Commit | Line | Data |
---|---|---|
3132f345 CW |
1 | ;;; calcalg2.el --- more algebraic functions for Calc |
2 | ||
bf77c646 | 3 | ;; Copyright (C) 1990, 1991, 1992, 1993, 2001 Free Software Foundation, Inc. |
3132f345 CW |
4 | |
5 | ;; Author: David Gillespie <daveg@synaptics.com> | |
a1506d29 | 6 | ;; Maintainers: D. Goel <deego@gnufans.org> |
6e1c888a | 7 | ;; Colin Walters <walters@debian.org> |
136211a9 EZ |
8 | |
9 | ;; This file is part of GNU Emacs. | |
10 | ||
11 | ;; GNU Emacs is distributed in the hope that it will be useful, | |
12 | ;; but WITHOUT ANY WARRANTY. No author or distributor | |
13 | ;; accepts responsibility to anyone for the consequences of using it | |
14 | ;; or for whether it serves any particular purpose or works at all, | |
15 | ;; unless he says so in writing. Refer to the GNU Emacs General Public | |
16 | ;; License for full details. | |
17 | ||
18 | ;; Everyone is granted permission to copy, modify and redistribute | |
19 | ;; GNU Emacs, but only under the conditions described in the | |
20 | ;; GNU Emacs General Public License. A copy of this license is | |
21 | ;; supposed to have been given to you along with GNU Emacs so you | |
22 | ;; can know your rights and responsibilities. It should be in a | |
23 | ;; file named COPYING. Among other things, the copyright notice | |
24 | ;; and this notice must be preserved on all copies. | |
25 | ||
3132f345 | 26 | ;;; Commentary: |
136211a9 | 27 | |
3132f345 | 28 | ;;; Code: |
136211a9 EZ |
29 | |
30 | ;; This file is autoloaded from calc-ext.el. | |
31 | (require 'calc-ext) | |
32 | ||
33 | (require 'calc-macs) | |
34 | ||
35 | (defun calc-Need-calc-alg-2 () nil) | |
36 | ||
37 | ||
38 | (defun calc-derivative (var num) | |
39 | (interactive "sDifferentiate with respect to: \np") | |
40 | (calc-slow-wrapper | |
3132f345 CW |
41 | (when (< num 0) |
42 | (error "Order of derivative must be positive")) | |
136211a9 EZ |
43 | (let ((func (if (calc-is-hyperbolic) 'calcFunc-tderiv 'calcFunc-deriv)) |
44 | n expr) | |
45 | (if (or (equal var "") (equal var "$")) | |
46 | (setq n 2 | |
47 | expr (calc-top-n 2) | |
48 | var (calc-top-n 1)) | |
49 | (setq var (math-read-expr var)) | |
3132f345 CW |
50 | (when (eq (car-safe var) 'error) |
51 | (error "Bad format in expression: %s" (nth 1 var))) | |
136211a9 EZ |
52 | (setq n 1 |
53 | expr (calc-top-n 1))) | |
54 | (while (>= (setq num (1- num)) 0) | |
55 | (setq expr (list func expr var))) | |
bf77c646 | 56 | (calc-enter-result n "derv" expr)))) |
136211a9 EZ |
57 | |
58 | (defun calc-integral (var) | |
59 | (interactive "sIntegration variable: ") | |
60 | (calc-slow-wrapper | |
61 | (if (or (equal var "") (equal var "$")) | |
62 | (calc-enter-result 2 "intg" (list 'calcFunc-integ | |
63 | (calc-top-n 2) | |
64 | (calc-top-n 1))) | |
65 | (let ((var (math-read-expr var))) | |
66 | (if (eq (car-safe var) 'error) | |
67 | (error "Bad format in expression: %s" (nth 1 var))) | |
68 | (calc-enter-result 1 "intg" (list 'calcFunc-integ | |
69 | (calc-top-n 1) | |
bf77c646 | 70 | var)))))) |
136211a9 EZ |
71 | |
72 | (defun calc-num-integral (&optional varname lowname highname) | |
73 | (interactive "sIntegration variable: ") | |
74 | (calc-tabular-command 'calcFunc-ninteg "Integration" "nint" | |
bf77c646 | 75 | nil varname lowname highname)) |
136211a9 EZ |
76 | |
77 | (defun calc-summation (arg &optional varname lowname highname) | |
78 | (interactive "P\nsSummation variable: ") | |
79 | (calc-tabular-command 'calcFunc-sum "Summation" "sum" | |
bf77c646 | 80 | arg varname lowname highname)) |
136211a9 EZ |
81 | |
82 | (defun calc-alt-summation (arg &optional varname lowname highname) | |
83 | (interactive "P\nsSummation variable: ") | |
84 | (calc-tabular-command 'calcFunc-asum "Summation" "asum" | |
bf77c646 | 85 | arg varname lowname highname)) |
136211a9 EZ |
86 | |
87 | (defun calc-product (arg &optional varname lowname highname) | |
88 | (interactive "P\nsIndex variable: ") | |
89 | (calc-tabular-command 'calcFunc-prod "Index" "prod" | |
bf77c646 | 90 | arg varname lowname highname)) |
136211a9 EZ |
91 | |
92 | (defun calc-tabulate (arg &optional varname lowname highname) | |
93 | (interactive "P\nsIndex variable: ") | |
94 | (calc-tabular-command 'calcFunc-table "Index" "tabl" | |
bf77c646 | 95 | arg varname lowname highname)) |
136211a9 EZ |
96 | |
97 | (defun calc-tabular-command (func prompt prefix arg varname lowname highname) | |
98 | (calc-slow-wrapper | |
99 | (let (var (low nil) (high nil) (step nil) stepname stepnum (num 1) expr) | |
100 | (if (consp arg) | |
101 | (setq stepnum 1) | |
102 | (setq stepnum 0)) | |
103 | (if (or (equal varname "") (equal varname "$") (null varname)) | |
104 | (setq high (calc-top-n (+ stepnum 1)) | |
105 | low (calc-top-n (+ stepnum 2)) | |
106 | var (calc-top-n (+ stepnum 3)) | |
107 | num (+ stepnum 4)) | |
108 | (setq var (if (stringp varname) (math-read-expr varname) varname)) | |
109 | (if (eq (car-safe var) 'error) | |
110 | (error "Bad format in expression: %s" (nth 1 var))) | |
111 | (or lowname | |
112 | (setq lowname (read-string (concat prompt " variable: " varname | |
113 | ", from: ")))) | |
114 | (if (or (equal lowname "") (equal lowname "$")) | |
115 | (setq high (calc-top-n (+ stepnum 1)) | |
116 | low (calc-top-n (+ stepnum 2)) | |
117 | num (+ stepnum 3)) | |
118 | (setq low (if (stringp lowname) (math-read-expr lowname) lowname)) | |
119 | (if (eq (car-safe low) 'error) | |
120 | (error "Bad format in expression: %s" (nth 1 low))) | |
121 | (or highname | |
122 | (setq highname (read-string (concat prompt " variable: " varname | |
123 | ", from: " lowname | |
124 | ", to: ")))) | |
125 | (if (or (equal highname "") (equal highname "$")) | |
126 | (setq high (calc-top-n (+ stepnum 1)) | |
127 | num (+ stepnum 2)) | |
128 | (setq high (if (stringp highname) (math-read-expr highname) | |
129 | highname)) | |
130 | (if (eq (car-safe high) 'error) | |
131 | (error "Bad format in expression: %s" (nth 1 high))) | |
132 | (if (consp arg) | |
133 | (progn | |
134 | (setq stepname (read-string (concat prompt " variable: " | |
135 | varname | |
136 | ", from: " lowname | |
137 | ", to: " highname | |
138 | ", step: "))) | |
139 | (if (or (equal stepname "") (equal stepname "$")) | |
140 | (setq step (calc-top-n 1) | |
141 | num 2) | |
142 | (setq step (math-read-expr stepname)) | |
143 | (if (eq (car-safe step) 'error) | |
144 | (error "Bad format in expression: %s" | |
145 | (nth 1 step))))))))) | |
146 | (or step | |
147 | (if (consp arg) | |
148 | (setq step (calc-top-n 1)) | |
149 | (if arg | |
150 | (setq step (prefix-numeric-value arg))))) | |
151 | (setq expr (calc-top-n num)) | |
152 | (calc-enter-result num prefix (append (list func expr var low high) | |
bf77c646 | 153 | (and step (list step))))))) |
136211a9 EZ |
154 | |
155 | (defun calc-solve-for (var) | |
156 | (interactive "sVariable to solve for: ") | |
157 | (calc-slow-wrapper | |
158 | (let ((func (if (calc-is-inverse) | |
159 | (if (calc-is-hyperbolic) 'calcFunc-ffinv 'calcFunc-finv) | |
160 | (if (calc-is-hyperbolic) 'calcFunc-fsolve 'calcFunc-solve)))) | |
161 | (if (or (equal var "") (equal var "$")) | |
162 | (calc-enter-result 2 "solv" (list func | |
163 | (calc-top-n 2) | |
164 | (calc-top-n 1))) | |
165 | (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var) | |
166 | (not (string-match "\\[" var))) | |
167 | (math-read-expr (concat "[" var "]")) | |
168 | (math-read-expr var)))) | |
169 | (if (eq (car-safe var) 'error) | |
170 | (error "Bad format in expression: %s" (nth 1 var))) | |
171 | (calc-enter-result 1 "solv" (list func | |
172 | (calc-top-n 1) | |
bf77c646 | 173 | var))))))) |
136211a9 EZ |
174 | |
175 | (defun calc-poly-roots (var) | |
176 | (interactive "sVariable to solve for: ") | |
177 | (calc-slow-wrapper | |
178 | (if (or (equal var "") (equal var "$")) | |
179 | (calc-enter-result 2 "prts" (list 'calcFunc-roots | |
180 | (calc-top-n 2) | |
181 | (calc-top-n 1))) | |
182 | (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var) | |
183 | (not (string-match "\\[" var))) | |
184 | (math-read-expr (concat "[" var "]")) | |
185 | (math-read-expr var)))) | |
186 | (if (eq (car-safe var) 'error) | |
187 | (error "Bad format in expression: %s" (nth 1 var))) | |
188 | (calc-enter-result 1 "prts" (list 'calcFunc-roots | |
189 | (calc-top-n 1) | |
bf77c646 | 190 | var)))))) |
136211a9 EZ |
191 | |
192 | (defun calc-taylor (var nterms) | |
193 | (interactive "sTaylor expansion variable: \nNNumber of terms: ") | |
194 | (calc-slow-wrapper | |
195 | (let ((var (math-read-expr var))) | |
196 | (if (eq (car-safe var) 'error) | |
197 | (error "Bad format in expression: %s" (nth 1 var))) | |
198 | (calc-enter-result 1 "tylr" (list 'calcFunc-taylor | |
199 | (calc-top-n 1) | |
200 | var | |
bf77c646 | 201 | (prefix-numeric-value nterms)))))) |
136211a9 EZ |
202 | |
203 | ||
204 | (defun math-derivative (expr) ; uses global values: deriv-var, deriv-total. | |
205 | (cond ((equal expr deriv-var) | |
206 | 1) | |
207 | ((or (Math-scalarp expr) | |
208 | (eq (car expr) 'sdev) | |
209 | (and (eq (car expr) 'var) | |
210 | (or (not deriv-total) | |
211 | (math-const-var expr) | |
212 | (progn | |
213 | (math-setup-declarations) | |
214 | (memq 'const (nth 1 (or (assq (nth 2 expr) | |
215 | math-decls-cache) | |
216 | math-decls-all))))))) | |
217 | 0) | |
218 | ((eq (car expr) '+) | |
219 | (math-add (math-derivative (nth 1 expr)) | |
220 | (math-derivative (nth 2 expr)))) | |
221 | ((eq (car expr) '-) | |
222 | (math-sub (math-derivative (nth 1 expr)) | |
223 | (math-derivative (nth 2 expr)))) | |
224 | ((memq (car expr) '(calcFunc-eq calcFunc-neq calcFunc-lt | |
225 | calcFunc-gt calcFunc-leq calcFunc-geq)) | |
226 | (list (car expr) | |
227 | (math-derivative (nth 1 expr)) | |
228 | (math-derivative (nth 2 expr)))) | |
229 | ((eq (car expr) 'neg) | |
230 | (math-neg (math-derivative (nth 1 expr)))) | |
231 | ((eq (car expr) '*) | |
232 | (math-add (math-mul (nth 2 expr) | |
233 | (math-derivative (nth 1 expr))) | |
234 | (math-mul (nth 1 expr) | |
235 | (math-derivative (nth 2 expr))))) | |
236 | ((eq (car expr) '/) | |
237 | (math-sub (math-div (math-derivative (nth 1 expr)) | |
238 | (nth 2 expr)) | |
239 | (math-div (math-mul (nth 1 expr) | |
240 | (math-derivative (nth 2 expr))) | |
241 | (math-sqr (nth 2 expr))))) | |
242 | ((eq (car expr) '^) | |
243 | (let ((du (math-derivative (nth 1 expr))) | |
244 | (dv (math-derivative (nth 2 expr)))) | |
245 | (or (Math-zerop du) | |
246 | (setq du (math-mul (nth 2 expr) | |
247 | (math-mul (math-normalize | |
248 | (list '^ | |
249 | (nth 1 expr) | |
250 | (math-add (nth 2 expr) -1))) | |
251 | du)))) | |
252 | (or (Math-zerop dv) | |
253 | (setq dv (math-mul (math-normalize | |
254 | (list 'calcFunc-ln (nth 1 expr))) | |
255 | (math-mul expr dv)))) | |
256 | (math-add du dv))) | |
257 | ((eq (car expr) '%) | |
258 | (math-derivative (nth 1 expr))) ; a reasonable definition | |
259 | ((eq (car expr) 'vec) | |
260 | (math-map-vec 'math-derivative expr)) | |
261 | ((and (memq (car expr) '(calcFunc-conj calcFunc-re calcFunc-im)) | |
262 | (= (length expr) 2)) | |
263 | (list (car expr) (math-derivative (nth 1 expr)))) | |
264 | ((and (memq (car expr) '(calcFunc-subscr calcFunc-mrow calcFunc-mcol)) | |
265 | (= (length expr) 3)) | |
266 | (let ((d (math-derivative (nth 1 expr)))) | |
267 | (if (math-numberp d) | |
268 | 0 ; assume x and x_1 are independent vars | |
269 | (list (car expr) d (nth 2 expr))))) | |
270 | (t (or (and (symbolp (car expr)) | |
271 | (if (= (length expr) 2) | |
272 | (let ((handler (get (car expr) 'math-derivative))) | |
273 | (and handler | |
274 | (let ((deriv (math-derivative (nth 1 expr)))) | |
275 | (if (Math-zerop deriv) | |
276 | deriv | |
277 | (math-mul (funcall handler (nth 1 expr)) | |
278 | deriv))))) | |
279 | (let ((handler (get (car expr) 'math-derivative-n))) | |
280 | (and handler | |
281 | (funcall handler expr))))) | |
282 | (and (not (eq deriv-symb 'pre-expand)) | |
283 | (let ((exp (math-expand-formula expr))) | |
284 | (and exp | |
285 | (or (let ((deriv-symb 'pre-expand)) | |
286 | (catch 'math-deriv (math-derivative expr))) | |
287 | (math-derivative exp))))) | |
288 | (if (or (Math-objvecp expr) | |
289 | (eq (car expr) 'var) | |
290 | (not (symbolp (car expr)))) | |
291 | (if deriv-symb | |
292 | (throw 'math-deriv nil) | |
293 | (list (if deriv-total 'calcFunc-tderiv 'calcFunc-deriv) | |
294 | expr | |
295 | deriv-var)) | |
296 | (let ((accum 0) | |
297 | (arg expr) | |
298 | (n 1) | |
299 | derv) | |
300 | (while (setq arg (cdr arg)) | |
301 | (or (Math-zerop (setq derv (math-derivative (car arg)))) | |
302 | (let ((func (intern (concat (symbol-name (car expr)) | |
303 | "'" | |
304 | (if (> n 1) | |
305 | (int-to-string n) | |
306 | "")))) | |
307 | (prop (cond ((= (length expr) 2) | |
308 | 'math-derivative-1) | |
309 | ((= (length expr) 3) | |
310 | 'math-derivative-2) | |
311 | ((= (length expr) 4) | |
312 | 'math-derivative-3) | |
313 | ((= (length expr) 5) | |
314 | 'math-derivative-4) | |
315 | ((= (length expr) 6) | |
316 | 'math-derivative-5)))) | |
317 | (setq accum | |
318 | (math-add | |
319 | accum | |
320 | (math-mul | |
321 | derv | |
322 | (let ((handler (get func prop))) | |
323 | (or (and prop handler | |
324 | (apply handler (cdr expr))) | |
325 | (if (and deriv-symb | |
326 | (not (get func | |
327 | 'calc-user-defn))) | |
328 | (throw 'math-deriv nil) | |
329 | (cons func (cdr expr)))))))))) | |
330 | (setq n (1+ n))) | |
bf77c646 | 331 | accum)))))) |
136211a9 EZ |
332 | |
333 | (defun calcFunc-deriv (expr deriv-var &optional deriv-value deriv-symb) | |
334 | (let* ((deriv-total nil) | |
335 | (res (catch 'math-deriv (math-derivative expr)))) | |
336 | (or (eq (car-safe res) 'calcFunc-deriv) | |
337 | (null res) | |
338 | (setq res (math-normalize res))) | |
339 | (and res | |
340 | (if deriv-value | |
341 | (math-expr-subst res deriv-var deriv-value) | |
bf77c646 | 342 | res)))) |
136211a9 EZ |
343 | |
344 | (defun calcFunc-tderiv (expr deriv-var &optional deriv-value deriv-symb) | |
345 | (math-setup-declarations) | |
346 | (let* ((deriv-total t) | |
347 | (res (catch 'math-deriv (math-derivative expr)))) | |
348 | (or (eq (car-safe res) 'calcFunc-tderiv) | |
349 | (null res) | |
350 | (setq res (math-normalize res))) | |
351 | (and res | |
352 | (if deriv-value | |
353 | (math-expr-subst res deriv-var deriv-value) | |
bf77c646 | 354 | res)))) |
136211a9 EZ |
355 | |
356 | (put 'calcFunc-inv\' 'math-derivative-1 | |
357 | (function (lambda (u) (math-neg (math-div 1 (math-sqr u)))))) | |
358 | ||
359 | (put 'calcFunc-sqrt\' 'math-derivative-1 | |
360 | (function (lambda (u) (math-div 1 (math-mul 2 (list 'calcFunc-sqrt u)))))) | |
361 | ||
362 | (put 'calcFunc-deg\' 'math-derivative-1 | |
363 | (function (lambda (u) (math-div-float '(float 18 1) (math-pi))))) | |
364 | ||
365 | (put 'calcFunc-rad\' 'math-derivative-1 | |
366 | (function (lambda (u) (math-pi-over-180)))) | |
367 | ||
368 | (put 'calcFunc-ln\' 'math-derivative-1 | |
369 | (function (lambda (u) (math-div 1 u)))) | |
370 | ||
371 | (put 'calcFunc-log10\' 'math-derivative-1 | |
372 | (function (lambda (u) | |
373 | (math-div (math-div 1 (math-normalize '(calcFunc-ln 10))) | |
374 | u)))) | |
375 | ||
376 | (put 'calcFunc-lnp1\' 'math-derivative-1 | |
377 | (function (lambda (u) (math-div 1 (math-add u 1))))) | |
378 | ||
379 | (put 'calcFunc-log\' 'math-derivative-2 | |
380 | (function (lambda (x b) | |
381 | (and (not (Math-zerop b)) | |
382 | (let ((lnv (math-normalize | |
383 | (list 'calcFunc-ln b)))) | |
384 | (math-div 1 (math-mul lnv x))))))) | |
385 | ||
386 | (put 'calcFunc-log\'2 'math-derivative-2 | |
387 | (function (lambda (x b) | |
388 | (let ((lnv (list 'calcFunc-ln b))) | |
389 | (math-neg (math-div (list 'calcFunc-log x b) | |
390 | (math-mul lnv b))))))) | |
391 | ||
392 | (put 'calcFunc-exp\' 'math-derivative-1 | |
393 | (function (lambda (u) (math-normalize (list 'calcFunc-exp u))))) | |
394 | ||
395 | (put 'calcFunc-expm1\' 'math-derivative-1 | |
396 | (function (lambda (u) (math-normalize (list 'calcFunc-expm1 u))))) | |
397 | ||
398 | (put 'calcFunc-sin\' 'math-derivative-1 | |
399 | (function (lambda (u) (math-to-radians-2 (math-normalize | |
400 | (list 'calcFunc-cos u)))))) | |
401 | ||
402 | (put 'calcFunc-cos\' 'math-derivative-1 | |
403 | (function (lambda (u) (math-neg (math-to-radians-2 | |
404 | (math-normalize | |
405 | (list 'calcFunc-sin u))))))) | |
406 | ||
407 | (put 'calcFunc-tan\' 'math-derivative-1 | |
408 | (function (lambda (u) (math-to-radians-2 | |
409 | (math-div 1 (math-sqr | |
410 | (math-normalize | |
