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3132f345 CW |
1 | ;;; calcalg2.el --- more algebraic functions for Calc |
2 | ||
58ba2f8f | 3 | ;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004, |
114f9c96 | 4 | ;; 2005, 2006, 2007, 2008, 2009, 2010 Free Software Foundation, Inc. |
3132f345 CW |
5 | |
6 | ;; Author: David Gillespie <daveg@synaptics.com> | |
e8fff8ed | 7 | ;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com> |
136211a9 EZ |
8 | |
9 | ;; This file is part of GNU Emacs. | |
10 | ||
662c9c64 | 11 | ;; GNU Emacs is free software: you can redistribute it and/or modify |
7c671b23 | 12 | ;; it under the terms of the GNU General Public License as published by |
662c9c64 GM |
13 | ;; the Free Software Foundation, either version 3 of the License, or |
14 | ;; (at your option) any later version. | |
7c671b23 | 15 | |
136211a9 | 16 | ;; GNU Emacs is distributed in the hope that it will be useful, |
7c671b23 GM |
17 | ;; but WITHOUT ANY WARRANTY; without even the implied warranty of |
18 | ;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
19 | ;; GNU General Public License for more details. | |
20 | ||
21 | ;; You should have received a copy of the GNU General Public License | |
662c9c64 | 22 | ;; along with GNU Emacs. If not, see <http://www.gnu.org/licenses/>. |
136211a9 | 23 | |
3132f345 | 24 | ;;; Commentary: |
136211a9 | 25 | |
3132f345 | 26 | ;;; Code: |
136211a9 EZ |
27 | |
28 | ;; This file is autoloaded from calc-ext.el. | |
136211a9 | 29 | |
59f2c6c0 | 30 | (require 'calc-ext) |
136211a9 EZ |
31 | (require 'calc-macs) |
32 | ||
136211a9 EZ |
33 | (defun calc-derivative (var num) |
34 | (interactive "sDifferentiate with respect to: \np") | |
35 | (calc-slow-wrapper | |
3132f345 CW |
36 | (when (< num 0) |
37 | (error "Order of derivative must be positive")) | |
136211a9 EZ |
38 | (let ((func (if (calc-is-hyperbolic) 'calcFunc-tderiv 'calcFunc-deriv)) |
39 | n expr) | |
40 | (if (or (equal var "") (equal var "$")) | |
41 | (setq n 2 | |
42 | expr (calc-top-n 2) | |
43 | var (calc-top-n 1)) | |
44 | (setq var (math-read-expr var)) | |
3132f345 CW |
45 | (when (eq (car-safe var) 'error) |
46 | (error "Bad format in expression: %s" (nth 1 var))) | |
136211a9 EZ |
47 | (setq n 1 |
48 | expr (calc-top-n 1))) | |
49 | (while (>= (setq num (1- num)) 0) | |
50 | (setq expr (list func expr var))) | |
bf77c646 | 51 | (calc-enter-result n "derv" expr)))) |
136211a9 | 52 | |
b682966a JB |
53 | (defun calc-integral (var &optional arg) |
54 | (interactive "sIntegration variable: \nP") | |
55 | (if arg | |
56 | (calc-tabular-command 'calcFunc-integ "Integration" "intg" nil var nil nil) | |
57 | (calc-slow-wrapper | |
58 | (if (or (equal var "") (equal var "$")) | |
59 | (calc-enter-result 2 "intg" (list 'calcFunc-integ | |
60 | (calc-top-n 2) | |
61 | (calc-top-n 1))) | |
62 | (let ((var (math-read-expr var))) | |
63 | (if (eq (car-safe var) 'error) | |
64 | (error "Bad format in expression: %s" (nth 1 var))) | |
65 | (calc-enter-result 1 "intg" (list 'calcFunc-integ | |
66 | (calc-top-n 1) | |
67 | var))))))) | |
136211a9 EZ |
68 | |
69 | (defun calc-num-integral (&optional varname lowname highname) | |
70 | (interactive "sIntegration variable: ") | |
71 | (calc-tabular-command 'calcFunc-ninteg "Integration" "nint" | |
bf77c646 | 72 | nil varname lowname highname)) |
136211a9 EZ |
73 | |
74 | (defun calc-summation (arg &optional varname lowname highname) | |
75 | (interactive "P\nsSummation variable: ") | |
76 | (calc-tabular-command 'calcFunc-sum "Summation" "sum" | |
bf77c646 | 77 | arg varname lowname highname)) |
136211a9 EZ |
78 | |
79 | (defun calc-alt-summation (arg &optional varname lowname highname) | |
80 | (interactive "P\nsSummation variable: ") | |
81 | (calc-tabular-command 'calcFunc-asum "Summation" "asum" | |
bf77c646 | 82 | arg varname lowname highname)) |
136211a9 EZ |
83 | |
84 | (defun calc-product (arg &optional varname lowname highname) | |
85 | (interactive "P\nsIndex variable: ") | |
86 | (calc-tabular-command 'calcFunc-prod "Index" "prod" | |
bf77c646 | 87 | arg varname lowname highname)) |
136211a9 EZ |
88 | |
89 | (defun calc-tabulate (arg &optional varname lowname highname) | |
90 | (interactive "P\nsIndex variable: ") | |
91 | (calc-tabular-command 'calcFunc-table "Index" "tabl" | |
bf77c646 | 92 | arg varname lowname highname)) |
136211a9 EZ |
93 | |
94 | (defun calc-tabular-command (func prompt prefix arg varname lowname highname) | |
95 | (calc-slow-wrapper | |
96 | (let (var (low nil) (high nil) (step nil) stepname stepnum (num 1) expr) | |
97 | (if (consp arg) | |
98 | (setq stepnum 1) | |
99 | (setq stepnum 0)) | |
100 | (if (or (equal varname "") (equal varname "$") (null varname)) | |
101 | (setq high (calc-top-n (+ stepnum 1)) | |
102 | low (calc-top-n (+ stepnum 2)) | |
103 | var (calc-top-n (+ stepnum 3)) | |
104 | num (+ stepnum 4)) | |
105 | (setq var (if (stringp varname) (math-read-expr varname) varname)) | |
106 | (if (eq (car-safe var) 'error) | |
107 | (error "Bad format in expression: %s" (nth 1 var))) | |
108 | (or lowname | |
109 | (setq lowname (read-string (concat prompt " variable: " varname | |
110 | ", from: ")))) | |
111 | (if (or (equal lowname "") (equal lowname "$")) | |
112 | (setq high (calc-top-n (+ stepnum 1)) | |
113 | low (calc-top-n (+ stepnum 2)) | |
114 | num (+ stepnum 3)) | |
115 | (setq low (if (stringp lowname) (math-read-expr lowname) lowname)) | |
116 | (if (eq (car-safe low) 'error) | |
117 | (error "Bad format in expression: %s" (nth 1 low))) | |
118 | (or highname | |
119 | (setq highname (read-string (concat prompt " variable: " varname | |
120 | ", from: " lowname | |
121 | ", to: ")))) | |
122 | (if (or (equal highname "") (equal highname "$")) | |
123 | (setq high (calc-top-n (+ stepnum 1)) | |
124 | num (+ stepnum 2)) | |
125 | (setq high (if (stringp highname) (math-read-expr highname) | |
126 | highname)) | |
127 | (if (eq (car-safe high) 'error) | |
128 | (error "Bad format in expression: %s" (nth 1 high))) | |
129 | (if (consp arg) | |
130 | (progn | |
131 | (setq stepname (read-string (concat prompt " variable: " | |
132 | varname | |
133 | ", from: " lowname | |
134 | ", to: " highname | |
135 | ", step: "))) | |
136 | (if (or (equal stepname "") (equal stepname "$")) | |
137 | (setq step (calc-top-n 1) | |
138 | num 2) | |
139 | (setq step (math-read-expr stepname)) | |
140 | (if (eq (car-safe step) 'error) | |
141 | (error "Bad format in expression: %s" | |
142 | (nth 1 step))))))))) | |
143 | (or step | |
144 | (if (consp arg) | |
145 | (setq step (calc-top-n 1)) | |
146 | (if arg | |
147 | (setq step (prefix-numeric-value arg))))) | |
148 | (setq expr (calc-top-n num)) | |
149 | (calc-enter-result num prefix (append (list func expr var low high) | |
bf77c646 | 150 | (and step (list step))))))) |
136211a9 EZ |
151 | |
152 | (defun calc-solve-for (var) | |
bba984aa | 153 | (interactive "sVariable(s) to solve for: ") |
136211a9 EZ |
154 | (calc-slow-wrapper |
155 | (let ((func (if (calc-is-inverse) | |
156 | (if (calc-is-hyperbolic) 'calcFunc-ffinv 'calcFunc-finv) | |
157 | (if (calc-is-hyperbolic) 'calcFunc-fsolve 'calcFunc-solve)))) | |
158 | (if (or (equal var "") (equal var "$")) | |
159 | (calc-enter-result 2 "solv" (list func | |
160 | (calc-top-n 2) | |
161 | (calc-top-n 1))) | |
162 | (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var) | |
163 | (not (string-match "\\[" var))) | |
164 | (math-read-expr (concat "[" var "]")) | |
165 | (math-read-expr var)))) | |
166 | (if (eq (car-safe var) 'error) | |
167 | (error "Bad format in expression: %s" (nth 1 var))) | |
168 | (calc-enter-result 1 "solv" (list func | |
169 | (calc-top-n 1) | |
bf77c646 | 170 | var))))))) |
136211a9 EZ |
171 | |
172 | (defun calc-poly-roots (var) | |
173 | (interactive "sVariable to solve for: ") | |
174 | (calc-slow-wrapper | |
175 | (if (or (equal var "") (equal var "$")) | |
176 | (calc-enter-result 2 "prts" (list 'calcFunc-roots | |
177 | (calc-top-n 2) | |
178 | (calc-top-n 1))) | |
179 | (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var) | |
180 | (not (string-match "\\[" var))) | |
181 | (math-read-expr (concat "[" var "]")) | |
182 | (math-read-expr var)))) | |
183 | (if (eq (car-safe var) 'error) | |
184 | (error "Bad format in expression: %s" (nth 1 var))) | |
185 | (calc-enter-result 1 "prts" (list 'calcFunc-roots | |
186 | (calc-top-n 1) | |
bf77c646 | 187 | var)))))) |
136211a9 EZ |
188 | |
189 | (defun calc-taylor (var nterms) | |
190 | (interactive "sTaylor expansion variable: \nNNumber of terms: ") | |
191 | (calc-slow-wrapper | |
192 | (let ((var (math-read-expr var))) | |
193 | (if (eq (car-safe var) 'error) | |
194 | (error "Bad format in expression: %s" (nth 1 var))) | |
195 | (calc-enter-result 1 "tylr" (list 'calcFunc-taylor | |
196 | (calc-top-n 1) | |
197 | var | |
bf77c646 | 198 | (prefix-numeric-value nterms)))))) |
136211a9 EZ |
199 | |
200 | ||
f3445fab | 201 | ;; The following are global variables used by math-derivative and some |
f7adda54 JB |
202 | ;; related functions |
203 | (defvar math-deriv-var) | |
204 | (defvar math-deriv-total) | |
205 | (defvar math-deriv-symb) | |
9d871e6d JB |
206 | (defvar math-decls-cache) |
207 | (defvar math-decls-all) | |
f7adda54 JB |
208 | |
209 | (defun math-derivative (expr) | |
210 | (cond ((equal expr math-deriv-var) | |
136211a9 EZ |
211 | 1) |
212 | ((or (Math-scalarp expr) | |
213 | (eq (car expr) 'sdev) | |
214 | (and (eq (car expr) 'var) | |
f7adda54 | 215 | (or (not math-deriv-total) |
136211a9 EZ |
216 | (math-const-var expr) |
217 | (progn | |
218 | (math-setup-declarations) | |
219 | (memq 'const (nth 1 (or (assq (nth 2 expr) | |
220 | math-decls-cache) | |
221 | math-decls-all))))))) | |
222 | 0) | |
223 | ((eq (car expr) '+) | |
224 | (math-add (math-derivative (nth 1 expr)) | |
225 | (math-derivative (nth 2 expr)))) | |
226 | ((eq (car expr) '-) | |
227 | (math-sub (math-derivative (nth 1 expr)) | |
228 | (math-derivative (nth 2 expr)))) | |
229 | ((memq (car expr) '(calcFunc-eq calcFunc-neq calcFunc-lt | |
230 | calcFunc-gt calcFunc-leq calcFunc-geq)) | |
231 | (list (car expr) | |
232 | (math-derivative (nth 1 expr)) | |
233 | (math-derivative (nth 2 expr)))) | |
234 | ((eq (car expr) 'neg) | |
235 | (math-neg (math-derivative (nth 1 expr)))) | |
236 | ((eq (car expr) '*) | |
237 | (math-add (math-mul (nth 2 expr) | |
238 | (math-derivative (nth 1 expr))) | |
239 | (math-mul (nth 1 expr) | |
240 | (math-derivative (nth 2 expr))))) | |
241 | ((eq (car expr) '/) | |
242 | (math-sub (math-div (math-derivative (nth 1 expr)) | |
243 | (nth 2 expr)) | |
244 | (math-div (math-mul (nth 1 expr) | |
245 | (math-derivative (nth 2 expr))) | |
246 | (math-sqr (nth 2 expr))))) | |
247 | ((eq (car expr) '^) | |
248 | (let ((du (math-derivative (nth 1 expr))) | |
249 | (dv (math-derivative (nth 2 expr)))) | |
250 | (or (Math-zerop du) | |
251 | (setq du (math-mul (nth 2 expr) | |
252 | (math-mul (math-normalize | |
253 | (list '^ | |
254 | (nth 1 expr) | |
255 | (math-add (nth 2 expr) -1))) | |
256 | du)))) | |
257 | (or (Math-zerop dv) | |
258 | (setq dv (math-mul (math-normalize | |
259 | (list 'calcFunc-ln (nth 1 expr))) | |
260 | (math-mul expr dv)))) | |
261 | (math-add du dv))) | |
262 | ((eq (car expr) '%) | |
263 | (math-derivative (nth 1 expr))) ; a reasonable definition | |
264 | ((eq (car expr) 'vec) | |
265 | (math-map-vec 'math-derivative expr)) | |
266 | ((and (memq (car expr) '(calcFunc-conj calcFunc-re calcFunc-im)) | |
267 | (= (length expr) 2)) | |
268 | (list (car expr) (math-derivative (nth 1 expr)))) | |
269 | ((and (memq (car expr) '(calcFunc-subscr calcFunc-mrow calcFunc-mcol)) | |
270 | (= (length expr) 3)) | |
271 | (let ((d (math-derivative (nth 1 expr)))) | |
272 | (if (math-numberp d) | |
273 | 0 ; assume x and x_1 are independent vars | |
274 | (list (car expr) d (nth 2 expr))))) | |
275 | (t (or (and (symbolp (car expr)) | |
276 | (if (= (length expr) 2) | |
277 | (let ((handler (get (car expr) 'math-derivative))) | |
278 | (and handler | |
279 | (let ((deriv (math-derivative (nth 1 expr)))) | |
280 | (if (Math-zerop deriv) | |
281 | deriv | |
282 | (math-mul (funcall handler (nth 1 expr)) | |
283 | deriv))))) | |
284 | (let ((handler (get (car expr) 'math-derivative-n))) | |
285 | (and handler | |
286 | (funcall handler expr))))) | |
f7adda54 | 287 | (and (not (eq math-deriv-symb 'pre-expand)) |
136211a9 EZ |
288 | (let ((exp (math-expand-formula expr))) |
289 | (and exp | |
f7adda54 | 290 | (or (let ((math-deriv-symb 'pre-expand)) |
136211a9 EZ |
291 | (catch 'math-deriv (math-derivative expr))) |
292 | (math-derivative exp))))) | |
293 | (if (or (Math-objvecp expr) | |
294 | (eq (car expr) 'var) | |
295 | (not (symbolp (car expr)))) | |
f7adda54 | 296 | (if math-deriv-symb |
136211a9 | 297 | (throw 'math-deriv nil) |
f7adda54 | 298 | (list (if math-deriv-total 'calcFunc-tderiv 'calcFunc-deriv) |
136211a9 | 299 | expr |
f7adda54 | 300 | math-deriv-var)) |
136211a9 EZ |
301 | (let ((accum 0) |
302 | (arg expr) | |
303 | (n 1) | |
304 | derv) | |
305 | (while (setq arg (cdr arg)) | |
306 | (or (Math-zerop (setq derv (math-derivative (car arg)))) | |
307 | (let ((func (intern (concat (symbol-name (car expr)) | |
308 | "'" | |
309 | (if (> n 1) | |
310 | (int-to-string n) | |
311 | "")))) | |
312 | (prop (cond ((= (length expr) 2) | |
313 | 'math-derivative-1) | |
314 | ((= (length expr) 3) | |
315 | 'math-derivative-2) | |
316 | ((= (length expr) 4) | |
317 | 'math-derivative-3) | |
318 | ((= (length expr) 5) | |
319 | 'math-derivative-4) | |
320 | ((= (length expr) 6) | |
321 | 'math-derivative-5)))) | |
322 | (setq accum | |
323 | (math-add | |
324 | accum | |
325 | (math-mul | |
326 | derv | |
327 | (let ((handler (get func prop))) | |
328 | (or (and prop handler | |
329 | (apply handler (cdr expr))) | |
f7adda54 | 330 | (if (and math-deriv-symb |
136211a9 EZ |
331 | (not (get func |
332 | 'calc-user-defn))) | |
333 | (throw 'math-deriv nil) | |
334 | (cons func (cdr expr)))))))))) | |
335 | (setq n (1+ n))) | |
bf77c646 | 336 | accum)))))) |
136211a9 | 337 | |
f7adda54 JB |
338 | (defun calcFunc-deriv (expr math-deriv-var &optional deriv-value math-deriv-symb) |
339 | (let* ((math-deriv-total nil) | |
136211a9 EZ |
340 | (res (catch 'math-deriv (math-derivative expr)))) |
341 | (or (eq (car-safe res) 'calcFunc-deriv) | |
342 | (null res) | |
343 | (setq res (math-normalize res))) | |
344 | (and res | |
345 | (if deriv-value | |
f7adda54 | 346 | (math-expr-subst res math-deriv-var deriv-value) |
bf77c646 | 347 | res)))) |
136211a9 | 348 | |
f7adda54 | 349 | (defun calcFunc-tderiv (expr math-deriv-var &optional deriv-value math-deriv-symb) |
136211a9 | 350 | (math-setup-declarations) |
f7adda54 | 351 | (let* ((math-deriv-total t) |
136211a9 EZ |
352 | (res (catch 'math-deriv (math-derivative expr)))) |
353 | (or (eq (car-safe res) 'calcFunc-tderiv) | |
354 | (null res) | |
355 | (setq res (math-normalize res))) | |
356 | (and res | |
357 | (if deriv-value | |
f7adda54 | 358 | (math-expr-subst res math-deriv-var deriv-value) |
bf77c646 | 359 | res)))) |
136211a9 EZ |
360 | |
361 | (put 'calcFunc-inv\' 'math-derivative-1 | |
362 | (function (lambda (u) (math-neg (math-div 1 (math-sqr u)))))) | |
363 | ||
364 | (put 'calcFunc-sqrt\' 'math-derivative-1 | |
365 | (function (lambda (u) (math-div 1 (math-mul 2 (list 'calcFunc-sqrt u)))))) | |
366 | ||
367 | (put 'calcFunc-deg\' 'math-derivative-1 | |
368 | (function (lambda (u) (math-div-float '(float 18 1) (math-pi))))) | |
369 | ||
370 | (put 'calcFunc-rad\' 'math-derivative-1 | |
371 | (function (lambda (u) (math-pi-over-180)))) | |
372 | ||
373 | (put 'calcFunc-ln\' 'math-derivative-1 | |
374 | (function (lambda (u) (math-div 1 u)))) | |
375 | ||
376 | (put 'calcFunc-log10\' 'math-derivative-1 | |
377 | (function (lambda (u) | |
378 | (math-div (math-div 1 (math-normalize '(calcFunc-ln 10))) | |
379 | u)))) | |
380 | ||
381 | (put 'calcFunc-lnp1\' 'math-derivative-1 | |
382 | (function (lambda (u) (math-div 1 (math-add u 1))))) | |
383 | ||
384 | (put 'calcFunc-log\' 'math-derivative-2 | |
385 | (function (lambda (x b) | |
386 | (and (not (Math-zerop b)) | |
387 | (let ((lnv (math-normalize | |
388 | (list 'calcFunc-ln b)))) | |
389 | (math-div 1 (math-mul lnv x))))))) | |
390 | ||
391 | (put 'calcFunc-log\'2 'math-derivative-2 | |
392 | (function (lambda (x b) | |
393 | (let ((lnv (list 'calcFunc-ln b))) | |
394 | (math-neg (math-div (list 'calcFunc-log x b) | |
395 | (math-mul lnv b))))))) | |
396 | ||
397 | (put 'calcFunc-exp\' 'math-derivative-1 | |
398 | (function (lambda (u) (math-normalize (list 'calcFunc-exp u))))) | |
399 | ||
400 | (put 'calcFunc-expm1\' 'math-derivative-1 | |
401 | (function (lambda (u) (math-normalize (list 'calcFunc-expm1 u))))) | |
402 | ||
403 | (put 'calcFunc-sin\' 'math-derivative-1 | |
404 | (function (lambda (u) (math-to-radians-2 (math-normalize | |
405 | (list 'calcFunc-cos u)))))) | |
406 | ||
407 | (put 'calcFunc-cos\' 'math-derivative-1 | |
408 | (function (lambda (u) (math-neg (math-to-radians-2 | |
409 | (math-normalize | |
410 | (list 'calcFunc-sin u))))))) | |
411 | ||
412 | (put 'calcFunc-tan\' 'math-derivative-1 | |
413 | (function (lambda (u) (math-to-radians-2 | |
dbf954fb JB |
414 | (math-sqr |
415 | (math-normalize | |
416 | (list 'calcFunc-sec u))))))) | |
136211a9 | 417 | |
9274224b | 418 | (put 'calcFunc-sec\' 'math-derivative-1 |
f3445fab | 419 | (function (lambda (u) (math-to-radians-2 |
9274224b JB |
420 | (math-mul |
421 | (math-normalize | |
422 | (list 'calcFunc-sec u)) | |
423 | (math-normalize | |
424 | (list 'calcFunc-tan u))))))) | |
425 | ||
426 | (put 'calcFunc-csc\' 'math-derivative-1 | |
f3445fab | 427 | (function (lambda (u) (math-neg |
9274224b JB |
428 | (math-to-radians-2 |
429 | (math-mul | |
430 | (math-normalize | |
431 | (list 'calcFunc-csc u)) | |
432 | (math-normalize | |
433 | (list 'calcFunc-cot u)))))))) | |
434 | ||
435 | (put 'calcFunc-cot\' 'math-derivative-1 | |
436 | (function (lambda (u) (math-neg | |
437 | (math-to-radians-2 | |
dbf954fb JB |
438 | (math-sqr |
439 | (math-normalize | |
440 | (list 'calcFunc-csc u)))))))) | |
9274224b | 441 | |
136211a9 EZ |
442 | (put 'calcFunc-arcsin\' 'math-derivative-1 |
443 | (function (lambda (u) | |
444 | (math-from-radians-2 | |
445 | (math-div 1 (math-normalize | |
446 | (list 'calcFunc-sqrt | |
447 | (math-sub 1 (math-sqr u))))))))) | |
448 | ||
449 | (put 'calcFunc-arccos\' 'math-derivative-1 | |
450 | (function (lambda (u) | |
451 | (math-from-radians-2 | |
452 | (math-div -1 (math-normalize | |
453 | (list 'calcFunc-sqrt | |
454 | (math-sub 1 (math-sqr u))))))))) | |
455 | ||
456 | (put 'calcFunc-arctan\' 'math-derivative-1 | |
457 | (function (lambda (u) (math-from-radians-2 | |
458 | (math-div 1 (math-add 1 (math-sqr u))))))) | |
459 | ||
460 | (put 'calcFunc-sinh\' 'math-derivative-1 | |
461 | (function (lambda (u) (math-normalize (list 'calcFunc-cosh u))))) | |
462 | ||
463 | (put 'calcFunc-cosh\' 'math-derivative-1 | |
464 | (function (lambda (u) (math-normalize (list 'calcFunc-sinh u))))) | |
465 | ||
466 | (put 'calcFunc-tanh\' 'math-derivative-1 | |
