const (nth 1 (or (assq (nth 2 expr) | |
222 | math-decls-cache) | |
223 | math-decls-all))))))) | |
224 | 0) | |
225 | ((eq (car expr) '+) | |
226 | (math-add (math-derivative (nth 1 expr)) | |
227 | (math-derivative (nth 2 expr)))) | |
228 | ((eq (car expr) '-) | |
229 | (math-sub (math-derivative (nth 1 expr)) | |
230 | (math-derivative (nth 2 expr)))) | |
231 | ((memq (car expr) '(calcFunc-eq calcFunc-neq calcFunc-lt | |
232 | calcFunc-gt calcFunc-leq calcFunc-geq)) | |
233 | (list (car expr) | |
234 | (math-derivative (nth 1 expr)) | |
235 | (math-derivative (nth 2 expr)))) | |
236 | ((eq (car expr) 'neg) | |
237 | (math-neg (math-derivative (nth 1 expr)))) | |
238 | ((eq (car expr) '*) | |
239 | (math-add (math-mul (nth 2 expr) | |
240 | (math-derivative (nth 1 expr))) | |
241 | (math-mul (nth 1 expr) | |
242 | (math-derivative (nth 2 expr))))) | |
243 | ((eq (car expr) '/) | |
244 | (math-sub (math-div (math-derivative (nth 1 expr)) | |
245 | (nth 2 expr)) | |
246 | (math-div (math-mul (nth 1 expr) | |
247 | (math-derivative (nth 2 expr))) | |
248 | (math-sqr (nth 2 expr))))) | |
249 | ((eq (car expr) '^) | |
250 | (let ((du (math-derivative (nth 1 expr))) | |
251 | (dv (math-derivative (nth 2 expr)))) | |
252 | (or (Math-zerop du) | |
253 | (setq du (math-mul (nth 2 expr) | |
254 | (math-mul (math-normalize | |
255 | (list '^ | |
256 | (nth 1 expr) | |
257 | (math-add (nth 2 expr) -1))) | |
258 | du)))) | |
259 | (or (Math-zerop dv) | |
260 | (setq dv (math-mul (math-normalize | |
261 | (list 'calcFunc-ln (nth 1 expr))) | |
262 | (math-mul expr dv)))) | |
263 | (math-add du dv))) | |
264 | ((eq (car expr) '%) | |
265 | (math-derivative (nth 1 expr))) ; a reasonable definition | |
266 | ((eq (car expr) 'vec) | |
267 | (math-map-vec 'math-derivative expr)) | |
268 | ((and (memq (car expr) '(calcFunc-conj calcFunc-re calcFunc-im)) | |
269 | (= (length expr) 2)) | |
270 | (list (car expr) (math-derivative (nth 1 expr)))) | |
271 | ((and (memq (car expr) '(calcFunc-subscr calcFunc-mrow calcFunc-mcol)) | |
272 | (= (length expr) 3)) | |
273 | (let ((d (math-derivative (nth 1 expr)))) | |
274 | (if (math-numberp d) | |
275 | 0 ; assume x and x_1 are independent vars | |
276 | (list (car expr) d (nth 2 expr))))) | |
277 | (t (or (and (symbolp (car expr)) | |
278 | (if (= (length expr) 2) | |
279 | (let ((handler (get (car expr) 'math-derivative))) | |
280 | (and handler | |
281 | (let ((deriv (math-derivative (nth 1 expr)))) | |
282 | (if (Math-zerop deriv) | |
283 | deriv | |
284 | (math-mul (funcall handler (nth 1 expr)) | |
285 | deriv))))) | |
286 | (let ((handler (get (car expr) 'math-derivative-n))) | |
287 | (and handler | |
288 | (funcall handler expr))))) | |
f7adda54 | 289 | (and (not (eq math-deriv-symb 'pre-expand)) |
136211a9 EZ |
290 | (let ((exp (math-expand-formula expr))) |
291 | (and exp | |
f7adda54 | 292 | (or (let ((math-deriv-symb 'pre-expand)) |
136211a9 EZ |
293 | (catch 'math-deriv (math-derivative expr))) |
294 | (math-derivative exp))))) | |
295 | (if (or (Math-objvecp expr) | |
296 | (eq (car expr) 'var) | |
297 | (not (symbolp (car expr)))) | |
f7adda54 | 298 | (if math-deriv-symb |
136211a9 | 299 | (throw 'math-deriv nil) |
f7adda54 | 300 | (list (if math-deriv-total 'calcFunc-tderiv 'calcFunc-deriv) |
136211a9 | 301 | expr |
f7adda54 | 302 | math-deriv-var)) |
136211a9 EZ |
303 | (let ((accum 0) |
304 | (arg expr) | |
305 | (n 1) | |
306 | derv) | |
307 | (while (setq arg (cdr arg)) | |
308 | (or (Math-zerop (setq derv (math-derivative (car arg)))) | |
309 | (let ((func (intern (concat (symbol-name (car expr)) | |
310 | "'" | |
311 | (if (> n 1) | |
312 | (int-to-string n) | |
313 | "")))) | |
314 | (prop (cond ((= (length expr) 2) | |
315 | 'math-derivative-1) | |
316 | ((= (length expr) 3) | |
317 | 'math-derivative-2) | |
318 | ((= (length expr) 4) | |
319 | 'math-derivative-3) | |
320 | ((= (length expr) 5) | |
321 | 'math-derivative-4) | |
322 | ((= (length expr) 6) | |
323 | 'math-derivative-5)))) | |
324 | (setq accum | |
325 | (math-add | |
326 | accum | |
327 | (math-mul | |
328 | derv | |
329 | (let ((handler (get func prop))) | |
330 | (or (and prop handler | |
331 | (apply handler (cdr expr))) | |
f7adda54 | 332 | (if (and math-deriv-symb |
136211a9 EZ |
333 | (not (get func |
334 | 'calc-user-defn))) | |
335 | (throw 'math-deriv nil) | |
336 | (cons func (cdr expr)))))))))) | |
337 | (setq n (1+ n))) | |
bf77c646 | 338 | accum)))))) |
136211a9 | 339 | |
f7adda54 JB |
340 | (defun calcFunc-deriv (expr math-deriv-var &optional deriv-value math-deriv-symb) |
341 | (let* ((math-deriv-total nil) | |
136211a9 EZ |
342 | (res (catch 'math-deriv (math-derivative expr)))) |
343 | (or (eq (car-safe res) 'calcFunc-deriv) | |
344 | (null res) | |
345 | (setq res (math-normalize res))) | |
346 | (and res | |
347 | (if deriv-value | |
f7adda54 | 348 | (math-expr-subst res math-deriv-var deriv-value) |
bf77c646 | 349 | res)))) |
136211a9 | 350 | |
f7adda54 | 351 | (defun calcFunc-tderiv (expr math-deriv-var &optional deriv-value math-deriv-symb) |
136211a9 | 352 | (math-setup-declarations) |
f7adda54 | 353 | (let* ((math-deriv-total t) |
136211a9 EZ |
354 | (res (catch 'math-deriv (math-derivative expr)))) |
355 | (or (eq (car-safe res) 'calcFunc-tderiv) | |
356 | (null res) | |
357 | (setq res (math-normalize res))) | |
358 | (and res | |
359 | (if deriv-value | |
f7adda54 | 360 | (math-expr-subst res math-deriv-var deriv-value) |
bf77c646 | 361 | res)))) |
136211a9 EZ |
362 | |
363 | (put 'calcFunc-inv\' 'math-derivative-1 | |
364 | (function (lambda (u) (math-neg (math-div 1 (math-sqr u)))))) | |
365 | ||
366 | (put 'calcFunc-sqrt\' 'math-derivative-1 | |
367 | (function (lambda (u) (math-div 1 (math-mul 2 (list 'calcFunc-sqrt u)))))) | |
368 | ||
369 | (put 'calcFunc-deg\' 'math-derivative-1 | |
370 | (function (lambda (u) (math-div-float '(float 18 1) (math-pi))))) | |
371 | ||
372 | (put 'calcFunc-rad\' 'math-derivative-1 | |
373 | (function (lambda (u) (math-pi-over-180)))) | |
374 | ||
375 | (put 'calcFunc-ln\' 'math-derivative-1 | |
376 | (function (lambda (u) (math-div 1 u)))) | |
377 | ||
378 | (put 'calcFunc-log10\' 'math-derivative-1 | |
379 | (function (lambda (u) | |
380 | (math-div (math-div 1 (math-normalize '(calcFunc-ln 10))) | |
381 | u)))) | |
382 | ||
383 | (put 'calcFunc-lnp1\' 'math-derivative-1 | |
384 | (function (lambda (u) (math-div 1 (math-add u 1))))) | |
385 | ||
386 | (put 'calcFunc-log\' 'math-derivative-2 | |
387 | (function (lambda (x b) | |
388 | (and (not (Math-zerop b)) | |
389 | (let ((lnv (math-normalize | |
390 | (list 'calcFunc-ln b)))) | |
391 | (math-div 1 (math-mul lnv x))))))) | |
392 | ||
393 | (put 'calcFunc-log\'2 'math-derivative-2 | |
394 | (function (lambda (x b) | |
395 | (let ((lnv (list 'calcFunc-ln b))) | |
396 | (math-neg (math-div (list 'calcFunc-log x b) | |
397 | (math-mul lnv b))))))) | |
398 | ||
399 | (put 'calcFunc-exp\' 'math-derivative-1 | |
400 | (function (lambda (u) (math-normalize (list 'calcFunc-exp u))))) | |
401 | ||
402 | (put 'calcFunc-expm1\' 'math-derivative-1 | |
403 | (function (lambda (u) (math-normalize (list 'calcFunc-expm1 u))))) | |
404 | ||
405 | (put 'calcFunc-sin\' 'math-derivative-1 | |
406 | (function (lambda (u) (math-to-radians-2 (math-normalize | |
407 | (list 'calcFunc-cos u)))))) | |
408 | ||
409 | (put 'calcFunc-cos\' 'math-derivative-1 | |
410 | (function (lambda (u) (math-neg (math-to-radians-2 | |
411 | (math-normalize | |
412 | (list 'calcFunc-sin u))))))) | |
413 | ||
414 | (put 'calcFunc-tan\' 'math-derivative-1 | |
415 | (function (lambda (u) (math-to-radians-2 | |
dbf954fb JB |
416 | (math-sqr |
417 | (math-normalize | |
418 | (list 'calcFunc-sec u))))))) | |
136211a9 | 419 | |
9274224b JB |
420 | (put 'calcFunc-sec\' 'math-derivative-1 |
421 | (function (lambda (u) (math-to-radians-2 | |
422 | (math-mul | |
423 | (math-normalize | |
424 | (list 'calcFunc-sec u)) | |
425 | (math-normalize | |
426 | (list 'calcFunc-tan u))))))) | |
427 | ||
428 | (put 'calcFunc-csc\' 'math-derivative-1 | |
429 | (function (lambda (u) (math-neg | |
430 | (math-to-radians-2 | |
431 | (math-mul | |
432 | (math-normalize | |
433 | (list 'calcFunc-csc u)) | |
434 | (math-normalize | |
435 | (list 'calcFunc-cot u)))))))) | |
436 | ||
437 | (put 'calcFunc-cot\' 'math-derivative-1 | |
438 | (function (lambda (u) (math-neg | |
439 | (math-to-radians-2 | |
dbf954fb JB |
440 | (math-sqr |
441 | (math-normalize | |
442 | (list 'calcFunc-csc u)))))))) | |
9274224b | 443 | |
136211a9 EZ |
444 | (put 'calcFunc-arcsin\' 'math-derivative-1 |
445 | (function (lambda (u) | |
446 | (math-from-radians-2 | |
447 | (math-div 1 (math-normalize | |
448 | (list 'calcFunc-sqrt | |
449 | (math-sub 1 (math-sqr u))))))))) | |
450 | ||
451 | (put 'calcFunc-arccos\' 'math-derivative-1 | |
452 | (function (lambda (u) | |
453 | (math-from-radians-2 | |
454 | (math-div -1 (math-normalize | |
455 | (list 'calcFunc-sqrt | |
456 | (math-sub 1 (math-sqr u))))))))) | |
457 | ||
458 | (put 'calcFunc-arctan\' 'math-derivative-1 | |
459 | (function (lambda (u) (math-from-radians-2 | |
460 | (math-div 1 (math-add 1 (math-sqr u))))))) | |
461 | ||
462 | (put 'calcFunc-sinh\' 'math-derivative-1 | |
463 | (function (lambda (u) (math-normalize (list 'calcFunc-cosh u))))) | |
464 | ||
465 | (put 'calcFunc-cosh\' 'math-derivative-1 | |
466 | (function (lambda (u) (math-normalize (list 'calcFunc-sinh u))))) | |
467 | ||
468 | (put 'calcFunc-tanh\' 'math-derivative-1 | |
dbf954fb JB |
469 | (function (lambda (u) (math-sqr |
470 | (math-normalize | |
471 | (list 'calcFunc-sech u)))))) | |
136211a9 | 472 | |
9274224b JB |
473 | (put 'calcFunc-sech\' 'math-derivative-1 |
474 | (function (lambda (u) (math-neg | |
475 | (math-mul | |
476 | (math-normalize (list 'calcFunc-sech u)) | |
477 | (math-normalize (list 'calcFunc-tanh u))))))) | |
478 | ||
479 | (put 'calcFunc-csch\' 'math-derivative-1 | |
480 | (function (lambda (u) (math-neg | |
481 | (math-mul | |
482 | (math-normalize (list 'calcFunc-csch u)) | |
483 | (math-normalize (list 'calcFunc-coth u))))))) | |
484 | ||
dbf954fb | 485 | (put 'calcFunc-coth\' 'math-derivative-1 |
9274224b | 486 | (function (lambda (u) (math-neg |
dbf954fb JB |
487 | (math-sqr |
488 | (math-normalize | |
489 | (list 'calcFunc-csch u))))))) | |
9274224b | 490 | |
136211a9 EZ |
491 | (put 'calcFunc-arcsinh\' 'math-derivative-1 |
492 | (function (lambda (u) | |
493 | (math-div 1 (math-normalize | |
494 | (list 'calcFunc-sqrt | |
495 | (math-add (math-sqr u) 1))))))) | |
496 | ||
497 | (put 'calcFunc-arccosh\' 'math-derivative-1 | |
498 | (function (lambda (u) | |
499 | (math-div 1 (math-normalize | |
500 | (list 'calcFunc-sqrt | |
501 | (math-add (math-sqr u) -1))))))) | |
502 | ||
503 | (put 'calcFunc-arctanh\' 'math-derivative-1 | |
504 | (function (lambda (u) (math-div 1 (math-sub 1 (math-sqr u)))))) | |
505 | ||
506 | (put 'calcFunc-bern\'2 'math-derivative-2 | |
507 | (function (lambda (n x) | |
508 | (math-mul n (list 'calcFunc-bern (math-add n -1) x))))) | |
509 | ||
510 | (put 'calcFunc-euler\'2 'math-derivative-2 | |
511 | (function (lambda (n x) | |
512 | (math-mul n (list 'calcFunc-euler (math-add n -1) x))))) | |
513 | ||
514 | (put 'calcFunc-gammag\'2 'math-derivative-2 | |
515 | (function (lambda (a x) (math-deriv-gamma a x 1)))) | |
516 | ||
517 | (put 'calcFunc-gammaG\'2 'math-derivative-2 | |
518 | (function (lambda (a x) (math-deriv-gamma a x -1)))) | |
519 | ||
520 | (put 'calcFunc-gammaP\'2 'math-derivative-2 | |
521 | (function (lambda (a x) (math-deriv-gamma a x | |
522 | (math-div | |
523 | 1 (math-normalize | |
524 | (list 'calcFunc-gamma | |
525 | a))))))) | |
526 | ||
527 | (put 'calcFunc-gammaQ\'2 'math-derivative-2 | |
528 | (function (lambda (a x) (math-deriv-gamma a x | |
529 | (math-div | |
530 | -1 (math-normalize | |
531 | (list 'calcFunc-gamma | |
532 | a))))))) | |
533 | ||
534 | (defun math-deriv-gamma (a x scale) | |
535 | (math-mul scale | |
536 | (math-mul (math-pow x (math-add a -1)) | |
bf77c646 | 537 | (list 'calcFunc-exp (math-neg x))))) |
136211a9 EZ |
538 | |
539 | (put 'calcFunc-betaB\' 'math-derivative-3 | |
540 | (function (lambda (x a b) (math-deriv-beta x a b 1)))) | |
541 | ||
542 | (put 'calcFunc-betaI\' 'math-derivative-3 | |
543 | (function (lambda (x a b) (math-deriv-beta x a b | |
544 | (math-div | |
545 | 1 (list 'calcFunc-beta | |
546 | a b)))))) | |
547 | ||
548 | (defun math-deriv-beta (x a b scale) | |
549 | (math-mul (math-mul (math-pow x (math-add a -1)) | |
550 | (math-pow (math-sub 1 x) (math-add b -1))) | |
bf77c646 | 551 | scale)) |
136211a9 EZ |
552 | |
553 | (put 'calcFunc-erf\' 'math-derivative-1 | |
554 | (function (lambda (x) (math-div 2 | |
555 | (math-mul (list 'calcFunc-exp | |
556 | (math-sqr x)) | |
557 | (if calc-symbolic-mode | |
558 | '(calcFunc-sqrt | |
559 | (var pi var-pi)) | |
560 | (math-sqrt-pi))))))) | |
561 | ||
562 | (put 'calcFunc-erfc\' 'math-derivative-1 | |
563 | (function (lambda (x) (math-div -2 | |
564 | (math-mul (list 'calcFunc-exp | |
565 | (math-sqr x)) | |
566 | (if calc-symbolic-mode | |
567 | '(calcFunc-sqrt | |
568 | (var pi var-pi)) | |
569 | (math-sqrt-pi))))))) | |
570 | ||
571 | (put 'calcFunc-besJ\'2 'math-derivative-2 | |
572 | (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besJ | |
573 | (math-add v -1) | |
574 | z) | |
575 | (list 'calcFunc-besJ | |
576 | (math-add v 1) | |
577 | z)) | |
578 | 2)))) | |
579 | ||
580 | (put 'calcFunc-besY\'2 'math-derivative-2 | |
581 | (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besY | |
582 | (math-add v -1) | |
583 | z) | |
584 | (list 'calcFunc-besY | |
585 | (math-add v 1) | |
586 | z)) | |
587 | 2)))) | |
588 | ||
589 | (put 'calcFunc-sum 'math-derivative-n | |
590 | (function | |
591 | (lambda (expr) | |
f7adda54 | 592 | (if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var) |
136211a9 EZ |
593 | (throw 'math-deriv nil) |
594 | (cons 'calcFunc-sum | |
595 | (cons (math-derivative (nth 1 expr)) | |
596 | (cdr (cdr expr)))))))) | |
597 | ||
598 | (put 'calcFunc-prod 'math-derivative-n | |
599 | (function | |
600 | (lambda (expr) | |
f7adda54 | 601 | (if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var) |
136211a9 EZ |
602 | (throw 'math-deriv nil) |
603 | (math-mul expr | |
604 | (cons 'calcFunc-sum | |
605 | (cons (math-div (math-derivative (nth 1 expr)) | |
606 | (nth 1 expr)) | |
607 | (cdr (cdr expr))))))))) | |
608 | ||
609 | (put 'calcFunc-integ 'math-derivative-n | |
610 | (function | |
611 | (lambda (expr) | |
612 | (if (= (length expr) 3) | |
f7adda54 | 613 | (if (equal (nth 2 expr) math-deriv-var) |
136211a9 EZ |
614 | (nth 1 expr) |
615 | (math-normalize | |
616 | (list 'calcFunc-integ | |
617 | (math-derivative (nth 1 expr)) | |
618 | (nth 2 expr)))) | |
619 | (if (= (length expr) 5) | |
620 | (let ((lower (math-expr-subst (nth 1 expr) (nth 2 expr) | |
621 | (nth 3 expr))) | |
622 | (upper (math-expr-subst (nth 1 expr) (nth 2 expr) | |
623 | (nth 4 expr)))) | |
624 | (math-add (math-sub (math-mul upper | |
625 | (math-derivative (nth 4 expr))) | |
626 | (math-mul lower | |
627 | (math-derivative (nth 3 expr)))) | |
f7adda54 | 628 | (if (equal (nth 2 expr) math-deriv-var) |
136211a9 EZ |
629 | 0 |
630 | (math-normalize | |
631 | (list 'calcFunc-integ | |
632 | (math-derivative (nth 1 expr)) (nth 2 expr) | |
633 | (nth 3 expr) (nth 4 expr))))))))))) | |
634 | ||
635 | (put 'calcFunc-if 'math-derivative-n | |
636 | (function | |
637 | (lambda (expr) | |
638 | (and (= (length expr) 4) | |
639 | (list 'calcFunc-if (nth 1 expr) | |
640 | (math-derivative (nth 2 expr)) | |
641 | (math-derivative (nth 3 expr))))))) | |
642 | ||
643 | (put 'calcFunc-subscr 'math-derivative-n | |
644 | (function | |
645 | (lambda (expr) | |
646 | (and (= (length expr) 3) | |
647 | (list 'calcFunc-subscr (nth 1 expr) | |
648 | (math-derivative (nth 2 expr))))))) | |
649 | ||
650 | ||
3132f345 CW |
651 | (defvar math-integ-var '(var X ---)) |
652 | (defvar math-integ-var-2 '(var Y ---)) | |
653 | (defvar math-integ-vars (list 'f math-integ-var math-integ-var-2)) | |
654 | (defvar math-integ-var-list (list math-integ-var)) | |
655 | (defvar math-integ-var-list-list (list math-integ-var-list)) | |
136211a9 | 656 | |
f7adda54 JB |
657 | ;; math-integ-depth is a local variable for math-try-integral, but is used |
658 | ;; by math-integral and math-tracing-integral | |
659 | ;; which are called (directly or indirectly) by math-try-integral. | |
660 | (defvar math-integ-depth) | |
661 | ;; math-integ-level is a local variable for math-try-integral, but is used | |
662 | ;; by math-integral, math-do-integral, math-tracing-integral, | |
663 | ;; math-sub-integration, math-integrate-by-parts and | |
664 | ;; math-integrate-by-substitution, which are called (directly or | |
665 | ;; indirectly) by math-try-integral. | |
666 | (defvar math-integ-level) | |
667 | ;; math-integral-limit is a local variable for calcFunc-integ, but is | |
668 | ;; used by math-tracing-integral, math-sub-integration and | |
669 | ;; math-try-integration. | |
670 | (defvar math-integral-limit) | |
671 | ||
136211a9 EZ |
672 | (defmacro math-tracing-integral (&rest parts) |
673 | (list 'and | |
674 | 'trace-buffer | |
675 | (list 'save-excursion | |
676 | '(set-buffer trace-buffer) | |
677 | '(goto-char (point-max)) | |
678 | (list 'and | |
679 | '(bolp) | |
680 | '(insert (make-string (- math-integral-limit | |
681 | math-integ-level) 32) | |
682 | (format "%2d " math-integ-depth) | |
683 | (make-string math-integ-level 32))) | |
684 | ;;(list 'condition-case 'err | |
685 | (cons 'insert parts) | |
686 | ;; '(error (insert (prin1-to-string err)))) | |
bf77c646 | 687 | '(sit-for 0)))) |
136211a9 EZ |
688 | |
689 | ;;; The following wrapper caches results and avoids infinite recursion. | |
690 | ;;; Each cache entry is: ( A B ) Integral of A is B; | |
