1 ;;; calcalg2.el --- more algebraic functions for Calc
3 ;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004,
4 ;; 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc.
6 ;; Author: David Gillespie <daveg@synaptics.com>
7 ;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com>
9 ;; This file is part of GNU Emacs.
11 ;; GNU Emacs is free software: you can redistribute it and/or modify
12 ;; it under the terms of the GNU General Public License as published by
13 ;; the Free Software Foundation, either version 3 of the License, or
14 ;; (at your option) any later version.
16 ;; GNU Emacs is distributed in the hope that it will be useful,
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.
21 ;; You should have received a copy of the GNU General Public License
22 ;; along with GNU Emacs. If not, see <http://www.gnu.org/licenses/>.
28 ;; This file is autoloaded from calc-ext.el.
33 (defun calc-derivative (var num)
34 (interactive "sDifferentiate with respect to: \np")
37 (error "Order of derivative must be positive"))
38 (let ((func (if (calc-is-hyperbolic) 'calcFunc-tderiv 'calcFunc-deriv))
40 (if (or (equal var "") (equal var "$"))
44 (setq var (math-read-expr var))
45 (when (eq (car-safe var) 'error)
46 (error "Bad format in expression: %s" (nth 1 var)))
49 (while (>= (setq num (1- num)) 0)
50 (setq expr (list func expr var)))
51 (calc-enter-result n "derv" expr))))
53 (defun calc-integral (var &optional arg)
54 (interactive "sIntegration variable: \nP")
56 (calc-tabular-command 'calcFunc-integ "Integration" "intg" nil var nil nil)
58 (if (or (equal var "") (equal var "$"))
59 (calc-enter-result 2 "intg" (list 'calcFunc-integ
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
69 (defun calc-num-integral (&optional varname lowname highname)
70 (interactive "sIntegration variable: ")
71 (calc-tabular-command 'calcFunc-ninteg "Integration" "nint"
72 nil varname lowname highname))
74 (defun calc-summation (arg &optional varname lowname highname)
75 (interactive "P\nsSummation variable: ")
76 (calc-tabular-command 'calcFunc-sum "Summation" "sum"
77 arg varname lowname highname))
79 (defun calc-alt-summation (arg &optional varname lowname highname)
80 (interactive "P\nsSummation variable: ")
81 (calc-tabular-command 'calcFunc-asum "Summation" "asum"
82 arg varname lowname highname))
84 (defun calc-product (arg &optional varname lowname highname)
85 (interactive "P\nsIndex variable: ")
86 (calc-tabular-command 'calcFunc-prod "Index" "prod"
87 arg varname lowname highname))
89 (defun calc-tabulate (arg &optional varname lowname highname)
90 (interactive "P\nsIndex variable: ")
91 (calc-tabular-command 'calcFunc-table "Index" "tabl"
92 arg varname lowname highname))
94 (defun calc-tabular-command (func prompt prefix arg varname lowname highname)
96 (let (var (low nil) (high nil) (step nil) stepname stepnum (num 1) expr)
100 (if (or (equal varname "") (equal varname "$") (null varname))
101 (setq high (calc-top-n (+ stepnum 1))
102 low (calc-top-n (+ stepnum 2))
103 var (calc-top-n (+ stepnum 3))
105 (setq var (if (stringp varname) (math-read-expr varname) varname))
106 (if (eq (car-safe var) 'error)
107 (error "Bad format in expression: %s" (nth 1 var)))
109 (setq lowname (read-string (concat prompt " variable: " varname
111 (if (or (equal lowname "") (equal lowname "$"))
112 (setq high (calc-top-n (+ stepnum 1))
113 low (calc-top-n (+ stepnum 2))
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)))
119 (setq highname (read-string (concat prompt " variable: " varname
122 (if (or (equal highname "") (equal highname "$"))
123 (setq high (calc-top-n (+ stepnum 1))
125 (setq high (if (stringp highname) (math-read-expr highname)
127 (if (eq (car-safe high) 'error)
128 (error "Bad format in expression: %s" (nth 1 high)))
131 (setq stepname (read-string (concat prompt " variable: "
136 (if (or (equal stepname "") (equal stepname "$"))
137 (setq step (calc-top-n 1)
139 (setq step (math-read-expr stepname))
140 (if (eq (car-safe step) 'error)
141 (error "Bad format in expression: %s"
145 (setq step (calc-top-n 1))
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)
150 (and step (list step)))))))
152 (defun calc-solve-for (var)
153 (interactive "sVariable(s) to solve for: ")
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 "$"))
159 (calc-enter-result 2 "solv" (list func
162 (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var)
163 (not (string-match "\\[" var)))
164 (math-read-expr (concat "[" var "]"))
165 (math-read-expr var))))
166 (if (eq (car-safe var) 'error)
167 (error "Bad format in expression: %s" (nth 1 var)))
168 (calc-enter-result 1 "solv" (list func
172 (defun calc-poly-roots (var)
173 (interactive "sVariable to solve for: ")
175 (if (or (equal var "") (equal var "$"))
176 (calc-enter-result 2 "prts" (list 'calcFunc-roots
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
189 (defun calc-taylor (var nterms)
190 (interactive "sTaylor expansion variable: \nNNumber of terms: ")
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
198 (prefix-numeric-value nterms))))))
201 ;; The following are global variables used by math-derivative and some
203 (defvar math-deriv-var)
204 (defvar math-deriv-total)
205 (defvar math-deriv-symb)
206 (defvar math-decls-cache)
207 (defvar math-decls-all)
209 (defun math-derivative (expr)
210 (cond ((equal expr math-deriv-var)
212 ((or (Math-scalarp expr)
213 (eq (car expr) 'sdev)
214 (and (eq (car expr) 'var)
215 (or (not math-deriv-total)
216 (math-const-var expr)
218 (math-setup-declarations)
219 (memq 'const (nth 1 (or (assq (nth 2 expr)
221 math-decls-all)))))))
224 (math-add (math-derivative (nth 1 expr))
225 (math-derivative (nth 2 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))
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))))
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)))))
242 (math-sub (math-div (math-derivative (nth 1 expr))
244 (math-div (math-mul (nth 1 expr)
245 (math-derivative (nth 2 expr)))
246 (math-sqr (nth 2 expr)))))
248 (let ((du (math-derivative (nth 1 expr)))
249 (dv (math-derivative (nth 2 expr))))
251 (setq du (math-mul (nth 2 expr)
252 (math-mul (math-normalize
255 (math-add (nth 2 expr) -1)))
258 (setq dv (math-mul (math-normalize
259 (list 'calcFunc-ln (nth 1 expr)))
260 (math-mul expr dv))))
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))
268 (list (car expr) (math-derivative (nth 1 expr))))
269 ((and (memq (car expr) '(calcFunc-subscr calcFunc-mrow calcFunc-mcol))
271 (let ((d (math-derivative (nth 1 expr))))
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)))
279 (let ((deriv (math-derivative (nth 1 expr))))
280 (if (Math-zerop deriv)
282 (math-mul (funcall handler (nth 1 expr))
284 (let ((handler (get (car expr) 'math-derivative-n)))
286 (funcall handler expr)))))
287 (and (not (eq math-deriv-symb 'pre-expand))
288 (let ((exp (math-expand-formula expr)))
290 (or (let ((math-deriv-symb 'pre-expand))
291 (catch 'math-deriv (math-derivative expr)))
292 (math-derivative exp)))))
293 (if (or (Math-objvecp expr)
295 (not (symbolp (car expr))))
297 (throw 'math-deriv nil)
298 (list (if math-deriv-total 'calcFunc-tderiv 'calcFunc-deriv)
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))
312 (prop (cond ((= (length expr) 2)
321 'math-derivative-5))))
327 (let ((handler (get func prop)))
328 (or (and prop handler
329 (apply handler (cdr expr)))
330 (if (and math-deriv-symb
333 (throw 'math-deriv nil)
334 (cons func (cdr expr))))))))))
338 (defun calcFunc-deriv (expr math-deriv-var &optional deriv-value math-deriv-symb)
339 (let* ((math-deriv-total nil)
340 (res (catch 'math-deriv (math-derivative expr))))
341 (or (eq (car-safe res) 'calcFunc-deriv)
343 (setq res (math-normalize res)))
346 (math-expr-subst res math-deriv-var deriv-value)
349 (defun calcFunc-tderiv (expr math-deriv-var &optional deriv-value math-deriv-symb)
350 (math-setup-declarations)
351 (let* ((math-deriv-total t)
352 (res (catch 'math-deriv (math-derivative expr))))
353 (or (eq (car-safe res) 'calcFunc-tderiv)
355 (setq res (math-normalize res)))
358 (math-expr-subst res math-deriv-var deriv-value)
361 (put 'calcFunc-inv\' 'math-derivative-1
362 (function (lambda (u) (math-neg (math-div 1 (math-sqr u))))))
