1 ;;; calcalg2.el --- more algebraic functions for Calc
3 ;; Copyright (C) 1990, 1991, 1992, 1993, 2001 Free Software Foundation, Inc.
5 ;; Author: David Gillespie <daveg@synaptics.com>
6 ;; Maintainer: Colin Walters <walters@debian.org>
8 ;; This file is part of GNU Emacs.
10 ;; GNU Emacs is distributed in the hope that it will be useful,
11 ;; but WITHOUT ANY WARRANTY. No author or distributor
12 ;; accepts responsibility to anyone for the consequences of using it
13 ;; or for whether it serves any particular purpose or works at all,
14 ;; unless he says so in writing. Refer to the GNU Emacs General Public
15 ;; License for full details.
17 ;; Everyone is granted permission to copy, modify and redistribute
18 ;; GNU Emacs, but only under the conditions described in the
19 ;; GNU Emacs General Public License. A copy of this license is
20 ;; supposed to have been given to you along with GNU Emacs so you
21 ;; can know your rights and responsibilities. It should be in a
22 ;; file named COPYING. Among other things, the copyright notice
23 ;; and this notice must be preserved on all copies.
29 ;; This file is autoloaded from calc-ext.el.
34 (defun calc-Need-calc-alg-2 () nil
)
37 (defun calc-derivative (var num
)
38 (interactive "sDifferentiate with respect to: \np")
41 (error "Order of derivative must be positive"))
42 (let ((func (if (calc-is-hyperbolic) 'calcFunc-tderiv
'calcFunc-deriv
))
44 (if (or (equal var
"") (equal var
"$"))
48 (setq var
(math-read-expr var
))
49 (when (eq (car-safe var
) 'error
)
50 (error "Bad format in expression: %s" (nth 1 var
)))
53 (while (>= (setq num
(1- num
)) 0)
54 (setq expr
(list func expr var
)))
55 (calc-enter-result n
"derv" expr
))))
57 (defun calc-integral (var)
58 (interactive "sIntegration variable: ")
60 (if (or (equal var
"") (equal var
"$"))
61 (calc-enter-result 2 "intg" (list 'calcFunc-integ
64 (let ((var (math-read-expr var
)))
65 (if (eq (car-safe var
) 'error
)
66 (error "Bad format in expression: %s" (nth 1 var
)))
67 (calc-enter-result 1 "intg" (list 'calcFunc-integ
71 (defun calc-num-integral (&optional varname lowname highname
)
72 (interactive "sIntegration variable: ")
73 (calc-tabular-command 'calcFunc-ninteg
"Integration" "nint"
74 nil varname lowname highname
))
76 (defun calc-summation (arg &optional varname lowname highname
)
77 (interactive "P\nsSummation variable: ")
78 (calc-tabular-command 'calcFunc-sum
"Summation" "sum"
79 arg varname lowname highname
))
81 (defun calc-alt-summation (arg &optional varname lowname highname
)
82 (interactive "P\nsSummation variable: ")
83 (calc-tabular-command 'calcFunc-asum
"Summation" "asum"
84 arg varname lowname highname
))
86 (defun calc-product (arg &optional varname lowname highname
)
87 (interactive "P\nsIndex variable: ")
88 (calc-tabular-command 'calcFunc-prod
"Index" "prod"
89 arg varname lowname highname
))
91 (defun calc-tabulate (arg &optional varname lowname highname
)
92 (interactive "P\nsIndex variable: ")
93 (calc-tabular-command 'calcFunc-table
"Index" "tabl"
94 arg varname lowname highname
))
96 (defun calc-tabular-command (func prompt prefix arg varname lowname highname
)
98 (let (var (low nil
) (high nil
) (step nil
) stepname stepnum
(num 1) expr
)
102 (if (or (equal varname
"") (equal varname
"$") (null varname
))
103 (setq high
(calc-top-n (+ stepnum
1))
104 low
(calc-top-n (+ stepnum
2))
105 var
(calc-top-n (+ stepnum
3))
107 (setq var
(if (stringp varname
) (math-read-expr varname
) varname
))
108 (if (eq (car-safe var
) 'error
)
109 (error "Bad format in expression: %s" (nth 1 var
)))
111 (setq lowname
(read-string (concat prompt
" variable: " varname
113 (if (or (equal lowname
"") (equal lowname
"$"))
114 (setq high
(calc-top-n (+ stepnum
1))
115 low
(calc-top-n (+ stepnum
2))
117 (setq low
(if (stringp lowname
) (math-read-expr lowname
) lowname
))
118 (if (eq (car-safe low
) 'error
)
119 (error "Bad format in expression: %s" (nth 1 low
)))
121 (setq highname
(read-string (concat prompt
" variable: " varname
124 (if (or (equal highname
"") (equal highname
"$"))
125 (setq high
(calc-top-n (+ stepnum
1))
127 (setq high
(if (stringp highname
) (math-read-expr highname
)
129 (if (eq (car-safe high
) 'error
)
130 (error "Bad format in expression: %s" (nth 1 high
)))
133 (setq stepname
(read-string (concat prompt
" variable: "
138 (if (or (equal stepname
"") (equal stepname
"$"))
139 (setq step
(calc-top-n 1)
141 (setq step
(math-read-expr stepname
))
142 (if (eq (car-safe step
) 'error
)
143 (error "Bad format in expression: %s"
147 (setq step
(calc-top-n 1))
149 (setq step
(prefix-numeric-value arg
)))))
150 (setq expr
(calc-top-n num
))
151 (calc-enter-result num prefix
(append (list func expr var low high
)
152 (and step
(list step
)))))))
154 (defun calc-solve-for (var)
155 (interactive "sVariable to solve for: ")
157 (let ((func (if (calc-is-inverse)
158 (if (calc-is-hyperbolic) 'calcFunc-ffinv
'calcFunc-finv
)
159 (if (calc-is-hyperbolic) 'calcFunc-fsolve
'calcFunc-solve
))))
160 (if (or (equal var
"") (equal var
"$"))
161 (calc-enter-result 2 "solv" (list func
164 (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var
)
165 (not (string-match "\\[" var
)))
166 (math-read-expr (concat "[" var
"]"))
167 (math-read-expr var
))))
168 (if (eq (car-safe var
) 'error
)
169 (error "Bad format in expression: %s" (nth 1 var
)))
170 (calc-enter-result 1 "solv" (list func
174 (defun calc-poly-roots (var)
175 (interactive "sVariable to solve for: ")
177 (if (or (equal var
"") (equal var
"$"))
178 (calc-enter-result 2 "prts" (list 'calcFunc-roots
181 (let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var
)
182 (not (string-match "\\[" var
)))
183 (math-read-expr (concat "[" var
"]"))
184 (math-read-expr var
))))
185 (if (eq (car-safe var
) 'error
)
186 (error "Bad format in expression: %s" (nth 1 var
)))
187 (calc-enter-result 1 "prts" (list 'calcFunc-roots
191 (defun calc-taylor (var nterms
)
192 (interactive "sTaylor expansion variable: \nNNumber of terms: ")
194 (let ((var (math-read-expr var
)))
195 (if (eq (car-safe var
) 'error
)
196 (error "Bad format in expression: %s" (nth 1 var
)))
197 (calc-enter-result 1 "tylr" (list 'calcFunc-taylor
200 (prefix-numeric-value nterms
))))))
203 (defun math-derivative (expr) ; uses global values: deriv-var, deriv-total.