dbf954fb JB |
467 | (function (lambda (u) (math-sqr |
468 | (math-normalize | |
469 | (list 'calcFunc-sech u)))))) | |
136211a9 | 470 | |
9274224b JB |
471 | (put 'calcFunc-sech\' 'math-derivative-1 |
472 | (function (lambda (u) (math-neg | |
473 | (math-mul | |
474 | (math-normalize (list 'calcFunc-sech u)) | |
475 | (math-normalize (list 'calcFunc-tanh u))))))) | |
476 | ||
477 | (put 'calcFunc-csch\' 'math-derivative-1 | |
478 | (function (lambda (u) (math-neg | |
479 | (math-mul | |
480 | (math-normalize (list 'calcFunc-csch u)) | |
481 | (math-normalize (list 'calcFunc-coth u))))))) | |
482 | ||
dbf954fb | 483 | (put 'calcFunc-coth\' 'math-derivative-1 |
9274224b | 484 | (function (lambda (u) (math-neg |
dbf954fb JB |
485 | (math-sqr |
486 | (math-normalize | |
487 | (list 'calcFunc-csch u))))))) | |
9274224b | 488 | |
136211a9 EZ |
489 | (put 'calcFunc-arcsinh\' 'math-derivative-1 |
490 | (function (lambda (u) | |
491 | (math-div 1 (math-normalize | |
492 | (list 'calcFunc-sqrt | |
493 | (math-add (math-sqr u) 1))))))) | |
494 | ||
495 | (put 'calcFunc-arccosh\' 'math-derivative-1 | |
496 | (function (lambda (u) | |
497 | (math-div 1 (math-normalize | |
498 | (list 'calcFunc-sqrt | |
499 | (math-add (math-sqr u) -1))))))) | |
500 | ||
501 | (put 'calcFunc-arctanh\' 'math-derivative-1 | |
502 | (function (lambda (u) (math-div 1 (math-sub 1 (math-sqr u)))))) | |
503 | ||
504 | (put 'calcFunc-bern\'2 'math-derivative-2 | |
505 | (function (lambda (n x) | |
506 | (math-mul n (list 'calcFunc-bern (math-add n -1) x))))) | |
507 | ||
508 | (put 'calcFunc-euler\'2 'math-derivative-2 | |
509 | (function (lambda (n x) | |
510 | (math-mul n (list 'calcFunc-euler (math-add n -1) x))))) | |
511 | ||
512 | (put 'calcFunc-gammag\'2 'math-derivative-2 | |
513 | (function (lambda (a x) (math-deriv-gamma a x 1)))) | |
514 | ||
515 | (put 'calcFunc-gammaG\'2 'math-derivative-2 | |
516 | (function (lambda (a x) (math-deriv-gamma a x -1)))) | |
517 | ||
518 | (put 'calcFunc-gammaP\'2 'math-derivative-2 | |
519 | (function (lambda (a x) (math-deriv-gamma a x | |
520 | (math-div | |
521 | 1 (math-normalize | |
522 | (list 'calcFunc-gamma | |
523 | a))))))) | |
524 | ||
525 | (put 'calcFunc-gammaQ\'2 'math-derivative-2 | |
526 | (function (lambda (a x) (math-deriv-gamma a x | |
527 | (math-div | |
528 | -1 (math-normalize | |
529 | (list 'calcFunc-gamma | |
530 | a))))))) | |
531 | ||
532 | (defun math-deriv-gamma (a x scale) | |
533 | (math-mul scale | |
534 | (math-mul (math-pow x (math-add a -1)) | |
bf77c646 | 535 | (list 'calcFunc-exp (math-neg x))))) |
136211a9 EZ |
536 | |
537 | (put 'calcFunc-betaB\' 'math-derivative-3 | |
538 | (function (lambda (x a b) (math-deriv-beta x a b 1)))) | |
539 | ||
540 | (put 'calcFunc-betaI\' 'math-derivative-3 | |
541 | (function (lambda (x a b) (math-deriv-beta x a b | |
542 | (math-div | |
543 | 1 (list 'calcFunc-beta | |
544 | a b)))))) | |
545 | ||
546 | (defun math-deriv-beta (x a b scale) | |
547 | (math-mul (math-mul (math-pow x (math-add a -1)) | |
548 | (math-pow (math-sub 1 x) (math-add b -1))) | |
bf77c646 | 549 | scale)) |
136211a9 EZ |
550 | |
551 | (put 'calcFunc-erf\' 'math-derivative-1 | |
552 | (function (lambda (x) (math-div 2 | |
553 | (math-mul (list 'calcFunc-exp | |
554 | (math-sqr x)) | |
555 | (if calc-symbolic-mode | |
556 | '(calcFunc-sqrt | |
557 | (var pi var-pi)) | |
558 | (math-sqrt-pi))))))) | |
559 | ||
560 | (put 'calcFunc-erfc\' 'math-derivative-1 | |
561 | (function (lambda (x) (math-div -2 | |
562 | (math-mul (list 'calcFunc-exp | |
563 | (math-sqr x)) | |
564 | (if calc-symbolic-mode | |
565 | '(calcFunc-sqrt | |
566 | (var pi var-pi)) | |
567 | (math-sqrt-pi))))))) | |
568 | ||
569 | (put 'calcFunc-besJ\'2 'math-derivative-2 | |
570 | (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besJ | |
571 | (math-add v -1) | |
572 | z) | |
573 | (list 'calcFunc-besJ | |
574 | (math-add v 1) | |
575 | z)) | |
576 | 2)))) | |
577 | ||
578 | (put 'calcFunc-besY\'2 'math-derivative-2 | |
579 | (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besY | |
580 | (math-add v -1) | |
581 | z) | |
582 | (list 'calcFunc-besY | |
583 | (math-add v 1) | |
584 | z)) | |
585 | 2)))) | |
586 | ||
587 | (put 'calcFunc-sum 'math-derivative-n | |
588 | (function | |
589 | (lambda (expr) | |
f7adda54 | 590 | (if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var) |
136211a9 EZ |
591 | (throw 'math-deriv nil) |
592 | (cons 'calcFunc-sum | |
593 | (cons (math-derivative (nth 1 expr)) | |
594 | (cdr (cdr expr)))))))) | |
595 | ||
596 | (put 'calcFunc-prod 'math-derivative-n | |
597 | (function | |
598 | (lambda (expr) | |
f7adda54 | 599 | (if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var) |
136211a9 EZ |
600 | (throw 'math-deriv nil) |
601 | (math-mul expr | |
602 | (cons 'calcFunc-sum | |
603 | (cons (math-div (math-derivative (nth 1 expr)) | |
604 | (nth 1 expr)) | |
605 | (cdr (cdr expr))))))))) | |
606 | ||
607 | (put 'calcFunc-integ 'math-derivative-n | |
608 | (function | |
609 | (lambda (expr) | |
610 | (if (= (length expr) 3) | |
f7adda54 | 611 | (if (equal (nth 2 expr) math-deriv-var) |
136211a9 EZ |
612 | (nth 1 expr) |
613 | (math-normalize | |
614 | (list 'calcFunc-integ | |
615 | (math-derivative (nth 1 expr)) | |
616 | (nth 2 expr)))) | |
617 | (if (= (length expr) 5) | |
618 | (let ((lower (math-expr-subst (nth 1 expr) (nth 2 expr) | |
619 | (nth 3 expr))) | |
620 | (upper (math-expr-subst (nth 1 expr) (nth 2 expr) | |
621 | (nth 4 expr)))) | |
622 | (math-add (math-sub (math-mul upper | |
623 | (math-derivative (nth 4 expr))) | |
624 | (math-mul lower | |
625 | (math-derivative (nth 3 expr)))) | |
f7adda54 | 626 | (if (equal (nth 2 expr) math-deriv-var) |
136211a9 EZ |
627 | 0 |
628 | (math-normalize | |
629 | (list 'calcFunc-integ | |
630 | (math-derivative (nth 1 expr)) (nth 2 expr) | |
631 | (nth 3 expr) (nth 4 expr))))))))))) | |
632 | ||
633 | (put 'calcFunc-if 'math-derivative-n | |
634 | (function | |
635 | (lambda (expr) | |
636 | (and (= (length expr) 4) | |
637 | (list 'calcFunc-if (nth 1 expr) | |
638 | (math-derivative (nth 2 expr)) | |
639 | (math-derivative (nth 3 expr))))))) | |
640 | ||
641 | (put 'calcFunc-subscr 'math-derivative-n | |
642 | (function | |
643 | (lambda (expr) | |
644 | (and (= (length expr) 3) | |
645 | (list 'calcFunc-subscr (nth 1 expr) | |
646 | (math-derivative (nth 2 expr))))))) | |
647 | ||
648 | ||
3132f345 CW |
649 | (defvar math-integ-var '(var X ---)) |
650 | (defvar math-integ-var-2 '(var Y ---)) | |
651 | (defvar math-integ-vars (list 'f math-integ-var math-integ-var-2)) | |
652 | (defvar math-integ-var-list (list math-integ-var)) | |
653 | (defvar math-integ-var-list-list (list math-integ-var-list)) | |
136211a9 | 654 | |
f7adda54 JB |
655 | ;; math-integ-depth is a local variable for math-try-integral, but is used |
656 | ;; by math-integral and math-tracing-integral | |
657 | ;; which are called (directly or indirectly) by math-try-integral. | |
658 | (defvar math-integ-depth) | |
659 | ;; math-integ-level is a local variable for math-try-integral, but is used | |
f3445fab GM |
660 | ;; by math-integral, math-do-integral, math-tracing-integral, |
661 | ;; math-sub-integration, math-integrate-by-parts and | |
662 | ;; math-integrate-by-substitution, which are called (directly or | |
f7adda54 JB |
663 | ;; indirectly) by math-try-integral. |
664 | (defvar math-integ-level) | |
665 | ;; math-integral-limit is a local variable for calcFunc-integ, but is | |
f3445fab GM |
666 | ;; used by math-tracing-integral, math-sub-integration and |
667 | ;; math-try-integration. | |
f7adda54 JB |
668 | (defvar math-integral-limit) |
669 | ||
136211a9 EZ |
670 | (defmacro math-tracing-integral (&rest parts) |
671 | (list 'and | |
672 | 'trace-buffer | |
673 | (list 'save-excursion | |
674 | '(set-buffer trace-buffer) | |
675 | '(goto-char (point-max)) | |
676 | (list 'and | |
677 | '(bolp) | |
678 | '(insert (make-string (- math-integral-limit | |
679 | math-integ-level) 32) | |
680 | (format "%2d " math-integ-depth) | |
681 | (make-string math-integ-level 32))) | |
682 | ;;(list 'condition-case 'err | |
683 | (cons 'insert parts) | |
684 | ;; '(error (insert (prin1-to-string err)))) | |
bf77c646 | 685 | '(sit-for 0)))) |
136211a9 EZ |
686 | |
687 | ;;; The following wrapper caches results and avoids infinite recursion. | |
688 | ;;; Each cache entry is: ( A B ) Integral of A is B; | |
689 | ;;; ( A N ) Integral of A failed at level N; | |
690 | ;;; ( A busy ) Currently working on integral of A; | |
691 | ;;; ( A parts ) Currently working, integ-by-parts; | |
692 | ;;; ( A parts2 ) Currently working, integ-by-parts; | |
693 | ;;; ( A cancelled ) Ignore this cache entry; | |
f7adda54 JB |
694 | ;;; ( A [B] ) Same result as for math-cur-record = B. |
695 | ||
696 | ;; math-cur-record is a local variable for math-try-integral, but is used | |
697 | ;; by math-integral, math-replace-integral-parts and math-integrate-by-parts | |
698 | ;; which are called (directly or indirectly) by math-try-integral, as well as | |
699 | ;; by calc-dump-integral-cache | |
700 | (defvar math-cur-record) | |
701 | ;; math-enable-subst and math-any-substs are local variables for | |
702 | ;; calcFunc-integ, but are used by math-integral and math-try-integral. | |
703 | (defvar math-enable-subst) | |
704 | (defvar math-any-substs) | |
705 | ||
706 | ;; math-integ-msg is a local variable for math-try-integral, but is | |
707 | ;; used (both locally and non-locally) by math-integral. | |
708 | (defvar math-integ-msg) | |
709 | ||
710 | (defvar math-integral-cache nil) | |
711 | (defvar math-integral-cache-state nil) | |
712 | ||
136211a9 | 713 | (defun math-integral (expr &optional simplify same-as-above) |
f7adda54 JB |
714 | (let* ((simp math-cur-record) |
715 | (math-cur-record (assoc expr math-integral-cache)) | |
136211a9 EZ |
716 | (math-integ-depth (1+ math-integ-depth)) |
717 | (val 'cancelled)) | |
718 | (math-tracing-integral "Integrating " | |
719 | (math-format-value expr 1000) | |
720 | "...\n") | |
f7adda54 | 721 | (and math-cur-record |
136211a9 EZ |
722 | (progn |
723 | (math-tracing-integral "Found " | |
f7adda54 JB |
724 | (math-format-value (nth 1 math-cur-record) 1000)) |
725 | (and (consp (nth 1 math-cur-record)) | |
726 | (math-replace-integral-parts math-cur-record)) | |
136211a9 | 727 | (math-tracing-integral " => " |
f7adda54 | 728 | (math-format-value (nth 1 math-cur-record) 1000) |
136211a9 | 729 | "\n"))) |
f7adda54 JB |
730 | (or (and math-cur-record |
731 | (not (eq (nth 1 math-cur-record) 'cancelled)) | |
732 | (or (not (integerp (nth 1 math-cur-record))) | |
733 | (>= (nth 1 math-cur-record) math-integ-level))) | |
136211a9 EZ |
734 | (and (math-integral-contains-parts expr) |
735 | (progn | |
736 | (setq val nil) | |
737 | t)) | |
738 | (unwind-protect | |
739 | (progn | |
740 | (let (math-integ-msg) | |
741 | (if (eq calc-display-working-message 'lots) | |
742 | (progn | |
743 | (calc-set-command-flag 'clear-message) | |
744 | (setq math-integ-msg (format | |
745 | "Working... Integrating %s" | |
746 | (math-format-flat-expr expr 0))) | |
274f1353 | 747 | (message "%s" math-integ-msg))) |
f7adda54 JB |
748 | (if math-cur-record |
749 | (setcar (cdr math-cur-record) | |
136211a9 | 750 | (if same-as-above (vector simp) 'busy)) |
f7adda54 | 751 | (setq math-cur-record |
136211a9 | 752 | (list expr (if same-as-above (vector simp) 'busy)) |
f7adda54 | 753 | math-integral-cache (cons math-cur-record |
136211a9 EZ |
754 | math-integral-cache))) |
755 | (if (eq simplify 'yes) | |
756 | (progn | |
757 | (math-tracing-integral "Simplifying...") | |
758 | (setq simp (math-simplify expr)) | |
759 | (setq val (if (equal simp expr) | |
760 | (progn | |
761 | (math-tracing-integral " no change\n") | |
762 | (math-do-integral expr)) | |
763 | (math-tracing-integral " simplified\n") | |
764 | (math-integral simp 'no t)))) | |
765 | (or (setq val (math-do-integral expr)) | |
766 | (eq simplify 'no) | |
767 | (let ((simp (math-simplify expr))) | |
768 | (or (equal simp expr) | |
769 | (progn | |
770 | (math-tracing-integral "Trying again after " | |
771 | "simplification...\n") | |
772 | (setq val (math-integral simp 'no t)))))))) | |
773 | (if (eq calc-display-working-message 'lots) | |
274f1353 | 774 | (message "%s" math-integ-msg))) |
f7adda54 | 775 | (setcar (cdr math-cur-record) (or val |
136211a9 EZ |
776 | (if (or math-enable-subst |
777 | (not math-any-substs)) | |
778 | math-integ-level | |
779 | 'cancelled))))) | |
f7adda54 | 780 | (setq val math-cur-record) |
136211a9 EZ |
781 | (while (vectorp (nth 1 val)) |
782 | (setq val (aref (nth 1 val) 0))) | |
783 | (setq val (if (memq (nth 1 val) '(parts parts2)) | |
784 | (progn | |
785 | (setcar (cdr val) 'parts2) | |
786 | (list 'var 'PARTS val)) | |
787 | (and (consp (nth 1 val)) | |
788 | (nth 1 val)))) | |
789 | (math-tracing-integral "Integral of " | |
790 | (math-format-value expr 1000) | |
791 | " is " | |
792 | (math-format-value val 1000) | |
793 | "\n") | |
bf77c646 | 794 | val)) |
136211a9 EZ |
795 | |
796 | (defun math-integral-contains-parts (expr) | |
797 | (if (Math-primp expr) | |
798 | (and (eq (car-safe expr) 'var) | |
799 | (eq (nth 1 expr) 'PARTS) | |
800 | (listp (nth 2 expr))) | |
801 | (while (and (setq expr (cdr expr)) | |
802 | (not (math-integral-contains-parts (car expr))))) | |
bf77c646 | 803 | expr)) |
136211a9 EZ |
804 | |
805 | (defun math-replace-integral-parts (expr) | |
806 | (or (Math-primp expr) | |
807 | (while (setq expr (cdr expr)) | |
808 | (and (consp (car expr)) | |
809 | (if (eq (car (car expr)) 'var) | |
810 | (and (eq (nth 1 (car expr)) 'PARTS) | |
811 | (consp (nth 2 (car expr))) | |
812 | (if (listp (nth 1 (nth 2 (car expr)))) | |
813 | (progn | |
814 | (setcar expr (nth 1 (nth 2 (car expr)))) | |
815 | (math-replace-integral-parts (cons 'foo expr))) | |
f7adda54 | 816 | (setcar (cdr math-cur-record) 'cancelled))) |
bf77c646 | 817 | (math-replace-integral-parts (car expr))))))) |
136211a9 | 818 | |
4710da05 JB |
819 | (defvar math-linear-subst-tried t |
820 | "Non-nil means that a linear substitution has been tried.") | |
821 | ||
f7adda54 JB |
822 | ;; The variable math-has-rules is a local variable for math-try-integral, |
823 | ;; but is used by math-do-integral, which is called (non-directly) by | |
824 | ;; math-try-integral. | |
825 | (defvar math-has-rules) | |
826 | ||
827 | ;; math-old-integ is a local variable for math-do-integral, but is | |
828 | ;; used by math-sub-integration. | |
829 | (defvar math-old-integ) | |
830 | ||
f3445fab | 831 | ;; The variables math-t1, math-t2 and math-t3 are local to |
f7adda54 | 832 | ;; math-do-integral, math-try-solve-for and math-decompose-poly, but |
f3445fab | 833 | ;; are used by functions they call (directly or indirectly); |
f7adda54 | 834 | ;; math-do-integral calls math-do-integral-methods; |
f3445fab | 835 | ;; math-try-solve-for calls math-try-solve-prod, |
f7adda54 JB |
836 | ;; math-solve-find-root-term and math-solve-find-root-in-prod; |
837 | ;; math-decompose-poly calls math-solve-poly-funny-powers and | |
838 | ;; math-solve-crunch-poly. | |
839 | (defvar math-t1) | |
840 | (defvar math-t2) | |
841 | (defvar math-t3) | |
842 | ||
136211a9 | 843 | (defun math-do-integral (expr) |
4710da05 | 844 | (let ((math-linear-subst-tried nil) |
f7adda54 | 845 | math-t1 math-t2) |
136211a9 EZ |
846 | (or (cond ((not (math-expr-contains expr math-integ-var)) |
847 | (math-mul expr math-integ-var)) | |
848 | ((equal expr math-integ-var) | |
849 | (math-div (math-sqr expr) 2)) | |
850 | ((eq (car expr) '+) | |
f7adda54 JB |
851 | (and (setq math-t1 (math-integral (nth 1 expr))) |
852 | (setq math-t2 (math-integral (nth 2 expr))) | |
853 | (math-add math-t1 math-t2))) | |
136211a9 | 854 | ((eq (car expr) '-) |
f7adda54 JB |
855 | (and (setq math-t1 (math-integral (nth 1 expr))) |
856 | (setq math-t2 (math-integral (nth 2 expr))) | |
857 | (math-sub math-t1 math-t2))) | |
136211a9 | 858 | ((eq (car expr) 'neg) |
f7adda54 JB |
859 | (and (setq math-t1 (math-integral (nth 1 expr))) |
860 | (math-neg math-t1))) | |
136211a9 EZ |
861 | ((eq (car expr) '*) |
862 | (cond ((not (math-expr-contains (nth 1 expr) math-integ-var)) | |
f7adda54 JB |
863 | (and (setq math-t1 (math-integral (nth 2 expr))) |
864 | (math-mul (nth 1 expr) math-t1))) | |
136211a9 | 865 | ((not (math-expr-contains (nth 2 expr) math-integ-var)) |
f7adda54 JB |
866 | (and (setq math-t1 (math-integral (nth 1 expr))) |
867 | (math-mul math-t1 (nth 2 expr)))) | |
136211a9 EZ |
868 | ((memq (car-safe (nth 1 expr)) '(+ -)) |
869 | (math-integral (list (car (nth 1 expr)) | |
870 | (math-mul (nth 1 (nth 1 expr)) | |
871 | (nth 2 expr)) | |
872 | (math-mul (nth 2 (nth 1 expr)) | |
873 | (nth 2 expr))) | |
874 | 'yes t)) | |
875 | ((memq (car-safe (nth 2 expr)) '(+ -)) | |
876 | (math-integral (list (car (nth 2 expr)) | |
877 | (math-mul (nth 1 (nth 2 expr)) | |
878 | (nth 1 expr)) | |
879 | (math-mul (nth 2 (nth 2 expr)) | |
880 | (nth 1 expr))) | |
881 | 'yes t)))) | |
882 | ((eq (car expr) '/) | |
883 | (cond ((and (not (math-expr-contains (nth 1 expr) | |
884 | math-integ-var)) | |
885 | (not (math-equal-int (nth 1 expr) 1))) | |
f7adda54 JB |
886 | (and (setq math-t1 (math-integral (math-div 1 (nth 2 expr)))) |
887 | (math-mul (nth 1 expr) math-t1))) | |
136211a9 | 888 | ((not (math-expr-contains (nth 2 expr) math-integ-var)) |
f7adda54 JB |
889 | (and (setq math-t1 (math-integral (nth 1 expr))) |
890 | (math-div math-t1 (nth 2 expr)))) | |
136211a9 EZ |
891 | ((and (eq (car-safe (nth 1 expr)) '*) |
892 | (not (math-expr-contains (nth 1 (nth 1 expr)) | |
893 | math-integ-var))) | |
f7adda54 | 894 | (and (setq math-t1 (math-integral |
136211a9 EZ |
895 | (math-div (nth 2 (nth 1 expr)) |
896 | (nth 2 expr)))) | |
f7adda54 | 897 | (math-mul math-t1 (nth 1 (nth 1 expr))))) |
136211a9 EZ |
898 | ((and (eq (car-safe (nth 1 expr)) '*) |
899 | (not (math-expr-contains (nth 2 (nth 1 expr)) | |
900 | math-integ-var))) | |
f7adda54 | 901 | (and (setq math-t1 (math-integral |
136211a9 EZ |
902 | (math-div (nth 1 (nth 1 expr)) |
903 | (nth 2 expr)))) | |
f7adda54 | 904 | (math-mul math-t1 (nth 2 (nth 1 expr))))) |
136211a9 EZ |
905 | ((and (eq (car-safe (nth 2 expr)) '*) |
906 | (not (math-expr-contains (nth 1 (nth 2 expr)) | |
907 | math-integ-var))) | |
f7adda54 | 908 | (and (setq math-t1 (math-integral |
136211a9 EZ |
909 | (math-div (nth 1 expr) |
910 | (nth 2 (nth 2 expr))))) | |
f7adda54 | 911 | (math-div math-t1 (nth 1 (nth 2 expr))))) |
136211a9 EZ |
912 | ((and (eq (car-safe (nth 2 expr)) '*) |
913 | (not (math-expr-contains (nth 2 (nth 2 expr)) | |
914 | math-integ-var))) | |
f7adda54 | 915 | (and (setq math-t1 (math-integral |
136211a9 EZ |
916 | (math-div (nth 1 expr) |
917 | (nth 1 (nth 2 expr))))) | |
f7adda54 | 918 | (math-div math-t1 (nth 2 (nth 2 expr))))) |
136211a9 EZ |
919 | ((eq (car-safe (nth 2 expr)) 'calcFunc-exp) |
920 | (math-integral | |
921 | (math-mul (nth 1 expr) | |
922 | (list 'calcFunc-exp | |
923 | (math-neg (nth 1 (nth 2 expr))))))))) | |
924 | ((eq (car expr) '^) | |