411 | (list 'calcFunc-cos u)))))))) | |
412 | ||
413 | (put 'calcFunc-arcsin\' 'math-derivative-1 | |
414 | (function (lambda (u) | |
415 | (math-from-radians-2 | |
416 | (math-div 1 (math-normalize | |
417 | (list 'calcFunc-sqrt | |
418 | (math-sub 1 (math-sqr u))))))))) | |
419 | ||
420 | (put 'calcFunc-arccos\' 'math-derivative-1 | |
421 | (function (lambda (u) | |
422 | (math-from-radians-2 | |
423 | (math-div -1 (math-normalize | |
424 | (list 'calcFunc-sqrt | |
425 | (math-sub 1 (math-sqr u))))))))) | |
426 | ||
427 | (put 'calcFunc-arctan\' 'math-derivative-1 | |
428 | (function (lambda (u) (math-from-radians-2 | |
429 | (math-div 1 (math-add 1 (math-sqr u))))))) | |
430 | ||
431 | (put 'calcFunc-sinh\' 'math-derivative-1 | |
432 | (function (lambda (u) (math-normalize (list 'calcFunc-cosh u))))) | |
433 | ||
434 | (put 'calcFunc-cosh\' 'math-derivative-1 | |
435 | (function (lambda (u) (math-normalize (list 'calcFunc-sinh u))))) | |
436 | ||
437 | (put 'calcFunc-tanh\' 'math-derivative-1 | |
438 | (function (lambda (u) (math-div 1 (math-sqr | |
439 | (math-normalize | |
440 | (list 'calcFunc-cosh u))))))) | |
441 | ||
442 | (put 'calcFunc-arcsinh\' 'math-derivative-1 | |
443 | (function (lambda (u) | |
444 | (math-div 1 (math-normalize | |
445 | (list 'calcFunc-sqrt | |
446 | (math-add (math-sqr u) 1))))))) | |
447 | ||
448 | (put 'calcFunc-arccosh\' 'math-derivative-1 | |
449 | (function (lambda (u) | |
450 | (math-div 1 (math-normalize | |
451 | (list 'calcFunc-sqrt | |
452 | (math-add (math-sqr u) -1))))))) | |
453 | ||
454 | (put 'calcFunc-arctanh\' 'math-derivative-1 | |
455 | (function (lambda (u) (math-div 1 (math-sub 1 (math-sqr u)))))) | |
456 | ||
457 | (put 'calcFunc-bern\'2 'math-derivative-2 | |
458 | (function (lambda (n x) | |
459 | (math-mul n (list 'calcFunc-bern (math-add n -1) x))))) | |
460 | ||
461 | (put 'calcFunc-euler\'2 'math-derivative-2 | |
462 | (function (lambda (n x) | |
463 | (math-mul n (list 'calcFunc-euler (math-add n -1) x))))) | |
464 | ||
465 | (put 'calcFunc-gammag\'2 'math-derivative-2 | |
466 | (function (lambda (a x) (math-deriv-gamma a x 1)))) | |
467 | ||
468 | (put 'calcFunc-gammaG\'2 'math-derivative-2 | |
469 | (function (lambda (a x) (math-deriv-gamma a x -1)))) | |
470 | ||
471 | (put 'calcFunc-gammaP\'2 'math-derivative-2 | |
472 | (function (lambda (a x) (math-deriv-gamma a x | |
473 | (math-div | |
474 | 1 (math-normalize | |
475 | (list 'calcFunc-gamma | |
476 | a))))))) | |
477 | ||
478 | (put 'calcFunc-gammaQ\'2 'math-derivative-2 | |
479 | (function (lambda (a x) (math-deriv-gamma a x | |
480 | (math-div | |
481 | -1 (math-normalize | |
482 | (list 'calcFunc-gamma | |
483 | a))))))) | |
484 | ||
485 | (defun math-deriv-gamma (a x scale) | |
486 | (math-mul scale | |
487 | (math-mul (math-pow x (math-add a -1)) | |
bf77c646 | 488 | (list 'calcFunc-exp (math-neg x))))) |
136211a9 EZ |
489 | |
490 | (put 'calcFunc-betaB\' 'math-derivative-3 | |
491 | (function (lambda (x a b) (math-deriv-beta x a b 1)))) | |
492 | ||
493 | (put 'calcFunc-betaI\' 'math-derivative-3 | |
494 | (function (lambda (x a b) (math-deriv-beta x a b | |
495 | (math-div | |
496 | 1 (list 'calcFunc-beta | |
497 | a b)))))) | |
498 | ||
499 | (defun math-deriv-beta (x a b scale) | |
500 | (math-mul (math-mul (math-pow x (math-add a -1)) | |
501 | (math-pow (math-sub 1 x) (math-add b -1))) | |
bf77c646 | 502 | scale)) |
136211a9 EZ |
503 | |
504 | (put 'calcFunc-erf\' 'math-derivative-1 | |
505 | (function (lambda (x) (math-div 2 | |
506 | (math-mul (list 'calcFunc-exp | |
507 | (math-sqr x)) | |
508 | (if calc-symbolic-mode | |
509 | '(calcFunc-sqrt | |
510 | (var pi var-pi)) | |
511 | (math-sqrt-pi))))))) | |
512 | ||
513 | (put 'calcFunc-erfc\' 'math-derivative-1 | |
514 | (function (lambda (x) (math-div -2 | |
515 | (math-mul (list 'calcFunc-exp | |
516 | (math-sqr x)) | |
517 | (if calc-symbolic-mode | |
518 | '(calcFunc-sqrt | |
519 | (var pi var-pi)) | |
520 | (math-sqrt-pi))))))) | |
521 | ||
522 | (put 'calcFunc-besJ\'2 'math-derivative-2 | |
523 | (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besJ | |
524 | (math-add v -1) | |
525 | z) | |
526 | (list 'calcFunc-besJ | |
527 | (math-add v 1) | |
528 | z)) | |
529 | 2)))) | |
530 | ||
531 | (put 'calcFunc-besY\'2 'math-derivative-2 | |
532 | (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besY | |
533 | (math-add v -1) | |
534 | z) | |
535 | (list 'calcFunc-besY | |
536 | (math-add v 1) | |
537 | z)) | |
538 | 2)))) | |
539 | ||
540 | (put 'calcFunc-sum 'math-derivative-n | |
541 | (function | |
542 | (lambda (expr) | |
543 | (if (math-expr-contains (cons 'vec (cdr (cdr expr))) deriv-var) | |
544 | (throw 'math-deriv nil) | |
545 | (cons 'calcFunc-sum | |
546 | (cons (math-derivative (nth 1 expr)) | |
547 | (cdr (cdr expr)))))))) | |
548 | ||
549 | (put 'calcFunc-prod 'math-derivative-n | |
550 | (function | |
551 | (lambda (expr) | |
552 | (if (math-expr-contains (cons 'vec (cdr (cdr expr))) deriv-var) | |
553 | (throw 'math-deriv nil) | |
554 | (math-mul expr | |
555 | (cons 'calcFunc-sum | |
556 | (cons (math-div (math-derivative (nth 1 expr)) | |
557 | (nth 1 expr)) | |
558 | (cdr (cdr expr))))))))) | |
559 | ||
560 | (put 'calcFunc-integ 'math-derivative-n | |
561 | (function | |
562 | (lambda (expr) | |
563 | (if (= (length expr) 3) | |
564 | (if (equal (nth 2 expr) deriv-var) | |
565 | (nth 1 expr) | |
566 | (math-normalize | |
567 | (list 'calcFunc-integ | |
568 | (math-derivative (nth 1 expr)) | |
569 | (nth 2 expr)))) | |
570 | (if (= (length expr) 5) | |
571 | (let ((lower (math-expr-subst (nth 1 expr) (nth 2 expr) | |
572 | (nth 3 expr))) | |
573 | (upper (math-expr-subst (nth 1 expr) (nth 2 expr) | |
574 | (nth 4 expr)))) | |
575 | (math-add (math-sub (math-mul upper | |
576 | (math-derivative (nth 4 expr))) | |
577 | (math-mul lower | |
578 | (math-derivative (nth 3 expr)))) | |
579 | (if (equal (nth 2 expr) deriv-var) | |
580 | 0 | |
581 | (math-normalize | |
582 | (list 'calcFunc-integ | |
583 | (math-derivative (nth 1 expr)) (nth 2 expr) | |
584 | (nth 3 expr) (nth 4 expr))))))))))) | |
585 | ||
586 | (put 'calcFunc-if 'math-derivative-n | |
587 | (function | |
588 | (lambda (expr) | |
589 | (and (= (length expr) 4) | |
590 | (list 'calcFunc-if (nth 1 expr) | |
591 | (math-derivative (nth 2 expr)) | |
592 | (math-derivative (nth 3 expr))))))) | |
593 | ||
594 | (put 'calcFunc-subscr 'math-derivative-n | |
595 | (function | |
596 | (lambda (expr) | |
597 | (and (= (length expr) 3) | |
598 | (list 'calcFunc-subscr (nth 1 expr) | |
599 | (math-derivative (nth 2 expr))))))) | |
600 | ||
601 | ||
3132f345 CW |
602 | (defvar math-integ-var '(var X ---)) |
603 | (defvar math-integ-var-2 '(var Y ---)) | |
604 | (defvar math-integ-vars (list 'f math-integ-var math-integ-var-2)) | |
605 | (defvar math-integ-var-list (list math-integ-var)) | |
606 | (defvar math-integ-var-list-list (list math-integ-var-list)) | |
136211a9 EZ |
607 | |
608 | (defmacro math-tracing-integral (&rest parts) | |
609 | (list 'and | |
610 | 'trace-buffer | |
611 | (list 'save-excursion | |
612 | '(set-buffer trace-buffer) | |
613 | '(goto-char (point-max)) | |
614 | (list 'and | |
615 | '(bolp) | |
616 | '(insert (make-string (- math-integral-limit | |
617 | math-integ-level) 32) | |
618 | (format "%2d " math-integ-depth) | |
619 | (make-string math-integ-level 32))) | |
620 | ;;(list 'condition-case 'err | |
621 | (cons 'insert parts) | |
622 | ;; '(error (insert (prin1-to-string err)))) | |
bf77c646 | 623 | '(sit-for 0)))) |
136211a9 EZ |
624 | |
625 | ;;; The following wrapper caches results and avoids infinite recursion. | |
626 | ;;; Each cache entry is: ( A B ) Integral of A is B; | |
627 | ;;; ( A N ) Integral of A failed at level N; | |
628 | ;;; ( A busy ) Currently working on integral of A; | |
629 | ;;; ( A parts ) Currently working, integ-by-parts; | |
630 | ;;; ( A parts2 ) Currently working, integ-by-parts; | |
631 | ;;; ( A cancelled ) Ignore this cache entry; | |
632 | ;;; ( A [B] ) Same result as for cur-record = B. | |
633 | (defun math-integral (expr &optional simplify same-as-above) | |
634 | (let* ((simp cur-record) | |
635 | (cur-record (assoc expr math-integral-cache)) | |
636 | (math-integ-depth (1+ math-integ-depth)) | |
637 | (val 'cancelled)) | |
638 | (math-tracing-integral "Integrating " | |
639 | (math-format-value expr 1000) | |
640 | "...\n") | |
641 | (and cur-record | |
642 | (progn | |
643 | (math-tracing-integral "Found " | |
644 | (math-format-value (nth 1 cur-record) 1000)) | |
645 | (and (consp (nth 1 cur-record)) | |
646 | (math-replace-integral-parts cur-record)) | |
647 | (math-tracing-integral " => " | |
648 | (math-format-value (nth 1 cur-record) 1000) | |
649 | "\n"))) | |
650 | (or (and cur-record | |
651 | (not (eq (nth 1 cur-record) 'cancelled)) | |
652 | (or (not (integerp (nth 1 cur-record))) | |
653 | (>= (nth 1 cur-record) math-integ-level))) | |
654 | (and (math-integral-contains-parts expr) | |
655 | (progn | |
656 | (setq val nil) | |
657 | t)) | |
658 | (unwind-protect | |
659 | (progn | |
660 | (let (math-integ-msg) | |
661 | (if (eq calc-display-working-message 'lots) | |
662 | (progn | |
663 | (calc-set-command-flag 'clear-message) | |
664 | (setq math-integ-msg (format | |
665 | "Working... Integrating %s" | |
666 | (math-format-flat-expr expr 0))) | |
667 | (message math-integ-msg))) | |
668 | (if cur-record | |
669 | (setcar (cdr cur-record) | |
670 | (if same-as-above (vector simp) 'busy)) | |
671 | (setq cur-record | |
672 | (list expr (if same-as-above (vector simp) 'busy)) | |
673 | math-integral-cache (cons cur-record | |
674 | math-integral-cache))) | |
675 | (if (eq simplify 'yes) | |
676 | (progn | |
677 | (math-tracing-integral "Simplifying...") | |
678 | (setq simp (math-simplify expr)) | |
679 | (setq val (if (equal simp expr) | |
680 | (progn | |
681 | (math-tracing-integral " no change\n") | |
682 | (math-do-integral expr)) | |
683 | (math-tracing-integral " simplified\n") | |
684 | (math-integral simp 'no t)))) | |
685 | (or (setq val (math-do-integral expr)) | |
686 | (eq simplify 'no) | |
687 | (let ((simp (math-simplify expr))) | |
688 | (or (equal simp expr) | |
689 | (progn | |
690 | (math-tracing-integral "Trying again after " | |
691 | "simplification...\n") | |
692 | (setq val (math-integral simp 'no t)))))))) | |
693 | (if (eq calc-display-working-message 'lots) | |
694 | (message math-integ-msg))) | |
695 | (setcar (cdr cur-record) (or val | |
696 | (if (or math-enable-subst | |
697 | (not math-any-substs)) | |
698 | math-integ-level | |
699 | 'cancelled))))) | |
700 | (setq val cur-record) | |
701 | (while (vectorp (nth 1 val)) | |
702 | (setq val (aref (nth 1 val) 0))) | |
703 | (setq val (if (memq (nth 1 val) '(parts parts2)) | |
704 | (progn | |
705 | (setcar (cdr val) 'parts2) | |
706 | (list 'var 'PARTS val)) | |
707 | (and (consp (nth 1 val)) | |
708 | (nth 1 val)))) | |
709 | (math-tracing-integral "Integral of " | |
710 | (math-format-value expr 1000) | |
711 | " is " | |
712 | (math-format-value val 1000) | |
713 | "\n") | |
bf77c646 | 714 | val)) |
136211a9 EZ |
715 | (defvar math-integral-cache nil) |
716 | (defvar math-integral-cache-state nil) | |
717 | ||
718 | (defun math-integral-contains-parts (expr) | |
719 | (if (Math-primp expr) | |
720 | (and (eq (car-safe expr) 'var) | |
721 | (eq (nth 1 expr) 'PARTS) | |
722 | (listp (nth 2 expr))) | |
723 | (while (and (setq expr (cdr expr)) | |
724 | (not (math-integral-contains-parts (car expr))))) | |
bf77c646 | 725 | expr)) |
136211a9 EZ |
726 | |
727 | (defun math-replace-integral-parts (expr) | |
728 | (or (Math-primp expr) | |
729 | (while (setq expr (cdr expr)) | |
730 | (and (consp (car expr)) | |
731 | (if (eq (car (car expr)) 'var) | |
732 | (and (eq (nth 1 (car expr)) 'PARTS) | |
733 | (consp (nth 2 (car expr))) | |
734 | (if (listp (nth 1 (nth 2 (car expr)))) | |
735 | (progn | |
736 | (setcar expr (nth 1 (nth 2 (car expr)))) | |
737 | (math-replace-integral-parts (cons 'foo expr))) | |
738 | (setcar (cdr cur-record) 'cancelled))) | |
bf77c646 | 739 | (math-replace-integral-parts (car expr))))))) |
136211a9 EZ |
740 | |
741 | (defun math-do-integral (expr) | |
742 | (let (t1 t2) | |
743 | (or (cond ((not (math-expr-contains expr math-integ-var)) | |
744 | (math-mul expr math-integ-var)) | |
745 | ((equal expr math-integ-var) | |
746 | (math-div (math-sqr expr) 2)) | |
747 | ((eq (car expr) '+) | |
748 | (and (setq t1 (math-integral (nth 1 expr))) | |
749 | (setq t2 (math-integral (nth 2 expr))) | |
750 | (math-add t1 t2))) | |
751 | ((eq (car expr) '-) | |
752 | (and (setq t1 (math-integral (nth 1 expr))) | |
753 | (setq t2 (math-integral (nth 2 expr))) | |
754 | (math-sub t1 t2))) | |
755 | ((eq (car expr) 'neg) | |
756 | (and (setq t1 (math-integral (nth 1 expr))) | |
757 | (math-neg t1))) | |
758 | ((eq (car expr) '*) | |
759 | (cond ((not (math-expr-contains (nth 1 expr) math-integ-var)) | |
760 | (and (setq t1 (math-integral (nth 2 expr))) | |
761 | (math-mul (nth 1 expr) t1))) | |
762 | ((not (math-expr-contains (nth 2 expr) math-integ-var)) | |
763 | (and (setq t1 (math-integral (nth 1 expr))) | |
764 | (math-mul t1 (nth 2 expr)))) | |
765 | ((memq (car-safe (nth 1 expr)) '(+ -)) | |
766 | (math-integral (list (car (nth 1 expr)) | |
767 | (math-mul (nth 1 (nth 1 expr)) | |
768 | (nth 2 expr)) | |
769 | (math-mul (nth 2 (nth 1 expr)) | |
770 | (nth 2 expr))) | |
771 | 'yes t)) | |
772 | ((memq (car-safe (nth 2 expr)) '(+ -)) | |
773 | (math-integral (list (car (nth 2 expr)) | |
774 | (math-mul (nth 1 (nth 2 expr)) | |
775 | (nth 1 expr)) | |
776 | (math-mul (nth 2 (nth 2 expr)) | |
777 | (nth 1 expr))) | |
778 | 'yes t)))) | |
779 | ((eq (car expr) '/) | |
780 | (cond ((and (not (math-expr-contains (nth 1 expr) | |
781 | math-integ-var)) | |
782 | (not (math-equal-int (nth 1 expr) 1))) | |
783 | (and (setq t1 (math-integral (math-div 1 (nth 2 expr)))) | |
784 | (math-mul (nth 1 expr) t1))) | |
785 | ((not (math-expr-contains (nth 2 expr) math-integ-var)) | |
786 | (and (setq t1 (math-integral (nth 1 expr))) | |
787 | (math-div t1 (nth 2 expr)))) | |
788 | ((and (eq (car-safe (nth 1 expr)) '*) | |
789 | (not (math-expr-contains (nth 1 (nth 1 expr)) | |
790 | math-integ-var))) | |
791 | (and (setq t1 (math-integral | |
792 | (math-div (nth 2 (nth 1 expr)) | |
793 | (nth 2 expr)))) | |
794 | (math-mul t1 (nth 1 (nth 1 expr))))) | |
795 | ((and (eq (car-safe (nth 1 expr)) '*) | |
796 | (not (math-expr-contains (nth 2 (nth 1 expr)) | |
797 | math-integ-var))) | |
798 | (and (setq t1 (math-integral | |
799 | (math-div (nth 1 (nth 1 expr)) | |
800 | (nth 2 expr)))) | |
801 | (math-mul t1 (nth 2 (nth 1 expr))))) | |
802 | ((and (eq (car-safe (nth 2 expr)) '*) | |
803 | (not (math-expr-contains (nth 1 (nth 2 expr)) | |
804 | math-integ-var))) | |
805 | (and (setq t1 (math-integral | |
806 | (math-div (nth 1 expr) | |
807 | (nth 2 (nth 2 expr))))) | |
808 | (math-div t1 (nth 1 (nth 2 expr))))) | |
809 | ((and (eq (car-safe (nth 2 expr)) '*) | |
810 | (not (math-expr-contains (nth 2 (nth 2 expr)) | |
811 | math-integ-var))) | |
812 | (and (setq t1 (math-integral | |
813 | (math-div (nth 1 expr) | |
814 | (nth 1 (nth 2 expr))))) | |
815 | (math-div t1 (nth 2 (nth 2 expr))))) | |
816 | ((eq (car-safe (nth 2 expr)) 'calcFunc-exp) | |
817 | (math-integral | |
818 | (math-mul (nth 1 expr) | |
819 | (list 'calcFunc-exp | |
820 | (math-neg (nth 1 (nth 2 expr))))))))) | |
821 | ((eq (car expr) '^) | |
822 | (cond ((not (math-expr-contains (nth 1 expr) math-integ-var)) | |
823 | (or (and (setq t1 (math-is-polynomial (nth 2 expr) | |
824 | math-integ-var 1)) | |
825 | (math-div expr | |
826 | (math-mul (nth 1 t1) | |
827 | (math-normalize | |
828 | (list 'calcFunc-ln | |
829 | (nth 1 expr)))))) | |
830 | (math-integral | |
831 | (list 'calcFunc-exp | |
832 | (math-mul (nth 2 expr) | |
833 | (math-normalize | |
834 | (list 'calcFunc-ln | |
835 | (nth 1 expr))))) | |
836 | 'yes t))) | |
837 | ((not (math-expr-contains (nth 2 expr) math-integ-var)) | |
838 | (if (and (integerp (nth 2 expr)) (< (nth 2 expr) 0)) | |
839 | (math-integral | |
840 | (list '/ 1 (math-pow (nth 1 expr) (- (nth 2 expr)))) | |
841 | nil t) | |
842 | (or (and (setq t1 (math-is-polynomial (nth 1 expr) | |
843 | math-integ-var | |
844 | 1)) | |
845 | (setq t2 (math-add (nth 2 expr) 1)) | |
846 | (math-div (math-pow (nth 1 expr) t2) | |
847 | (math-mul t2 (nth 1 t1)))) | |
848 | (and (Math-negp (nth 2 expr)) | |
849 | (math-integral | |
850 | (math-div 1 | |
851 | (math-pow (nth 1 expr) | |
852 | (math-neg | |
853 | (nth 2 expr)))) | |
854 | nil t)) | |
855 | nil)))))) | |
856 | ||