204 (cond ((equal expr deriv-var
)
206 ((or (Math-scalarp expr
)
207 (eq (car expr
) 'sdev
)
208 (and (eq (car expr
) 'var
)
209 (or (not deriv-total
)
210 (math-const-var expr
)
212 (math-setup-declarations)
213 (memq 'const
(nth 1 (or (assq (nth 2 expr
)
215 math-decls-all
)))))))
218 (math-add (math-derivative (nth 1 expr
))
219 (math-derivative (nth 2 expr
))))
221 (math-sub (math-derivative (nth 1 expr
))
222 (math-derivative (nth 2 expr
))))
223 ((memq (car expr
) '(calcFunc-eq calcFunc-neq calcFunc-lt
224 calcFunc-gt calcFunc-leq calcFunc-geq
))
226 (math-derivative (nth 1 expr
))
227 (math-derivative (nth 2 expr
))))
228 ((eq (car expr
) 'neg
)
229 (math-neg (math-derivative (nth 1 expr
))))
231 (math-add (math-mul (nth 2 expr
)
232 (math-derivative (nth 1 expr
)))
233 (math-mul (nth 1 expr
)
234 (math-derivative (nth 2 expr
)))))
236 (math-sub (math-div (math-derivative (nth 1 expr
))
238 (math-div (math-mul (nth 1 expr
)
239 (math-derivative (nth 2 expr
)))
240 (math-sqr (nth 2 expr
)))))
242 (let ((du (math-derivative (nth 1 expr
)))
243 (dv (math-derivative (nth 2 expr
))))
245 (setq du
(math-mul (nth 2 expr
)
246 (math-mul (math-normalize
249 (math-add (nth 2 expr
) -
1)))
252 (setq dv
(math-mul (math-normalize
253 (list 'calcFunc-ln
(nth 1 expr
)))
254 (math-mul expr dv
))))
257 (math-derivative (nth 1 expr
))) ; a reasonable definition
258 ((eq (car expr
) 'vec
)
259 (math-map-vec 'math-derivative expr
))
260 ((and (memq (car expr
) '(calcFunc-conj calcFunc-re calcFunc-im
))
262 (list (car expr
) (math-derivative (nth 1 expr
))))
263 ((and (memq (car expr
) '(calcFunc-subscr calcFunc-mrow calcFunc-mcol
))
265 (let ((d (math-derivative (nth 1 expr
))))
267 0 ; assume x and x_1 are independent vars
268 (list (car expr
) d
(nth 2 expr
)))))
269 (t (or (and (symbolp (car expr
))
270 (if (= (length expr
) 2)
271 (let ((handler (get (car expr
) 'math-derivative
)))
273 (let ((deriv (math-derivative (nth 1 expr
))))
274 (if (Math-zerop deriv
)
276 (math-mul (funcall handler
(nth 1 expr
))
278 (let ((handler (get (car expr
) 'math-derivative-n
)))
280 (funcall handler expr
)))))
281 (and (not (eq deriv-symb
'pre-expand
))
282 (let ((exp (math-expand-formula expr
)))
284 (or (let ((deriv-symb 'pre-expand
))
285 (catch 'math-deriv
(math-derivative expr
)))
286 (math-derivative exp
)))))
287 (if (or (Math-objvecp expr
)
289 (not (symbolp (car expr
))))
291 (throw 'math-deriv nil
)
292 (list (if deriv-total
'calcFunc-tderiv
'calcFunc-deriv
)
299 (while (setq arg
(cdr arg
))
300 (or (Math-zerop (setq derv
(math-derivative (car arg
))))
301 (let ((func (intern (concat (symbol-name (car expr
))
306 (prop (cond ((= (length expr
) 2)
315 'math-derivative-5
))))
321 (let ((handler (get func prop
)))
322 (or (and prop handler
323 (apply handler
(cdr expr
)))
327 (throw 'math-deriv nil
)
328 (cons func
(cdr expr
))))))))))
332 (defun calcFunc-deriv (expr deriv-var
&optional deriv-value deriv-symb
)
333 (let* ((deriv-total nil
)
334 (res (catch 'math-deriv
(math-derivative expr
))))
335 (or (eq (car-safe res
) 'calcFunc-deriv
)
337 (setq res
(math-normalize res
)))
340 (math-expr-subst res deriv-var deriv-value
)
343 (defun calcFunc-tderiv (expr deriv-var
&optional deriv-value deriv-symb
)
344 (math-setup-declarations)
345 (let* ((deriv-total t
)
346 (res (catch 'math-deriv
(math-derivative expr
))))
347 (or (eq (car-safe res
) 'calcFunc-tderiv
)
349 (setq res
(math-normalize res
)))
352 (math-expr-subst res deriv-var deriv-value
)
355 (put 'calcFunc-inv
\' 'math-derivative-1
356 (function (lambda (u) (math-neg (math-div 1 (math-sqr u
))))))
358 (put 'calcFunc-sqrt
\' 'math-derivative-1
359 (function (lambda (u) (math-div 1 (math-mul 2 (list 'calcFunc-sqrt u
))))))
361 (put 'calcFunc-deg
\' 'math-derivative-1
362 (function (lambda (u) (math-div-float '(float 18 1) (math-pi)))))
364 (put 'calcFunc-rad
\' 'math-derivative-1
365 (function (lambda (u) (math-pi-over-180))))
367 (put 'calcFunc-ln
\' 'math-derivative-1
368 (function (lambda (u) (math-div 1 u
))))
370 (put 'calcFunc-log10
\' 'math-derivative-1
371 (function (lambda (u)
372 (math-div (math-div 1 (math-normalize '(calcFunc-ln 10)))
375 (put 'calcFunc-lnp1
\' 'math-derivative-1
376 (function (lambda (u) (math-div 1 (math-add u
1)))))
378 (put 'calcFunc-log
\' 'math-derivative-2
379 (function (lambda (x b
)
380 (and (not (Math-zerop b
))
381 (let ((lnv (math-normalize
382 (list 'calcFunc-ln b
))))
383 (math-div 1 (math-mul lnv x
)))))))
385 (put 'calcFunc-log
\'2 'math-derivative-2
386 (function (lambda (x b
)
387 (let ((lnv (list 'calcFunc-ln b
)))
388 (math-neg (math-div (list 'calcFunc-log x b
)
389 (math-mul lnv b
)))))))
391 (put 'calcFunc-exp
\' 'math-derivative-1
392 (function (lambda (u) (math-normalize (list 'calcFunc-exp u
)))))
394 (put 'calcFunc-expm1
\' 'math-derivative-1
395 (function (lambda (u) (math-normalize (list 'calcFunc-expm1 u
)))))
397 (put 'calcFunc-sin
\' 'math-derivative-1
398 (function (lambda (u) (math-to-radians-2 (math-normalize
399 (list 'calcFunc-cos u
))))))
401 (put 'calcFunc-cos
\' 'math-derivative-1
402 (function (lambda (u) (math-neg (math-to-radians-2
404 (list 'calcFunc-sin u
)))))))
406 (put 'calcFunc-tan