925 | (cond ((not (math-expr-contains (nth 1 expr) math-integ-var)) | |
f7adda54 | 926 | (or (and (setq math-t1 (math-is-polynomial (nth 2 expr) |
136211a9 EZ |
927 | math-integ-var 1)) |
928 | (math-div expr | |
f7adda54 | 929 | (math-mul (nth 1 math-t1) |
136211a9 EZ |
930 | (math-normalize |
931 | (list 'calcFunc-ln | |
932 | (nth 1 expr)))))) | |
933 | (math-integral | |
934 | (list 'calcFunc-exp | |
935 | (math-mul (nth 2 expr) | |
936 | (math-normalize | |
937 | (list 'calcFunc-ln | |
938 | (nth 1 expr))))) | |
939 | 'yes t))) | |
940 | ((not (math-expr-contains (nth 2 expr) math-integ-var)) | |
941 | (if (and (integerp (nth 2 expr)) (< (nth 2 expr) 0)) | |
942 | (math-integral | |
943 | (list '/ 1 (math-pow (nth 1 expr) (- (nth 2 expr)))) | |
944 | nil t) | |
f7adda54 | 945 | (or (and (setq math-t1 (math-is-polynomial (nth 1 expr) |
136211a9 EZ |
946 | math-integ-var |
947 | 1)) | |
f7adda54 JB |
948 | (setq math-t2 (math-add (nth 2 expr) 1)) |
949 | (math-div (math-pow (nth 1 expr) math-t2) | |
950 | (math-mul math-t2 (nth 1 math-t1)))) | |
136211a9 EZ |
951 | (and (Math-negp (nth 2 expr)) |
952 | (math-integral | |
953 | (math-div 1 | |
954 | (math-pow (nth 1 expr) | |
955 | (math-neg | |
956 | (nth 2 expr)))) | |
957 | nil t)) | |
958 | nil)))))) | |
959 | ||
960 | ;; Integral of a polynomial. | |
f7adda54 | 961 | (and (setq math-t1 (math-is-polynomial expr math-integ-var 20)) |
136211a9 EZ |
962 | (let ((accum 0) |
963 | (n 1)) | |
f7adda54 | 964 | (while math-t1 |
136211a9 | 965 | (if (setq accum (math-add accum |
f7adda54 | 966 | (math-div (math-mul (car math-t1) |
136211a9 EZ |
967 | (math-pow |
968 | math-integ-var | |
969 | n)) | |
970 | n)) | |
f7adda54 | 971 | math-t1 (cdr math-t1)) |
136211a9 EZ |
972 | (setq n (1+ n)))) |
973 | accum)) | |
974 | ||
975 | ;; Try looking it up! | |
976 | (cond ((= (length expr) 2) | |
977 | (and (symbolp (car expr)) | |
f7adda54 | 978 | (setq math-t1 (get (car expr) 'math-integral)) |
136211a9 | 979 | (progn |
f7adda54 JB |
980 | (while (and math-t1 |
981 | (not (setq math-t2 (funcall (car math-t1) | |
136211a9 | 982 | (nth 1 expr))))) |
f7adda54 JB |
983 | (setq math-t1 (cdr math-t1))) |
984 | (and math-t2 (math-normalize math-t2))))) | |
136211a9 EZ |
985 | ((= (length expr) 3) |
986 | (and (symbolp (car expr)) | |
f7adda54 | 987 | (setq math-t1 (get (car expr) 'math-integral-2)) |
136211a9 | 988 | (progn |
f7adda54 JB |
989 | (while (and math-t1 |
990 | (not (setq math-t2 (funcall (car math-t1) | |
136211a9 EZ |
991 | (nth 1 expr) |
992 | (nth 2 expr))))) | |
f7adda54 JB |
993 | (setq math-t1 (cdr math-t1))) |
994 | (and math-t2 (math-normalize math-t2)))))) | |
136211a9 EZ |
995 | |
996 | ;; Integral of a rational function. | |
997 | (and (math-ratpoly-p expr math-integ-var) | |
f7adda54 JB |
998 | (setq math-t1 (calcFunc-apart expr math-integ-var)) |
999 | (not (equal math-t1 expr)) | |
1000 | (math-integral math-t1)) | |
136211a9 EZ |
1001 | |
1002 | ;; Try user-defined integration rules. | |
f7adda54 | 1003 | (and math-has-rules |
136211a9 EZ |
1004 | (let ((math-old-integ (symbol-function 'calcFunc-integ)) |
1005 | (input (list 'calcFunc-integtry expr math-integ-var)) | |
1006 | res part) | |
1007 | (unwind-protect | |
1008 | (progn | |
1009 | (fset 'calcFunc-integ 'math-sub-integration) | |
1010 | (setq res (math-rewrite input | |
1011 | '(var IntegRules var-IntegRules) | |
1012 | 1)) | |
1013 | (fset 'calcFunc-integ math-old-integ) | |
1014 | (and (not (equal res input)) | |
1015 | (if (setq part (math-expr-calls | |
1016 | res '(calcFunc-integsubst))) | |
1017 | (and (memq (length part) '(3 4 5)) | |
1018 | (let ((parts (mapcar | |
1019 | (function | |
1020 | (lambda (x) | |
1021 | (math-expr-subst | |
1022 | x (nth 2 part) | |
1023 | math-integ-var))) | |
1024 | (cdr part)))) | |
1025 | (math-integrate-by-substitution | |
1026 | expr (car parts) t | |
1027 | (or (nth 2 parts) | |
1028 | (list 'calcFunc-integfailed | |
1029 | math-integ-var)) | |
1030 | (nth 3 parts)))) | |
1031 | (if (not (math-expr-calls res | |
1032 | '(calcFunc-integtry | |
1033 | calcFunc-integfailed))) | |
1034 | res)))) | |
1035 | (fset 'calcFunc-integ math-old-integ)))) | |
1036 | ||
1037 | ;; See if the function is a symbolic derivative. | |
1038 | (and (string-match "'" (symbol-name (car expr))) | |
1039 | (let ((name (symbol-name (car expr))) | |
1040 | (p expr) (n 0) (which nil) (bad nil)) | |
1041 | (while (setq n (1+ n) p (cdr p)) | |
1042 | (if (equal (car p) math-integ-var) | |
1043 | (if which (setq bad t) (setq which n)) | |
1044 | (if (math-expr-contains (car p) math-integ-var) | |
1045 | (setq bad t)))) | |
1046 | (and which (not bad) | |
1047 | (let ((prime (if (= which 1) "'" (format "'%d" which)))) | |
1048 | (and (string-match (concat prime "\\('['0-9]*\\|$\\)") | |
1049 | name) | |
1050 | (cons (intern | |
1051 | (concat | |
1052 | (substring name 0 (match-beginning 0)) | |
1053 | (substring name (+ (match-beginning 0) | |
1054 | (length prime))))) | |
1055 | (cdr expr))))))) | |
1056 | ||
1057 | ;; Try transformation methods (parts, substitutions). | |
1058 | (and (> math-integ-level 0) | |
1059 | (math-do-integral-methods expr)) | |
1060 | ||
1061 | ;; Try expanding the function's definition. | |
1062 | (let ((res (math-expand-formula expr))) | |
1063 | (and res | |
bf77c646 | 1064 | (math-integral res)))))) |
136211a9 EZ |
1065 | |
1066 | (defun math-sub-integration (expr &rest rest) | |
1067 | (or (if (or (not rest) | |
1068 | (and (< math-integ-level math-integral-limit) | |
1069 | (eq (car rest) math-integ-var))) | |
1070 | (math-integral expr) | |
1071 | (let ((res (apply math-old-integ expr rest))) | |
1072 | (and (or (= math-integ-level math-integral-limit) | |
1073 | (not (math-expr-calls res 'calcFunc-integ))) | |
1074 | res))) | |
bf77c646 | 1075 | (list 'calcFunc-integfailed expr))) |
136211a9 | 1076 | |
f7adda54 | 1077 | ;; math-so-far is a local variable for math-do-integral-methods, but |
f3445fab | 1078 | ;; is used by math-integ-try-linear-substitutions and |
f7adda54 JB |
1079 | ;; math-integ-try-substitutions. |
1080 | (defvar math-so-far) | |
1081 | ||
1082 | ;; math-integ-expr is a local variable for math-do-integral-methods, | |
f3445fab | 1083 | ;; but is used by math-integ-try-linear-substitutions and |
f7adda54 JB |
1084 | ;; math-integ-try-substitutions. |
1085 | (defvar math-integ-expr) | |
1086 | ||
1087 | (defun math-do-integral-methods (math-integ-expr) | |
1088 | (let ((math-so-far math-integ-var-list-list) | |
136211a9 EZ |
1089 | rat-in) |
1090 | ||
1091 | ;; Integration by substitution, for various likely sub-expressions. | |
1092 | ;; (In first pass, we look only for sub-exprs that are linear in X.) | |
f7adda54 JB |
1093 | (or (math-integ-try-linear-substitutions math-integ-expr) |
1094 | (math-integ-try-substitutions math-integ-expr) | |
136211a9 EZ |
1095 | |
1096 | ;; If function has sines and cosines, try tan(x/2) substitution. | |
f7adda54 | 1097 | (and (let ((p (setq rat-in (math-expr-rational-in math-integ-expr)))) |
136211a9 EZ |
1098 | (while (and p |
1099 | (memq (car (car p)) '(calcFunc-sin | |
1100 | calcFunc-cos | |
9274224b JB |
1101 | calcFunc-tan |
1102 | calcFunc-sec | |
1103 | calcFunc-csc | |
1104 | calcFunc-cot)) | |
136211a9 EZ |
1105 | (equal (nth 1 (car p)) math-integ-var)) |
1106 | (setq p (cdr p))) | |
1107 | (null p)) | |
f7adda54 JB |
1108 | (or (and (math-integ-parts-easy math-integ-expr) |
1109 | (math-integ-try-parts math-integ-expr t)) | |
136211a9 | 1110 | (math-integrate-by-good-substitution |
f7adda54 | 1111 | math-integ-expr (list 'calcFunc-tan (math-div math-integ-var 2))))) |
136211a9 EZ |
1112 | |
1113 | ;; If function has sinh and cosh, try tanh(x/2) substitution. | |
1114 | (and (let ((p rat-in)) | |
1115 | (while (and p | |
1116 | (memq (car (car p)) '(calcFunc-sinh | |
1117 | calcFunc-cosh | |
1118 | calcFunc-tanh | |
9274224b JB |
1119 | calcFunc-sech |
1120 | calcFunc-csch | |
1121 | calcFunc-coth | |
136211a9 EZ |
1122 | calcFunc-exp)) |
1123 | (equal (nth 1 (car p)) math-integ-var)) | |
1124 | (setq p (cdr p))) | |
1125 | (null p)) | |
f7adda54 JB |
1126 | (or (and (math-integ-parts-easy math-integ-expr) |
1127 | (math-integ-try-parts math-integ-expr t)) | |
136211a9 | 1128 | (math-integrate-by-good-substitution |
f7adda54 | 1129 | math-integ-expr (list 'calcFunc-tanh (math-div math-integ-var 2))))) |
136211a9 EZ |
1130 | |
1131 | ;; If function has square roots, try sin, tan, or sec substitution. | |
1132 | (and (let ((p rat-in)) | |
f7adda54 | 1133 | (setq math-t1 nil) |
136211a9 EZ |
1134 | (while (and p |
1135 | (or (equal (car p) math-integ-var) | |
1136 | (and (eq (car (car p)) 'calcFunc-sqrt) | |
f7adda54 JB |
1137 | (setq math-t1 (math-is-polynomial |
1138 | (nth 1 (setq math-t2 (car p))) | |
136211a9 EZ |
1139 | math-integ-var 2))))) |
1140 | (setq p (cdr p))) | |
f7adda54 JB |
1141 | (and (null p) math-t1)) |
1142 | (if (cdr (cdr math-t1)) | |
1143 | (if (math-guess-if-neg (nth 2 math-t1)) | |
1144 | (let* ((c (math-sqrt (math-neg (nth 2 math-t1)))) | |
1145 | (d (math-div (nth 1 math-t1) (math-mul -2 c))) | |
1146 | (a (math-sqrt (math-add (car math-t1) (math-sqr d))))) | |
136211a9 | 1147 | (math-integrate-by-good-substitution |
f7adda54 | 1148 | math-integ-expr (list 'calcFunc-arcsin |
136211a9 EZ |
1149 | (math-div-thru |
1150 | (math-add (math-mul c math-integ-var) d) | |
1151 | a)))) | |
f7adda54 JB |
1152 | (let* ((c (math-sqrt (nth 2 math-t1))) |
1153 | (d (math-div (nth 1 math-t1) (math-mul 2 c))) | |
1154 | (aa (math-sub (car math-t1) (math-sqr d)))) | |
136211a9 EZ |
1155 | (if (and nil (not (and (eq d 0) (eq c 1)))) |
1156 | (math-integrate-by-good-substitution | |
f7adda54 | 1157 | math-integ-expr (math-add (math-mul c math-integ-var) d)) |
136211a9 EZ |
1158 | (if (math-guess-if-neg aa) |
1159 | (math-integrate-by-good-substitution | |
f7adda54 | 1160 | math-integ-expr (list 'calcFunc-arccosh |
136211a9 EZ |
1161 | (math-div-thru |
1162 | (math-add (math-mul c math-integ-var) | |
1163 | d) | |
1164 | (math-sqrt (math-neg aa))))) | |
1165 | (math-integrate-by-good-substitution | |
f7adda54 | 1166 | math-integ-expr (list 'calcFunc-arcsinh |
136211a9 EZ |
1167 | (math-div-thru |
1168 | (math-add (math-mul c math-integ-var) | |
1169 | d) | |
1170 | (math-sqrt aa)))))))) | |
f7adda54 | 1171 | (math-integrate-by-good-substitution math-integ-expr math-t2)) ) |
136211a9 EZ |
1172 | |
1173 | ;; Try integration by parts. | |
f7adda54 | 1174 | (math-integ-try-parts math-integ-expr) |
136211a9 EZ |
1175 | |
1176 | ;; Give up. | |
bf77c646 | 1177 | nil))) |
136211a9 EZ |
1178 | |
1179 | (defun math-integ-parts-easy (expr) | |
1180 | (cond ((Math-primp expr) t) | |
1181 | ((memq (car expr) '(+ - *)) | |
1182 | (and (math-integ-parts-easy (nth 1 expr)) | |
1183 | (math-integ-parts-easy (nth 2 expr)))) | |
1184 | ((eq (car expr) '/) | |
1185 | (and (math-integ-parts-easy (nth 1 expr)) | |
1186 | (math-atomic-factorp (nth 2 expr)))) | |
1187 | ((eq (car expr) '^) | |
1188 | (and (natnump (nth 2 expr)) | |
1189 | (math-integ-parts-easy (nth 1 expr)))) | |
1190 | ((eq (car expr) 'neg) | |
1191 | (math-integ-parts-easy (nth 1 expr))) | |
bf77c646 | 1192 | (t t))) |
136211a9 | 1193 | |
f7adda54 JB |
1194 | ;; math-prev-parts-v is local to calcFunc-integ (as well as |
1195 | ;; math-integrate-by-parts), but is used by math-integ-try-parts. | |
1196 | (defvar math-prev-parts-v) | |
1197 | ||
1198 | ;; math-good-parts is local to calcFunc-integ (as well as | |
1199 | ;; math-integ-try-parts), but is used by math-integrate-by-parts. | |
1200 | (defvar math-good-parts) | |
1201 | ||
1202 | ||
136211a9 EZ |
1203 | (defun math-integ-try-parts (expr &optional math-good-parts) |
1204 | ;; Integration by parts: | |
1205 | ;; integ(f(x) g(x),x) = f(x) h(x) - integ(h(x) f'(x),x) | |
1206 | ;; where h(x) = integ(g(x),x). | |
1207 | (or (let ((exp (calcFunc-expand expr))) | |
1208 | (and (not (equal exp expr)) | |
1209 | (math-integral exp))) | |
1210 | (and (eq (car expr) '*) | |
1211 | (let ((first-bad (or (math-polynomial-p (nth 1 expr) | |
1212 | math-integ-var) | |
1213 | (equal (nth 2 expr) math-prev-parts-v)))) | |
1214 | (or (and first-bad ; so try this one first | |
1215 | (math-integrate-by-parts (nth 1 expr) (nth 2 expr))) | |
1216 | (math-integrate-by-parts (nth 2 expr) (nth 1 expr)) | |
1217 | (and (not first-bad) | |
1218 | (math-integrate-by-parts (nth 1 expr) (nth 2 expr)))))) | |
1219 | (and (eq (car expr) '/) | |
1220 | (math-expr-contains (nth 1 expr) math-integ-var) | |
1221 | (let ((recip (math-div 1 (nth 2 expr)))) | |
1222 | (or (math-integrate-by-parts (nth 1 expr) recip) | |
1223 | (math-integrate-by-parts recip (nth 1 expr))))) | |
1224 | (and (eq (car expr) '^) | |
1225 | (math-integrate-by-parts (math-pow (nth 1 expr) | |
1226 | (math-sub (nth 2 expr) 1)) | |
bf77c646 | 1227 | (nth 1 expr))))) |
136211a9 EZ |
1228 | |
1229 | (defun math-integrate-by-parts (u vprime) | |
1230 | (let ((math-integ-level (if (or math-good-parts | |
1231 | (math-polynomial-p u math-integ-var)) | |
1232 | math-integ-level | |
1233 | (1- math-integ-level))) | |
1234 | (math-doing-parts t) | |
1235 | v temp) | |
1236 | (and (>= math-integ-level 0) | |
1237 | (unwind-protect | |
1238 | (progn | |
f7adda54 | 1239 | (setcar (cdr math-cur-record) 'parts) |
136211a9 EZ |
1240 | (math-tracing-integral "Integrating by parts, u = " |
1241 | (math-format-value u 1000) | |
1242 | ", v' = " | |
1243 | (math-format-value vprime 1000) | |
1244 | "\n") | |
1245 | (and (setq v (math-integral vprime)) | |
1246 | (setq temp (calcFunc-deriv u math-integ-var nil t)) | |
1247 | (setq temp (let ((math-prev-parts-v v)) | |
1248 | (math-integral (math-mul v temp) 'yes))) | |
1249 | (setq temp (math-sub (math-mul u v) temp)) | |
f7adda54 | 1250 | (if (eq (nth 1 math-cur-record) 'parts) |
136211a9 | 1251 | (calcFunc-expand temp) |
f7adda54 | 1252 | (setq v (list 'var 'PARTS math-cur-record) |
136211a9 | 1253 | temp (let (calc-next-why) |
63f7edac JB |
1254 | (math-simplify-extended |
1255 | (math-solve-for (math-sub v temp) 0 v nil))) | |
f3445fab GM |
1256 | temp (if (and (eq (car-safe temp) '/) |
1257 | (math-zerop (nth 2 temp))) | |
63f7edac | 1258 | nil temp))))) |
f7adda54 | 1259 | (setcar (cdr math-cur-record) 'busy))))) |
136211a9 EZ |
1260 | |
1261 | ;;; This tries two different formulations, hoping the algebraic simplifier | |
1262 | ;;; will be strong enough to handle at least one. | |
1263 | (defun math-integrate-by-substitution (expr u &optional user uinv uinvprime) | |
1264 | (and (> math-integ-level 0) | |
1265 | (let ((math-integ-level (max (- math-integ-level 2) 0))) | |
bf77c646 | 1266 | (math-integrate-by-good-substitution expr u user uinv uinvprime)))) |
136211a9 EZ |
1267 | |
1268 | (defun math-integrate-by-good-substitution (expr u &optional user | |
1269 | uinv uinvprime) | |
1270 | (let ((math-living-dangerously t) | |
1271 | deriv temp) | |
1272 | (and (setq uinv (if uinv | |
1273 | (math-expr-subst uinv math-integ-var | |
1274 | math-integ-var-2) | |
1275 | (let (calc-next-why) | |
1276 | (math-solve-for u | |
1277 | math-integ-var-2 | |
1278 | math-integ-var nil)))) | |
1279 | (progn | |
1280 | (math-tracing-integral "Integrating by substitution, u = " | |
1281 | (math-format-value u 1000) | |
1282 | "\n") | |
1283 | (or (and (setq deriv (calcFunc-deriv u | |
1284 | math-integ-var nil | |
1285 | (not user))) | |
1286 | (setq temp (math-integral (math-expr-subst | |
1287 | (math-expr-subst | |
1288 | (math-expr-subst | |
1289 | (math-div expr deriv) | |
1290 | u | |
1291 | math-integ-var-2) | |
1292 | math-integ-var | |
1293 | uinv) | |
1294 | math-integ-var-2 | |
1295 | math-integ-var) | |
1296 | 'yes))) | |
1297 | (and (setq deriv (or uinvprime | |
1298 | (calcFunc-deriv uinv | |
1299 | math-integ-var-2 | |
1300 | math-integ-var | |
1301 | (not user)))) | |
1302 | (setq temp (math-integral (math-mul | |
1303 | (math-expr-subst | |
1304 | (math-expr-subst | |
1305 | (math-expr-subst | |
1306 | expr | |
1307 | u | |
1308 | math-integ-var-2) | |
1309 | math-integ-var | |
1310 | uinv) | |
1311 | math-integ-var-2 | |
1312 | math-integ-var) | |
1313 | deriv) | |
1314 | 'yes))))) | |
1315 | (math-simplify-extended | |
bf77c646 | 1316 | (math-expr-subst temp math-integ-var u))))) |
136211a9 EZ |
1317 | |
1318 | ;;; Look for substitutions of the form u = a x + b. | |
1319 | (defun math-integ-try-linear-substitutions (sub-expr) | |
4710da05 | 1320 | (setq math-linear-subst-tried t) |
136211a9 EZ |
1321 | (and (not (Math-primp sub-expr)) |
1322 | (or (and (not (memq (car sub-expr) '(+ - * / neg))) | |
1323 | (not (and (eq (car sub-expr) '^) | |
1324 | (integerp (nth 2 sub-expr)))) | |
1325 | (math-expr-contains sub-expr math-integ-var) | |
1326 | (let ((res nil)) | |
1327 | (while (and (setq sub-expr (cdr sub-expr)) | |
1328 | (or (not (math-linear-in (car sub-expr) | |
1329 | math-integ-var)) | |
f7adda54 | 1330 | (assoc (car sub-expr) math-so-far) |
136211a9 | 1331 | (progn |
f7adda54 JB |
1332 | (setq math-so-far (cons (list (car sub-expr)) |
1333 | math-so-far)) | |
136211a9 EZ |
1334 | (not (setq res |
1335 | (math-integrate-by-substitution | |
f7adda54 | 1336 | math-integ-expr (car sub-expr)))))))) |
136211a9 EZ |
1337 | res)) |
1338 | (let ((res nil)) | |
1339 | (while (and (setq sub-expr (cdr sub-expr)) | |
1340 | (not (setq res (math-integ-try-linear-substitutions | |
1341 | (car sub-expr)))))) | |
bf77c646 | 1342 | res)))) |
136211a9 EZ |
1343 | |
1344 | ;;; Recursively try different substitutions based on various sub-expressions. | |
1345 | (defun math-integ-try-substitutions (sub-expr &optional allow-rat) | |
1346 | (and (not (Math-primp sub-expr)) | |
f7adda54 | 1347 | (not (assoc sub-expr math-so-far)) |
136211a9 EZ |
1348 | (math-expr-contains sub-expr math-integ-var) |
1349 | (or (and (if (and (not (memq (car sub-expr) '(+ - * / neg))) | |
1350 | (not (and (eq (car sub-expr) '^) | |
1351 | (integerp (nth 2 sub-expr))))) | |
1352 | (setq allow-rat t) | |
1353 | (prog1 allow-rat (setq allow-rat nil))) | |
f7adda54 JB |
1354 | (not (eq sub-expr math-integ-expr)) |
1355 | (or (math-integrate-by-substitution math-integ-expr sub-expr) | |
136211a9 EZ |
1356 | (and (eq (car sub-expr) '^) |
1357 | (integerp (nth 2 sub-expr)) | |
1358 | (< (nth 2 sub-expr) 0) | |
1359 | (math-integ-try-substitutions | |
1360 | (math-pow (nth 1 sub-expr) (- (nth 2 sub-expr))) | |
1361 | t)))) | |
1362 | (let ((res nil)) | |
f7adda54 | 1363 | (setq math-so-far (cons (list sub-expr) math-so-far)) |
136211a9 EZ |
1364 | (while (and (setq sub-expr (cdr sub-expr)) |
1365 | (not (setq res (math-integ-try-substitutions | |
1366 | (car sub-expr) allow-rat))))) | |
bf77c646 | 1367 | res)))) |
136211a9 | 1368 | |
f7adda54 JB |
1369 | ;; The variable math-expr-parts is local to math-expr-rational-in, |
1370 | ;; but is used by math-expr-rational-in-rec | |
79d2746f | 1371 | (defvar math-expr-parts) |
f7adda54 | 1372 | |
136211a9 | 1373 | (defun math-expr-rational-in (expr) |
f7adda54 | 1374 | (let ((math-expr-parts nil)) |
136211a9 | 1375 | (math-expr-rational-in-rec expr) |
f7adda54 | 1376 | (mapcar 'car math-expr-parts))) |
136211a9 EZ |