691 | ;;; ( A N ) Integral of A failed at level N; | |
692 | ;;; ( A busy ) Currently working on integral of A; | |
693 | ;;; ( A parts ) Currently working, integ-by-parts; | |
694 | ;;; ( A parts2 ) Currently working, integ-by-parts; | |
695 | ;;; ( A cancelled ) Ignore this cache entry; | |
f7adda54 JB |
696 | ;;; ( A [B] ) Same result as for math-cur-record = B. |
697 | ||
698 | ;; math-cur-record is a local variable for math-try-integral, but is used | |
699 | ;; by math-integral, math-replace-integral-parts and math-integrate-by-parts | |
700 | ;; which are called (directly or indirectly) by math-try-integral, as well as | |
701 | ;; by calc-dump-integral-cache | |
702 | (defvar math-cur-record) | |
703 | ;; math-enable-subst and math-any-substs are local variables for | |
704 | ;; calcFunc-integ, but are used by math-integral and math-try-integral. | |
705 | (defvar math-enable-subst) | |
706 | (defvar math-any-substs) | |
707 | ||
708 | ;; math-integ-msg is a local variable for math-try-integral, but is | |
709 | ;; used (both locally and non-locally) by math-integral. | |
710 | (defvar math-integ-msg) | |
711 | ||
712 | (defvar math-integral-cache nil) | |
713 | (defvar math-integral-cache-state nil) | |
714 | ||
136211a9 | 715 | (defun math-integral (expr &optional simplify same-as-above) |
f7adda54 JB |
716 | (let* ((simp math-cur-record) |
717 | (math-cur-record (assoc expr math-integral-cache)) | |
136211a9 EZ |
718 | (math-integ-depth (1+ math-integ-depth)) |
719 | (val 'cancelled)) | |
720 | (math-tracing-integral "Integrating " | |
721 | (math-format-value expr 1000) | |
722 | "...\n") | |
f7adda54 | 723 | (and math-cur-record |
136211a9 EZ |
724 | (progn |
725 | (math-tracing-integral "Found " | |
f7adda54 JB |
726 | (math-format-value (nth 1 math-cur-record) 1000)) |
727 | (and (consp (nth 1 math-cur-record)) | |
728 | (math-replace-integral-parts math-cur-record)) | |
136211a9 | 729 | (math-tracing-integral " => " |
f7adda54 | 730 | (math-format-value (nth 1 math-cur-record) 1000) |
136211a9 | 731 | "\n"))) |
f7adda54 JB |
732 | (or (and math-cur-record |
733 | (not (eq (nth 1 math-cur-record) 'cancelled)) | |
734 | (or (not (integerp (nth 1 math-cur-record))) | |
735 | (>= (nth 1 math-cur-record) math-integ-level))) | |
136211a9 EZ |
736 | (and (math-integral-contains-parts expr) |
737 | (progn | |
738 | (setq val nil) | |
739 | t)) | |
740 | (unwind-protect | |
741 | (progn | |
742 | (let (math-integ-msg) | |
743 | (if (eq calc-display-working-message 'lots) | |
744 | (progn | |
745 | (calc-set-command-flag 'clear-message) | |
746 | (setq math-integ-msg (format | |
747 | "Working... Integrating %s" | |
748 | (math-format-flat-expr expr 0))) | |
274f1353 | 749 | (message "%s" math-integ-msg))) |
f7adda54 JB |
750 | (if math-cur-record |
751 | (setcar (cdr math-cur-record) | |
136211a9 | 752 | (if same-as-above (vector simp) 'busy)) |
f7adda54 | 753 | (setq math-cur-record |
136211a9 | 754 | (list expr (if same-as-above (vector simp) 'busy)) |
f7adda54 | 755 | math-integral-cache (cons math-cur-record |
136211a9 EZ |
756 | math-integral-cache))) |
757 | (if (eq simplify 'yes) | |
758 | (progn | |
759 | (math-tracing-integral "Simplifying...") | |
760 | (setq simp (math-simplify expr)) | |
761 | (setq val (if (equal simp expr) | |
762 | (progn | |
763 | (math-tracing-integral " no change\n") | |
764 | (math-do-integral expr)) | |
765 | (math-tracing-integral " simplified\n") | |
766 | (math-integral simp 'no t)))) | |
767 | (or (setq val (math-do-integral expr)) | |
768 | (eq simplify 'no) | |
769 | (let ((simp (math-simplify expr))) | |
770 | (or (equal simp expr) | |
771 | (progn | |
772 | (math-tracing-integral "Trying again after " | |
773 | "simplification...\n") | |
774 | (setq val (math-integral simp 'no t)))))))) | |
775 | (if (eq calc-display-working-message 'lots) | |
274f1353 | 776 | (message "%s" math-integ-msg))) |
f7adda54 | 777 | (setcar (cdr math-cur-record) (or val |
136211a9 EZ |
778 | (if (or math-enable-subst |
779 | (not math-any-substs)) | |
780 | math-integ-level | |
781 | 'cancelled))))) | |
f7adda54 | 782 | (setq val math-cur-record) |
136211a9 EZ |
783 | (while (vectorp (nth 1 val)) |
784 | (setq val (aref (nth 1 val) 0))) | |
785 | (setq val (if (memq (nth 1 val) '(parts parts2)) | |
786 | (progn | |
787 | (setcar (cdr val) 'parts2) | |
788 | (list 'var 'PARTS val)) | |
789 | (and (consp (nth 1 val)) | |
790 | (nth 1 val)))) | |
791 | (math-tracing-integral "Integral of " | |
792 | (math-format-value expr 1000) | |
793 | " is " | |
794 | (math-format-value val 1000) | |
795 | "\n") | |
bf77c646 | 796 | val)) |
136211a9 EZ |
797 | |
798 | (defun math-integral-contains-parts (expr) | |
799 | (if (Math-primp expr) | |
800 | (and (eq (car-safe expr) 'var) | |
801 | (eq (nth 1 expr) 'PARTS) | |
802 | (listp (nth 2 expr))) | |
803 | (while (and (setq expr (cdr expr)) | |
804 | (not (math-integral-contains-parts (car expr))))) | |
bf77c646 | 805 | expr)) |
136211a9 EZ |
806 | |
807 | (defun math-replace-integral-parts (expr) | |
808 | (or (Math-primp expr) | |
809 | (while (setq expr (cdr expr)) | |
810 | (and (consp (car expr)) | |
811 | (if (eq (car (car expr)) 'var) | |
812 | (and (eq (nth 1 (car expr)) 'PARTS) | |
813 | (consp (nth 2 (car expr))) | |
814 | (if (listp (nth 1 (nth 2 (car expr)))) | |
815 | (progn | |
816 | (setcar expr (nth 1 (nth 2 (car expr)))) | |
817 | (math-replace-integral-parts (cons 'foo expr))) | |
f7adda54 | 818 | (setcar (cdr math-cur-record) 'cancelled))) |
bf77c646 | 819 | (math-replace-integral-parts (car expr))))))) |
136211a9 | 820 | |
4710da05 JB |
821 | (defvar math-linear-subst-tried t |
822 | "Non-nil means that a linear substitution has been tried.") | |
823 | ||
f7adda54 JB |
824 | ;; The variable math-has-rules is a local variable for math-try-integral, |
825 | ;; but is used by math-do-integral, which is called (non-directly) by | |
826 | ;; math-try-integral. | |
827 | (defvar math-has-rules) | |
828 | ||
829 | ;; math-old-integ is a local variable for math-do-integral, but is | |
830 | ;; used by math-sub-integration. | |
831 | (defvar math-old-integ) | |
832 | ||
833 | ;; The variables math-t1, math-t2 and math-t3 are local to | |
834 | ;; math-do-integral, math-try-solve-for and math-decompose-poly, but | |
835 | ;; are used by functions they call (directly or indirectly); | |
836 | ;; math-do-integral calls math-do-integral-methods; | |
837 | ;; math-try-solve-for calls math-try-solve-prod, | |
838 | ;; math-solve-find-root-term and math-solve-find-root-in-prod; | |
839 | ;; math-decompose-poly calls math-solve-poly-funny-powers and | |
840 | ;; math-solve-crunch-poly. | |
841 | (defvar math-t1) | |
842 | (defvar math-t2) | |
843 | (defvar math-t3) | |
844 | ||
136211a9 | 845 | (defun math-do-integral (expr) |
4710da05 | 846 | (let ((math-linear-subst-tried nil) |
f7adda54 | 847 | math-t1 math-t2) |
136211a9 EZ |
848 | (or (cond ((not (math-expr-contains expr math-integ-var)) |
849 | (math-mul expr math-integ-var)) | |
850 | ((equal expr math-integ-var) | |
851 | (math-div (math-sqr expr) 2)) | |
852 | ((eq (car expr) '+) | |
f7adda54 JB |
853 | (and (setq math-t1 (math-integral (nth 1 expr))) |
854 | (setq math-t2 (math-integral (nth 2 expr))) | |
855 | (math-add math-t1 math-t2))) | |
136211a9 | 856 | ((eq (car expr) '-) |
f7adda54 JB |
857 | (and (setq math-t1 (math-integral (nth 1 expr))) |
858 | (setq math-t2 (math-integral (nth 2 expr))) | |
859 | (math-sub math-t1 math-t2))) | |
136211a9 | 860 | ((eq (car expr) 'neg) |
f7adda54 JB |
861 | (and (setq math-t1 (math-integral (nth 1 expr))) |
862 | (math-neg math-t1))) | |
136211a9 EZ |
863 | ((eq (car expr) '*) |
864 | (cond ((not (math-expr-contains (nth 1 expr) math-integ-var)) | |
f7adda54 JB |
865 | (and (setq math-t1 (math-integral (nth 2 expr))) |
866 | (math-mul (nth 1 expr) math-t1))) | |
136211a9 | 867 | ((not (math-expr-contains (nth 2 expr) math-integ-var)) |
f7adda54 JB |
868 | (and (setq math-t1 (math-integral (nth 1 expr))) |
869 | (math-mul math-t1 (nth 2 expr)))) | |
136211a9 EZ |
870 | ((memq (car-safe (nth 1 expr)) '(+ -)) |
871 | (math-integral (list (car (nth 1 expr)) | |
872 | (math-mul (nth 1 (nth 1 expr)) | |
873 | (nth 2 expr)) | |
874 | (math-mul (nth 2 (nth 1 expr)) | |
875 | (nth 2 expr))) | |
876 | 'yes t)) | |
877 | ((memq (car-safe (nth 2 expr)) '(+ -)) | |
878 | (math-integral (list (car (nth 2 expr)) | |
879 | (math-mul (nth 1 (nth 2 expr)) | |
880 | (nth 1 expr)) | |
881 | (math-mul (nth 2 (nth 2 expr)) | |
882 | (nth 1 expr))) | |
883 | 'yes t)))) | |
884 | ((eq (car expr) '/) | |
885 | (cond ((and (not (math-expr-contains (nth 1 expr) | |
886 | math-integ-var)) | |
887 | (not (math-equal-int (nth 1 expr) 1))) | |
f7adda54 JB |
888 | (and (setq math-t1 (math-integral (math-div 1 (nth 2 expr)))) |
889 | (math-mul (nth 1 expr) math-t1))) | |
136211a9 | 890 | ((not (math-expr-contains (nth 2 expr) math-integ-var)) |
f7adda54 JB |
891 | (and (setq math-t1 (math-integral (nth 1 expr))) |
892 | (math-div math-t1 (nth 2 expr)))) | |
136211a9 EZ |
893 | ((and (eq (car-safe (nth 1 expr)) '*) |
894 | (not (math-expr-contains (nth 1 (nth 1 expr)) | |
895 | math-integ-var))) | |
f7adda54 | 896 | (and (setq math-t1 (math-integral |
136211a9 EZ |
897 | (math-div (nth 2 (nth 1 expr)) |
898 | (nth 2 expr)))) | |
f7adda54 | 899 | (math-mul math-t1 (nth 1 (nth 1 expr))))) |
136211a9 EZ |
900 | ((and (eq (car-safe (nth 1 expr)) '*) |
901 | (not (math-expr-contains (nth 2 (nth 1 expr)) | |
902 | math-integ-var))) | |
f7adda54 | 903 | (and (setq math-t1 (math-integral |
136211a9 EZ |
904 | (math-div (nth 1 (nth 1 expr)) |
905 | (nth 2 expr)))) | |
f7adda54 | 906 | (math-mul math-t1 (nth 2 (nth 1 expr))))) |
136211a9 EZ |
907 | ((and (eq (car-safe (nth 2 expr)) '*) |
908 | (not (math-expr-contains (nth 1 (nth 2 expr)) | |
909 | math-integ-var))) | |
f7adda54 | 910 | (and (setq math-t1 (math-integral |
136211a9 EZ |
911 | (math-div (nth 1 expr) |
912 | (nth 2 (nth 2 expr))))) | |
f7adda54 | 913 | (math-div math-t1 (nth 1 (nth 2 expr))))) |
136211a9 EZ |
914 | ((and (eq (car-safe (nth 2 expr)) '*) |
915 | (not (math-expr-contains (nth 2 (nth 2 expr)) | |
916 | math-integ-var))) | |
f7adda54 | 917 | (and (setq math-t1 (math-integral |
136211a9 EZ |
918 | (math-div (nth 1 expr) |
919 | (nth 1 (nth 2 expr))))) | |
f7adda54 | 920 | (math-div math-t1 (nth 2 (nth 2 expr))))) |
136211a9 EZ |
921 | ((eq (car-safe (nth 2 expr)) 'calcFunc-exp) |
922 | (math-integral | |
923 | (math-mul (nth 1 expr) | |
924 | (list 'calcFunc-exp | |
925 | (math-neg (nth 1 (nth 2 expr))))))))) | |
926 | ((eq (car expr) '^) | |
927 | (cond ((not (math-expr-contains (nth 1 expr) math-integ-var)) | |
f7adda54 | 928 | (or (and (setq math-t1 (math-is-polynomial (nth 2 expr) |
136211a9 EZ |
929 | math-integ-var 1)) |
930 | (math-div expr | |
f7adda54 | 931 | (math-mul (nth 1 math-t1) |
136211a9 EZ |
932 | (math-normalize |
933 | (list 'calcFunc-ln | |
934 | (nth 1 expr)))))) | |
935 | (math-integral | |
936 | (list 'calcFunc-exp | |
937 | (math-mul (nth 2 expr) | |
938 | (math-normalize | |
939 | (list 'calcFunc-ln | |
940 | (nth 1 expr))))) | |
941 | 'yes t))) | |
942 | ((not (math-expr-contains (nth 2 expr) math-integ-var)) | |
943 | (if (and (integerp (nth 2 expr)) (< (nth 2 expr) 0)) | |
944 | (math-integral | |
945 | (list '/ 1 (math-pow (nth 1 expr) (- (nth 2 expr)))) | |
946 | nil t) | |
f7adda54 | 947 | (or (and (setq math-t1 (math-is-polynomial (nth 1 expr) |
136211a9 EZ |
948 | math-integ-var |
949 | 1)) | |
f7adda54 JB |
950 | (setq math-t2 (math-add (nth 2 expr) 1)) |
951 | (math-div (math-pow (nth 1 expr) math-t2) | |
952 | (math-mul math-t2 (nth 1 math-t1)))) | |
136211a9 EZ |
953 | (and (Math-negp (nth 2 expr)) |
954 | (math-integral | |
955 | (math-div 1 | |
956 | (math-pow (nth 1 expr) | |
957 | (math-neg | |
958 | (nth 2 expr)))) | |
959 | nil t)) | |
960 | nil)))))) | |
961 | ||
962 | ;; Integral of a polynomial. | |
f7adda54 | 963 | (and (setq math-t1 (math-is-polynomial expr math-integ-var 20)) |
136211a9 EZ |
964 | (let ((accum 0) |
965 | (n 1)) | |
f7adda54 | 966 | (while math-t1 |
136211a9 | 967 | (if (setq accum (math-add accum |
f7adda54 | 968 | (math-div (math-mul (car math-t1) |
136211a9 EZ |
969 | (math-pow |
970 | math-integ-var | |
971 | n)) | |
972 | n)) | |
f7adda54 | 973 | math-t1 (cdr math-t1)) |
136211a9 EZ |
974 | (setq n (1+ n)))) |
975 | accum)) | |
976 | ||
977 | ;; Try looking it up! | |
978 | (cond ((= (length expr) 2) | |
979 | (and (symbolp (car expr)) | |
f7adda54 | 980 | (setq math-t1 (get (car expr) 'math-integral)) |
136211a9 | 981 | (progn |
f7adda54 JB |
982 | (while (and math-t1 |
983 | (not (setq math-t2 (funcall (car math-t1) | |
136211a9 | 984 | (nth 1 expr))))) |
f7adda54 JB |
985 | (setq math-t1 (cdr math-t1))) |
986 | (and math-t2 (math-normalize math-t2))))) | |
136211a9 EZ |
987 | ((= (length expr) 3) |
988 | (and (symbolp (car expr)) | |
f7adda54 | 989 | (setq math-t1 (get (car expr) 'math-integral-2)) |
136211a9 | 990 | (progn |
f7adda54 JB |
991 | (while (and math-t1 |
992 | (not (setq math-t2 (funcall (car math-t1) | |
136211a9 EZ |
993 | (nth 1 expr) |
994 | (nth 2 expr))))) | |
f7adda54 JB |
995 | (setq math-t1 (cdr math-t1))) |
996 | (and math-t2 (math-normalize math-t2)))))) | |
136211a9 EZ |
997 | |
998 | ;; Integral of a rational function. | |
999 | (and (math-ratpoly-p expr math-integ-var) | |
f7adda54 JB |
1000 | (setq math-t1 (calcFunc-apart expr math-integ-var)) |
1001 | (not (equal math-t1 expr)) | |
1002 | (math-integral math-t1)) | |
136211a9 EZ |
1003 | |
1004 | ;; Try user-defined integration rules. | |
f7adda54 | 1005 | (and math-has-rules |
136211a9 EZ |
1006 | (let ((math-old-integ (symbol-function 'calcFunc-integ)) |
1007 | (input (list 'calcFunc-integtry expr math-integ-var)) | |
1008 | res part) | |
1009 | (unwind-protect | |
1010 | (progn | |
1011 | (fset 'calcFunc-integ 'math-sub-integration) | |
1012 | (setq res (math-rewrite input | |
1013 | '(var IntegRules var-IntegRules) | |
1014 | 1)) | |
1015 | (fset 'calcFunc-integ math-old-integ) | |
1016 | (and (not (equal res input)) | |
1017 | (if (setq part (math-expr-calls | |
1018 | res '(calcFunc-integsubst))) | |
1019 | (and (memq (length part) '(3 4 5)) | |
1020 | (let ((parts (mapcar | |
1021 | (function | |
1022 | (lambda (x) | |
1023 | (math-expr-subst | |
1024 | x (nth 2 part) | |
1025 | math-integ-var))) | |
1026 | (cdr part)))) | |
1027 | (math-integrate-by-substitution | |
1028 | expr (car parts) t | |
1029 | (or (nth 2 parts) | |
1030 | (list 'calcFunc-integfailed | |
1031 | math-integ-var)) | |
1032 | (nth 3 parts)))) | |
1033 | (if (not (math-expr-calls res | |
1034 | '(calcFunc-integtry | |
1035 | calcFunc-integfailed))) | |
1036 | res)))) | |
1037 | (fset 'calcFunc-integ math-old-integ)))) | |
1038 | ||
1039 | ;; See if the function is a symbolic derivative. | |
1040 | (and (string-match "'" (symbol-name (car expr))) | |
1041 | (let ((name (symbol-name (car expr))) | |
1042 | (p expr) (n 0) (which nil) (bad nil)) | |
1043 | (while (setq n (1+ n) p (cdr p)) | |
1044 | (if (equal (car p) math-integ-var) | |
1045 | (if which (setq bad t) (setq which n)) | |
1046 | (if (math-expr-contains (car p) math-integ-var) | |
1047 | (setq bad t)))) | |
1048 | (and which (not bad) | |
1049 | (let ((prime (if (= which 1) "'" (format "'%d" which)))) | |
1050 | (and (string-match (concat prime "\\('['0-9]*\\|$\\)") | |
1051 | name) | |
1052 | (cons (intern | |
1053 | (concat | |
1054 | (substring name 0 (match-beginning 0)) | |
1055 | (substring name (+ (match-beginning 0) | |
1056 | (length prime))))) | |
1057 | (cdr expr))))))) | |
1058 | ||
1059 | ;; Try transformation methods (parts, substitutions). | |
1060 | (and (> math-integ-level 0) | |
1061 | (math-do-integral-methods expr)) | |
1062 | ||
1063 | ;; Try expanding the function's definition. | |
1064 | (let ((res (math-expand-formula expr))) | |
1065 | (and res | |
bf77c646 | 1066 | (math-integral res)))))) |
136211a9 EZ |
1067 | |
1068 | (defun math-sub-integration (expr &rest rest) | |
1069 | (or (if (or (not rest) | |
1070 | (and (< math-integ-level math-integral-limit) | |
1071 | (eq (car rest) math-integ-var))) | |
1072 | (math-integral expr) | |
1073 | (let ((res (apply math-old-integ expr rest))) | |
1074 | (and (or (= math-integ-level math-integral-limit) | |
1075 | (not (math-expr-calls res 'calcFunc-integ))) | |
1076 | res))) | |
bf77c646 | 1077 | (list 'calcFunc-integfailed expr))) |
136211a9 | 1078 | |
f7adda54 JB |
1079 | ;; math-so-far is a local variable for math-do-integral-methods, but |
1080 | ;; is used by math-integ-try-linear-substitutions and | |
1081 | ;; math-integ-try-substitutions. | |
1082 | (defvar math-so-far) | |
1083 | ||
1084 | ;; math-integ-expr is a local variable for math-do-integral-methods, | |
1085 | ;; but is used by math-integ-try-linear-substitutions and | |
1086 | ;; math-integ-try-substitutions. | |
1087 | (defvar math-integ-expr) | |
1088 | ||
1089 | (defun math-do-integral-methods (math-integ-expr) | |
1090 | (let ((math-so-far math-integ-var-list-list) | |
136211a9 EZ |
1091 | rat-in) |
1092 | ||
1093 | ;; Integration by substitution, for various likely sub-expressions. | |
1094 | ;; (In first pass, we look only for sub-exprs that are linear in X.) | |
f7adda54 JB |
1095 | (or (math-integ-try-linear-substitutions math-integ-expr) |
1096 | (math-integ-try-substitutions math-integ-expr) | |
136211a9 EZ |
1097 | |
1098 | ;; If function has sines and cosines, try tan(x/2) substitution. | |
f7adda54 | 1099 | (and (let ((p (setq rat-in (math-expr-rational-in math-integ-expr)))) |
136211a9 EZ |
1100 | (while (and p |
1101 | (memq (car (car p)) '(calcFunc-sin | |
1102 | calcFunc-cos | |
9274224b JB |
1103 | calcFunc-tan |
1104 | calcFunc-sec | |
1105 | calcFunc-csc | |
1106 | calcFunc-cot)) | |