857 | ;; Integral of a polynomial. | |
858 | (and (setq t1 (math-is-polynomial expr math-integ-var 20)) | |
859 | (let ((accum 0) | |
860 | (n 1)) | |
861 | (while t1 | |
862 | (if (setq accum (math-add accum | |
863 | (math-div (math-mul (car t1) | |
864 | (math-pow | |
865 | math-integ-var | |
866 | n)) | |
867 | n)) | |
868 | t1 (cdr t1)) | |
869 | (setq n (1+ n)))) | |
870 | accum)) | |
871 | ||
872 | ;; Try looking it up! | |
873 | (cond ((= (length expr) 2) | |
874 | (and (symbolp (car expr)) | |
875 | (setq t1 (get (car expr) 'math-integral)) | |
876 | (progn | |
877 | (while (and t1 | |
878 | (not (setq t2 (funcall (car t1) | |
879 | (nth 1 expr))))) | |
880 | (setq t1 (cdr t1))) | |
881 | (and t2 (math-normalize t2))))) | |
882 | ((= (length expr) 3) | |
883 | (and (symbolp (car expr)) | |
884 | (setq t1 (get (car expr) 'math-integral-2)) | |
885 | (progn | |
886 | (while (and t1 | |
887 | (not (setq t2 (funcall (car t1) | |
888 | (nth 1 expr) | |
889 | (nth 2 expr))))) | |
890 | (setq t1 (cdr t1))) | |
891 | (and t2 (math-normalize t2)))))) | |
892 | ||
893 | ;; Integral of a rational function. | |
894 | (and (math-ratpoly-p expr math-integ-var) | |
895 | (setq t1 (calcFunc-apart expr math-integ-var)) | |
896 | (not (equal t1 expr)) | |
897 | (math-integral t1)) | |
898 | ||
899 | ;; Try user-defined integration rules. | |
900 | (and has-rules | |
901 | (let ((math-old-integ (symbol-function 'calcFunc-integ)) | |
902 | (input (list 'calcFunc-integtry expr math-integ-var)) | |
903 | res part) | |
904 | (unwind-protect | |
905 | (progn | |
906 | (fset 'calcFunc-integ 'math-sub-integration) | |
907 | (setq res (math-rewrite input | |
908 | '(var IntegRules var-IntegRules) | |
909 | 1)) | |
910 | (fset 'calcFunc-integ math-old-integ) | |
911 | (and (not (equal res input)) | |
912 | (if (setq part (math-expr-calls | |
913 | res '(calcFunc-integsubst))) | |
914 | (and (memq (length part) '(3 4 5)) | |
915 | (let ((parts (mapcar | |
916 | (function | |
917 | (lambda (x) | |
918 | (math-expr-subst | |
919 | x (nth 2 part) | |
920 | math-integ-var))) | |
921 | (cdr part)))) | |
922 | (math-integrate-by-substitution | |
923 | expr (car parts) t | |
924 | (or (nth 2 parts) | |
925 | (list 'calcFunc-integfailed | |
926 | math-integ-var)) | |
927 | (nth 3 parts)))) | |
928 | (if (not (math-expr-calls res | |
929 | '(calcFunc-integtry | |
930 | calcFunc-integfailed))) | |
931 | res)))) | |
932 | (fset 'calcFunc-integ math-old-integ)))) | |
933 | ||
934 | ;; See if the function is a symbolic derivative. | |
935 | (and (string-match "'" (symbol-name (car expr))) | |
936 | (let ((name (symbol-name (car expr))) | |
937 | (p expr) (n 0) (which nil) (bad nil)) | |
938 | (while (setq n (1+ n) p (cdr p)) | |
939 | (if (equal (car p) math-integ-var) | |
940 | (if which (setq bad t) (setq which n)) | |
941 | (if (math-expr-contains (car p) math-integ-var) | |
942 | (setq bad t)))) | |
943 | (and which (not bad) | |
944 | (let ((prime (if (= which 1) "'" (format "'%d" which)))) | |
945 | (and (string-match (concat prime "\\('['0-9]*\\|$\\)") | |
946 | name) | |
947 | (cons (intern | |
948 | (concat | |
949 | (substring name 0 (match-beginning 0)) | |
950 | (substring name (+ (match-beginning 0) | |
951 | (length prime))))) | |
952 | (cdr expr))))))) | |
953 | ||
954 | ;; Try transformation methods (parts, substitutions). | |
955 | (and (> math-integ-level 0) | |
956 | (math-do-integral-methods expr)) | |
957 | ||
958 | ;; Try expanding the function's definition. | |
959 | (let ((res (math-expand-formula expr))) | |
960 | (and res | |
bf77c646 | 961 | (math-integral res)))))) |
136211a9 EZ |
962 | |
963 | (defun math-sub-integration (expr &rest rest) | |
964 | (or (if (or (not rest) | |
965 | (and (< math-integ-level math-integral-limit) | |
966 | (eq (car rest) math-integ-var))) | |
967 | (math-integral expr) | |
968 | (let ((res (apply math-old-integ expr rest))) | |
969 | (and (or (= math-integ-level math-integral-limit) | |
970 | (not (math-expr-calls res 'calcFunc-integ))) | |
971 | res))) | |
bf77c646 | 972 | (list 'calcFunc-integfailed expr))) |
136211a9 EZ |
973 | |
974 | (defun math-do-integral-methods (expr) | |
975 | (let ((so-far math-integ-var-list-list) | |
976 | rat-in) | |
977 | ||
978 | ;; Integration by substitution, for various likely sub-expressions. | |
979 | ;; (In first pass, we look only for sub-exprs that are linear in X.) | |
980 | (or (if math-enable-subst | |
981 | (math-integ-try-substitutions expr) | |
982 | (math-integ-try-linear-substitutions expr)) | |
983 | ||
984 | ;; If function has sines and cosines, try tan(x/2) substitution. | |
985 | (and (let ((p (setq rat-in (math-expr-rational-in expr)))) | |
986 | (while (and p | |
987 | (memq (car (car p)) '(calcFunc-sin | |
988 | calcFunc-cos | |
989 | calcFunc-tan)) | |
990 | (equal (nth 1 (car p)) math-integ-var)) | |
991 | (setq p (cdr p))) | |
992 | (null p)) | |
993 | (or (and (math-integ-parts-easy expr) | |
994 | (math-integ-try-parts expr t)) | |
995 | (math-integrate-by-good-substitution | |
996 | expr (list 'calcFunc-tan (math-div math-integ-var 2))))) | |
997 | ||
998 | ;; If function has sinh and cosh, try tanh(x/2) substitution. | |
999 | (and (let ((p rat-in)) | |
1000 | (while (and p | |
1001 | (memq (car (car p)) '(calcFunc-sinh | |
1002 | calcFunc-cosh | |
1003 | calcFunc-tanh | |
1004 | calcFunc-exp)) | |
1005 | (equal (nth 1 (car p)) math-integ-var)) | |
1006 | (setq p (cdr p))) | |
1007 | (null p)) | |
1008 | (or (and (math-integ-parts-easy expr) | |
1009 | (math-integ-try-parts expr t)) | |
1010 | (math-integrate-by-good-substitution | |
1011 | expr (list 'calcFunc-tanh (math-div math-integ-var 2))))) | |
1012 | ||
1013 | ;; If function has square roots, try sin, tan, or sec substitution. | |
1014 | (and (let ((p rat-in)) | |
1015 | (setq t1 nil) | |
1016 | (while (and p | |
1017 | (or (equal (car p) math-integ-var) | |
1018 | (and (eq (car (car p)) 'calcFunc-sqrt) | |
1019 | (setq t1 (math-is-polynomial | |
1020 | (nth 1 (setq t2 (car p))) | |
1021 | math-integ-var 2))))) | |
1022 | (setq p (cdr p))) | |
1023 | (and (null p) t1)) | |
1024 | (if (cdr (cdr t1)) | |
1025 | (if (math-guess-if-neg (nth 2 t1)) | |
1026 | (let* ((c (math-sqrt (math-neg (nth 2 t1)))) | |
1027 | (d (math-div (nth 1 t1) (math-mul -2 c))) | |
1028 | (a (math-sqrt (math-add (car t1) (math-sqr d))))) | |
1029 | (math-integrate-by-good-substitution | |
1030 | expr (list 'calcFunc-arcsin | |
1031 | (math-div-thru | |
1032 | (math-add (math-mul c math-integ-var) d) | |
1033 | a)))) | |
1034 | (let* ((c (math-sqrt (nth 2 t1))) | |
1035 | (d (math-div (nth 1 t1) (math-mul 2 c))) | |
1036 | (aa (math-sub (car t1) (math-sqr d)))) | |
1037 | (if (and nil (not (and (eq d 0) (eq c 1)))) | |
1038 | (math-integrate-by-good-substitution | |
1039 | expr (math-add (math-mul c math-integ-var) d)) | |
1040 | (if (math-guess-if-neg aa) | |
1041 | (math-integrate-by-good-substitution | |
1042 | expr (list 'calcFunc-arccosh | |
1043 | (math-div-thru | |
1044 | (math-add (math-mul c math-integ-var) | |
1045 | d) | |
1046 | (math-sqrt (math-neg aa))))) | |
1047 | (math-integrate-by-good-substitution | |
1048 | expr (list 'calcFunc-arcsinh | |
1049 | (math-div-thru | |
1050 | (math-add (math-mul c math-integ-var) | |
1051 | d) | |
1052 | (math-sqrt aa)))))))) | |
1053 | (math-integrate-by-good-substitution expr t2)) ) | |
1054 | ||
1055 | ;; Try integration by parts. | |
1056 | (math-integ-try-parts expr) | |
1057 | ||
1058 | ;; Give up. | |
bf77c646 | 1059 | nil))) |
136211a9 EZ |
1060 | |
1061 | (defun math-integ-parts-easy (expr) | |
1062 | (cond ((Math-primp expr) t) | |
1063 | ((memq (car expr) '(+ - *)) | |
1064 | (and (math-integ-parts-easy (nth 1 expr)) | |
1065 | (math-integ-parts-easy (nth 2 expr)))) | |
1066 | ((eq (car expr) '/) | |
1067 | (and (math-integ-parts-easy (nth 1 expr)) | |
1068 | (math-atomic-factorp (nth 2 expr)))) | |
1069 | ((eq (car expr) '^) | |
1070 | (and (natnump (nth 2 expr)) | |
1071 | (math-integ-parts-easy (nth 1 expr)))) | |
1072 | ((eq (car expr) 'neg) | |
1073 | (math-integ-parts-easy (nth 1 expr))) | |
bf77c646 | 1074 | (t t))) |
136211a9 EZ |
1075 | |
1076 | (defun math-integ-try-parts (expr &optional math-good-parts) | |
1077 | ;; Integration by parts: | |
1078 | ;; integ(f(x) g(x),x) = f(x) h(x) - integ(h(x) f'(x),x) | |
1079 | ;; where h(x) = integ(g(x),x). | |
1080 | (or (let ((exp (calcFunc-expand expr))) | |
1081 | (and (not (equal exp expr)) | |
1082 | (math-integral exp))) | |
1083 | (and (eq (car expr) '*) | |
1084 | (let ((first-bad (or (math-polynomial-p (nth 1 expr) | |
1085 | math-integ-var) | |
1086 | (equal (nth 2 expr) math-prev-parts-v)))) | |
1087 | (or (and first-bad ; so try this one first | |
1088 | (math-integrate-by-parts (nth 1 expr) (nth 2 expr))) | |
1089 | (math-integrate-by-parts (nth 2 expr) (nth 1 expr)) | |
1090 | (and (not first-bad) | |
1091 | (math-integrate-by-parts (nth 1 expr) (nth 2 expr)))))) | |
1092 | (and (eq (car expr) '/) | |
1093 | (math-expr-contains (nth 1 expr) math-integ-var) | |
1094 | (let ((recip (math-div 1 (nth 2 expr)))) | |
1095 | (or (math-integrate-by-parts (nth 1 expr) recip) | |
1096 | (math-integrate-by-parts recip (nth 1 expr))))) | |
1097 | (and (eq (car expr) '^) | |
1098 | (math-integrate-by-parts (math-pow (nth 1 expr) | |
1099 | (math-sub (nth 2 expr) 1)) | |
bf77c646 | 1100 | (nth 1 expr))))) |
136211a9 EZ |
1101 | |
1102 | (defun math-integrate-by-parts (u vprime) | |
1103 | (let ((math-integ-level (if (or math-good-parts | |
1104 | (math-polynomial-p u math-integ-var)) | |
1105 | math-integ-level | |
1106 | (1- math-integ-level))) | |
1107 | (math-doing-parts t) | |
1108 | v temp) | |
1109 | (and (>= math-integ-level 0) | |
1110 | (unwind-protect | |
1111 | (progn | |
1112 | (setcar (cdr cur-record) 'parts) | |
1113 | (math-tracing-integral "Integrating by parts, u = " | |
1114 | (math-format-value u 1000) | |
1115 | ", v' = " | |
1116 | (math-format-value vprime 1000) | |
1117 | "\n") | |
1118 | (and (setq v (math-integral vprime)) | |
1119 | (setq temp (calcFunc-deriv u math-integ-var nil t)) | |
1120 | (setq temp (let ((math-prev-parts-v v)) | |
1121 | (math-integral (math-mul v temp) 'yes))) | |
1122 | (setq temp (math-sub (math-mul u v) temp)) | |
1123 | (if (eq (nth 1 cur-record) 'parts) | |
1124 | (calcFunc-expand temp) | |
1125 | (setq v (list 'var 'PARTS cur-record) | |
1126 | var-thing (list 'vec (math-sub v temp) v) | |
1127 | temp (let (calc-next-why) | |
1128 | (math-solve-for (math-sub v temp) 0 v nil))) | |
1129 | (and temp (not (integerp temp)) | |
1130 | (math-simplify-extended temp))))) | |
bf77c646 | 1131 | (setcar (cdr cur-record) 'busy))))) |
136211a9 EZ |
1132 | |
1133 | ;;; This tries two different formulations, hoping the algebraic simplifier | |
1134 | ;;; will be strong enough to handle at least one. | |
1135 | (defun math-integrate-by-substitution (expr u &optional user uinv uinvprime) | |
1136 | (and (> math-integ-level 0) | |
1137 | (let ((math-integ-level (max (- math-integ-level 2) 0))) | |
bf77c646 | 1138 | (math-integrate-by-good-substitution expr u user uinv uinvprime)))) |
136211a9 EZ |
1139 | |
1140 | (defun math-integrate-by-good-substitution (expr u &optional user | |
1141 | uinv uinvprime) | |
1142 | (let ((math-living-dangerously t) | |
1143 | deriv temp) | |
1144 | (and (setq uinv (if uinv | |
1145 | (math-expr-subst uinv math-integ-var | |
1146 | math-integ-var-2) | |
1147 | (let (calc-next-why) | |
1148 | (math-solve-for u | |
1149 | math-integ-var-2 | |
1150 | math-integ-var nil)))) | |
1151 | (progn | |
1152 | (math-tracing-integral "Integrating by substitution, u = " | |
1153 | (math-format-value u 1000) | |
1154 | "\n") | |
1155 | (or (and (setq deriv (calcFunc-deriv u | |
1156 | math-integ-var nil | |
1157 | (not user))) | |
1158 | (setq temp (math-integral (math-expr-subst | |
1159 | (math-expr-subst | |
1160 | (math-expr-subst | |
1161 | (math-div expr deriv) | |
1162 | u | |
1163 | math-integ-var-2) | |
1164 | math-integ-var | |
1165 | uinv) | |
1166 | math-integ-var-2 | |
1167 | math-integ-var) | |
1168 | 'yes))) | |
1169 | (and (setq deriv (or uinvprime | |
1170 | (calcFunc-deriv uinv | |
1171 | math-integ-var-2 | |
1172 | math-integ-var | |
1173 | (not user)))) | |
1174 | (setq temp (math-integral (math-mul | |
1175 | (math-expr-subst | |
1176 | (math-expr-subst | |
1177 | (math-expr-subst | |
1178 | expr | |
1179 | u | |
1180 | math-integ-var-2) | |
1181 | math-integ-var | |
1182 | uinv) | |
1183 | math-integ-var-2 | |
1184 | math-integ-var) | |
1185 | deriv) | |
1186 | 'yes))))) | |
1187 | (math-simplify-extended | |
bf77c646 | 1188 | (math-expr-subst temp math-integ-var u))))) |
136211a9 EZ |
1189 | |
1190 | ;;; Look for substitutions of the form u = a x + b. | |
1191 | (defun math-integ-try-linear-substitutions (sub-expr) | |
1192 | (and (not (Math-primp sub-expr)) | |
1193 | (or (and (not (memq (car sub-expr) '(+ - * / neg))) | |
1194 | (not (and (eq (car sub-expr) '^) | |
1195 | (integerp (nth 2 sub-expr)))) | |
1196 | (math-expr-contains sub-expr math-integ-var) | |
1197 | (let ((res nil)) | |
1198 | (while (and (setq sub-expr (cdr sub-expr)) | |
1199 | (or (not (math-linear-in (car sub-expr) | |
1200 | math-integ-var)) | |
1201 | (assoc (car sub-expr) so-far) | |
1202 | (progn | |
1203 | (setq so-far (cons (list (car sub-expr)) | |
1204 | so-far)) | |
1205 | (not (setq res | |
1206 | (math-integrate-by-substitution | |
1207 | expr (car sub-expr)))))))) | |
1208 | res)) | |
1209 | (let ((res nil)) | |
1210 | (while (and (setq sub-expr (cdr sub-expr)) | |
1211 | (not (setq res (math-integ-try-linear-substitutions | |
1212 | (car sub-expr)))))) | |
bf77c646 | 1213 | res)))) |
136211a9 EZ |
1214 | |
1215 | ;;; Recursively try different substitutions based on various sub-expressions. | |
1216 | (defun math-integ-try-substitutions (sub-expr &optional allow-rat) | |
1217 | (and (not (Math-primp sub-expr)) | |
1218 | (not (assoc sub-expr so-far)) | |
1219 | (math-expr-contains sub-expr math-integ-var) | |
1220 | (or (and (if (and (not (memq (car sub-expr) '(+ - * / neg))) | |
1221 | (not (and (eq (car sub-expr) '^) | |
1222 | (integerp (nth 2 sub-expr))))) | |
1223 | (setq allow-rat t) | |
1224 | (prog1 allow-rat (setq allow-rat nil))) | |
1225 | (not (eq sub-expr expr)) | |
1226 | (or (math-integrate-by-substitution expr sub-expr) | |
1227 | (and (eq (car sub-expr) '^) | |
1228 | (integerp (nth 2 sub-expr)) | |
1229 | (< (nth 2 sub-expr) 0) | |
1230 | (math-integ-try-substitutions | |
1231 | (math-pow (nth 1 sub-expr) (- (nth 2 sub-expr))) | |
1232 | t)))) | |
1233 | (let ((res nil)) | |
1234 | (setq so-far (cons (list sub-expr) so-far)) | |
1235 | (while (and (setq sub-expr (cdr sub-expr)) | |
1236 | (not (setq res (math-integ-try-substitutions | |
1237 | (car sub-expr) allow-rat))))) | |
bf77c646 | 1238 | res)))) |
136211a9 EZ |
1239 | |
1240 | (defun math-expr-rational-in (expr) | |
1241 | (let ((parts nil)) | |
1242 | (math-expr-rational-in-rec expr) | |
bf77c646 | 1243 | (mapcar 'car parts))) |
136211a9 EZ |
1244 | |
1245 | (defun math-expr-rational-in-rec (expr) | |
1246 | (cond ((Math-primp expr) | |
1247 | (and (equal expr math-integ-var) | |
1248 | (not (assoc expr parts)) | |
1249 | (setq parts (cons (list expr) parts)))) | |
1250 | ((or (memq (car expr) '(+ - * / neg)) | |
1251 | (and (eq (car expr) '^) (integerp (nth 2 expr)))) | |
1252 | (math-expr-rational-in-rec (nth 1 expr)) | |
1253 | (and (nth 2 expr) (math-expr-rational-in-rec (nth 2 expr)))) | |
1254 | ((and (eq (car expr) '^) | |
1255 | (eq (math-quarter-integer (nth 2 expr)) 2)) | |
1256 | (math-expr-rational-in-rec (list 'calcFunc-sqrt (nth 1 expr)))) | |
1257 | (t | |
1258 | (and (not (assoc expr parts)) | |
1259 | (math-expr-contains expr math-integ-var) | |
bf77c646 | 1260 | (setq parts (cons (list expr) parts)))))) |
136211a9 EZ |
1261 | |
1262 | (defun math-expr-calls (expr funcs &optional arg-contains) | |
1263 | (if (consp expr) | |
1264 | (if (or (memq (car expr) funcs) | |
1265 | (and (eq (car expr) '^) (eq (car funcs) 'calcFunc-sqrt) | |
1266 | (eq (math-quarter-integer (nth 2 expr)) 2))) | |
1267 | (and (or (not arg-contains) | |
1268 | (math-expr-contains expr arg-contains)) | |
1269 | expr) | |
1270 | (and (not (Math-primp expr)) | |
1271 | (let ((res nil)) | |
1272 | (while (and (setq expr (cdr expr)) | |
1273 | (not (setq res (math-expr-calls | |
1274 | (car expr) funcs arg-contains))))) | |
bf77c646 | 1275 | res))))) |
136211a9 EZ |
1276 | |
1277 | (defun math-fix-const-terms (expr except-vars) | |
1278 | (cond ((not (math-expr-depends expr except-vars)) 0) | |
1279 | ((Math-primp expr) expr) | |