364 (put 'calcFunc-sqrt\' 'math-derivative-1
365 (function (lambda (u) (math-div 1 (math-mul 2 (list 'calcFunc-sqrt u))))))
367 (put 'calcFunc-deg\' 'math-derivative-1
368 (function (lambda (u) (math-div-float '(float 18 1) (math-pi)))))
370 (put 'calcFunc-rad\' 'math-derivative-1
371 (function (lambda (u) (math-pi-over-180))))
373 (put 'calcFunc-ln\' 'math-derivative-1
374 (function (lambda (u) (math-div 1 u))))
376 (put 'calcFunc-log10\' 'math-derivative-1
377 (function (lambda (u)
378 (math-div (math-div 1 (math-normalize '(calcFunc-ln 10)))
381 (put 'calcFunc-lnp1\' 'math-derivative-1
382 (function (lambda (u) (math-div 1 (math-add u 1)))))
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)))))))
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)))))))
397 (put 'calcFunc-exp\' 'math-derivative-1
398 (function (lambda (u) (math-normalize (list 'calcFunc-exp u)))))
400 (put 'calcFunc-expm1\' 'math-derivative-1
401 (function (lambda (u) (math-normalize (list 'calcFunc-expm1 u)))))
403 (put 'calcFunc-sin\' 'math-derivative-1
404 (function (lambda (u) (math-to-radians-2 (math-normalize
405 (list 'calcFunc-cos u))))))
407 (put 'calcFunc-cos\' 'math-derivative-1
408 (function (lambda (u) (math-neg (math-to-radians-2
410 (list 'calcFunc-sin u)))))))
412 (put 'calcFunc-tan\' 'math-derivative-1
413 (function (lambda (u) (math-to-radians-2
416 (list 'calcFunc-sec u)))))))
418 (put 'calcFunc-sec\' 'math-derivative-1
419 (function (lambda (u) (math-to-radians-2
422 (list 'calcFunc-sec u))
424 (list 'calcFunc-tan u)))))))
426 (put 'calcFunc-csc\' 'math-derivative-1
427 (function (lambda (u) (math-neg
431 (list 'calcFunc-csc u))
433 (list 'calcFunc-cot u))))))))
435 (put 'calcFunc-cot\' 'math-derivative-1
436 (function (lambda (u) (math-neg
440 (list 'calcFunc-csc u))))))))
442 (put 'calcFunc-arcsin\' 'math-derivative-1
443 (function (lambda (u)
445 (math-div 1 (math-normalize
447 (math-sub 1 (math-sqr u)))))))))
449 (put 'calcFunc-arccos\' 'math-derivative-1
450 (function (lambda (u)
452 (math-div -1 (math-normalize
454 (math-sub 1 (math-sqr u)))))))))
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)))))))
460 (put 'calcFunc-sinh\' 'math-derivative-1
461 (function (lambda (u) (math-normalize (list 'calcFunc-cosh u)))))
463 (put 'calcFunc-cosh\' 'math-derivative-1
464 (function (lambda (u) (math-normalize (list 'calcFunc-sinh u)))))
466 (put 'calcFunc-tanh\' 'math-derivative-1
467 (function (lambda (u) (math-sqr
469 (list 'calcFunc-sech u))))))
471 (put 'calcFunc-sech\' 'math-derivative-1
472 (function (lambda (u) (math-neg
474 (math-normalize (list 'calcFunc-sech u))
475 (math-normalize (list 'calcFunc-tanh u)))))))
477 (put 'calcFunc-csch\' 'math-derivative-1
478 (function (lambda (u) (math-neg
480 (math-normalize (list 'calcFunc-csch u))
481 (math-normalize (list 'calcFunc-coth u)))))))
483 (put 'calcFunc-coth\' 'math-derivative-1
484 (function (lambda (u) (math-neg
487 (list 'calcFunc-csch u)))))))
489 (put 'calcFunc-arcsinh\' 'math-derivative-1
490 (function (lambda (u)
491 (math-div 1 (math-normalize
493 (math-add (math-sqr u) 1)))))))
495 (put 'calcFunc-arccosh\' 'math-derivative-1
496 (function (lambda (u)
497 (math-div 1 (math-normalize
499 (math-add (math-sqr u) -1)))))))
501 (put 'calcFunc-arctanh\' 'math-derivative-1
502 (function (lambda (u) (math-div 1 (math-sub 1 (math-sqr u))))))
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)))))
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)))))
512 (put 'calcFunc-gammag\'2 'math-derivative-2
513 (function (lambda (a x) (math-deriv-gamma a x 1))))
515 (put 'calcFunc-gammaG\'2 'math-derivative-2
516 (function (lambda (a x) (math-deriv-gamma a x -1))))
518 (put 'calcFunc-gammaP\'2 'math-derivative-2
519 (function (lambda (a x) (math-deriv-gamma a x
522 (list 'calcFunc-gamma
525 (put 'calcFunc-gammaQ\'2 'math-derivative-2
526 (function (lambda (a x) (math-deriv-gamma a x
529 (list 'calcFunc-gamma
532 (defun math-deriv-gamma (a x scale)
534 (math-mul (math-pow x (math-add a -1))
535 (list 'calcFunc-exp (math-neg x)))))
537 (put 'calcFunc-betaB\' 'math-derivative-3
538 (function (lambda (x a b) (math-deriv-beta x a b 1))))
540 (put 'calcFunc-betaI\' 'math-derivative-3
541 (function (lambda (x a b) (math-deriv-beta x a b
543 1 (list 'calcFunc-beta
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)))
551 (put 'calcFunc-erf\' 'math-derivative-1
552 (function (lambda (x) (math-div 2
553 (math-mul (list 'calcFunc-exp
555 (if calc-symbolic-mode
560 (put 'calcFunc-erfc\' 'math-derivative-1
561 (function (lambda (x) (math-div -2
562 (math-mul (list 'calcFunc-exp
564 (if calc-symbolic-mode
569 (put 'calcFunc-besJ\'2 'math-derivative-2
570 (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besJ
578 (put 'calcFunc-besY\'2 'math-derivative-2
579 (function (lambda (v z) (math-div (math-sub (list 'calcFunc-besY
587 (put 'calcFunc-sum 'math-derivative-n
590 (if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var)
591 (throw 'math-deriv nil)
593 (cons (math-derivative (nth 1 expr))
594 (cdr (cdr expr))))))))
596 (put 'calcFunc-prod 'math-derivative-n
599 (if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var)
600 (throw 'math-deriv nil)
603 (cons (math-div (math-derivative (nth 1 expr))
605 (cdr (cdr expr)))))))))
607 (put 'calcFunc-integ 'math-derivative-n
610 (if (= (length expr) 3)
611 (if (equal (nth 2 expr) math-deriv-var)
614 (list 'calcFunc-integ
615 (math-derivative (nth 1 expr))
617 (if (= (length expr) 5)
618 (let ((lower (math-expr-subst (nth 1 expr) (nth 2 expr)
620 (upper (math-expr-subst (nth 1 expr) (nth 2 expr)
622 (math-add (math-sub (math-mul upper
623 (math-derivative (nth 4 expr)))
625 (math-derivative (nth 3 expr))))
626 (if (equal (nth 2 expr) math-deriv-var)
629 (list 'calcFunc-integ
630 (math-derivative (nth 1 expr)) (nth 2 expr)
631 (nth 3 expr) (nth 4 expr)))))))))))
633 (put 'calcFunc-if 'math-derivative-n
636 (and (= (length expr) 4)
637 (list 'calcFunc-if (nth 1 expr)
638 (math-derivative (nth 2 expr))
639 (math-derivative (nth 3 expr)))))))
641 (put 'calcFunc-subscr 'math-derivative-n
644 (and (= (length expr) 3)
645 (list 'calcFunc-subscr (nth 1 expr)
646 (math-derivative (nth 2 expr)))))))
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))
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
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
663 ;; indirectly) by math-try-integral.
664 (defvar math-integ-level)
665 ;; math-integral-limit is a local variable for calcFunc-integ, but is
666 ;; used by math-tracing-integral, math-sub-integration and
667 ;; math-try-integration.
668 (defvar math-integral-limit)
670 (defmacro math-tracing-integral (&rest parts)
673 (list 'with-current-buffer
675 '(goto-char (point-max))
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
684 ;; '(error (insert (prin1-to-string err))))
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;
694 ;;; ( A [B] ) Same result as for math-cur-record = B.
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)
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)
710 (defvar math-integral-cache nil)
711 (defvar math-integral-cache-state nil)
713 (defun math-integral (expr &optional simplify same-as-above)
714 (let* ((simp math-cur-record)
715 (math-cur-record (assoc expr math-integral-cache))
716 (math-integ-depth (1+ math-integ-depth))
718 (math-tracing-integral "Integrating "
719 (math-format-value expr 1000)
723 (math-tracing-integral "Found "
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))
727 (math-tracing-integral " => "
728 (math-format-value (nth 1 math-cur-record) 1000)
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)))
734 (and (math-integral-contains-parts expr)
740 (let (math-integ-msg)
741 (if (eq calc-display-working-message 'lots)
743 (calc-set-command-flag 'clear-message)
744 (setq math-integ-msg (format
745 "Working... Integrating %s"
746 (math-format-flat-expr expr 0)))
747 (message "%s" math-integ-msg)))
749 (setcar (cdr math-cur-record)
750 (if same-as-above (vector simp) 'busy))
751 (setq math-cur-record
752 (list expr (if same-as-above (vector simp) 'busy))
753 math-integral-cache (cons math-cur-record
754 math-integral-cache)))
755 (if (eq simplify 'yes)
757 (math-tracing-integral "Simplifying...")