1377 | |
1378 | (defun math-expr-rational-in-rec (expr) | |
1379 | (cond ((Math-primp expr) | |
1380 | (and (equal expr math-integ-var) | |
f7adda54 JB |
1381 | (not (assoc expr math-expr-parts)) |
1382 | (setq math-expr-parts (cons (list expr) math-expr-parts)))) | |
136211a9 EZ |
1383 | ((or (memq (car expr) '(+ - * / neg)) |
1384 | (and (eq (car expr) '^) (integerp (nth 2 expr)))) | |
1385 | (math-expr-rational-in-rec (nth 1 expr)) | |
1386 | (and (nth 2 expr) (math-expr-rational-in-rec (nth 2 expr)))) | |
1387 | ((and (eq (car expr) '^) | |
1388 | (eq (math-quarter-integer (nth 2 expr)) 2)) | |
1389 | (math-expr-rational-in-rec (list 'calcFunc-sqrt (nth 1 expr)))) | |
1390 | (t | |
f7adda54 | 1391 | (and (not (assoc expr math-expr-parts)) |
136211a9 | 1392 | (math-expr-contains expr math-integ-var) |
f7adda54 | 1393 | (setq math-expr-parts (cons (list expr) math-expr-parts)))))) |
136211a9 EZ |
1394 | |
1395 | (defun math-expr-calls (expr funcs &optional arg-contains) | |
1396 | (if (consp expr) | |
1397 | (if (or (memq (car expr) funcs) | |
1398 | (and (eq (car expr) '^) (eq (car funcs) 'calcFunc-sqrt) | |
1399 | (eq (math-quarter-integer (nth 2 expr)) 2))) | |
1400 | (and (or (not arg-contains) | |
1401 | (math-expr-contains expr arg-contains)) | |
1402 | expr) | |
1403 | (and (not (Math-primp expr)) | |
1404 | (let ((res nil)) | |
1405 | (while (and (setq expr (cdr expr)) | |
1406 | (not (setq res (math-expr-calls | |
1407 | (car expr) funcs arg-contains))))) | |
bf77c646 | 1408 | res))))) |
136211a9 EZ |
1409 | |
1410 | (defun math-fix-const-terms (expr except-vars) | |
1411 | (cond ((not (math-expr-depends expr except-vars)) 0) | |
1412 | ((Math-primp expr) expr) | |
1413 | ((eq (car expr) '+) | |
1414 | (math-add (math-fix-const-terms (nth 1 expr) except-vars) | |
1415 | (math-fix-const-terms (nth 2 expr) except-vars))) | |
1416 | ((eq (car expr) '-) | |
1417 | (math-sub (math-fix-const-terms (nth 1 expr) except-vars) | |
1418 | (math-fix-const-terms (nth 2 expr) except-vars))) | |
bf77c646 | 1419 | (t expr))) |
136211a9 EZ |
1420 | |
1421 | ;; Command for debugging the Calculator's symbolic integrator. | |
1422 | (defun calc-dump-integral-cache (&optional arg) | |
1423 | (interactive "P") | |
1424 | (let ((buf (current-buffer))) | |
1425 | (unwind-protect | |
1426 | (let ((p math-integral-cache) | |
f7adda54 | 1427 | math-cur-record) |
a1506d29 | 1428 | (display-buffer (get-buffer-create "*Integral Cache*")) |
136211a9 EZ |
1429 | (set-buffer (get-buffer "*Integral Cache*")) |
1430 | (erase-buffer) | |
1431 | (while p | |
f7adda54 JB |
1432 | (setq math-cur-record (car p)) |
1433 | (or arg (math-replace-integral-parts math-cur-record)) | |
1434 | (insert (math-format-flat-expr (car math-cur-record) 0) | |
136211a9 | 1435 | " --> " |
f7adda54 JB |
1436 | (if (symbolp (nth 1 math-cur-record)) |
1437 | (concat "(" (symbol-name (nth 1 math-cur-record)) ")") | |
1438 | (math-format-flat-expr (nth 1 math-cur-record) 0)) | |
136211a9 EZ |
1439 | "\n") |
1440 | (setq p (cdr p))) | |
1441 | (goto-char (point-min))) | |
bf77c646 | 1442 | (set-buffer buf)))) |
136211a9 | 1443 | |
f7adda54 JB |
1444 | ;; The variable math-max-integral-limit is local to calcFunc-integ, |
1445 | ;; but is used by math-try-integral. | |
1446 | (defvar math-max-integral-limit) | |
1447 | ||
136211a9 EZ |
1448 | (defun math-try-integral (expr) |
1449 | (let ((math-integ-level math-integral-limit) | |
1450 | (math-integ-depth 0) | |
1451 | (math-integ-msg "Working...done") | |
f7adda54 | 1452 | (math-cur-record nil) ; a technicality |
136211a9 EZ |
1453 | (math-integrating t) |
1454 | (calc-prefer-frac t) | |
1455 | (calc-symbolic-mode t) | |
f7adda54 | 1456 | (math-has-rules (calc-has-rules 'var-IntegRules))) |
136211a9 EZ |
1457 | (or (math-integral expr 'yes) |
1458 | (and math-any-substs | |
1459 | (setq math-enable-subst t) | |
1460 | (math-integral expr 'yes)) | |
1461 | (and (> math-max-integral-limit math-integral-limit) | |
1462 | (setq math-integral-limit math-max-integral-limit | |
1463 | math-integ-level math-integral-limit) | |
bf77c646 | 1464 | (math-integral expr 'yes))))) |
136211a9 | 1465 | |
f7adda54 JB |
1466 | (defvar var-IntegLimit nil) |
1467 | ||
136211a9 EZ |
1468 | (defun calcFunc-integ (expr var &optional low high) |
1469 | (cond | |
1470 | ;; Do these even if the parts turn out not to be integrable. | |
1471 | ((eq (car-safe expr) '+) | |
1472 | (math-add (calcFunc-integ (nth 1 expr) var low high) | |
1473 | (calcFunc-integ (nth 2 expr) var low high))) | |
1474 | ((eq (car-safe expr) '-) | |
1475 | (math-sub (calcFunc-integ (nth 1 expr) var low high) | |
1476 | (calcFunc-integ (nth 2 expr) var low high))) | |
1477 | ((eq (car-safe expr) 'neg) | |
1478 | (math-neg (calcFunc-integ (nth 1 expr) var low high))) | |
1479 | ((and (eq (car-safe expr) '*) | |
1480 | (not (math-expr-contains (nth 1 expr) var))) | |
1481 | (math-mul (nth 1 expr) (calcFunc-integ (nth 2 expr) var low high))) | |
1482 | ((and (eq (car-safe expr) '*) | |
1483 | (not (math-expr-contains (nth 2 expr) var))) | |
1484 | (math-mul (calcFunc-integ (nth 1 expr) var low high) (nth 2 expr))) | |
1485 | ((and (eq (car-safe expr) '/) | |
1486 | (not (math-expr-contains (nth 1 expr) var)) | |
1487 | (not (math-equal-int (nth 1 expr) 1))) | |
1488 | (math-mul (nth 1 expr) | |
1489 | (calcFunc-integ (math-div 1 (nth 2 expr)) var low high))) | |
1490 | ((and (eq (car-safe expr) '/) | |
1491 | (not (math-expr-contains (nth 2 expr) var))) | |
1492 | (math-div (calcFunc-integ (nth 1 expr) var low high) (nth 2 expr))) | |
1493 | ((and (eq (car-safe expr) '/) | |
1494 | (eq (car-safe (nth 1 expr)) '*) | |
1495 | (not (math-expr-contains (nth 1 (nth 1 expr)) var))) | |
1496 | (math-mul (nth 1 (nth 1 expr)) | |
1497 | (calcFunc-integ (math-div (nth 2 (nth 1 expr)) (nth 2 expr)) | |
1498 | var low high))) | |
1499 | ((and (eq (car-safe expr) '/) | |
1500 | (eq (car-safe (nth 1 expr)) '*) | |
1501 | (not (math-expr-contains (nth 2 (nth 1 expr)) var))) | |
1502 | (math-mul (nth 2 (nth 1 expr)) | |
1503 | (calcFunc-integ (math-div (nth 1 (nth 1 expr)) (nth 2 expr)) | |
1504 | var low high))) | |
1505 | ((and (eq (car-safe expr) '/) | |
1506 | (eq (car-safe (nth 2 expr)) '*) | |
1507 | (not (math-expr-contains (nth 1 (nth 2 expr)) var))) | |
1508 | (math-div (calcFunc-integ (math-div (nth 1 expr) (nth 2 (nth 2 expr))) | |
1509 | var low high) | |
1510 | (nth 1 (nth 2 expr)))) | |
1511 | ((and (eq (car-safe expr) '/) | |
1512 | (eq (car-safe (nth 2 expr)) '*) | |
1513 | (not (math-expr-contains (nth 2 (nth 2 expr)) var))) | |
1514 | (math-div (calcFunc-integ (math-div (nth 1 expr) (nth 1 (nth 2 expr))) | |
1515 | var low high) | |
1516 | (nth 2 (nth 2 expr)))) | |
1517 | ((eq (car-safe expr) 'vec) | |
1518 | (cons 'vec (mapcar (function (lambda (x) (calcFunc-integ x var low high))) | |
1519 | (cdr expr)))) | |
1520 | (t | |
1521 | (let ((state (list calc-angle-mode | |
1522 | ;;calc-symbolic-mode | |
1523 | ;;calc-prefer-frac | |
1524 | calc-internal-prec | |
1525 | (calc-var-value 'var-IntegRules) | |
1526 | (calc-var-value 'var-IntegSimpRules)))) | |
1527 | (or (equal state math-integral-cache-state) | |
1528 | (setq math-integral-cache-state state | |
1529 | math-integral-cache nil))) | |
f7adda54 | 1530 | (let* ((math-max-integral-limit (or (and (natnump var-IntegLimit) |
136211a9 EZ |
1531 | var-IntegLimit) |
1532 | 3)) | |
1533 | (math-integral-limit 1) | |
1534 | (sexpr (math-expr-subst expr var math-integ-var)) | |
1535 | (trace-buffer (get-buffer "*Trace*")) | |
1536 | (calc-language (if (eq calc-language 'big) nil calc-language)) | |
1537 | (math-any-substs t) | |
1538 | (math-enable-subst nil) | |
1539 | (math-prev-parts-v nil) | |
1540 | (math-doing-parts nil) | |
1541 | (math-good-parts nil) | |
1542 | (res | |
1543 | (if trace-buffer | |
1544 | (let ((calcbuf (current-buffer)) | |
1545 | (calcwin (selected-window))) | |
1546 | (unwind-protect | |
1547 | (progn | |
1548 | (if (get-buffer-window trace-buffer) | |
1549 | (select-window (get-buffer-window trace-buffer))) | |
1550 | (set-buffer trace-buffer) | |
1551 | (goto-char (point-max)) | |
1552 | (or (assq 'scroll-stop (buffer-local-variables)) | |
1553 | (progn | |
1554 | (make-local-variable 'scroll-step) | |
1555 | (setq scroll-step 3))) | |
1556 | (insert "\n\n\n") | |
1557 | (set-buffer calcbuf) | |
1558 | (math-try-integral sexpr)) | |
1559 | (select-window calcwin) | |
1560 | (set-buffer calcbuf))) | |
1561 | (math-try-integral sexpr)))) | |
1562 | (if res | |
1563 | (progn | |
1564 | (if (calc-has-rules 'var-IntegAfterRules) | |
1565 | (setq res (math-rewrite res '(var IntegAfterRules | |
1566 | var-IntegAfterRules)))) | |
1567 | (math-simplify | |
1568 | (if (and low high) | |
1569 | (math-sub (math-expr-subst res math-integ-var high) | |
1570 | (math-expr-subst res math-integ-var low)) | |
1571 | (setq res (math-fix-const-terms res math-integ-vars)) | |
1572 | (if low | |
1573 | (math-expr-subst res math-integ-var low) | |
1574 | (math-expr-subst res math-integ-var var))))) | |
1575 | (append (list 'calcFunc-integ expr var) | |
1576 | (and low (list low)) | |
bf77c646 | 1577 | (and high (list high)))))))) |
136211a9 EZ |
1578 | |
1579 | ||
1580 | (math-defintegral calcFunc-inv | |
1581 | (math-integral (math-div 1 u))) | |
1582 | ||
1583 | (math-defintegral calcFunc-conj | |
1584 | (let ((int (math-integral u))) | |
1585 | (and int | |
1586 | (list 'calcFunc-conj int)))) | |
1587 | ||
1588 | (math-defintegral calcFunc-deg | |
1589 | (let ((int (math-integral u))) | |
1590 | (and int | |
1591 | (list 'calcFunc-deg int)))) | |
1592 | ||
1593 | (math-defintegral calcFunc-rad | |
1594 | (let ((int (math-integral u))) | |
1595 | (and int | |
1596 | (list 'calcFunc-rad int)))) | |
1597 | ||
1598 | (math-defintegral calcFunc-re | |
1599 | (let ((int (math-integral u))) | |
1600 | (and int | |
1601 | (list 'calcFunc-re int)))) | |
1602 | ||
1603 | (math-defintegral calcFunc-im | |
1604 | (let ((int (math-integral u))) | |
1605 | (and int | |
1606 | (list 'calcFunc-im int)))) | |
1607 | ||
1608 | (math-defintegral calcFunc-sqrt | |
1609 | (and (equal u math-integ-var) | |
1610 | (math-mul '(frac 2 3) | |
1611 | (list 'calcFunc-sqrt (math-pow u 3))))) | |
1612 | ||
1613 | (math-defintegral calcFunc-exp | |
1614 | (or (and (equal u math-integ-var) | |
1615 | (list 'calcFunc-exp u)) | |
1616 | (let ((p (math-is-polynomial u math-integ-var 2))) | |
1617 | (and (nth 2 p) | |
1618 | (let ((sqa (math-sqrt (math-neg (nth 2 p))))) | |
1619 | (math-div | |
1620 | (math-mul | |
1621 | (math-mul (math-div (list 'calcFunc-sqrt '(var pi var-pi)) | |
1622 | sqa) | |
1623 | (math-normalize | |
1624 | (list 'calcFunc-exp | |
1625 | (math-div (math-sub (math-mul (car p) | |
1626 | (nth 2 p)) | |
1627 | (math-div | |
1628 | (math-sqr (nth 1 p)) | |
1629 | 4)) | |
1630 | (nth 2 p))))) | |
1631 | (list 'calcFunc-erf | |
1632 | (math-sub (math-mul sqa math-integ-var) | |
1633 | (math-div (nth 1 p) (math-mul 2 sqa))))) | |
1634 | 2)))))) | |
1635 | ||
1636 | (math-defintegral calcFunc-ln | |
1637 | (or (and (equal u math-integ-var) | |
1638 | (math-sub (math-mul u (list 'calcFunc-ln u)) u)) | |
1639 | (and (eq (car u) '*) | |
1640 | (math-integral (math-add (list 'calcFunc-ln (nth 1 u)) | |
1641 | (list 'calcFunc-ln (nth 2 u))))) | |
1642 | (and (eq (car u) '/) | |
1643 | (math-integral (math-sub (list 'calcFunc-ln (nth 1 u)) | |
1644 | (list 'calcFunc-ln (nth 2 u))))) | |
1645 | (and (eq (car u) '^) | |
1646 | (math-integral (math-mul (nth 2 u) | |
1647 | (list 'calcFunc-ln (nth 1 u))))))) | |
1648 | ||
1649 | (math-defintegral calcFunc-log10 | |
1650 | (and (equal u math-integ-var) | |
1651 | (math-sub (math-mul u (list 'calcFunc-ln u)) | |
1652 | (math-div u (list 'calcFunc-ln 10))))) | |
1653 | ||
1654 | (math-defintegral-2 calcFunc-log | |
1655 | (math-integral (math-div (list 'calcFunc-ln u) | |
1656 | (list 'calcFunc-ln v)))) | |
1657 | ||
1658 | (math-defintegral calcFunc-sin | |
1659 | (or (and (equal u math-integ-var) | |
1660 | (math-neg (math-from-radians-2 (list 'calcFunc-cos u)))) | |
1661 | (and (nth 2 (math-is-polynomial u math-integ-var 2)) | |
1662 | (math-integral (math-to-exponentials (list 'calcFunc-sin u)))))) | |
1663 | ||
1664 | (math-defintegral calcFunc-cos | |
1665 | (or (and (equal u math-integ-var) | |
1666 | (math-from-radians-2 (list 'calcFunc-sin u))) | |
1667 | (and (nth 2 (math-is-polynomial u math-integ-var 2)) | |
1668 | (math-integral (math-to-exponentials (list 'calcFunc-cos u)))))) | |
1669 | ||
1670 | (math-defintegral calcFunc-tan | |
1671 | (and (equal u math-integ-var) | |
dbf954fb JB |
1672 | (math-from-radians-2 |
1673 | (list 'calcFunc-ln (list 'calcFunc-sec u))))) | |
136211a9 | 1674 | |
9274224b JB |
1675 | (math-defintegral calcFunc-sec |
1676 | (and (equal u math-integ-var) | |
1677 | (math-from-radians-2 | |
f3445fab | 1678 | (list 'calcFunc-ln |
9274224b JB |
1679 | (math-add |
1680 | (list 'calcFunc-sec u) | |
1681 | (list 'calcFunc-tan u)))))) | |
1682 | ||
1683 | (math-defintegral calcFunc-csc | |
1684 | (and (equal u math-integ-var) | |
1685 | (math-from-radians-2 | |
f3445fab | 1686 | (list 'calcFunc-ln |
9274224b JB |
1687 | (math-sub |
1688 | (list 'calcFunc-csc u) | |
1689 | (list 'calcFunc-cot u)))))) | |
1690 | ||
1691 | (math-defintegral calcFunc-cot | |
1692 | (and (equal u math-integ-var) | |
1693 | (math-from-radians-2 | |
1694 | (list 'calcFunc-ln (list 'calcFunc-sin u))))) | |
1695 | ||
136211a9 EZ |
1696 | (math-defintegral calcFunc-arcsin |
1697 | (and (equal u math-integ-var) | |
1698 | (math-add (math-mul u (list 'calcFunc-arcsin u)) | |
1699 | (math-from-radians-2 | |
1700 | (list 'calcFunc-sqrt (math-sub 1 (math-sqr u))))))) | |
1701 | ||
1702 | (math-defintegral calcFunc-arccos | |
1703 | (and (equal u math-integ-var) | |
1704 | (math-sub (math-mul u (list 'calcFunc-arccos u)) | |
1705 | (math-from-radians-2 | |
1706 | (list 'calcFunc-sqrt (math-sub 1 (math-sqr u))))))) | |
1707 | ||
1708 | (math-defintegral calcFunc-arctan | |
1709 | (and (equal u math-integ-var) | |
1710 | (math-sub (math-mul u (list 'calcFunc-arctan u)) | |
1711 | (math-from-radians-2 | |
1712 | (math-div (list 'calcFunc-ln (math-add 1 (math-sqr u))) | |
1713 | 2))))) | |
1714 | ||
1715 | (math-defintegral calcFunc-sinh | |
1716 | (and (equal u math-integ-var) | |
1717 | (list 'calcFunc-cosh u))) | |
1718 | ||
1719 | (math-defintegral calcFunc-cosh | |
1720 | (and (equal u math-integ-var) | |
1721 | (list 'calcFunc-sinh u))) | |
1722 | ||
1723 | (math-defintegral calcFunc-tanh | |
1724 | (and (equal u math-integ-var) | |
1725 | (list 'calcFunc-ln (list 'calcFunc-cosh u)))) | |
1726 | ||
9274224b JB |
1727 | (math-defintegral calcFunc-sech |
1728 | (and (equal u math-integ-var) | |
1729 | (list 'calcFunc-arctan (list 'calcFunc-sinh u)))) | |
1730 | ||
1731 | (math-defintegral calcFunc-csch | |
1732 | (and (equal u math-integ-var) | |
1733 | (list 'calcFunc-ln (list 'calcFunc-tanh (math-div u 2))))) | |
1734 | ||
1735 | (math-defintegral calcFunc-coth | |
1736 | (and (equal u math-integ-var) | |
1737 | (list 'calcFunc-ln (list 'calcFunc-sinh u)))) | |
1738 | ||
136211a9 EZ |
1739 | (math-defintegral calcFunc-arcsinh |
1740 | (and (equal u math-integ-var) | |
1741 | (math-sub (math-mul u (list 'calcFunc-arcsinh u)) | |
1742 | (list 'calcFunc-sqrt (math-add (math-sqr u) 1))))) | |
1743 | ||
1744 | (math-defintegral calcFunc-arccosh | |
1745 | (and (equal u math-integ-var) | |
1746 | (math-sub (math-mul u (list 'calcFunc-arccosh u)) | |
1747 | (list 'calcFunc-sqrt (math-sub 1 (math-sqr u)))))) | |
1748 | ||
1749 | (math-defintegral calcFunc-arctanh | |
1750 | (and (equal u math-integ-var) | |
1751 | (math-sub (math-mul u (list 'calcFunc-arctan u)) | |
1752 | (math-div (list 'calcFunc-ln | |
1753 | (math-add 1 (math-sqr u))) | |
1754 | 2)))) | |
1755 | ||
1756 | ;;; (Ax + B) / (ax^2 + bx + c)^n forms. | |
1757 | (math-defintegral-2 / | |
1758 | (math-integral-rational-funcs u v)) | |
1759 | ||
1760 | (defun math-integral-rational-funcs (u v) | |
1761 | (let ((pu (math-is-polynomial u math-integ-var 1)) | |
1762 | (vpow 1) pv) | |
1763 | (and pu | |
1764 | (catch 'int-rat | |
1765 | (if (and (eq (car-safe v) '^) (natnump (nth 2 v))) | |
1766 | (setq vpow (nth 2 v) | |
1767 | v (nth 1 v))) | |
1768 | (and (setq pv (math-is-polynomial v math-integ-var 2)) | |
1769 | (let ((int (math-mul-thru | |
1770 | (car pu) | |
1771 | (math-integral-q02 (car pv) (nth 1 pv) | |
1772 | (nth 2 pv) v vpow)))) | |
1773 | (if (cdr pu) | |
1774 | (setq int (math-add int | |
1775 | (math-mul-thru | |
1776 | (nth 1 pu) | |
1777 | (math-integral-q12 | |
1778 | (car pv) (nth 1 pv) | |
1779 | (nth 2 pv) v vpow))))) | |
1780 | int)))))) | |
1781 | ||
1782 | (defun math-integral-q12 (a b c v vpow) | |
1783 | (let (q) | |
1784 | (cond ((not c) | |
1785 | (cond ((= vpow 1) | |
1786 | (math-sub (math-div math-integ-var b) | |
1787 | (math-mul (math-div a (math-sqr b)) | |
1788 | (list 'calcFunc-ln v)))) | |
1789 | ((= vpow 2) | |
1790 | (math-div (math-add (list 'calcFunc-ln v) | |
1791 | (math-div a v)) | |
1792 | (math-sqr b))) | |
1793 | (t | |
1794 | (let ((nm1 (math-sub vpow 1)) | |
1795 | (nm2 (math-sub vpow 2))) | |
1796 | (math-div (math-sub | |
1797 | (math-div a (math-mul nm1 (math-pow v nm1))) | |
1798 | (math-div 1 (math-mul nm2 (math-pow v nm2)))) | |
1799 | (math-sqr b)))))) | |
1800 | ((math-zerop | |
1801 | (setq q (math-sub (math-mul 4 (math-mul a c)) (math-sqr b)))) | |
1802 | (let ((part (math-div b (math-mul 2 c)))) | |
1803 | (math-mul-thru (math-pow c vpow) | |
1804 | (math-integral-q12 part 1 nil | |
1805 | (math-add math-integ-var part) | |
1806 | (* vpow 2))))) | |
1807 | ((= vpow 1) | |
1808 | (and (math-ratp q) (math-negp q) | |
1809 | (let ((calc-symbolic-mode t)) | |
1810 | (math-ratp (math-sqrt (math-neg q)))) | |
1811 | (throw 'int-rat nil)) ; should have used calcFunc-apart first | |
1812 | (math-sub (math-div (list 'calcFunc-ln v) (math-mul 2 c)) | |
1813 | (math-mul-thru (math-div b (math-mul 2 c)) | |
1814 | (math-integral-q02 a b c v 1)))) | |
1815 | (t | |
1816 | (let ((n (1- vpow))) | |
1817 | (math-sub (math-neg (math-div | |
1818 | (math-add (math-mul b math-integ-var) | |
1819 | (math-mul 2 a)) | |
1820 | (math-mul n (math-mul q (math-pow v n))))) | |
1821 | (math-mul-thru (math-div (math-mul b (1- (* 2 n))) | |
1822 | (math-mul n q)) | |
bf77c646 | 1823 | (math-integral-q02 a b c v n)))))))) |
136211a9 EZ |
1824 | |
1825 | (defun math-integral-q02 (a b c v vpow) | |
1826 | (let (q rq part) | |
1827 | (cond ((not c) | |
1828 | (cond ((= vpow 1) | |
1829 | (math-div (list 'calcFunc-ln v) b)) | |
1830 | (t | |
1831 | (math-div (math-pow v (- 1 vpow)) | |
1832 | (math-mul (- 1 vpow) b))))) | |
1833 | ((math-zerop | |
1834 | (setq q (math-sub (math-mul 4 (math-mul a c)) (math-sqr b)))) | |
1835 | (let ((part (math-div b (math-mul 2 c)))) | |
1836 | (math-mul-thru (math-pow c vpow) | |
1837 | (math-integral-q02 part 1 nil | |
1838 | (math-add math-integ-var part) | |
1839 | (* vpow 2))))) | |
1840 | ((progn | |
1841 | (setq part (math-add (math-mul 2 (math-mul c math-integ-var)) b)) | |