1280 | ((eq (car expr) '+) | |
1281 | (math-add (math-fix-const-terms (nth 1 expr) except-vars) | |
1282 | (math-fix-const-terms (nth 2 expr) except-vars))) | |
1283 | ((eq (car expr) '-) | |
1284 | (math-sub (math-fix-const-terms (nth 1 expr) except-vars) | |
1285 | (math-fix-const-terms (nth 2 expr) except-vars))) | |
bf77c646 | 1286 | (t expr))) |
136211a9 EZ |
1287 | |
1288 | ;; Command for debugging the Calculator's symbolic integrator. | |
1289 | (defun calc-dump-integral-cache (&optional arg) | |
1290 | (interactive "P") | |
1291 | (let ((buf (current-buffer))) | |
1292 | (unwind-protect | |
1293 | (let ((p math-integral-cache) | |
1294 | cur-record) | |
a1506d29 | 1295 | (display-buffer (get-buffer-create "*Integral Cache*")) |
136211a9 EZ |
1296 | (set-buffer (get-buffer "*Integral Cache*")) |
1297 | (erase-buffer) | |
1298 | (while p | |
1299 | (setq cur-record (car p)) | |
1300 | (or arg (math-replace-integral-parts cur-record)) | |
1301 | (insert (math-format-flat-expr (car cur-record) 0) | |
1302 | " --> " | |
1303 | (if (symbolp (nth 1 cur-record)) | |
1304 | (concat "(" (symbol-name (nth 1 cur-record)) ")") | |
1305 | (math-format-flat-expr (nth 1 cur-record) 0)) | |
1306 | "\n") | |
1307 | (setq p (cdr p))) | |
1308 | (goto-char (point-min))) | |
bf77c646 | 1309 | (set-buffer buf)))) |
136211a9 EZ |
1310 | |
1311 | (defun math-try-integral (expr) | |
1312 | (let ((math-integ-level math-integral-limit) | |
1313 | (math-integ-depth 0) | |
1314 | (math-integ-msg "Working...done") | |
1315 | (cur-record nil) ; a technicality | |
1316 | (math-integrating t) | |
1317 | (calc-prefer-frac t) | |
1318 | (calc-symbolic-mode t) | |
1319 | (has-rules (calc-has-rules 'var-IntegRules))) | |
1320 | (or (math-integral expr 'yes) | |
1321 | (and math-any-substs | |
1322 | (setq math-enable-subst t) | |
1323 | (math-integral expr 'yes)) | |
1324 | (and (> math-max-integral-limit math-integral-limit) | |
1325 | (setq math-integral-limit math-max-integral-limit | |
1326 | math-integ-level math-integral-limit) | |
bf77c646 | 1327 | (math-integral expr 'yes))))) |
136211a9 EZ |
1328 | |
1329 | (defun calcFunc-integ (expr var &optional low high) | |
1330 | (cond | |
1331 | ;; Do these even if the parts turn out not to be integrable. | |
1332 | ((eq (car-safe expr) '+) | |
1333 | (math-add (calcFunc-integ (nth 1 expr) var low high) | |
1334 | (calcFunc-integ (nth 2 expr) var low high))) | |
1335 | ((eq (car-safe expr) '-) | |
1336 | (math-sub (calcFunc-integ (nth 1 expr) var low high) | |
1337 | (calcFunc-integ (nth 2 expr) var low high))) | |
1338 | ((eq (car-safe expr) 'neg) | |
1339 | (math-neg (calcFunc-integ (nth 1 expr) var low high))) | |
1340 | ((and (eq (car-safe expr) '*) | |
1341 | (not (math-expr-contains (nth 1 expr) var))) | |
1342 | (math-mul (nth 1 expr) (calcFunc-integ (nth 2 expr) var low high))) | |
1343 | ((and (eq (car-safe expr) '*) | |
1344 | (not (math-expr-contains (nth 2 expr) var))) | |
1345 | (math-mul (calcFunc-integ (nth 1 expr) var low high) (nth 2 expr))) | |
1346 | ((and (eq (car-safe expr) '/) | |
1347 | (not (math-expr-contains (nth 1 expr) var)) | |
1348 | (not (math-equal-int (nth 1 expr) 1))) | |
1349 | (math-mul (nth 1 expr) | |
1350 | (calcFunc-integ (math-div 1 (nth 2 expr)) var low high))) | |
1351 | ((and (eq (car-safe expr) '/) | |
1352 | (not (math-expr-contains (nth 2 expr) var))) | |
1353 | (math-div (calcFunc-integ (nth 1 expr) var low high) (nth 2 expr))) | |
1354 | ((and (eq (car-safe expr) '/) | |
1355 | (eq (car-safe (nth 1 expr)) '*) | |
1356 | (not (math-expr-contains (nth 1 (nth 1 expr)) var))) | |
1357 | (math-mul (nth 1 (nth 1 expr)) | |
1358 | (calcFunc-integ (math-div (nth 2 (nth 1 expr)) (nth 2 expr)) | |
1359 | var low high))) | |
1360 | ((and (eq (car-safe expr) '/) | |
1361 | (eq (car-safe (nth 1 expr)) '*) | |
1362 | (not (math-expr-contains (nth 2 (nth 1 expr)) var))) | |
1363 | (math-mul (nth 2 (nth 1 expr)) | |
1364 | (calcFunc-integ (math-div (nth 1 (nth 1 expr)) (nth 2 expr)) | |
1365 | var low high))) | |
1366 | ((and (eq (car-safe expr) '/) | |
1367 | (eq (car-safe (nth 2 expr)) '*) | |
1368 | (not (math-expr-contains (nth 1 (nth 2 expr)) var))) | |
1369 | (math-div (calcFunc-integ (math-div (nth 1 expr) (nth 2 (nth 2 expr))) | |
1370 | var low high) | |
1371 | (nth 1 (nth 2 expr)))) | |
1372 | ((and (eq (car-safe expr) '/) | |
1373 | (eq (car-safe (nth 2 expr)) '*) | |
1374 | (not (math-expr-contains (nth 2 (nth 2 expr)) var))) | |
1375 | (math-div (calcFunc-integ (math-div (nth 1 expr) (nth 1 (nth 2 expr))) | |
1376 | var low high) | |
1377 | (nth 2 (nth 2 expr)))) | |
1378 | ((eq (car-safe expr) 'vec) | |
1379 | (cons 'vec (mapcar (function (lambda (x) (calcFunc-integ x var low high))) | |
1380 | (cdr expr)))) | |
1381 | (t | |
1382 | (let ((state (list calc-angle-mode | |
1383 | ;;calc-symbolic-mode | |
1384 | ;;calc-prefer-frac | |
1385 | calc-internal-prec | |
1386 | (calc-var-value 'var-IntegRules) | |
1387 | (calc-var-value 'var-IntegSimpRules)))) | |
1388 | (or (equal state math-integral-cache-state) | |
1389 | (setq math-integral-cache-state state | |
1390 | math-integral-cache nil))) | |
1391 | (let* ((math-max-integral-limit (or (and (boundp 'var-IntegLimit) | |
1392 | (natnump var-IntegLimit) | |
1393 | var-IntegLimit) | |
1394 | 3)) | |
1395 | (math-integral-limit 1) | |
1396 | (sexpr (math-expr-subst expr var math-integ-var)) | |
1397 | (trace-buffer (get-buffer "*Trace*")) | |
1398 | (calc-language (if (eq calc-language 'big) nil calc-language)) | |
1399 | (math-any-substs t) | |
1400 | (math-enable-subst nil) | |
1401 | (math-prev-parts-v nil) | |
1402 | (math-doing-parts nil) | |
1403 | (math-good-parts nil) | |
1404 | (res | |
1405 | (if trace-buffer | |
1406 | (let ((calcbuf (current-buffer)) | |
1407 | (calcwin (selected-window))) | |
1408 | (unwind-protect | |
1409 | (progn | |
1410 | (if (get-buffer-window trace-buffer) | |
1411 | (select-window (get-buffer-window trace-buffer))) | |
1412 | (set-buffer trace-buffer) | |
1413 | (goto-char (point-max)) | |
1414 | (or (assq 'scroll-stop (buffer-local-variables)) | |
1415 | (progn | |
1416 | (make-local-variable 'scroll-step) | |
1417 | (setq scroll-step 3))) | |
1418 | (insert "\n\n\n") | |
1419 | (set-buffer calcbuf) | |
1420 | (math-try-integral sexpr)) | |
1421 | (select-window calcwin) | |
1422 | (set-buffer calcbuf))) | |
1423 | (math-try-integral sexpr)))) | |
1424 | (if res | |
1425 | (progn | |
1426 | (if (calc-has-rules 'var-IntegAfterRules) | |
1427 | (setq res (math-rewrite res '(var IntegAfterRules | |
1428 | var-IntegAfterRules)))) | |
1429 | (math-simplify | |
1430 | (if (and low high) | |
1431 | (math-sub (math-expr-subst res math-integ-var high) | |
1432 | (math-expr-subst res math-integ-var low)) | |
1433 | (setq res (math-fix-const-terms res math-integ-vars)) | |
1434 | (if low | |
1435 | (math-expr-subst res math-integ-var low) | |
1436 | (math-expr-subst res math-integ-var var))))) | |
1437 | (append (list 'calcFunc-integ expr var) | |
1438 | (and low (list low)) | |
bf77c646 | 1439 | (and high (list high)))))))) |
136211a9 EZ |
1440 | |
1441 | ||
1442 | (math-defintegral calcFunc-inv | |
1443 | (math-integral (math-div 1 u))) | |
1444 | ||
1445 | (math-defintegral calcFunc-conj | |
1446 | (let ((int (math-integral u))) | |
1447 | (and int | |
1448 | (list 'calcFunc-conj int)))) | |
1449 | ||
1450 | (math-defintegral calcFunc-deg | |
1451 | (let ((int (math-integral u))) | |
1452 | (and int | |
1453 | (list 'calcFunc-deg int)))) | |
1454 | ||
1455 | (math-defintegral calcFunc-rad | |
1456 | (let ((int (math-integral u))) | |
1457 | (and int | |
1458 | (list 'calcFunc-rad int)))) | |
1459 | ||
1460 | (math-defintegral calcFunc-re | |
1461 | (let ((int (math-integral u))) | |
1462 | (and int | |
1463 | (list 'calcFunc-re int)))) | |
1464 | ||
1465 | (math-defintegral calcFunc-im | |
1466 | (let ((int (math-integral u))) | |
1467 | (and int | |
1468 | (list 'calcFunc-im int)))) | |
1469 | ||
1470 | (math-defintegral calcFunc-sqrt | |
1471 | (and (equal u math-integ-var) | |
1472 | (math-mul '(frac 2 3) | |
1473 | (list 'calcFunc-sqrt (math-pow u 3))))) | |
1474 | ||
1475 | (math-defintegral calcFunc-exp | |
1476 | (or (and (equal u math-integ-var) | |
1477 | (list 'calcFunc-exp u)) | |
1478 | (let ((p (math-is-polynomial u math-integ-var 2))) | |
1479 | (and (nth 2 p) | |
1480 | (let ((sqa (math-sqrt (math-neg (nth 2 p))))) | |
1481 | (math-div | |
1482 | (math-mul | |
1483 | (math-mul (math-div (list 'calcFunc-sqrt '(var pi var-pi)) | |
1484 | sqa) | |
1485 | (math-normalize | |
1486 | (list 'calcFunc-exp | |
1487 | (math-div (math-sub (math-mul (car p) | |
1488 | (nth 2 p)) | |
1489 | (math-div | |
1490 | (math-sqr (nth 1 p)) | |
1491 | 4)) | |
1492 | (nth 2 p))))) | |
1493 | (list 'calcFunc-erf | |
1494 | (math-sub (math-mul sqa math-integ-var) | |
1495 | (math-div (nth 1 p) (math-mul 2 sqa))))) | |
1496 | 2)))))) | |
1497 | ||
1498 | (math-defintegral calcFunc-ln | |
1499 | (or (and (equal u math-integ-var) | |
1500 | (math-sub (math-mul u (list 'calcFunc-ln u)) u)) | |
1501 | (and (eq (car u) '*) | |
1502 | (math-integral (math-add (list 'calcFunc-ln (nth 1 u)) | |
1503 | (list 'calcFunc-ln (nth 2 u))))) | |
1504 | (and (eq (car u) '/) | |
1505 | (math-integral (math-sub (list 'calcFunc-ln (nth 1 u)) | |
1506 | (list 'calcFunc-ln (nth 2 u))))) | |
1507 | (and (eq (car u) '^) | |
1508 | (math-integral (math-mul (nth 2 u) | |
1509 | (list 'calcFunc-ln (nth 1 u))))))) | |
1510 | ||
1511 | (math-defintegral calcFunc-log10 | |
1512 | (and (equal u math-integ-var) | |
1513 | (math-sub (math-mul u (list 'calcFunc-ln u)) | |
1514 | (math-div u (list 'calcFunc-ln 10))))) | |
1515 | ||
1516 | (math-defintegral-2 calcFunc-log | |
1517 | (math-integral (math-div (list 'calcFunc-ln u) | |
1518 | (list 'calcFunc-ln v)))) | |
1519 | ||
1520 | (math-defintegral calcFunc-sin | |
1521 | (or (and (equal u math-integ-var) | |
1522 | (math-neg (math-from-radians-2 (list 'calcFunc-cos u)))) | |
1523 | (and (nth 2 (math-is-polynomial u math-integ-var 2)) | |
1524 | (math-integral (math-to-exponentials (list 'calcFunc-sin u)))))) | |
1525 | ||
1526 | (math-defintegral calcFunc-cos | |
1527 | (or (and (equal u math-integ-var) | |
1528 | (math-from-radians-2 (list 'calcFunc-sin u))) | |
1529 | (and (nth 2 (math-is-polynomial u math-integ-var 2)) | |
1530 | (math-integral (math-to-exponentials (list 'calcFunc-cos u)))))) | |
1531 | ||
1532 | (math-defintegral calcFunc-tan | |
1533 | (and (equal u math-integ-var) | |
1534 | (math-neg (math-from-radians-2 | |
1535 | (list 'calcFunc-ln (list 'calcFunc-cos u)))))) | |
1536 | ||
1537 | (math-defintegral calcFunc-arcsin | |
1538 | (and (equal u math-integ-var) | |
1539 | (math-add (math-mul u (list 'calcFunc-arcsin u)) | |
1540 | (math-from-radians-2 | |
1541 | (list 'calcFunc-sqrt (math-sub 1 (math-sqr u))))))) | |
1542 | ||
1543 | (math-defintegral calcFunc-arccos | |
1544 | (and (equal u math-integ-var) | |
1545 | (math-sub (math-mul u (list 'calcFunc-arccos u)) | |
1546 | (math-from-radians-2 | |
1547 | (list 'calcFunc-sqrt (math-sub 1 (math-sqr u))))))) | |
1548 | ||
1549 | (math-defintegral calcFunc-arctan | |
1550 | (and (equal u math-integ-var) | |
1551 | (math-sub (math-mul u (list 'calcFunc-arctan u)) | |
1552 | (math-from-radians-2 | |
1553 | (math-div (list 'calcFunc-ln (math-add 1 (math-sqr u))) | |
1554 | 2))))) | |
1555 | ||
1556 | (math-defintegral calcFunc-sinh | |
1557 | (and (equal u math-integ-var) | |
1558 | (list 'calcFunc-cosh u))) | |
1559 | ||
1560 | (math-defintegral calcFunc-cosh | |
1561 | (and (equal u math-integ-var) | |
1562 | (list 'calcFunc-sinh u))) | |
1563 | ||
1564 | (math-defintegral calcFunc-tanh | |
1565 | (and (equal u math-integ-var) | |
1566 | (list 'calcFunc-ln (list 'calcFunc-cosh u)))) | |
1567 | ||
1568 | (math-defintegral calcFunc-arcsinh | |
1569 | (and (equal u math-integ-var) | |
1570 | (math-sub (math-mul u (list 'calcFunc-arcsinh u)) | |
1571 | (list 'calcFunc-sqrt (math-add (math-sqr u) 1))))) | |
1572 | ||
1573 | (math-defintegral calcFunc-arccosh | |
1574 | (and (equal u math-integ-var) | |
1575 | (math-sub (math-mul u (list 'calcFunc-arccosh u)) | |
1576 | (list 'calcFunc-sqrt (math-sub 1 (math-sqr u)))))) | |
1577 | ||
1578 | (math-defintegral calcFunc-arctanh | |
1579 | (and (equal u math-integ-var) | |
1580 | (math-sub (math-mul u (list 'calcFunc-arctan u)) | |
1581 | (math-div (list 'calcFunc-ln | |
1582 | (math-add 1 (math-sqr u))) | |
1583 | 2)))) | |
1584 | ||
1585 | ;;; (Ax + B) / (ax^2 + bx + c)^n forms. | |
1586 | (math-defintegral-2 / | |
1587 | (math-integral-rational-funcs u v)) | |
1588 | ||
1589 | (defun math-integral-rational-funcs (u v) | |
1590 | (let ((pu (math-is-polynomial u math-integ-var 1)) | |
1591 | (vpow 1) pv) | |
1592 | (and pu | |
1593 | (catch 'int-rat | |
1594 | (if (and (eq (car-safe v) '^) (natnump (nth 2 v))) | |
1595 | (setq vpow (nth 2 v) | |
1596 | v (nth 1 v))) | |
1597 | (and (setq pv (math-is-polynomial v math-integ-var 2)) | |
1598 | (let ((int (math-mul-thru | |
1599 | (car pu) | |
1600 | (math-integral-q02 (car pv) (nth 1 pv) | |
1601 | (nth 2 pv) v vpow)))) | |
1602 | (if (cdr pu) | |
1603 | (setq int (math-add int | |
1604 | (math-mul-thru | |
1605 | (nth 1 pu) | |
1606 | (math-integral-q12 | |
1607 | (car pv) (nth 1 pv) | |
1608 | (nth 2 pv) v vpow))))) | |
1609 | int)))))) | |
1610 | ||
1611 | (defun math-integral-q12 (a b c v vpow) | |
1612 | (let (q) | |
1613 | (cond ((not c) | |
1614 | (cond ((= vpow 1) | |
1615 | (math-sub (math-div math-integ-var b) | |
1616 | (math-mul (math-div a (math-sqr b)) | |
1617 | (list 'calcFunc-ln v)))) | |
1618 | ((= vpow 2) | |
1619 | (math-div (math-add (list 'calcFunc-ln v) | |
1620 | (math-div a v)) | |
1621 | (math-sqr b))) | |
1622 | (t | |
1623 | (let ((nm1 (math-sub vpow 1)) | |
1624 | (nm2 (math-sub vpow 2))) | |
1625 | (math-div (math-sub | |
1626 | (math-div a (math-mul nm1 (math-pow v nm1))) | |
1627 | (math-div 1 (math-mul nm2 (math-pow v nm2)))) | |
1628 | (math-sqr b)))))) | |
1629 | ((math-zerop | |
1630 | (setq q (math-sub (math-mul 4 (math-mul a c)) (math-sqr b)))) | |
1631 | (let ((part (math-div b (math-mul 2 c)))) | |
1632 | (math-mul-thru (math-pow c vpow) | |
1633 | (math-integral-q12 part 1 nil | |
1634 | (math-add math-integ-var part) | |
1635 | (* vpow 2))))) | |
1636 | ((= vpow 1) | |
1637 | (and (math-ratp q) (math-negp q) | |
1638 | (let ((calc-symbolic-mode t)) | |
1639 | (math-ratp (math-sqrt (math-neg q)))) | |
1640 | (throw 'int-rat nil)) ; should have used calcFunc-apart first | |
1641 | (math-sub (math-div (list 'calcFunc-ln v) (math-mul 2 c)) | |
1642 | (math-mul-thru (math-div b (math-mul 2 c)) | |
1643 | (math-integral-q02 a b c v 1)))) | |
1644 | (t | |
1645 | (let ((n (1- vpow))) | |
1646 | (math-sub (math-neg (math-div | |
1647 | (math-add (math-mul b math-integ-var) | |
1648 | (math-mul 2 a)) | |
1649 | (math-mul n (math-mul q (math-pow v n))))) | |
1650 | (math-mul-thru (math-div (math-mul b (1- (* 2 n))) | |
1651 | (math-mul n q)) | |
bf77c646 | 1652 | (math-integral-q02 a b c v n)))))))) |
136211a9 EZ |
1653 | |
1654 | (defun math-integral-q02 (a b c v vpow) | |
1655 | (let (q rq part) | |
1656 | (cond ((not c) | |
1657 | (cond ((= vpow 1) | |
1658 | (math-div (list 'calcFunc-ln v) b)) | |
1659 | (t | |
1660 | (math-div (math-pow v (- 1 vpow)) | |
1661 | (math-mul (- 1 vpow) b))))) | |
1662 | ((math-zerop | |
1663 | (setq q (math-sub (math-mul 4 (math-mul a c)) (math-sqr b)))) | |
1664 | (let ((part (math-div b (math-mul 2 c)))) | |
1665 | (math-mul-thru (math-pow c vpow) | |
1666 | (math-integral-q02 part 1 nil | |
1667 | (math-add math-integ-var part) | |
1668 | (* vpow 2))))) | |
1669 | ((progn | |
1670 | (setq part (math-add (math-mul 2 (math-mul c math-integ-var)) b)) | |
1671 | (> vpow 1)) | |
1672 | (let ((n (1- vpow))) | |
1673 | (math-add (math-div part (math-mul n (math-mul q (math-pow v n)))) | |
1674 | (math-mul-thru (math-div (math-mul (- (* 4 n) 2) c) | |
1675 | (math-mul n q)) | |
1676 | (math-integral-q02 a b c v n))))) | |
1677 | ((math-guess-if-neg q) | |
1678 | (setq rq (list 'calcFunc-sqrt (math-neg q))) | |
1679 | ;;(math-div-thru (list 'calcFunc-ln | |
1680 | ;; (math-div (math-sub part rq) | |
1681 | ;; (math-add part rq))) | |
1682 | ;; rq) | |
1683 | (math-div (math-mul -2 (list 'calcFunc-arctanh | |
1684 | (math-div part rq))) | |
1685 | rq)) | |
1686 | (t | |
1687 | (setq rq (list 'calcFunc-sqrt q)) | |
1688 | (math-div (math-mul 2 (math-to-radians-2 | |
1689 | (list 'calcFunc-arctan | |
1690 | (math-div part rq)))) | |
bf77c646 | 1691 | rq))))) |
136211a9 EZ |
1692 | |
1693 | ||
1694 | (math-defintegral calcFunc-erf | |