758 (setq simp (math-simplify expr))
759 (setq val (if (equal simp expr)
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))
767 (let ((simp (math-simplify expr)))
768 (or (equal simp expr)
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)
774 (message "%s" math-integ-msg)))
775 (setcar (cdr math-cur-record) (or val
776 (if (or math-enable-subst
777 (not math-any-substs))
780 (setq val math-cur-record)
781 (while (vectorp (nth 1 val))
782 (setq val (aref (nth 1 val) 0)))
783 (setq val (if (memq (nth 1 val) '(parts parts2))
785 (setcar (cdr val) 'parts2)
786 (list 'var 'PARTS val))
787 (and (consp (nth 1 val))
789 (math-tracing-integral "Integral of "
790 (math-format-value expr 1000)
792 (math-format-value val 1000)
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)))))
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))))
814 (setcar expr (nth 1 (nth 2 (car expr))))
815 (math-replace-integral-parts (cons 'foo expr)))
816 (setcar (cdr math-cur-record) 'cancelled)))
817 (math-replace-integral-parts (car expr)))))))
819 (defvar math-linear-subst-tried t
820 "Non-nil means that a linear substitution has been tried.")
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)
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)
831 ;; The variables math-t1, math-t2 and math-t3 are local to
832 ;; math-do-integral, math-try-solve-for and math-decompose-poly, but
833 ;; are used by functions they call (directly or indirectly);
834 ;; math-do-integral calls math-do-integral-methods;
835 ;; math-try-solve-for calls math-try-solve-prod,
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.
843 (defun math-do-integral (expr)
844 (let ((math-linear-subst-tried nil)
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))
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)))
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)))
858 ((eq (car expr) 'neg)
859 (and (setq math-t1 (math-integral (nth 1 expr)))
862 (cond ((not (math-expr-contains (nth 1 expr) math-integ-var))
863 (and (setq math-t1 (math-integral (nth 2 expr)))
864 (math-mul (nth 1 expr) math-t1)))
865 ((not (math-expr-contains (nth 2 expr) math-integ-var))
866 (and (setq math-t1 (math-integral (nth 1 expr)))
867 (math-mul math-t1 (nth 2 expr))))
868 ((memq (car-safe (nth 1 expr)) '(+ -))
869 (math-integral (list (car (nth 1 expr))
870 (math-mul (nth 1 (nth 1 expr))
872 (math-mul (nth 2 (nth 1 expr))
875 ((memq (car-safe (nth 2 expr)) '(+ -))
876 (math-integral (list (car (nth 2 expr))
877 (math-mul (nth 1 (nth 2 expr))
879 (math-mul (nth 2 (nth 2 expr))
883 (cond ((and (not (math-expr-contains (nth 1 expr)
885 (not (math-equal-int (nth 1 expr) 1)))
886 (and (setq math-t1 (math-integral (math-div 1 (nth 2 expr))))
887 (math-mul (nth 1 expr) math-t1)))
888 ((not (math-expr-contains (nth 2 expr) math-integ-var))
889 (and (setq math-t1 (math-integral (nth 1 expr)))
890 (math-div math-t1 (nth 2 expr))))
891 ((and (eq (car-safe (nth 1 expr)) '*)
892 (not (math-expr-contains (nth 1 (nth 1 expr))
894 (and (setq math-t1 (math-integral
895 (math-div (nth 2 (nth 1 expr))
897 (math-mul math-t1 (nth 1 (nth 1 expr)))))
898 ((and (eq (car-safe (nth 1 expr)) '*)
899 (not (math-expr-contains (nth 2 (nth 1 expr))
901 (and (setq math-t1 (math-integral
902 (math-div (nth 1 (nth 1 expr))
904 (math-mul math-t1 (nth 2 (nth 1 expr)))))
905 ((and (eq (car-safe (nth 2 expr)) '*)
906 (not (math-expr-contains (nth 1 (nth 2 expr))
908 (and (setq math-t1 (math-integral
909 (math-div (nth 1 expr)
910 (nth 2 (nth 2 expr)))))
911 (math-div math-t1 (nth 1 (nth 2 expr)))))
912 ((and (eq (car-safe (nth 2 expr)) '*)
913 (not (math-expr-contains (nth 2 (nth 2 expr))
915 (and (setq math-t1 (math-integral
916 (math-div (nth 1 expr)
917 (nth 1 (nth 2 expr)))))
918 (math-div math-t1 (nth 2 (nth 2 expr)))))
919 ((eq (car-safe (nth 2 expr)) 'calcFunc-exp)
921 (math-mul (nth 1 expr)
923 (math-neg (nth 1 (nth 2 expr)))))))))
925 (cond ((not (math-expr-contains (nth 1 expr) math-integ-var))
926 (or (and (setq math-t1 (math-is-polynomial (nth 2 expr)
929 (math-mul (nth 1 math-t1)
935 (math-mul (nth 2 expr)
940 ((not (math-expr-contains (nth 2 expr) math-integ-var))
941 (if (and (integerp (nth 2 expr)) (< (nth 2 expr) 0))
943 (list '/ 1 (math-pow (nth 1 expr) (- (nth 2 expr))))
945 (or (and (setq math-t1 (math-is-polynomial (nth 1 expr)
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))))
951 (and (Math-negp (nth 2 expr))
954 (math-pow (nth 1 expr)
960 ;; Integral of a polynomial.
961 (and (setq math-t1 (math-is-polynomial expr math-integ-var 20))
965 (if (setq accum (math-add accum
966 (math-div (math-mul (car math-t1)
971 math-t1 (cdr math-t1))
975 ;; Try looking it up!
976 (cond ((= (length expr) 2)
977 (and (symbolp (car expr))
978 (setq math-t1 (get (car expr) 'math-integral))
981 (not (setq math-t2 (funcall (car math-t1)
983 (setq math-t1 (cdr math-t1)))
984 (and math-t2 (math-normalize math-t2)))))
986 (and (symbolp (car expr))
987 (setq math-t1 (get (car expr) 'math-integral-2))
990 (not (setq math-t2 (funcall (car math-t1)
993 (setq math-t1 (cdr math-t1)))
994 (and math-t2 (math-normalize math-t2))))))
996 ;; Integral of a rational function.
997 (and (math-ratpoly-p expr math-integ-var)
998 (setq math-t1 (calcFunc-apart expr math-integ-var))
999 (not (equal math-t1 expr))
1000 (math-integral math-t1))
1002 ;; Try user-defined integration rules.
1004 (let ((math-old-integ (symbol-function 'calcFunc-integ))
1005 (input (list 'calcFunc-integtry expr math-integ-var))
1009 (fset 'calcFunc-integ 'math-sub-integration)
1010 (setq res (math-rewrite input
1011 '(var IntegRules var-IntegRules)
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
1025 (math-integrate-by-substitution
1028 (list 'calcFunc-integfailed
1031 (if (not (math-expr-calls res
1033 calcFunc-integfailed)))
1035 (fset 'calcFunc-integ math-old-integ))))
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)
1046 (and which (not bad)
1047 (let ((prime (if (= which 1) "'" (format "'%d" which))))
1048 (and (string-match (concat prime "\\('['0-9]*\\|$\\)")
1052 (substring name 0 (match-beginning 0))
1053 (substring name (+ (match-beginning 0)
1057 ;; Try transformation methods (parts, substitutions).
1058 (and (> math-integ-level 0)
1059 (math-do-integral-methods expr))
1061 ;; Try expanding the function's definition.
1062 (let ((res (math-expand-formula expr)))
1064 (math-integral res))))))
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)))
1075 (list 'calcFunc-integfailed expr)))
1077 ;; math-so-far is a local variable for math-do-integral-methods, but
1078 ;; is used by math-integ-try-linear-substitutions and
1079 ;; math-integ-try-substitutions.
1080 (defvar math-so-far)
1082 ;; math-integ-expr is a local variable for math-do-integral-methods,
1083 ;; but is used by math-integ-try-linear-substitutions and
1084 ;; math-integ-try-substitutions.
1085 (defvar math-integ-expr)
1087 (defun math-do-integral-methods (math-integ-expr)
1088 (let ((math-so-far math-integ-var-list-list)
1091 ;; Integration by substitution, for various likely sub-expressions.
1092 ;; (In first pass, we look only for sub-exprs that are linear in X.)
1093 (or (math-integ-try-linear-substitutions math-integ-expr)
1094 (math-integ-try-substitutions math-integ-expr)
1096 ;; If function has sines and cosines, try tan(x/2) substitution.
1097 (and (let ((p (setq rat-in (math-expr-rational-in math-integ-expr))))
1099 (memq (car (car p)) '(calcFunc-sin
1105 (equal (nth 1 (car p)) math-integ-var))
1108 (or (and (math-integ-parts-easy math-integ-expr)
1109 (math-integ-try-parts math-integ-expr t))
1110 (math-integrate-by-good-substitution
1111 math-integ-expr (list 'calcFunc-tan (math-div math-integ-var 2)))))
1113 ;; If function has sinh and cosh, try tanh(x/2) substitution.
1114 (and (let ((p rat-in))
1116 (memq (car (car p)) '(calcFunc-sinh
1123 (equal (nth 1 (car p)) math-integ-var))
1126 (or (and (math-integ-parts-easy math-integ-expr)
1127 (math-integ-try-parts math-integ-expr t))
1128 (math-integrate-by-good-substitution
1129 math-integ-expr (list 'calcFunc-tanh (math-div math-integ-var 2)))))
1131 ;; If function has square roots, try sin, tan, or sec substitution.