136211a9 EZ |
1107 | (equal (nth 1 (car p)) math-integ-var)) |
1108 | (setq p (cdr p))) | |
1109 | (null p)) | |
f7adda54 JB |
1110 | (or (and (math-integ-parts-easy math-integ-expr) |
1111 | (math-integ-try-parts math-integ-expr t)) | |
136211a9 | 1112 | (math-integrate-by-good-substitution |
f7adda54 | 1113 | math-integ-expr (list 'calcFunc-tan (math-div math-integ-var 2))))) |
136211a9 EZ |
1114 | |
1115 | ;; If function has sinh and cosh, try tanh(x/2) substitution. | |
1116 | (and (let ((p rat-in)) | |
1117 | (while (and p | |
1118 | (memq (car (car p)) '(calcFunc-sinh | |
1119 | calcFunc-cosh | |
1120 | calcFunc-tanh | |
9274224b JB |
1121 | calcFunc-sech |
1122 | calcFunc-csch | |
1123 | calcFunc-coth | |
136211a9 EZ |
1124 | calcFunc-exp)) |
1125 | (equal (nth 1 (car p)) math-integ-var)) | |
1126 | (setq p (cdr p))) | |
1127 | (null p)) | |
f7adda54 JB |
1128 | (or (and (math-integ-parts-easy math-integ-expr) |
1129 | (math-integ-try-parts math-integ-expr t)) | |
136211a9 | 1130 | (math-integrate-by-good-substitution |
f7adda54 | 1131 | math-integ-expr (list 'calcFunc-tanh (math-div math-integ-var 2))))) |
136211a9 EZ |
1132 | |
1133 | ;; If function has square roots, try sin, tan, or sec substitution. | |
1134 | (and (let ((p rat-in)) | |
f7adda54 | 1135 | (setq math-t1 nil) |
136211a9 EZ |
1136 | (while (and p |
1137 | (or (equal (car p) math-integ-var) | |
1138 | (and (eq (car (car p)) 'calcFunc-sqrt) | |
f7adda54 JB |
1139 | (setq math-t1 (math-is-polynomial |
1140 | (nth 1 (setq math-t2 (car p))) | |
136211a9 EZ |
1141 | math-integ-var 2))))) |
1142 | (setq p (cdr p))) | |
f7adda54 JB |
1143 | (and (null p) math-t1)) |
1144 | (if (cdr (cdr math-t1)) | |
1145 | (if (math-guess-if-neg (nth 2 math-t1)) | |
1146 | (let* ((c (math-sqrt (math-neg (nth 2 math-t1)))) | |
1147 | (d (math-div (nth 1 math-t1) (math-mul -2 c))) | |
1148 | (a (math-sqrt (math-add (car math-t1) (math-sqr d))))) | |
136211a9 | 1149 | (math-integrate-by-good-substitution |
f7adda54 | 1150 | math-integ-expr (list 'calcFunc-arcsin |
136211a9 EZ |
1151 | (math-div-thru |
1152 | (math-add (math-mul c math-integ-var) d) | |
1153 | a)))) | |
f7adda54 JB |
1154 | (let* ((c (math-sqrt (nth 2 math-t1))) |
1155 | (d (math-div (nth 1 math-t1) (math-mul 2 c))) | |
1156 | (aa (math-sub (car math-t1) (math-sqr d)))) | |
136211a9 EZ |
1157 | (if (and nil (not (and (eq d 0) (eq c 1)))) |
1158 | (math-integrate-by-good-substitution | |
f7adda54 | 1159 | math-integ-expr (math-add (math-mul c math-integ-var) d)) |
136211a9 EZ |
1160 | (if (math-guess-if-neg aa) |
1161 | (math-integrate-by-good-substitution | |
f7adda54 | 1162 | math-integ-expr (list 'calcFunc-arccosh |
136211a9 EZ |
1163 | (math-div-thru |
1164 | (math-add (math-mul c math-integ-var) | |
1165 | d) | |
1166 | (math-sqrt (math-neg aa))))) | |
1167 | (math-integrate-by-good-substitution | |
f7adda54 | 1168 | math-integ-expr (list 'calcFunc-arcsinh |
136211a9 EZ |
1169 | (math-div-thru |
1170 | (math-add (math-mul c math-integ-var) | |
1171 | d) | |
1172 | (math-sqrt aa)))))))) | |
f7adda54 | 1173 | (math-integrate-by-good-substitution math-integ-expr math-t2)) ) |
136211a9 EZ |
1174 | |
1175 | ;; Try integration by parts. | |
f7adda54 | 1176 | (math-integ-try-parts math-integ-expr) |
136211a9 EZ |
1177 | |
1178 | ;; Give up. | |
bf77c646 | 1179 | nil))) |
136211a9 EZ |
1180 | |
1181 | (defun math-integ-parts-easy (expr) | |
1182 | (cond ((Math-primp expr) t) | |
1183 | ((memq (car expr) '(+ - *)) | |
1184 | (and (math-integ-parts-easy (nth 1 expr)) | |
1185 | (math-integ-parts-easy (nth 2 expr)))) | |
1186 | ((eq (car expr) '/) | |
1187 | (and (math-integ-parts-easy (nth 1 expr)) | |
1188 | (math-atomic-factorp (nth 2 expr)))) | |
1189 | ((eq (car expr) '^) | |
1190 | (and (natnump (nth 2 expr)) | |
1191 | (math-integ-parts-easy (nth 1 expr)))) | |
1192 | ((eq (car expr) 'neg) | |
1193 | (math-integ-parts-easy (nth 1 expr))) | |
bf77c646 | 1194 | (t t))) |
136211a9 | 1195 | |
f7adda54 JB |
1196 | ;; math-prev-parts-v is local to calcFunc-integ (as well as |
1197 | ;; math-integrate-by-parts), but is used by math-integ-try-parts. | |
1198 | (defvar math-prev-parts-v) | |
1199 | ||
1200 | ;; math-good-parts is local to calcFunc-integ (as well as | |
1201 | ;; math-integ-try-parts), but is used by math-integrate-by-parts. | |
1202 | (defvar math-good-parts) | |
1203 | ||
1204 | ||
136211a9 EZ |
1205 | (defun math-integ-try-parts (expr &optional math-good-parts) |
1206 | ;; Integration by parts: | |
1207 | ;; integ(f(x) g(x),x) = f(x) h(x) - integ(h(x) f'(x),x) | |
1208 | ;; where h(x) = integ(g(x),x). | |
1209 | (or (let ((exp (calcFunc-expand expr))) | |
1210 | (and (not (equal exp expr)) | |
1211 | (math-integral exp))) | |
1212 | (and (eq (car expr) '*) | |
1213 | (let ((first-bad (or (math-polynomial-p (nth 1 expr) | |
1214 | math-integ-var) | |
1215 | (equal (nth 2 expr) math-prev-parts-v)))) | |
1216 | (or (and first-bad ; so try this one first | |
1217 | (math-integrate-by-parts (nth 1 expr) (nth 2 expr))) | |
1218 | (math-integrate-by-parts (nth 2 expr) (nth 1 expr)) | |
1219 | (and (not first-bad) | |
1220 | (math-integrate-by-parts (nth 1 expr) (nth 2 expr)))))) | |
1221 | (and (eq (car expr) '/) | |
1222 | (math-expr-contains (nth 1 expr) math-integ-var) | |
1223 | (let ((recip (math-div 1 (nth 2 expr)))) | |
1224 | (or (math-integrate-by-parts (nth 1 expr) recip) | |
1225 | (math-integrate-by-parts recip (nth 1 expr))))) | |
1226 | (and (eq (car expr) '^) | |
1227 | (math-integrate-by-parts (math-pow (nth 1 expr) | |
1228 | (math-sub (nth 2 expr) 1)) | |
bf77c646 | 1229 | (nth 1 expr))))) |
136211a9 EZ |
1230 | |
1231 | (defun math-integrate-by-parts (u vprime) | |
1232 | (let ((math-integ-level (if (or math-good-parts | |
1233 | (math-polynomial-p u math-integ-var)) | |
1234 | math-integ-level | |
1235 | (1- math-integ-level))) | |
1236 | (math-doing-parts t) | |
1237 | v temp) | |
1238 | (and (>= math-integ-level 0) | |
1239 | (unwind-protect | |
1240 | (progn | |
f7adda54 | 1241 | (setcar (cdr math-cur-record) 'parts) |
136211a9 EZ |
1242 | (math-tracing-integral "Integrating by parts, u = " |
1243 | (math-format-value u 1000) | |
1244 | ", v' = " | |
1245 | (math-format-value vprime 1000) | |
1246 | "\n") | |
1247 | (and (setq v (math-integral vprime)) | |
1248 | (setq temp (calcFunc-deriv u math-integ-var nil t)) | |
1249 | (setq temp (let ((math-prev-parts-v v)) | |
1250 | (math-integral (math-mul v temp) 'yes))) | |
1251 | (setq temp (math-sub (math-mul u v) temp)) | |
f7adda54 | 1252 | (if (eq (nth 1 math-cur-record) 'parts) |
136211a9 | 1253 | (calcFunc-expand temp) |
f7adda54 | 1254 | (setq v (list 'var 'PARTS math-cur-record) |
136211a9 | 1255 | temp (let (calc-next-why) |
63f7edac JB |
1256 | (math-simplify-extended |
1257 | (math-solve-for (math-sub v temp) 0 v nil))) | |
1258 | temp (if (and (eq (car-safe temp) '/) | |
1259 | (math-zerop (nth 2 temp))) | |
1260 | nil temp))))) | |
f7adda54 | 1261 | (setcar (cdr math-cur-record) 'busy))))) |
136211a9 EZ |
1262 | |
1263 | ;;; This tries two different formulations, hoping the algebraic simplifier | |
1264 | ;;; will be strong enough to handle at least one. | |
1265 | (defun math-integrate-by-substitution (expr u &optional user uinv uinvprime) | |
1266 | (and (> math-integ-level 0) | |
1267 | (let ((math-integ-level (max (- math-integ-level 2) 0))) | |
bf77c646 | 1268 | (math-integrate-by-good-substitution expr u user uinv uinvprime)))) |
136211a9 EZ |
1269 | |
1270 | (defun math-integrate-by-good-substitution (expr u &optional user | |
1271 | uinv uinvprime) | |
1272 | (let ((math-living-dangerously t) | |
1273 | deriv temp) | |
1274 | (and (setq uinv (if uinv | |
1275 | (math-expr-subst uinv math-integ-var | |
1276 | math-integ-var-2) | |
1277 | (let (calc-next-why) | |
1278 | (math-solve-for u | |
1279 | math-integ-var-2 | |
1280 | math-integ-var nil)))) | |
1281 | (progn | |
1282 | (math-tracing-integral "Integrating by substitution, u = " | |
1283 | (math-format-value u 1000) | |
1284 | "\n") | |
1285 | (or (and (setq deriv (calcFunc-deriv u | |
1286 | math-integ-var nil | |
1287 | (not user))) | |
1288 | (setq temp (math-integral (math-expr-subst | |
1289 | (math-expr-subst | |
1290 | (math-expr-subst | |
1291 | (math-div expr deriv) | |
1292 | u | |
1293 | math-integ-var-2) | |
1294 | math-integ-var | |
1295 | uinv) | |
1296 | math-integ-var-2 | |
1297 | math-integ-var) | |
1298 | 'yes))) | |
1299 | (and (setq deriv (or uinvprime | |
1300 | (calcFunc-deriv uinv | |
1301 | math-integ-var-2 | |
1302 | math-integ-var | |
1303 | (not user)))) | |
1304 | (setq temp (math-integral (math-mul | |
1305 | (math-expr-subst | |
1306 | (math-expr-subst | |
1307 | (math-expr-subst | |
1308 | expr | |
1309 | u | |
1310 | math-integ-var-2) | |
1311 | math-integ-var | |
1312 | uinv) | |
1313 | math-integ-var-2 | |
1314 | math-integ-var) | |
1315 | deriv) | |
1316 | 'yes))))) | |
1317 | (math-simplify-extended | |
bf77c646 | 1318 | (math-expr-subst temp math-integ-var u))))) |
136211a9 EZ |
1319 | |
1320 | ;;; Look for substitutions of the form u = a x + b. | |
1321 | (defun math-integ-try-linear-substitutions (sub-expr) | |
4710da05 | 1322 | (setq math-linear-subst-tried t) |
136211a9 EZ |
1323 | (and (not (Math-primp sub-expr)) |
1324 | (or (and (not (memq (car sub-expr) '(+ - * / neg))) | |
1325 | (not (and (eq (car sub-expr) '^) | |
1326 | (integerp (nth 2 sub-expr)))) | |
1327 | (math-expr-contains sub-expr math-integ-var) | |
1328 | (let ((res nil)) | |
1329 | (while (and (setq sub-expr (cdr sub-expr)) | |
1330 | (or (not (math-linear-in (car sub-expr) | |
1331 | math-integ-var)) | |
f7adda54 | 1332 | (assoc (car sub-expr) math-so-far) |
136211a9 | 1333 | (progn |
f7adda54 JB |
1334 | (setq math-so-far (cons (list (car sub-expr)) |
1335 | math-so-far)) | |
136211a9 EZ |
1336 | (not (setq res |
1337 | (math-integrate-by-substitution | |
f7adda54 | 1338 | math-integ-expr (car sub-expr)))))))) |
136211a9 EZ |
1339 | res)) |
1340 | (let ((res nil)) | |
1341 | (while (and (setq sub-expr (cdr sub-expr)) | |
1342 | (not (setq res (math-integ-try-linear-substitutions | |
1343 | (car sub-expr)))))) | |
bf77c646 | 1344 | res)))) |
136211a9 EZ |
1345 | |
1346 | ;;; Recursively try different substitutions based on various sub-expressions. | |
1347 | (defun math-integ-try-substitutions (sub-expr &optional allow-rat) | |
1348 | (and (not (Math-primp sub-expr)) | |
f7adda54 | 1349 | (not (assoc sub-expr math-so-far)) |
136211a9 EZ |
1350 | (math-expr-contains sub-expr math-integ-var) |
1351 | (or (and (if (and (not (memq (car sub-expr) '(+ - * / neg))) | |
1352 | (not (and (eq (car sub-expr) '^) | |
1353 | (integerp (nth 2 sub-expr))))) | |
1354 | (setq allow-rat t) | |
1355 | (prog1 allow-rat (setq allow-rat nil))) | |
f7adda54 JB |
1356 | (not (eq sub-expr math-integ-expr)) |
1357 | (or (math-integrate-by-substitution math-integ-expr sub-expr) | |
136211a9 EZ |
1358 | (and (eq (car sub-expr) '^) |
1359 | (integerp (nth 2 sub-expr)) | |
1360 | (< (nth 2 sub-expr) 0) | |
1361 | (math-integ-try-substitutions | |
1362 | (math-pow (nth 1 sub-expr) (- (nth 2 sub-expr))) | |
1363 | t)))) | |
1364 | (let ((res nil)) | |
f7adda54 | 1365 | (setq math-so-far (cons (list sub-expr) math-so-far)) |
136211a9 EZ |
1366 | (while (and (setq sub-expr (cdr sub-expr)) |
1367 | (not (setq res (math-integ-try-substitutions | |
1368 | (car sub-expr) allow-rat))))) | |
bf77c646 | 1369 | res)))) |
136211a9 | 1370 | |
f7adda54 JB |
1371 | ;; The variable math-expr-parts is local to math-expr-rational-in, |
1372 | ;; but is used by math-expr-rational-in-rec | |
79d2746f | 1373 | (defvar math-expr-parts) |
f7adda54 | 1374 | |
136211a9 | 1375 | (defun math-expr-rational-in (expr) |
f7adda54 | 1376 | (let ((math-expr-parts nil)) |
136211a9 | 1377 | (math-expr-rational-in-rec expr) |
f7adda54 | 1378 | (mapcar 'car math-expr-parts))) |
136211a9 EZ |
1379 | |
1380 | (defun math-expr-rational-in-rec (expr) | |
1381 | (cond ((Math-primp expr) | |
1382 | (and (equal expr math-integ-var) | |
f7adda54 JB |
1383 | (not (assoc expr math-expr-parts)) |
1384 | (setq math-expr-parts (cons (list expr) math-expr-parts)))) | |
136211a9 EZ |
1385 | ((or (memq (car expr) '(+ - * / neg)) |
1386 | (and (eq (car expr) '^) (integerp (nth 2 expr)))) | |
1387 | (math-expr-rational-in-rec (nth 1 expr)) | |
1388 | (and (nth 2 expr) (math-expr-rational-in-rec (nth 2 expr)))) | |
1389 | ((and (eq (car expr) '^) | |
1390 | (eq (math-quarter-integer (nth 2 expr)) 2)) | |
1391 | (math-expr-rational-in-rec (list 'calcFunc-sqrt (nth 1 expr)))) | |
1392 | (t | |
f7adda54 | 1393 | (and (not (assoc expr math-expr-parts)) |
136211a9 | 1394 | (math-expr-contains expr math-integ-var) |
f7adda54 | 1395 | (setq math-expr-parts (cons (list expr) math-expr-parts)))))) |
136211a9 EZ |
1396 | |
1397 | (defun math-expr-calls (expr funcs &optional arg-contains) | |
1398 | (if (consp expr) | |
1399 | (if (or (memq (car expr) funcs) | |
1400 | (and (eq (car expr) '^) (eq (car funcs) 'calcFunc-sqrt) | |
1401 | (eq (math-quarter-integer (nth 2 expr)) 2))) | |
1402 | (and (or (not arg-contains) | |
1403 | (math-expr-contains expr arg-contains)) | |
1404 | expr) | |
1405 | (and (not (Math-primp expr)) | |
1406 | (let ((res nil)) | |
1407 | (while (and (setq expr (cdr expr)) | |
1408 | (not (setq res (math-expr-calls | |
1409 | (car expr) funcs arg-contains))))) | |
bf77c646 | 1410 | res))))) |
136211a9 EZ |
1411 | |
1412 | (defun math-fix-const-terms (expr except-vars) | |
1413 | (cond ((not (math-expr-depends expr except-vars)) 0) | |
1414 | ((Math-primp expr) expr) | |
1415 | ((eq (car expr) '+) | |
1416 | (math-add (math-fix-const-terms (nth 1 expr) except-vars) | |
1417 | (math-fix-const-terms (nth 2 expr) except-vars))) | |
1418 | ((eq (car expr) '-) | |
1419 | (math-sub (math-fix-const-terms (nth 1 expr) except-vars) | |
1420 | (math-fix-const-terms (nth 2 expr) except-vars))) | |
bf77c646 | 1421 | (t expr))) |
136211a9 EZ |
1422 | |
1423 | ;; Command for debugging the Calculator's symbolic integrator. | |
1424 | (defun calc-dump-integral-cache (&optional arg) | |
1425 | (interactive "P") | |
1426 | (let ((buf (current-buffer))) | |
1427 | (unwind-protect | |
1428 | (let ((p math-integral-cache) | |
f7adda54 | 1429 | math-cur-record) |
a1506d29 | 1430 | (display-buffer (get-buffer-create "*Integral Cache*")) |
136211a9 EZ |
1431 | (set-buffer (get-buffer "*Integral Cache*")) |
1432 | (erase-buffer) | |
1433 | (while p | |
f7adda54 JB |
1434 | (setq math-cur-record (car p)) |
1435 | (or arg (math-replace-integral-parts math-cur-record)) | |
1436 | (insert (math-format-flat-expr (car math-cur-record) 0) | |
136211a9 | 1437 | " --> " |
f7adda54 JB |
1438 | (if (symbolp (nth 1 math-cur-record)) |
1439 | (concat "(" (symbol-name (nth 1 math-cur-record)) ")") | |
1440 | (math-format-flat-expr (nth 1 math-cur-record) 0)) | |
136211a9 EZ |
1441 | "\n") |
1442 | (setq p (cdr p))) | |
1443 | (goto-char (point-min))) | |
bf77c646 | 1444 | (set-buffer buf)))) |
136211a9 | 1445 | |
f7adda54 JB |
1446 | ;; The variable math-max-integral-limit is local to calcFunc-integ, |
1447 | ;; but is used by math-try-integral. | |
1448 | (defvar math-max-integral-limit) | |
1449 | ||
136211a9 EZ |
1450 | (defun math-try-integral (expr) |
1451 | (let ((math-integ-level math-integral-limit) | |
1452 | (math-integ-depth 0) | |
1453 | (math-integ-msg "Working...done") | |
f7adda54 | 1454 | (math-cur-record nil) ; a technicality |
136211a9 EZ |
1455 | (math-integrating t) |
1456 | (calc-prefer-frac t) | |
1457 | (calc-symbolic-mode t) | |
f7adda54 | 1458 | (math-has-rules (calc-has-rules 'var-IntegRules))) |
136211a9 EZ |
1459 | (or (math-integral expr 'yes) |
1460 | (and math-any-substs | |
1461 | (setq math-enable-subst t) | |
1462 | (math-integral expr 'yes)) | |
1463 | (and (> math-max-integral-limit math-integral-limit) | |
1464 | (setq math-integral-limit math-max-integral-limit | |
1465 | math-integ-level math-integral-limit) | |
bf77c646 | 1466 | (math-integral expr 'yes))))) |
136211a9 | 1467 | |
f7adda54 JB |
1468 | (defvar var-IntegLimit nil) |
1469 | ||
136211a9 EZ |
1470 | (defun calcFunc-integ (expr var &optional low high) |
1471 | (cond | |
1472 | ;; Do these even if the parts turn out not to be integrable. | |
1473 | ((eq (car-safe expr) '+) | |
1474 | (math-add (calcFunc-integ (nth 1 expr) var low high) | |
1475 | (calcFunc-integ (nth 2 expr) var low high))) | |
1476 | ((eq (car-safe expr) '-) | |
1477 | (math-sub (calcFunc-integ (nth 1 expr) var low high) | |
1478 | (calcFunc-integ (nth 2 expr) var low high))) | |
1479 | ((eq (car-safe expr) 'neg) | |
1480 | (math-neg (calcFunc-integ (nth 1 expr) var low high))) | |
1481 | ((and (eq (car-safe expr) '*) | |
1482 | (not (math-expr-contains (nth 1 expr) var))) | |
1483 | (math-mul (nth 1 expr) (calcFunc-integ (nth 2 expr) var low high))) | |
1484 | ((and (eq (car-safe expr) '*) | |
1485 | (not (math-expr-contains (nth 2 expr) var))) | |
1486 | (math-mul (calcFunc-integ (nth 1 expr) var low high) (nth 2 expr))) | |
1487 | ((and (eq (car-safe expr) '/) | |
1488 | (not (math-expr-contains (nth 1 expr) var)) | |
1489 | (not (math-equal-int (nth 1 expr) 1))) | |
1490 | (math-mul (nth 1 expr) | |
1491 | (calcFunc-integ (math-div 1 (nth 2 expr)) var low high))) | |
1492 | ((and (eq (car-safe expr) '/) | |
1493 | (not (math-expr-contains (nth 2 expr) var))) | |
1494 | (math-div (calcFunc-integ (nth 1 expr) var low high) (nth 2 expr))) | |
1495 | ((and (eq (car-safe expr) '/) | |
1496 | (eq (car-safe (nth 1 expr)) '*) | |
1497 | (not (math-expr-contains (nth 1 (nth 1 expr)) var))) | |
1498 | (math-mul (nth 1 (nth 1 expr)) | |
1499 | (calcFunc-integ (math-div (nth 2 (nth 1 expr)) (nth 2 expr)) | |
1500 | var low high))) | |
1501 | ((and (eq (car-safe expr) '/) | |
1502 | (eq (car-safe (nth 1 expr)) '*) | |