\' 'math-derivative-1
407 (function (lambda (u) (math-to-radians-2
408 (math-div 1 (math-sqr
410 (list 'calcFunc-cos u
))))))))
412 (put 'calcFunc-arcsin
\' 'math-derivative-1
413 (function (lambda (u)
415 (math-div 1 (math-normalize
417 (math-sub 1 (math-sqr u
)))))))))
419 (put 'calcFunc-arccos
\' 'math-derivative-1
420 (function (lambda (u)
422 (math-div -
1 (math-normalize
424 (math-sub 1 (math-sqr u
)))))))))
426 (put 'calcFunc-arctan
\' 'math-derivative-1
427 (function (lambda (u) (math-from-radians-2
428 (math-div 1 (math-add 1 (math-sqr u
)))))))
430 (put 'calcFunc-sinh
\' 'math-derivative-1
431 (function (lambda (u) (math-normalize (list 'calcFunc-cosh u
)))))
433 (put 'calcFunc-cosh
\' 'math-derivative-1
434 (function (lambda (u) (math-normalize (list 'calcFunc-sinh u
)))))
436 (put 'calcFunc-tanh
\' 'math-derivative-1
437 (function (lambda (u) (math-div 1 (math-sqr
439 (list 'calcFunc-cosh u
)))))))
441 (put 'calcFunc-arcsinh
\' 'math-derivative-1
442 (function (lambda (u)
443 (math-div 1 (math-normalize
445 (math-add (math-sqr u
) 1)))))))
447 (put 'calcFunc-arccosh
\' 'math-derivative-1
448 (function (lambda (u)
449 (math-div 1 (math-normalize
451 (math-add (math-sqr u
) -
1)))))))
453 (put 'calcFunc-arctanh
\' 'math-derivative-1
454 (function (lambda (u) (math-div 1 (math-sub 1 (math-sqr u
))))))
456 (put 'calcFunc-bern
\'2 'math-derivative-2
457 (function (lambda (n x
)
458 (math-mul n
(list 'calcFunc-bern
(math-add n -
1) x
)))))
460 (put 'calcFunc-euler
\'2 'math-derivative-2
461 (function (lambda (n x
)
462 (math-mul n
(list 'calcFunc-euler
(math-add n -
1) x
)))))
464 (put 'calcFunc-gammag
\'2 'math-derivative-2
465 (function (lambda (a x
) (math-deriv-gamma a x
1))))
467 (put 'calcFunc-gammaG
\'2 'math-derivative-2
468 (function (lambda (a x
) (math-deriv-gamma a x -
1))))
470 (put 'calcFunc-gammaP
\'2 'math-derivative-2
471 (function (lambda (a x
) (math-deriv-gamma a x
474 (list 'calcFunc-gamma
477 (put 'calcFunc-gammaQ
\'2 'math-derivative-2
478 (function (lambda (a x
) (math-deriv-gamma a x
481 (list 'calcFunc-gamma
484 (defun math-deriv-gamma (a x scale
)
486 (math-mul (math-pow x
(math-add a -
1))
487 (list 'calcFunc-exp
(math-neg x
)))))
489 (put 'calcFunc-betaB
\' 'math-derivative-3
490 (function (lambda (x a b
) (math-deriv-beta x a b
1))))
492 (put 'calcFunc-betaI
\' 'math-derivative-3
493 (function (lambda (x a b
) (math-deriv-beta x a b
495 1 (list 'calcFunc-beta
498 (defun math-deriv-beta (x a b scale
)
499 (math-mul (math-mul (math-pow x
(math-add a -
1))
500 (math-pow (math-sub 1 x
) (math-add b -
1)))
503 (put 'calcFunc-erf
\' 'math-derivative-1
504 (function (lambda (x) (math-div 2
505 (math-mul (list 'calcFunc-exp
507 (if calc-symbolic-mode
512 (put 'calcFunc-erfc
\' 'math-derivative-1
513 (function (lambda (x) (math-div -
2
514 (math-mul (list 'calcFunc-exp
516 (if calc-symbolic-mode
521 (put 'calcFunc-besJ
\'2 'math-derivative-2
522 (function (lambda (v z
) (math-div (math-sub (list 'calcFunc-besJ
530 (put 'calcFunc-besY
\'2 'math-derivative-2
531 (function (lambda (v z
) (math-div (math-sub (list 'calcFunc-besY
539 (put 'calcFunc-sum
'math-derivative-n
542 (if (math-expr-contains (cons 'vec
(cdr (cdr expr
))) deriv-var
)
543 (throw 'math-deriv nil
)
545 (cons (math-derivative (nth 1 expr
))
546 (cdr (cdr expr
))))))))
548 (put 'calcFunc-prod
'math-derivative-n
551 (if (math-expr-contains (cons 'vec
(cdr (cdr expr
))) deriv-var
)
552 (throw 'math-deriv nil
)
555 (cons (math-div (math-derivative (nth 1 expr
))
557 (cdr (cdr expr
)))))))))
559 (put 'calcFunc-integ
'math-derivative-n
562 (if (= (length expr
) 3)
563 (if (equal (nth 2 expr
) deriv-var
)
566 (list 'calcFunc-integ
567 (math-derivative (nth 1 expr
))
569 (if (= (length expr
) 5)
570 (let ((lower (math-expr-subst (nth 1 expr
) (nth 2 expr
)
572 (upper (math-expr-subst (nth 1 expr
) (nth 2 expr
)
574 (math-add (math-sub (math-mul upper
575 (math-derivative (nth 4 expr
)))
577 (math-derivative (nth 3 expr
))))
578 (if (equal (nth 2 expr
) deriv-var
)
581 (list 'calcFunc-integ
582 (math-derivative (nth 1 expr
)) (nth 2 expr
)
583 (nth 3 expr
) (nth 4 expr
)))))))))))
585 (put 'calcFunc-if
'math-derivative-n
588 (and (= (length expr
) 4)
589 (list 'calcFunc-if
(nth 1 expr
)
590 (math-derivative (nth 2 expr
))
591 (math-derivative (nth 3 expr
)))))))
593 (put 'calcFunc-subscr
'math-derivative-n
596 (and (= (length expr
) 3)
597 (list 'calcFunc-subscr
(nth 1 expr
)
598 (math-derivative (nth 2 expr
)))))))
601 (defvar math-integ-var
'(var X ---
))
602 (defvar math-integ-var-2
'(var Y ---
))
603 (defvar math-integ-vars
(list 'f math-integ-var math-integ-var-2
))
604 (defvar math-integ-var-list
(list math-integ-var
))
605 (defvar math-integ-var-list-list
(list math-integ-var-list
))
607 (defmacro math-tracing-integral
(&rest parts
)
610 (list 'save-excursion
611 '(set-buffer trace-buffer
)
612 '(goto-char (point-max))
615 '(insert (make-string (- math-integral-limit
616 math-integ-level
) 32)
617 (format "%2d " math-integ-depth
)
618 (make-string math-integ-level
32)))
619 ;;(list 'condition-case 'err
621 ;; '(error (insert (prin1-to-string err))))
624 ;;; The following wrapper caches results and avoids infinite recursion.