1695 | (and (equal u math-integ-var) | |
1696 | (math-add (math-mul u (list 'calcFunc-erf u)) | |
1697 | (math-div 1 (math-mul (list 'calcFunc-exp (math-sqr u)) | |
1698 | (list 'calcFunc-sqrt | |
1699 | '(var pi var-pi))))))) | |
1700 | ||
1701 | (math-defintegral calcFunc-erfc | |
1702 | (and (equal u math-integ-var) | |
1703 | (math-sub (math-mul u (list 'calcFunc-erfc u)) | |
1704 | (math-div 1 (math-mul (list 'calcFunc-exp (math-sqr u)) | |
1705 | (list 'calcFunc-sqrt | |
1706 | '(var pi var-pi))))))) | |
1707 | ||
1708 | ||
1709 | ||
1710 | ||
3132f345 CW |
1711 | (defvar math-tabulate-initial nil) |
1712 | (defvar math-tabulate-function nil) | |
136211a9 EZ |
1713 | (defun calcFunc-table (expr var &optional low high step) |
1714 | (or low (setq low '(neg (var inf var-inf)) high '(var inf var-inf))) | |
1715 | (or high (setq high low low 1)) | |
1716 | (and (or (math-infinitep low) (math-infinitep high)) | |
1717 | (not step) | |
1718 | (math-scan-for-limits expr)) | |
1719 | (and step (math-zerop step) (math-reject-arg step 'nonzerop)) | |
1720 | (let ((known (+ (if (Math-objectp low) 1 0) | |
1721 | (if (Math-objectp high) 1 0) | |
1722 | (if (or (null step) (Math-objectp step)) 1 0))) | |
1723 | (count '(var inf var-inf)) | |
1724 | vec) | |
1725 | (or (= known 2) ; handy optimization | |
1726 | (equal high '(var inf var-inf)) | |
1727 | (progn | |
1728 | (setq count (math-div (math-sub high low) (or step 1))) | |
1729 | (or (Math-objectp count) | |
1730 | (setq count (math-simplify count))) | |
1731 | (if (Math-messy-integerp count) | |
1732 | (setq count (math-trunc count))))) | |
1733 | (if (Math-negp count) | |
1734 | (setq count -1)) | |
1735 | (if (integerp count) | |
1736 | (let ((var-DUMMY nil) | |
1737 | (vec math-tabulate-initial) | |
1738 | (math-working-step-2 (1+ count)) | |
1739 | (math-working-step 0)) | |
1740 | (setq expr (math-evaluate-expr | |
1741 | (math-expr-subst expr var '(var DUMMY var-DUMMY)))) | |
1742 | (while (>= count 0) | |
1743 | (setq math-working-step (1+ math-working-step) | |
1744 | var-DUMMY low | |
1745 | vec (cond ((eq math-tabulate-function 'calcFunc-sum) | |
1746 | (math-add vec (math-evaluate-expr expr))) | |
1747 | ((eq math-tabulate-function 'calcFunc-prod) | |
1748 | (math-mul vec (math-evaluate-expr expr))) | |
1749 | (t | |
1750 | (cons (math-evaluate-expr expr) vec))) | |
1751 | low (math-add low (or step 1)) | |
1752 | count (1- count))) | |
1753 | (if math-tabulate-function | |
1754 | vec | |
1755 | (cons 'vec (nreverse vec)))) | |
1756 | (if (Math-integerp count) | |
1757 | (calc-record-why 'fixnump high) | |
1758 | (if (Math-num-integerp low) | |
1759 | (if (Math-num-integerp high) | |
1760 | (calc-record-why 'integerp step) | |
1761 | (calc-record-why 'integerp high)) | |
1762 | (calc-record-why 'integerp low))) | |
1763 | (append (list (or math-tabulate-function 'calcFunc-table) | |
1764 | expr var) | |
1765 | (and (not (and (equal low '(neg (var inf var-inf))) | |
1766 | (equal high '(var inf var-inf)))) | |
1767 | (list low high)) | |
bf77c646 | 1768 | (and step (list step)))))) |
136211a9 | 1769 | |
136211a9 EZ |
1770 | (defun math-scan-for-limits (x) |
1771 | (cond ((Math-primp x)) | |
1772 | ((and (eq (car x) 'calcFunc-subscr) | |
1773 | (Math-vectorp (nth 1 x)) | |
1774 | (math-expr-contains (nth 2 x) var)) | |
1775 | (let* ((calc-next-why nil) | |
1776 | (low-val (math-solve-for (nth 2 x) 1 var nil)) | |
1777 | (high-val (math-solve-for (nth 2 x) (1- (length (nth 1 x))) | |
1778 | var nil)) | |
1779 | temp) | |
1780 | (and low-val (math-realp low-val) | |
1781 | high-val (math-realp high-val)) | |
1782 | (and (Math-lessp high-val low-val) | |
1783 | (setq temp low-val low-val high-val high-val temp)) | |
1784 | (setq low (math-max low (math-ceiling low-val)) | |
1785 | high (math-min high (math-floor high-val))))) | |
1786 | (t | |
1787 | (while (setq x (cdr x)) | |
bf77c646 | 1788 | (math-scan-for-limits (car x)))))) |
136211a9 EZ |
1789 | |
1790 | ||
3132f345 | 1791 | (defvar math-disable-sums nil) |
136211a9 EZ |
1792 | (defun calcFunc-sum (expr var &optional low high step) |
1793 | (if math-disable-sums (math-reject-arg)) | |
1794 | (let* ((res (let* ((calc-internal-prec (+ calc-internal-prec 2))) | |
1795 | (math-sum-rec expr var low high step))) | |
1796 | (math-disable-sums t)) | |
bf77c646 | 1797 | (math-normalize res))) |
136211a9 EZ |
1798 | |
1799 | (defun math-sum-rec (expr var &optional low high step) | |
1800 | (or low (setq low '(neg (var inf var-inf)) high '(var inf var-inf))) | |
1801 | (and low (not high) (setq high low low 1)) | |
1802 | (let (t1 t2 val) | |
1803 | (setq val | |
1804 | (cond | |
1805 | ((not (math-expr-contains expr var)) | |
1806 | (math-mul expr (math-add (math-div (math-sub high low) (or step 1)) | |
1807 | 1))) | |
1808 | ((and step (not (math-equal-int step 1))) | |
1809 | (if (math-negp step) | |
1810 | (math-sum-rec expr var high low (math-neg step)) | |
1811 | (let ((lo (math-simplify (math-div low step)))) | |
1812 | (if (math-known-num-integerp lo) | |
1813 | (math-sum-rec (math-normalize | |
1814 | (math-expr-subst expr var | |
1815 | (math-mul step var))) | |
1816 | var lo (math-simplify (math-div high step))) | |
1817 | (math-sum-rec (math-normalize | |
1818 | (math-expr-subst expr var | |
1819 | (math-add (math-mul step var) | |
1820 | low))) | |
1821 | var 0 | |
1822 | (math-simplify (math-div (math-sub high low) | |
1823 | step))))))) | |
1824 | ((memq (setq t1 (math-compare low high)) '(0 1)) | |
1825 | (if (eq t1 0) | |
1826 | (math-expr-subst expr var low) | |
1827 | 0)) | |
1828 | ((setq t1 (math-is-polynomial expr var 20)) | |
1829 | (let ((poly nil) | |
1830 | (n 0)) | |
1831 | (while t1 | |
1832 | (setq poly (math-poly-mix poly 1 | |
1833 | (math-sum-integer-power n) (car t1)) | |
1834 | n (1+ n) | |
1835 | t1 (cdr t1))) | |
1836 | (setq n (math-build-polynomial-expr poly high)) | |
1837 | (if (memq low '(0 1)) | |
1838 | n | |
1839 | (math-sub n (math-build-polynomial-expr poly | |
1840 | (math-sub low 1)))))) | |
1841 | ((and (memq (car expr) '(+ -)) | |
1842 | (setq t1 (math-sum-rec (nth 1 expr) var low high) | |
1843 | t2 (math-sum-rec (nth 2 expr) var low high)) | |
1844 | (not (and (math-expr-calls t1 '(calcFunc-sum)) | |
1845 | (math-expr-calls t2 '(calcFunc-sum))))) | |
1846 | (list (car expr) t1 t2)) | |
1847 | ((and (eq (car expr) '*) | |
1848 | (setq t1 (math-sum-const-factors expr var))) | |
1849 | (math-mul (car t1) (math-sum-rec (cdr t1) var low high))) | |
1850 | ((and (eq (car expr) '*) (memq (car-safe (nth 1 expr)) '(+ -))) | |
1851 | (math-sum-rec (math-add-or-sub (math-mul (nth 1 (nth 1 expr)) | |
1852 | (nth 2 expr)) | |
1853 | (math-mul (nth 2 (nth 1 expr)) | |
1854 | (nth 2 expr)) | |
1855 | nil (eq (car (nth 1 expr)) '-)) | |
1856 | var low high)) | |
1857 | ((and (eq (car expr) '*) (memq (car-safe (nth 2 expr)) '(+ -))) | |
1858 | (math-sum-rec (math-add-or-sub (math-mul (nth 1 expr) | |
1859 | (nth 1 (nth 2 expr))) | |
1860 | (math-mul (nth 1 expr) | |
1861 | (nth 2 (nth 2 expr))) | |
1862 | nil (eq (car (nth 2 expr)) '-)) | |
1863 | var low high)) | |
1864 | ((and (eq (car expr) '/) | |
1865 | (not (math-primp (nth 1 expr))) | |
1866 | (setq t1 (math-sum-const-factors (nth 1 expr) var))) | |
1867 | (math-mul (car t1) | |
1868 | (math-sum-rec (math-div (cdr t1) (nth 2 expr)) | |
1869 | var low high))) | |
1870 | ((and (eq (car expr) '/) | |
1871 | (setq t1 (math-sum-const-factors (nth 2 expr) var))) | |
1872 | (math-div (math-sum-rec (math-div (nth 1 expr) (cdr t1)) | |
1873 | var low high) | |
1874 | (car t1))) | |
1875 | ((eq (car expr) 'neg) | |
1876 | (math-neg (math-sum-rec (nth 1 expr) var low high))) | |
1877 | ((and (eq (car expr) '^) | |
1878 | (not (math-expr-contains (nth 1 expr) var)) | |
1879 | (setq t1 (math-is-polynomial (nth 2 expr) var 1))) | |
1880 | (let ((x (math-pow (nth 1 expr) (nth 1 t1)))) | |
1881 | (math-div (math-mul (math-sub (math-pow x (math-add 1 high)) | |
1882 | (math-pow x low)) | |
1883 | (math-pow (nth 1 expr) (car t1))) | |
1884 | (math-sub x 1)))) | |
1885 | ((and (setq t1 (math-to-exponentials expr)) | |
1886 | (setq t1 (math-sum-rec t1 var low high)) | |
1887 | (not (math-expr-calls t1 '(calcFunc-sum)))) | |
1888 | (math-to-exps t1)) | |
1889 | ((memq (car expr) '(calcFunc-ln calcFunc-log10)) | |
1890 | (list (car expr) (calcFunc-prod (nth 1 expr) var low high))) | |
1891 | ((and (eq (car expr) 'calcFunc-log) | |
1892 | (= (length expr) 3) | |
1893 | (not (math-expr-contains (nth 2 expr) var))) | |
1894 | (list 'calcFunc-log | |
1895 | (calcFunc-prod (nth 1 expr) var low high) | |
1896 | (nth 2 expr))))) | |
1897 | (if (equal val '(var nan var-nan)) (setq val nil)) | |
1898 | (or val | |
1899 | (let* ((math-tabulate-initial 0) | |
1900 | (math-tabulate-function 'calcFunc-sum)) | |
bf77c646 | 1901 | (calcFunc-table expr var low high))))) |
136211a9 EZ |
1902 | |
1903 | (defun calcFunc-asum (expr var low &optional high step no-mul-flag) | |
1904 | (or high (setq high low low 1)) | |
1905 | (if (and step (not (math-equal-int step 1))) | |
1906 | (if (math-negp step) | |
1907 | (math-mul (math-pow -1 low) | |
1908 | (calcFunc-asum expr var high low (math-neg step) t)) | |
1909 | (let ((lo (math-simplify (math-div low step)))) | |
1910 | (if (math-num-integerp lo) | |
1911 | (calcFunc-asum (math-normalize | |
1912 | (math-expr-subst expr var | |
1913 | (math-mul step var))) | |
1914 | var lo (math-simplify (math-div high step))) | |
1915 | (calcFunc-asum (math-normalize | |
1916 | (math-expr-subst expr var | |
1917 | (math-add (math-mul step var) | |
1918 | low))) | |
1919 | var 0 | |
1920 | (math-simplify (math-div (math-sub high low) | |
1921 | step)))))) | |
1922 | (math-mul (if no-mul-flag 1 (math-pow -1 low)) | |
bf77c646 | 1923 | (calcFunc-sum (math-mul (math-pow -1 var) expr) var low high)))) |
136211a9 EZ |
1924 | |
1925 | (defun math-sum-const-factors (expr var) | |
1926 | (let ((const nil) | |
1927 | (not-const nil) | |
1928 | (p expr)) | |
1929 | (while (eq (car-safe p) '*) | |
1930 | (if (math-expr-contains (nth 1 p) var) | |
1931 | (setq not-const (cons (nth 1 p) not-const)) | |
1932 | (setq const (cons (nth 1 p) const))) | |
1933 | (setq p (nth 2 p))) | |
1934 | (if (math-expr-contains p var) | |
1935 | (setq not-const (cons p not-const)) | |
1936 | (setq const (cons p const))) | |
1937 | (and const | |
1938 | (cons (let ((temp (car const))) | |
1939 | (while (setq const (cdr const)) | |
1940 | (setq temp (list '* (car const) temp))) | |
1941 | temp) | |
1942 | (let ((temp (or (car not-const) 1))) | |
1943 | (while (setq not-const (cdr not-const)) | |
1944 | (setq temp (list '* (car not-const) temp))) | |
bf77c646 | 1945 | temp))))) |
136211a9 | 1946 | |
3132f345 | 1947 | (defvar math-sum-int-pow-cache (list '(0 1))) |
136211a9 EZ |
1948 | ;; Following is from CRC Math Tables, 27th ed, pp. 52-53. |
1949 | (defun math-sum-integer-power (pow) | |
1950 | (let ((calc-prefer-frac t) | |
1951 | (n (length math-sum-int-pow-cache))) | |
1952 | (while (<= n pow) | |
1953 | (let* ((new (list 0 0)) | |
1954 | (lin new) | |
1955 | (pp (cdr (nth (1- n) math-sum-int-pow-cache))) | |
1956 | (p 2) | |
1957 | (sum 0) | |
1958 | q) | |
1959 | (while pp | |
1960 | (setq q (math-div (car pp) p) | |
1961 | new (cons (math-mul q n) new) | |
1962 | sum (math-add sum q) | |
1963 | p (1+ p) | |
1964 | pp (cdr pp))) | |
1965 | (setcar lin (math-sub 1 (math-mul n sum))) | |
1966 | (setq math-sum-int-pow-cache | |
1967 | (nconc math-sum-int-pow-cache (list (nreverse new))) | |
1968 | n (1+ n)))) | |
bf77c646 | 1969 | (nth pow math-sum-int-pow-cache))) |
136211a9 EZ |
1970 | |
1971 | (defun math-to-exponentials (expr) | |
1972 | (and (consp expr) | |
1973 | (= (length expr) 2) | |
1974 | (let ((x (nth 1 expr)) | |
1975 | (pi (if calc-symbolic-mode '(var pi var-pi) (math-pi))) | |
1976 | (i (if calc-symbolic-mode '(var i var-i) '(cplx 0 1)))) | |
1977 | (cond ((eq (car expr) 'calcFunc-exp) | |
1978 | (list '^ '(var e var-e) x)) | |
1979 | ((eq (car expr) 'calcFunc-sin) | |
1980 | (or (eq calc-angle-mode 'rad) | |
1981 | (setq x (list '/ (list '* x pi) 180))) | |
1982 | (list '/ (list '- | |
1983 | (list '^ '(var e var-e) (list '* x i)) | |
1984 | (list '^ '(var e var-e) | |
1985 | (list 'neg (list '* x i)))) | |
1986 | (list '* 2 i))) | |
1987 | ((eq (car expr) 'calcFunc-cos) | |
1988 | (or (eq calc-angle-mode 'rad) | |
1989 | (setq x (list '/ (list '* x pi) 180))) | |
1990 | (list '/ (list '+ | |
1991 | (list '^ '(var e var-e) | |
1992 | (list '* x i)) | |
1993 | (list '^ '(var e var-e) | |
1994 | (list 'neg (list '* x i)))) | |
1995 | 2)) | |
1996 | ((eq (car expr) 'calcFunc-sinh) | |
1997 | (list '/ (list '- | |
1998 | (list '^ '(var e var-e) x) | |
1999 | (list '^ '(var e var-e) (list 'neg x))) | |
2000 | 2)) | |
2001 | ((eq (car expr) 'calcFunc-cosh) | |
2002 | (list '/ (list '+ | |
2003 | (list '^ '(var e var-e) x) | |
2004 | (list '^ '(var e var-e) (list 'neg x))) | |
2005 | 2)) | |
bf77c646 | 2006 | (t nil))))) |
136211a9 EZ |
2007 | |
2008 | (defun math-to-exps (expr) | |
2009 | (cond (calc-symbolic-mode expr) | |
2010 | ((Math-primp expr) | |
2011 | (if (equal expr '(var e var-e)) (math-e) expr)) | |
2012 | ((and (eq (car expr) '^) | |
2013 | (equal (nth 1 expr) '(var e var-e))) | |
2014 | (list 'calcFunc-exp (nth 2 expr))) | |
2015 | (t | |
bf77c646 | 2016 | (cons (car expr) (mapcar 'math-to-exps (cdr expr)))))) |
136211a9 EZ |
2017 | |
2018 | ||
3132f345 | 2019 | (defvar math-disable-prods nil) |
136211a9 EZ |
2020 | (defun calcFunc-prod (expr var &optional low high step) |
2021 | (if math-disable-prods (math-reject-arg)) | |
2022 | (let* ((res (let* ((calc-internal-prec (+ calc-internal-prec 2))) | |
2023 | (math-prod-rec expr var low high step))) | |
2024 | (math-disable-prods t)) | |
bf77c646 | 2025 | (math-normalize res))) |
136211a9 EZ |
2026 | |
2027 | (defun math-prod-rec (expr var &optional low high step) | |
2028 | (or low (setq low '(neg (var inf var-inf)) high '(var inf var-inf))) | |
2029 | (and low (not high) (setq high '(var inf var-inf))) | |
2030 | (let (t1 t2 t3 val) | |
2031 | (setq val | |
2032 | (cond | |
2033 | ((not (math-expr-contains expr var)) | |
2034 | (math-pow expr (math-add (math-div (math-sub high low) (or step 1)) | |
2035 | 1))) | |
2036 | ((and step (not (math-equal-int step 1))) | |
2037 | (if (math-negp step) | |
2038 | (math-prod-rec expr var high low (math-neg step)) | |
2039 | (let ((lo (math-simplify (math-div low step)))) | |
2040 | (if (math-known-num-integerp lo) | |
2041 | (math-prod-rec (math-normalize | |
2042 | (math-expr-subst expr var | |
2043 | (math-mul step var))) | |
2044 | var lo (math-simplify (math-div high step))) | |
2045 | (math-prod-rec (math-normalize | |
2046 | (math-expr-subst expr var | |
2047 | (math-add (math-mul step | |
2048 | var) | |
2049 | low))) | |
2050 | var 0 | |
2051 | (math-simplify (math-div (math-sub high low) | |
2052 | step))))))) | |
2053 | ((and (memq (car expr) '(* /)) | |
2054 | (setq t1 (math-prod-rec (nth 1 expr) var low high) | |
2055 | t2 (math-prod-rec (nth 2 expr) var low high)) | |
2056 | (not (and (math-expr-calls t1 '(calcFunc-prod)) | |
2057 | (math-expr-calls t2 '(calcFunc-prod))))) | |
2058 | (list (car expr) t1 t2)) | |
2059 | ((and (eq (car expr) '^) | |
2060 | (not (math-expr-contains (nth 2 expr) var))) | |
2061 | (math-pow (math-prod-rec (nth 1 expr) var low high) | |
2062 | (nth 2 expr))) | |
2063 | ((and (eq (car expr) '^) | |
2064 | (not (math-expr-contains (nth 1 expr) var))) | |
2065 | (math-pow (nth 1 expr) | |
2066 | (calcFunc-sum (nth 2 expr) var low high))) | |
2067 | ((eq (car expr) 'sqrt) | |
2068 | (math-normalize (list 'calcFunc-sqrt | |
2069 | (list 'calcFunc-prod (nth 1 expr) | |
2070 | var low high)))) | |
2071 | ((eq (car expr) 'neg) | |
2072 | (math-mul (math-pow -1 (math-add (math-sub high low) 1)) | |
2073 | (math-prod-rec (nth 1 expr) var low high))) | |
2074 | ((eq (car expr) 'calcFunc-exp) | |
2075 | (list 'calcFunc-exp (calcFunc-sum (nth 1 expr) var low high))) | |
2076 | ((and (setq t1 (math-is-polynomial expr var 1)) | |
2077 | (setq t2 | |
2078 | (cond | |
2079 | ((or (and (math-equal-int (nth 1 t1) 1) | |
2080 | (setq low (math-simplify | |
2081 | (math-add low (car t1))) | |
2082 | high (math-simplify | |
2083 | (math-add high (car t1))))) | |
2084 | (and (math-equal-int (nth 1 t1) -1) | |
2085 | (setq t2 low | |
2086 | low (math-simplify | |
2087 | (math-sub (car t1) high)) | |
2088 | high (math-simplify | |
2089 | (math-sub (car t1) t2))))) | |
2090 | (if (or (math-zerop low) (math-zerop high)) | |
2091 | 0 | |
2092 | (if (and (or (math-negp low) (math-negp high)) | |