1132 (and (let ((p rat-in))
1135 (or (equal (car p) math-integ-var)
1136 (and (eq (car (car p)) 'calcFunc-sqrt)
1137 (setq math-t1 (math-is-polynomial
1138 (nth 1 (setq math-t2 (car p)))
1139 math-integ-var 2)))))
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)))))
1147 (math-integrate-by-good-substitution
1148 math-integ-expr (list 'calcFunc-arcsin
1150 (math-add (math-mul c math-integ-var) d)
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))))
1155 (if (and nil (not (and (eq d 0) (eq c 1))))
1156 (math-integrate-by-good-substitution
1157 math-integ-expr (math-add (math-mul c math-integ-var) d))
1158 (if (math-guess-if-neg aa)
1159 (math-integrate-by-good-substitution
1160 math-integ-expr (list 'calcFunc-arccosh
1162 (math-add (math-mul c math-integ-var)
1164 (math-sqrt (math-neg aa)))))
1165 (math-integrate-by-good-substitution
1166 math-integ-expr (list 'calcFunc-arcsinh
1168 (math-add (math-mul c math-integ-var)
1170 (math-sqrt aa))))))))
1171 (math-integrate-by-good-substitution math-integ-expr math-t2)) )
1173 ;; Try integration by parts.
1174 (math-integ-try-parts math-integ-expr)
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))))
1185 (and (math-integ-parts-easy (nth 1 expr))
1186 (math-atomic-factorp (nth 2 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)))
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)
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)
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)
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))
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))
1233 (1- math-integ-level)))
1234 (math-doing-parts t)
1236 (and (>= math-integ-level 0)
1239 (setcar (cdr math-cur-record) 'parts)
1240 (math-tracing-integral "Integrating by parts, u = "
1241 (math-format-value u 1000)
1243 (math-format-value vprime 1000)
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))
1250 (if (eq (nth 1 math-cur-record) 'parts)
1251 (calcFunc-expand temp)
1252 (setq v (list 'var 'PARTS math-cur-record)
1253 temp (let (calc-next-why)
1254 (math-simplify-extended
1255 (math-solve-for (math-sub v temp) 0 v nil)))
1256 temp (if (and (eq (car-safe temp) '/)
1257 (math-zerop (nth 2 temp)))
1259 (setcar (cdr math-cur-record) 'busy)))))
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)))
1266 (math-integrate-by-good-substitution expr u user uinv uinvprime))))
1268 (defun math-integrate-by-good-substitution (expr u &optional user
1270 (let ((math-living-dangerously t)
1272 (and (setq uinv (if uinv
1273 (math-expr-subst uinv math-integ-var
1275 (let (calc-next-why)
1278 math-integ-var nil))))
1280 (math-tracing-integral "Integrating by substitution, u = "
1281 (math-format-value u 1000)
1283 (or (and (setq deriv (calcFunc-deriv u
1286 (setq temp (math-integral (math-expr-subst
1289 (math-div expr deriv)
1297 (and (setq deriv (or uinvprime
1298 (calcFunc-deriv uinv
1302 (setq temp (math-integral (math-mul
1315 (math-simplify-extended
1316 (math-expr-subst temp math-integ-var u)))))
1318 ;;; Look for substitutions of the form u = a x + b.
1319 (defun math-integ-try-linear-substitutions (sub-expr)
1320 (setq math-linear-subst-tried t)
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)
1327 (while (and (setq sub-expr (cdr sub-expr))
1328 (or (not (math-linear-in (car sub-expr)
1330 (assoc (car sub-expr) math-so-far)
1332 (setq math-so-far (cons (list (car sub-expr))
1335 (math-integrate-by-substitution
1336 math-integ-expr (car sub-expr))))))))
1339 (while (and (setq sub-expr (cdr sub-expr))
1340 (not (setq res (math-integ-try-linear-substitutions
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))
1347 (not (assoc sub-expr math-so-far))
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)))))
1353 (prog1 allow-rat (setq allow-rat nil)))
1354 (not (eq sub-expr math-integ-expr))
1355 (or (math-integrate-by-substitution math-integ-expr sub-expr)
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)))
1363 (setq math-so-far (cons (list sub-expr) math-so-far))
1364 (while (and (setq sub-expr (cdr sub-expr))
1365 (not (setq res (math-integ-try-substitutions
1366 (car sub-expr) allow-rat)))))
1369 ;; The variable math-expr-parts is local to math-expr-rational-in,
1370 ;; but is used by math-expr-rational-in-rec
1371 (defvar math-expr-parts)
1373 (defun math-expr-rational-in (expr)
1374 (let ((math-expr-parts nil))
1375 (math-expr-rational-in-rec expr)
1376 (mapcar 'car math-expr-parts)))
1378 (defun math-expr-rational-in-rec (expr)
1379 (cond ((Math-primp expr)
1380 (and (equal expr math-integ-var)
1381 (not (assoc expr math-expr-parts))
1382 (setq math-expr-parts (cons (list expr) math-expr-parts))))
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))))
1391 (and (not (assoc expr math-expr-parts))
1392 (math-expr-contains expr math-integ-var)
1393 (setq math-expr-parts (cons (list expr) math-expr-parts))))))
1395 (defun math-expr-calls (expr funcs &optional arg-contains)
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))
1403 (and (not (Math-primp expr))
1405 (while (and (setq expr (cdr expr))
1406 (not (setq res (math-expr-calls
1407 (car expr) funcs arg-contains)))))
1410 (defun math-fix-const-terms (expr except-vars)
1411 (cond ((not (math-expr-depends expr except-vars)) 0)
1412 ((Math-primp expr) expr)
1414 (math-add (math-fix-const-terms (nth 1 expr) except-vars)
1415 (math-fix-const-terms (nth 2 expr) except-vars)))
1417 (math-sub (math-fix-const-terms (nth 1 expr) except-vars)
1418 (math-fix-const-terms (nth 2 expr) except-vars)))
1421 ;; Command for debugging the Calculator's symbolic integrator.
1422 (defun calc-dump-integral-cache (&optional arg)
1424 (let ((buf (current-buffer)))
1426 (let ((p math-integral-cache)
1428 (display-buffer (get-buffer-create "*Integral Cache*"))
1429 (set-buffer (get-buffer "*Integral Cache*"))
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)
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))
1441 (goto-char (point-min)))
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)
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")
1452 (math-cur-record nil) ; a technicality
1453 (math-integrating t)
1454 (calc-prefer-frac t)
1455 (calc-symbolic-mode t)
1456 (math-has-rules (calc-has-rules 'var-IntegRules)))
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)
1464 (math-integral expr 'yes)))))
1466 (defvar var-IntegLimit nil)
1468 (defun calcFunc-integ (expr var &optional low high)
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))
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))
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)))
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)))
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)))
1521 (let ((state (list calc-angle-mode
1522 ;;calc-symbolic-mode
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)))
1530 (let* ((math-max-integral-limit (or (and (natnump var-IntegLimit)
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))
1538 (math-enable-subst nil)
1539 (math-prev-parts-v nil)
1540 (math-doing-parts nil)
1541 (math-good-parts nil)
1544 (let ((calcbuf (current-buffer))
1545 (calcwin (selected-window)))
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))
1554 (make-local-variable 'scroll-step)
1555 (setq scroll-step 3)))
1557 (set-buffer calcbuf)
1558 (math-try-integral sexpr))
1559 (select-window calcwin)
1560 (set-buffer calcbuf)))
1561 (math-try-integral sexpr))))
1564 (if (calc-has-rules 'var-IntegAfterRules)
1565 (setq res (math-rewrite res '(var IntegAfterRules
1566 var-IntegAfterRules))))
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))
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))
1577 (and high (list high))))))))
1580 (math-defintegral calcFunc-inv
1581 (math-integral (math-div 1 u)))
1583 (math-defintegral calcFunc-conj
1584 (let ((int (math-integral u)))
1586 (list 'calcFunc-conj int))))
1588 (math-defintegral calcFunc-deg
1589 (let ((int (math-integral u)))
1591 (list 'calcFunc-deg int))))
1593 (math-defintegral calcFunc-rad
1594 (let ((int (math-integral u)))
1596 (list 'calcFunc-rad int))))
1598 (math-defintegral calcFunc-re
1599 (let ((int (math-integral u)))
1601 (list 'calcFunc-re int))))
1603 (math-defintegral calcFunc-im
1604 (let ((int (math-integral u)))
1606 (list 'calcFunc-im int))))
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)))))
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)))
1618 (let ((sqa (math-sqrt (math-neg (nth 2 p)))))
1621 (math-mul (math-div (list 'calcFunc-sqrt '(var pi var-pi))
1625 (math-div (math-sub (math-mul (car p)
1628 (math-sqr (nth 1 p))
1632 (math-sub (math-mul sqa math-integ-var)
1633 (math-div (nth 1 p) (math-mul 2 sqa)))))
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)))))))
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)))))
1654 (math-defintegral-2 calcFunc-log
1655 (math-integral (math-div (list 'calcFunc-ln u)
1656 (list 'calcFunc-ln v))))
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))))))
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))))))
1670 (math-defintegral calcFunc-tan
1671 (and (equal u math-integ-var)
1672 (math-from-radians-2
1673 (list 'calcFunc-ln (list 'calcFunc-sec u)))))
1675 (math-defintegral calcFunc-sec
1676 (and (equal u math-integ-var)
1677 (math-from-radians-2
1680 (list 'calcFunc-sec u)
1681 (list 'calcFunc-tan u))))))
1683 (math-defintegral calcFunc-csc
1684 (and (equal u math-integ-var)
1685 (math-from-radians-2
1688 (list 'calcFunc-csc u)
1689 (list 'calcFunc-cot u))))))
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)))))
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)))))))
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)))))))
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)))
1715 (math-defintegral calcFunc-sinh
1716 (and (equal u math-integ-var)
1717 (list 'calcFunc-cosh u)))
1719 (math-defintegral calcFunc-cosh
1720 (and (equal u math-integ-var)
1721 (list 'calcFunc-sinh u)))
1723 (math-defintegral calcFunc-tanh
1724 (and (equal u math-integ-var)
1725 (list 'calcFunc-ln (list 'calcFunc-cosh u))))
1727 (math-defintegral calcFunc-sech
1728 (and (equal u math-integ-var)
1729 (list 'calcFunc-arctan (list 'calcFunc-sinh u))))
1731 (math-defintegral calcFunc-csch
1732 (and (equal u math-integ-var)
1733 (list 'calcFunc-ln (list 'calcFunc-tanh (math-div u 2)))))
1735 (math-defintegral calcFunc-coth
1736 (and (equal u math-integ-var)
1737 (list 'calcFunc-ln (list 'calcFunc-sinh u))))
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)))))
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))))))
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)))
1756 ;;; (Ax + B) / (ax^2 + bx + c)^n forms.