1842 | (> vpow 1)) | |
1843 | (let ((n (1- vpow))) | |
1844 | (math-add (math-div part (math-mul n (math-mul q (math-pow v n)))) | |
1845 | (math-mul-thru (math-div (math-mul (- (* 4 n) 2) c) | |
1846 | (math-mul n q)) | |
1847 | (math-integral-q02 a b c v n))))) | |
1848 | ((math-guess-if-neg q) | |
1849 | (setq rq (list 'calcFunc-sqrt (math-neg q))) | |
1850 | ;;(math-div-thru (list 'calcFunc-ln | |
1851 | ;; (math-div (math-sub part rq) | |
1852 | ;; (math-add part rq))) | |
1853 | ;; rq) | |
1854 | (math-div (math-mul -2 (list 'calcFunc-arctanh | |
1855 | (math-div part rq))) | |
1856 | rq)) | |
1857 | (t | |
1858 | (setq rq (list 'calcFunc-sqrt q)) | |
1859 | (math-div (math-mul 2 (math-to-radians-2 | |
1860 | (list 'calcFunc-arctan | |
1861 | (math-div part rq)))) | |
bf77c646 | 1862 | rq))))) |
136211a9 EZ |
1863 | |
1864 | ||
1865 | (math-defintegral calcFunc-erf | |
1866 | (and (equal u math-integ-var) | |
1867 | (math-add (math-mul u (list 'calcFunc-erf u)) | |
1868 | (math-div 1 (math-mul (list 'calcFunc-exp (math-sqr u)) | |
1869 | (list 'calcFunc-sqrt | |
1870 | '(var pi var-pi))))))) | |
1871 | ||
1872 | (math-defintegral calcFunc-erfc | |
1873 | (and (equal u math-integ-var) | |
1874 | (math-sub (math-mul u (list 'calcFunc-erfc u)) | |
1875 | (math-div 1 (math-mul (list 'calcFunc-exp (math-sqr u)) | |
1876 | (list 'calcFunc-sqrt | |
1877 | '(var pi var-pi))))))) | |
1878 | ||
1879 | ||
1880 | ||
1881 | ||
3132f345 CW |
1882 | (defvar math-tabulate-initial nil) |
1883 | (defvar math-tabulate-function nil) | |
f7adda54 | 1884 | |
f3445fab GM |
1885 | ;; These variables are local to calcFunc-table, but are used by |
1886 | ;; math-scan-for-limits. | |
f7adda54 JB |
1887 | (defvar calc-low) |
1888 | (defvar calc-high) | |
f3445fab | 1889 | (defvar var) |
f7adda54 JB |
1890 | |
1891 | (defun calcFunc-table (expr var &optional calc-low calc-high step) | |
f3445fab | 1892 | (or calc-low |
f7adda54 JB |
1893 | (setq calc-low '(neg (var inf var-inf)) calc-high '(var inf var-inf))) |
1894 | (or calc-high (setq calc-high calc-low calc-low 1)) | |
1895 | (and (or (math-infinitep calc-low) (math-infinitep calc-high)) | |
136211a9 EZ |
1896 | (not step) |
1897 | (math-scan-for-limits expr)) | |
1898 | (and step (math-zerop step) (math-reject-arg step 'nonzerop)) | |
f7adda54 JB |
1899 | (let ((known (+ (if (Math-objectp calc-low) 1 0) |
1900 | (if (Math-objectp calc-high) 1 0) | |
136211a9 EZ |
1901 | (if (or (null step) (Math-objectp step)) 1 0))) |
1902 | (count '(var inf var-inf)) | |
1903 | vec) | |
1904 | (or (= known 2) ; handy optimization | |
f7adda54 | 1905 | (equal calc-high '(var inf var-inf)) |
136211a9 | 1906 | (progn |
f7adda54 | 1907 | (setq count (math-div (math-sub calc-high calc-low) (or step 1))) |
136211a9 EZ |
1908 | (or (Math-objectp count) |
1909 | (setq count (math-simplify count))) | |
1910 | (if (Math-messy-integerp count) | |
1911 | (setq count (math-trunc count))))) | |
1912 | (if (Math-negp count) | |
1913 | (setq count -1)) | |
1914 | (if (integerp count) | |
1915 | (let ((var-DUMMY nil) | |
1916 | (vec math-tabulate-initial) | |
1917 | (math-working-step-2 (1+ count)) | |
1918 | (math-working-step 0)) | |
1919 | (setq expr (math-evaluate-expr | |
1920 | (math-expr-subst expr var '(var DUMMY var-DUMMY)))) | |
1921 | (while (>= count 0) | |
1922 | (setq math-working-step (1+ math-working-step) | |
f7adda54 | 1923 | var-DUMMY calc-low |
136211a9 EZ |
1924 | vec (cond ((eq math-tabulate-function 'calcFunc-sum) |
1925 | (math-add vec (math-evaluate-expr expr))) | |
1926 | ((eq math-tabulate-function 'calcFunc-prod) | |
1927 | (math-mul vec (math-evaluate-expr expr))) | |
1928 | (t | |
1929 | (cons (math-evaluate-expr expr) vec))) | |
f7adda54 | 1930 | calc-low (math-add calc-low (or step 1)) |
136211a9 EZ |
1931 | count (1- count))) |
1932 | (if math-tabulate-function | |
1933 | vec | |
1934 | (cons 'vec (nreverse vec)))) | |
1935 | (if (Math-integerp count) | |
f7adda54 JB |
1936 | (calc-record-why 'fixnump calc-high) |
1937 | (if (Math-num-integerp calc-low) | |
1938 | (if (Math-num-integerp calc-high) | |
136211a9 | 1939 | (calc-record-why 'integerp step) |
f7adda54 JB |
1940 | (calc-record-why 'integerp calc-high)) |
1941 | (calc-record-why 'integerp calc-low))) | |
136211a9 EZ |
1942 | (append (list (or math-tabulate-function 'calcFunc-table) |
1943 | expr var) | |
f7adda54 JB |
1944 | (and (not (and (equal calc-low '(neg (var inf var-inf))) |
1945 | (equal calc-high '(var inf var-inf)))) | |
1946 | (list calc-low calc-high)) | |
bf77c646 | 1947 | (and step (list step)))))) |
136211a9 | 1948 | |
136211a9 EZ |
1949 | (defun math-scan-for-limits (x) |
1950 | (cond ((Math-primp x)) | |
1951 | ((and (eq (car x) 'calcFunc-subscr) | |
1952 | (Math-vectorp (nth 1 x)) | |
1953 | (math-expr-contains (nth 2 x) var)) | |
1954 | (let* ((calc-next-why nil) | |
1955 | (low-val (math-solve-for (nth 2 x) 1 var nil)) | |
1956 | (high-val (math-solve-for (nth 2 x) (1- (length (nth 1 x))) | |
1957 | var nil)) | |
1958 | temp) | |
1959 | (and low-val (math-realp low-val) | |
1960 | high-val (math-realp high-val)) | |
1961 | (and (Math-lessp high-val low-val) | |
1962 | (setq temp low-val low-val high-val high-val temp)) | |
f7adda54 JB |
1963 | (setq calc-low (math-max calc-low (math-ceiling low-val)) |
1964 | calc-high (math-min calc-high (math-floor high-val))))) | |
136211a9 EZ |
1965 | (t |
1966 | (while (setq x (cdr x)) | |
bf77c646 | 1967 | (math-scan-for-limits (car x)))))) |
136211a9 EZ |
1968 | |
1969 | ||
3132f345 | 1970 | (defvar math-disable-sums nil) |
136211a9 EZ |
1971 | (defun calcFunc-sum (expr var &optional low high step) |
1972 | (if math-disable-sums (math-reject-arg)) | |
1973 | (let* ((res (let* ((calc-internal-prec (+ calc-internal-prec 2))) | |
1974 | (math-sum-rec expr var low high step))) | |
1975 | (math-disable-sums t)) | |
bf77c646 | 1976 | (math-normalize res))) |
136211a9 EZ |
1977 | |
1978 | (defun math-sum-rec (expr var &optional low high step) | |
1979 | (or low (setq low '(neg (var inf var-inf)) high '(var inf var-inf))) | |
1980 | (and low (not high) (setq high low low 1)) | |
1981 | (let (t1 t2 val) | |
1982 | (setq val | |
1983 | (cond | |
1984 | ((not (math-expr-contains expr var)) | |
1985 | (math-mul expr (math-add (math-div (math-sub high low) (or step 1)) | |
1986 | 1))) | |
1987 | ((and step (not (math-equal-int step 1))) | |
1988 | (if (math-negp step) | |
1989 | (math-sum-rec expr var high low (math-neg step)) | |
1990 | (let ((lo (math-simplify (math-div low step)))) | |
1991 | (if (math-known-num-integerp lo) | |
1992 | (math-sum-rec (math-normalize | |
1993 | (math-expr-subst expr var | |
1994 | (math-mul step var))) | |
1995 | var lo (math-simplify (math-div high step))) | |
1996 | (math-sum-rec (math-normalize | |
1997 | (math-expr-subst expr var | |
1998 | (math-add (math-mul step var) | |
1999 | low))) | |
2000 | var 0 | |
2001 | (math-simplify (math-div (math-sub high low) | |
2002 | step))))))) | |
2003 | ((memq (setq t1 (math-compare low high)) '(0 1)) | |
2004 | (if (eq t1 0) | |
2005 | (math-expr-subst expr var low) | |
2006 | 0)) | |
2007 | ((setq t1 (math-is-polynomial expr var 20)) | |
2008 | (let ((poly nil) | |
2009 | (n 0)) | |
2010 | (while t1 | |
2011 | (setq poly (math-poly-mix poly 1 | |
2012 | (math-sum-integer-power n) (car t1)) | |
2013 | n (1+ n) | |
2014 | t1 (cdr t1))) | |
2015 | (setq n (math-build-polynomial-expr poly high)) | |
a6c72dc3 | 2016 | (if (= low 1) |
136211a9 EZ |
2017 | n |
2018 | (math-sub n (math-build-polynomial-expr poly | |
2019 | (math-sub low 1)))))) | |
2020 | ((and (memq (car expr) '(+ -)) | |
2021 | (setq t1 (math-sum-rec (nth 1 expr) var low high) | |
2022 | t2 (math-sum-rec (nth 2 expr) var low high)) | |
2023 | (not (and (math-expr-calls t1 '(calcFunc-sum)) | |
2024 | (math-expr-calls t2 '(calcFunc-sum))))) | |
2025 | (list (car expr) t1 t2)) | |
2026 | ((and (eq (car expr) '*) | |
2027 | (setq t1 (math-sum-const-factors expr var))) | |
2028 | (math-mul (car t1) (math-sum-rec (cdr t1) var low high))) | |
2029 | ((and (eq (car expr) '*) (memq (car-safe (nth 1 expr)) '(+ -))) | |
2030 | (math-sum-rec (math-add-or-sub (math-mul (nth 1 (nth 1 expr)) | |
2031 | (nth 2 expr)) | |
2032 | (math-mul (nth 2 (nth 1 expr)) | |
2033 | (nth 2 expr)) | |
2034 | nil (eq (car (nth 1 expr)) '-)) | |
2035 | var low high)) | |
2036 | ((and (eq (car expr) '*) (memq (car-safe (nth 2 expr)) '(+ -))) | |
2037 | (math-sum-rec (math-add-or-sub (math-mul (nth 1 expr) | |
2038 | (nth 1 (nth 2 expr))) | |
2039 | (math-mul (nth 1 expr) | |
2040 | (nth 2 (nth 2 expr))) | |
2041 | nil (eq (car (nth 2 expr)) '-)) | |
2042 | var low high)) | |
2043 | ((and (eq (car expr) '/) | |
2044 | (not (math-primp (nth 1 expr))) | |
2045 | (setq t1 (math-sum-const-factors (nth 1 expr) var))) | |
2046 | (math-mul (car t1) | |
2047 | (math-sum-rec (math-div (cdr t1) (nth 2 expr)) | |
2048 | var low high))) | |
2049 | ((and (eq (car expr) '/) | |
2050 | (setq t1 (math-sum-const-factors (nth 2 expr) var))) | |
2051 | (math-div (math-sum-rec (math-div (nth 1 expr) (cdr t1)) | |
2052 | var low high) | |
2053 | (car t1))) | |
2054 | ((eq (car expr) 'neg) | |
2055 | (math-neg (math-sum-rec (nth 1 expr) var low high))) | |
2056 | ((and (eq (car expr) '^) | |
2057 | (not (math-expr-contains (nth 1 expr) var)) | |
2058 | (setq t1 (math-is-polynomial (nth 2 expr) var 1))) | |
2059 | (let ((x (math-pow (nth 1 expr) (nth 1 t1)))) | |
2060 | (math-div (math-mul (math-sub (math-pow x (math-add 1 high)) | |
2061 | (math-pow x low)) | |
2062 | (math-pow (nth 1 expr) (car t1))) | |
2063 | (math-sub x 1)))) | |
2064 | ((and (setq t1 (math-to-exponentials expr)) | |
2065 | (setq t1 (math-sum-rec t1 var low high)) | |
2066 | (not (math-expr-calls t1 '(calcFunc-sum)))) | |
2067 | (math-to-exps t1)) | |
2068 | ((memq (car expr) '(calcFunc-ln calcFunc-log10)) | |
2069 | (list (car expr) (calcFunc-prod (nth 1 expr) var low high))) | |
2070 | ((and (eq (car expr) 'calcFunc-log) | |
2071 | (= (length expr) 3) | |
2072 | (not (math-expr-contains (nth 2 expr) var))) | |
2073 | (list 'calcFunc-log | |
2074 | (calcFunc-prod (nth 1 expr) var low high) | |
2075 | (nth 2 expr))))) | |
2076 | (if (equal val '(var nan var-nan)) (setq val nil)) | |
2077 | (or val | |
2078 | (let* ((math-tabulate-initial 0) | |
2079 | (math-tabulate-function 'calcFunc-sum)) | |
bf77c646 | 2080 | (calcFunc-table expr var low high))))) |
136211a9 EZ |
2081 | |
2082 | (defun calcFunc-asum (expr var low &optional high step no-mul-flag) | |
2083 | (or high (setq high low low 1)) | |
2084 | (if (and step (not (math-equal-int step 1))) | |
2085 | (if (math-negp step) | |
2086 | (math-mul (math-pow -1 low) | |
2087 | (calcFunc-asum expr var high low (math-neg step) t)) | |
2088 | (let ((lo (math-simplify (math-div low step)))) | |
2089 | (if (math-num-integerp lo) | |
2090 | (calcFunc-asum (math-normalize | |
2091 | (math-expr-subst expr var | |
2092 | (math-mul step var))) | |
2093 | var lo (math-simplify (math-div high step))) | |
2094 | (calcFunc-asum (math-normalize | |
2095 | (math-expr-subst expr var | |
2096 | (math-add (math-mul step var) | |
2097 | low))) | |
2098 | var 0 | |
2099 | (math-simplify (math-div (math-sub high low) | |
2100 | step)))))) | |
2101 | (math-mul (if no-mul-flag 1 (math-pow -1 low)) | |
bf77c646 | 2102 | (calcFunc-sum (math-mul (math-pow -1 var) expr) var low high)))) |
136211a9 EZ |
2103 | |
2104 | (defun math-sum-const-factors (expr var) | |
2105 | (let ((const nil) | |
2106 | (not-const nil) | |
2107 | (p expr)) | |
2108 | (while (eq (car-safe p) '*) | |
2109 | (if (math-expr-contains (nth 1 p) var) | |
2110 | (setq not-const (cons (nth 1 p) not-const)) | |
2111 | (setq const (cons (nth 1 p) const))) | |
2112 | (setq p (nth 2 p))) | |
2113 | (if (math-expr-contains p var) | |
2114 | (setq not-const (cons p not-const)) | |
2115 | (setq const (cons p const))) | |
2116 | (and const | |
2117 | (cons (let ((temp (car const))) | |
2118 | (while (setq const (cdr const)) | |
2119 | (setq temp (list '* (car const) temp))) | |
2120 | temp) | |
2121 | (let ((temp (or (car not-const) 1))) | |
2122 | (while (setq not-const (cdr not-const)) | |
2123 | (setq temp (list '* (car not-const) temp))) | |
bf77c646 | 2124 | temp))))) |
136211a9 | 2125 | |
3132f345 | 2126 | (defvar math-sum-int-pow-cache (list '(0 1))) |
136211a9 EZ |
2127 | ;; Following is from CRC Math Tables, 27th ed, pp. 52-53. |
2128 | (defun math-sum-integer-power (pow) | |
2129 | (let ((calc-prefer-frac t) | |
2130 | (n (length math-sum-int-pow-cache))) | |
2131 | (while (<= n pow) | |
2132 | (let* ((new (list 0 0)) | |
2133 | (lin new) | |
2134 | (pp (cdr (nth (1- n) math-sum-int-pow-cache))) | |
2135 | (p 2) | |
2136 | (sum 0) | |
2137 | q) | |
2138 | (while pp | |
2139 | (setq q (math-div (car pp) p) | |
2140 | new (cons (math-mul q n) new) | |
2141 | sum (math-add sum q) | |
2142 | p (1+ p) | |
2143 | pp (cdr pp))) | |
2144 | (setcar lin (math-sub 1 (math-mul n sum))) | |
2145 | (setq math-sum-int-pow-cache | |
2146 | (nconc math-sum-int-pow-cache (list (nreverse new))) | |
2147 | n (1+ n)))) | |
bf77c646 | 2148 | (nth pow math-sum-int-pow-cache))) |
136211a9 EZ |
2149 | |
2150 | (defun math-to-exponentials (expr) | |
2151 | (and (consp expr) | |
2152 | (= (length expr) 2) | |
2153 | (let ((x (nth 1 expr)) | |
2154 | (pi (if calc-symbolic-mode '(var pi var-pi) (math-pi))) | |
2155 | (i (if calc-symbolic-mode '(var i var-i) '(cplx 0 1)))) | |
2156 | (cond ((eq (car expr) 'calcFunc-exp) | |
2157 | (list '^ '(var e var-e) x)) | |
2158 | ((eq (car expr) 'calcFunc-sin) | |
2159 | (or (eq calc-angle-mode 'rad) | |
2160 | (setq x (list '/ (list '* x pi) 180))) | |
2161 | (list '/ (list '- | |
2162 | (list '^ '(var e var-e) (list '* x i)) | |
2163 | (list '^ '(var e var-e) | |
2164 | (list 'neg (list '* x i)))) | |
2165 | (list '* 2 i))) | |
2166 | ((eq (car expr) 'calcFunc-cos) | |
2167 | (or (eq calc-angle-mode 'rad) | |
2168 | (setq x (list '/ (list '* x pi) 180))) | |
2169 | (list '/ (list '+ | |
2170 | (list '^ '(var e var-e) | |
2171 | (list '* x i)) | |
2172 | (list '^ '(var e var-e) | |
2173 | (list 'neg (list '* x i)))) | |
2174 | 2)) | |
2175 | ((eq (car expr) 'calcFunc-sinh) | |
2176 | (list '/ (list '- | |
2177 | (list '^ '(var e var-e) x) | |
2178 | (list '^ '(var e var-e) (list 'neg x))) | |
2179 | 2)) | |
2180 | ((eq (car expr) 'calcFunc-cosh) | |
2181 | (list '/ (list '+ | |
2182 | (list '^ '(var e var-e) x) | |
2183 | (list '^ '(var e var-e) (list 'neg x))) | |
2184 | 2)) | |
bf77c646 | 2185 | (t nil))))) |
136211a9 EZ |
2186 | |
2187 | (defun math-to-exps (expr) | |
2188 | (cond (calc-symbolic-mode expr) | |
2189 | ((Math-primp expr) | |
2190 | (if (equal expr '(var e var-e)) (math-e) expr)) | |
2191 | ((and (eq (car expr) '^) | |
2192 | (equal (nth 1 expr) '(var e var-e))) | |
2193 | (list 'calcFunc-exp (nth 2 expr))) | |
2194 | (t | |
bf77c646 | 2195 | (cons (car expr) (mapcar 'math-to-exps (cdr expr)))))) |
136211a9 EZ |
2196 | |
2197 | ||
3132f345 | 2198 | (defvar math-disable-prods nil) |
136211a9 EZ |
2199 | (defun calcFunc-prod (expr var &optional low high step) |
2200 | (if math-disable-prods (math-reject-arg)) | |
2201 | (let* ((res (let* ((calc-internal-prec (+ calc-internal-prec 2))) | |
2202 | (math-prod-rec expr var low high step))) | |
2203 | (math-disable-prods t)) | |
bf77c646 | 2204 | (math-normalize res))) |
136211a9 EZ |
2205 | |
2206 | (defun math-prod-rec (expr var &optional low high step) | |
2207 | (or low (setq low '(neg (var inf var-inf)) high '(var inf var-inf))) | |
2208 | (and low (not high) (setq high '(var inf var-inf))) | |
2209 | (let (t1 t2 t3 val) | |
2210 | (setq val | |
2211 | (cond | |
2212 | ((not (math-expr-contains expr var)) | |
2213 | (math-pow expr (math-add (math-div (math-sub high low) (or step 1)) | |
2214 | 1))) | |
2215 | ((and step (not (math-equal-int step 1))) | |
2216 | (if (math-negp step) | |
2217 | (math-prod-rec expr var high low (math-neg step)) | |
2218 | (let ((lo (math-simplify (math-div low step)))) | |
2219 | (if (math-known-num-integerp lo) | |
2220 | (math-prod-rec (math-normalize | |
2221 | (math-expr-subst expr var | |
2222 | (math-mul step var))) | |
2223 | var lo (math-simplify (math-div high step))) | |
2224 | (math-prod-rec (math-normalize | |
2225 | (math-expr-subst expr var | |
2226 | (math-add (math-mul step | |
2227 | var) | |
2228 | low))) | |
2229 | var 0 | |
2230 | (math-simplify (math-div (math-sub high low) | |
2231 | step))))))) | |
2232 | ((and (memq (car expr) '(* /)) | |
2233 | (setq t1 (math-prod-rec (nth 1 expr) var low high) | |
2234 | t2 (math-prod-rec (nth 2 expr) var low high)) | |
2235 | (not (and (math-expr-calls t1 '(calcFunc-prod)) | |
2236 | (math-expr-calls t2 '(calcFunc-prod))))) | |
2237 | (list (car expr) t1 t2)) | |
2238 | ((and (eq (car expr) '^) | |
2239 | (not (math-expr-contains (nth 2 expr) var))) | |
2240 | (math-pow (math-prod-rec (nth 1 expr) var low high) | |
2241 | (nth 2 expr))) | |
2242 | ((and (eq (car expr) '^) | |
2243 | (not (math-expr-contains (nth 1 expr) var))) | |
2244 | (math-pow (nth 1 expr) | |
2245 | (calcFunc-sum (nth 2 expr) var low high))) | |
2246 | ((eq (car expr) 'sqrt) | |
2247 | (math-normalize (list 'calcFunc-sqrt | |
2248 | (list 'calcFunc-prod (nth 1 expr) | |
2249 | var low high)))) | |
2250 | ((eq (car expr) 'neg) | |
2251 | (math-mul (math-pow -1 (math-add (math-sub high low) 1)) | |
2252 | (math-prod-rec (nth 1 expr) var low high))) | |
2253 | ((eq (car expr) 'calcFunc-exp) | |
2254 | (list 'calcFunc-exp (calcFunc-sum (nth 1 expr) var low high))) | |
2255 | ((and (setq t1 (math-is-polynomial expr var 1)) | |
2256 | (setq t2 | |
2257 | (cond | |
2258 | ((or (and (math-equal-int (nth 1 t1) 1) | |
2259 | (setq low (math-simplify | |
2260 | (math-add low (car t1))) | |
2261 | high (math-simplify | |
2262 | (math-add high (car t1))))) | |
2263 | (and (math-equal-int (nth 1 t1) -1) | |
2264 | (setq t2 low | |
2265 | low (math-simplify | |
2266 | (math-sub (car t1) high)) | |
2267 | high (math-simplify | |
2268 | (math-sub (car t1) t2))))) | |
2269 | (if (or (math-zerop low) (math-zerop high)) | |
2270 | 0 | |
2271 | (if (and (or (math-negp low) (math-negp high)) | |
2272 | (or (math-num-integerp low) | |
2273 | (math-num-integerp high))) | |
2274 | (if (math-posp high) | |
2275 | 0 | |
2276 | (math-mul (math-pow -1 | |
2277 | (math-add | |
2278 | (math-add low high) 1)) | |
2279 | (list '/ | |
2280 | (list 'calcFunc-fact | |
2281 | (math-neg low)) | |
2282 | (list 'calcFunc-fact | |
2283 | (math-sub -1 high))))) | |
2284 | (list '/ | |
2285 | (list 'calcFunc-fact high) | |
2286 | (list 'calcFunc-fact (math-sub low 1)))))) | |
2287 | ((and (or (and (math-equal-int (nth 1 t1) 2) | |
2288 | (setq t2 (math-simplify | |
2289 | (math-add (math-mul low 2) | |
2290 | (car t1))) | |
2291 | t3 (math-simplify | |
2292 | (math-add (math-mul high 2) | |
2293 | (car t1))))) | |
2294 | (and (math-equal-int (nth 1 t1) -2) | |
2295 | (setq t2 (math-simplify | |
2296 | (math-sub (car t1) | |
2297 | (math-mul high 2))) | |
a1506d29 | 2298 | t3 (math-simplify |
136211a9 EZ |
2299 | (math-sub (car t1) |