625 ;;; Each cache entry is: ( A B ) Integral of A is B;
626 ;;; ( A N ) Integral of A failed at level N;
627 ;;; ( A busy ) Currently working on integral of A;
628 ;;; ( A parts ) Currently working, integ-by-parts;
629 ;;; ( A parts2 ) Currently working, integ-by-parts;
630 ;;; ( A cancelled ) Ignore this cache entry;
631 ;;; ( A [B] ) Same result as for cur-record = B.
632 (defun math-integral (expr &optional simplify same-as-above
)
633 (let* ((simp cur-record
)
634 (cur-record (assoc expr math-integral-cache
))
635 (math-integ-depth (1+ math-integ-depth
))
637 (math-tracing-integral "Integrating "
638 (math-format-value expr
1000)
642 (math-tracing-integral "Found "
643 (math-format-value (nth 1 cur-record
) 1000))
644 (and (consp (nth 1 cur-record
))
645 (math-replace-integral-parts cur-record
))
646 (math-tracing-integral " => "
647 (math-format-value (nth 1 cur-record
) 1000)
650 (not (eq (nth 1 cur-record
) 'cancelled
))
651 (or (not (integerp (nth 1 cur-record
)))
652 (>= (nth 1 cur-record
) math-integ-level
)))
653 (and (math-integral-contains-parts expr
)
659 (let (math-integ-msg)
660 (if (eq calc-display-working-message
'lots
)
662 (calc-set-command-flag 'clear-message
)
663 (setq math-integ-msg
(format
664 "Working... Integrating %s"
665 (math-format-flat-expr expr
0)))
666 (message math-integ-msg
)))
668 (setcar (cdr cur-record
)
669 (if same-as-above
(vector simp
) 'busy
))
671 (list expr
(if same-as-above
(vector simp
) 'busy
))
672 math-integral-cache
(cons cur-record
673 math-integral-cache
)))
674 (if (eq simplify
'yes
)
676 (math-tracing-integral "Simplifying...")
677 (setq simp
(math-simplify expr
))
678 (setq val
(if (equal simp expr
)
680 (math-tracing-integral " no change\n")
681 (math-do-integral expr
))
682 (math-tracing-integral " simplified\n")
683 (math-integral simp
'no t
))))
684 (or (setq val
(math-do-integral expr
))
686 (let ((simp (math-simplify expr
)))
687 (or (equal simp expr
)
689 (math-tracing-integral "Trying again after "
690 "simplification...\n")
691 (setq val
(math-integral simp
'no t
))))))))
692 (if (eq calc-display-working-message
'lots
)
693 (message math-integ-msg
)))
694 (setcar (cdr cur-record
) (or val
695 (if (or math-enable-subst
696 (not math-any-substs
))
699 (setq val cur-record
)
700 (while (vectorp (nth 1 val
))
701 (setq val
(aref (nth 1 val
) 0)))
702 (setq val
(if (memq (nth 1 val
) '(parts parts2
))
704 (setcar (cdr val
) 'parts2
)
705 (list 'var
'PARTS val
))
706 (and (consp (nth 1 val
))
708 (math-tracing-integral "Integral of "
709 (math-format-value expr
1000)
711 (math-format-value val
1000)
714 (defvar math-integral-cache nil
)
715 (defvar math-integral-cache-state nil
)
717 (defun math-integral-contains-parts (expr)
718 (if (Math-primp expr
)
719 (and (eq (car-safe expr
) 'var
)
720 (eq (nth 1 expr
) 'PARTS
)
721 (listp (nth 2 expr
)))
722 (while (and (setq expr
(cdr expr
))
723 (not (math-integral-contains-parts (car expr
)))))
726 (defun math-replace-integral-parts (expr)
727 (or (Math-primp expr
)
728 (while (setq expr
(cdr expr
))
729 (and (consp (car expr
))
730 (if (eq (car (car expr
)) 'var
)
731 (and (eq (nth 1 (car expr
)) 'PARTS
)
732 (consp (nth 2 (car expr
)))
733 (if (listp (nth 1 (nth 2 (car expr
))))
735 (setcar expr
(nth 1 (nth 2 (car expr
))))
736 (math-replace-integral-parts (cons 'foo expr
)))
737 (setcar (cdr cur-record
) 'cancelled
)))
738 (math-replace-integral-parts (car expr
)))))))
740 (defun math-do-integral (expr)
742 (or (cond ((not (math-expr-contains expr math-integ-var
))
743 (math-mul expr math-integ-var
))
744 ((equal expr math-integ-var
)
745 (math-div (math-sqr expr
) 2))
747 (and (setq t1
(math-integral (nth 1 expr
)))
748 (setq t2
(math-integral (nth 2 expr
)))
751 (and (setq t1
(math-integral (nth 1 expr
)))
752 (setq t2
(math-integral (nth 2 expr
)))
754 ((eq (car expr
) 'neg
)
755 (and (setq t1
(math-integral (nth 1 expr
)))
758 (cond ((not (math-expr-contains (nth 1 expr
) math-integ-var
))
759 (and (setq t1
(math-integral (nth 2 expr
)))
760 (math-mul (nth 1 expr
) t1
)))
761 ((not (math-expr-contains (nth 2 expr
) math-integ-var
))
762 (and (setq t1
(math-integral (nth 1 expr
)))
763 (math-mul t1
(nth 2 expr
))))
764 ((memq (car-safe (nth 1 expr
)) '(+ -
))
765 (math-integral (list (car (nth 1 expr
))
766 (math-mul (nth 1 (nth 1 expr
))
768 (math-mul (nth 2 (nth 1 expr
))
771 ((memq (car-safe (nth 2 expr
)) '(+ -
))
772 (math-integral (list (car (nth 2 expr
))
773 (math-mul (nth 1 (nth 2 expr
))
775 (math-mul (nth 2 (nth 2 expr
))
779 (cond ((and (not (math-expr-contains (nth 1 expr
)
781 (not (math-equal-int (nth 1 expr
) 1)))
782 (and (setq t1
(math-integral (math-div 1 (nth 2 expr
))))
783 (math-mul (nth 1 expr
) t1
)))
784 ((not (math-expr-contains (nth 2 expr
) math-integ-var
))
785 (and (setq t1
(math-integral (nth 1 expr
)))
786 (math-div t1
(nth 2 expr
))))
787 ((and (eq (car-safe (nth 1 expr
)) '*)
788 (not (math-expr-contains (nth 1 (nth 1 expr
))
790 (and (setq t1
(math-integral
791 (math-div (nth 2 (nth 1 expr
))
793 (math-mul t1
(nth 1 (nth 1 expr
)))))
794 ((and (eq (car-safe (nth 1 expr
)) '*)
795 (not (math-expr-contains (nth 2 (nth 1 expr
))
797 (and (setq t1
(math-integral
798 (math-div (nth 1 (nth 1 expr
))
800 (math-mul t1
(nth 2 (nth 1 expr
)))))
801 ((and (eq (car-safe (nth 2 expr
)) '*)
802 (not (math-expr-contains (nth 1 (nth 2 expr
))
804 (and (setq t1
(math-integral
805 (math-div (nth 1 expr
)
806 (nth 2 (nth 2 expr
)))))
807 (math-div t1
(nth 1 (nth 2 expr
)))))
808 ((and (eq (car-safe (nth 2 expr
)) '*)
809 (not (math-expr-contains (nth 2 (nth 2 expr
))
811 (and (setq t1
(math-integral
812 (math-div (nth 1 expr
)
813 (nth 1 (nth 2 expr
)))))
814 (math-div t1
(nth 2 (nth 2 expr
)))))
815 ((eq (car-safe (nth 2 expr
)) 'calcFunc-exp
)
817 (math-mul (nth 1 expr
)
819 (math-neg (nth 1 (nth 2 expr
)))))))))
821 (cond ((not (math-expr-contains (nth 1 expr
) math-integ-var
))
822 (or (and (setq t1
(math-is-polynomial (nth 2 expr
)
831 (math-mul (nth 2 expr
)
836 ((not (math-expr-contains (nth 2 expr
) math-integ-var
))
837 (if (and (integerp (nth 2 expr
)) (< (nth 2 expr
) 0))
839 (list '/ 1 (math-pow (nth 1 expr
) (- (nth 2 expr
))))
841 (or (and (setq t1
(math-is-polynomial (nth 1 expr
)
844 (setq t2
(math-add (nth 2 expr
) 1))
845 (math-div (math-pow (nth 1 expr
) t2
)
846 (math-mul t2
(nth 1 t1
))))
847 (and (Math-negp (nth 2 expr
))
850 (math-pow (nth 1 expr
)
856 ;; Integral of a polynomial.