1757 (math-defintegral-2 /
1758 (math-integral-rational-funcs u v))
1760 (defun math-integral-rational-funcs (u v)
1761 (let ((pu (math-is-polynomial u math-integ-var 1))
1765 (if (and (eq (car-safe v) '^) (natnump (nth 2 v)))
1766 (setq vpow (nth 2 v)
1768 (and (setq pv (math-is-polynomial v math-integ-var 2))
1769 (let ((int (math-mul-thru
1771 (math-integral-q02 (car pv) (nth 1 pv)
1772 (nth 2 pv) v vpow))))
1774 (setq int (math-add int
1779 (nth 2 pv) v vpow)))))
1782 (defun math-integral-q12 (a b c v vpow)
1786 (math-sub (math-div math-integ-var b)
1787 (math-mul (math-div a (math-sqr b))
1788 (list 'calcFunc-ln v))))
1790 (math-div (math-add (list 'calcFunc-ln v)
1794 (let ((nm1 (math-sub vpow 1))
1795 (nm2 (math-sub vpow 2)))
1797 (math-div a (math-mul nm1 (math-pow v nm1)))
1798 (math-div 1 (math-mul nm2 (math-pow v nm2))))
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)
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))))
1816 (let ((n (1- vpow)))
1817 (math-sub (math-neg (math-div
1818 (math-add (math-mul b math-integ-var)
1820 (math-mul n (math-mul q (math-pow v n)))))
1821 (math-mul-thru (math-div (math-mul b (1- (* 2 n)))
1823 (math-integral-q02 a b c v n))))))))
1825 (defun math-integral-q02 (a b c v vpow)
1829 (math-div (list 'calcFunc-ln v) b))
1831 (math-div (math-pow v (- 1 vpow))
1832 (math-mul (- 1 vpow) b)))))
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)
1841 (setq part (math-add (math-mul 2 (math-mul c math-integ-var)) b))
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)
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)))
1854 (math-div (math-mul -2 (list 'calcFunc-arctanh
1855 (math-div part rq)))
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))))
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)))))))
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)))))))
1882 (defvar math-tabulate-initial nil)
1883 (defvar math-tabulate-function nil)
1885 ;; These variables are local to calcFunc-table, but are used by
1886 ;; math-scan-for-limits.
1891 (defun calcFunc-table (expr var &optional calc-low calc-high step)
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))
1897 (math-scan-for-limits expr))
1898 (and step (math-zerop step) (math-reject-arg step 'nonzerop))
1899 (let ((known (+ (if (Math-objectp calc-low) 1 0)
1900 (if (Math-objectp calc-high) 1 0)
1901 (if (or (null step) (Math-objectp step)) 1 0)))
1902 (count '(var inf var-inf))
1904 (or (= known 2) ; handy optimization
1905 (equal calc-high '(var inf var-inf))
1907 (setq count (math-div (math-sub calc-high calc-low) (or step 1)))
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)
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))))
1922 (setq math-working-step (1+ math-working-step)
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)))
1929 (cons (math-evaluate-expr expr) vec)))
1930 calc-low (math-add calc-low (or step 1))
1932 (if math-tabulate-function
1934 (cons 'vec (nreverse vec))))
1935 (if (Math-integerp count)
1936 (calc-record-why 'fixnump calc-high)
1937 (if (Math-num-integerp calc-low)
1938 (if (Math-num-integerp calc-high)
1939 (calc-record-why 'integerp step)
1940 (calc-record-why 'integerp calc-high))
1941 (calc-record-why 'integerp calc-low)))
1942 (append (list (or math-tabulate-function 'calcFunc-table)
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))
1947 (and step (list step))))))
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)))
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))
1963 (setq calc-low (math-max calc-low (math-ceiling low-val))
1964 calc-high (math-min calc-high (math-floor high-val)))))
1966 (while (setq x (cdr x))
1967 (math-scan-for-limits (car x))))))
1970 (defvar math-disable-sums nil)
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))
1976 (math-normalize res)))
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))
1984 ((not (math-expr-contains expr var))
1985 (math-mul expr (math-add (math-div (math-sub high low) (or step 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)
2001 (math-simplify (math-div (math-sub high low)
2003 ((memq (setq t1 (math-compare low high)) '(0 1))
2005 (math-expr-subst expr var low)
2007 ((setq t1 (math-is-polynomial expr var 20))
2011 (setq poly (math-poly-mix poly 1
2012 (math-sum-integer-power n) (car t1))
2015 (setq n (math-build-polynomial-expr poly high))
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))
2032 (math-mul (nth 2 (nth 1 expr))
2034 nil (eq (car (nth 1 expr)) '-))
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)) '-))
2043 ((and (eq (car expr) '/)
2044 (not (math-primp (nth 1 expr)))
2045 (setq t1 (math-sum-const-factors (nth 1 expr) var)))
2047 (math-sum-rec (math-div (cdr t1) (nth 2 expr))
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))
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))
2062 (math-pow (nth 1 expr) (car t1)))
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))))
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)
2072 (not (math-expr-contains (nth 2 expr) var)))
2074 (calcFunc-prod (nth 1 expr) var low high)
2076 (if (equal val '(var nan var-nan)) (setq val nil))
2078 (let* ((math-tabulate-initial 0)
2079 (math-tabulate-function 'calcFunc-sum))
2080 (calcFunc-table expr var low high)))))
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)
2099 (math-simplify (math-div (math-sub high low)
2101 (math-mul (if no-mul-flag 1 (math-pow -1 low))
2102 (calcFunc-sum (math-mul (math-pow -1 var) expr) var low high))))
2104 (defun math-sum-const-factors (expr var)
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)))
2113 (if (math-expr-contains p var)
2114 (setq not-const (cons p not-const))
2115 (setq const (cons p const)))
2117 (cons (let ((temp (car const)))
2118 (while (setq const (cdr const))
2119 (setq temp (list '* (car const) 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)))
2126 (defvar math-sum-int-pow-cache (list '(0 1)))
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)))
2132 (let* ((new (list 0 0))
2134 (pp (cdr (nth (1- n) math-sum-int-pow-cache)))
2139 (setq q (math-div (car pp) p)
2140 new (cons (math-mul q n) new)
2141 sum (math-add sum q)
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)))
2148 (nth pow math-sum-int-pow-cache)))
2150 (defun math-to-exponentials (expr)
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)))
2162 (list '^ '(var e var-e) (list '* x i))
2163 (list '^ '(var e var-e)
2164 (list 'neg (list '* x i))))
2166 ((eq (car expr) 'calcFunc-cos)
2167 (or (eq calc-angle-mode 'rad)
2168 (setq x (list '/ (list '* x pi) 180)))
2170 (list '^ '(var e var-e)
2172 (list '^ '(var e var-e)
2173 (list 'neg (list '* x i))))
2175 ((eq (car expr) 'calcFunc-sinh)
2177 (list '^ '(var e var-e) x)
2178 (list '^ '(var e var-e) (list 'neg x)))
2180 ((eq (car expr) 'calcFunc-cosh)
2182 (list '^ '(var e var-e) x)
2183 (list '^ '(var e var-e) (list 'neg x)))
2187 (defun math-to-exps (expr)
2188 (cond (calc-symbolic-mode 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)))
2195 (cons (car expr) (mapcar 'math-to-exps (cdr expr))))))
2198 (defvar math-disable-prods nil)
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))
2204 (math-normalize res)))
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)))
2212 ((not (math-expr-contains expr var))
2213 (math-pow expr (math-add (math-div (math-sub high low) (or step 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
2230 (math-simplify (math-div (math-sub high low)
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)
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)
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))
2258 ((or (and (math-equal-int (nth 1 t1) 1)
2259 (setq low (math-simplify
2260 (math-add low (car t1)))
2262 (math-add high (car t1)))))
2263 (and (math-equal-int (nth 1 t1) -1)
2266 (math-sub (car t1) high))
2268 (math-sub (car t1) t2)))))
2269 (if (or (math-zerop low) (math-zerop high))
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)
2276 (math-mul (math-pow -1
2278 (math-add low high) 1))
2280 (list 'calcFunc-fact
2282 (list 'calcFunc-fact
2283 (math-sub -1 high)))))
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)
2292 (math-add (math-mul high 2)
2294 (and (math-equal-int (nth 1 t1) -2)
2295 (setq t2 (math-simplify
2302 (or (math-integerp t2)
2303 (and (math-messy-integerp t2)
2304 (setq t2 (math-trunc t2)))
2306 (and (math-messy-integerp t3)
2307 (setq t3 (math-trunc t3)))))
2308 (if (or (math-zerop t2) (math-zerop t3))
2310 (if (or (math-evenp t2) (math-evenp t3))
2311 (if (or (math-negp t2) (math-negp t3))
2312 (if (math-posp high)
2315 (list 'calcFunc-dfact
2317 (list 'calcFunc-dfact
2320 (list 'calcFunc-dfact t3)
2321 (list 'calcFunc-dfact
2326 (list '/ (list '- (list '- t2 t3)
2330 (list 'calcFunc-dfact
2332 (list 'calcFunc-dfact
2336 (list 'calcFunc-dfact t3)
2337 (list 'calcFunc-dfact
2341 (if (equal val '(var nan var-nan)) (setq val nil))
2343 (let* ((math-tabulate-initial 1)
2344 (math-tabulate-function 'calcFunc-prod))
2345 (calcFunc-table expr var low high)))))
2350 (defvar math-solve-ranges nil)
2351 (defvar math-solve-sign)
2352 ;;; Attempt to reduce math-solve-lhs = math-solve-rhs to
2353 ;;; math-solve-var = math-solve-rhs', where math-solve-var appears
2354 ;;; in math-solve-lhs but not in math-solve-rhs or math-solve-rhs';
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)
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
2363 ;; math-decompose-poly, but used by math-solve-poly-funny-powers.)