2300 | (math-mul low | |
2301 | 2)))))) | |
2302 | (or (math-integerp t2) | |
2303 | (and (math-messy-integerp t2) | |
2304 | (setq t2 (math-trunc t2))) | |
2305 | (math-integerp t3) | |
2306 | (and (math-messy-integerp t3) | |
2307 | (setq t3 (math-trunc t3))))) | |
2308 | (if (or (math-zerop t2) (math-zerop t3)) | |
2309 | 0 | |
2310 | (if (or (math-evenp t2) (math-evenp t3)) | |
2311 | (if (or (math-negp t2) (math-negp t3)) | |
2312 | (if (math-posp high) | |
2313 | 0 | |
2314 | (list '/ | |
2315 | (list 'calcFunc-dfact | |
2316 | (math-neg t2)) | |
2317 | (list 'calcFunc-dfact | |
2318 | (math-sub -2 t3)))) | |
2319 | (list '/ | |
2320 | (list 'calcFunc-dfact t3) | |
2321 | (list 'calcFunc-dfact | |
2322 | (math-sub t2 2)))) | |
2323 | (if (math-negp t3) | |
2324 | (list '* | |
2325 | (list '^ -1 | |
2326 | (list '/ (list '- (list '- t2 t3) | |
2327 | 2) | |
2328 | 2)) | |
2329 | (list '/ | |
2330 | (list 'calcFunc-dfact | |
2331 | (math-neg t2)) | |
2332 | (list 'calcFunc-dfact | |
2333 | (math-sub -2 t3)))) | |
2334 | (if (math-posp t2) | |
2335 | (list '/ | |
2336 | (list 'calcFunc-dfact t3) | |
2337 | (list 'calcFunc-dfact | |
2338 | (math-sub t2 2))) | |
2339 | nil)))))))) | |
2340 | t2))) | |
2341 | (if (equal val '(var nan var-nan)) (setq val nil)) | |
2342 | (or val | |
2343 | (let* ((math-tabulate-initial 1) | |
2344 | (math-tabulate-function 'calcFunc-prod)) | |
bf77c646 | 2345 | (calcFunc-table expr var low high))))) |
136211a9 EZ |
2346 | |
2347 | ||
2348 | ||
2349 | ||
3132f345 | 2350 | (defvar math-solve-ranges nil) |
f7adda54 | 2351 | (defvar math-solve-sign) |
f3445fab | 2352 | ;;; Attempt to reduce math-solve-lhs = math-solve-rhs to |
f7adda54 | 2353 | ;;; math-solve-var = math-solve-rhs', where math-solve-var appears |
f3445fab | 2354 | ;;; in math-solve-lhs but not in math-solve-rhs or math-solve-rhs'; |
f7adda54 JB |
2355 | ;;; return math-solve-rhs'. |
2356 | ;;; Uses global values: math-solve-var, math-solve-full. | |
2357 | (defvar math-solve-var) | |
2358 | (defvar math-solve-full) | |
2359 | ||
f3445fab GM |
2360 | ;; The variables math-solve-lhs, math-solve-rhs and math-try-solve-sign |
2361 | ;; are local to math-try-solve-for, but are used by math-try-solve-prod. | |
2362 | ;; (math-solve-lhs and math-solve-rhs are is also local to | |
f7adda54 JB |
2363 | ;; math-decompose-poly, but used by math-solve-poly-funny-powers.) |
2364 | (defvar math-solve-lhs) | |
2365 | (defvar math-solve-rhs) | |
79d2746f | 2366 | (defvar math-try-solve-sign) |
f7adda54 | 2367 | |
f3445fab | 2368 | (defun math-try-solve-for |
f7adda54 JB |
2369 | (math-solve-lhs math-solve-rhs &optional math-try-solve-sign no-poly) |
2370 | (let (math-t1 math-t2 math-t3) | |
2371 | (cond ((equal math-solve-lhs math-solve-var) | |
2372 | (setq math-solve-sign math-try-solve-sign) | |
2373 | (if (eq math-solve-full 'all) | |
2374 | (let ((vec (list 'vec (math-evaluate-expr math-solve-rhs))) | |
136211a9 EZ |
2375 | newvec var p) |
2376 | (while math-solve-ranges | |
2377 | (setq p (car math-solve-ranges) | |
2378 | var (car p) | |
2379 | newvec (list 'vec)) | |
2380 | (while (setq p (cdr p)) | |
2381 | (setq newvec (nconc newvec | |
2382 | (cdr (math-expr-subst | |
2383 | vec var (car p)))))) | |
2384 | (setq vec newvec | |
2385 | math-solve-ranges (cdr math-solve-ranges))) | |
2386 | (math-normalize vec)) | |
f7adda54 JB |
2387 | math-solve-rhs)) |
2388 | ((Math-primp math-solve-lhs) | |
136211a9 | 2389 | nil) |
f7adda54 JB |
2390 | ((and (eq (car math-solve-lhs) '-) |
2391 | (eq (car-safe (nth 1 math-solve-lhs)) (car-safe (nth 2 math-solve-lhs))) | |
2392 | (Math-zerop math-solve-rhs) | |
2393 | (= (length (nth 1 math-solve-lhs)) 2) | |
2394 | (= (length (nth 2 math-solve-lhs)) 2) | |
2395 | (setq math-t1 (get (car (nth 1 math-solve-lhs)) 'math-inverse)) | |
2396 | (setq math-t2 (funcall math-t1 '(var SOLVEDUM SOLVEDUM))) | |
2397 | (eq (math-expr-contains-count math-t2 '(var SOLVEDUM SOLVEDUM)) 1) | |
2398 | (setq math-t3 (math-solve-above-dummy math-t2)) | |
f3445fab | 2399 | (setq math-t1 (math-try-solve-for |
f7adda54 JB |
2400 | (math-sub (nth 1 (nth 1 math-solve-lhs)) |
2401 | (math-expr-subst | |
2402 | math-t2 math-t3 | |
2403 | (nth 1 (nth 2 math-solve-lhs)))) | |
2404 | 0))) | |
2405 | math-t1) | |
2406 | ((eq (car math-solve-lhs) 'neg) | |
2407 | (math-try-solve-for (nth 1 math-solve-lhs) (math-neg math-solve-rhs) | |
2408 | (and math-try-solve-sign (- math-try-solve-sign)))) | |
2409 | ((and (not (eq math-solve-full 't)) (math-try-solve-prod))) | |
136211a9 | 2410 | ((and (not no-poly) |
f3445fab GM |
2411 | (setq math-t2 |
2412 | (math-decompose-poly math-solve-lhs | |
f7adda54 JB |
2413 | math-solve-var 15 math-solve-rhs))) |
2414 | (setq math-t1 (cdr (nth 1 math-t2)) | |
2415 | math-t1 (let ((math-solve-ranges math-solve-ranges)) | |
2416 | (cond ((= (length math-t1) 5) | |
2417 | (apply 'math-solve-quartic (car math-t2) math-t1)) | |
2418 | ((= (length math-t1) 4) | |
2419 | (apply 'math-solve-cubic (car math-t2) math-t1)) | |
2420 | ((= (length math-t1) 3) | |
2421 | (apply 'math-solve-quadratic (car math-t2) math-t1)) | |
2422 | ((= (length math-t1) 2) | |
f3445fab | 2423 | (apply 'math-solve-linear |
f7adda54 JB |
2424 | (car math-t2) math-try-solve-sign math-t1)) |
2425 | (math-solve-full | |
2426 | (math-poly-all-roots (car math-t2) math-t1)) | |
136211a9 EZ |
2427 | (calc-symbolic-mode nil) |
2428 | (t | |
2429 | (math-try-solve-for | |
f7adda54 JB |
2430 | (car math-t2) |
2431 | (math-poly-any-root (reverse math-t1) 0 t) | |
136211a9 | 2432 | nil t))))) |
f7adda54 JB |
2433 | (if math-t1 |
2434 | (if (eq (nth 2 math-t2) 1) | |
2435 | math-t1 | |
2436 | (math-solve-prod math-t1 (math-try-solve-for (nth 2 math-t2) 0 nil t))) | |
136211a9 EZ |
2437 | (calc-record-why "*Unable to find a symbolic solution") |
2438 | nil)) | |
f7adda54 JB |
2439 | ((and (math-solve-find-root-term math-solve-lhs nil) |
2440 | (eq (math-expr-contains-count math-solve-lhs math-t1) 1)) ; just in case | |
136211a9 | 2441 | (math-try-solve-for (math-simplify |
f7adda54 JB |
2442 | (math-sub (if (or math-t3 (math-evenp math-t2)) |
2443 | (math-pow math-t1 math-t2) | |
2444 | (math-neg (math-pow math-t1 math-t2))) | |
136211a9 EZ |
2445 | (math-expand-power |
2446 | (math-sub (math-normalize | |
2447 | (math-expr-subst | |
f7adda54 JB |
2448 | math-solve-lhs math-t1 0)) |
2449 | math-solve-rhs) | |
2450 | math-t2 math-solve-var))) | |
136211a9 | 2451 | 0)) |
f7adda54 JB |
2452 | ((eq (car math-solve-lhs) '+) |
2453 | (cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var)) | |
2454 | (math-try-solve-for (nth 2 math-solve-lhs) | |
2455 | (math-sub math-solve-rhs (nth 1 math-solve-lhs)) | |
2456 | math-try-solve-sign)) | |
2457 | ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var)) | |
2458 | (math-try-solve-for (nth 1 math-solve-lhs) | |
2459 | (math-sub math-solve-rhs (nth 2 math-solve-lhs)) | |
2460 | math-try-solve-sign)))) | |
2461 | ((eq (car math-solve-lhs) 'calcFunc-eq) | |
2462 | (math-try-solve-for (math-sub (nth 1 math-solve-lhs) (nth 2 math-solve-lhs)) | |
2463 | math-solve-rhs math-try-solve-sign no-poly)) | |
2464 | ((eq (car math-solve-lhs) '-) | |
2465 | (cond ((or (and (eq (car-safe (nth 1 math-solve-lhs)) 'calcFunc-sin) | |
2466 | (eq (car-safe (nth 2 math-solve-lhs)) 'calcFunc-cos)) | |
2467 | (and (eq (car-safe (nth 1 math-solve-lhs)) 'calcFunc-cos) | |
2468 | (eq (car-safe (nth 2 math-solve-lhs)) 'calcFunc-sin))) | |
2469 | (math-try-solve-for (math-sub (nth 1 math-solve-lhs) | |
2470 | (list (car (nth 1 math-solve-lhs)) | |
136211a9 EZ |
2471 | (math-sub |
2472 | (math-quarter-circle t) | |
f7adda54 JB |
2473 | (nth 1 (nth 2 math-solve-lhs))))) |
2474 | math-solve-rhs)) | |
2475 | ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var)) | |
2476 | (math-try-solve-for (nth 2 math-solve-lhs) | |
2477 | (math-sub (nth 1 math-solve-lhs) math-solve-rhs) | |
f3445fab | 2478 | (and math-try-solve-sign |
f7adda54 JB |
2479 | (- math-try-solve-sign)))) |
2480 | ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var)) | |
2481 | (math-try-solve-for (nth 1 math-solve-lhs) | |
2482 | (math-add math-solve-rhs (nth 2 math-solve-lhs)) | |
2483 | math-try-solve-sign)))) | |
2484 | ((and (eq math-solve-full 't) (math-try-solve-prod))) | |
2485 | ((and (eq (car math-solve-lhs) '%) | |
2486 | (not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var))) | |
2487 | (math-try-solve-for (nth 1 math-solve-lhs) (math-add math-solve-rhs | |
136211a9 | 2488 | (math-solve-get-int |
f7adda54 JB |
2489 | (nth 2 math-solve-lhs))))) |
2490 | ((eq (car math-solve-lhs) 'calcFunc-log) | |
2491 | (cond ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var)) | |
f3445fab | 2492 | (math-try-solve-for (nth 1 math-solve-lhs) |
f7adda54 JB |
2493 | (math-pow (nth 2 math-solve-lhs) math-solve-rhs))) |
2494 | ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var)) | |
2495 | (math-try-solve-for (nth 2 math-solve-lhs) (math-pow | |
2496 | (nth 1 math-solve-lhs) | |
2497 | (math-div 1 math-solve-rhs)))))) | |
2498 | ((and (= (length math-solve-lhs) 2) | |
2499 | (symbolp (car math-solve-lhs)) | |
2500 | (setq math-t1 (get (car math-solve-lhs) 'math-inverse)) | |
2501 | (setq math-t2 (funcall math-t1 math-solve-rhs))) | |
2502 | (setq math-t1 (get (car math-solve-lhs) 'math-inverse-sign)) | |
2503 | (math-try-solve-for (nth 1 math-solve-lhs) (math-normalize math-t2) | |
2504 | (and math-try-solve-sign math-t1 | |
2505 | (if (integerp math-t1) | |
2506 | (* math-t1 math-try-solve-sign) | |
f3445fab | 2507 | (funcall math-t1 math-solve-lhs |
f7adda54 JB |
2508 | math-try-solve-sign))))) |
2509 | ((and (symbolp (car math-solve-lhs)) | |
2510 | (setq math-t1 (get (car math-solve-lhs) 'math-inverse-n)) | |
2511 | (setq math-t2 (funcall math-t1 math-solve-lhs math-solve-rhs))) | |
2512 | math-t2) | |
2513 | ((setq math-t1 (math-expand-formula math-solve-lhs)) | |
2514 | (math-try-solve-for math-t1 math-solve-rhs math-try-solve-sign)) | |
136211a9 | 2515 | (t |
f7adda54 | 2516 | (calc-record-why "*No inverse known" math-solve-lhs) |
bf77c646 | 2517 | nil)))) |
136211a9 | 2518 | |
136211a9 EZ |
2519 | |
2520 | (defun math-try-solve-prod () | |
f7adda54 JB |
2521 | (cond ((eq (car math-solve-lhs) '*) |
2522 | (cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var)) | |
2523 | (math-try-solve-for (nth 2 math-solve-lhs) | |
2524 | (math-div math-solve-rhs (nth 1 math-solve-lhs)) | |
f3445fab | 2525 | (math-solve-sign math-try-solve-sign |
f7adda54 JB |
2526 | (nth 1 math-solve-lhs)))) |
2527 | ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var)) | |
2528 | (math-try-solve-for (nth 1 math-solve-lhs) | |
2529 | (math-div math-solve-rhs (nth 2 math-solve-lhs)) | |
f3445fab | 2530 | (math-solve-sign math-try-solve-sign |
f7adda54 JB |
2531 | (nth 2 math-solve-lhs)))) |
2532 | ((Math-zerop math-solve-rhs) | |
136211a9 | 2533 | (math-solve-prod (let ((math-solve-ranges math-solve-ranges)) |
f7adda54 JB |
2534 | (math-try-solve-for (nth 2 math-solve-lhs) 0)) |
2535 | (math-try-solve-for (nth 1 math-solve-lhs) 0))))) | |
2536 | ((eq (car math-solve-lhs) '/) | |
2537 | (cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var)) | |
2538 | (math-try-solve-for (nth 2 math-solve-lhs) | |
2539 | (math-div (nth 1 math-solve-lhs) math-solve-rhs) | |
f3445fab | 2540 | (math-solve-sign math-try-solve-sign |
f7adda54 JB |
2541 | (nth 1 math-solve-lhs)))) |
2542 | ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var)) | |
2543 | (math-try-solve-for (nth 1 math-solve-lhs) | |
2544 | (math-mul math-solve-rhs (nth 2 math-solve-lhs)) | |
f3445fab | 2545 | (math-solve-sign math-try-solve-sign |
f7adda54 JB |
2546 | (nth 2 math-solve-lhs)))) |
2547 | ((setq math-t1 (math-try-solve-for (math-sub (nth 1 math-solve-lhs) | |
2548 | (math-mul (nth 2 math-solve-lhs) | |
2549 | math-solve-rhs)) | |
136211a9 | 2550 | 0)) |
f7adda54 JB |
2551 | math-t1))) |
2552 | ((eq (car math-solve-lhs) '^) | |
2553 | (cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var)) | |
136211a9 | 2554 | (math-try-solve-for |
f7adda54 | 2555 | (nth 2 math-solve-lhs) |
136211a9 | 2556 | (math-add (math-normalize |
f7adda54 | 2557 | (list 'calcFunc-log math-solve-rhs (nth 1 math-solve-lhs))) |
136211a9 EZ |
2558 | (math-div |
2559 | (math-mul 2 | |
2560 | (math-mul '(var pi var-pi) | |
2561 | (math-solve-get-int | |
2562 | '(var i var-i)))) | |
2563 | (math-normalize | |
f7adda54 JB |
2564 | (list 'calcFunc-ln (nth 1 math-solve-lhs))))))) |
2565 | ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var)) | |
2566 | (cond ((and (integerp (nth 2 math-solve-lhs)) | |
2567 | (>= (nth 2 math-solve-lhs) 2) | |
2568 | (setq math-t1 (math-integer-log2 (nth 2 math-solve-lhs)))) | |
2569 | (setq math-t2 math-solve-rhs) | |
2570 | (if (and (eq math-solve-full t) | |
2571 | (math-known-realp (nth 1 math-solve-lhs))) | |
136211a9 | 2572 | (progn |
f7adda54 JB |
2573 | (while (>= (setq math-t1 (1- math-t1)) 0) |
2574 | (setq math-t2 (list 'calcFunc-sqrt math-t2))) | |
2575 | (setq math-t2 (math-solve-get-sign math-t2))) | |
2576 | (while (>= (setq math-t1 (1- math-t1)) 0) | |
2577 | (setq math-t2 (math-solve-get-sign | |
136211a9 | 2578 | (math-normalize |
f7adda54 | 2579 | (list 'calcFunc-sqrt math-t2)))))) |
136211a9 | 2580 | (math-try-solve-for |
f7adda54 JB |
2581 | (nth 1 math-solve-lhs) |
2582 | (math-normalize math-t2))) | |
2583 | ((math-looks-negp (nth 2 math-solve-lhs)) | |
136211a9 | 2584 | (math-try-solve-for |
f3445fab | 2585 | (list '^ (nth 1 math-solve-lhs) |
f7adda54 JB |
2586 | (math-neg (nth 2 math-solve-lhs))) |
2587 | (math-div 1 math-solve-rhs))) | |
2588 | ((and (eq math-solve-full t) | |
2589 | (Math-integerp (nth 2 math-solve-lhs)) | |
2590 | (math-known-realp (nth 1 math-solve-lhs))) | |
2591 | (setq math-t1 (math-normalize | |
f3445fab | 2592 | (list 'calcFunc-nroot math-solve-rhs |
f7adda54 JB |
2593 | (nth 2 math-solve-lhs)))) |
2594 | (if (math-evenp (nth 2 math-solve-lhs)) | |
2595 | (setq math-t1 (math-solve-get-sign math-t1))) | |
136211a9 | 2596 | (math-try-solve-for |
f7adda54 JB |
2597 | (nth 1 math-solve-lhs) math-t1 |
2598 | (and math-try-solve-sign | |
2599 | (math-oddp (nth 2 math-solve-lhs)) | |
f3445fab | 2600 | (math-solve-sign math-try-solve-sign |
f7adda54 | 2601 | (nth 2 math-solve-lhs))))) |
136211a9 | 2602 | (t (math-try-solve-for |
f7adda54 | 2603 | (nth 1 math-solve-lhs) |
136211a9 EZ |
2604 | (math-mul |
2605 | (math-normalize | |
2606 | (list 'calcFunc-exp | |
f7adda54 | 2607 | (if (Math-realp (nth 2 math-solve-lhs)) |
136211a9 EZ |
2608 | (math-div (math-mul |
2609 | '(var pi var-pi) | |
2610 | (math-solve-get-int | |
2611 | '(var i var-i) | |
f7adda54 | 2612 | (and (integerp (nth 2 math-solve-lhs)) |
136211a9 | 2613 | (math-abs |
f7adda54 JB |
2614 | (nth 2 math-solve-lhs))))) |
2615 | (math-div (nth 2 math-solve-lhs) 2)) | |
136211a9 EZ |
2616 | (math-div (math-mul |
2617 | 2 | |
2618 | (math-mul | |
2619 | '(var pi var-pi) | |
2620 | (math-solve-get-int | |
2621 | '(var i var-i) | |
f7adda54 | 2622 | (and (integerp (nth 2 math-solve-lhs)) |
136211a9 | 2623 | (math-abs |
f7adda54 JB |
2624 | (nth 2 math-solve-lhs)))))) |
2625 | (nth 2 math-solve-lhs))))) | |
136211a9 EZ |
2626 | (math-normalize |
2627 | (list 'calcFunc-nroot | |
f7adda54 JB |
2628 | math-solve-rhs |
2629 | (nth 2 math-solve-lhs)))) | |
2630 | (and math-try-solve-sign | |
2631 | (math-oddp (nth 2 math-solve-lhs)) | |
f3445fab | 2632 | (math-solve-sign math-try-solve-sign |
f7adda54 | 2633 | (nth 2 math-solve-lhs))))))))) |
bf77c646 | 2634 | (t nil))) |
136211a9 EZ |
2635 | |
2636 | (defun math-solve-prod (lsoln rsoln) | |
2637 | (cond ((null lsoln) | |
2638 | rsoln) | |
2639 | ((null rsoln) | |
2640 | lsoln) | |
f7adda54 | 2641 | ((eq math-solve-full 'all) |
136211a9 | 2642 | (cons 'vec (append (cdr lsoln) (cdr rsoln)))) |
f7adda54 | 2643 | (math-solve-full |
136211a9 EZ |
2644 | (list 'calcFunc-if |
2645 | (list 'calcFunc-gt (math-solve-get-sign 1) 0) | |
2646 | lsoln | |
2647 | rsoln)) | |
bf77c646 | 2648 | (t lsoln))) |
136211a9 EZ |
2649 | |
2650 | ;;; This deals with negative, fractional, and symbolic powers of "x". | |
f7adda54 JB |
2651 | ;; The variable math-solve-b is local to math-decompose-poly, |
2652 | ;; but is used by math-solve-poly-funny-powers. | |
79d2746f | 2653 | (defvar math-solve-b) |
f7adda54 | 2654 | |
136211a9 | 2655 | (defun math-solve-poly-funny-powers (sub-rhs) ; uses "t1", "t2" |
f7adda54 | 2656 | (setq math-t1 math-solve-lhs) |
136211a9 EZ |
2657 | (let ((pp math-poly-neg-powers) |
2658 | fac) | |
2659 | (while pp | |
2660 | (setq fac (math-pow (car pp) (or math-poly-mult-powers 1)) | |
f7adda54 JB |
2661 | math-t1 (math-mul math-t1 fac) |
2662 | math-solve-rhs (math-mul math-solve-rhs fac) | |
136211a9 | 2663 | pp (cdr pp)))) |
f7adda54 | 2664 | (if sub-rhs (setq math-t1 (math-sub math-t1 math-solve-rhs))) |
136211a9 | 2665 | (let ((math-poly-neg-powers nil)) |
f7adda54 | 2666 | (setq math-t2 (math-mul (or math-poly-mult-powers 1) |
136211a9 EZ |
2667 | (let ((calc-prefer-frac t)) |
2668 | (math-div 1 math-poly-frac-powers))) | |
f3445fab | 2669 | math-t1 (math-is-polynomial |
f7adda54 | 2670 | (math-simplify (calcFunc-expand math-t1)) math-solve-b 50)))) |
136211a9 EZ |
2671 | |
2672 | ;;; This converts "a x^8 + b x^5 + c x^2" to "(a (x^3)^2 + b (x^3) + c) * x^2". | |
2673 | (defun math-solve-crunch-poly (max-degree) ; uses "t1", "t3" | |
2674 | (let ((count 0)) | |
f7adda54 JB |
2675 | (while (and math-t1 (Math-zerop (car math-t1))) |
2676 | (setq math-t1 (cdr math-t1) | |
136211a9 | 2677 | count (1+ count))) |
f7adda54 JB |
2678 | (and math-t1 |
2679 | (let* ((degree (1- (length math-t1))) | |
136211a9 | 2680 | (scale degree)) |
f7adda54 | 2681 | (while (and (> scale 1) (= (car math-t3) 1)) |
136211a9 | 2682 | (and (= (% degree scale) 0) |
f7adda54 | 2683 | (let ((p math-t1) |
136211a9 EZ |
2684 | (n 0) |
2685 | (new-t1 nil) | |
2686 | (okay t)) | |
2687 | (while (and p okay) | |
2688 | (if (= (% n scale) 0) | |
2689 | (setq new-t1 (nconc new-t1 (list (car p)))) | |
2690 | (or (Math-zerop (car p)) | |
2691 | (setq okay nil))) | |
2692 | (setq p (cdr p) | |
2693 | n (1+ n))) | |
2694 | (if okay | |
f7adda54 JB |
2695 | (setq math-t3 (cons scale (cdr math-t3)) |
2696 | math-t1 new-t1)))) | |
136211a9 | 2697 | (setq scale (1- scale))) |
f3445fab | 2698 | (setq math-t3 (list (math-mul (car math-t3) math-t2) |
f7adda54 JB |
2699 | (math-mul count math-t2))) |
2700 | (<= (1- (length math-t1)) max-degree))))) | |
136211a9 EZ |
2701 | |
2702 | (defun calcFunc-poly (expr var &optional degree) | |
2703 | (if degree | |
2704 | (or (natnump degree) (math-reject-arg degree 'fixnatnump)) | |
2705 | (setq degree 50)) | |
2706 | (let ((p (math-is-polynomial expr var degree 'gen))) | |
2707 | (if p | |
2708 | (if (equal p '(0)) | |
2709 | (list 'vec) | |
2710 | (cons 'vec p)) | |
bf77c646 | 2711 | (math-reject-arg expr "Expected a polynomial")))) |
136211a9 EZ |
2712 | |
2713 | (defun calcFunc-gpoly (expr var &optional degree) | |
2714 | (if degree | |
2715 | (or (natnump degree) (math-reject-arg degree 'fixnatnump)) | |
2716 | (setq degree 50)) | |