857 (and (setq t1
(math-is-polynomial expr math-integ-var
20))
861 (if (setq accum
(math-add accum
862 (math-div (math-mul (car t1
)
871 ;; Try looking it up!
872 (cond ((= (length expr
) 2)
873 (and (symbolp (car expr
))
874 (setq t1
(get (car expr
) 'math-integral
))
877 (not (setq t2
(funcall (car t1
)
880 (and t2
(math-normalize t2
)))))
882 (and (symbolp (car expr
))
883 (setq t1
(get (car expr
) 'math-integral-2
))
886 (not (setq t2
(funcall (car t1
)
890 (and t2
(math-normalize t2
))))))
892 ;; Integral of a rational function.
893 (and (math-ratpoly-p expr math-integ-var
)
894 (setq t1
(calcFunc-apart expr math-integ-var
))
895 (not (equal t1 expr
))
898 ;; Try user-defined integration rules.
900 (let ((math-old-integ (symbol-function 'calcFunc-integ
))
901 (input (list 'calcFunc-integtry expr math-integ-var
))
905 (fset 'calcFunc-integ
'math-sub-integration
)
906 (setq res
(math-rewrite input
907 '(var IntegRules var-IntegRules
)
909 (fset 'calcFunc-integ math-old-integ
)
910 (and (not (equal res input
))
911 (if (setq part
(math-expr-calls
912 res
'(calcFunc-integsubst)))
913 (and (memq (length part
) '(3 4 5))
921 (math-integrate-by-substitution
924 (list 'calcFunc-integfailed
927 (if (not (math-expr-calls res
929 calcFunc-integfailed
)))
931 (fset 'calcFunc-integ math-old-integ
))))
933 ;; See if the function is a symbolic derivative.
934 (and (string-match "'" (symbol-name (car expr
)))
935 (let ((name (symbol-name (car expr
)))
936 (p expr
) (n 0) (which nil
) (bad nil
))
937 (while (setq n
(1+ n
) p
(cdr p
))
938 (if (equal (car p
) math-integ-var
)
939 (if which
(setq bad t
) (setq which n
))
940 (if (math-expr-contains (car p
) math-integ-var
)
943 (let ((prime (if (= which
1) "'" (format "'%d" which
))))
944 (and (string-match (concat prime
"\\('['0-9]*\\|$\\)")
948 (substring name
0 (match-beginning 0))
949 (substring name
(+ (match-beginning 0)
953 ;; Try transformation methods (parts, substitutions).
954 (and (> math-integ-level
0)
955 (math-do-integral-methods expr
))
957 ;; Try expanding the function's definition.
958 (let ((res (math-expand-formula expr
)))
960 (math-integral res
))))))
962 (defun math-sub-integration (expr &rest rest
)
963 (or (if (or (not rest
)
964 (and (< math-integ-level math-integral-limit
)
965 (eq (car rest
) math-integ-var
)))
967 (let ((res (apply math-old-integ expr rest
)))
968 (and (or (= math-integ-level math-integral-limit
)
969 (not (math-expr-calls res
'calcFunc-integ
)))
971 (list 'calcFunc-integfailed expr
)))
973 (defun math-do-integral-methods (expr)
974 (let ((so-far math-integ-var-list-list
)
977 ;; Integration by substitution, for various likely sub-expressions.
978 ;; (In first pass, we look only for sub-exprs that are linear in X.)
979 (or (if math-enable-subst
980 (math-integ-try-substitutions expr
)
981 (math-integ-try-linear-substitutions expr
))
983 ;; If function has sines and cosines, try tan(x/2) substitution.
984 (and (let ((p (setq rat-in
(math-expr-rational-in expr
))))
986 (memq (car (car p
)) '(calcFunc-sin
989 (equal (nth 1 (car p
)) math-integ-var
))
992 (or (and (math-integ-parts-easy expr
)
993 (math-integ-try-parts expr t
))
994 (math-integrate-by-good-substitution
995 expr
(list 'calcFunc-tan
(math-div math-integ-var
2)))))
997 ;; If function has sinh and cosh, try tanh(x/2) substitution.
998 (and (let ((p rat-in
))
1000 (memq (car (car p
)) '(calcFunc-sinh
1004 (equal (nth 1 (car p
)) math-integ-var
))
1007 (or (and (math-integ-parts-easy expr
)
1008 (math-integ-try-parts expr t
))
1009 (math-integrate-by-good-substitution
1010 expr
(list 'calcFunc-tanh
(math-div math-integ-var
2)))))
1012 ;; If function has square roots, try sin, tan, or sec substitution.