1503 | (not (math-expr-contains (nth 2 (nth 1 expr)) var))) | |
1504 | (math-mul (nth 2 (nth 1 expr)) | |
1505 | (calcFunc-integ (math-div (nth 1 (nth 1 expr)) (nth 2 expr)) | |
1506 | var low high))) | |
1507 | ((and (eq (car-safe expr) '/) | |
1508 | (eq (car-safe (nth 2 expr)) '*) | |
1509 | (not (math-expr-contains (nth 1 (nth 2 expr)) var))) | |
1510 | (math-div (calcFunc-integ (math-div (nth 1 expr) (nth 2 (nth 2 expr))) | |
1511 | var low high) | |
1512 | (nth 1 (nth 2 expr)))) | |
1513 | ((and (eq (car-safe expr) '/) | |
1514 | (eq (car-safe (nth 2 expr)) '*) | |
1515 | (not (math-expr-contains (nth 2 (nth 2 expr)) var))) | |
1516 | (math-div (calcFunc-integ (math-div (nth 1 expr) (nth 1 (nth 2 expr))) | |
1517 | var low high) | |
1518 | (nth 2 (nth 2 expr)))) | |
1519 | ((eq (car-safe expr) 'vec) | |
1520 | (cons 'vec (mapcar (function (lambda (x) (calcFunc-integ x var low high))) | |
1521 | (cdr expr)))) | |
1522 | (t | |
1523 | (let ((state (list calc-angle-mode | |
1524 | ;;calc-symbolic-mode | |
1525 | ;;calc-prefer-frac | |
1526 | calc-internal-prec | |
1527 | (calc-var-value 'var-IntegRules) | |
1528 | (calc-var-value 'var-IntegSimpRules)))) | |
1529 | (or (equal state math-integral-cache-state) | |
1530 | (setq math-integral-cache-state state | |
1531 | math-integral-cache nil))) | |
f7adda54 | 1532 | (let* ((math-max-integral-limit (or (and (natnump var-IntegLimit) |
136211a9 EZ |
1533 | var-IntegLimit) |
1534 | 3)) | |
1535 | (math-integral-limit 1) | |
1536 | (sexpr (math-expr-subst expr var math-integ-var)) | |
1537 | (trace-buffer (get-buffer "*Trace*")) | |
1538 | (calc-language (if (eq calc-language 'big) nil calc-language)) | |
1539 | (math-any-substs t) | |
1540 | (math-enable-subst nil) | |
1541 | (math-prev-parts-v nil) | |
1542 | (math-doing-parts nil) | |
1543 | (math-good-parts nil) | |
1544 | (res | |
1545 | (if trace-buffer | |
1546 | (let ((calcbuf (current-buffer)) | |
1547 | (calcwin (selected-window))) | |
1548 | (unwind-protect | |
1549 | (progn | |
1550 | (if (get-buffer-window trace-buffer) | |
1551 | (select-window (get-buffer-window trace-buffer))) | |
1552 | (set-buffer trace-buffer) | |
1553 | (goto-char (point-max)) | |
1554 | (or (assq 'scroll-stop (buffer-local-variables)) | |
1555 | (progn | |
1556 | (make-local-variable 'scroll-step) | |
1557 | (setq scroll-step 3))) | |
1558 | (insert "\n\n\n") | |
1559 | (set-buffer calcbuf) | |
1560 | (math-try-integral sexpr)) | |
1561 | (select-window calcwin) | |
1562 | (set-buffer calcbuf))) | |
1563 | (math-try-integral sexpr)))) | |
1564 | (if res | |
1565 | (progn | |
1566 | (if (calc-has-rules 'var-IntegAfterRules) | |
1567 | (setq res (math-rewrite res '(var IntegAfterRules | |
1568 | var-IntegAfterRules)))) | |
1569 | (math-simplify | |
1570 | (if (and low high) | |
1571 | (math-sub (math-expr-subst res math-integ-var high) | |
1572 | (math-expr-subst res math-integ-var low)) | |
1573 | (setq res (math-fix-const-terms res math-integ-vars)) | |
1574 | (if low | |
1575 | (math-expr-subst res math-integ-var low) | |
1576 | (math-expr-subst res math-integ-var var))))) | |
1577 | (append (list 'calcFunc-integ expr var) | |
1578 | (and low (list low)) | |
bf77c646 | 1579 | (and high (list high)))))))) |
136211a9 EZ |
1580 | |
1581 | ||
1582 | (math-defintegral calcFunc-inv | |
1583 | (math-integral (math-div 1 u))) | |
1584 | ||
1585 | (math-defintegral calcFunc-conj | |
1586 | (let ((int (math-integral u))) | |
1587 | (and int | |
1588 | (list 'calcFunc-conj int)))) | |
1589 | ||
1590 | (math-defintegral calcFunc-deg | |
1591 | (let ((int (math-integral u))) | |
1592 | (and int | |
1593 | (list 'calcFunc-deg int)))) | |
1594 | ||
1595 | (math-defintegral calcFunc-rad | |
1596 | (let ((int (math-integral u))) | |
1597 | (and int | |
1598 | (list 'calcFunc-rad int)))) | |
1599 | ||
1600 | (math-defintegral calcFunc-re | |
1601 | (let ((int (math-integral u))) | |
1602 | (and int | |
1603 | (list 'calcFunc-re int)))) | |
1604 | ||
1605 | (math-defintegral calcFunc-im | |
1606 | (let ((int (math-integral u))) | |
1607 | (and int | |
1608 | (list 'calcFunc-im int)))) | |
1609 | ||
1610 | (math-defintegral calcFunc-sqrt | |
1611 | (and (equal u math-integ-var) | |
1612 | (math-mul '(frac 2 3) | |
1613 | (list 'calcFunc-sqrt (math-pow u 3))))) | |
1614 | ||
1615 | (math-defintegral calcFunc-exp | |
1616 | (or (and (equal u math-integ-var) | |
1617 | (list 'calcFunc-exp u)) | |
1618 | (let ((p (math-is-polynomial u math-integ-var 2))) | |
1619 | (and (nth 2 p) | |
1620 | (let ((sqa (math-sqrt (math-neg (nth 2 p))))) | |
1621 | (math-div | |
1622 | (math-mul | |
1623 | (math-mul (math-div (list 'calcFunc-sqrt '(var pi var-pi)) | |
1624 | sqa) | |
1625 | (math-normalize | |
1626 | (list 'calcFunc-exp | |
1627 | (math-div (math-sub (math-mul (car p) | |
1628 | (nth 2 p)) | |
1629 | (math-div | |
1630 | (math-sqr (nth 1 p)) | |
1631 | 4)) | |
1632 | (nth 2 p))))) | |
1633 | (list 'calcFunc-erf | |
1634 | (math-sub (math-mul sqa math-integ-var) | |
1635 | (math-div (nth 1 p) (math-mul 2 sqa))))) | |
1636 | 2)))))) | |
1637 | ||
1638 | (math-defintegral calcFunc-ln | |
1639 | (or (and (equal u math-integ-var) | |
1640 | (math-sub (math-mul u (list 'calcFunc-ln u)) u)) | |
1641 | (and (eq (car u) '*) | |
1642 | (math-integral (math-add (list 'calcFunc-ln (nth 1 u)) | |
1643 | (list 'calcFunc-ln (nth 2 u))))) | |
1644 | (and (eq (car u) '/) | |
1645 | (math-integral (math-sub (list 'calcFunc-ln (nth 1 u)) | |
1646 | (list 'calcFunc-ln (nth 2 u))))) | |
1647 | (and (eq (car u) '^) | |
1648 | (math-integral (math-mul (nth 2 u) | |
1649 | (list 'calcFunc-ln (nth 1 u))))))) | |
1650 | ||
1651 | (math-defintegral calcFunc-log10 | |
1652 | (and (equal u math-integ-var) | |
1653 | (math-sub (math-mul u (list 'calcFunc-ln u)) | |
1654 | (math-div u (list 'calcFunc-ln 10))))) | |
1655 | ||
1656 | (math-defintegral-2 calcFunc-log | |
1657 | (math-integral (math-div (list 'calcFunc-ln u) | |
1658 | (list 'calcFunc-ln v)))) | |
1659 | ||
1660 | (math-defintegral calcFunc-sin | |
1661 | (or (and (equal u math-integ-var) | |
1662 | (math-neg (math-from-radians-2 (list 'calcFunc-cos u)))) | |
1663 | (and (nth 2 (math-is-polynomial u math-integ-var 2)) | |
1664 | (math-integral (math-to-exponentials (list 'calcFunc-sin u)))))) | |
1665 | ||
1666 | (math-defintegral calcFunc-cos | |
1667 | (or (and (equal u math-integ-var) | |
1668 | (math-from-radians-2 (list 'calcFunc-sin u))) | |
1669 | (and (nth 2 (math-is-polynomial u math-integ-var 2)) | |
1670 | (math-integral (math-to-exponentials (list 'calcFunc-cos u)))))) | |
1671 | ||
1672 | (math-defintegral calcFunc-tan | |
1673 | (and (equal u math-integ-var) | |
dbf954fb JB |
1674 | (math-from-radians-2 |
1675 | (list 'calcFunc-ln (list 'calcFunc-sec u))))) | |
136211a9 | 1676 | |
9274224b JB |
1677 | (math-defintegral calcFunc-sec |
1678 | (and (equal u math-integ-var) | |
1679 | (math-from-radians-2 | |
1680 | (list 'calcFunc-ln | |
1681 | (math-add | |
1682 | (list 'calcFunc-sec u) | |
1683 | (list 'calcFunc-tan u)))))) | |
1684 | ||
1685 | (math-defintegral calcFunc-csc | |
1686 | (and (equal u math-integ-var) | |
1687 | (math-from-radians-2 | |
1688 | (list 'calcFunc-ln | |
1689 | (math-sub | |
1690 | (list 'calcFunc-csc u) | |
1691 | (list 'calcFunc-cot u)))))) | |
1692 | ||
1693 | (math-defintegral calcFunc-cot | |
1694 | (and (equal u math-integ-var) | |
1695 | (math-from-radians-2 | |
1696 | (list 'calcFunc-ln (list 'calcFunc-sin u))))) | |
1697 | ||
136211a9 EZ |
1698 | (math-defintegral calcFunc-arcsin |
1699 | (and (equal u math-integ-var) | |
1700 | (math-add (math-mul u (list 'calcFunc-arcsin u)) | |
1701 | (math-from-radians-2 | |
1702 | (list 'calcFunc-sqrt (math-sub 1 (math-sqr u))))))) | |
1703 | ||
1704 | (math-defintegral calcFunc-arccos | |
1705 | (and (equal u math-integ-var) | |
1706 | (math-sub (math-mul u (list 'calcFunc-arccos u)) | |
1707 | (math-from-radians-2 | |
1708 | (list 'calcFunc-sqrt (math-sub 1 (math-sqr u))))))) | |
1709 | ||
1710 | (math-defintegral calcFunc-arctan | |
1711 | (and (equal u math-integ-var) | |
1712 | (math-sub (math-mul u (list 'calcFunc-arctan u)) | |
1713 | (math-from-radians-2 | |
1714 | (math-div (list 'calcFunc-ln (math-add 1 (math-sqr u))) | |
1715 | 2))))) | |
1716 | ||
1717 | (math-defintegral calcFunc-sinh | |
1718 | (and (equal u math-integ-var) | |
1719 | (list 'calcFunc-cosh u))) | |
1720 | ||
1721 | (math-defintegral calcFunc-cosh | |
1722 | (and (equal u math-integ-var) | |
1723 | (list 'calcFunc-sinh u))) | |
1724 | ||
1725 | (math-defintegral calcFunc-tanh | |
1726 | (and (equal u math-integ-var) | |
1727 | (list 'calcFunc-ln (list 'calcFunc-cosh u)))) | |
1728 | ||
9274224b JB |
1729 | (math-defintegral calcFunc-sech |
1730 | (and (equal u math-integ-var) | |
1731 | (list 'calcFunc-arctan (list 'calcFunc-sinh u)))) | |
1732 | ||
1733 | (math-defintegral calcFunc-csch | |
1734 | (and (equal u math-integ-var) | |
1735 | (list 'calcFunc-ln (list 'calcFunc-tanh (math-div u 2))))) | |
1736 | ||
1737 | (math-defintegral calcFunc-coth | |
1738 | (and (equal u math-integ-var) | |
1739 | (list 'calcFunc-ln (list 'calcFunc-sinh u)))) | |
1740 | ||
136211a9 EZ |
1741 | (math-defintegral calcFunc-arcsinh |
1742 | (and (equal u math-integ-var) | |
1743 | (math-sub (math-mul u (list 'calcFunc-arcsinh u)) | |
1744 | (list 'calcFunc-sqrt (math-add (math-sqr u) 1))))) | |
1745 | ||
1746 | (math-defintegral calcFunc-arccosh | |
1747 | (and (equal u math-integ-var) | |
1748 | (math-sub (math-mul u (list 'calcFunc-arccosh u)) | |
1749 | (list 'calcFunc-sqrt (math-sub 1 (math-sqr u)))))) | |
1750 | ||
1751 | (math-defintegral calcFunc-arctanh | |
1752 | (and (equal u math-integ-var) | |
1753 | (math-sub (math-mul u (list 'calcFunc-arctan u)) | |
1754 | (math-div (list 'calcFunc-ln | |
1755 | (math-add 1 (math-sqr u))) | |
1756 | 2)))) | |
1757 | ||
1758 | ;;; (Ax + B) / (ax^2 + bx + c)^n forms. | |
1759 | (math-defintegral-2 / | |
1760 | (math-integral-rational-funcs u v)) | |
1761 | ||
1762 | (defun math-integral-rational-funcs (u v) | |
1763 | (let ((pu (math-is-polynomial u math-integ-var 1)) | |
1764 | (vpow 1) pv) | |
1765 | (and pu | |
1766 | (catch 'int-rat | |
1767 | (if (and (eq (car-safe v) '^) (natnump (nth 2 v))) | |
1768 | (setq vpow (nth 2 v) | |
1769 | v (nth 1 v))) | |
1770 | (and (setq pv (math-is-polynomial v math-integ-var 2)) | |
1771 | (let ((int (math-mul-thru | |
1772 | (car pu) | |
1773 | (math-integral-q02 (car pv) (nth 1 pv) | |
1774 | (nth 2 pv) v vpow)))) | |
1775 | (if (cdr pu) | |
1776 | (setq int (math-add int | |
1777 | (math-mul-thru | |
1778 | (nth 1 pu) | |
1779 | (math-integral-q12 | |
1780 | (car pv) (nth 1 pv) | |
1781 | (nth 2 pv) v vpow))))) | |
1782 | int)))))) | |
1783 | ||
1784 | (defun math-integral-q12 (a b c v vpow) | |
1785 | (let (q) | |
1786 | (cond ((not c) | |
1787 | (cond ((= vpow 1) | |
1788 | (math-sub (math-div math-integ-var b) | |
1789 | (math-mul (math-div a (math-sqr b)) | |
1790 | (list 'calcFunc-ln v)))) | |
1791 | ((= vpow 2) | |
1792 | (math-div (math-add (list 'calcFunc-ln v) | |
1793 | (math-div a v)) | |
1794 | (math-sqr b))) | |
1795 | (t | |
1796 | (let ((nm1 (math-sub vpow 1)) | |
1797 | (nm2 (math-sub vpow 2))) | |
1798 | (math-div (math-sub | |
1799 | (math-div a (math-mul nm1 (math-pow v nm1))) | |
1800 | (math-div 1 (math-mul nm2 (math-pow v nm2)))) | |
1801 | (math-sqr b)))))) | |
1802 | ((math-zerop | |
1803 | (setq q (math-sub (math-mul 4 (math-mul a c)) (math-sqr b)))) | |
1804 | (let ((part (math-div b (math-mul 2 c)))) | |
1805 | (math-mul-thru (math-pow c vpow) | |
1806 | (math-integral-q12 part 1 nil | |
1807 | (math-add math-integ-var part) | |
1808 | (* vpow 2))))) | |
1809 | ((= vpow 1) | |
1810 | (and (math-ratp q) (math-negp q) | |
1811 | (let ((calc-symbolic-mode t)) | |
1812 | (math-ratp (math-sqrt (math-neg q)))) | |
1813 | (throw 'int-rat nil)) ; should have used calcFunc-apart first | |
1814 | (math-sub (math-div (list 'calcFunc-ln v) (math-mul 2 c)) | |
1815 | (math-mul-thru (math-div b (math-mul 2 c)) | |
1816 | (math-integral-q02 a b c v 1)))) | |
1817 | (t | |
1818 | (let ((n (1- vpow))) | |
1819 | (math-sub (math-neg (math-div | |
1820 | (math-add (math-mul b math-integ-var) | |
1821 | (math-mul 2 a)) | |
1822 | (math-mul n (math-mul q (math-pow v n))))) | |
1823 | (math-mul-thru (math-div (math-mul b (1- (* 2 n))) | |
1824 | (math-mul n q)) | |
bf77c646 | 1825 | (math-integral-q02 a b c v n)))))))) |
136211a9 EZ |
1826 | |
1827 | (defun math-integral-q02 (a b c v vpow) | |
1828 | (let (q rq part) | |
1829 | (cond ((not c) | |
1830 | (cond ((= vpow 1) | |
1831 | (math-div (list 'calcFunc-ln v) b)) | |
1832 | (t | |
1833 | (math-div (math-pow v (- 1 vpow)) | |
1834 | (math-mul (- 1 vpow) b))))) | |
1835 | ((math-zerop | |
1836 | (setq q (math-sub (math-mul 4 (math-mul a c)) (math-sqr b)))) | |
1837 | (let ((part (math-div b (math-mul 2 c)))) | |
1838 | (math-mul-thru (math-pow c vpow) | |
1839 | (math-integral-q02 part 1 nil | |
1840 | (math-add math-integ-var part) | |
1841 | (* vpow 2))))) | |
1842 | ((progn | |
1843 | (setq part (math-add (math-mul 2 (math-mul c math-integ-var)) b)) | |
1844 | (> vpow 1)) | |
1845 | (let ((n (1- vpow))) | |
1846 | (math-add (math-div part (math-mul n (math-mul q (math-pow v n)))) | |
1847 | (math-mul-thru (math-div (math-mul (- (* 4 n) 2) c) | |
1848 | (math-mul n q)) | |
1849 | (math-integral-q02 a b c v n))))) | |
1850 | ((math-guess-if-neg q) | |
1851 | (setq rq (list 'calcFunc-sqrt (math-neg q))) | |
1852 | ;;(math-div-thru (list 'calcFunc-ln | |
1853 | ;; (math-div (math-sub part rq) | |
1854 | ;; (math-add part rq))) | |
1855 | ;; rq) | |
1856 | (math-div (math-mul -2 (list 'calcFunc-arctanh | |
1857 | (math-div part rq))) | |
1858 | rq)) | |
1859 | (t | |
1860 | (setq rq (list 'calcFunc-sqrt q)) | |
1861 | (math-div (math-mul 2 (math-to-radians-2 | |
1862 | (list 'calcFunc-arctan | |
1863 | (math-div part rq)))) | |
bf77c646 | 1864 | rq))))) |
136211a9 EZ |
1865 | |
1866 | ||
1867 | (math-defintegral calcFunc-erf | |
1868 | (and (equal u math-integ-var) | |
1869 | (math-add (math-mul u (list 'calcFunc-erf u)) | |
1870 | (math-div 1 (math-mul (list 'calcFunc-exp (math-sqr u)) | |
1871 | (list 'calcFunc-sqrt | |
1872 | '(var pi var-pi))))))) | |
1873 | ||
1874 | (math-defintegral calcFunc-erfc | |
1875 | (and (equal u math-integ-var) | |
1876 | (math-sub (math-mul u (list 'calcFunc-erfc u)) | |
1877 | (math-div 1 (math-mul (list 'calcFunc-exp (math-sqr u)) | |
1878 | (list 'calcFunc-sqrt | |
1879 | '(var pi var-pi))))))) | |
1880 | ||
1881 | ||
1882 | ||
1883 | ||
3132f345 CW |
1884 | (defvar math-tabulate-initial nil) |
1885 | (defvar math-tabulate-function nil) | |
f7adda54 JB |
1886 | |
1887 | ;; The variables calc-low and calc-high are local to calcFunc-table, | |
1888 | ;; but are used by math-scan-for-limits. | |
1889 | (defvar calc-low) | |
1890 | (defvar calc-high) | |
1891 | ||
1892 | (defun calcFunc-table (expr var &optional calc-low calc-high step) | |
1893 | (or calc-low | |
1894 | (setq calc-low '(neg (var inf var-inf)) calc-high '(var inf var-inf))) | |
1895 | (or calc-high (setq calc-high calc-low calc-low 1)) | |
1896 | (and (or (math-infinitep calc-low) (math-infinitep calc-high)) | |
136211a9 EZ |
1897 | (not step) |
1898 | (math-scan-for-limits expr)) | |
1899 | (and step (math-zerop step) (math-reject-arg step 'nonzerop)) | |
f7adda54 JB |
1900 | (let ((known (+ (if (Math-objectp calc-low) 1 0) |
1901 | (if (Math-objectp calc-high) 1 0) | |
136211a9 EZ |
1902 | (if (or (null step) (Math-objectp step)) 1 0))) |
1903 | (count '(var inf var-inf)) | |
1904 | vec) | |
1905 | (or (= known 2) ; handy optimization | |
f7adda54 | 1906 | (equal calc-high '(var inf var-inf)) |
136211a9 | 1907 | (progn |
f7adda54 | 1908 | (setq count (math-div (math-sub calc-high calc-low) (or step 1))) |
136211a9 EZ |
1909 | (or (Math-objectp count) |
1910 | (setq count (math-simplify count))) | |
1911 | (if (Math-messy-integerp count) | |
1912 | (setq count (math-trunc count))))) | |
1913 | (if (Math-negp count) | |
1914 | (setq count -1)) | |
1915 | (if (integerp count) | |
1916 | (let ((var-DUMMY nil) | |
1917 | (vec math-tabulate-initial) | |
1918 | (math-working-step-2 (1+ count)) | |
1919 | (math-working-step 0)) | |
1920 | (setq expr (math-evaluate-expr | |
1921 | (math-expr-subst expr var '(var DUMMY var-DUMMY)))) | |
1922 | (while (>= count 0) | |
1923 | (setq math-working-step (1+ math-working-step) | |
f7adda54 | 1924 | var-DUMMY calc-low |
136211a9 EZ |
1925 | vec (cond ((eq math-tabulate-function 'calcFunc-sum) |
1926 | (math-add vec (math-evaluate-expr expr))) | |
1927 | ((eq math-tabulate-function 'calcFunc-prod) | |
1928 | (math-mul vec (math-evaluate-expr expr))) | |
1929 | (t | |
1930 | (cons (math-evaluate-expr expr) vec))) | |
f7adda54 | 1931 | calc-low (math-add calc-low (or step 1)) |
136211a9 EZ |
1932 | count (1- count))) |
1933 | (if math-tabulate-function | |
1934 | vec | |
1935 | (cons 'vec (nreverse vec)))) | |
1936 | (if (Math-integerp count) | |
f7adda54 JB |
1937 | (calc-record-why 'fixnump calc-high) |
1938 | (if (Math-num-integerp calc-low) | |
1939 | (if (Math-num-integerp calc-high) | |
136211a9 | 1940 | (calc-record-why 'integerp step) |
f7adda54 JB |
1941 | (calc-record-why 'integerp calc-high)) |
1942 | (calc-record-why 'integerp calc-low))) | |
136211a9 EZ |
1943 | (append (list (or math-tabulate-function 'calcFunc-table) |
1944 | expr var) | |
f7adda54 JB |
1945 | (and (not (and (equal calc-low '(neg (var inf var-inf))) |
1946 | (equal calc-high '(var inf var-inf)))) | |
1947 | (list calc-low calc-high)) | |
bf77c646 | 1948 | (and step (list step)))))) |
136211a9 | 1949 | |
136211a9 EZ |
1950 | (defun math-scan-for-limits (x) |
1951 | (cond ((Math-primp x)) | |
1952 | ((and (eq (car x) 'calcFunc-subscr) | |
1953 | (Math-vectorp (nth 1 x)) | |