2093 | (or (math-num-integerp low) | |
2094 | (math-num-integerp high))) | |
2095 | (if (math-posp high) | |
2096 | 0 | |
2097 | (math-mul (math-pow -1 | |
2098 | (math-add | |
2099 | (math-add low high) 1)) | |
2100 | (list '/ | |
2101 | (list 'calcFunc-fact | |
2102 | (math-neg low)) | |
2103 | (list 'calcFunc-fact | |
2104 | (math-sub -1 high))))) | |
2105 | (list '/ | |
2106 | (list 'calcFunc-fact high) | |
2107 | (list 'calcFunc-fact (math-sub low 1)))))) | |
2108 | ((and (or (and (math-equal-int (nth 1 t1) 2) | |
2109 | (setq t2 (math-simplify | |
2110 | (math-add (math-mul low 2) | |
2111 | (car t1))) | |
2112 | t3 (math-simplify | |
2113 | (math-add (math-mul high 2) | |
2114 | (car t1))))) | |
2115 | (and (math-equal-int (nth 1 t1) -2) | |
2116 | (setq t2 (math-simplify | |
2117 | (math-sub (car t1) | |
2118 | (math-mul high 2))) | |
a1506d29 | 2119 | t3 (math-simplify |
136211a9 EZ |
2120 | (math-sub (car t1) |
2121 | (math-mul low | |
2122 | 2)))))) | |
2123 | (or (math-integerp t2) | |
2124 | (and (math-messy-integerp t2) | |
2125 | (setq t2 (math-trunc t2))) | |
2126 | (math-integerp t3) | |
2127 | (and (math-messy-integerp t3) | |
2128 | (setq t3 (math-trunc t3))))) | |
2129 | (if (or (math-zerop t2) (math-zerop t3)) | |
2130 | 0 | |
2131 | (if (or (math-evenp t2) (math-evenp t3)) | |
2132 | (if (or (math-negp t2) (math-negp t3)) | |
2133 | (if (math-posp high) | |
2134 | 0 | |
2135 | (list '/ | |
2136 | (list 'calcFunc-dfact | |
2137 | (math-neg t2)) | |
2138 | (list 'calcFunc-dfact | |
2139 | (math-sub -2 t3)))) | |
2140 | (list '/ | |
2141 | (list 'calcFunc-dfact t3) | |
2142 | (list 'calcFunc-dfact | |
2143 | (math-sub t2 2)))) | |
2144 | (if (math-negp t3) | |
2145 | (list '* | |
2146 | (list '^ -1 | |
2147 | (list '/ (list '- (list '- t2 t3) | |
2148 | 2) | |
2149 | 2)) | |
2150 | (list '/ | |
2151 | (list 'calcFunc-dfact | |
2152 | (math-neg t2)) | |
2153 | (list 'calcFunc-dfact | |
2154 | (math-sub -2 t3)))) | |
2155 | (if (math-posp t2) | |
2156 | (list '/ | |
2157 | (list 'calcFunc-dfact t3) | |
2158 | (list 'calcFunc-dfact | |
2159 | (math-sub t2 2))) | |
2160 | nil)))))))) | |
2161 | t2))) | |
2162 | (if (equal val '(var nan var-nan)) (setq val nil)) | |
2163 | (or val | |
2164 | (let* ((math-tabulate-initial 1) | |
2165 | (math-tabulate-function 'calcFunc-prod)) | |
bf77c646 | 2166 | (calcFunc-table expr var low high))))) |
136211a9 EZ |
2167 | |
2168 | ||
2169 | ||
2170 | ||
3132f345 | 2171 | (defvar math-solve-ranges nil) |
136211a9 EZ |
2172 | ;;; Attempt to reduce lhs = rhs to solve-var = rhs', where solve-var appears |
2173 | ;;; in lhs but not in rhs or rhs'; return rhs'. | |
2174 | ;;; Uses global values: solve-*. | |
2175 | (defun math-try-solve-for (lhs rhs &optional sign no-poly) | |
2176 | (let (t1 t2 t3) | |
2177 | (cond ((equal lhs solve-var) | |
2178 | (setq math-solve-sign sign) | |
2179 | (if (eq solve-full 'all) | |
2180 | (let ((vec (list 'vec (math-evaluate-expr rhs))) | |
2181 | newvec var p) | |
2182 | (while math-solve-ranges | |
2183 | (setq p (car math-solve-ranges) | |
2184 | var (car p) | |
2185 | newvec (list 'vec)) | |
2186 | (while (setq p (cdr p)) | |
2187 | (setq newvec (nconc newvec | |
2188 | (cdr (math-expr-subst | |
2189 | vec var (car p)))))) | |
2190 | (setq vec newvec | |
2191 | math-solve-ranges (cdr math-solve-ranges))) | |
2192 | (math-normalize vec)) | |
2193 | rhs)) | |
2194 | ((Math-primp lhs) | |
2195 | nil) | |
2196 | ((and (eq (car lhs) '-) | |
2197 | (eq (car-safe (nth 1 lhs)) (car-safe (nth 2 lhs))) | |
2198 | (Math-zerop rhs) | |
2199 | (= (length (nth 1 lhs)) 2) | |
2200 | (= (length (nth 2 lhs)) 2) | |
2201 | (setq t1 (get (car (nth 1 lhs)) 'math-inverse)) | |
2202 | (setq t2 (funcall t1 '(var SOLVEDUM SOLVEDUM))) | |
2203 | (eq (math-expr-contains-count t2 '(var SOLVEDUM SOLVEDUM)) 1) | |
2204 | (setq t3 (math-solve-above-dummy t2)) | |
2205 | (setq t1 (math-try-solve-for (math-sub (nth 1 (nth 1 lhs)) | |
2206 | (math-expr-subst | |
2207 | t2 t3 | |
2208 | (nth 1 (nth 2 lhs)))) | |
2209 | 0))) | |
2210 | t1) | |
2211 | ((eq (car lhs) 'neg) | |
2212 | (math-try-solve-for (nth 1 lhs) (math-neg rhs) | |
2213 | (and sign (- sign)))) | |
2214 | ((and (not (eq solve-full 't)) (math-try-solve-prod))) | |
2215 | ((and (not no-poly) | |
2216 | (setq t2 (math-decompose-poly lhs solve-var 15 rhs))) | |
2217 | (setq t1 (cdr (nth 1 t2)) | |
2218 | t1 (let ((math-solve-ranges math-solve-ranges)) | |
2219 | (cond ((= (length t1) 5) | |
2220 | (apply 'math-solve-quartic (car t2) t1)) | |
2221 | ((= (length t1) 4) | |
2222 | (apply 'math-solve-cubic (car t2) t1)) | |
2223 | ((= (length t1) 3) | |
2224 | (apply 'math-solve-quadratic (car t2) t1)) | |
2225 | ((= (length t1) 2) | |
2226 | (apply 'math-solve-linear (car t2) sign t1)) | |
2227 | (solve-full | |
2228 | (math-poly-all-roots (car t2) t1)) | |
2229 | (calc-symbolic-mode nil) | |
2230 | (t | |
2231 | (math-try-solve-for | |
2232 | (car t2) | |
2233 | (math-poly-any-root (reverse t1) 0 t) | |
2234 | nil t))))) | |
2235 | (if t1 | |
2236 | (if (eq (nth 2 t2) 1) | |
2237 | t1 | |
2238 | (math-solve-prod t1 (math-try-solve-for (nth 2 t2) 0 nil t))) | |
2239 | (calc-record-why "*Unable to find a symbolic solution") | |
2240 | nil)) | |
2241 | ((and (math-solve-find-root-term lhs nil) | |
2242 | (eq (math-expr-contains-count lhs t1) 1)) ; just in case | |
2243 | (math-try-solve-for (math-simplify | |
2244 | (math-sub (if (or t3 (math-evenp t2)) | |
2245 | (math-pow t1 t2) | |
2246 | (math-neg (math-pow t1 t2))) | |
2247 | (math-expand-power | |
2248 | (math-sub (math-normalize | |
2249 | (math-expr-subst | |
2250 | lhs t1 0)) | |
2251 | rhs) | |
2252 | t2 solve-var))) | |
2253 | 0)) | |
2254 | ((eq (car lhs) '+) | |
2255 | (cond ((not (math-expr-contains (nth 1 lhs) solve-var)) | |
2256 | (math-try-solve-for (nth 2 lhs) | |
2257 | (math-sub rhs (nth 1 lhs)) | |
2258 | sign)) | |
2259 | ((not (math-expr-contains (nth 2 lhs) solve-var)) | |
2260 | (math-try-solve-for (nth 1 lhs) | |
2261 | (math-sub rhs (nth 2 lhs)) | |
2262 | sign)))) | |
2263 | ((eq (car lhs) 'calcFunc-eq) | |
2264 | (math-try-solve-for (math-sub (nth 1 lhs) (nth 2 lhs)) | |
2265 | rhs sign no-poly)) | |
2266 | ((eq (car lhs) '-) | |
2267 | (cond ((or (and (eq (car-safe (nth 1 lhs)) 'calcFunc-sin) | |
2268 | (eq (car-safe (nth 2 lhs)) 'calcFunc-cos)) | |
2269 | (and (eq (car-safe (nth 1 lhs)) 'calcFunc-cos) | |
2270 | (eq (car-safe (nth 2 lhs)) 'calcFunc-sin))) | |
2271 | (math-try-solve-for (math-sub (nth 1 lhs) | |
2272 | (list (car (nth 1 lhs)) | |
2273 | (math-sub | |
2274 | (math-quarter-circle t) | |
2275 | (nth 1 (nth 2 lhs))))) | |
2276 | rhs)) | |
2277 | ((not (math-expr-contains (nth 1 lhs) solve-var)) | |
2278 | (math-try-solve-for (nth 2 lhs) | |
2279 | (math-sub (nth 1 lhs) rhs) | |
2280 | (and sign (- sign)))) | |
2281 | ((not (math-expr-contains (nth 2 lhs) solve-var)) | |
2282 | (math-try-solve-for (nth 1 lhs) | |
2283 | (math-add rhs (nth 2 lhs)) | |
2284 | sign)))) | |
2285 | ((and (eq solve-full 't) (math-try-solve-prod))) | |
2286 | ((and (eq (car lhs) '%) | |
2287 | (not (math-expr-contains (nth 2 lhs) solve-var))) | |
2288 | (math-try-solve-for (nth 1 lhs) (math-add rhs | |
2289 | (math-solve-get-int | |
2290 | (nth 2 lhs))))) | |
2291 | ((eq (car lhs) 'calcFunc-log) | |
2292 | (cond ((not (math-expr-contains (nth 2 lhs) solve-var)) | |
2293 | (math-try-solve-for (nth 1 lhs) (math-pow (nth 2 lhs) rhs))) | |
2294 | ((not (math-expr-contains (nth 1 lhs) solve-var)) | |
2295 | (math-try-solve-for (nth 2 lhs) (math-pow | |
2296 | (nth 1 lhs) | |
2297 | (math-div 1 rhs)))))) | |
2298 | ((and (= (length lhs) 2) | |
2299 | (symbolp (car lhs)) | |
2300 | (setq t1 (get (car lhs) 'math-inverse)) | |
2301 | (setq t2 (funcall t1 rhs))) | |
2302 | (setq t1 (get (car lhs) 'math-inverse-sign)) | |
2303 | (math-try-solve-for (nth 1 lhs) (math-normalize t2) | |
2304 | (and sign t1 | |
2305 | (if (integerp t1) | |
2306 | (* t1 sign) | |
2307 | (funcall t1 lhs sign))))) | |
2308 | ((and (symbolp (car lhs)) | |
2309 | (setq t1 (get (car lhs) 'math-inverse-n)) | |
2310 | (setq t2 (funcall t1 lhs rhs))) | |
2311 | t2) | |
2312 | ((setq t1 (math-expand-formula lhs)) | |
2313 | (math-try-solve-for t1 rhs sign)) | |
2314 | (t | |
2315 | (calc-record-why "*No inverse known" lhs) | |
bf77c646 | 2316 | nil)))) |
136211a9 | 2317 | |
136211a9 EZ |
2318 | |
2319 | (defun math-try-solve-prod () | |
2320 | (cond ((eq (car lhs) '*) | |
2321 | (cond ((not (math-expr-contains (nth 1 lhs) solve-var)) | |
2322 | (math-try-solve-for (nth 2 lhs) | |
2323 | (math-div rhs (nth 1 lhs)) | |
2324 | (math-solve-sign sign (nth 1 lhs)))) | |
2325 | ((not (math-expr-contains (nth 2 lhs) solve-var)) | |
2326 | (math-try-solve-for (nth 1 lhs) | |
2327 | (math-div rhs (nth 2 lhs)) | |
2328 | (math-solve-sign sign (nth 2 lhs)))) | |
2329 | ((Math-zerop rhs) | |
2330 | (math-solve-prod (let ((math-solve-ranges math-solve-ranges)) | |
2331 | (math-try-solve-for (nth 2 lhs) 0)) | |
2332 | (math-try-solve-for (nth 1 lhs) 0))))) | |
2333 | ((eq (car lhs) '/) | |
2334 | (cond ((not (math-expr-contains (nth 1 lhs) solve-var)) | |
2335 | (math-try-solve-for (nth 2 lhs) | |
2336 | (math-div (nth 1 lhs) rhs) | |
2337 | (math-solve-sign sign (nth 1 lhs)))) | |
2338 | ((not (math-expr-contains (nth 2 lhs) solve-var)) | |
2339 | (math-try-solve-for (nth 1 lhs) | |
2340 | (math-mul rhs (nth 2 lhs)) | |
2341 | (math-solve-sign sign (nth 2 lhs)))) | |
2342 | ((setq t1 (math-try-solve-for (math-sub (nth 1 lhs) | |
2343 | (math-mul (nth 2 lhs) | |
2344 | rhs)) | |
2345 | 0)) | |
2346 | t1))) | |
2347 | ((eq (car lhs) '^) | |
2348 | (cond ((not (math-expr-contains (nth 1 lhs) solve-var)) | |
2349 | (math-try-solve-for | |
2350 | (nth 2 lhs) | |
2351 | (math-add (math-normalize | |
2352 | (list 'calcFunc-log rhs (nth 1 lhs))) | |
2353 | (math-div | |
2354 | (math-mul 2 | |
2355 | (math-mul '(var pi var-pi) | |
2356 | (math-solve-get-int | |
2357 | '(var i var-i)))) | |
2358 | (math-normalize | |
2359 | (list 'calcFunc-ln (nth 1 lhs))))))) | |
2360 | ((not (math-expr-contains (nth 2 lhs) solve-var)) | |
2361 | (cond ((and (integerp (nth 2 lhs)) | |
2362 | (>= (nth 2 lhs) 2) | |
2363 | (setq t1 (math-integer-log2 (nth 2 lhs)))) | |
2364 | (setq t2 rhs) | |
2365 | (if (and (eq solve-full t) | |
2366 | (math-known-realp (nth 1 lhs))) | |
2367 | (progn | |
2368 | (while (>= (setq t1 (1- t1)) 0) | |
2369 | (setq t2 (list 'calcFunc-sqrt t2))) | |
2370 | (setq t2 (math-solve-get-sign t2))) | |
2371 | (while (>= (setq t1 (1- t1)) 0) | |
2372 | (setq t2 (math-solve-get-sign | |
2373 | (math-normalize | |
2374 | (list 'calcFunc-sqrt t2)))))) | |
2375 | (math-try-solve-for | |
2376 | (nth 1 lhs) | |
2377 | (math-normalize t2))) | |
2378 | ((math-looks-negp (nth 2 lhs)) | |
2379 | (math-try-solve-for | |
2380 | (list '^ (nth 1 lhs) (math-neg (nth 2 lhs))) | |
2381 | (math-div 1 rhs))) | |
2382 | ((and (eq solve-full t) | |
2383 | (Math-integerp (nth 2 lhs)) | |
2384 | (math-known-realp (nth 1 lhs))) | |
2385 | (setq t1 (math-normalize | |
2386 | (list 'calcFunc-nroot rhs (nth 2 lhs)))) | |
2387 | (if (math-evenp (nth 2 lhs)) | |
2388 | (setq t1 (math-solve-get-sign t1))) | |
2389 | (math-try-solve-for | |
2390 | (nth 1 lhs) t1 | |
2391 | (and sign | |
2392 | (math-oddp (nth 2 lhs)) | |
2393 | (math-solve-sign sign (nth 2 lhs))))) | |
2394 | (t (math-try-solve-for | |
2395 | (nth 1 lhs) | |
2396 | (math-mul | |
2397 | (math-normalize | |
2398 | (list 'calcFunc-exp | |
2399 | (if (Math-realp (nth 2 lhs)) | |
2400 | (math-div (math-mul | |
2401 | '(var pi var-pi) | |
2402 | (math-solve-get-int | |
2403 | '(var i var-i) | |
2404 | (and (integerp (nth 2 lhs)) | |
2405 | (math-abs | |
2406 | (nth 2 lhs))))) | |
2407 | (math-div (nth 2 lhs) 2)) | |
2408 | (math-div (math-mul | |
2409 | 2 | |
2410 | (math-mul | |
2411 | '(var pi var-pi) | |
2412 | (math-solve-get-int | |
2413 | '(var i var-i) | |
2414 | (and (integerp (nth 2 lhs)) | |
2415 | (math-abs | |
2416 | (nth 2 lhs)))))) | |
2417 | (nth 2 lhs))))) | |
2418 | (math-normalize | |
2419 | (list 'calcFunc-nroot | |
2420 | rhs | |
2421 | (nth 2 lhs)))) | |
2422 | (and sign | |
2423 | (math-oddp (nth 2 lhs)) | |
2424 | (math-solve-sign sign (nth 2 lhs))))))))) | |
bf77c646 | 2425 | (t nil))) |
136211a9 EZ |
2426 | |
2427 | (defun math-solve-prod (lsoln rsoln) | |
2428 | (cond ((null lsoln) | |
2429 | rsoln) | |
2430 | ((null rsoln) | |
2431 | lsoln) | |
2432 | ((eq solve-full 'all) | |
2433 | (cons 'vec (append (cdr lsoln) (cdr rsoln)))) | |
2434 | (solve-full | |
2435 | (list 'calcFunc-if | |
2436 | (list 'calcFunc-gt (math-solve-get-sign 1) 0) | |
2437 | lsoln | |
2438 | rsoln)) | |
bf77c646 | 2439 | (t lsoln))) |
136211a9 EZ |
2440 | |
2441 | ;;; This deals with negative, fractional, and symbolic powers of "x". | |
2442 | (defun math-solve-poly-funny-powers (sub-rhs) ; uses "t1", "t2" | |
2443 | (setq t1 lhs) | |
2444 | (let ((pp math-poly-neg-powers) | |
2445 | fac) | |
2446 | (while pp | |
2447 | (setq fac (math-pow (car pp) (or math-poly-mult-powers 1)) | |
2448 | t1 (math-mul t1 fac) | |
2449 | rhs (math-mul rhs fac) | |
2450 | pp (cdr pp)))) | |
2451 | (if sub-rhs (setq t1 (math-sub t1 rhs))) | |
2452 | (let ((math-poly-neg-powers nil)) | |
2453 | (setq t2 (math-mul (or math-poly-mult-powers 1) | |
2454 | (let ((calc-prefer-frac t)) | |
2455 | (math-div 1 math-poly-frac-powers))) | |
bf77c646 | 2456 | t1 (math-is-polynomial (math-simplify (calcFunc-expand t1)) b 50)))) |
136211a9 EZ |
2457 | |
2458 | ;;; This converts "a x^8 + b x^5 + c x^2" to "(a (x^3)^2 + b (x^3) + c) * x^2". | |
2459 | (defun math-solve-crunch-poly (max-degree) ; uses "t1", "t3" | |
2460 | (let ((count 0)) | |
2461 | (while (and t1 (Math-zerop (car t1))) | |
2462 | (setq t1 (cdr t1) | |
2463 | count (1+ count))) | |
2464 | (and t1 | |
2465 | (let* ((degree (1- (length t1))) | |
2466 | (scale degree)) | |
2467 | (while (and (> scale 1) (= (car t3) 1)) | |
2468 | (and (= (% degree scale) 0) | |
2469 | (let ((p t1) | |
2470 | (n 0) | |
2471 | (new-t1 nil) | |
2472 | (okay t)) | |
2473 | (while (and p okay) | |
2474 | (if (= (% n scale) 0) | |
2475 | (setq new-t1 (nconc new-t1 (list (car p)))) | |
2476 | (or (Math-zerop (car p)) | |
2477 | (setq okay nil))) | |
2478 | (setq p (cdr p) | |
2479 | n (1+ n))) | |
2480 | (if okay | |
2481 | (setq t3 (cons scale (cdr t3)) | |
2482 | t1 new-t1)))) | |
2483 | (setq scale (1- scale))) | |
2484 | (setq t3 (list (math-mul (car t3) t2) (math-mul count t2))) | |
bf77c646 | 2485 | (<= (1- (length t1)) max-degree))))) |
136211a9 EZ |
2486 | |
2487 | (defun calcFunc-poly (expr var &optional degree) | |
2488 | (if degree | |
2489 | (or (natnump degree) (math-reject-arg degree 'fixnatnump)) | |
2490 | (setq degree 50)) | |
2491 | (let ((p (math-is-polynomial expr var degree 'gen))) | |
2492 | (if p | |
2493 | (if (equal p '(0)) | |
2494 | (list 'vec) | |
2495 | (cons 'vec p)) | |
bf77c646 | 2496 | (math-reject-arg expr "Expected a polynomial")))) |
136211a9 EZ |
2497 | |
2498 | (defun calcFunc-gpoly (expr var &optional degree) | |
2499 | (if degree | |
2500 | (or (natnump degree) (math-reject-arg degree 'fixnatnump)) | |
2501 | (setq degree 50)) | |
2502 | (let* ((math-poly-base-variable var) | |
2503 | (d (math-decompose-poly expr var degree nil))) | |
2504 | (if d | |
2505 | (cons 'vec d) | |
bf77c646 | 2506 | (math-reject-arg expr "Expected a polynomial")))) |
136211a9 EZ |
2507 | |
2508 | (defun math-decompose-poly (lhs solve-var degree sub-rhs) | |
2509 | (let ((rhs (or sub-rhs 1)) | |
2510 | t1 t2 t3) | |
2511 | (setq t2 (math-polynomial-base | |
2512 | lhs | |
2513 | (function | |
2514 | (lambda (b) | |
2515 | (let ((math-poly-neg-powers '(1)) | |
2516 | (math-poly-mult-powers nil) | |
2517 | (math-poly-frac-powers 1) | |
2518 | (math-poly-exp-base t)) | |
2519 | (and (not (equal b lhs)) | |
2520 | (or (not (memq (car-safe b) '(+ -))) sub-rhs) | |
2521 | (setq t3 '(1 0) t2 1 | |
2522 | t1 (math-is-polynomial lhs b 50)) | |
2523 | (if (and (equal math-poly-neg-powers '(1)) | |
2524 | (memq math-poly-mult-powers '(nil 1)) | |
2525 | (eq math-poly-frac-powers 1) | |
2526 | sub-rhs) | |
2527 | (setq t1 (cons (math-sub (car t1) rhs) | |
2528 | (cdr t1))) | |
2529 | (math-solve-poly-funny-powers sub-rhs)) | |
2530 | (math-solve-crunch-poly degree) | |
2531 | (or (math-expr-contains b solve-var) | |
2532 | (math-expr-contains (car t3) solve-var)))))))) | |
2533 | (if t2 | |
2534 | (list (math-pow t2 (car t3)) | |
2535 | (cons 'vec t1) | |
2536 | (if sub-rhs | |
2537 | (math-pow t2 (nth 1 t3)) | |
bf77c646 | 2538 | (math-div (math-pow t2 (nth 1 t3)) rhs)))))) |
136211a9 EZ |
2539 | |