2717 | (let* ((math-poly-base-variable var) | |
2718 | (d (math-decompose-poly expr var degree nil))) | |
2719 | (if d | |
2720 | (cons 'vec d) | |
bf77c646 | 2721 | (math-reject-arg expr "Expected a polynomial")))) |
136211a9 | 2722 | |
f7adda54 JB |
2723 | (defun math-decompose-poly (math-solve-lhs math-solve-var degree sub-rhs) |
2724 | (let ((math-solve-rhs (or sub-rhs 1)) | |
2725 | math-t1 math-t2 math-t3) | |
2726 | (setq math-t2 (math-polynomial-base | |
2727 | math-solve-lhs | |
136211a9 | 2728 | (function |
f7adda54 | 2729 | (lambda (math-solve-b) |
136211a9 EZ |
2730 | (let ((math-poly-neg-powers '(1)) |
2731 | (math-poly-mult-powers nil) | |
2732 | (math-poly-frac-powers 1) | |
2733 | (math-poly-exp-base t)) | |
f7adda54 JB |
2734 | (and (not (equal math-solve-b math-solve-lhs)) |
2735 | (or (not (memq (car-safe math-solve-b) '(+ -))) sub-rhs) | |
2736 | (setq math-t3 '(1 0) math-t2 1 | |
f3445fab | 2737 | math-t1 (math-is-polynomial math-solve-lhs |
f7adda54 | 2738 | math-solve-b 50)) |
136211a9 EZ |
2739 | (if (and (equal math-poly-neg-powers '(1)) |
2740 | (memq math-poly-mult-powers '(nil 1)) | |
2741 | (eq math-poly-frac-powers 1) | |
2742 | sub-rhs) | |
f7adda54 JB |
2743 | (setq math-t1 (cons (math-sub (car math-t1) math-solve-rhs) |
2744 | (cdr math-t1))) | |
136211a9 EZ |
2745 | (math-solve-poly-funny-powers sub-rhs)) |
2746 | (math-solve-crunch-poly degree) | |
f7adda54 JB |
2747 | (or (math-expr-contains math-solve-b math-solve-var) |
2748 | (math-expr-contains (car math-t3) math-solve-var)))))))) | |
2749 | (if math-t2 | |
2750 | (list (math-pow math-t2 (car math-t3)) | |
2751 | (cons 'vec math-t1) | |
136211a9 | 2752 | (if sub-rhs |
f7adda54 JB |
2753 | (math-pow math-t2 (nth 1 math-t3)) |
2754 | (math-div (math-pow math-t2 (nth 1 math-t3)) math-solve-rhs)))))) | |
136211a9 EZ |
2755 | |
2756 | (defun math-solve-linear (var sign b a) | |
2757 | (math-try-solve-for var | |
2758 | (math-div (math-neg b) a) | |
2759 | (math-solve-sign sign a) | |
bf77c646 | 2760 | t)) |
136211a9 EZ |
2761 | |
2762 | (defun math-solve-quadratic (var c b a) | |
2763 | (math-try-solve-for | |
2764 | var | |
2765 | (if (math-looks-evenp b) | |
2766 | (let ((halfb (math-div b 2))) | |
2767 | (math-div | |
2768 | (math-add | |
2769 | (math-neg halfb) | |
2770 | (math-solve-get-sign | |
2771 | (math-normalize | |
2772 | (list 'calcFunc-sqrt | |
2773 | (math-add (math-sqr halfb) | |
2774 | (math-mul (math-neg c) a)))))) | |
2775 | a)) | |
2776 | (math-div | |
2777 | (math-add | |
2778 | (math-neg b) | |
2779 | (math-solve-get-sign | |
2780 | (math-normalize | |
2781 | (list 'calcFunc-sqrt | |
2782 | (math-add (math-sqr b) | |
2783 | (math-mul 4 (math-mul (math-neg c) a))))))) | |
2784 | (math-mul 2 a))) | |
bf77c646 | 2785 | nil t)) |
136211a9 EZ |
2786 | |
2787 | (defun math-solve-cubic (var d c b a) | |
2788 | (let* ((p (math-div b a)) | |
2789 | (q (math-div c a)) | |
2790 | (r (math-div d a)) | |
2791 | (psqr (math-sqr p)) | |
2792 | (aa (math-sub q (math-div psqr 3))) | |
2793 | (bb (math-add r | |
2794 | (math-div (math-sub (math-mul 2 (math-mul psqr p)) | |
2795 | (math-mul 9 (math-mul p q))) | |
2796 | 27))) | |
2797 | m) | |
2798 | (if (Math-zerop aa) | |
2799 | (math-try-solve-for (math-pow (math-add var (math-div p 3)) 3) | |
2800 | (math-neg bb) nil t) | |
2801 | (if (Math-zerop bb) | |
2802 | (math-try-solve-for | |
2803 | (math-mul (math-add var (math-div p 3)) | |
2804 | (math-add (math-sqr (math-add var (math-div p 3))) | |
2805 | aa)) | |
2806 | 0 nil t) | |
2807 | (setq m (math-mul 2 (list 'calcFunc-sqrt (math-div aa -3)))) | |
2808 | (math-try-solve-for | |
2809 | var | |
2810 | (math-sub | |
2811 | (math-normalize | |
2812 | (math-mul | |
2813 | m | |
2814 | (list 'calcFunc-cos | |
2815 | (math-div | |
2816 | (math-sub (list 'calcFunc-arccos | |
2817 | (math-div (math-mul 3 bb) | |
2818 | (math-mul aa m))) | |
2819 | (math-mul 2 | |
2820 | (math-mul | |
2821 | (math-add 1 (math-solve-get-int | |
2822 | 1 3)) | |
2823 | (math-half-circle | |
2824 | calc-symbolic-mode)))) | |
2825 | 3)))) | |
2826 | (math-div p 3)) | |
bf77c646 | 2827 | nil t))))) |
136211a9 EZ |
2828 | |
2829 | (defun math-solve-quartic (var d c b a aa) | |
2830 | (setq a (math-div a aa)) | |
2831 | (setq b (math-div b aa)) | |
2832 | (setq c (math-div c aa)) | |
2833 | (setq d (math-div d aa)) | |
2834 | (math-try-solve-for | |
2835 | var | |
2836 | (let* ((asqr (math-sqr a)) | |
2837 | (asqr4 (math-div asqr 4)) | |
f7adda54 | 2838 | (y (let ((math-solve-full nil) |
136211a9 | 2839 | calc-next-why) |
f7adda54 | 2840 | (math-solve-cubic math-solve-var |
136211a9 EZ |
2841 | (math-sub (math-sub |
2842 | (math-mul 4 (math-mul b d)) | |
2843 | (math-mul asqr d)) | |
2844 | (math-sqr c)) | |
2845 | (math-sub (math-mul a c) | |
2846 | (math-mul 4 d)) | |
2847 | (math-neg b) | |
2848 | 1))) | |
2849 | (rsqr (math-add (math-sub asqr4 b) y)) | |
2850 | (r (list 'calcFunc-sqrt rsqr)) | |
2851 | (sign1 (math-solve-get-sign 1)) | |
2852 | (de (list 'calcFunc-sqrt | |
2853 | (math-add | |
2854 | (math-sub (math-mul 3 asqr4) | |
2855 | (math-mul 2 b)) | |
2856 | (if (Math-zerop rsqr) | |
2857 | (math-mul | |
2858 | 2 | |
2859 | (math-mul sign1 | |
2860 | (list 'calcFunc-sqrt | |
2861 | (math-sub (math-sqr y) | |
2862 | (math-mul 4 d))))) | |
2863 | (math-sub | |
2864 | (math-mul sign1 | |
2865 | (math-div | |
2866 | (math-sub (math-sub | |
2867 | (math-mul 4 (math-mul a b)) | |
2868 | (math-mul 8 c)) | |
2869 | (math-mul asqr a)) | |
2870 | (math-mul 4 r))) | |
2871 | rsqr)))))) | |
2872 | (math-normalize | |
2873 | (math-sub (math-add (math-mul sign1 (math-div r 2)) | |
2874 | (math-solve-get-sign (math-div de 2))) | |
2875 | (math-div a 4)))) | |
bf77c646 | 2876 | nil t)) |
136211a9 | 2877 | |
3132f345 CW |
2878 | (defvar math-symbolic-solve nil) |
2879 | (defvar math-int-coefs nil) | |
f7adda54 JB |
2880 | |
2881 | ;; The variable math-int-threshold is local to math-poly-all-roots, | |
2882 | ;; but is used by math-poly-newton-root. | |
2883 | (defvar math-int-threshold) | |
2884 | ;; The variables math-int-scale, math-int-factors and math-double-roots | |
2885 | ;; are local to math-poly-all-roots, but are used by math-poly-integer-root. | |
2886 | (defvar math-int-scale) | |
79d2746f JB |
2887 | (defvar math-int-factors) |
2888 | (defvar math-double-roots) | |
f7adda54 | 2889 | |
136211a9 EZ |
2890 | (defun math-poly-all-roots (var p &optional math-factoring) |
2891 | (catch 'ouch | |
2892 | (let* ((math-symbolic-solve calc-symbolic-mode) | |
2893 | (roots nil) | |
2894 | (deg (1- (length p))) | |
2895 | (orig-p (reverse p)) | |
2896 | (math-int-coefs nil) | |
2897 | (math-int-scale nil) | |
2898 | (math-double-roots nil) | |
2899 | (math-int-factors nil) | |
2900 | (math-int-threshold nil) | |
2901 | (pp p)) | |
2902 | ;; If rational coefficients, look for exact rational factors. | |
2903 | (while (and pp (Math-ratp (car pp))) | |
2904 | (setq pp (cdr pp))) | |
2905 | (if pp | |
2906 | (if (or math-factoring math-symbolic-solve) | |
2907 | (throw 'ouch nil)) | |
2908 | (let ((lead (car orig-p)) | |
2909 | (calc-prefer-frac t) | |
2910 | (scale (apply 'math-lcm-denoms p))) | |
2911 | (setq math-int-scale (math-abs (math-mul scale lead)) | |
2912 | math-int-threshold (math-div '(float 5 -2) math-int-scale) | |
2913 | math-int-coefs (cdr (math-div (cons 'vec orig-p) lead))))) | |
2914 | (if (> deg 4) | |
2915 | (let ((calc-prefer-frac nil) | |
2916 | (calc-symbolic-mode nil) | |
2917 | (pp p) | |
2918 | (def-p (copy-sequence orig-p))) | |
2919 | (while pp | |
2920 | (if (Math-numberp (car pp)) | |
2921 | (setq pp (cdr pp)) | |
2922 | (throw 'ouch nil))) | |
2923 | (while (> deg (if math-symbolic-solve 2 4)) | |
2924 | (let* ((x (math-poly-any-root def-p '(float 0 0) nil)) | |
2925 | b c pp) | |
2926 | (if (and (eq (car-safe x) 'cplx) | |
2927 | (math-nearly-zerop (nth 2 x) (nth 1 x))) | |
2928 | (setq x (calcFunc-re x))) | |
2929 | (or math-factoring | |
2930 | (setq roots (cons x roots))) | |
2931 | (or (math-numberp x) | |
2932 | (setq x (math-evaluate-expr x))) | |
2933 | (setq pp def-p | |
2934 | b (car def-p)) | |
2935 | (while (setq pp (cdr pp)) | |
2936 | (setq c (car pp)) | |
2937 | (setcar pp b) | |
2938 | (setq b (math-add (math-mul x b) c))) | |
2939 | (setq def-p (cdr def-p) | |
2940 | deg (1- deg)))) | |
2941 | (setq p (reverse def-p)))) | |
2942 | (if (> deg 1) | |
f7adda54 | 2943 | (let ((math-solve-var '(var DUMMY var-DUMMY)) |
136211a9 EZ |
2944 | (math-solve-sign nil) |
2945 | (math-solve-ranges nil) | |
f7adda54 | 2946 | (math-solve-full 'all)) |
136211a9 EZ |
2947 | (if (= (length p) (length math-int-coefs)) |
2948 | (setq p (reverse math-int-coefs))) | |
2949 | (setq roots (append (cdr (apply (cond ((= deg 2) | |
2950 | 'math-solve-quadratic) | |
2951 | ((= deg 3) | |
2952 | 'math-solve-cubic) | |
2953 | (t | |
2954 | 'math-solve-quartic)) | |
f7adda54 | 2955 | math-solve-var p)) |
136211a9 EZ |
2956 | roots))) |
2957 | (if (> deg 0) | |
2958 | (setq roots (cons (math-div (math-neg (car p)) (nth 1 p)) | |
2959 | roots)))) | |
2960 | (if math-factoring | |
2961 | (progn | |
2962 | (while roots | |
2963 | (math-poly-integer-root (car roots)) | |
2964 | (setq roots (cdr roots))) | |
2965 | (list math-int-factors (nreverse math-int-coefs) math-int-scale)) | |
2966 | (let ((vec nil) res) | |
2967 | (while roots | |
2968 | (let ((root (car roots)) | |
f7adda54 | 2969 | (math-solve-full (and math-solve-full 'all))) |
136211a9 EZ |
2970 | (if (math-floatp root) |
2971 | (setq root (math-poly-any-root orig-p root t))) | |
2972 | (setq vec (append vec | |
2973 | (cdr (or (math-try-solve-for var root nil t) | |
2974 | (throw 'ouch nil)))))) | |
2975 | (setq roots (cdr roots))) | |
2976 | (setq vec (cons 'vec (nreverse vec))) | |
2977 | (if math-symbolic-solve | |
2978 | (setq vec (math-normalize vec))) | |
f7adda54 | 2979 | (if (eq math-solve-full t) |
136211a9 EZ |
2980 | (list 'calcFunc-subscr |
2981 | vec | |
2982 | (math-solve-get-int 1 (1- (length orig-p)) 1)) | |
bf77c646 | 2983 | vec)))))) |
136211a9 EZ |
2984 | |
2985 | (defun math-lcm-denoms (&rest fracs) | |
2986 | (let ((den 1)) | |
2987 | (while fracs | |
2988 | (if (eq (car-safe (car fracs)) 'frac) | |
2989 | (setq den (calcFunc-lcm den (nth 2 (car fracs))))) | |
2990 | (setq fracs (cdr fracs))) | |
bf77c646 | 2991 | den)) |
136211a9 EZ |
2992 | |
2993 | (defun math-poly-any-root (p x polish) ; p is a reverse poly coeff list | |
2994 | (let* ((newt (if (math-zerop x) | |
2995 | (math-poly-newton-root | |
2996 | p '(cplx (float 123 -6) (float 1 -4)) 4) | |
2997 | (math-poly-newton-root p x 4))) | |
2998 | (res (if (math-zerop (cdr newt)) | |
2999 | (car newt) | |
3000 | (if (and (math-lessp (cdr newt) '(float 1 -3)) (not polish)) | |
3001 | (setq newt (math-poly-newton-root p (car newt) 30))) | |
3002 | (if (math-zerop (cdr newt)) | |
3003 | (car newt) | |
3004 | (math-poly-laguerre-root p x polish))))) | |
3005 | (and math-symbolic-solve (math-floatp res) | |
3006 | (throw 'ouch nil)) | |
bf77c646 | 3007 | res)) |
136211a9 EZ |
3008 | |
3009 | (defun math-poly-newton-root (p x iters) | |
3010 | (let* ((calc-prefer-frac nil) | |
3011 | (calc-symbolic-mode nil) | |
3012 | (try-integer math-int-coefs) | |
3013 | (dx x) b d) | |
3014 | (while (and (> (setq iters (1- iters)) 0) | |
3015 | (let ((pp p)) | |
3016 | (math-working "newton" x) | |
3017 | (setq b (car p) | |
3018 | d 0) | |
3019 | (while (setq pp (cdr pp)) | |
3020 | (setq d (math-add (math-mul x d) b) | |
3021 | b (math-add (math-mul x b) (car pp)))) | |
3022 | (not (math-zerop d))) | |
3023 | (progn | |
3024 | (setq dx (math-div b d) | |
3025 | x (math-sub x dx)) | |
3026 | (if try-integer | |
3027 | (let ((adx (math-abs-approx dx))) | |
3028 | (and (math-lessp adx math-int-threshold) | |
3029 | (let ((iroot (math-poly-integer-root x))) | |
3030 | (if iroot | |
3031 | (setq x iroot dx 0) | |
3032 | (setq try-integer nil)))))) | |
3033 | (or (not (or (eq dx 0) | |
3034 | (math-nearly-zerop dx (math-abs-approx x)))) | |
3035 | (progn (setq dx 0) nil))))) | |
3036 | (cons x (if (math-zerop x) | |
bf77c646 | 3037 | 1 (math-div (math-abs-approx dx) (math-abs-approx x)))))) |
136211a9 EZ |
3038 | |
3039 | (defun math-poly-integer-root (x) | |
3040 | (and (math-lessp (calcFunc-xpon (math-abs-approx x)) calc-internal-prec) | |
3041 | math-int-coefs | |
3042 | (let* ((calc-prefer-frac t) | |
3043 | (xre (calcFunc-re x)) | |
3044 | (xim (calcFunc-im x)) | |
3045 | (xresq (math-sqr xre)) | |
3046 | (ximsq (math-sqr xim))) | |
3047 | (if (math-lessp ximsq (calcFunc-scf xresq -1)) | |
3048 | ;; Look for linear factor | |
3049 | (let* ((rnd (math-div (math-round (math-mul xre math-int-scale)) | |
3050 | math-int-scale)) | |
3051 | (icp math-int-coefs) | |
3052 | (rem (car icp)) | |
3053 | (newcoef nil)) | |
3054 | (while (setq icp (cdr icp)) | |
3055 | (setq newcoef (cons rem newcoef) | |
3056 | rem (math-add (car icp) | |
3057 | (math-mul rem rnd)))) | |
3058 | (and (math-zerop rem) | |
3059 | (progn | |
3060 | (setq math-int-coefs (nreverse newcoef) | |
3061 | math-int-factors (cons (list (math-neg rnd)) | |
3062 | math-int-factors)) | |
3063 | rnd))) | |
3064 | ;; Look for irreducible quadratic factor | |
3065 | (let* ((rnd1 (math-div (math-round | |
3066 | (math-mul xre (math-mul -2 math-int-scale))) | |
3067 | math-int-scale)) | |
3068 | (sqscale (math-sqr math-int-scale)) | |
3069 | (rnd0 (math-div (math-round (math-mul (math-add xresq ximsq) | |
3070 | sqscale)) | |
3071 | sqscale)) | |
3072 | (rem1 (car math-int-coefs)) | |
3073 | (icp (cdr math-int-coefs)) | |
3074 | (rem0 (car icp)) | |
3075 | (newcoef nil) | |
3076 | (found (assoc (list rnd0 rnd1 (math-posp xim)) | |
3077 | math-double-roots)) | |
3078 | this) | |
3079 | (if found | |
3080 | (setq math-double-roots (delq found math-double-roots) | |
3081 | rem0 0 rem1 0) | |
3082 | (while (setq icp (cdr icp)) | |
3083 | (setq this rem1 | |
3084 | newcoef (cons rem1 newcoef) | |
3085 | rem1 (math-sub rem0 (math-mul this rnd1)) | |
3086 | rem0 (math-sub (car icp) (math-mul this rnd0))))) | |
3087 | (and (math-zerop rem0) | |
3088 | (math-zerop rem1) | |
3089 | (let ((aa (math-div rnd1 -2))) | |
3090 | (or found (setq math-int-coefs (reverse newcoef) | |
3091 | math-double-roots (cons (list | |
3092 | (list | |
3093 | rnd0 rnd1 | |
3094 | (math-negp xim))) | |
3095 | math-double-roots) | |
3096 | math-int-factors (cons (cons rnd0 rnd1) | |
3097 | math-int-factors))) | |
3098 | (math-add aa | |
3099 | (let ((calc-symbolic-mode math-symbolic-solve)) | |
3100 | (math-mul (math-sqrt (math-sub (math-sqr aa) | |
3101 | rnd0)) | |
bf77c646 | 3102 | (if (math-negp xim) -1 1))))))))))) |
136211a9 EZ |
3103 | |
3104 | ;;; The following routine is from Numerical Recipes, section 9.5. | |
3105 | (defun math-poly-laguerre-root (p x polish) | |
3106 | (let* ((calc-prefer-frac nil) | |
3107 | (calc-symbolic-mode nil) | |
3108 | (iters 0) | |
3109 | (m (1- (length p))) | |
3110 | (try-newt (not polish)) | |
3111 | (tried-newt nil) | |
3112 | b d f x1 dx dxold) | |
3113 | (while | |
3114 | (and (or (< (setq iters (1+ iters)) 50) | |
3115 | (math-reject-arg x "*Laguerre's method failed to converge")) | |
3116 | (let ((err (math-abs-approx (car p))) | |
3117 | (abx (math-abs-approx x)) | |
3118 | (pp p)) | |
3119 | (setq b (car p) | |
3120 | d 0 f 0) | |
3121 | (while (setq pp (cdr pp)) | |
3122 | (setq f (math-add (math-mul x f) d) | |
3123 | d (math-add (math-mul x d) b) | |
3124 | b (math-add (math-mul x b) (car pp)) | |
3125 | err (math-add (math-abs-approx b) (math-mul abx err)))) | |
3126 | (math-lessp (calcFunc-scf err (- -2 calc-internal-prec)) | |
3127 | (math-abs-approx b))) | |
3128 | (or (not (math-zerop d)) | |
3129 | (not (math-zerop f)) | |
3130 | (progn | |
3131 | (setq x (math-pow (math-neg b) (list 'frac 1 m))) | |
3132 | nil)) | |
3133 | (let* ((g (math-div d b)) | |
3134 | (g2 (math-sqr g)) | |
3135 | (h (math-sub g2 (math-mul 2 (math-div f b)))) | |
3136 | (sq (math-sqrt | |
3137 | (math-mul (1- m) (math-sub (math-mul m h) g2)))) | |
3138 | (gp (math-add g sq)) | |
3139 | (gm (math-sub g sq))) | |
3140 | (if (math-lessp (calcFunc-abssqr gp) (calcFunc-abssqr gm)) | |
3141 | (setq gp gm)) | |
3142 | (setq dx (math-div m gp) | |
3143 | x1 (math-sub x dx)) | |
3144 | (if (and try-newt | |
3145 | (math-lessp (math-abs-approx dx) | |
3146 | (calcFunc-scf (math-abs-approx x) -3))) | |
3147 | (let ((newt (math-poly-newton-root p x1 7))) | |
3148 | (setq tried-newt t | |
3149 | try-newt nil) | |
3150 | (if (math-zerop (cdr newt)) | |
3151 | (setq x (car newt) x1 x) | |
3152 | (if (math-lessp (cdr newt) '(float 1 -6)) | |
3153 | (let ((newt2 (math-poly-newton-root | |
3154 | p (car newt) 20))) | |
3155 | (if (math-zerop (cdr newt2)) | |
3156 | (setq x (car newt2) x1 x) | |
3157 | (setq x (car newt)))))))) | |
3158 | (not (or (eq x x1) | |
3159 | (math-nearly-equal x x1)))) | |
3160 | (let ((cdx (math-abs-approx dx))) | |
3161 | (setq x x1 | |
3162 | tried-newt nil) | |
3163 | (prog1 | |
3164 | (or (<= iters 6) | |
3165 | (math-lessp cdx dxold) | |
3166 | (progn | |
3167 | (if polish | |
3168 | (let ((digs (calcFunc-xpon | |
3169 | (math-div (math-abs-approx x) cdx)))) | |
3170 | (calc-record-why | |
3171 | "*Could not attain full precision") | |
3172 | (if (natnump digs) | |
3173 | (let ((calc-internal-prec (max 3 digs))) | |
3174 | (setq x (math-normalize x)))))) | |
3175 | nil)) | |
3176 | (setq dxold cdx))) | |
3177 | (or polish | |
3178 | (math-lessp (calcFunc-scf (math-abs-approx x) | |
3179 | (- calc-internal-prec)) | |
3180 | dxold)))) | |
3181 | (or (and (math-floatp x) | |
3182 | (math-poly-integer-root x)) | |
bf77c646 | 3183 | x))) |
136211a9 EZ |
3184 | |
3185 | (defun math-solve-above-dummy (x) | |
3186 | (and (not (Math-primp x)) | |
3187 | (if (and (equal (nth 1 x) '(var SOLVEDUM SOLVEDUM)) | |
3188 | (= (length x) 2)) | |
3189 | x | |
3190 | (let ((res nil)) | |
3191 | (while (and (setq x (cdr x)) | |
3192 | (not (setq res (math-solve-above-dummy (car x)))))) | |
bf77c646 | 3193 | res)))) |
136211a9 EZ |
3194 | |
3195 | (defun math-solve-find-root-term (x neg) ; sets "t2", "t3" | |
3196 | (if (math-solve-find-root-in-prod x) | |
f7adda54 JB |
3197 | (setq math-t3 neg |
3198 | math-t1 x) | |
136211a9 EZ |
3199 | (and (memq (car-safe x) '(+ -)) |
3200 | (or (math-solve-find-root-term (nth 1 x) neg) | |
3201 | (math-solve-find-root-term (nth 2 x) | |
bf77c646 | 3202 | (if (eq (car x) '-) (not neg) neg)))))) |
136211a9 EZ |
3203 | |
3204 | (defun math-solve-find-root-in-prod (x) | |
3205 | (and (consp x) | |
f7adda54 | 3206 | (math-expr-contains x math-solve-var) |
136211a9 | 3207 | (or (and (eq (car x) 'calcFunc-sqrt) |
f7adda54 | 3208 | (setq math-t2 2)) |
136211a9 EZ |
3209 | (and (eq (car x) '^) |
3210 | (or (and (memq (math-quarter-integer (nth 2 x)) '(1 2 3)) | |
f7adda54 | 3211 | (setq math-t2 2)) |
136211a9 EZ |
3212 | (and (eq (car-safe (nth 2 x)) 'frac) |
3213 | (eq (nth 2 (nth 2 x)) 3) | |
f7adda54 | 3214 | (setq math-t2 3)))) |
136211a9 | 3215 | (and (memq (car x) '(* /)) |
f7adda54 | 3216 | (or (and (not (math-expr-contains (nth 1 x) math-solve-var)) |