1954 | (math-expr-contains (nth 2 x) var)) | |
1955 | (let* ((calc-next-why nil) | |
1956 | (low-val (math-solve-for (nth 2 x) 1 var nil)) | |
1957 | (high-val (math-solve-for (nth 2 x) (1- (length (nth 1 x))) | |
1958 | var nil)) | |
1959 | temp) | |
1960 | (and low-val (math-realp low-val) | |
1961 | high-val (math-realp high-val)) | |
1962 | (and (Math-lessp high-val low-val) | |
1963 | (setq temp low-val low-val high-val high-val temp)) | |
f7adda54 JB |
1964 | (setq calc-low (math-max calc-low (math-ceiling low-val)) |
1965 | calc-high (math-min calc-high (math-floor high-val))))) | |
136211a9 EZ |
1966 | (t |
1967 | (while (setq x (cdr x)) | |
bf77c646 | 1968 | (math-scan-for-limits (car x)))))) |
136211a9 EZ |
1969 | |
1970 | ||
3132f345 | 1971 | (defvar math-disable-sums nil) |
136211a9 EZ |
1972 | (defun calcFunc-sum (expr var &optional low high step) |
1973 | (if math-disable-sums (math-reject-arg)) | |
1974 | (let* ((res (let* ((calc-internal-prec (+ calc-internal-prec 2))) | |
1975 | (math-sum-rec expr var low high step))) | |
1976 | (math-disable-sums t)) | |
bf77c646 | 1977 | (math-normalize res))) |
136211a9 EZ |
1978 | |
1979 | (defun math-sum-rec (expr var &optional low high step) | |
1980 | (or low (setq low '(neg (var inf var-inf)) high '(var inf var-inf))) | |
1981 | (and low (not high) (setq high low low 1)) | |
1982 | (let (t1 t2 val) | |
1983 | (setq val | |
1984 | (cond | |
1985 | ((not (math-expr-contains expr var)) | |
1986 | (math-mul expr (math-add (math-div (math-sub high low) (or step 1)) | |
1987 | 1))) | |
1988 | ((and step (not (math-equal-int step 1))) | |
1989 | (if (math-negp step) | |
1990 | (math-sum-rec expr var high low (math-neg step)) | |
1991 | (let ((lo (math-simplify (math-div low step)))) | |
1992 | (if (math-known-num-integerp lo) | |
1993 | (math-sum-rec (math-normalize | |
1994 | (math-expr-subst expr var | |
1995 | (math-mul step var))) | |
1996 | var lo (math-simplify (math-div high step))) | |
1997 | (math-sum-rec (math-normalize | |
1998 | (math-expr-subst expr var | |
1999 | (math-add (math-mul step var) | |
2000 | low))) | |
2001 | var 0 | |
2002 | (math-simplify (math-div (math-sub high low) | |
2003 | step))))))) | |
2004 | ((memq (setq t1 (math-compare low high)) '(0 1)) | |
2005 | (if (eq t1 0) | |
2006 | (math-expr-subst expr var low) | |
2007 | 0)) | |
2008 | ((setq t1 (math-is-polynomial expr var 20)) | |
2009 | (let ((poly nil) | |
2010 | (n 0)) | |
2011 | (while t1 | |
2012 | (setq poly (math-poly-mix poly 1 | |
2013 | (math-sum-integer-power n) (car t1)) | |
2014 | n (1+ n) | |
2015 | t1 (cdr t1))) | |
2016 | (setq n (math-build-polynomial-expr poly high)) | |
a6c72dc3 | 2017 | (if (= low 1) |
136211a9 EZ |
2018 | n |
2019 | (math-sub n (math-build-polynomial-expr poly | |
2020 | (math-sub low 1)))))) | |
2021 | ((and (memq (car expr) '(+ -)) | |
2022 | (setq t1 (math-sum-rec (nth 1 expr) var low high) | |
2023 | t2 (math-sum-rec (nth 2 expr) var low high)) | |
2024 | (not (and (math-expr-calls t1 '(calcFunc-sum)) | |
2025 | (math-expr-calls t2 '(calcFunc-sum))))) | |
2026 | (list (car expr) t1 t2)) | |
2027 | ((and (eq (car expr) '*) | |
2028 | (setq t1 (math-sum-const-factors expr var))) | |
2029 | (math-mul (car t1) (math-sum-rec (cdr t1) var low high))) | |
2030 | ((and (eq (car expr) '*) (memq (car-safe (nth 1 expr)) '(+ -))) | |
2031 | (math-sum-rec (math-add-or-sub (math-mul (nth 1 (nth 1 expr)) | |
2032 | (nth 2 expr)) | |
2033 | (math-mul (nth 2 (nth 1 expr)) | |
2034 | (nth 2 expr)) | |
2035 | nil (eq (car (nth 1 expr)) '-)) | |
2036 | var low high)) | |
2037 | ((and (eq (car expr) '*) (memq (car-safe (nth 2 expr)) '(+ -))) | |
2038 | (math-sum-rec (math-add-or-sub (math-mul (nth 1 expr) | |
2039 | (nth 1 (nth 2 expr))) | |
2040 | (math-mul (nth 1 expr) | |
2041 | (nth 2 (nth 2 expr))) | |
2042 | nil (eq (car (nth 2 expr)) '-)) | |
2043 | var low high)) | |
2044 | ((and (eq (car expr) '/) | |
2045 | (not (math-primp (nth 1 expr))) | |
2046 | (setq t1 (math-sum-const-factors (nth 1 expr) var))) | |
2047 | (math-mul (car t1) | |
2048 | (math-sum-rec (math-div (cdr t1) (nth 2 expr)) | |
2049 | var low high))) | |
2050 | ((and (eq (car expr) '/) | |
2051 | (setq t1 (math-sum-const-factors (nth 2 expr) var))) | |
2052 | (math-div (math-sum-rec (math-div (nth 1 expr) (cdr t1)) | |
2053 | var low high) | |
2054 | (car t1))) | |
2055 | ((eq (car expr) 'neg) | |
2056 | (math-neg (math-sum-rec (nth 1 expr) var low high))) | |
2057 | ((and (eq (car expr) '^) | |
2058 | (not (math-expr-contains (nth 1 expr) var)) | |
2059 | (setq t1 (math-is-polynomial (nth 2 expr) var 1))) | |
2060 | (let ((x (math-pow (nth 1 expr) (nth 1 t1)))) | |
2061 | (math-div (math-mul (math-sub (math-pow x (math-add 1 high)) | |
2062 | (math-pow x low)) | |
2063 | (math-pow (nth 1 expr) (car t1))) | |
2064 | (math-sub x 1)))) | |
2065 | ((and (setq t1 (math-to-exponentials expr)) | |
2066 | (setq t1 (math-sum-rec t1 var low high)) | |
2067 | (not (math-expr-calls t1 '(calcFunc-sum)))) | |
2068 | (math-to-exps t1)) | |
2069 | ((memq (car expr) '(calcFunc-ln calcFunc-log10)) | |
2070 | (list (car expr) (calcFunc-prod (nth 1 expr) var low high))) | |
2071 | ((and (eq (car expr) 'calcFunc-log) | |
2072 | (= (length expr) 3) | |
2073 | (not (math-expr-contains (nth 2 expr) var))) | |
2074 | (list 'calcFunc-log | |
2075 | (calcFunc-prod (nth 1 expr) var low high) | |
2076 | (nth 2 expr))))) | |
2077 | (if (equal val '(var nan var-nan)) (setq val nil)) | |
2078 | (or val | |
2079 | (let* ((math-tabulate-initial 0) | |
2080 | (math-tabulate-function 'calcFunc-sum)) | |
bf77c646 | 2081 | (calcFunc-table expr var low high))))) |
136211a9 EZ |
2082 | |
2083 | (defun calcFunc-asum (expr var low &optional high step no-mul-flag) | |
2084 | (or high (setq high low low 1)) | |
2085 | (if (and step (not (math-equal-int step 1))) | |
2086 | (if (math-negp step) | |
2087 | (math-mul (math-pow -1 low) | |
2088 | (calcFunc-asum expr var high low (math-neg step) t)) | |
2089 | (let ((lo (math-simplify (math-div low step)))) | |
2090 | (if (math-num-integerp lo) | |
2091 | (calcFunc-asum (math-normalize | |
2092 | (math-expr-subst expr var | |
2093 | (math-mul step var))) | |
2094 | var lo (math-simplify (math-div high step))) | |
2095 | (calcFunc-asum (math-normalize | |
2096 | (math-expr-subst expr var | |
2097 | (math-add (math-mul step var) | |
2098 | low))) | |
2099 | var 0 | |
2100 | (math-simplify (math-div (math-sub high low) | |
2101 | step)))))) | |
2102 | (math-mul (if no-mul-flag 1 (math-pow -1 low)) | |
bf77c646 | 2103 | (calcFunc-sum (math-mul (math-pow -1 var) expr) var low high)))) |
136211a9 EZ |
2104 | |
2105 | (defun math-sum-const-factors (expr var) | |
2106 | (let ((const nil) | |
2107 | (not-const nil) | |
2108 | (p expr)) | |
2109 | (while (eq (car-safe p) '*) | |
2110 | (if (math-expr-contains (nth 1 p) var) | |
2111 | (setq not-const (cons (nth 1 p) not-const)) | |
2112 | (setq const (cons (nth 1 p) const))) | |
2113 | (setq p (nth 2 p))) | |
2114 | (if (math-expr-contains p var) | |
2115 | (setq not-const (cons p not-const)) | |
2116 | (setq const (cons p const))) | |
2117 | (and const | |
2118 | (cons (let ((temp (car const))) | |
2119 | (while (setq const (cdr const)) | |
2120 | (setq temp (list '* (car const) temp))) | |
2121 | temp) | |
2122 | (let ((temp (or (car not-const) 1))) | |
2123 | (while (setq not-const (cdr not-const)) | |
2124 | (setq temp (list '* (car not-const) temp))) | |
bf77c646 | 2125 | temp))))) |
136211a9 | 2126 | |
3132f345 | 2127 | (defvar math-sum-int-pow-cache (list '(0 1))) |
136211a9 EZ |
2128 | ;; Following is from CRC Math Tables, 27th ed, pp. 52-53. |
2129 | (defun math-sum-integer-power (pow) | |
2130 | (let ((calc-prefer-frac t) | |
2131 | (n (length math-sum-int-pow-cache))) | |
2132 | (while (<= n pow) | |
2133 | (let* ((new (list 0 0)) | |
2134 | (lin new) | |
2135 | (pp (cdr (nth (1- n) math-sum-int-pow-cache))) | |
2136 | (p 2) | |
2137 | (sum 0) | |
2138 | q) | |
2139 | (while pp | |
2140 | (setq q (math-div (car pp) p) | |
2141 | new (cons (math-mul q n) new) | |
2142 | sum (math-add sum q) | |
2143 | p (1+ p) | |
2144 | pp (cdr pp))) | |
2145 | (setcar lin (math-sub 1 (math-mul n sum))) | |
2146 | (setq math-sum-int-pow-cache | |
2147 | (nconc math-sum-int-pow-cache (list (nreverse new))) | |
2148 | n (1+ n)))) | |
bf77c646 | 2149 | (nth pow math-sum-int-pow-cache))) |
136211a9 EZ |
2150 | |
2151 | (defun math-to-exponentials (expr) | |
2152 | (and (consp expr) | |
2153 | (= (length expr) 2) | |
2154 | (let ((x (nth 1 expr)) | |
2155 | (pi (if calc-symbolic-mode '(var pi var-pi) (math-pi))) | |
2156 | (i (if calc-symbolic-mode '(var i var-i) '(cplx 0 1)))) | |
2157 | (cond ((eq (car expr) 'calcFunc-exp) | |
2158 | (list '^ '(var e var-e) x)) | |
2159 | ((eq (car expr) 'calcFunc-sin) | |
2160 | (or (eq calc-angle-mode 'rad) | |
2161 | (setq x (list '/ (list '* x pi) 180))) | |
2162 | (list '/ (list '- | |
2163 | (list '^ '(var e var-e) (list '* x i)) | |
2164 | (list '^ '(var e var-e) | |
2165 | (list 'neg (list '* x i)))) | |
2166 | (list '* 2 i))) | |
2167 | ((eq (car expr) 'calcFunc-cos) | |
2168 | (or (eq calc-angle-mode 'rad) | |
2169 | (setq x (list '/ (list '* x pi) 180))) | |
2170 | (list '/ (list '+ | |
2171 | (list '^ '(var e var-e) | |
2172 | (list '* x i)) | |
2173 | (list '^ '(var e var-e) | |
2174 | (list 'neg (list '* x i)))) | |
2175 | 2)) | |
2176 | ((eq (car expr) 'calcFunc-sinh) | |
2177 | (list '/ (list '- | |
2178 | (list '^ '(var e var-e) x) | |
2179 | (list '^ '(var e var-e) (list 'neg x))) | |
2180 | 2)) | |
2181 | ((eq (car expr) 'calcFunc-cosh) | |
2182 | (list '/ (list '+ | |
2183 | (list '^ '(var e var-e) x) | |
2184 | (list '^ '(var e var-e) (list 'neg x))) | |
2185 | 2)) | |
bf77c646 | 2186 | (t nil))))) |
136211a9 EZ |
2187 | |
2188 | (defun math-to-exps (expr) | |
2189 | (cond (calc-symbolic-mode expr) | |
2190 | ((Math-primp expr) | |
2191 | (if (equal expr '(var e var-e)) (math-e) expr)) | |
2192 | ((and (eq (car expr) '^) | |
2193 | (equal (nth 1 expr) '(var e var-e))) | |
2194 | (list 'calcFunc-exp (nth 2 expr))) | |
2195 | (t | |
bf77c646 | 2196 | (cons (car expr) (mapcar 'math-to-exps (cdr expr)))))) |
136211a9 EZ |
2197 | |
2198 | ||
3132f345 | 2199 | (defvar math-disable-prods nil) |
136211a9 EZ |
2200 | (defun calcFunc-prod (expr var &optional low high step) |
2201 | (if math-disable-prods (math-reject-arg)) | |
2202 | (let* ((res (let* ((calc-internal-prec (+ calc-internal-prec 2))) | |
2203 | (math-prod-rec expr var low high step))) | |
2204 | (math-disable-prods t)) | |
bf77c646 | 2205 | (math-normalize res))) |
136211a9 EZ |
2206 | |
2207 | (defun math-prod-rec (expr var &optional low high step) | |
2208 | (or low (setq low '(neg (var inf var-inf)) high '(var inf var-inf))) | |
2209 | (and low (not high) (setq high '(var inf var-inf))) | |
2210 | (let (t1 t2 t3 val) | |
2211 | (setq val | |
2212 | (cond | |
2213 | ((not (math-expr-contains expr var)) | |
2214 | (math-pow expr (math-add (math-div (math-sub high low) (or step 1)) | |
2215 | 1))) | |
2216 | ((and step (not (math-equal-int step 1))) | |
2217 | (if (math-negp step) | |
2218 | (math-prod-rec expr var high low (math-neg step)) | |
2219 | (let ((lo (math-simplify (math-div low step)))) | |
2220 | (if (math-known-num-integerp lo) | |
2221 | (math-prod-rec (math-normalize | |
2222 | (math-expr-subst expr var | |
2223 | (math-mul step var))) | |
2224 | var lo (math-simplify (math-div high step))) | |
2225 | (math-prod-rec (math-normalize | |
2226 | (math-expr-subst expr var | |
2227 | (math-add (math-mul step | |
2228 | var) | |
2229 | low))) | |
2230 | var 0 | |
2231 | (math-simplify (math-div (math-sub high low) | |
2232 | step))))))) | |
2233 | ((and (memq (car expr) '(* /)) | |
2234 | (setq t1 (math-prod-rec (nth 1 expr) var low high) | |
2235 | t2 (math-prod-rec (nth 2 expr) var low high)) | |
2236 | (not (and (math-expr-calls t1 '(calcFunc-prod)) | |
2237 | (math-expr-calls t2 '(calcFunc-prod))))) | |
2238 | (list (car expr) t1 t2)) | |
2239 | ((and (eq (car expr) '^) | |
2240 | (not (math-expr-contains (nth 2 expr) var))) | |
2241 | (math-pow (math-prod-rec (nth 1 expr) var low high) | |
2242 | (nth 2 expr))) | |
2243 | ((and (eq (car expr) '^) | |
2244 | (not (math-expr-contains (nth 1 expr) var))) | |
2245 | (math-pow (nth 1 expr) | |
2246 | (calcFunc-sum (nth 2 expr) var low high))) | |
2247 | ((eq (car expr) 'sqrt) | |
2248 | (math-normalize (list 'calcFunc-sqrt | |
2249 | (list 'calcFunc-prod (nth 1 expr) | |
2250 | var low high)))) | |
2251 | ((eq (car expr) 'neg) | |
2252 | (math-mul (math-pow -1 (math-add (math-sub high low) 1)) | |
2253 | (math-prod-rec (nth 1 expr) var low high))) | |
2254 | ((eq (car expr) 'calcFunc-exp) | |
2255 | (list 'calcFunc-exp (calcFunc-sum (nth 1 expr) var low high))) | |
2256 | ((and (setq t1 (math-is-polynomial expr var 1)) | |
2257 | (setq t2 | |
2258 | (cond | |
2259 | ((or (and (math-equal-int (nth 1 t1) 1) | |
2260 | (setq low (math-simplify | |
2261 | (math-add low (car t1))) | |
2262 | high (math-simplify | |
2263 | (math-add high (car t1))))) | |
2264 | (and (math-equal-int (nth 1 t1) -1) | |
2265 | (setq t2 low | |
2266 | low (math-simplify | |
2267 | (math-sub (car t1) high)) | |
2268 | high (math-simplify | |
2269 | (math-sub (car t1) t2))))) | |
2270 | (if (or (math-zerop low) (math-zerop high)) | |
2271 | 0 | |
2272 | (if (and (or (math-negp low) (math-negp high)) | |
2273 | (or (math-num-integerp low) | |
2274 | (math-num-integerp high))) | |
2275 | (if (math-posp high) | |
2276 | 0 | |
2277 | (math-mul (math-pow -1 | |
2278 | (math-add | |
2279 | (math-add low high) 1)) | |
2280 | (list '/ | |
2281 | (list 'calcFunc-fact | |
2282 | (math-neg low)) | |
2283 | (list 'calcFunc-fact | |
2284 | (math-sub -1 high))))) | |
2285 | (list '/ | |
2286 | (list 'calcFunc-fact high) | |
2287 | (list 'calcFunc-fact (math-sub low 1)))))) | |
2288 | ((and (or (and (math-equal-int (nth 1 t1) 2) | |
2289 | (setq t2 (math-simplify | |
2290 | (math-add (math-mul low 2) | |
2291 | (car t1))) | |
2292 | t3 (math-simplify | |
2293 | (math-add (math-mul high 2) | |
2294 | (car t1))))) | |
2295 | (and (math-equal-int (nth 1 t1) -2) | |
2296 | (setq t2 (math-simplify | |
2297 | (math-sub (car t1) | |
2298 | (math-mul high 2))) | |
a1506d29 | 2299 | t3 (math-simplify |
136211a9 EZ |
2300 | (math-sub (car t1) |
2301 | (math-mul low | |
2302 | 2)))))) | |
2303 | (or (math-integerp t2) | |
2304 | (and (math-messy-integerp t2) | |
2305 | (setq t2 (math-trunc t2))) | |
2306 | (math-integerp t3) | |
2307 | (and (math-messy-integerp t3) | |
2308 | (setq t3 (math-trunc t3))))) | |
2309 | (if (or (math-zerop t2) (math-zerop t3)) | |
2310 | 0 | |
2311 | (if (or (math-evenp t2) (math-evenp t3)) | |
2312 | (if (or (math-negp t2) (math-negp t3)) | |
2313 | (if (math-posp high) | |
2314 | 0 | |
2315 | (list '/ | |
2316 | (list 'calcFunc-dfact | |
2317 | (math-neg t2)) | |
2318 | (list 'calcFunc-dfact | |
2319 | (math-sub -2 t3)))) | |
2320 | (list '/ | |
2321 | (list 'calcFunc-dfact t3) | |
2322 | (list 'calcFunc-dfact | |
2323 | (math-sub t2 2)))) | |
2324 | (if (math-negp t3) | |
2325 | (list '* | |
2326 | (list '^ -1 | |
2327 | (list '/ (list '- (list '- t2 t3) | |
2328 | 2) | |
2329 | 2)) | |
2330 | (list '/ | |
2331 | (list 'calcFunc-dfact | |
2332 | (math-neg t2)) | |
2333 | (list 'calcFunc-dfact | |
2334 | (math-sub -2 t3)))) | |
2335 | (if (math-posp t2) | |
2336 | (list '/ | |
2337 | (list 'calcFunc-dfact t3) | |
2338 | (list 'calcFunc-dfact | |
2339 | (math-sub t2 2))) | |
2340 | nil)))))))) | |
2341 | t2))) | |
2342 | (if (equal val '(var nan var-nan)) (setq val nil)) | |
2343 | (or val | |
2344 | (let* ((math-tabulate-initial 1) | |
2345 | (math-tabulate-function 'calcFunc-prod)) | |
bf77c646 | 2346 | (calcFunc-table expr var low high))))) |
136211a9 EZ |
2347 | |
2348 | ||
2349 | ||
2350 | ||
3132f345 | 2351 | (defvar math-solve-ranges nil) |
f7adda54 JB |
2352 | (defvar math-solve-sign) |
2353 | ;;; Attempt to reduce math-solve-lhs = math-solve-rhs to | |
2354 | ;;; math-solve-var = math-solve-rhs', where math-solve-var appears | |
2355 | ;;; in math-solve-lhs but not in math-solve-rhs or math-solve-rhs'; | |
2356 | ;;; return math-solve-rhs'. | |
2357 | ;;; Uses global values: math-solve-var, math-solve-full. | |
2358 | (defvar math-solve-var) | |
2359 | (defvar math-solve-full) | |
2360 | ||
2361 | ;; The variables math-solve-lhs, math-solve-rhs and math-try-solve-sign | |
2362 | ;; are local to math-try-solve-for, but are used by math-try-solve-prod. | |
2363 | ;; (math-solve-lhs and math-solve-rhs are is also local to | |
2364 | ;; math-decompose-poly, but used by math-solve-poly-funny-powers.) | |
2365 | (defvar math-solve-lhs) | |
2366 | (defvar math-solve-rhs) | |
79d2746f | 2367 | (defvar math-try-solve-sign) |
f7adda54 JB |
2368 | |
2369 | (defun math-try-solve-for | |
2370 | (math-solve-lhs math-solve-rhs &optional math-try-solve-sign no-poly) | |
2371 | (let (math-t1 math-t2 math-t3) | |
2372 | (cond ((equal math-solve-lhs math-solve-var) | |
2373 | (setq math-solve-sign math-try-solve-sign) | |
2374 | (if (eq math-solve-full 'all) | |
2375 | (let ((vec (list 'vec (math-evaluate-expr math-solve-rhs))) | |
136211a9 EZ |
2376 | newvec var p) |
2377 | (while math-solve-ranges | |
2378 | (setq p (car math-solve-ranges) | |
2379 | var (car p) | |
2380 | newvec (list 'vec)) | |
2381 | (while (setq p (cdr p)) | |
2382 | (setq newvec (nconc newvec | |
2383 | (cdr (math-expr-subst | |
2384 | vec var (car p)))))) | |
2385 | (setq vec newvec | |
2386 | math-solve-ranges (cdr math-solve-ranges))) | |
2387 | (math-normalize vec)) | |
f7adda54 JB |
2388 | math-solve-rhs)) |
2389 | ((Math-primp math-solve-lhs) | |
136211a9 | 2390 | nil) |
f7adda54 JB |
2391 | ((and (eq (car math-solve-lhs) '-) |