2364 (defvar math-solve-lhs)
2365 (defvar math-solve-rhs)
2366 (defvar math-try-solve-sign)
2368 (defun math-try-solve-for
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)))
2376 (while math-solve-ranges
2377 (setq p (car math-solve-ranges)
2380 (while (setq p (cdr p))
2381 (setq newvec (nconc newvec
2382 (cdr (math-expr-subst
2383 vec var (car p))))))
2385 math-solve-ranges (cdr math-solve-ranges)))
2386 (math-normalize vec))
2388 ((Math-primp math-solve-lhs)
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))
2399 (setq math-t1 (math-try-solve-for
2400 (math-sub (nth 1 (nth 1 math-solve-lhs))
2403 (nth 1 (nth 2 math-solve-lhs))))
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)))
2412 (math-decompose-poly math-solve-lhs
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)
2423 (apply 'math-solve-linear
2424 (car math-t2) math-try-solve-sign math-t1))
2426 (math-poly-all-roots (car math-t2) math-t1))
2427 (calc-symbolic-mode nil)
2431 (math-poly-any-root (reverse math-t1) 0 t)
2434 (if (eq (nth 2 math-t2) 1)
2436 (math-solve-prod math-t1 (math-try-solve-for (nth 2 math-t2) 0 nil t)))
2437 (calc-record-why "*Unable to find a symbolic solution")
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
2441 (math-try-solve-for (math-simplify
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)))
2446 (math-sub (math-normalize
2448 math-solve-lhs math-t1 0))
2450 math-t2 math-solve-var)))
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))
2472 (math-quarter-circle t)
2473 (nth 1 (nth 2 math-solve-lhs)))))
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)
2478 (and math-try-solve-sign
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
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))
2492 (math-try-solve-for (nth 1 math-solve-lhs)
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)
2507 (funcall math-t1 math-solve-lhs
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)))
2513 ((setq math-t1 (math-expand-formula math-solve-lhs))
2514 (math-try-solve-for math-t1 math-solve-rhs math-try-solve-sign))
2516 (calc-record-why "*No inverse known" math-solve-lhs)
2520 (defun math-try-solve-prod ()
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))
2525 (math-solve-sign math-try-solve-sign
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))
2530 (math-solve-sign math-try-solve-sign
2531 (nth 2 math-solve-lhs))))
2532 ((Math-zerop math-solve-rhs)
2533 (math-solve-prod (let ((math-solve-ranges math-solve-ranges))
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)
2540 (math-solve-sign math-try-solve-sign
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))
2545 (math-solve-sign math-try-solve-sign
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)
2552 ((eq (car math-solve-lhs) '^)
2553 (cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var))
2555 (nth 2 math-solve-lhs)
2556 (math-add (math-normalize
2557 (list 'calcFunc-log math-solve-rhs (nth 1 math-solve-lhs)))
2560 (math-mul '(var pi var-pi)
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)))
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
2579 (list 'calcFunc-sqrt math-t2))))))
2581 (nth 1 math-solve-lhs)
2582 (math-normalize math-t2)))
2583 ((math-looks-negp (nth 2 math-solve-lhs))
2585 (list '^ (nth 1 math-solve-lhs)
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
2592 (list 'calcFunc-nroot math-solve-rhs
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)))
2597 (nth 1 math-solve-lhs) math-t1
2598 (and math-try-solve-sign
2599 (math-oddp (nth 2 math-solve-lhs))
2600 (math-solve-sign math-try-solve-sign
2601 (nth 2 math-solve-lhs)))))
2602 (t (math-try-solve-for
2603 (nth 1 math-solve-lhs)
2607 (if (Math-realp (nth 2 math-solve-lhs))
2612 (and (integerp (nth 2 math-solve-lhs))
2614 (nth 2 math-solve-lhs)))))
2615 (math-div (nth 2 math-solve-lhs) 2))
2622 (and (integerp (nth 2 math-solve-lhs))
2624 (nth 2 math-solve-lhs))))))
2625 (nth 2 math-solve-lhs)))))
2627 (list 'calcFunc-nroot
2629 (nth 2 math-solve-lhs))))
2630 (and math-try-solve-sign
2631 (math-oddp (nth 2 math-solve-lhs))
2632 (math-solve-sign math-try-solve-sign
2633 (nth 2 math-solve-lhs)))))))))
2636 (defun math-solve-prod (lsoln rsoln)
2641 ((eq math-solve-full 'all)
2642 (cons 'vec (append (cdr lsoln) (cdr rsoln))))
2645 (list 'calcFunc-gt (math-solve-get-sign 1) 0)
2650 ;;; This deals with negative, fractional, and symbolic powers of "x".
2651 ;; The variable math-solve-b is local to math-decompose-poly,
2652 ;; but is used by math-solve-poly-funny-powers.
2653 (defvar math-solve-b)
2655 (defun math-solve-poly-funny-powers (sub-rhs) ; uses "t1", "t2"
2656 (setq math-t1 math-solve-lhs)
2657 (let ((pp math-poly-neg-powers)
2660 (setq fac (math-pow (car pp) (or math-poly-mult-powers 1))
2661 math-t1 (math-mul math-t1 fac)
2662 math-solve-rhs (math-mul math-solve-rhs fac)
2664 (if sub-rhs (setq math-t1 (math-sub math-t1 math-solve-rhs)))
2665 (let ((math-poly-neg-powers nil))
2666 (setq math-t2 (math-mul (or math-poly-mult-powers 1)
2667 (let ((calc-prefer-frac t))
2668 (math-div 1 math-poly-frac-powers)))
2669 math-t1 (math-is-polynomial
2670 (math-simplify (calcFunc-expand math-t1)) math-solve-b 50))))
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"
2675 (while (and math-t1 (Math-zerop (car math-t1)))
2676 (setq math-t1 (cdr math-t1)
2679 (let* ((degree (1- (length math-t1)))
2681 (while (and (> scale 1) (= (car math-t3) 1))
2682 (and (= (% degree scale) 0)
2688 (if (= (% n scale) 0)
2689 (setq new-t1 (nconc new-t1 (list (car p))))
2690 (or (Math-zerop (car p))
2695 (setq math-t3 (cons scale (cdr math-t3))
2697 (setq scale (1- scale)))
2698 (setq math-t3 (list (math-mul (car math-t3) math-t2)
2699 (math-mul count math-t2)))
2700 (<= (1- (length math-t1)) max-degree)))))
2702 (defun calcFunc-poly (expr var &optional degree)
2704 (or (natnump degree) (math-reject-arg degree 'fixnatnump))
2706 (let ((p (math-is-polynomial expr var degree 'gen)))
2711 (math-reject-arg expr "Expected a polynomial"))))
2713 (defun calcFunc-gpoly (expr var &optional degree)
2715 (or (natnump degree) (math-reject-arg degree 'fixnatnump))
2717 (let* ((math-poly-base-variable var)
2718 (d (math-decompose-poly expr var degree nil)))
2721 (math-reject-arg expr "Expected a polynomial"))))
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
2729 (lambda (math-solve-b)
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))
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
2737 math-t1 (math-is-polynomial math-solve-lhs
2739 (if (and (equal math-poly-neg-powers '(1))
2740 (memq math-poly-mult-powers '(nil 1))
2741 (eq math-poly-frac-powers 1)
2743 (setq math-t1 (cons (math-sub (car math-t1) math-solve-rhs)
2745 (math-solve-poly-funny-powers sub-rhs))
2746 (math-solve-crunch-poly degree)
2747 (or (math-expr-contains math-solve-b math-solve-var)
2748 (math-expr-contains (car math-t3) math-solve-var))))))))
2750 (list (math-pow math-t2 (car math-t3))
2753 (math-pow math-t2 (nth 1 math-t3))
2754 (math-div (math-pow math-t2 (nth 1 math-t3)) math-solve-rhs))))))
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)
2762 (defun math-solve-quadratic (var c b a)
2765 (if (math-looks-evenp b)
2766 (let ((halfb (math-div b 2)))
2770 (math-solve-get-sign
2772 (list 'calcFunc-sqrt
2773 (math-add (math-sqr halfb)
2774 (math-mul (math-neg c) a))))))
2779 (math-solve-get-sign
2781 (list 'calcFunc-sqrt
2782 (math-add (math-sqr b)
2783 (math-mul 4 (math-mul (math-neg c) a)))))))
2787 (defun math-solve-cubic (var d c b a)
2788 (let* ((p (math-div b a))
2792 (aa (math-sub q (math-div psqr 3)))
2794 (math-div (math-sub (math-mul 2 (math-mul psqr p))
2795 (math-mul 9 (math-mul p q)))
2799 (math-try-solve-for (math-pow (math-add var (math-div p 3)) 3)
2800 (math-neg bb) nil t)
2803 (math-mul (math-add var (math-div p 3))
2804 (math-add (math-sqr (math-add var (math-div p 3)))
2807 (setq m (math-mul 2 (list 'calcFunc-sqrt (math-div aa -3))))
2816 (math-sub (list 'calcFunc-arccos
2817 (math-div (math-mul 3 bb)
2821 (math-add 1 (math-solve-get-int
2824 calc-symbolic-mode))))
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))
2836 (let* ((asqr (math-sqr a))
2837 (asqr4 (math-div asqr 4))
2838 (y (let ((math-solve-full nil)
2840 (math-solve-cubic math-solve-var
2842 (math-mul 4 (math-mul b d))
2845 (math-sub (math-mul a c)
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
2854 (math-sub (math-mul 3 asqr4)
2856 (if (Math-zerop rsqr)
2860 (list 'calcFunc-sqrt
2861 (math-sub (math-sqr y)
2867 (math-mul 4 (math-mul a b))
2873 (math-sub (math-add (math-mul sign1 (math-div r 2))
2874 (math-solve-get-sign (math-div de 2)))
2878 (defvar math-symbolic-solve nil)
2879 (defvar math-int-coefs nil)
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)
2887 (defvar math-int-factors)
2888 (defvar math-double-roots)
2890 (defun math-poly-all-roots (var p &optional math-factoring)
2892 (let* ((math-symbolic-solve calc-symbolic-mode)
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)
2902 ;; If rational coefficients, look for exact rational factors.