2540 | (defun math-solve-linear (var sign b a) | |
2541 | (math-try-solve-for var | |
2542 | (math-div (math-neg b) a) | |
2543 | (math-solve-sign sign a) | |
bf77c646 | 2544 | t)) |
136211a9 EZ |
2545 | |
2546 | (defun math-solve-quadratic (var c b a) | |
2547 | (math-try-solve-for | |
2548 | var | |
2549 | (if (math-looks-evenp b) | |
2550 | (let ((halfb (math-div b 2))) | |
2551 | (math-div | |
2552 | (math-add | |
2553 | (math-neg halfb) | |
2554 | (math-solve-get-sign | |
2555 | (math-normalize | |
2556 | (list 'calcFunc-sqrt | |
2557 | (math-add (math-sqr halfb) | |
2558 | (math-mul (math-neg c) a)))))) | |
2559 | a)) | |
2560 | (math-div | |
2561 | (math-add | |
2562 | (math-neg b) | |
2563 | (math-solve-get-sign | |
2564 | (math-normalize | |
2565 | (list 'calcFunc-sqrt | |
2566 | (math-add (math-sqr b) | |
2567 | (math-mul 4 (math-mul (math-neg c) a))))))) | |
2568 | (math-mul 2 a))) | |
bf77c646 | 2569 | nil t)) |
136211a9 EZ |
2570 | |
2571 | (defun math-solve-cubic (var d c b a) | |
2572 | (let* ((p (math-div b a)) | |
2573 | (q (math-div c a)) | |
2574 | (r (math-div d a)) | |
2575 | (psqr (math-sqr p)) | |
2576 | (aa (math-sub q (math-div psqr 3))) | |
2577 | (bb (math-add r | |
2578 | (math-div (math-sub (math-mul 2 (math-mul psqr p)) | |
2579 | (math-mul 9 (math-mul p q))) | |
2580 | 27))) | |
2581 | m) | |
2582 | (if (Math-zerop aa) | |
2583 | (math-try-solve-for (math-pow (math-add var (math-div p 3)) 3) | |
2584 | (math-neg bb) nil t) | |
2585 | (if (Math-zerop bb) | |
2586 | (math-try-solve-for | |
2587 | (math-mul (math-add var (math-div p 3)) | |
2588 | (math-add (math-sqr (math-add var (math-div p 3))) | |
2589 | aa)) | |
2590 | 0 nil t) | |
2591 | (setq m (math-mul 2 (list 'calcFunc-sqrt (math-div aa -3)))) | |
2592 | (math-try-solve-for | |
2593 | var | |
2594 | (math-sub | |
2595 | (math-normalize | |
2596 | (math-mul | |
2597 | m | |
2598 | (list 'calcFunc-cos | |
2599 | (math-div | |
2600 | (math-sub (list 'calcFunc-arccos | |
2601 | (math-div (math-mul 3 bb) | |
2602 | (math-mul aa m))) | |
2603 | (math-mul 2 | |
2604 | (math-mul | |
2605 | (math-add 1 (math-solve-get-int | |
2606 | 1 3)) | |
2607 | (math-half-circle | |
2608 | calc-symbolic-mode)))) | |
2609 | 3)))) | |
2610 | (math-div p 3)) | |
bf77c646 | 2611 | nil t))))) |
136211a9 EZ |
2612 | |
2613 | (defun math-solve-quartic (var d c b a aa) | |
2614 | (setq a (math-div a aa)) | |
2615 | (setq b (math-div b aa)) | |
2616 | (setq c (math-div c aa)) | |
2617 | (setq d (math-div d aa)) | |
2618 | (math-try-solve-for | |
2619 | var | |
2620 | (let* ((asqr (math-sqr a)) | |
2621 | (asqr4 (math-div asqr 4)) | |
2622 | (y (let ((solve-full nil) | |
2623 | calc-next-why) | |
2624 | (math-solve-cubic solve-var | |
2625 | (math-sub (math-sub | |
2626 | (math-mul 4 (math-mul b d)) | |
2627 | (math-mul asqr d)) | |
2628 | (math-sqr c)) | |
2629 | (math-sub (math-mul a c) | |
2630 | (math-mul 4 d)) | |
2631 | (math-neg b) | |
2632 | 1))) | |
2633 | (rsqr (math-add (math-sub asqr4 b) y)) | |
2634 | (r (list 'calcFunc-sqrt rsqr)) | |
2635 | (sign1 (math-solve-get-sign 1)) | |
2636 | (de (list 'calcFunc-sqrt | |
2637 | (math-add | |
2638 | (math-sub (math-mul 3 asqr4) | |
2639 | (math-mul 2 b)) | |
2640 | (if (Math-zerop rsqr) | |
2641 | (math-mul | |
2642 | 2 | |
2643 | (math-mul sign1 | |
2644 | (list 'calcFunc-sqrt | |
2645 | (math-sub (math-sqr y) | |
2646 | (math-mul 4 d))))) | |
2647 | (math-sub | |
2648 | (math-mul sign1 | |
2649 | (math-div | |
2650 | (math-sub (math-sub | |
2651 | (math-mul 4 (math-mul a b)) | |
2652 | (math-mul 8 c)) | |
2653 | (math-mul asqr a)) | |
2654 | (math-mul 4 r))) | |
2655 | rsqr)))))) | |
2656 | (math-normalize | |
2657 | (math-sub (math-add (math-mul sign1 (math-div r 2)) | |
2658 | (math-solve-get-sign (math-div de 2))) | |
2659 | (math-div a 4)))) | |
bf77c646 | 2660 | nil t)) |
136211a9 | 2661 | |
3132f345 CW |
2662 | (defvar math-symbolic-solve nil) |
2663 | (defvar math-int-coefs nil) | |
136211a9 EZ |
2664 | (defun math-poly-all-roots (var p &optional math-factoring) |
2665 | (catch 'ouch | |
2666 | (let* ((math-symbolic-solve calc-symbolic-mode) | |
2667 | (roots nil) | |
2668 | (deg (1- (length p))) | |
2669 | (orig-p (reverse p)) | |
2670 | (math-int-coefs nil) | |
2671 | (math-int-scale nil) | |
2672 | (math-double-roots nil) | |
2673 | (math-int-factors nil) | |
2674 | (math-int-threshold nil) | |
2675 | (pp p)) | |
2676 | ;; If rational coefficients, look for exact rational factors. | |
2677 | (while (and pp (Math-ratp (car pp))) | |
2678 | (setq pp (cdr pp))) | |
2679 | (if pp | |
2680 | (if (or math-factoring math-symbolic-solve) | |
2681 | (throw 'ouch nil)) | |
2682 | (let ((lead (car orig-p)) | |
2683 | (calc-prefer-frac t) | |
2684 | (scale (apply 'math-lcm-denoms p))) | |
2685 | (setq math-int-scale (math-abs (math-mul scale lead)) | |
2686 | math-int-threshold (math-div '(float 5 -2) math-int-scale) | |
2687 | math-int-coefs (cdr (math-div (cons 'vec orig-p) lead))))) | |
2688 | (if (> deg 4) | |
2689 | (let ((calc-prefer-frac nil) | |
2690 | (calc-symbolic-mode nil) | |
2691 | (pp p) | |
2692 | (def-p (copy-sequence orig-p))) | |
2693 | (while pp | |
2694 | (if (Math-numberp (car pp)) | |
2695 | (setq pp (cdr pp)) | |
2696 | (throw 'ouch nil))) | |
2697 | (while (> deg (if math-symbolic-solve 2 4)) | |
2698 | (let* ((x (math-poly-any-root def-p '(float 0 0) nil)) | |
2699 | b c pp) | |
2700 | (if (and (eq (car-safe x) 'cplx) | |
2701 | (math-nearly-zerop (nth 2 x) (nth 1 x))) | |
2702 | (setq x (calcFunc-re x))) | |
2703 | (or math-factoring | |
2704 | (setq roots (cons x roots))) | |
2705 | (or (math-numberp x) | |
2706 | (setq x (math-evaluate-expr x))) | |
2707 | (setq pp def-p | |
2708 | b (car def-p)) | |
2709 | (while (setq pp (cdr pp)) | |
2710 | (setq c (car pp)) | |
2711 | (setcar pp b) | |
2712 | (setq b (math-add (math-mul x b) c))) | |
2713 | (setq def-p (cdr def-p) | |
2714 | deg (1- deg)))) | |
2715 | (setq p (reverse def-p)))) | |
2716 | (if (> deg 1) | |
2717 | (let ((solve-var '(var DUMMY var-DUMMY)) | |
2718 | (math-solve-sign nil) | |
2719 | (math-solve-ranges nil) | |
2720 | (solve-full 'all)) | |
2721 | (if (= (length p) (length math-int-coefs)) | |
2722 | (setq p (reverse math-int-coefs))) | |
2723 | (setq roots (append (cdr (apply (cond ((= deg 2) | |
2724 | 'math-solve-quadratic) | |
2725 | ((= deg 3) | |
2726 | 'math-solve-cubic) | |
2727 | (t | |
2728 | 'math-solve-quartic)) | |
2729 | solve-var p)) | |
2730 | roots))) | |
2731 | (if (> deg 0) | |
2732 | (setq roots (cons (math-div (math-neg (car p)) (nth 1 p)) | |
2733 | roots)))) | |
2734 | (if math-factoring | |
2735 | (progn | |
2736 | (while roots | |
2737 | (math-poly-integer-root (car roots)) | |
2738 | (setq roots (cdr roots))) | |
2739 | (list math-int-factors (nreverse math-int-coefs) math-int-scale)) | |
2740 | (let ((vec nil) res) | |
2741 | (while roots | |
2742 | (let ((root (car roots)) | |
2743 | (solve-full (and solve-full 'all))) | |
2744 | (if (math-floatp root) | |
2745 | (setq root (math-poly-any-root orig-p root t))) | |
2746 | (setq vec (append vec | |
2747 | (cdr (or (math-try-solve-for var root nil t) | |
2748 | (throw 'ouch nil)))))) | |
2749 | (setq roots (cdr roots))) | |
2750 | (setq vec (cons 'vec (nreverse vec))) | |
2751 | (if math-symbolic-solve | |
2752 | (setq vec (math-normalize vec))) | |
2753 | (if (eq solve-full t) | |
2754 | (list 'calcFunc-subscr | |
2755 | vec | |
2756 | (math-solve-get-int 1 (1- (length orig-p)) 1)) | |
bf77c646 | 2757 | vec)))))) |
136211a9 EZ |
2758 | |
2759 | (defun math-lcm-denoms (&rest fracs) | |
2760 | (let ((den 1)) | |
2761 | (while fracs | |
2762 | (if (eq (car-safe (car fracs)) 'frac) | |
2763 | (setq den (calcFunc-lcm den (nth 2 (car fracs))))) | |
2764 | (setq fracs (cdr fracs))) | |
bf77c646 | 2765 | den)) |
136211a9 EZ |
2766 | |
2767 | (defun math-poly-any-root (p x polish) ; p is a reverse poly coeff list | |
2768 | (let* ((newt (if (math-zerop x) | |
2769 | (math-poly-newton-root | |
2770 | p '(cplx (float 123 -6) (float 1 -4)) 4) | |
2771 | (math-poly-newton-root p x 4))) | |
2772 | (res (if (math-zerop (cdr newt)) | |
2773 | (car newt) | |
2774 | (if (and (math-lessp (cdr newt) '(float 1 -3)) (not polish)) | |
2775 | (setq newt (math-poly-newton-root p (car newt) 30))) | |
2776 | (if (math-zerop (cdr newt)) | |
2777 | (car newt) | |
2778 | (math-poly-laguerre-root p x polish))))) | |
2779 | (and math-symbolic-solve (math-floatp res) | |
2780 | (throw 'ouch nil)) | |
bf77c646 | 2781 | res)) |
136211a9 EZ |
2782 | |
2783 | (defun math-poly-newton-root (p x iters) | |
2784 | (let* ((calc-prefer-frac nil) | |
2785 | (calc-symbolic-mode nil) | |
2786 | (try-integer math-int-coefs) | |
2787 | (dx x) b d) | |
2788 | (while (and (> (setq iters (1- iters)) 0) | |
2789 | (let ((pp p)) | |
2790 | (math-working "newton" x) | |
2791 | (setq b (car p) | |
2792 | d 0) | |
2793 | (while (setq pp (cdr pp)) | |
2794 | (setq d (math-add (math-mul x d) b) | |
2795 | b (math-add (math-mul x b) (car pp)))) | |
2796 | (not (math-zerop d))) | |
2797 | (progn | |
2798 | (setq dx (math-div b d) | |
2799 | x (math-sub x dx)) | |
2800 | (if try-integer | |
2801 | (let ((adx (math-abs-approx dx))) | |
2802 | (and (math-lessp adx math-int-threshold) | |
2803 | (let ((iroot (math-poly-integer-root x))) | |
2804 | (if iroot | |
2805 | (setq x iroot dx 0) | |
2806 | (setq try-integer nil)))))) | |
2807 | (or (not (or (eq dx 0) | |
2808 | (math-nearly-zerop dx (math-abs-approx x)))) | |
2809 | (progn (setq dx 0) nil))))) | |
2810 | (cons x (if (math-zerop x) | |
bf77c646 | 2811 | 1 (math-div (math-abs-approx dx) (math-abs-approx x)))))) |
136211a9 EZ |
2812 | |
2813 | (defun math-poly-integer-root (x) | |
2814 | (and (math-lessp (calcFunc-xpon (math-abs-approx x)) calc-internal-prec) | |
2815 | math-int-coefs | |
2816 | (let* ((calc-prefer-frac t) | |
2817 | (xre (calcFunc-re x)) | |
2818 | (xim (calcFunc-im x)) | |
2819 | (xresq (math-sqr xre)) | |
2820 | (ximsq (math-sqr xim))) | |
2821 | (if (math-lessp ximsq (calcFunc-scf xresq -1)) | |
2822 | ;; Look for linear factor | |
2823 | (let* ((rnd (math-div (math-round (math-mul xre math-int-scale)) | |
2824 | math-int-scale)) | |
2825 | (icp math-int-coefs) | |
2826 | (rem (car icp)) | |
2827 | (newcoef nil)) | |
2828 | (while (setq icp (cdr icp)) | |
2829 | (setq newcoef (cons rem newcoef) | |
2830 | rem (math-add (car icp) | |
2831 | (math-mul rem rnd)))) | |
2832 | (and (math-zerop rem) | |
2833 | (progn | |
2834 | (setq math-int-coefs (nreverse newcoef) | |
2835 | math-int-factors (cons (list (math-neg rnd)) | |
2836 | math-int-factors)) | |
2837 | rnd))) | |
2838 | ;; Look for irreducible quadratic factor | |
2839 | (let* ((rnd1 (math-div (math-round | |
2840 | (math-mul xre (math-mul -2 math-int-scale))) | |
2841 | math-int-scale)) | |
2842 | (sqscale (math-sqr math-int-scale)) | |
2843 | (rnd0 (math-div (math-round (math-mul (math-add xresq ximsq) | |
2844 | sqscale)) | |
2845 | sqscale)) | |
2846 | (rem1 (car math-int-coefs)) | |
2847 | (icp (cdr math-int-coefs)) | |
2848 | (rem0 (car icp)) | |
2849 | (newcoef nil) | |
2850 | (found (assoc (list rnd0 rnd1 (math-posp xim)) | |
2851 | math-double-roots)) | |
2852 | this) | |
2853 | (if found | |
2854 | (setq math-double-roots (delq found math-double-roots) | |
2855 | rem0 0 rem1 0) | |
2856 | (while (setq icp (cdr icp)) | |
2857 | (setq this rem1 | |
2858 | newcoef (cons rem1 newcoef) | |
2859 | rem1 (math-sub rem0 (math-mul this rnd1)) | |
2860 | rem0 (math-sub (car icp) (math-mul this rnd0))))) | |
2861 | (and (math-zerop rem0) | |
2862 | (math-zerop rem1) | |
2863 | (let ((aa (math-div rnd1 -2))) | |
2864 | (or found (setq math-int-coefs (reverse newcoef) | |
2865 | math-double-roots (cons (list | |
2866 | (list | |
2867 | rnd0 rnd1 | |
2868 | (math-negp xim))) | |
2869 | math-double-roots) | |
2870 | math-int-factors (cons (cons rnd0 rnd1) | |
2871 | math-int-factors))) | |
2872 | (math-add aa | |
2873 | (let ((calc-symbolic-mode math-symbolic-solve)) | |
2874 | (math-mul (math-sqrt (math-sub (math-sqr aa) | |
2875 | rnd0)) | |
bf77c646 | 2876 | (if (math-negp xim) -1 1))))))))))) |
136211a9 EZ |
2877 | |
2878 | ;;; The following routine is from Numerical Recipes, section 9.5. | |
2879 | (defun math-poly-laguerre-root (p x polish) | |
2880 | (let* ((calc-prefer-frac nil) | |
2881 | (calc-symbolic-mode nil) | |
2882 | (iters 0) | |
2883 | (m (1- (length p))) | |
2884 | (try-newt (not polish)) | |
2885 | (tried-newt nil) | |
2886 | b d f x1 dx dxold) | |
2887 | (while | |
2888 | (and (or (< (setq iters (1+ iters)) 50) | |
2889 | (math-reject-arg x "*Laguerre's method failed to converge")) | |
2890 | (let ((err (math-abs-approx (car p))) | |
2891 | (abx (math-abs-approx x)) | |
2892 | (pp p)) | |
2893 | (setq b (car p) | |
2894 | d 0 f 0) | |
2895 | (while (setq pp (cdr pp)) | |
2896 | (setq f (math-add (math-mul x f) d) | |
2897 | d (math-add (math-mul x d) b) | |
2898 | b (math-add (math-mul x b) (car pp)) | |
2899 | err (math-add (math-abs-approx b) (math-mul abx err)))) | |
2900 | (math-lessp (calcFunc-scf err (- -2 calc-internal-prec)) | |
2901 | (math-abs-approx b))) | |
2902 | (or (not (math-zerop d)) | |
2903 | (not (math-zerop f)) | |
2904 | (progn | |
2905 | (setq x (math-pow (math-neg b) (list 'frac 1 m))) | |
2906 | nil)) | |
2907 | (let* ((g (math-div d b)) | |
2908 | (g2 (math-sqr g)) | |
2909 | (h (math-sub g2 (math-mul 2 (math-div f b)))) | |
2910 | (sq (math-sqrt | |
2911 | (math-mul (1- m) (math-sub (math-mul m h) g2)))) | |
2912 | (gp (math-add g sq)) | |
2913 | (gm (math-sub g sq))) | |
2914 | (if (math-lessp (calcFunc-abssqr gp) (calcFunc-abssqr gm)) | |
2915 | (setq gp gm)) | |
2916 | (setq dx (math-div m gp) | |
2917 | x1 (math-sub x dx)) | |
2918 | (if (and try-newt | |
2919 | (math-lessp (math-abs-approx dx) | |
2920 | (calcFunc-scf (math-abs-approx x) -3))) | |
2921 | (let ((newt (math-poly-newton-root p x1 7))) | |
2922 | (setq tried-newt t | |
2923 | try-newt nil) | |
2924 | (if (math-zerop (cdr newt)) | |
2925 | (setq x (car newt) x1 x) | |
2926 | (if (math-lessp (cdr newt) '(float 1 -6)) | |
2927 | (let ((newt2 (math-poly-newton-root | |
2928 | p (car newt) 20))) | |
2929 | (if (math-zerop (cdr newt2)) | |
2930 | (setq x (car newt2) x1 x) | |
2931 | (setq x (car newt)))))))) | |
2932 | (not (or (eq x x1) | |
2933 | (math-nearly-equal x x1)))) | |
2934 | (let ((cdx (math-abs-approx dx))) | |
2935 | (setq x x1 | |
2936 | tried-newt nil) | |
2937 | (prog1 | |
2938 | (or (<= iters 6) | |
2939 | (math-lessp cdx dxold) | |
2940 | (progn | |
2941 | (if polish | |
2942 | (let ((digs (calcFunc-xpon | |
2943 | (math-div (math-abs-approx x) cdx)))) | |
2944 | (calc-record-why | |
2945 | "*Could not attain full precision") | |
2946 | (if (natnump digs) | |
2947 | (let ((calc-internal-prec (max 3 digs))) | |
2948 | (setq x (math-normalize x)))))) | |
2949 | nil)) | |
2950 | (setq dxold cdx))) | |
2951 | (or polish | |
2952 | (math-lessp (calcFunc-scf (math-abs-approx x) | |
2953 | (- calc-internal-prec)) | |
2954 | dxold)))) | |
2955 | (or (and (math-floatp x) | |
2956 | (math-poly-integer-root x)) | |
bf77c646 | 2957 | x))) |
136211a9 EZ |
2958 | |
2959 | (defun math-solve-above-dummy (x) | |
2960 | (and (not (Math-primp x)) | |
2961 | (if (and (equal (nth 1 x) '(var SOLVEDUM SOLVEDUM)) | |
2962 | (= (length x) 2)) | |
2963 | x | |
2964 | (let ((res nil)) | |
2965 | (while (and (setq x (cdr x)) | |
2966 | (not (setq res (math-solve-above-dummy (car x)))))) | |
bf77c646 | 2967 | res)))) |
136211a9 EZ |
2968 | |
2969 | (defun math-solve-find-root-term (x neg) ; sets "t2", "t3" | |
2970 | (if (math-solve-find-root-in-prod x) | |
2971 | (setq t3 neg | |
2972 | t1 x) | |
2973 | (and (memq (car-safe x) '(+ -)) | |
2974 | (or (math-solve-find-root-term (nth 1 x) neg) | |
2975 | (math-solve-find-root-term (nth 2 x) | |
bf77c646 | 2976 | (if (eq (car x) '-) (not neg) neg)))))) |
136211a9 EZ |
2977 | |
2978 | (defun math-solve-find-root-in-prod (x) | |
2979 | (and (consp x) | |
2980 | (math-expr-contains x solve-var) | |
2981 | (or (and (eq (car x) 'calcFunc-sqrt) | |
2982 | (setq t2 2)) | |
2983 | (and (eq (car x) '^) | |
2984 | (or (and (memq (math-quarter-integer (nth 2 x)) '(1 2 3)) | |
2985 | (setq t2 2)) | |
2986 | (and (eq (car-safe (nth 2 x)) 'frac) | |
2987 | (eq (nth 2 (nth 2 x)) 3) | |
2988 | (setq t2 3)))) | |
2989 | (and (memq (car x) '(* /)) | |
2990 | (or (and (not (math-expr-contains (nth 1 x) solve-var)) | |
2991 | (math-solve-find-root-in-prod (nth 2 x))) | |
2992 | (and (not (math-expr-contains (nth 2 x) solve-var)) | |
bf77c646 | 2993 | (math-solve-find-root-in-prod (nth 1 x)))))))) |
136211a9 EZ |
2994 | |