2392 | (eq (car-safe (nth 1 math-solve-lhs)) (car-safe (nth 2 math-solve-lhs))) | |
2393 | (Math-zerop math-solve-rhs) | |
2394 | (= (length (nth 1 math-solve-lhs)) 2) | |
2395 | (= (length (nth 2 math-solve-lhs)) 2) | |
2396 | (setq math-t1 (get (car (nth 1 math-solve-lhs)) 'math-inverse)) | |
2397 | (setq math-t2 (funcall math-t1 '(var SOLVEDUM SOLVEDUM))) | |
2398 | (eq (math-expr-contains-count math-t2 '(var SOLVEDUM SOLVEDUM)) 1) | |
2399 | (setq math-t3 (math-solve-above-dummy math-t2)) | |
2400 | (setq math-t1 (math-try-solve-for | |
2401 | (math-sub (nth 1 (nth 1 math-solve-lhs)) | |
2402 | (math-expr-subst | |
2403 | math-t2 math-t3 | |
2404 | (nth 1 (nth 2 math-solve-lhs)))) | |
2405 | 0))) | |
2406 | math-t1) | |
2407 | ((eq (car math-solve-lhs) 'neg) | |
2408 | (math-try-solve-for (nth 1 math-solve-lhs) (math-neg math-solve-rhs) | |
2409 | (and math-try-solve-sign (- math-try-solve-sign)))) | |
2410 | ((and (not (eq math-solve-full 't)) (math-try-solve-prod))) | |
136211a9 | 2411 | ((and (not no-poly) |
f7adda54 JB |
2412 | (setq math-t2 |
2413 | (math-decompose-poly math-solve-lhs | |
2414 | math-solve-var 15 math-solve-rhs))) | |
2415 | (setq math-t1 (cdr (nth 1 math-t2)) | |
2416 | math-t1 (let ((math-solve-ranges math-solve-ranges)) | |
2417 | (cond ((= (length math-t1) 5) | |
2418 | (apply 'math-solve-quartic (car math-t2) math-t1)) | |
2419 | ((= (length math-t1) 4) | |
2420 | (apply 'math-solve-cubic (car math-t2) math-t1)) | |
2421 | ((= (length math-t1) 3) | |
2422 | (apply 'math-solve-quadratic (car math-t2) math-t1)) | |
2423 | ((= (length math-t1) 2) | |
2424 | (apply 'math-solve-linear | |
2425 | (car math-t2) math-try-solve-sign math-t1)) | |
2426 | (math-solve-full | |
2427 | (math-poly-all-roots (car math-t2) math-t1)) | |
136211a9 EZ |
2428 | (calc-symbolic-mode nil) |
2429 | (t | |
2430 | (math-try-solve-for | |
f7adda54 JB |
2431 | (car math-t2) |
2432 | (math-poly-any-root (reverse math-t1) 0 t) | |
136211a9 | 2433 | nil t))))) |
f7adda54 JB |
2434 | (if math-t1 |
2435 | (if (eq (nth 2 math-t2) 1) | |
2436 | math-t1 | |
2437 | (math-solve-prod math-t1 (math-try-solve-for (nth 2 math-t2) 0 nil t))) | |
136211a9 EZ |
2438 | (calc-record-why "*Unable to find a symbolic solution") |
2439 | nil)) | |
f7adda54 JB |
2440 | ((and (math-solve-find-root-term math-solve-lhs nil) |
2441 | (eq (math-expr-contains-count math-solve-lhs math-t1) 1)) ; just in case | |
136211a9 | 2442 | (math-try-solve-for (math-simplify |
f7adda54 JB |
2443 | (math-sub (if (or math-t3 (math-evenp math-t2)) |
2444 | (math-pow math-t1 math-t2) | |
2445 | (math-neg (math-pow math-t1 math-t2))) | |
136211a9 EZ |
2446 | (math-expand-power |
2447 | (math-sub (math-normalize | |
2448 | (math-expr-subst | |
f7adda54 JB |
2449 | math-solve-lhs math-t1 0)) |
2450 | math-solve-rhs) | |
2451 | math-t2 math-solve-var))) | |
136211a9 | 2452 | 0)) |
f7adda54 JB |
2453 | ((eq (car math-solve-lhs) '+) |
2454 | (cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var)) | |
2455 | (math-try-solve-for (nth 2 math-solve-lhs) | |
2456 | (math-sub math-solve-rhs (nth 1 math-solve-lhs)) | |
2457 | math-try-solve-sign)) | |
2458 | ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var)) | |
2459 | (math-try-solve-for (nth 1 math-solve-lhs) | |
2460 | (math-sub math-solve-rhs (nth 2 math-solve-lhs)) | |
2461 | math-try-solve-sign)))) | |
2462 | ((eq (car math-solve-lhs) 'calcFunc-eq) | |
2463 | (math-try-solve-for (math-sub (nth 1 math-solve-lhs) (nth 2 math-solve-lhs)) | |
2464 | math-solve-rhs math-try-solve-sign no-poly)) | |
2465 | ((eq (car math-solve-lhs) '-) | |
2466 | (cond ((or (and (eq (car-safe (nth 1 math-solve-lhs)) 'calcFunc-sin) | |
2467 | (eq (car-safe (nth 2 math-solve-lhs)) 'calcFunc-cos)) | |
2468 | (and (eq (car-safe (nth 1 math-solve-lhs)) 'calcFunc-cos) | |
2469 | (eq (car-safe (nth 2 math-solve-lhs)) 'calcFunc-sin))) | |
2470 | (math-try-solve-for (math-sub (nth 1 math-solve-lhs) | |
2471 | (list (car (nth 1 math-solve-lhs)) | |
136211a9 EZ |
2472 | (math-sub |
2473 | (math-quarter-circle t) | |
f7adda54 JB |
2474 | (nth 1 (nth 2 math-solve-lhs))))) |
2475 | math-solve-rhs)) | |
2476 | ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var)) | |
2477 | (math-try-solve-for (nth 2 math-solve-lhs) | |
2478 | (math-sub (nth 1 math-solve-lhs) math-solve-rhs) | |
2479 | (and math-try-solve-sign | |
2480 | (- math-try-solve-sign)))) | |
2481 | ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var)) | |
2482 | (math-try-solve-for (nth 1 math-solve-lhs) | |
2483 | (math-add math-solve-rhs (nth 2 math-solve-lhs)) | |
2484 | math-try-solve-sign)))) | |
2485 | ((and (eq math-solve-full 't) (math-try-solve-prod))) | |
2486 | ((and (eq (car math-solve-lhs) '%) | |
2487 | (not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var))) | |
2488 | (math-try-solve-for (nth 1 math-solve-lhs) (math-add math-solve-rhs | |
136211a9 | 2489 | (math-solve-get-int |
f7adda54 JB |
2490 | (nth 2 math-solve-lhs))))) |
2491 | ((eq (car math-solve-lhs) 'calcFunc-log) | |
2492 | (cond ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var)) | |
2493 | (math-try-solve-for (nth 1 math-solve-lhs) | |
2494 | (math-pow (nth 2 math-solve-lhs) math-solve-rhs))) | |
2495 | ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var)) | |
2496 | (math-try-solve-for (nth 2 math-solve-lhs) (math-pow | |
2497 | (nth 1 math-solve-lhs) | |
2498 | (math-div 1 math-solve-rhs)))))) | |
2499 | ((and (= (length math-solve-lhs) 2) | |
2500 | (symbolp (car math-solve-lhs)) | |
2501 | (setq math-t1 (get (car math-solve-lhs) 'math-inverse)) | |
2502 | (setq math-t2 (funcall math-t1 math-solve-rhs))) | |
2503 | (setq math-t1 (get (car math-solve-lhs) 'math-inverse-sign)) | |
2504 | (math-try-solve-for (nth 1 math-solve-lhs) (math-normalize math-t2) | |
2505 | (and math-try-solve-sign math-t1 | |
2506 | (if (integerp math-t1) | |
2507 | (* math-t1 math-try-solve-sign) | |
2508 | (funcall math-t1 math-solve-lhs | |
2509 | math-try-solve-sign))))) | |
2510 | ((and (symbolp (car math-solve-lhs)) | |
2511 | (setq math-t1 (get (car math-solve-lhs) 'math-inverse-n)) | |
2512 | (setq math-t2 (funcall math-t1 math-solve-lhs math-solve-rhs))) | |
2513 | math-t2) | |
2514 | ((setq math-t1 (math-expand-formula math-solve-lhs)) | |
2515 | (math-try-solve-for math-t1 math-solve-rhs math-try-solve-sign)) | |
136211a9 | 2516 | (t |
f7adda54 | 2517 | (calc-record-why "*No inverse known" math-solve-lhs) |
bf77c646 | 2518 | nil)))) |
136211a9 | 2519 | |
136211a9 EZ |
2520 | |
2521 | (defun math-try-solve-prod () | |
f7adda54 JB |
2522 | (cond ((eq (car math-solve-lhs) '*) |
2523 | (cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var)) | |
2524 | (math-try-solve-for (nth 2 math-solve-lhs) | |
2525 | (math-div math-solve-rhs (nth 1 math-solve-lhs)) | |
2526 | (math-solve-sign math-try-solve-sign | |
2527 | (nth 1 math-solve-lhs)))) | |
2528 | ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var)) | |
2529 | (math-try-solve-for (nth 1 math-solve-lhs) | |
2530 | (math-div math-solve-rhs (nth 2 math-solve-lhs)) | |
2531 | (math-solve-sign math-try-solve-sign | |
2532 | (nth 2 math-solve-lhs)))) | |
2533 | ((Math-zerop math-solve-rhs) | |
136211a9 | 2534 | (math-solve-prod (let ((math-solve-ranges math-solve-ranges)) |
f7adda54 JB |
2535 | (math-try-solve-for (nth 2 math-solve-lhs) 0)) |
2536 | (math-try-solve-for (nth 1 math-solve-lhs) 0))))) | |
2537 | ((eq (car math-solve-lhs) '/) | |
2538 | (cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var)) | |
2539 | (math-try-solve-for (nth 2 math-solve-lhs) | |
2540 | (math-div (nth 1 math-solve-lhs) math-solve-rhs) | |
2541 | (math-solve-sign math-try-solve-sign | |
2542 | (nth 1 math-solve-lhs)))) | |
2543 | ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var)) | |
2544 | (math-try-solve-for (nth 1 math-solve-lhs) | |
2545 | (math-mul math-solve-rhs (nth 2 math-solve-lhs)) | |
2546 | (math-solve-sign math-try-solve-sign | |
2547 | (nth 2 math-solve-lhs)))) | |
2548 | ((setq math-t1 (math-try-solve-for (math-sub (nth 1 math-solve-lhs) | |
2549 | (math-mul (nth 2 math-solve-lhs) | |
2550 | math-solve-rhs)) | |
136211a9 | 2551 | 0)) |
f7adda54 JB |
2552 | math-t1))) |
2553 | ((eq (car math-solve-lhs) '^) | |
2554 | (cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var)) | |
136211a9 | 2555 | (math-try-solve-for |
f7adda54 | 2556 | (nth 2 math-solve-lhs) |
136211a9 | 2557 | (math-add (math-normalize |
f7adda54 | 2558 | (list 'calcFunc-log math-solve-rhs (nth 1 math-solve-lhs))) |
136211a9 EZ |
2559 | (math-div |
2560 | (math-mul 2 | |
2561 | (math-mul '(var pi var-pi) | |
2562 | (math-solve-get-int | |
2563 | '(var i var-i)))) | |
2564 | (math-normalize | |
f7adda54 JB |
2565 | (list 'calcFunc-ln (nth 1 math-solve-lhs))))))) |
2566 | ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var)) | |
2567 | (cond ((and (integerp (nth 2 math-solve-lhs)) | |
2568 | (>= (nth 2 math-solve-lhs) 2) | |
2569 | (setq math-t1 (math-integer-log2 (nth 2 math-solve-lhs)))) | |
2570 | (setq math-t2 math-solve-rhs) | |
2571 | (if (and (eq math-solve-full t) | |
2572 | (math-known-realp (nth 1 math-solve-lhs))) | |
136211a9 | 2573 | (progn |
f7adda54 JB |
2574 | (while (>= (setq math-t1 (1- math-t1)) 0) |
2575 | (setq math-t2 (list 'calcFunc-sqrt math-t2))) | |
2576 | (setq math-t2 (math-solve-get-sign math-t2))) | |
2577 | (while (>= (setq math-t1 (1- math-t1)) 0) | |
2578 | (setq math-t2 (math-solve-get-sign | |
136211a9 | 2579 | (math-normalize |
f7adda54 | 2580 | (list 'calcFunc-sqrt math-t2)))))) |
136211a9 | 2581 | (math-try-solve-for |
f7adda54 JB |
2582 | (nth 1 math-solve-lhs) |
2583 | (math-normalize math-t2))) | |
2584 | ((math-looks-negp (nth 2 math-solve-lhs)) | |
136211a9 | 2585 | (math-try-solve-for |
f7adda54 JB |
2586 | (list '^ (nth 1 math-solve-lhs) |
2587 | (math-neg (nth 2 math-solve-lhs))) | |
2588 | (math-div 1 math-solve-rhs))) | |
2589 | ((and (eq math-solve-full t) | |
2590 | (Math-integerp (nth 2 math-solve-lhs)) | |
2591 | (math-known-realp (nth 1 math-solve-lhs))) | |
2592 | (setq math-t1 (math-normalize | |
2593 | (list 'calcFunc-nroot math-solve-rhs | |
2594 | (nth 2 math-solve-lhs)))) | |
2595 | (if (math-evenp (nth 2 math-solve-lhs)) | |
2596 | (setq math-t1 (math-solve-get-sign math-t1))) | |
136211a9 | 2597 | (math-try-solve-for |
f7adda54 JB |
2598 | (nth 1 math-solve-lhs) math-t1 |
2599 | (and math-try-solve-sign | |
2600 | (math-oddp (nth 2 math-solve-lhs)) | |
2601 | (math-solve-sign math-try-solve-sign | |
2602 | (nth 2 math-solve-lhs))))) | |
136211a9 | 2603 | (t (math-try-solve-for |
f7adda54 | 2604 | (nth 1 math-solve-lhs) |
136211a9 EZ |
2605 | (math-mul |
2606 | (math-normalize | |
2607 | (list 'calcFunc-exp | |
f7adda54 | 2608 | (if (Math-realp (nth 2 math-solve-lhs)) |
136211a9 EZ |
2609 | (math-div (math-mul |
2610 | '(var pi var-pi) | |
2611 | (math-solve-get-int | |
2612 | '(var i var-i) | |
f7adda54 | 2613 | (and (integerp (nth 2 math-solve-lhs)) |
136211a9 | 2614 | (math-abs |
f7adda54 JB |
2615 | (nth 2 math-solve-lhs))))) |
2616 | (math-div (nth 2 math-solve-lhs) 2)) | |
136211a9 EZ |
2617 | (math-div (math-mul |
2618 | 2 | |
2619 | (math-mul | |
2620 | '(var pi var-pi) | |
2621 | (math-solve-get-int | |
2622 | '(var i var-i) | |
f7adda54 | 2623 | (and (integerp (nth 2 math-solve-lhs)) |
136211a9 | 2624 | (math-abs |
f7adda54 JB |
2625 | (nth 2 math-solve-lhs)))))) |
2626 | (nth 2 math-solve-lhs))))) | |
136211a9 EZ |
2627 | (math-normalize |
2628 | (list 'calcFunc-nroot | |
f7adda54 JB |
2629 | math-solve-rhs |
2630 | (nth 2 math-solve-lhs)))) | |
2631 | (and math-try-solve-sign | |
2632 | (math-oddp (nth 2 math-solve-lhs)) | |
2633 | (math-solve-sign math-try-solve-sign | |
2634 | (nth 2 math-solve-lhs))))))))) | |
bf77c646 | 2635 | (t nil))) |
136211a9 EZ |
2636 | |
2637 | (defun math-solve-prod (lsoln rsoln) | |
2638 | (cond ((null lsoln) | |
2639 | rsoln) | |
2640 | ((null rsoln) | |
2641 | lsoln) | |
f7adda54 | 2642 | ((eq math-solve-full 'all) |
136211a9 | 2643 | (cons 'vec (append (cdr lsoln) (cdr rsoln)))) |
f7adda54 | 2644 | (math-solve-full |
136211a9 EZ |
2645 | (list 'calcFunc-if |
2646 | (list 'calcFunc-gt (math-solve-get-sign 1) 0) | |
2647 | lsoln | |
2648 | rsoln)) | |
bf77c646 | 2649 | (t lsoln))) |
136211a9 EZ |
2650 | |
2651 | ;;; This deals with negative, fractional, and symbolic powers of "x". | |
f7adda54 JB |
2652 | ;; The variable math-solve-b is local to math-decompose-poly, |
2653 | ;; but is used by math-solve-poly-funny-powers. | |
79d2746f | 2654 | (defvar math-solve-b) |
f7adda54 | 2655 | |
136211a9 | 2656 | (defun math-solve-poly-funny-powers (sub-rhs) ; uses "t1", "t2" |
f7adda54 | 2657 | (setq math-t1 math-solve-lhs) |
136211a9 EZ |
2658 | (let ((pp math-poly-neg-powers) |
2659 | fac) | |
2660 | (while pp | |
2661 | (setq fac (math-pow (car pp) (or math-poly-mult-powers 1)) | |
f7adda54 JB |
2662 | math-t1 (math-mul math-t1 fac) |
2663 | math-solve-rhs (math-mul math-solve-rhs fac) | |
136211a9 | 2664 | pp (cdr pp)))) |
f7adda54 | 2665 | (if sub-rhs (setq math-t1 (math-sub math-t1 math-solve-rhs))) |
136211a9 | 2666 | (let ((math-poly-neg-powers nil)) |
f7adda54 | 2667 | (setq math-t2 (math-mul (or math-poly-mult-powers 1) |
136211a9 EZ |
2668 | (let ((calc-prefer-frac t)) |
2669 | (math-div 1 math-poly-frac-powers))) | |
f7adda54 JB |
2670 | math-t1 (math-is-polynomial |
2671 | (math-simplify (calcFunc-expand math-t1)) math-solve-b 50)))) | |
136211a9 EZ |
2672 | |
2673 | ;;; This converts "a x^8 + b x^5 + c x^2" to "(a (x^3)^2 + b (x^3) + c) * x^2". | |
2674 | (defun math-solve-crunch-poly (max-degree) ; uses "t1", "t3" | |
2675 | (let ((count 0)) | |
f7adda54 JB |
2676 | (while (and math-t1 (Math-zerop (car math-t1))) |
2677 | (setq math-t1 (cdr math-t1) | |
136211a9 | 2678 | count (1+ count))) |
f7adda54 JB |
2679 | (and math-t1 |
2680 | (let* ((degree (1- (length math-t1))) | |
136211a9 | 2681 | (scale degree)) |
f7adda54 | 2682 | (while (and (> scale 1) (= (car math-t3) 1)) |
136211a9 | 2683 | (and (= (% degree scale) 0) |
f7adda54 | 2684 | (let ((p math-t1) |
136211a9 EZ |
2685 | (n 0) |
2686 | (new-t1 nil) | |
2687 | (okay t)) | |
2688 | (while (and p okay) | |
2689 | (if (= (% n scale) 0) | |
2690 | (setq new-t1 (nconc new-t1 (list (car p)))) | |
2691 | (or (Math-zerop (car p)) | |
2692 | (setq okay nil))) | |
2693 | (setq p (cdr p) | |
2694 | n (1+ n))) | |
2695 | (if okay | |
f7adda54 JB |
2696 | (setq math-t3 (cons scale (cdr math-t3)) |
2697 | math-t1 new-t1)))) | |
136211a9 | 2698 | (setq scale (1- scale))) |
f7adda54 JB |
2699 | (setq math-t3 (list (math-mul (car math-t3) math-t2) |
2700 | (math-mul count math-t2))) | |
2701 | (<= (1- (length math-t1)) max-degree))))) | |
136211a9 EZ |
2702 | |
2703 | (defun calcFunc-poly (expr var &optional degree) | |
2704 | (if degree | |
2705 | (or (natnump degree) (math-reject-arg degree 'fixnatnump)) | |
2706 | (setq degree 50)) | |
2707 | (let ((p (math-is-polynomial expr var degree 'gen))) | |
2708 | (if p | |
2709 | (if (equal p '(0)) | |
2710 | (list 'vec) | |
2711 | (cons 'vec p)) | |
bf77c646 | 2712 | (math-reject-arg expr "Expected a polynomial")))) |
136211a9 EZ |
2713 | |
2714 | (defun calcFunc-gpoly (expr var &optional degree) | |
2715 | (if degree | |
2716 | (or (natnump degree) (math-reject-arg degree 'fixnatnump)) | |
2717 | (setq degree 50)) | |
2718 | (let* ((math-poly-base-variable var) | |
2719 | (d (math-decompose-poly expr var degree nil))) | |
2720 | (if d | |
2721 | (cons 'vec d) | |
bf77c646 | 2722 | (math-reject-arg expr "Expected a polynomial")))) |
136211a9 | 2723 | |
f7adda54 JB |
2724 | (defun math-decompose-poly (math-solve-lhs math-solve-var degree sub-rhs) |
2725 | (let ((math-solve-rhs (or sub-rhs 1)) | |
2726 | math-t1 math-t2 math-t3) | |
2727 | (setq math-t2 (math-polynomial-base | |
2728 | math-solve-lhs | |
136211a9 | 2729 | (function |
f7adda54 | 2730 | (lambda (math-solve-b) |
136211a9 EZ |
2731 | (let ((math-poly-neg-powers '(1)) |
2732 | (math-poly-mult-powers nil) | |
2733 | (math-poly-frac-powers 1) | |
2734 | (math-poly-exp-base t)) | |
f7adda54 JB |
2735 | (and (not (equal math-solve-b math-solve-lhs)) |
2736 | (or (not (memq (car-safe math-solve-b) '(+ -))) sub-rhs) | |
2737 | (setq math-t3 '(1 0) math-t2 1 | |
2738 | math-t1 (math-is-polynomial math-solve-lhs | |
2739 | math-solve-b 50)) | |
136211a9 EZ |
2740 | (if (and (equal math-poly-neg-powers '(1)) |
2741 | (memq math-poly-mult-powers '(nil 1)) | |
2742 | (eq math-poly-frac-powers 1) | |
2743 | sub-rhs) | |
f7adda54 JB |
2744 | (setq math-t1 (cons (math-sub (car math-t1) math-solve-rhs) |
2745 | (cdr math-t1))) | |
136211a9 EZ |
2746 | (math-solve-poly-funny-powers sub-rhs)) |
2747 | (math-solve-crunch-poly degree) | |
f7adda54 JB |
2748 | (or (math-expr-contains math-solve-b math-solve-var) |
2749 | (math-expr-contains (car math-t3) math-solve-var)))))))) | |
2750 | (if math-t2 | |
2751 | (list (math-pow math-t2 (car math-t3)) | |
2752 | (cons 'vec math-t1) | |
136211a9 | 2753 | (if sub-rhs |
f7adda54 JB |
2754 | (math-pow math-t2 (nth 1 math-t3)) |
2755 | (math-div (math-pow math-t2 (nth 1 math-t3)) math-solve-rhs)))))) | |
136211a9 EZ |
2756 | |
2757 | (defun math-solve-linear (var sign b a) | |
2758 | (math-try-solve-for var | |
2759 | (math-div (math-neg b) a) | |
2760 | (math-solve-sign sign a) | |
bf77c646 | 2761 | t)) |
136211a9 EZ |
2762 | |
2763 | (defun math-solve-quadratic (var c b a) | |
2764 | (math-try-solve-for | |
2765 | var | |
2766 | (if (math-looks-evenp b) | |
2767 | (let ((halfb (math-div b 2))) | |
2768 | (math-div | |
2769 | (math-add | |
2770 | (math-neg halfb) | |
2771 | (math-solve-get-sign | |
2772 | (math-normalize | |
2773 | (list 'calcFunc-sqrt | |
2774 | (math-add (math-sqr halfb) | |
2775 | (math-mul (math-neg c) a)))))) | |
2776 | a)) | |
2777 | (math-div | |
2778 | (math-add | |
2779 | (math-neg b) | |
2780 | (math-solve-get-sign | |