1013 (and (let ((p rat-in
))
1016 (or (equal (car p
) math-integ-var
)
1017 (and (eq (car (car p
)) 'calcFunc-sqrt
)
1018 (setq t1
(math-is-polynomial
1019 (nth 1 (setq t2
(car p
)))
1020 math-integ-var
2)))))
1024 (if (math-guess-if-neg (nth 2 t1
))
1025 (let* ((c (math-sqrt (math-neg (nth 2 t1
))))
1026 (d (math-div (nth 1 t1
) (math-mul -
2 c
)))
1027 (a (math-sqrt (math-add (car t1
) (math-sqr d
)))))
1028 (math-integrate-by-good-substitution
1029 expr
(list 'calcFunc-arcsin
1031 (math-add (math-mul c math-integ-var
) d
)
1033 (let* ((c (math-sqrt (nth 2 t1
)))
1034 (d (math-div (nth 1 t1
) (math-mul 2 c
)))
1035 (aa (math-sub (car t1
) (math-sqr d
))))
1036 (if (and nil
(not (and (eq d
0) (eq c
1))))
1037 (math-integrate-by-good-substitution
1038 expr
(math-add (math-mul c math-integ-var
) d
))
1039 (if (math-guess-if-neg aa
)
1040 (math-integrate-by-good-substitution
1041 expr
(list 'calcFunc-arccosh
1043 (math-add (math-mul c math-integ-var
)
1045 (math-sqrt (math-neg aa
)))))
1046 (math-integrate-by-good-substitution
1047 expr
(list 'calcFunc-arcsinh
1049 (math-add (math-mul c math-integ-var
)
1051 (math-sqrt aa
))))))))
1052 (math-integrate-by-good-substitution expr t2
)) )
1054 ;; Try integration by parts.
1055 (math-integ-try-parts expr
)
1060 (defun math-integ-parts-easy (expr)
1061 (cond ((Math-primp expr
) t
)
1062 ((memq (car expr
) '(+ -
*))
1063 (and (math-integ-parts-easy (nth 1 expr
))
1064 (math-integ-parts-easy (nth 2 expr
))))
1066 (and (math-integ-parts-easy (nth 1 expr
))
1067 (math-atomic-factorp (nth 2 expr
))))
1069 (and (natnump (nth 2 expr
))
1070 (math-integ-parts-easy (nth 1 expr
))))
1071 ((eq (car expr
) 'neg
)
1072 (math-integ-parts-easy (nth 1 expr
)))
1075 (defun math-integ-try-parts (expr &optional math-good-parts
)
1076 ;; Integration by parts:
1077 ;; integ(f(x) g(x),x) = f(x) h(x) - integ(h(x) f'(x),x)
1078 ;; where h(x) = integ(g(x),x).
1079 (or (let ((exp (calcFunc-expand expr
)))
1080 (and (not (equal exp expr
))
1081 (math-integral exp
)))
1082 (and (eq (car expr
) '*)
1083 (let ((first-bad (or (math-polynomial-p (nth 1 expr
)
1085 (equal (nth 2 expr
) math-prev-parts-v
))))
1086 (or (and first-bad
; so try this one first
1087 (math-integrate-by-parts (nth 1 expr
) (nth 2 expr
)))
1088 (math-integrate-by-parts (nth 2 expr
) (nth 1 expr
))
1089 (and (not first-bad
)
1090 (math-integrate-by-parts (nth 1 expr
) (nth 2 expr
))))))
1091 (and (eq (car expr
) '/)
1092 (math-expr-contains (nth 1 expr
) math-integ-var
)
1093 (let ((recip (math-div 1 (nth 2 expr
))))
1094 (or (math-integrate-by-parts (nth 1 expr
) recip
)
1095 (math-integrate-by-parts recip
(nth 1 expr
)))))
1096 (and (eq (car expr
) '^
)
1097 (math-integrate-by-parts (math-pow (nth 1 expr
)
1098 (math-sub (nth 2 expr
) 1))
1101 (defun math-integrate-by-parts (u vprime
)
1102 (let ((math-integ-level (if (or math-good-parts
1103 (math-polynomial-p u math-integ-var
))
1105 (1- math-integ-level
)))
1106 (math-doing-parts t
)
1108 (and (>= math-integ-level
0)
1111 (setcar (cdr cur-record
) 'parts
)
1112 (math-tracing-integral "Integrating by parts, u = "
1113 (math-format-value u
1000)
1115 (math-format-value vprime
1000)
1117 (and (setq v
(math-integral vprime
))
1118 (setq temp
(calcFunc-deriv u math-integ-var nil t
))
1119 (setq temp
(let ((math-prev-parts-v v
))
1120 (math-integral (math-mul v temp
) 'yes
)))
1121 (setq temp
(math-sub (math-mul u v
) temp
))
1122 (if (eq (nth 1 cur-record
) 'parts
)
1123 (calcFunc-expand temp
)
1124 (setq v
(list 'var
'PARTS cur-record
)
1125 var-thing
(list 'vec
(math-sub v temp
) v
)
1126 temp
(let (calc-next-why)
1127 (math-solve-for (math-sub v temp
) 0 v nil
)))
1128 (and temp
(not (integerp temp
))
1129 (math-simplify-extended temp
)))))
1130 (setcar (cdr cur-record
) 'busy
)))))
1132 ;;; This tries two different formulations, hoping the algebraic simplifier
1133 ;;; will be strong enough to handle at least one.
1134 (defun math-integrate-by-substitution (expr u
&optional user uinv uinvprime
)
1135 (and (> math-integ-level
0)
1136 (let ((math-integ-level (max (- math-integ-level
2) 0)))
1137 (math-integrate-by-good-substitution expr u user uinv uinvprime
))))
1139 (defun math-integrate-by-good-substitution (expr u
&optional user
1141 (let ((math-living-dangerously t
)
1143 (and (setq uinv
(if uinv
1144 (math-expr-subst uinv math-integ-var
1146 (let (calc-next-why)
1149 math-integ-var nil
))))
1151 (math-tracing-integral "Integrating by substitution, u = "
1152 (math-format-value u
1000)
1154 (or (and (setq deriv
(calcFunc-deriv u
1157 (setq temp
(math-integral (math-expr-subst
1160 (math-div expr deriv
)
1168 (and (setq deriv
(or uinvprime
1169 (calcFunc-deriv uinv
1173 (setq temp
(math-integral (math-mul
1186 (math-simplify-extended
1187 (math-expr-subst temp math-integ-var u
)))))
1189 ;;; Look for substitutions of the form u = a x + b.
1190 (defun math-integ-try-linear-substitutions (sub-expr)
1191 (and (not (Math-primp sub-expr
))
1192 (or (and (not (memq (car sub-expr
) '(+ -
* / neg
)))
1193 (not (and (eq (car sub-expr
) '^
)
1194 (integerp (nth 2 sub-expr
))))
1195 (math-expr-contains sub-expr math-integ-var
)
1197 (while (and (setq sub-expr
(cdr sub-expr
))
1198 (or (not (math-linear-in (car sub-expr
)
1200 (assoc (car sub-expr
) so-far
)
1202 (setq so-far
(cons (list (car sub-expr
))
1205 (math-integrate-by-substitution
1206 expr
(car sub-expr
))))))))
1209 (while (and (setq sub-expr
(cdr sub-expr
))
1210 (not (setq res
(math-integ-try-linear-substitutions
1214 ;;; Recursively try different substitutions based on various sub-expressions.