2903 (while (and pp (Math-ratp (car pp)))
2906 (if (or math-factoring math-symbolic-solve)
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)))))
2915 (let ((calc-prefer-frac nil)
2916 (calc-symbolic-mode nil)
2918 (def-p (copy-sequence orig-p)))
2920 (if (Math-numberp (car pp))
2923 (while (> deg (if math-symbolic-solve 2 4))
2924 (let* ((x (math-poly-any-root def-p '(float 0 0) nil))
2926 (if (and (eq (car-safe x) 'cplx)
2927 (math-nearly-zerop (nth 2 x) (nth 1 x)))
2928 (setq x (calcFunc-re x)))
2930 (setq roots (cons x roots)))
2931 (or (math-numberp x)
2932 (setq x (math-evaluate-expr x)))
2935 (while (setq pp (cdr pp))
2938 (setq b (math-add (math-mul x b) c)))
2939 (setq def-p (cdr def-p)
2941 (setq p (reverse def-p))))
2943 (let ((math-solve-var '(var DUMMY var-DUMMY))
2944 (math-solve-sign nil)
2945 (math-solve-ranges nil)
2946 (math-solve-full 'all))
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)
2954 'math-solve-quartic))
2958 (setq roots (cons (math-div (math-neg (car p)) (nth 1 p))
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)
2968 (let ((root (car roots))
2969 (math-solve-full (and math-solve-full 'all)))
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)))
2979 (if (eq math-solve-full t)
2980 (list 'calcFunc-subscr
2982 (math-solve-get-int 1 (1- (length orig-p)) 1))
2985 (defun math-lcm-denoms (&rest fracs)
2988 (if (eq (car-safe (car fracs)) 'frac)
2989 (setq den (calcFunc-lcm den (nth 2 (car fracs)))))
2990 (setq fracs (cdr fracs)))
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))
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))
3004 (math-poly-laguerre-root p x polish)))))
3005 (and math-symbolic-solve (math-floatp res)
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)
3014 (while (and (> (setq iters (1- iters)) 0)
3016 (math-working "newton" x)
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)))
3024 (setq dx (math-div b d)
3027 (let ((adx (math-abs-approx dx)))
3028 (and (math-lessp adx math-int-threshold)
3029 (let ((iroot (math-poly-integer-root x)))
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)
3037 1 (math-div (math-abs-approx dx) (math-abs-approx x))))))
3039 (defun math-poly-integer-root (x)
3040 (and (math-lessp (calcFunc-xpon (math-abs-approx x)) calc-internal-prec)
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))
3051 (icp math-int-coefs)
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)
3060 (setq math-int-coefs (nreverse newcoef)
3061 math-int-factors (cons (list (math-neg rnd))
3064 ;; Look for irreducible quadratic factor
3065 (let* ((rnd1 (math-div (math-round
3066 (math-mul xre (math-mul -2 math-int-scale)))
3068 (sqscale (math-sqr math-int-scale))
3069 (rnd0 (math-div (math-round (math-mul (math-add xresq ximsq)
3072 (rem1 (car math-int-coefs))
3073 (icp (cdr math-int-coefs))
3076 (found (assoc (list rnd0 rnd1 (math-posp xim))
3080 (setq math-double-roots (delq found math-double-roots)
3082 (while (setq icp (cdr icp))
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)
3089 (let ((aa (math-div rnd1 -2)))
3090 (or found (setq math-int-coefs (reverse newcoef)
3091 math-double-roots (cons (list
3096 math-int-factors (cons (cons rnd0 rnd1)
3099 (let ((calc-symbolic-mode math-symbolic-solve))
3100 (math-mul (math-sqrt (math-sub (math-sqr aa)
3102 (if (math-negp xim) -1 1)))))))))))
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)
3110 (try-newt (not polish))
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))
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))
3131 (setq x (math-pow (math-neg b) (list 'frac 1 m)))
3133 (let* ((g (math-div d b))
3135 (h (math-sub g2 (math-mul 2 (math-div f b))))
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))
3142 (setq dx (math-div m gp)
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)))
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
3155 (if (math-zerop (cdr newt2))
3156 (setq x (car newt2) x1 x)
3157 (setq x (car newt))))))))
3159 (math-nearly-equal x x1))))
3160 (let ((cdx (math-abs-approx dx)))
3165 (math-lessp cdx dxold)
3168 (let ((digs (calcFunc-xpon
3169 (math-div (math-abs-approx x) cdx))))
3171 "*Could not attain full precision")
3173 (let ((calc-internal-prec (max 3 digs)))
3174 (setq x (math-normalize x))))))
3178 (math-lessp (calcFunc-scf (math-abs-approx x)
3179 (- calc-internal-prec))
3181 (or (and (math-floatp x)
3182 (math-poly-integer-root x))
3185 (defun math-solve-above-dummy (x)
3186 (and (not (Math-primp x))
3187 (if (and (equal (nth 1 x) '(var SOLVEDUM SOLVEDUM))
3191 (while (and (setq x (cdr x))
3192 (not (setq res (math-solve-above-dummy (car x))))))
3195 (defun math-solve-find-root-term (x neg) ; sets "t2", "t3"
3196 (if (math-solve-find-root-in-prod x)
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)
3202 (if (eq (car x) '-) (not neg) neg))))))
3204 (defun math-solve-find-root-in-prod (x)
3206 (math-expr-contains x math-solve-var)
3207 (or (and (eq (car x) 'calcFunc-sqrt)
3209 (and (eq (car x) '^)
3210 (or (and (memq (math-quarter-integer (nth 2 x)) '(1 2 3))
3212 (and (eq (car-safe (nth 2 x)) 'frac)
3213 (eq (nth 2 (nth 2 x)) 3)
3215 (and (memq (car x) '(* /))
3216 (or (and (not (math-expr-contains (nth 1 x) math-solve-var))
3217 (math-solve-find-root-in-prod (nth 2 x)))
3218 (and (not (math-expr-contains (nth 2 x) math-solve-var))
3219 (math-solve-find-root-in-prod (nth 1 x))))))))
3221 ;; The variable math-solve-vars is local to math-solve-system,
3222 ;; but is used by math-solve-system-rec.
3223 (defvar math-solve-vars)
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.
3227 (defvar math-solve-simplifying)
3229 (defun math-solve-system (exprs math-solve-vars math-solve-full)
3230 (setq exprs (mapcar 'list (if (Math-vectorp exprs)
3233 math-solve-vars (if (Math-vectorp math-solve-vars)
3234 (cdr math-solve-vars)
3235 (list math-solve-vars)))
3236 (or (let ((math-solve-simplifying nil))
3237 (math-solve-system-rec exprs math-solve-vars nil))
3238 (let ((math-solve-simplifying t))
3239 (math-solve-system-rec exprs math-solve-vars nil))))
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.
3249 ;;; To support calcFunc-roots, entries in eqn-list and solns are
3250 ;;; actually lists of equations.
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)
3258 (defun math-solve-system-rec (eqn-list var-list solns)
3261 (math-solve-system-res nil))
3263 ;; Try each variable in turn.
3267 (let* ((math-solve-system-vv (car v))
3269 (elim (eq (car-safe math-solve-system-vv) 'calcFunc-elim)))
3271 (setq math-solve-system-vv (nth 1 math-solve-system-vv)))
3273 ;; Try each equation in turn.
3280 (setq math-solve-system-res nil)
3282 ;; Try to solve for math-solve-system-vv the list of equations e2.
3284 (setq res2 (or (and (eq (car e2) eprev)
3286 (math-solve-for (car e2) 0
3287 math-solve-system-vv
3289 (setq eprev (car e2)
3290 math-solve-system-res (cons (if (eq math-solve-full 'all)
3293 math-solve-system-res)
3296 (setq math-solve-system-res nil)
3298 ;; Found a solution. Now try other variables.