2781 | (math-normalize | |
2782 | (list 'calcFunc-sqrt | |
2783 | (math-add (math-sqr b) | |
2784 | (math-mul 4 (math-mul (math-neg c) a))))))) | |
2785 | (math-mul 2 a))) | |
bf77c646 | 2786 | nil t)) |
136211a9 EZ |
2787 | |
2788 | (defun math-solve-cubic (var d c b a) | |
2789 | (let* ((p (math-div b a)) | |
2790 | (q (math-div c a)) | |
2791 | (r (math-div d a)) | |
2792 | (psqr (math-sqr p)) | |
2793 | (aa (math-sub q (math-div psqr 3))) | |
2794 | (bb (math-add r | |
2795 | (math-div (math-sub (math-mul 2 (math-mul psqr p)) | |
2796 | (math-mul 9 (math-mul p q))) | |
2797 | 27))) | |
2798 | m) | |
2799 | (if (Math-zerop aa) | |
2800 | (math-try-solve-for (math-pow (math-add var (math-div p 3)) 3) | |
2801 | (math-neg bb) nil t) | |
2802 | (if (Math-zerop bb) | |
2803 | (math-try-solve-for | |
2804 | (math-mul (math-add var (math-div p 3)) | |
2805 | (math-add (math-sqr (math-add var (math-div p 3))) | |
2806 | aa)) | |
2807 | 0 nil t) | |
2808 | (setq m (math-mul 2 (list 'calcFunc-sqrt (math-div aa -3)))) | |
2809 | (math-try-solve-for | |
2810 | var | |
2811 | (math-sub | |
2812 | (math-normalize | |
2813 | (math-mul | |
2814 | m | |
2815 | (list 'calcFunc-cos | |
2816 | (math-div | |
2817 | (math-sub (list 'calcFunc-arccos | |
2818 | (math-div (math-mul 3 bb) | |
2819 | (math-mul aa m))) | |
2820 | (math-mul 2 | |
2821 | (math-mul | |
2822 | (math-add 1 (math-solve-get-int | |
2823 | 1 3)) | |
2824 | (math-half-circle | |
2825 | calc-symbolic-mode)))) | |
2826 | 3)))) | |
2827 | (math-div p 3)) | |
bf77c646 | 2828 | nil t))))) |
136211a9 EZ |
2829 | |
2830 | (defun math-solve-quartic (var d c b a aa) | |
2831 | (setq a (math-div a aa)) | |
2832 | (setq b (math-div b aa)) | |
2833 | (setq c (math-div c aa)) | |
2834 | (setq d (math-div d aa)) | |
2835 | (math-try-solve-for | |
2836 | var | |
2837 | (let* ((asqr (math-sqr a)) | |
2838 | (asqr4 (math-div asqr 4)) | |
f7adda54 | 2839 | (y (let ((math-solve-full nil) |
136211a9 | 2840 | calc-next-why) |
f7adda54 | 2841 | (math-solve-cubic math-solve-var |
136211a9 EZ |
2842 | (math-sub (math-sub |
2843 | (math-mul 4 (math-mul b d)) | |
2844 | (math-mul asqr d)) | |
2845 | (math-sqr c)) | |
2846 | (math-sub (math-mul a c) | |
2847 | (math-mul 4 d)) | |
2848 | (math-neg b) | |
2849 | 1))) | |
2850 | (rsqr (math-add (math-sub asqr4 b) y)) | |
2851 | (r (list 'calcFunc-sqrt rsqr)) | |
2852 | (sign1 (math-solve-get-sign 1)) | |
2853 | (de (list 'calcFunc-sqrt | |
2854 | (math-add | |
2855 | (math-sub (math-mul 3 asqr4) | |
2856 | (math-mul 2 b)) | |
2857 | (if (Math-zerop rsqr) | |
2858 | (math-mul | |
2859 | 2 | |
2860 | (math-mul sign1 | |
2861 | (list 'calcFunc-sqrt | |
2862 | (math-sub (math-sqr y) | |
2863 | (math-mul 4 d))))) | |
2864 | (math-sub | |
2865 | (math-mul sign1 | |
2866 | (math-div | |
2867 | (math-sub (math-sub | |
2868 | (math-mul 4 (math-mul a b)) | |
2869 | (math-mul 8 c)) | |
2870 | (math-mul asqr a)) | |
2871 | (math-mul 4 r))) | |
2872 | rsqr)))))) | |
2873 | (math-normalize | |
2874 | (math-sub (math-add (math-mul sign1 (math-div r 2)) | |
2875 | (math-solve-get-sign (math-div de 2))) | |
2876 | (math-div a 4)))) | |
bf77c646 | 2877 | nil t)) |
136211a9 | 2878 | |
3132f345 CW |
2879 | (defvar math-symbolic-solve nil) |
2880 | (defvar math-int-coefs nil) | |
f7adda54 JB |
2881 | |
2882 | ;; The variable math-int-threshold is local to math-poly-all-roots, | |
2883 | ;; but is used by math-poly-newton-root. | |
2884 | (defvar math-int-threshold) | |
2885 | ;; The variables math-int-scale, math-int-factors and math-double-roots | |
2886 | ;; are local to math-poly-all-roots, but are used by math-poly-integer-root. | |
2887 | (defvar math-int-scale) | |
79d2746f JB |
2888 | (defvar math-int-factors) |
2889 | (defvar math-double-roots) | |
f7adda54 | 2890 | |
136211a9 EZ |
2891 | (defun math-poly-all-roots (var p &optional math-factoring) |
2892 | (catch 'ouch | |
2893 | (let* ((math-symbolic-solve calc-symbolic-mode) | |
2894 | (roots nil) | |
2895 | (deg (1- (length p))) | |
2896 | (orig-p (reverse p)) | |
2897 | (math-int-coefs nil) | |
2898 | (math-int-scale nil) | |
2899 | (math-double-roots nil) | |
2900 | (math-int-factors nil) | |
2901 | (math-int-threshold nil) | |
2902 | (pp p)) | |
2903 | ;; If rational coefficients, look for exact rational factors. | |
2904 | (while (and pp (Math-ratp (car pp))) | |
2905 | (setq pp (cdr pp))) | |
2906 | (if pp | |
2907 | (if (or math-factoring math-symbolic-solve) | |
2908 | (throw 'ouch nil)) | |
2909 | (let ((lead (car orig-p)) | |
2910 | (calc-prefer-frac t) | |
2911 | (scale (apply 'math-lcm-denoms p))) | |
2912 | (setq math-int-scale (math-abs (math-mul scale lead)) | |
2913 | math-int-threshold (math-div '(float 5 -2) math-int-scale) | |
2914 | math-int-coefs (cdr (math-div (cons 'vec orig-p) lead))))) | |
2915 | (if (> deg 4) | |
2916 | (let ((calc-prefer-frac nil) | |
2917 | (calc-symbolic-mode nil) | |
2918 | (pp p) | |
2919 | (def-p (copy-sequence orig-p))) | |
2920 | (while pp | |
2921 | (if (Math-numberp (car pp)) | |
2922 | (setq pp (cdr pp)) | |
2923 | (throw 'ouch nil))) | |
2924 | (while (> deg (if math-symbolic-solve 2 4)) | |
2925 | (let* ((x (math-poly-any-root def-p '(float 0 0) nil)) | |
2926 | b c pp) | |
2927 | (if (and (eq (car-safe x) 'cplx) | |
2928 | (math-nearly-zerop (nth 2 x) (nth 1 x))) | |
2929 | (setq x (calcFunc-re x))) | |
2930 | (or math-factoring | |
2931 | (setq roots (cons x roots))) | |
2932 | (or (math-numberp x) | |
2933 | (setq x (math-evaluate-expr x))) | |
2934 | (setq pp def-p | |
2935 | b (car def-p)) | |
2936 | (while (setq pp (cdr pp)) | |
2937 | (setq c (car pp)) | |
2938 | (setcar pp b) | |
2939 | (setq b (math-add (math-mul x b) c))) | |
2940 | (setq def-p (cdr def-p) | |
2941 | deg (1- deg)))) | |
2942 | (setq p (reverse def-p)))) | |
2943 | (if (> deg 1) | |
f7adda54 | 2944 | (let ((math-solve-var '(var DUMMY var-DUMMY)) |
136211a9 EZ |
2945 | (math-solve-sign nil) |
2946 | (math-solve-ranges nil) | |
f7adda54 | 2947 | (math-solve-full 'all)) |
136211a9 EZ |
2948 | (if (= (length p) (length math-int-coefs)) |
2949 | (setq p (reverse math-int-coefs))) | |
2950 | (setq roots (append (cdr (apply (cond ((= deg 2) | |
2951 | 'math-solve-quadratic) | |
2952 | ((= deg 3) | |
2953 | 'math-solve-cubic) | |
2954 | (t | |
2955 | 'math-solve-quartic)) | |
f7adda54 | 2956 | math-solve-var p)) |
136211a9 EZ |
2957 | roots))) |
2958 | (if (> deg 0) | |
2959 | (setq roots (cons (math-div (math-neg (car p)) (nth 1 p)) | |
2960 | roots)))) | |
2961 | (if math-factoring | |
2962 | (progn | |
2963 | (while roots | |
2964 | (math-poly-integer-root (car roots)) | |
2965 | (setq roots (cdr roots))) | |
2966 | (list math-int-factors (nreverse math-int-coefs) math-int-scale)) | |
2967 | (let ((vec nil) res) | |
2968 | (while roots | |
2969 | (let ((root (car roots)) | |
f7adda54 | 2970 | (math-solve-full (and math-solve-full 'all))) |
136211a9 EZ |
2971 | (if (math-floatp root) |
2972 | (setq root (math-poly-any-root orig-p root t))) | |
2973 | (setq vec (append vec | |
2974 | (cdr (or (math-try-solve-for var root nil t) | |
2975 | (throw 'ouch nil)))))) | |
2976 | (setq roots (cdr roots))) | |
2977 | (setq vec (cons 'vec (nreverse vec))) | |
2978 | (if math-symbolic-solve | |
2979 | (setq vec (math-normalize vec))) | |
f7adda54 | 2980 | (if (eq math-solve-full t) |
136211a9 EZ |
2981 | (list 'calcFunc-subscr |
2982 | vec | |
2983 | (math-solve-get-int 1 (1- (length orig-p)) 1)) | |
bf77c646 | 2984 | vec)))))) |
136211a9 EZ |
2985 | |
2986 | (defun math-lcm-denoms (&rest fracs) | |
2987 | (let ((den 1)) | |
2988 | (while fracs | |
2989 | (if (eq (car-safe (car fracs)) 'frac) | |
2990 | (setq den (calcFunc-lcm den (nth 2 (car fracs))))) | |
2991 | (setq fracs (cdr fracs))) | |
bf77c646 | 2992 | den)) |
136211a9 EZ |
2993 | |
2994 | (defun math-poly-any-root (p x polish) ; p is a reverse poly coeff list | |
2995 | (let* ((newt (if (math-zerop x) | |
2996 | (math-poly-newton-root | |
2997 | p '(cplx (float 123 -6) (float 1 -4)) 4) | |
2998 | (math-poly-newton-root p x 4))) | |
2999 | (res (if (math-zerop (cdr newt)) | |
3000 | (car newt) | |
3001 | (if (and (math-lessp (cdr newt) '(float 1 -3)) (not polish)) | |
3002 | (setq newt (math-poly-newton-root p (car newt) 30))) | |
3003 | (if (math-zerop (cdr newt)) | |
3004 | (car newt) | |
3005 | (math-poly-laguerre-root p x polish))))) | |
3006 | (and math-symbolic-solve (math-floatp res) | |
3007 | (throw 'ouch nil)) | |
bf77c646 | 3008 | res)) |
136211a9 EZ |
3009 | |
3010 | (defun math-poly-newton-root (p x iters) | |
3011 | (let* ((calc-prefer-frac nil) | |
3012 | (calc-symbolic-mode nil) | |
3013 | (try-integer math-int-coefs) | |
3014 | (dx x) b d) | |
3015 | (while (and (> (setq iters (1- iters)) 0) | |
3016 | (let ((pp p)) | |
3017 | (math-working "newton" x) | |
3018 | (setq b (car p) | |
3019 | d 0) | |
3020 | (while (setq pp (cdr pp)) | |
3021 | (setq d (math-add (math-mul x d) b) | |
3022 | b (math-add (math-mul x b) (car pp)))) | |
3023 | (not (math-zerop d))) | |
3024 | (progn | |
3025 | (setq dx (math-div b d) | |
3026 | x (math-sub x dx)) | |
3027 | (if try-integer | |
3028 | (let ((adx (math-abs-approx dx))) | |
3029 | (and (math-lessp adx math-int-threshold) | |
3030 | (let ((iroot (math-poly-integer-root x))) | |
3031 | (if iroot | |
3032 | (setq x iroot dx 0) | |
3033 | (setq try-integer nil)))))) | |
3034 | (or (not (or (eq dx 0) | |
3035 | (math-nearly-zerop dx (math-abs-approx x)))) | |
3036 | (progn (setq dx 0) nil))))) | |
3037 | (cons x (if (math-zerop x) | |
bf77c646 | 3038 | 1 (math-div (math-abs-approx dx) (math-abs-approx x)))))) |
136211a9 EZ |
3039 | |
3040 | (defun math-poly-integer-root (x) | |
3041 | (and (math-lessp (calcFunc-xpon (math-abs-approx x)) calc-internal-prec) | |
3042 | math-int-coefs | |
3043 | (let* ((calc-prefer-frac t) | |
3044 | (xre (calcFunc-re x)) | |
3045 | (xim (calcFunc-im x)) | |
3046 | (xresq (math-sqr xre)) | |
3047 | (ximsq (math-sqr xim))) | |
3048 | (if (math-lessp ximsq (calcFunc-scf xresq -1)) | |
3049 | ;; Look for linear factor | |
3050 | (let* ((rnd (math-div (math-round (math-mul xre math-int-scale)) | |
3051 | math-int-scale)) | |
3052 | (icp math-int-coefs) | |
3053 | (rem (car icp)) | |
3054 | (newcoef nil)) | |
3055 | (while (setq icp (cdr icp)) | |
3056 | (setq newcoef (cons rem newcoef) | |
3057 | rem (math-add (car icp) | |
3058 | (math-mul rem rnd)))) | |
3059 | (and (math-zerop rem) | |
3060 | (progn | |
3061 | (setq math-int-coefs (nreverse newcoef) | |
3062 | math-int-factors (cons (list (math-neg rnd)) | |
3063 | math-int-factors)) | |
3064 | rnd))) | |
3065 | ;; Look for irreducible quadratic factor | |
3066 | (let* ((rnd1 (math-div (math-round | |
3067 | (math-mul xre (math-mul -2 math-int-scale))) | |
3068 | math-int-scale)) | |
3069 | (sqscale (math-sqr math-int-scale)) | |
3070 | (rnd0 (math-div (math-round (math-mul (math-add xresq ximsq) | |
3071 | sqscale)) | |
3072 | sqscale)) | |
3073 | (rem1 (car math-int-coefs)) | |
3074 | (icp (cdr math-int-coefs)) | |
3075 | (rem0 (car icp)) | |
3076 | (newcoef nil) | |
3077 | (found (assoc (list rnd0 rnd1 (math-posp xim)) | |
3078 | math-double-roots)) | |
3079 | this) | |
3080 | (if found | |
3081 | (setq math-double-roots (delq found math-double-roots) | |
3082 | rem0 0 rem1 0) | |
3083 | (while (setq icp (cdr icp)) | |
3084 | (setq this rem1 | |
3085 | newcoef (cons rem1 newcoef) | |
3086 | rem1 (math-sub rem0 (math-mul this rnd1)) | |
3087 | rem0 (math-sub (car icp) (math-mul this rnd0))))) | |
3088 | (and (math-zerop rem0) | |
3089 | (math-zerop rem1) | |
3090 | (let ((aa (math-div rnd1 -2))) | |
3091 | (or found (setq math-int-coefs (reverse newcoef) | |
3092 | math-double-roots (cons (list | |
3093 | (list | |
3094 | rnd0 rnd1 | |
3095 | (math-negp xim))) | |
3096 | math-double-roots) | |
3097 | math-int-factors (cons (cons rnd0 rnd1) | |
3098 | math-int-factors))) | |
3099 | (math-add aa | |
3100 | (let ((calc-symbolic-mode math-symbolic-solve)) | |
3101 | (math-mul (math-sqrt (math-sub (math-sqr aa) | |
3102 | rnd0)) | |
bf77c646 | 3103 | (if (math-negp xim) -1 1))))))))))) |
136211a9 EZ |
3104 | |
3105 | ;;; The following routine is from Numerical Recipes, section 9.5. | |
3106 | (defun math-poly-laguerre-root (p x polish) | |
3107 | (let* ((calc-prefer-frac nil) | |
3108 | (calc-symbolic-mode nil) | |
3109 | (iters 0) | |
3110 | (m (1- (length p))) | |
3111 | (try-newt (not polish)) | |
3112 | (tried-newt nil) | |
3113 | b d f x1 dx dxold) | |
3114 | (while | |
3115 | (and (or (< (setq iters (1+ iters)) 50) | |
3116 | (math-reject-arg x "*Laguerre's method failed to converge")) | |
3117 | (let ((err (math-abs-approx (car p))) | |
3118 | (abx (math-abs-approx x)) | |
3119 | (pp p)) | |
3120 | (setq b (car p) | |
3121 | d 0 f 0) | |
3122 | (while (setq pp (cdr pp)) | |
3123 | (setq f (math-add (math-mul x f) d) | |
3124 | d (math-add (math-mul x d) b) | |
3125 | b (math-add (math-mul x b) (car pp)) | |
3126 | err (math-add (math-abs-approx b) (math-mul abx err)))) | |
3127 | (math-lessp (calcFunc-scf err (- -2 calc-internal-prec)) | |
3128 | (math-abs-approx b))) | |
3129 | (or (not (math-zerop d)) | |
3130 | (not (math-zerop f)) | |
3131 | (progn | |
3132 | (setq x (math-pow (math-neg b) (list 'frac 1 m))) | |
3133 | nil)) | |
3134 | (let* ((g (math-div d b)) | |
3135 | (g2 (math-sqr g)) | |
3136 | (h (math-sub g2 (math-mul 2 (math-div f b)))) | |
3137 | (sq (math-sqrt | |
3138 | (math-mul (1- m) (math-sub (math-mul m h) g2)))) | |
3139 | (gp (math-add g sq)) | |
3140 | (gm (math-sub g sq))) | |
3141 | (if (math-lessp (calcFunc-abssqr gp) (calcFunc-abssqr gm)) | |
3142 | (setq gp gm)) | |
3143 | (setq dx (math-div m gp) | |
3144 | x1 (math-sub x dx)) | |
3145 | (if (and try-newt | |
3146 | (math-lessp (math-abs-approx dx) | |
3147 | (calcFunc-scf (math-abs-approx x) -3))) | |
3148 | (let ((newt (math-poly-newton-root p x1 7))) | |
3149 | (setq tried-newt t | |
3150 | try-newt nil) | |
3151 | (if (math-zerop (cdr newt)) | |
3152 | (setq x (car newt) x1 x) | |
3153 | (if (math-lessp (cdr newt) '(float 1 -6)) | |
3154 | (let ((newt2 (math-poly-newton-root | |
3155 | p (car newt) 20))) | |
3156 | (if (math-zerop (cdr newt2)) | |
3157 | (setq x (car newt2) x1 x) | |
3158 | (setq x (car newt)))))))) | |
3159 | (not (or (eq x x1) | |
3160 | (math-nearly-equal x x1)))) | |
3161 | (let ((cdx (math-abs-approx dx))) | |
3162 | (setq x x1 | |
3163 | tried-newt nil) | |
3164 | (prog1 | |
3165 | (or (<= iters 6) | |
3166 | (math-lessp cdx dxold) | |
3167 | (progn | |
3168 | (if polish | |
3169 | (let ((digs (calcFunc-xpon | |
3170 | (math-div (math-abs-approx x) cdx)))) | |
3171 | (calc-record-why | |
3172 | "*Could not attain full precision") | |
3173 | (if (natnump digs) | |
3174 | (let ((calc-internal-prec (max 3 digs))) | |
3175 | (setq x (math-normalize x)))))) | |
3176 | nil)) | |
3177 | (setq dxold cdx))) | |
3178 | (or polish | |
3179 | (math-lessp (calcFunc-scf (math-abs-approx x) | |
3180 | (- calc-internal-prec)) | |
3181 | dxold)))) | |
3182 | (or (and (math-floatp x) | |
3183 | (math-poly-integer-root x)) | |
bf77c646 | 3184 | x))) |
136211a9 EZ |
3185 | |
3186 | (defun math-solve-above-dummy (x) | |
3187 | (and (not (Math-primp x)) | |
3188 | (if (and (equal (nth 1 x) '(var SOLVEDUM SOLVEDUM)) | |
3189 | (= (length x) 2)) | |
3190 | x | |
3191 | (let ((res nil)) | |
3192 | (while (and (setq x (cdr x)) | |
3193 | (not (setq res (math-solve-above-dummy (car x)))))) | |
bf77c646 | 3194 | res)))) |
136211a9 EZ |
3195 | |
3196 | (defun math-solve-find-root-term (x neg) ; sets "t2", "t3" | |
3197 | (if (math-solve-find-root-in-prod x) | |
f7adda54 JB |
3198 | (setq math-t3 neg |
3199 | math-t1 x) | |
136211a9 EZ |
3200 | (and (memq (car-safe x) '(+ -)) |
3201 | (or (math-solve-find-root-term (nth 1 x) neg) | |
3202 | (math-solve-find-root-term (nth 2 x) | |
bf77c646 | 3203 | (if (eq (car x) '-) (not neg) neg)))))) |
136211a9 EZ |
3204 | |
3205 | (defun math-solve-find-root-in-prod (x) | |
3206 | (and (consp x) | |
f7adda54 | 3207 | (math-expr-contains x math-solve-var) |
136211a9 | 3208 | (or (and (eq (car x) 'calcFunc-sqrt) |
f7adda54 | 3209 | (setq math-t2 2)) |
136211a9 EZ |
3210 | (and (eq (car x) '^) |
3211 | (or (and (memq (math-quarter-integer (nth 2 x)) '(1 2 3)) | |
f7adda54 | 3212 | (setq math-t2 2)) |
136211a9 EZ |
3213 | (and (eq (car-safe (nth 2 x)) 'frac) |
3214 | (eq (nth 2 (nth 2 x)) 3) | |
f7adda54 | 3215 | (setq math-t2 3)))) |
136211a9 | 3216 | (and (memq (car x) '(* /)) |
f7adda54 | 3217 | (or (and (not (math-expr-contains (nth 1 x) math-solve-var)) |
136211a9 | 3218 | (math-solve-find-root-in-prod (nth 2 x))) |
f7adda54 | 3219 | (and (not (math-expr-contains (nth 2 x) math-solve-var)) |
bf77c646 | 3220 | (math-solve-find-root-in-prod (nth 1 x)))))))) |
136211a9 | 3221 | |
f7adda54 JB |
3222 | ;; The variable math-solve-vars is local to math-solve-system, |
3223 | ;; but is used by math-solve-system-rec. | |
3224 | (defvar math-solve-vars) | |
136211a9 | 3225 | |
f7adda54 JB |
3226 | ;; The variable math-solve-simplifying is local to math-solve-system |
3227 | ;; and math-solve-system-rec, but is used by math-solve-system-subst. | |
79d2746f | 3228 | (defvar math-solve-simplifying) |
f7adda54 JB |
3229 | |
3230 | (defun math-solve-system (exprs math-solve-vars math-solve-full) | |
136211a9 EZ |
3231 | (setq exprs (mapcar 'list (if (Math-vectorp exprs) |
3232 | (cdr exprs) | |
3233 | (list exprs))) | |
f7adda54 JB |
3234 | math-solve-vars (if (Math-vectorp math-solve-vars) |
3235 | (cdr math-solve-vars) | |
3236 | (list math-solve-vars))) | |
136211a9 | 3237 | (or (let ((math-solve-simplifying nil)) |
f7adda54 | 3238 | (math-solve-system-rec exprs math-solve-vars nil)) |
136211a9 | 3239 | (let ((math-solve-simplifying t)) |
f7adda54 | 3240 | (math-solve-system-rec exprs math-solve-vars nil)))) |