2995 | ||
2996 | (defun math-solve-system (exprs solve-vars solve-full) | |
2997 | (setq exprs (mapcar 'list (if (Math-vectorp exprs) | |
2998 | (cdr exprs) | |
2999 | (list exprs))) | |
3000 | solve-vars (if (Math-vectorp solve-vars) | |
3001 | (cdr solve-vars) | |
3002 | (list solve-vars))) | |
3003 | (or (let ((math-solve-simplifying nil)) | |
3004 | (math-solve-system-rec exprs solve-vars nil)) | |
3005 | (let ((math-solve-simplifying t)) | |
bf77c646 | 3006 | (math-solve-system-rec exprs solve-vars nil)))) |
136211a9 EZ |
3007 | |
3008 | ;;; The following backtracking solver works by choosing a variable | |
3009 | ;;; and equation, and trying to solve the equation for the variable. | |
3010 | ;;; If it succeeds it calls itself recursively with that variable and | |
3011 | ;;; equation removed from their respective lists, and with the solution | |
3012 | ;;; added to solns as well as being substituted into all existing | |
3013 | ;;; equations. The algorithm terminates when any solution path | |
3014 | ;;; manages to remove all the variables from var-list. | |
3015 | ||
3016 | ;;; To support calcFunc-roots, entries in eqn-list and solns are | |
3017 | ;;; actually lists of equations. | |
3018 | ||
3019 | (defun math-solve-system-rec (eqn-list var-list solns) | |
3020 | (if var-list | |
3021 | (let ((v var-list) | |
3022 | (res nil)) | |
3023 | ||
3024 | ;; Try each variable in turn. | |
3025 | (while | |
3026 | (and | |
3027 | v | |
3028 | (let* ((vv (car v)) | |
3029 | (e eqn-list) | |
3030 | (elim (eq (car-safe vv) 'calcFunc-elim))) | |
3031 | (if elim | |
3032 | (setq vv (nth 1 vv))) | |
3033 | ||
3034 | ;; Try each equation in turn. | |
3035 | (while | |
3036 | (and | |
3037 | e | |
3038 | (let ((e2 (car e)) | |
3039 | (eprev nil) | |
3040 | res2) | |
3041 | (setq res nil) | |
3042 | ||
3043 | ;; Try to solve for vv the list of equations e2. | |
3044 | (while (and e2 | |
3045 | (setq res2 (or (and (eq (car e2) eprev) | |
3046 | res2) | |
3047 | (math-solve-for (car e2) 0 vv | |
3048 | solve-full)))) | |
3049 | (setq eprev (car e2) | |
3050 | res (cons (if (eq solve-full 'all) | |
3051 | (cdr res2) | |
3052 | (list res2)) | |
3053 | res) | |
3054 | e2 (cdr e2))) | |
3055 | (if e2 | |
3056 | (setq res nil) | |
3057 | ||
3058 | ;; Found a solution. Now try other variables. | |
3059 | (setq res (nreverse res) | |
3060 | res (math-solve-system-rec | |
3061 | (mapcar | |
3062 | 'math-solve-system-subst | |
3063 | (delq (car e) | |
3064 | (copy-sequence eqn-list))) | |
3065 | (delq (car v) (copy-sequence var-list)) | |
3066 | (let ((math-solve-simplifying nil) | |
3067 | (s (mapcar | |
3068 | (function | |
3069 | (lambda (x) | |
3070 | (cons | |
3071 | (car x) | |
3072 | (math-solve-system-subst | |
3073 | (cdr x))))) | |
3074 | solns))) | |
3075 | (if elim | |
3076 | s | |
3077 | (cons (cons vv (apply 'append res)) | |
3078 | s))))) | |
3079 | (not res)))) | |
3080 | (setq e (cdr e))) | |
3081 | (not res))) | |
3082 | (setq v (cdr v))) | |
3083 | res) | |
3084 | ||
3085 | ;; Eliminated all variables, so now put solution into the proper format. | |
3086 | (setq solns (sort solns | |
3087 | (function | |
3088 | (lambda (x y) | |
3089 | (not (memq (car x) (memq (car y) solve-vars))))))) | |
3090 | (if (eq solve-full 'all) | |
3091 | (math-transpose | |
3092 | (math-normalize | |
3093 | (cons 'vec | |
3094 | (if solns | |
3095 | (mapcar (function (lambda (x) (cons 'vec (cdr x)))) solns) | |
3096 | (mapcar (function (lambda (x) (cons 'vec x))) eqn-list))))) | |
3097 | (math-normalize | |
a1506d29 | 3098 | (cons 'vec |
136211a9 EZ |
3099 | (if solns |
3100 | (mapcar (function (lambda (x) (cons 'calcFunc-eq x))) solns) | |
bf77c646 | 3101 | (mapcar 'car eqn-list))))))) |
136211a9 EZ |
3102 | |
3103 | (defun math-solve-system-subst (x) ; uses "res" and "v" | |
3104 | (let ((accum nil) | |
3105 | (res2 res)) | |
3106 | (while x | |
3107 | (setq accum (nconc accum | |
3108 | (mapcar (function | |
3109 | (lambda (r) | |
3110 | (if math-solve-simplifying | |
3111 | (math-simplify | |
3112 | (math-expr-subst (car x) vv r)) | |
3113 | (math-expr-subst (car x) vv r)))) | |
3114 | (car res2))) | |
3115 | x (cdr x) | |
3116 | res2 (cdr res2))) | |
bf77c646 | 3117 | accum)) |
136211a9 EZ |
3118 | |
3119 | ||
3120 | (defun math-get-from-counter (name) | |
3121 | (let ((ctr (assq name calc-command-flags))) | |
3122 | (if ctr | |
3123 | (setcdr ctr (1+ (cdr ctr))) | |
3124 | (setq ctr (cons name 1) | |
3125 | calc-command-flags (cons ctr calc-command-flags))) | |
bf77c646 | 3126 | (cdr ctr))) |
136211a9 EZ |
3127 | |
3128 | (defun math-solve-get-sign (val) | |
3129 | (setq val (math-simplify val)) | |
3130 | (if (and (eq (car-safe val) '*) | |
3131 | (Math-numberp (nth 1 val))) | |
3132 | (list '* (nth 1 val) (math-solve-get-sign (nth 2 val))) | |
3133 | (and (eq (car-safe val) 'calcFunc-sqrt) | |
3134 | (eq (car-safe (nth 1 val)) '^) | |
3135 | (setq val (math-normalize (list '^ | |
3136 | (nth 1 (nth 1 val)) | |
3137 | (math-div (nth 2 (nth 1 val)) 2))))) | |
3138 | (if solve-full | |
3139 | (if (and (calc-var-value 'var-GenCount) | |
3140 | (Math-natnump var-GenCount) | |
3141 | (not (eq solve-full 'all))) | |
3142 | (prog1 | |
3143 | (math-mul (list 'calcFunc-as var-GenCount) val) | |
3144 | (setq var-GenCount (math-add var-GenCount 1)) | |
3145 | (calc-refresh-evaltos 'var-GenCount)) | |
a9c76ab2 | 3146 | (let* ((var (concat "s" (int-to-string (math-get-from-counter 'solve-sign)))) |
136211a9 EZ |
3147 | (var2 (list 'var (intern var) (intern (concat "var-" var))))) |
3148 | (if (eq solve-full 'all) | |
3149 | (setq math-solve-ranges (cons (list var2 1 -1) | |
3150 | math-solve-ranges))) | |
3151 | (math-mul var2 val))) | |
3152 | (calc-record-why "*Choosing positive solution") | |
bf77c646 | 3153 | val))) |
136211a9 EZ |
3154 | |
3155 | (defun math-solve-get-int (val &optional range first) | |
3156 | (if solve-full | |
3157 | (if (and (calc-var-value 'var-GenCount) | |
3158 | (Math-natnump var-GenCount) | |
3159 | (not (eq solve-full 'all))) | |
3160 | (prog1 | |
3161 | (math-mul val (list 'calcFunc-an var-GenCount)) | |
3162 | (setq var-GenCount (math-add var-GenCount 1)) | |
3163 | (calc-refresh-evaltos 'var-GenCount)) | |
88258edf CW |
3164 | (let* ((var (concat "n" (int-to-string |
3165 | (math-get-from-counter 'solve-int)))) | |
136211a9 EZ |
3166 | (var2 (list 'var (intern var) (intern (concat "var-" var))))) |
3167 | (if (and range (eq solve-full 'all)) | |
3168 | (setq math-solve-ranges (cons (cons var2 | |
3169 | (cdr (calcFunc-index | |
3170 | range (or first 0)))) | |
3171 | math-solve-ranges))) | |
3172 | (math-mul val var2))) | |
3173 | (calc-record-why "*Choosing 0 for arbitrary integer in solution") | |
bf77c646 | 3174 | 0)) |
136211a9 EZ |
3175 | |
3176 | (defun math-solve-sign (sign expr) | |
3177 | (and sign | |
3178 | (let ((s1 (math-possible-signs expr))) | |
3179 | (cond ((memq s1 '(4 6)) | |
3180 | sign) | |
3181 | ((memq s1 '(1 3)) | |
bf77c646 | 3182 | (- sign)))))) |
136211a9 EZ |
3183 | |
3184 | (defun math-looks-evenp (expr) | |
3185 | (if (Math-integerp expr) | |
3186 | (math-evenp expr) | |
3187 | (if (memq (car expr) '(* /)) | |
bf77c646 | 3188 | (math-looks-evenp (nth 1 expr))))) |
136211a9 EZ |
3189 | |
3190 | (defun math-solve-for (lhs rhs solve-var solve-full &optional sign) | |
3191 | (if (math-expr-contains rhs solve-var) | |
3192 | (math-solve-for (math-sub lhs rhs) 0 solve-var solve-full) | |
3193 | (and (math-expr-contains lhs solve-var) | |
3194 | (math-with-extra-prec 1 | |
3195 | (let* ((math-poly-base-variable solve-var) | |
3196 | (res (math-try-solve-for lhs rhs sign))) | |
3197 | (if (and (eq solve-full 'all) | |
3198 | (math-known-realp solve-var)) | |
3199 | (let ((old-len (length res)) | |
3200 | new-len) | |
3201 | (setq res (delq nil | |
3202 | (mapcar (function | |
3203 | (lambda (x) | |
3204 | (and (not (memq (car-safe x) | |
3205 | '(cplx polar))) | |
3206 | x))) | |
3207 | res)) | |
3208 | new-len (length res)) | |
3209 | (if (< new-len old-len) | |
3210 | (calc-record-why (if (= new-len 1) | |
3211 | "*All solutions were complex" | |
3212 | (format | |
3213 | "*Omitted %d complex solutions" | |
3214 | (- old-len new-len))))))) | |
bf77c646 | 3215 | res))))) |
136211a9 EZ |
3216 | |
3217 | (defun math-solve-eqn (expr var full) | |
3218 | (if (memq (car-safe expr) '(calcFunc-neq calcFunc-lt calcFunc-gt | |
3219 | calcFunc-leq calcFunc-geq)) | |
3220 | (let ((res (math-solve-for (cons '- (cdr expr)) | |
3221 | 0 var full | |
3222 | (if (eq (car expr) 'calcFunc-neq) nil 1)))) | |
3223 | (and res | |
3224 | (if (eq math-solve-sign 1) | |
3225 | (list (car expr) var res) | |
3226 | (if (eq math-solve-sign -1) | |
3227 | (list (car expr) res var) | |
3228 | (or (eq (car expr) 'calcFunc-neq) | |
3229 | (calc-record-why | |
3230 | "*Can't determine direction of inequality")) | |
3231 | (and (memq (car expr) '(calcFunc-neq calcFunc-lt calcFunc-gt)) | |
3232 | (list 'calcFunc-neq var res)))))) | |
3233 | (let ((res (math-solve-for expr 0 var full))) | |
3234 | (and res | |
bf77c646 | 3235 | (list 'calcFunc-eq var res))))) |
136211a9 EZ |
3236 | |
3237 | (defun math-reject-solution (expr var func) | |
3238 | (if (math-expr-contains expr var) | |
3239 | (or (equal (car calc-next-why) '(* "Unable to find a symbolic solution")) | |
3240 | (calc-record-why "*Unable to find a solution"))) | |
bf77c646 | 3241 | (list func expr var)) |
136211a9 EZ |
3242 | |
3243 | (defun calcFunc-solve (expr var) | |
3244 | (or (if (or (Math-vectorp expr) (Math-vectorp var)) | |
3245 | (math-solve-system expr var nil) | |
3246 | (math-solve-eqn expr var nil)) | |
bf77c646 | 3247 | (math-reject-solution expr var 'calcFunc-solve))) |
136211a9 EZ |
3248 | |
3249 | (defun calcFunc-fsolve (expr var) | |
3250 | (or (if (or (Math-vectorp expr) (Math-vectorp var)) | |
3251 | (math-solve-system expr var t) | |
3252 | (math-solve-eqn expr var t)) | |
bf77c646 | 3253 | (math-reject-solution expr var 'calcFunc-fsolve))) |
136211a9 EZ |
3254 | |
3255 | (defun calcFunc-roots (expr var) | |
3256 | (let ((math-solve-ranges nil)) | |
3257 | (or (if (or (Math-vectorp expr) (Math-vectorp var)) | |
3258 | (math-solve-system expr var 'all) | |
3259 | (math-solve-for expr 0 var 'all)) | |
bf77c646 | 3260 | (math-reject-solution expr var 'calcFunc-roots)))) |
136211a9 EZ |
3261 | |
3262 | (defun calcFunc-finv (expr var) | |
3263 | (let ((res (math-solve-for expr math-integ-var var nil))) | |
3264 | (if res | |
3265 | (math-normalize (math-expr-subst res math-integ-var var)) | |
bf77c646 | 3266 | (math-reject-solution expr var 'calcFunc-finv)))) |
136211a9 EZ |
3267 | |
3268 | (defun calcFunc-ffinv (expr var) | |
3269 | (let ((res (math-solve-for expr math-integ-var var t))) | |
3270 | (if res | |
3271 | (math-normalize (math-expr-subst res math-integ-var var)) | |
bf77c646 | 3272 | (math-reject-solution expr var 'calcFunc-finv)))) |
136211a9 EZ |
3273 | |
3274 | ||
3275 | (put 'calcFunc-inv 'math-inverse | |
3276 | (function (lambda (x) (math-div 1 x)))) | |
3277 | (put 'calcFunc-inv 'math-inverse-sign -1) | |
3278 | ||
3279 | (put 'calcFunc-sqrt 'math-inverse | |
3280 | (function (lambda (x) (math-sqr x)))) | |
3281 | ||
3282 | (put 'calcFunc-conj 'math-inverse | |
3283 | (function (lambda (x) (list 'calcFunc-conj x)))) | |
3284 | ||
3285 | (put 'calcFunc-abs 'math-inverse | |
3286 | (function (lambda (x) (math-solve-get-sign x)))) | |
3287 | ||
3288 | (put 'calcFunc-deg 'math-inverse | |
3289 | (function (lambda (x) (list 'calcFunc-rad x)))) | |
3290 | (put 'calcFunc-deg 'math-inverse-sign 1) | |
3291 | ||
3292 | (put 'calcFunc-rad 'math-inverse | |
3293 | (function (lambda (x) (list 'calcFunc-deg x)))) | |
3294 | (put 'calcFunc-rad 'math-inverse-sign 1) | |
3295 | ||
3296 | (put 'calcFunc-ln 'math-inverse | |
3297 | (function (lambda (x) (list 'calcFunc-exp x)))) | |
3298 | (put 'calcFunc-ln 'math-inverse-sign 1) | |
3299 | ||
3300 | (put 'calcFunc-log10 'math-inverse | |
3301 | (function (lambda (x) (list 'calcFunc-exp10 x)))) | |
3302 | (put 'calcFunc-log10 'math-inverse-sign 1) | |
3303 | ||
3304 | (put 'calcFunc-lnp1 'math-inverse | |
3305 | (function (lambda (x) (list 'calcFunc-expm1 x)))) | |
3306 | (put 'calcFunc-lnp1 'math-inverse-sign 1) | |
3307 | ||
3308 | (put 'calcFunc-exp 'math-inverse | |
3309 | (function (lambda (x) (math-add (math-normalize (list 'calcFunc-ln x)) | |
3310 | (math-mul 2 | |
3311 | (math-mul '(var pi var-pi) | |
3312 | (math-solve-get-int | |
3313 | '(var i var-i)))))))) | |
3314 | (put 'calcFunc-exp 'math-inverse-sign 1) | |
3315 | ||
3316 | (put 'calcFunc-expm1 'math-inverse | |
3317 | (function (lambda (x) (math-add (math-normalize (list 'calcFunc-lnp1 x)) | |
3318 | (math-mul 2 | |
3319 | (math-mul '(var pi var-pi) | |
3320 | (math-solve-get-int | |
3321 | '(var i var-i)))))))) | |
3322 | (put 'calcFunc-expm1 'math-inverse-sign 1) | |
3323 | ||
3324 | (put 'calcFunc-sin 'math-inverse | |
3325 | (function (lambda (x) (let ((n (math-solve-get-int 1))) | |
3326 | (math-add (math-mul (math-normalize | |
3327 | (list 'calcFunc-arcsin x)) | |
3328 | (math-pow -1 n)) | |
3329 | (math-mul (math-half-circle t) | |
3330 | n)))))) | |
3331 | ||
3332 | (put 'calcFunc-cos 'math-inverse | |
3333 | (function (lambda (x) (math-add (math-solve-get-sign | |
3334 | (math-normalize | |
3335 | (list 'calcFunc-arccos x))) | |
3336 | (math-solve-get-int | |
3337 | (math-full-circle t)))))) | |
3338 | ||
3339 | (put 'calcFunc-tan 'math-inverse | |
3340 | (function (lambda (x) (math-add (math-normalize (list 'calcFunc-arctan x)) | |
3341 | (math-solve-get-int | |
3342 | (math-half-circle t)))))) | |
3343 | ||
3344 | (put 'calcFunc-arcsin 'math-inverse | |
3345 | (function (lambda (x) (math-normalize (list 'calcFunc-sin x))))) | |
3346 | ||
3347 | (put 'calcFunc-arccos 'math-inverse | |
3348 | (function (lambda (x) (math-normalize (list 'calcFunc-cos x))))) | |
3349 | ||
3350 | (put 'calcFunc-arctan 'math-inverse | |
3351 | (function (lambda (x) (math-normalize (list 'calcFunc-tan x))))) | |
3352 | ||
3353 | (put 'calcFunc-sinh 'math-inverse | |
3354 | (function (lambda (x) (let ((n (math-solve-get-int 1))) | |
3355 | (math-add (math-mul (math-normalize | |
3356 | (list 'calcFunc-arcsinh x)) | |
3357 | (math-pow -1 n)) | |
3358 | (math-mul (math-half-circle t) | |
3359 | (math-mul | |
3360 | '(var i var-i) | |
3361 | n))))))) | |
3362 | (put 'calcFunc-sinh 'math-inverse-sign 1) | |
3363 | ||
3364 | (put 'calcFunc-cosh 'math-inverse | |
3365 | (function (lambda (x) (math-add (math-solve-get-sign | |
3366 | (math-normalize | |
3367 | (list 'calcFunc-arccosh x))) | |
3368 | (math-mul (math-full-circle t) | |
3369 | (math-solve-get-int | |
3370 | '(var i var-i))))))) | |
3371 | ||
3372 | (put 'calcFunc-tanh 'math-inverse | |
3373 | (function (lambda (x) (math-add (math-normalize | |
3374 | (list 'calcFunc-arctanh x)) | |
3375 | (math-mul (math-half-circle t) | |
3376 | (math-solve-get-int | |
3377 | '(var i var-i))))))) | |
3378 | (put 'calcFunc-tanh 'math-inverse-sign 1) | |
3379 | ||
3380 | (put 'calcFunc-arcsinh 'math-inverse | |
3381 | (function (lambda (x) (math-normalize (list 'calcFunc-sinh x))))) | |
3382 | (put 'calcFunc-arcsinh 'math-inverse-sign 1) | |
3383 | ||
3384 | (put 'calcFunc-arccosh 'math-inverse | |
3385 | (function (lambda (x) (math-normalize (list 'calcFunc-cosh x))))) | |
3386 | ||
3387 | (put 'calcFunc-arctanh 'math-inverse | |
3388 | (function (lambda (x) (math-normalize (list 'calcFunc-tanh x))))) | |
3389 | (put 'calcFunc-arctanh 'math-inverse-sign 1) | |
3390 | ||
3391 | ||
3392 | ||
3393 | (defun calcFunc-taylor (expr var num) | |
3394 | (let ((x0 0) (v var)) | |
3395 | (if (memq (car-safe var) '(+ - calcFunc-eq)) | |
3396 | (setq x0 (if (eq (car var) '+) (math-neg (nth 2 var)) (nth 2 var)) | |
3397 | v (nth 1 var))) | |
3398 | (or (and (eq (car-safe v) 'var) | |
3399 | (math-expr-contains expr v) | |
3400 | (natnump num) | |
3401 | (let ((accum (math-expr-subst expr v x0)) | |
3402 | (var2 (if (eq (car var) 'calcFunc-eq) | |
3403 | (cons '- (cdr var)) | |
3404 | var)) | |
3405 | (n 0) | |
3406 | (nfac 1) | |
3407 | (fprime expr)) | |
3408 | (while (and (<= (setq n (1+ n)) num) | |
3409 | (setq fprime (calcFunc-deriv fprime v nil t))) | |
3410 | (setq fprime (math-simplify fprime) | |
3411 | nfac (math-mul nfac n) | |
3412 | accum (math-add accum | |
3413 | (math-div (math-mul (math-pow var2 n) | |
3414 | (math-expr-subst | |
3415 | fprime v x0)) | |
3416 | nfac)))) | |
3417 | (and fprime | |
3418 | (math-normalize accum)))) | |
bf77c646 | 3419 | (list 'calcFunc-taylor expr var num)))) |
136211a9 | 3420 | |
bf77c646 | 3421 | ;;; calcalg2.el ends here |