1215 (defun math-integ-try-substitutions (sub-expr &optional allow-rat
)
1216 (and (not (Math-primp sub-expr
))
1217 (not (assoc sub-expr so-far
))
1218 (math-expr-contains sub-expr math-integ-var
)
1219 (or (and (if (and (not (memq (car sub-expr
) '(+ -
* / neg
)))
1220 (not (and (eq (car sub-expr
) '^
)
1221 (integerp (nth 2 sub-expr
)))))
1223 (prog1 allow-rat
(setq allow-rat nil
)))
1224 (not (eq sub-expr expr
))
1225 (or (math-integrate-by-substitution expr sub-expr
)
1226 (and (eq (car sub-expr
) '^
)
1227 (integerp (nth 2 sub-expr
))
1228 (< (nth 2 sub-expr
) 0)
1229 (math-integ-try-substitutions
1230 (math-pow (nth 1 sub-expr
) (- (nth 2 sub-expr
)))
1233 (setq so-far
(cons (list sub-expr
) so-far
))
1234 (while (and (setq sub-expr
(cdr sub-expr
))
1235 (not (setq res
(math-integ-try-substitutions
1236 (car sub-expr
) allow-rat
)))))
1239 (defun math-expr-rational-in (expr)
1241 (math-expr-rational-in-rec expr
)
1242 (mapcar 'car parts
)))
1244 (defun math-expr-rational-in-rec (expr)
1245 (cond ((Math-primp expr
)
1246 (and (equal expr math-integ-var
)
1247 (not (assoc expr parts
))
1248 (setq parts
(cons (list expr
) parts
))))
1249 ((or (memq (car expr
) '(+ -
* / neg
))
1250 (and (eq (car expr
) '^
) (integerp (nth 2 expr
))))
1251 (math-expr-rational-in-rec (nth 1 expr
))
1252 (and (nth 2 expr
) (math-expr-rational-in-rec (nth 2 expr
))))
1253 ((and (eq (car expr
) '^
)
1254 (eq (math-quarter-integer (nth 2 expr
)) 2))
1255 (math-expr-rational-in-rec (list 'calcFunc-sqrt
(nth 1 expr
))))
1257 (and (not (assoc expr parts
))
1258 (math-expr-contains expr math-integ-var
)
1259 (setq parts
(cons (list expr
) parts
))))))
1261 (defun math-expr-calls (expr funcs
&optional arg-contains
)
1263 (if (or (memq (car expr
) funcs
)
1264 (and (eq (car expr
) '^
) (eq (car funcs
) 'calcFunc-sqrt
)
1265 (eq (math-quarter-integer (nth 2 expr
)) 2)))
1266 (and (or (not arg-contains
)
1267 (math-expr-contains expr arg-contains
))
1269 (and (not (Math-primp expr
))
1271 (while (and (setq expr
(cdr expr
))
1272 (not (setq res
(math-expr-calls
1273 (car expr
) funcs arg-contains
)))))
1276 (defun math-fix-const-terms (expr except-vars
)
1277 (cond ((not (math-expr-depends expr except-vars
)) 0)
1278 ((Math-primp expr
) expr
)
1280 (math-add (math-fix-const-terms (nth 1 expr
) except-vars
)
1281 (math-fix-const-terms (nth 2 expr
) except-vars
)))
1283 (math-sub (math-fix-const-terms (nth 1 expr
) except-vars
)
1284 (math-fix-const-terms (nth 2 expr
) except-vars
)))
1287 ;; Command for debugging the Calculator's symbolic integrator.
1288 (defun calc-dump-integral-cache (&optional arg
)
1290 (let ((buf (current-buffer)))
1292 (let ((p math-integral-cache
)
1294 (display-buffer (get-buffer-create "*Integral Cache*"))
1295 (set-buffer (get-buffer "*Integral Cache*"))
1298 (setq cur-record
(car p
))
1299 (or arg
(math-replace-integral-parts cur-record
))
1300 (insert (math-format-flat-expr (car cur-record
) 0)
1302 (if (symbolp (nth 1 cur-record
))
1303 (concat "(" (symbol-name (nth 1 cur-record
)) ")")
1304 (math-format-flat-expr (nth 1 cur-record
) 0))
1307 (goto-char (point-min)))
1310 (defun math-try-integral (expr)
1311 (let ((math-integ-level math-integral-limit
)
1312 (math-integ-depth 0)
1313 (math-integ-msg "Working...done")
1314 (cur-record nil
) ; a technicality
1315 (math-integrating t
)
1316 (calc-prefer-frac t
)
1317 (calc-symbolic-mode t
)
1318 (has-rules (calc-has-rules 'var-IntegRules
)))
1319 (or (math-integral expr
'yes
)
1320 (and math-any-substs
1321 (setq math-enable-subst t
)
1322 (math-integral expr
'yes
))
1323 (and (> math-max-integral-limit math-integral-limit
)
1324 (setq math-integral-limit math-max-integral-limit
1325 math-integ-level math-integral-limit
)
1326 (math-integral expr
'yes
)))))
1328 (defun calcFunc-integ (expr var
&optional low high
)
1330 ;; Do these even if the parts turn out not to be integrable.
1331 ((eq (car-safe expr
) '+)
1332 (math-add (calcFunc-integ (nth 1 expr
) var low high
)
1333 (calcFunc-integ (nth 2 expr
) var low high
)))
1334 ((eq (car-safe expr
) '-
)
1335 (math-sub (calcFunc-integ (nth 1 expr
) var low high
)
1336 (calcFunc-integ (nth 2 expr
) var low high
)))
1337 ((eq (car-safe expr
) 'neg
)
1338 (math-neg (calcFunc-integ (nth 1 expr
) var low high
)))
1339 ((and (eq (car-safe expr
) '*)
1340 (not (math-expr-contains (nth 1 expr
) var
)))
1341 (math-mul (nth 1 expr
) (calcFunc-integ (nth 2 expr
) var low high
)))
1342 ((and (eq (car-safe expr
) '*)
1343 (not (math-expr-contains (nth 2 expr
) var
)))
1344 (math-mul (calcFunc-integ (nth 1 expr
) var low high
) (nth 2 expr
)))
1345 ((and (eq (car-safe expr
) '/)
1346 (not (math-expr-contains (nth 1 expr
) var
))
1347 (not (math-equal-int (nth 1 expr
) 1)))
1348 (math-mul (nth 1 expr
)
1349 (calcFunc-integ (math-div 1 (nth 2 expr
)) var low high
)))
1350 ((and (eq (car-safe expr
) '/)
1351 (not (math-expr-contains (nth 2 expr
) var
)))
1352 (math-div (calcFunc-integ (nth 1 expr
) var low high
) (nth 2 expr
)))
1353 ((and (eq (car-safe expr
) '/)
1354 (eq (car-safe (nth 1 expr
)) '*)
1355 (not (math-expr-contains (nth 1 (nth 1 expr
)) var
)))
1356 (math-mul (nth 1 (nth 1 expr
))
1357 (calcFunc-integ (math-div (nth 2 (nth 1 expr
)) (nth 2 expr
))
1359 ((and (eq (car-safe expr
) '/)
1360 (eq (car-safe (nth 1 expr
)) '*)