136211a9 | 3217 | (math-solve-find-root-in-prod (nth 2 x))) |
f7adda54 | 3218 | (and (not (math-expr-contains (nth 2 x) math-solve-var)) |
bf77c646 | 3219 | (math-solve-find-root-in-prod (nth 1 x)))))))) |
136211a9 | 3220 | |
f3445fab | 3221 | ;; The variable math-solve-vars is local to math-solve-system, |
f7adda54 JB |
3222 | ;; but is used by math-solve-system-rec. |
3223 | (defvar math-solve-vars) | |
136211a9 | 3224 | |
f7adda54 JB |
3225 | ;; The variable math-solve-simplifying is local to math-solve-system |
3226 | ;; and math-solve-system-rec, but is used by math-solve-system-subst. | |
79d2746f | 3227 | (defvar math-solve-simplifying) |
f7adda54 JB |
3228 | |
3229 | (defun math-solve-system (exprs math-solve-vars math-solve-full) | |
136211a9 EZ |
3230 | (setq exprs (mapcar 'list (if (Math-vectorp exprs) |
3231 | (cdr exprs) | |
3232 | (list exprs))) | |
f7adda54 JB |
3233 | math-solve-vars (if (Math-vectorp math-solve-vars) |
3234 | (cdr math-solve-vars) | |
3235 | (list math-solve-vars))) | |
136211a9 | 3236 | (or (let ((math-solve-simplifying nil)) |
f7adda54 | 3237 | (math-solve-system-rec exprs math-solve-vars nil)) |
136211a9 | 3238 | (let ((math-solve-simplifying t)) |
f7adda54 | 3239 | (math-solve-system-rec exprs math-solve-vars nil)))) |
136211a9 EZ |
3240 | |
3241 | ;;; The following backtracking solver works by choosing a variable | |
3242 | ;;; and equation, and trying to solve the equation for the variable. | |
3243 | ;;; If it succeeds it calls itself recursively with that variable and | |
3244 | ;;; equation removed from their respective lists, and with the solution | |
3245 | ;;; added to solns as well as being substituted into all existing | |
3246 | ;;; equations. The algorithm terminates when any solution path | |
3247 | ;;; manages to remove all the variables from var-list. | |
3248 | ||
3249 | ;;; To support calcFunc-roots, entries in eqn-list and solns are | |
3250 | ;;; actually lists of equations. | |
3251 | ||
f7adda54 JB |
3252 | ;; The variables math-solve-system-res and math-solve-system-vv are |
3253 | ;; local to math-solve-system-rec, but are used by math-solve-system-subst. | |
3254 | (defvar math-solve-system-vv) | |
3255 | (defvar math-solve-system-res) | |
3256 | ||
3257 | ||
136211a9 EZ |
3258 | (defun math-solve-system-rec (eqn-list var-list solns) |
3259 | (if var-list | |
3260 | (let ((v var-list) | |
f7adda54 | 3261 | (math-solve-system-res nil)) |
136211a9 EZ |
3262 | |
3263 | ;; Try each variable in turn. | |
3264 | (while | |
3265 | (and | |
3266 | v | |
f7adda54 | 3267 | (let* ((math-solve-system-vv (car v)) |
136211a9 | 3268 | (e eqn-list) |
f7adda54 | 3269 | (elim (eq (car-safe math-solve-system-vv) 'calcFunc-elim))) |
136211a9 | 3270 | (if elim |
f7adda54 | 3271 | (setq math-solve-system-vv (nth 1 math-solve-system-vv))) |
136211a9 EZ |
3272 | |
3273 | ;; Try each equation in turn. | |
3274 | (while | |
3275 | (and | |
3276 | e | |
3277 | (let ((e2 (car e)) | |
3278 | (eprev nil) | |
3279 | res2) | |
f7adda54 | 3280 | (setq math-solve-system-res nil) |
136211a9 | 3281 | |
f7adda54 | 3282 | ;; Try to solve for math-solve-system-vv the list of equations e2. |
136211a9 EZ |
3283 | (while (and e2 |
3284 | (setq res2 (or (and (eq (car e2) eprev) | |
3285 | res2) | |
f3445fab | 3286 | (math-solve-for (car e2) 0 |
f7adda54 JB |
3287 | math-solve-system-vv |
3288 | math-solve-full)))) | |
136211a9 | 3289 | (setq eprev (car e2) |
f7adda54 | 3290 | math-solve-system-res (cons (if (eq math-solve-full 'all) |
136211a9 EZ |
3291 | (cdr res2) |
3292 | (list res2)) | |
f7adda54 | 3293 | math-solve-system-res) |
136211a9 EZ |
3294 | e2 (cdr e2))) |
3295 | (if e2 | |
f7adda54 | 3296 | (setq math-solve-system-res nil) |
136211a9 EZ |
3297 | |
3298 | ;; Found a solution. Now try other variables. | |
f7adda54 JB |
3299 | (setq math-solve-system-res (nreverse math-solve-system-res) |
3300 | math-solve-system-res (math-solve-system-rec | |
136211a9 EZ |
3301 | (mapcar |
3302 | 'math-solve-system-subst | |
3303 | (delq (car e) | |
3304 | (copy-sequence eqn-list))) | |
3305 | (delq (car v) (copy-sequence var-list)) | |
3306 | (let ((math-solve-simplifying nil) | |
3307 | (s (mapcar | |
3308 | (function | |
3309 | (lambda (x) | |
3310 | (cons | |
3311 | (car x) | |
3312 | (math-solve-system-subst | |
3313 | (cdr x))))) | |
3314 | solns))) | |
3315 | (if elim | |
3316 | s | |
f3445fab GM |
3317 | (cons (cons |
3318 | math-solve-system-vv | |
f7adda54 | 3319 | (apply 'append math-solve-system-res)) |
136211a9 | 3320 | s))))) |
f7adda54 | 3321 | (not math-solve-system-res)))) |
136211a9 | 3322 | (setq e (cdr e))) |
f7adda54 | 3323 | (not math-solve-system-res))) |
136211a9 | 3324 | (setq v (cdr v))) |
f7adda54 | 3325 | math-solve-system-res) |
136211a9 EZ |
3326 | |
3327 | ;; Eliminated all variables, so now put solution into the proper format. | |
3328 | (setq solns (sort solns | |
3329 | (function | |
3330 | (lambda (x y) | |
f7adda54 JB |
3331 | (not (memq (car x) (memq (car y) math-solve-vars))))))) |
3332 | (if (eq math-solve-full 'all) | |
136211a9 EZ |
3333 | (math-transpose |
3334 | (math-normalize | |
3335 | (cons 'vec | |
3336 | (if solns | |
3337 | (mapcar (function (lambda (x) (cons 'vec (cdr x)))) solns) | |
3338 | (mapcar (function (lambda (x) (cons 'vec x))) eqn-list))))) | |
3339 | (math-normalize | |
a1506d29 | 3340 | (cons 'vec |
136211a9 EZ |
3341 | (if solns |
3342 | (mapcar (function (lambda (x) (cons 'calcFunc-eq x))) solns) | |
bf77c646 | 3343 | (mapcar 'car eqn-list))))))) |
136211a9 EZ |
3344 | |
3345 | (defun math-solve-system-subst (x) ; uses "res" and "v" | |
3346 | (let ((accum nil) | |
f7adda54 | 3347 | (res2 math-solve-system-res)) |
136211a9 EZ |
3348 | (while x |
3349 | (setq accum (nconc accum | |
3350 | (mapcar (function | |
3351 | (lambda (r) | |
3352 | (if math-solve-simplifying | |
3353 | (math-simplify | |
f3445fab | 3354 | (math-expr-subst |
f7adda54 | 3355 | (car x) math-solve-system-vv r)) |
f3445fab | 3356 | (math-expr-subst |
f7adda54 | 3357 | (car x) math-solve-system-vv r)))) |
136211a9 EZ |
3358 | (car res2))) |
3359 | x (cdr x) | |
3360 | res2 (cdr res2))) | |
bf77c646 | 3361 | accum)) |
136211a9 EZ |
3362 | |
3363 | ||
f7adda54 JB |
3364 | ;; calc-command-flags is declared in calc.el |
3365 | (defvar calc-command-flags) | |
3366 | ||
136211a9 EZ |
3367 | (defun math-get-from-counter (name) |
3368 | (let ((ctr (assq name calc-command-flags))) | |
3369 | (if ctr | |
3370 | (setcdr ctr (1+ (cdr ctr))) | |
3371 | (setq ctr (cons name 1) | |
3372 | calc-command-flags (cons ctr calc-command-flags))) | |
bf77c646 | 3373 | (cdr ctr))) |
136211a9 | 3374 | |
f7adda54 JB |
3375 | (defvar var-GenCount) |
3376 | ||
136211a9 EZ |
3377 | (defun math-solve-get-sign (val) |
3378 | (setq val (math-simplify val)) | |
3379 | (if (and (eq (car-safe val) '*) | |
3380 | (Math-numberp (nth 1 val))) | |
3381 | (list '* (nth 1 val) (math-solve-get-sign (nth 2 val))) | |
3382 | (and (eq (car-safe val) 'calcFunc-sqrt) | |
3383 | (eq (car-safe (nth 1 val)) '^) | |
3384 | (setq val (math-normalize (list '^ | |
3385 | (nth 1 (nth 1 val)) | |
3386 | (math-div (nth 2 (nth 1 val)) 2))))) | |
f7adda54 | 3387 | (if math-solve-full |
136211a9 EZ |
3388 | (if (and (calc-var-value 'var-GenCount) |
3389 | (Math-natnump var-GenCount) | |
f7adda54 | 3390 | (not (eq math-solve-full 'all))) |
136211a9 EZ |
3391 | (prog1 |
3392 | (math-mul (list 'calcFunc-as var-GenCount) val) | |
3393 | (setq var-GenCount (math-add var-GenCount 1)) | |
3394 | (calc-refresh-evaltos 'var-GenCount)) | |
a9c76ab2 | 3395 | (let* ((var (concat "s" (int-to-string (math-get-from-counter 'solve-sign)))) |
136211a9 | 3396 | (var2 (list 'var (intern var) (intern (concat "var-" var))))) |
f7adda54 | 3397 | (if (eq math-solve-full 'all) |
136211a9 EZ |
3398 | (setq math-solve-ranges (cons (list var2 1 -1) |
3399 | math-solve-ranges))) | |
3400 | (math-mul var2 val))) | |
3401 | (calc-record-why "*Choosing positive solution") | |
bf77c646 | 3402 | val))) |
136211a9 EZ |
3403 | |
3404 | (defun math-solve-get-int (val &optional range first) | |
f7adda54 | 3405 | (if math-solve-full |
136211a9 EZ |
3406 | (if (and (calc-var-value 'var-GenCount) |
3407 | (Math-natnump var-GenCount) | |
f7adda54 | 3408 | (not (eq math-solve-full 'all))) |
136211a9 EZ |
3409 | (prog1 |
3410 | (math-mul val (list 'calcFunc-an var-GenCount)) | |
3411 | (setq var-GenCount (math-add var-GenCount 1)) | |
3412 | (calc-refresh-evaltos 'var-GenCount)) | |
88258edf CW |
3413 | (let* ((var (concat "n" (int-to-string |
3414 | (math-get-from-counter 'solve-int)))) | |
136211a9 | 3415 | (var2 (list 'var (intern var) (intern (concat "var-" var))))) |
f7adda54 | 3416 | (if (and range (eq math-solve-full 'all)) |
136211a9 EZ |
3417 | (setq math-solve-ranges (cons (cons var2 |
3418 | (cdr (calcFunc-index | |
3419 | range (or first 0)))) | |
3420 | math-solve-ranges))) | |
3421 | (math-mul val var2))) | |
3422 | (calc-record-why "*Choosing 0 for arbitrary integer in solution") | |
bf77c646 | 3423 | 0)) |
136211a9 EZ |
3424 | |
3425 | (defun math-solve-sign (sign expr) | |
3426 | (and sign | |
3427 | (let ((s1 (math-possible-signs expr))) | |
3428 | (cond ((memq s1 '(4 6)) | |
3429 | sign) | |
3430 | ((memq s1 '(1 3)) | |
bf77c646 | 3431 | (- sign)))))) |
136211a9 EZ |
3432 | |
3433 | (defun math-looks-evenp (expr) | |
3434 | (if (Math-integerp expr) | |
3435 | (math-evenp expr) | |
3436 | (if (memq (car expr) '(* /)) | |
bf77c646 | 3437 | (math-looks-evenp (nth 1 expr))))) |
136211a9 | 3438 | |
f7adda54 JB |
3439 | (defun math-solve-for (lhs rhs math-solve-var math-solve-full &optional sign) |
3440 | (if (math-expr-contains rhs math-solve-var) | |
3441 | (math-solve-for (math-sub lhs rhs) 0 math-solve-var math-solve-full) | |
3442 | (and (math-expr-contains lhs math-solve-var) | |
136211a9 | 3443 | (math-with-extra-prec 1 |
f7adda54 | 3444 | (let* ((math-poly-base-variable math-solve-var) |
136211a9 | 3445 | (res (math-try-solve-for lhs rhs sign))) |
f7adda54 JB |
3446 | (if (and (eq math-solve-full 'all) |
3447 | (math-known-realp math-solve-var)) | |
136211a9 EZ |
3448 | (let ((old-len (length res)) |
3449 | new-len) | |
3450 | (setq res (delq nil | |
3451 | (mapcar (function | |
3452 | (lambda (x) | |
3453 | (and (not (memq (car-safe x) | |
3454 | '(cplx polar))) | |
3455 | x))) | |
3456 | res)) | |
3457 | new-len (length res)) | |
3458 | (if (< new-len old-len) | |
3459 | (calc-record-why (if (= new-len 1) | |
3460 | "*All solutions were complex" | |
3461 | (format | |
3462 | "*Omitted %d complex solutions" | |
3463 | (- old-len new-len))))))) | |
bf77c646 | 3464 | res))))) |
136211a9 EZ |
3465 | |
3466 | (defun math-solve-eqn (expr var full) | |
3467 | (if (memq (car-safe expr) '(calcFunc-neq calcFunc-lt calcFunc-gt | |
3468 | calcFunc-leq calcFunc-geq)) | |
3469 | (let ((res (math-solve-for (cons '- (cdr expr)) | |
3470 | 0 var full | |
3471 | (if (eq (car expr) 'calcFunc-neq) nil 1)))) | |
3472 | (and res | |
3473 | (if (eq math-solve-sign 1) | |
3474 | (list (car expr) var res) | |
3475 | (if (eq math-solve-sign -1) | |
3476 | (list (car expr) res var) | |
3477 | (or (eq (car expr) 'calcFunc-neq) | |
3478 | (calc-record-why | |
3479 | "*Can't determine direction of inequality")) | |
3480 | (and (memq (car expr) '(calcFunc-neq calcFunc-lt calcFunc-gt)) | |
3481 | (list 'calcFunc-neq var res)))))) | |
3482 | (let ((res (math-solve-for expr 0 var full))) | |
3483 | (and res | |
bf77c646 | 3484 | (list 'calcFunc-eq var res))))) |
136211a9 EZ |
3485 | |
3486 | (defun math-reject-solution (expr var func) | |
3487 | (if (math-expr-contains expr var) | |
3488 | (or (equal (car calc-next-why) '(* "Unable to find a symbolic solution")) | |
3489 | (calc-record-why "*Unable to find a solution"))) | |
bf77c646 | 3490 | (list func expr var)) |
136211a9 EZ |
3491 | |
3492 | (defun calcFunc-solve (expr var) | |
3493 | (or (if (or (Math-vectorp expr) (Math-vectorp var)) | |
3494 | (math-solve-system expr var nil) | |
3495 | (math-solve-eqn expr var nil)) | |
bf77c646 | 3496 | (math-reject-solution expr var 'calcFunc-solve))) |
136211a9 EZ |
3497 | |
3498 | (defun calcFunc-fsolve (expr var) | |
3499 | (or (if (or (Math-vectorp expr) (Math-vectorp var)) | |
3500 | (math-solve-system expr var t) | |
3501 | (math-solve-eqn expr var t)) | |
bf77c646 | 3502 | (math-reject-solution expr var 'calcFunc-fsolve))) |
136211a9 EZ |
3503 | |
3504 | (defun calcFunc-roots (expr var) | |
3505 | (let ((math-solve-ranges nil)) | |
3506 | (or (if (or (Math-vectorp expr) (Math-vectorp var)) | |
3507 | (math-solve-system expr var 'all) | |
3508 | (math-solve-for expr 0 var 'all)) | |
bf77c646 | 3509 | (math-reject-solution expr var 'calcFunc-roots)))) |
136211a9 EZ |
3510 | |
3511 | (defun calcFunc-finv (expr var) | |
3512 | (let ((res (math-solve-for expr math-integ-var var nil))) | |
3513 | (if res | |
3514 | (math-normalize (math-expr-subst res math-integ-var var)) | |
bf77c646 | 3515 | (math-reject-solution expr var 'calcFunc-finv)))) |
136211a9 EZ |
3516 | |
3517 | (defun calcFunc-ffinv (expr var) | |
3518 | (let ((res (math-solve-for expr math-integ-var var t))) | |
3519 | (if res | |
3520 | (math-normalize (math-expr-subst res math-integ-var var)) | |
bf77c646 | 3521 | (math-reject-solution expr var 'calcFunc-finv)))) |
136211a9 EZ |
3522 | |
3523 | ||
3524 | (put 'calcFunc-inv 'math-inverse | |
3525 | (function (lambda (x) (math-div 1 x)))) | |
3526 | (put 'calcFunc-inv 'math-inverse-sign -1) | |
3527 | ||
3528 | (put 'calcFunc-sqrt 'math-inverse | |
3529 | (function (lambda (x) (math-sqr x)))) | |
3530 | ||
3531 | (put 'calcFunc-conj 'math-inverse | |
3532 | (function (lambda (x) (list 'calcFunc-conj x)))) | |
3533 | ||
3534 | (put 'calcFunc-abs 'math-inverse | |
3535 | (function (lambda (x) (math-solve-get-sign x)))) | |
3536 | ||
3537 | (put 'calcFunc-deg 'math-inverse | |
3538 | (function (lambda (x) (list 'calcFunc-rad x)))) | |
3539 | (put 'calcFunc-deg 'math-inverse-sign 1) | |
3540 | ||
3541 | (put 'calcFunc-rad 'math-inverse | |
3542 | (function (lambda (x) (list 'calcFunc-deg x)))) | |
3543 | (put 'calcFunc-rad 'math-inverse-sign 1) | |
3544 | ||
3545 | (put 'calcFunc-ln 'math-inverse | |
3546 | (function (lambda (x) (list 'calcFunc-exp x)))) | |
3547 | (put 'calcFunc-ln 'math-inverse-sign 1) | |
3548 | ||
3549 | (put 'calcFunc-log10 'math-inverse | |
3550 | (function (lambda (x) (list 'calcFunc-exp10 x)))) | |
3551 | (put 'calcFunc-log10 'math-inverse-sign 1) | |
3552 | ||
3553 | (put 'calcFunc-lnp1 'math-inverse | |
3554 | (function (lambda (x) (list 'calcFunc-expm1 x)))) | |
3555 | (put 'calcFunc-lnp1 'math-inverse-sign 1) | |
3556 | ||
3557 | (put 'calcFunc-exp 'math-inverse | |
3558 | (function (lambda (x) (math-add (math-normalize (list 'calcFunc-ln x)) | |
3559 | (math-mul 2 | |
3560 | (math-mul '(var pi var-pi) | |
3561 | (math-solve-get-int | |
3562 | '(var i var-i)))))))) | |
3563 | (put 'calcFunc-exp 'math-inverse-sign 1) | |
3564 | ||
3565 | (put 'calcFunc-expm1 'math-inverse | |
3566 | (function (lambda (x) (math-add (math-normalize (list 'calcFunc-lnp1 x)) | |
3567 | (math-mul 2 | |
3568 | (math-mul '(var pi var-pi) | |
3569 | (math-solve-get-int | |
3570 | '(var i var-i)))))))) | |
3571 | (put 'calcFunc-expm1 'math-inverse-sign 1) | |
3572 | ||
3573 | (put 'calcFunc-sin 'math-inverse | |
3574 | (function (lambda (x) (let ((n (math-solve-get-int 1))) | |
3575 | (math-add (math-mul (math-normalize | |
3576 | (list 'calcFunc-arcsin x)) | |
3577 | (math-pow -1 n)) | |
3578 | (math-mul (math-half-circle t) | |
3579 | n)))))) | |
3580 | ||
3581 | (put 'calcFunc-cos 'math-inverse | |
3582 | (function (lambda (x) (math-add (math-solve-get-sign | |
3583 | (math-normalize | |
3584 | (list 'calcFunc-arccos x))) | |
3585 | (math-solve-get-int | |
3586 | (math-full-circle t)))))) | |
3587 | ||
3588 | (put 'calcFunc-tan 'math-inverse | |
3589 | (function (lambda (x) (math-add (math-normalize (list 'calcFunc-arctan x)) | |
3590 | (math-solve-get-int | |
3591 | (math-half-circle t)))))) | |
3592 | ||
3593 | (put 'calcFunc-arcsin 'math-inverse | |
3594 | (function (lambda (x) (math-normalize (list 'calcFunc-sin x))))) | |
3595 | ||
3596 | (put 'calcFunc-arccos 'math-inverse | |
3597 | (function (lambda (x) (math-normalize (list 'calcFunc-cos x))))) | |
3598 | ||
3599 | (put 'calcFunc-arctan 'math-inverse | |
3600 | (function (lambda (x) (math-normalize (list 'calcFunc-tan x))))) | |
3601 | ||
3602 | (put 'calcFunc-sinh 'math-inverse | |
3603 | (function (lambda (x) (let ((n (math-solve-get-int 1))) | |
3604 | (math-add (math-mul (math-normalize | |
3605 | (list 'calcFunc-arcsinh x)) | |
3606 | (math-pow -1 n)) | |
3607 | (math-mul (math-half-circle t) | |
3608 | (math-mul | |
3609 | '(var i var-i) | |
3610 | n))))))) | |
3611 | (put 'calcFunc-sinh 'math-inverse-sign 1) | |
3612 | ||
3613 | (put 'calcFunc-cosh 'math-inverse | |
3614 | (function (lambda (x) (math-add (math-solve-get-sign | |
3615 | (math-normalize | |
3616 | (list 'calcFunc-arccosh x))) | |
3617 | (math-mul (math-full-circle t) | |
3618 | (math-solve-get-int | |
3619 | '(var i var-i))))))) | |
3620 | ||
3621 | (put 'calcFunc-tanh 'math-inverse | |
3622 | (function (lambda (x) (math-add (math-normalize | |
3623 | (list 'calcFunc-arctanh x)) | |
3624 | (math-mul (math-half-circle t) | |
3625 | (math-solve-get-int | |
3626 | '(var i var-i))))))) | |
3627 | (put 'calcFunc-tanh 'math-inverse-sign 1) | |
3628 | ||
3629 | (put 'calcFunc-arcsinh 'math-inverse | |
3630 | (function (lambda (x) (math-normalize (list 'calcFunc-sinh x))))) | |
3631 | (put 'calcFunc-arcsinh 'math-inverse-sign 1) | |
3632 | ||
3633 | (put 'calcFunc-arccosh 'math-inverse | |
3634 | (function (lambda (x) (math-normalize (list 'calcFunc-cosh x))))) | |
3635 | ||
3636 | (put 'calcFunc-arctanh 'math-inverse | |
3637 | (function (lambda (x) (math-normalize (list 'calcFunc-tanh x))))) | |
3638 | (put 'calcFunc-arctanh 'math-inverse-sign 1) | |
3639 | ||
3640 | ||
3641 | ||
3642 | (defun calcFunc-taylor (expr var num) | |
3643 | (let ((x0 0) (v var)) | |
3644 | (if (memq (car-safe var) '(+ - calcFunc-eq)) | |
3645 | (setq x0 (if (eq (car var) '+) (math-neg (nth 2 var)) (nth 2 var)) | |
3646 | v (nth 1 var))) | |
3647 | (or (and (eq (car-safe v) 'var) | |
3648 | (math-expr-contains expr v) | |
3649 | (natnump num) | |
3650 | (let ((accum (math-expr-subst expr v x0)) | |
3651 | (var2 (if (eq (car var) 'calcFunc-eq) | |
3652 | (cons '- (cdr var)) | |
3653 | var)) | |
3654 | (n 0) | |
3655 | (nfac 1) | |
3656 | (fprime expr)) | |
3657 | (while (and (<= (setq n (1+ n)) num) | |
3658 | (setq fprime (calcFunc-deriv fprime v nil t))) | |
3659 | (setq fprime (math-simplify fprime) | |
3660 | nfac (math-mul nfac n) | |
3661 | accum (math-add accum | |
3662 | (math-div (math-mul (math-pow var2 n) | |
3663 | (math-expr-subst | |
3664 | fprime v x0)) | |
3665 | nfac)))) | |
3666 | (and fprime | |
3667 | (math-normalize accum)))) | |
bf77c646 | 3668 | (list 'calcFunc-taylor expr var num)))) |
136211a9 | 3669 | |
59f2c6c0 JB |
3670 | (provide 'calcalg2) |
3671 | ||
cbee283d | 3672 | ;; arch-tag: f2932ec8-dd63-418b-a542-11a644b9d4c4 |
bf77c646 | 3673 | ;;; calcalg2.el ends here |