3299 (setq math-solve-system-res (nreverse math-solve-system-res)
3300 math-solve-system-res (math-solve-system-rec
3302 'math-solve-system-subst
3304 (copy-sequence eqn-list)))
3305 (delq (car v) (copy-sequence var-list))
3306 (let ((math-solve-simplifying nil)
3312 (math-solve-system-subst
3318 math-solve-system-vv
3319 (apply 'append math-solve-system-res))
3321 (not math-solve-system-res))))
3323 (not math-solve-system-res)))
3325 math-solve-system-res)
3327 ;; Eliminated all variables, so now put solution into the proper format.
3328 (setq solns (sort solns
3331 (not (memq (car x) (memq (car y) math-solve-vars)))))))
3332 (if (eq math-solve-full 'all)
3337 (mapcar (function (lambda (x) (cons 'vec (cdr x)))) solns)
3338 (mapcar (function (lambda (x) (cons 'vec x))) eqn-list)))))
3342 (mapcar (function (lambda (x) (cons 'calcFunc-eq x))) solns)
3343 (mapcar 'car eqn-list)))))))
3345 (defun math-solve-system-subst (x) ; uses "res" and "v"
3347 (res2 math-solve-system-res))
3349 (setq accum (nconc accum
3352 (if math-solve-simplifying
3355 (car x) math-solve-system-vv r))
3357 (car x) math-solve-system-vv r))))
3364 ;; calc-command-flags is declared in calc.el
3365 (defvar calc-command-flags)
3367 (defun math-get-from-counter (name)
3368 (let ((ctr (assq name calc-command-flags)))
3370 (setcdr ctr (1+ (cdr ctr)))
3371 (setq ctr (cons name 1)
3372 calc-command-flags (cons ctr calc-command-flags)))
3375 (defvar var-GenCount)
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 '^
3386 (math-div (nth 2 (nth 1 val)) 2)))))
3388 (if (and (calc-var-value 'var-GenCount)
3389 (Math-natnump var-GenCount)
3390 (not (eq math-solve-full 'all)))
3392 (math-mul (list 'calcFunc-as var-GenCount) val)
3393 (setq var-GenCount (math-add var-GenCount 1))
3394 (calc-refresh-evaltos 'var-GenCount))
3395 (let* ((var (concat "s" (int-to-string (math-get-from-counter 'solve-sign))))
3396 (var2 (list 'var (intern var) (intern (concat "var-" var)))))
3397 (if (eq math-solve-full 'all)
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")
3404 (defun math-solve-get-int (val &optional range first)
3406 (if (and (calc-var-value 'var-GenCount)
3407 (Math-natnump var-GenCount)
3408 (not (eq math-solve-full 'all)))
3410 (math-mul val (list 'calcFunc-an var-GenCount))
3411 (setq var-GenCount (math-add var-GenCount 1))
3412 (calc-refresh-evaltos 'var-GenCount))
3413 (let* ((var (concat "n" (int-to-string
3414 (math-get-from-counter 'solve-int))))
3415 (var2 (list 'var (intern var) (intern (concat "var-" var)))))
3416 (if (and range (eq math-solve-full 'all))
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")
3425 (defun math-solve-sign (sign expr)
3427 (let ((s1 (math-possible-signs expr)))
3428 (cond ((memq s1 '(4 6))
3433 (defun math-looks-evenp (expr)
3434 (if (Math-integerp expr)
3436 (if (memq (car expr) '(* /))
3437 (math-looks-evenp (nth 1 expr)))))
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)
3443 (math-with-extra-prec 1
3444 (let* ((math-poly-base-variable math-solve-var)
3445 (res (math-try-solve-for lhs rhs sign)))
3446 (if (and (eq math-solve-full 'all)
3447 (math-known-realp math-solve-var))
3448 (let ((old-len (length res))
3453 (and (not (memq (car-safe x)
3457 new-len (length res))
3458 (if (< new-len old-len)
3459 (calc-record-why (if (= new-len 1)
3460 "*All solutions were complex"
3462 "*Omitted %d complex solutions"
3463 (- old-len new-len)))))))
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))
3471 (if (eq (car expr) 'calcFunc-neq) nil 1))))
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)
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)))
3484 (list 'calcFunc-eq var res)))))
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")))
3490 (list func expr var))
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))
3496 (math-reject-solution expr var 'calcFunc-solve)))
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))
3502 (math-reject-solution expr var 'calcFunc-fsolve)))
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))
3509 (math-reject-solution expr var 'calcFunc-roots))))
3511 (defun calcFunc-finv (expr var)
3512 (let ((res (math-solve-for expr math-integ-var var nil)))
3514 (math-normalize (math-expr-subst res math-integ-var var))
3515 (math-reject-solution expr var 'calcFunc-finv))))
3517 (defun calcFunc-ffinv (expr var)
3518 (let ((res (math-solve-for expr math-integ-var var t)))
3520 (math-normalize (math-expr-subst res math-integ-var var))
3521 (math-reject-solution expr var 'calcFunc-finv))))
3524 (put 'calcFunc-inv 'math-inverse
3525 (function (lambda (x) (math-div 1 x))))
3526 (put 'calcFunc-inv 'math-inverse-sign -1)
3528 (put 'calcFunc-sqrt 'math-inverse
3529 (function (lambda (x) (math-sqr x))))
3531 (put 'calcFunc-conj 'math-inverse
3532 (function (lambda (x) (list 'calcFunc-conj x))))
3534 (put 'calcFunc-abs 'math-inverse
3535 (function (lambda (x) (math-solve-get-sign x))))
3537 (put 'calcFunc-deg 'math-inverse
3538 (function (lambda (x) (list 'calcFunc-rad x))))
3539 (put 'calcFunc-deg 'math-inverse-sign 1)
3541 (put 'calcFunc-rad 'math-inverse
3542 (function (lambda (x) (list 'calcFunc-deg x))))
3543 (put 'calcFunc-rad 'math-inverse-sign 1)
3545 (put 'calcFunc-ln 'math-inverse
3546 (function (lambda (x) (list 'calcFunc-exp x))))
3547 (put 'calcFunc-ln 'math-inverse-sign 1)
3549 (put 'calcFunc-log10 'math-inverse
3550 (function (lambda (x) (list 'calcFunc-exp10 x))))
3551 (put 'calcFunc-log10 'math-inverse-sign 1)
3553 (put 'calcFunc-lnp1 'math-inverse
3554 (function (lambda (x) (list 'calcFunc-expm1 x))))
3555 (put 'calcFunc-lnp1 'math-inverse-sign 1)
3557 (put 'calcFunc-exp 'math-inverse
3558 (function (lambda (x) (math-add (math-normalize (list 'calcFunc-ln x))
3560 (math-mul '(var pi var-pi)
3562 '(var i var-i))))))))
3563 (put 'calcFunc-exp 'math-inverse-sign 1)
3565 (put 'calcFunc-expm1 'math-inverse
3566 (function (lambda (x) (math-add (math-normalize (list 'calcFunc-lnp1 x))
3568 (math-mul '(var pi var-pi)
3570 '(var i var-i))))))))
3571 (put 'calcFunc-expm1 'math-inverse-sign 1)
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))
3578 (math-mul (math-half-circle t)
3581 (put 'calcFunc-cos 'math-inverse
3582 (function (lambda (x) (math-add (math-solve-get-sign
3584 (list 'calcFunc-arccos x)))
3586 (math-full-circle t))))))
3588 (put 'calcFunc-tan 'math-inverse
3589 (function (lambda (x) (math-add (math-normalize (list 'calcFunc-arctan x))
3591 (math-half-circle t))))))
3593 (put 'calcFunc-arcsin 'math-inverse
3594 (function (lambda (x) (math-normalize (list 'calcFunc-sin x)))))
3596 (put 'calcFunc-arccos 'math-inverse
3597 (function (lambda (x) (math-normalize (list 'calcFunc-cos x)))))
3599 (put 'calcFunc-arctan 'math-inverse
3600 (function (lambda (x) (math-normalize (list 'calcFunc-tan x)))))
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))
3607 (math-mul (math-half-circle t)
3611 (put 'calcFunc-sinh 'math-inverse-sign 1)
3613 (put 'calcFunc-cosh 'math-inverse
3614 (function (lambda (x) (math-add (math-solve-get-sign
3616 (list 'calcFunc-arccosh x)))
3617 (math-mul (math-full-circle t)
3619 '(var i var-i)))))))
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)
3626 '(var i var-i)))))))
3627 (put 'calcFunc-tanh 'math-inverse-sign 1)
3629 (put 'calcFunc-arcsinh 'math-inverse
3630 (function (lambda (x) (math-normalize (list 'calcFunc-sinh x)))))
3631 (put 'calcFunc-arcsinh 'math-inverse-sign 1)
3633 (put 'calcFunc-arccosh 'math-inverse
3634 (function (lambda (x) (math-normalize (list 'calcFunc-cosh x)))))
3636 (put 'calcFunc-arctanh 'math-inverse
3637 (function (lambda (x) (math-normalize (list 'calcFunc-tanh x)))))
3638 (put 'calcFunc-arctanh 'math-inverse-sign 1)
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))
3647 (or (and (eq (car-safe v) 'var)
3648 (math-expr-contains expr v)
3650 (let ((accum (math-expr-subst expr v x0))
3651 (var2 (if (eq (car var) 'calcFunc-eq)
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)
3667 (math-normalize accum))))
3668 (list 'calcFunc-taylor expr var num))))
3672 ;; arch-tag: f2932ec8-dd63-418b-a542-11a644b9d4c4
3673 ;;; calcalg2.el ends here