136211a9 EZ |
3241 | |
3242 | ;;; The following backtracking solver works by choosing a variable | |
3243 | ;;; and equation, and trying to solve the equation for the variable. | |
3244 | ;;; If it succeeds it calls itself recursively with that variable and | |
3245 | ;;; equation removed from their respective lists, and with the solution | |
3246 | ;;; added to solns as well as being substituted into all existing | |
3247 | ;;; equations. The algorithm terminates when any solution path | |
3248 | ;;; manages to remove all the variables from var-list. | |
3249 | ||
3250 | ;;; To support calcFunc-roots, entries in eqn-list and solns are | |
3251 | ;;; actually lists of equations. | |
3252 | ||
f7adda54 JB |
3253 | ;; The variables math-solve-system-res and math-solve-system-vv are |
3254 | ;; local to math-solve-system-rec, but are used by math-solve-system-subst. | |
3255 | (defvar math-solve-system-vv) | |
3256 | (defvar math-solve-system-res) | |
3257 | ||
3258 | ||
136211a9 EZ |
3259 | (defun math-solve-system-rec (eqn-list var-list solns) |
3260 | (if var-list | |
3261 | (let ((v var-list) | |
f7adda54 | 3262 | (math-solve-system-res nil)) |
136211a9 EZ |
3263 | |
3264 | ;; Try each variable in turn. | |
3265 | (while | |
3266 | (and | |
3267 | v | |
f7adda54 | 3268 | (let* ((math-solve-system-vv (car v)) |
136211a9 | 3269 | (e eqn-list) |
f7adda54 | 3270 | (elim (eq (car-safe math-solve-system-vv) 'calcFunc-elim))) |
136211a9 | 3271 | (if elim |
f7adda54 | 3272 | (setq math-solve-system-vv (nth 1 math-solve-system-vv))) |
136211a9 EZ |
3273 | |
3274 | ;; Try each equation in turn. | |
3275 | (while | |
3276 | (and | |
3277 | e | |
3278 | (let ((e2 (car e)) | |
3279 | (eprev nil) | |
3280 | res2) | |
f7adda54 | 3281 | (setq math-solve-system-res nil) |
136211a9 | 3282 | |
f7adda54 | 3283 | ;; Try to solve for math-solve-system-vv the list of equations e2. |
136211a9 EZ |
3284 | (while (and e2 |
3285 | (setq res2 (or (and (eq (car e2) eprev) | |
3286 | res2) | |
f7adda54 JB |
3287 | (math-solve-for (car e2) 0 |
3288 | math-solve-system-vv | |
3289 | math-solve-full)))) | |
136211a9 | 3290 | (setq eprev (car e2) |
f7adda54 | 3291 | math-solve-system-res (cons (if (eq math-solve-full 'all) |
136211a9 EZ |
3292 | (cdr res2) |
3293 | (list res2)) | |
f7adda54 | 3294 | math-solve-system-res) |
136211a9 EZ |
3295 | e2 (cdr e2))) |
3296 | (if e2 | |
f7adda54 | 3297 | (setq math-solve-system-res nil) |
136211a9 EZ |
3298 | |
3299 | ;; Found a solution. Now try other variables. | |
f7adda54 JB |
3300 | (setq math-solve-system-res (nreverse math-solve-system-res) |
3301 | math-solve-system-res (math-solve-system-rec | |
136211a9 EZ |
3302 | (mapcar |
3303 | 'math-solve-system-subst | |
3304 | (delq (car e) | |
3305 | (copy-sequence eqn-list))) | |
3306 | (delq (car v) (copy-sequence var-list)) | |
3307 | (let ((math-solve-simplifying nil) | |
3308 | (s (mapcar | |
3309 | (function | |
3310 | (lambda (x) | |
3311 | (cons | |
3312 | (car x) | |
3313 | (math-solve-system-subst | |
3314 | (cdr x))))) | |
3315 | solns))) | |
3316 | (if elim | |
3317 | s | |
f7adda54 JB |
3318 | (cons (cons |
3319 | math-solve-system-vv | |
3320 | (apply 'append math-solve-system-res)) | |
136211a9 | 3321 | s))))) |
f7adda54 | 3322 | (not math-solve-system-res)))) |
136211a9 | 3323 | (setq e (cdr e))) |
f7adda54 | 3324 | (not math-solve-system-res))) |
136211a9 | 3325 | (setq v (cdr v))) |
f7adda54 | 3326 | math-solve-system-res) |
136211a9 EZ |
3327 | |
3328 | ;; Eliminated all variables, so now put solution into the proper format. | |
3329 | (setq solns (sort solns | |
3330 | (function | |
3331 | (lambda (x y) | |
f7adda54 JB |
3332 | (not (memq (car x) (memq (car y) math-solve-vars))))))) |
3333 | (if (eq math-solve-full 'all) | |
136211a9 EZ |
3334 | (math-transpose |
3335 | (math-normalize | |
3336 | (cons 'vec | |
3337 | (if solns | |
3338 | (mapcar (function (lambda (x) (cons 'vec (cdr x)))) solns) | |
3339 | (mapcar (function (lambda (x) (cons 'vec x))) eqn-list))))) | |
3340 | (math-normalize | |
a1506d29 | 3341 | (cons 'vec |
136211a9 EZ |
3342 | (if solns |
3343 | (mapcar (function (lambda (x) (cons 'calcFunc-eq x))) solns) | |
bf77c646 | 3344 | (mapcar 'car eqn-list))))))) |
136211a9 EZ |
3345 | |
3346 | (defun math-solve-system-subst (x) ; uses "res" and "v" | |
3347 | (let ((accum nil) | |
f7adda54 | 3348 | (res2 math-solve-system-res)) |
136211a9 EZ |
3349 | (while x |
3350 | (setq accum (nconc accum | |
3351 | (mapcar (function | |
3352 | (lambda (r) | |
3353 | (if math-solve-simplifying | |
3354 | (math-simplify | |
f7adda54 JB |
3355 | (math-expr-subst |
3356 | (car x) math-solve-system-vv r)) | |
3357 | (math-expr-subst | |
3358 | (car x) math-solve-system-vv r)))) | |
136211a9 EZ |
3359 | (car res2))) |
3360 | x (cdr x) | |
3361 | res2 (cdr res2))) | |
bf77c646 | 3362 | accum)) |
136211a9 EZ |
3363 | |
3364 | ||
f7adda54 JB |
3365 | ;; calc-command-flags is declared in calc.el |
3366 | (defvar calc-command-flags) | |
3367 | ||
136211a9 EZ |
3368 | (defun math-get-from-counter (name) |
3369 | (let ((ctr (assq name calc-command-flags))) | |
3370 | (if ctr | |
3371 | (setcdr ctr (1+ (cdr ctr))) | |
3372 | (setq ctr (cons name 1) | |
3373 | calc-command-flags (cons ctr calc-command-flags))) | |
bf77c646 | 3374 | (cdr ctr))) |
136211a9 | 3375 | |
f7adda54 JB |
3376 | (defvar var-GenCount) |
3377 | ||
136211a9 EZ |
3378 | (defun math-solve-get-sign (val) |
3379 | (setq val (math-simplify val)) | |
3380 | (if (and (eq (car-safe val) '*) | |
3381 | (Math-numberp (nth 1 val))) | |
3382 | (list '* (nth 1 val) (math-solve-get-sign (nth 2 val))) | |
3383 | (and (eq (car-safe val) 'calcFunc-sqrt) | |
3384 | (eq (car-safe (nth 1 val)) '^) | |
3385 | (setq val (math-normalize (list '^ | |
3386 | (nth 1 (nth 1 val)) | |
3387 | (math-div (nth 2 (nth 1 val)) 2))))) | |
f7adda54 | 3388 | (if math-solve-full |
136211a9 EZ |
3389 | (if (and (calc-var-value 'var-GenCount) |
3390 | (Math-natnump var-GenCount) | |
f7adda54 | 3391 | (not (eq math-solve-full 'all))) |
136211a9 EZ |
3392 | (prog1 |
3393 | (math-mul (list 'calcFunc-as var-GenCount) val) | |
3394 | (setq var-GenCount (math-add var-GenCount 1)) | |
3395 | (calc-refresh-evaltos 'var-GenCount)) | |
a9c76ab2 | 3396 | (let* ((var (concat "s" (int-to-string (math-get-from-counter 'solve-sign)))) |
136211a9 | 3397 | (var2 (list 'var (intern var) (intern (concat "var-" var))))) |
f7adda54 | 3398 | (if (eq math-solve-full 'all) |
136211a9 EZ |
3399 | (setq math-solve-ranges (cons (list var2 1 -1) |
3400 | math-solve-ranges))) | |
3401 | (math-mul var2 val))) | |
3402 | (calc-record-why "*Choosing positive solution") | |
bf77c646 | 3403 | val))) |
136211a9 EZ |
3404 | |
3405 | (defun math-solve-get-int (val &optional range first) | |
f7adda54 | 3406 | (if math-solve-full |
136211a9 EZ |
3407 | (if (and (calc-var-value 'var-GenCount) |
3408 | (Math-natnump var-GenCount) | |
f7adda54 | 3409 | (not (eq math-solve-full 'all))) |
136211a9 EZ |
3410 | (prog1 |
3411 | (math-mul val (list 'calcFunc-an var-GenCount)) | |
3412 | (setq var-GenCount (math-add var-GenCount 1)) | |
3413 | (calc-refresh-evaltos 'var-GenCount)) | |
88258edf CW |
3414 | (let* ((var (concat "n" (int-to-string |
3415 | (math-get-from-counter 'solve-int)))) | |
136211a9 | 3416 | (var2 (list 'var (intern var) (intern (concat "var-" var))))) |
f7adda54 | 3417 | (if (and range (eq math-solve-full 'all)) |
136211a9 EZ |
3418 | (setq math-solve-ranges (cons (cons var2 |
3419 | (cdr (calcFunc-index | |
3420 | range (or first 0)))) | |
3421 | math-solve-ranges))) | |
3422 | (math-mul val var2))) | |
3423 | (calc-record-why "*Choosing 0 for arbitrary integer in solution") | |
bf77c646 | 3424 | 0)) |
136211a9 EZ |
3425 | |
3426 | (defun math-solve-sign (sign expr) | |
3427 | (and sign | |
3428 | (let ((s1 (math-possible-signs expr))) | |
3429 | (cond ((memq s1 '(4 6)) | |
3430 | sign) | |
3431 | ((memq s1 '(1 3)) | |
bf77c646 | 3432 | (- sign)))))) |
136211a9 EZ |
3433 | |
3434 | (defun math-looks-evenp (expr) | |
3435 | (if (Math-integerp expr) | |
3436 | (math-evenp expr) | |
3437 | (if (memq (car expr) '(* /)) | |
bf77c646 | 3438 | (math-looks-evenp (nth 1 expr))))) |
136211a9 | 3439 | |
f7adda54 JB |
3440 | (defun math-solve-for (lhs rhs math-solve-var math-solve-full &optional sign) |
3441 | (if (math-expr-contains rhs math-solve-var) | |
3442 | (math-solve-for (math-sub lhs rhs) 0 math-solve-var math-solve-full) | |
3443 | (and (math-expr-contains lhs math-solve-var) | |
136211a9 | 3444 | (math-with-extra-prec 1 |
f7adda54 | 3445 | (let* ((math-poly-base-variable math-solve-var) |
136211a9 | 3446 | (res (math-try-solve-for lhs rhs sign))) |
f7adda54 JB |
3447 | (if (and (eq math-solve-full 'all) |
3448 | (math-known-realp math-solve-var)) | |
136211a9 EZ |
3449 | (let ((old-len (length res)) |
3450 | new-len) | |
3451 | (setq res (delq nil | |
3452 | (mapcar (function | |
3453 | (lambda (x) | |
3454 | (and (not (memq (car-safe x) | |
3455 | '(cplx polar))) | |
3456 | x))) | |
3457 | res)) | |
3458 | new-len (length res)) | |
3459 | (if (< new-len old-len) | |
3460 | (calc-record-why (if (= new-len 1) | |
3461 | "*All solutions were complex" | |
3462 | (format | |
3463 | "*Omitted %d complex solutions" | |
3464 | (- old-len new-len))))))) | |
bf77c646 | 3465 | res))))) |
136211a9 EZ |
3466 | |
3467 | (defun math-solve-eqn (expr var full) | |
3468 | (if (memq (car-safe expr) '(calcFunc-neq calcFunc-lt calcFunc-gt | |
3469 | calcFunc-leq calcFunc-geq)) | |
3470 | (let ((res (math-solve-for (cons '- (cdr expr)) | |
3471 | 0 var full | |
3472 | (if (eq (car expr) 'calcFunc-neq) nil 1)))) | |
3473 | (and res | |
3474 | (if (eq math-solve-sign 1) | |
3475 | (list (car expr) var res) | |
3476 | (if (eq math-solve-sign -1) | |
3477 | (list (car expr) res var) | |
3478 | (or (eq (car expr) 'calcFunc-neq) | |
3479 | (calc-record-why | |
3480 | "*Can't determine direction of inequality")) | |
3481 | (and (memq (car expr) '(calcFunc-neq calcFunc-lt calcFunc-gt)) | |
3482 | (list 'calcFunc-neq var res)))))) | |
3483 | (let ((res (math-solve-for expr 0 var full))) | |
3484 | (and res | |
bf77c646 | 3485 | (list 'calcFunc-eq var res))))) |
136211a9 EZ |
3486 | |
3487 | (defun math-reject-solution (expr var func) | |
3488 | (if (math-expr-contains expr var) | |
3489 | (or (equal (car calc-next-why) '(* "Unable to find a symbolic solution")) | |
3490 | (calc-record-why "*Unable to find a solution"))) | |
bf77c646 | 3491 | (list func expr var)) |
136211a9 EZ |
3492 | |
3493 | (defun calcFunc-solve (expr var) | |
3494 | (or (if (or (Math-vectorp expr) (Math-vectorp var)) | |
3495 | (math-solve-system expr var nil) | |
3496 | (math-solve-eqn expr var nil)) | |
bf77c646 | 3497 | (math-reject-solution expr var 'calcFunc-solve))) |
136211a9 EZ |
3498 | |
3499 | (defun calcFunc-fsolve (expr var) | |
3500 | (or (if (or (Math-vectorp expr) (Math-vectorp var)) | |
3501 | (math-solve-system expr var t) | |
3502 | (math-solve-eqn expr var t)) | |
bf77c646 | 3503 | (math-reject-solution expr var 'calcFunc-fsolve))) |
136211a9 EZ |
3504 | |
3505 | (defun calcFunc-roots (expr var) | |
3506 | (let ((math-solve-ranges nil)) | |
3507 | (or (if (or (Math-vectorp expr) (Math-vectorp var)) | |
3508 | (math-solve-system expr var 'all) | |
3509 | (math-solve-for expr 0 var 'all)) | |
bf77c646 | 3510 | (math-reject-solution expr var 'calcFunc-roots)))) |
136211a9 EZ |
3511 | |
3512 | (defun calcFunc-finv (expr var) | |
3513 | (let ((res (math-solve-for expr math-integ-var var nil))) | |
3514 | (if res | |
3515 | (math-normalize (math-expr-subst res math-integ-var var)) | |
bf77c646 | 3516 | (math-reject-solution expr var 'calcFunc-finv)))) |
136211a9 EZ |
3517 | |
3518 | (defun calcFunc-ffinv (expr var) | |
3519 | (let ((res (math-solve-for expr math-integ-var var t))) | |
3520 | (if res | |
3521 | (math-normalize (math-expr-subst res math-integ-var var)) | |
bf77c646 | 3522 | (math-reject-solution expr var 'calcFunc-finv)))) |
136211a9 EZ |
3523 | |
3524 | ||
3525 | (put 'calcFunc-inv 'math-inverse | |
3526 | (function (lambda (x) (math-div 1 x)))) | |
3527 | (put 'calcFunc-inv 'math-inverse-sign -1) | |
3528 | ||
3529 | (put 'calcFunc-sqrt 'math-inverse | |
3530 | (function (lambda (x) (math-sqr x)))) | |
3531 | ||
3532 | (put 'calcFunc-conj 'math-inverse | |
3533 | (function (lambda (x) (list 'calcFunc-conj x)))) | |
3534 | ||
3535 | (put 'calcFunc-abs 'math-inverse | |
3536 | (function (lambda (x) (math-solve-get-sign x)))) | |
3537 | ||
3538 | (put 'calcFunc-deg 'math-inverse | |
3539 | (function (lambda (x) (list 'calcFunc-rad x)))) | |
3540 | (put 'calcFunc-deg 'math-inverse-sign 1) | |
3541 | ||
3542 | (put 'calcFunc-rad 'math-inverse | |
3543 | (function (lambda (x) (list 'calcFunc-deg x)))) | |
3544 | (put 'calcFunc-rad 'math-inverse-sign 1) | |
3545 | ||
3546 | (put 'calcFunc-ln 'math-inverse | |
3547 | (function (lambda (x) (list 'calcFunc-exp x)))) | |
3548 | (put 'calcFunc-ln 'math-inverse-sign 1) | |
3549 | ||
3550 | (put 'calcFunc-log10 'math-inverse | |
3551 | (function (lambda (x) (list 'calcFunc-exp10 x)))) | |
3552 | (put 'calcFunc-log10 'math-inverse-sign 1) | |
3553 | ||
3554 | (put 'calcFunc-lnp1 'math-inverse | |
3555 | (function (lambda (x) (list 'calcFunc-expm1 x)))) | |
3556 | (put 'calcFunc-lnp1 'math-inverse-sign 1) | |
3557 | ||
3558 | (put 'calcFunc-exp 'math-inverse | |
3559 | (function (lambda (x) (math-add (math-normalize (list 'calcFunc-ln x)) | |
3560 | (math-mul 2 | |
3561 | (math-mul '(var pi var-pi) | |
3562 | (math-solve-get-int | |
3563 | '(var i var-i)))))))) | |
3564 | (put 'calcFunc-exp 'math-inverse-sign 1) | |
3565 | ||
3566 | (put 'calcFunc-expm1 'math-inverse | |
3567 | (function (lambda (x) (math-add (math-normalize (list 'calcFunc-lnp1 x)) | |
3568 | (math-mul 2 | |
3569 | (math-mul '(var pi var-pi) | |
3570 | (math-solve-get-int | |
3571 | '(var i var-i)))))))) | |
3572 | (put 'calcFunc-expm1 'math-inverse-sign 1) | |
3573 | ||
3574 | (put 'calcFunc-sin 'math-inverse | |
3575 | (function (lambda (x) (let ((n (math-solve-get-int 1))) | |
3576 | (math-add (math-mul (math-normalize | |
3577 | (list 'calcFunc-arcsin x)) | |
3578 | (math-pow -1 n)) | |
3579 | (math-mul (math-half-circle t) | |
3580 | n)))))) | |
3581 | ||
3582 | (put 'calcFunc-cos 'math-inverse | |
3583 | (function (lambda (x) (math-add (math-solve-get-sign | |
3584 | (math-normalize | |
3585 | (list 'calcFunc-arccos x))) | |
3586 | (math-solve-get-int | |
3587 | (math-full-circle t)))))) | |
3588 | ||
3589 | (put 'calcFunc-tan 'math-inverse | |
3590 | (function (lambda (x) (math-add (math-normalize (list 'calcFunc-arctan x)) | |
3591 | (math-solve-get-int | |
3592 | (math-half-circle t)))))) | |
3593 | ||
3594 | (put 'calcFunc-arcsin 'math-inverse | |
3595 | (function (lambda (x) (math-normalize (list 'calcFunc-sin x))))) | |
3596 | ||
3597 | (put 'calcFunc-arccos 'math-inverse | |
3598 | (function (lambda (x) (math-normalize (list 'calcFunc-cos x))))) | |
3599 | ||
3600 | (put 'calcFunc-arctan 'math-inverse | |
3601 | (function (lambda (x) (math-normalize (list 'calcFunc-tan x))))) | |
3602 | ||
3603 | (put 'calcFunc-sinh 'math-inverse | |
3604 | (function (lambda (x) (let ((n (math-solve-get-int 1))) | |
3605 | (math-add (math-mul (math-normalize | |
3606 | (list 'calcFunc-arcsinh x)) | |
3607 | (math-pow -1 n)) | |
3608 | (math-mul (math-half-circle t) | |
3609 | (math-mul | |
3610 | '(var i var-i) | |
3611 | n))))))) | |
3612 | (put 'calcFunc-sinh 'math-inverse-sign 1) | |
3613 | ||
3614 | (put 'calcFunc-cosh 'math-inverse | |
3615 | (function (lambda (x) (math-add (math-solve-get-sign | |
3616 | (math-normalize | |
3617 | (list 'calcFunc-arccosh x))) | |
3618 | (math-mul (math-full-circle t) | |
3619 | (math-solve-get-int | |
3620 | '(var i var-i))))))) | |
3621 | ||
3622 | (put 'calcFunc-tanh 'math-inverse | |
3623 | (function (lambda (x) (math-add (math-normalize | |
3624 | (list 'calcFunc-arctanh x)) | |
3625 | (math-mul (math-half-circle t) | |
3626 | (math-solve-get-int | |
3627 | '(var i var-i))))))) | |
3628 | (put 'calcFunc-tanh 'math-inverse-sign 1) | |
3629 | ||
3630 | (put 'calcFunc-arcsinh 'math-inverse | |
3631 | (function (lambda (x) (math-normalize (list 'calcFunc-sinh x))))) | |
3632 | (put 'calcFunc-arcsinh 'math-inverse-sign 1) | |
3633 | ||
3634 | (put 'calcFunc-arccosh 'math-inverse | |
3635 | (function (lambda (x) (math-normalize (list 'calcFunc-cosh x))))) | |
3636 | ||
3637 | (put 'calcFunc-arctanh 'math-inverse | |
3638 | (function (lambda (x) (math-normalize (list 'calcFunc-tanh x))))) | |
3639 | (put 'calcFunc-arctanh 'math-inverse-sign 1) | |
3640 | ||
3641 | ||
3642 | ||
3643 | (defun calcFunc-taylor (expr var num) | |
3644 | (let ((x0 0) (v var)) | |
3645 | (if (memq (car-safe var) '(+ - calcFunc-eq)) | |
3646 | (setq x0 (if (eq (car var) '+) (math-neg (nth 2 var)) (nth 2 var)) | |
3647 | v (nth 1 var))) | |
3648 | (or (and (eq (car-safe v) 'var) | |
3649 | (math-expr-contains expr v) | |
3650 | (natnump num) | |
3651 | (let ((accum (math-expr-subst expr v x0)) | |
3652 | (var2 (if (eq (car var) 'calcFunc-eq) | |
3653 | (cons '- (cdr var)) | |
3654 | var)) | |
3655 | (n 0) | |
3656 | (nfac 1) | |
3657 | (fprime expr)) | |
3658 | (while (and (<= (setq n (1+ n)) num) | |
3659 | (setq fprime (calcFunc-deriv fprime v nil t))) | |
3660 | (setq fprime (math-simplify fprime) | |
3661 | nfac (math-mul nfac n) | |
3662 | accum (math-add accum | |
3663 | (math-div (math-mul (math-pow var2 n) | |
3664 | (math-expr-subst | |
3665 | fprime v x0)) | |
3666 | nfac)))) | |
3667 | (and fprime | |
3668 | (math-normalize accum)))) | |
bf77c646 | 3669 | (list 'calcFunc-taylor expr var num)))) |
136211a9 | 3670 | |
59f2c6c0 JB |
3671 | (provide 'calcalg2) |
3672 | ||
cbee283d | 3673 | ;; arch-tag: f2932ec8-dd63-418b-a542-11a644b9d4c4 |
bf77c646 | 3674 | ;;; calcalg2.el ends here |