1361 (not (math-expr-contains (nth 2 (nth 1 expr
)) var
)))
1362 (math-mul (nth 2 (nth 1 expr
))
1363 (calcFunc-integ (math-div (nth 1 (nth 1 expr
)) (nth 2 expr
))
1365 ((and (eq (car-safe expr
) '/)
1366 (eq (car-safe (nth 2 expr
)) '*)
1367 (not (math-expr-contains (nth 1 (nth 2 expr
)) var
)))
1368 (math-div (calcFunc-integ (math-div (nth 1 expr
) (nth 2 (nth 2 expr
)))
1370 (nth 1 (nth 2 expr
))))
1371 ((and (eq (car-safe expr
) '/)
1372 (eq (car-safe (nth 2 expr
)) '*)
1373 (not (math-expr-contains (nth 2 (nth 2 expr
)) var
)))
1374 (math-div (calcFunc-integ (math-div (nth 1 expr
) (nth 1 (nth 2 expr
)))
1376 (nth 2 (nth 2 expr
))))
1377 ((eq (car-safe expr
) 'vec
)
1378 (cons 'vec
(mapcar (function (lambda (x) (calcFunc-integ x var low high
)))
1381 (let ((state (list calc-angle-mode
1382 ;;calc-symbolic-mode
1385 (calc-var-value 'var-IntegRules
)
1386 (calc-var-value 'var-IntegSimpRules
))))
1387 (or (equal state math-integral-cache-state
)
1388 (setq math-integral-cache-state state
1389 math-integral-cache nil
)))
1390 (let* ((math-max-integral-limit (or (and (boundp 'var-IntegLimit
)
1391 (natnump var-IntegLimit
)
1394 (math-integral-limit 1)
1395 (sexpr (math-expr-subst expr var math-integ-var
))
1396 (trace-buffer (get-buffer "*Trace*"))
1397 (calc-language (if (eq calc-language
'big
) nil calc-language
))
1399 (math-enable-subst nil
)
1400 (math-prev-parts-v nil
)
1401 (math-doing-parts nil
)
1402 (math-good-parts nil
)
1405 (let ((calcbuf (current-buffer))
1406 (calcwin (selected-window)))
1409 (if (get-buffer-window trace-buffer
)
1410 (select-window (get-buffer-window trace-buffer
)))
1411 (set-buffer trace-buffer
)
1412 (goto-char (point-max))
1413 (or (assq 'scroll-stop
(buffer-local-variables))
1415 (make-local-variable 'scroll-step
)
1416 (setq scroll-step
3)))
1418 (set-buffer calcbuf
)
1419 (math-try-integral sexpr
))
1420 (select-window calcwin
)
1421 (set-buffer calcbuf
)))
1422 (math-try-integral sexpr
))))
1425 (if (calc-has-rules 'var-IntegAfterRules
)
1426 (setq res
(math-rewrite res
'(var IntegAfterRules
1427 var-IntegAfterRules
))))
1430 (math-sub (math-expr-subst res math-integ-var high
)
1431 (math-expr-subst res math-integ-var low
))
1432 (setq res
(math-fix-const-terms res math-integ-vars
))
1434 (math-expr-subst res math-integ-var low
)
1435 (math-expr-subst res math-integ-var var
)))))
1436 (append (list 'calcFunc-integ expr var
)
1437 (and low
(list low
))
1438 (and high
(list high
))))))))
1441 (math-defintegral calcFunc-inv
1442 (math-integral (math-div 1 u
)))
1444 (math-defintegral calcFunc-conj
1445 (let ((int (math-integral u
)))
1447 (list 'calcFunc-conj int
))))
1449 (math-defintegral calcFunc-deg
1450 (let ((int (math-integral u
)))
1452 (list 'calcFunc-deg int
))))
1454 (math-defintegral calcFunc-rad
1455 (let ((int (math-integral u
)))
1457 (list 'calcFunc-rad int
))))
1459 (math-defintegral calcFunc-re
1460 (let ((int (math-integral u
)))
1462 (list 'calcFunc-re int
))))
1464 (math-defintegral calcFunc-im
1465 (let ((int (math-integral u
)))
1467 (list 'calcFunc-im int
))))
1469 (math-defintegral calcFunc-sqrt
1470 (and (equal u math-integ-var
)
1471 (math-mul '(frac 2 3)
1472 (list 'calcFunc-sqrt
(math-pow u
3)))))
1474 (math-defintegral calcFunc-exp
1475 (or (and (equal u math-integ-var
)
1476 (list 'calcFunc-exp u
))
1477 (let ((p (math-is-polynomial u math-integ-var
2)))
1479 (let ((sqa (math-sqrt (math-neg (nth 2 p
)))))
1482 (math-mul (math-div (list 'calcFunc-sqrt
'(var pi var-pi
))
1486 (math-div (math-sub (math-mul (car p
)
1489 (math-sqr (nth 1 p
))
1493 (math-sub (math-mul sqa math-integ-var
)
1494 (math-div (nth 1 p
) (math-mul 2 sqa
)))))
1497 (math-defintegral calcFunc-ln
1498 (or (and (equal u math-integ-var
)
1499 (math-sub (math-mul u
(list 'calcFunc-ln u
)) u
))
1500 (and (eq (car u
) '*)
1501 (math-integral (math-add (list 'calcFunc-ln
(nth 1 u
))
1502 (list 'calcFunc-ln
(nth 2 u
)))))
1503 (and (eq (car u
) '/)
1504 (math-integral (math-sub (list 'calcFunc-ln
(nth 1 u
))
1505 (list 'calcFunc-ln
(nth 2 u
)))))
1506 (and (eq (car u
) '^
)
1507 (math-integral (math-mul (nth 2 u
)
1508 (list 'calcFunc-ln
(nth 1 u
)))))))
1510 (math-defintegral calcFunc-log10
1511 (and (equal u math-integ-var
)
1512 (math-sub (math-mul u
(list 'calcFunc-ln u
))
1513 (math-div u
(list 'calcFunc-ln
10)))))
1515 (math-defintegral-2 calcFunc-log
1516 (math-integral (math-div (list 'calcFunc-ln u
)
1517 (list 'calcFunc-ln v
))))
1519 (math-defintegral calcFunc-sin
1520 (or (and (equal u math-integ-var
)
1521 (math-neg (math-from-radians-2 (list 'calcFunc-cos u
))))
1522 (and (nth 2 (math-is-polynomial u math-integ-var
2))
1523 (math-integral (math-to-exponentials (list 'calcFunc-sin u
))))))
1525 (math-defintegral calcFunc-cos
1526 (or (and (equal u math-integ-var
)
1527 (math-from-radians-2 (list 'calcFunc-sin u
)))
1528 (and (nth 2 (math-is-polynomial u math-integ-var
2))
1529 (math-integral (math-to-exponentials (list 'calcFunc-cos u
))))))
1531 (math-defintegral calcFunc-tan
1532 (and (equal u math-integ-var
)
1533 (math-neg (math-from-radians-2
1534 (list 'calcFunc-ln
(list 'calcFunc-cos u
))))))
1536 (math-defintegral calcFunc-arcsin
1537 (and (equal u math-integ-var
)
1538 (math-add (math-mul u
(list 'calcFunc-arcsin u
))
1539 (math-from-radians-2
1540 (list 'calcFunc-sqrt
(math-sub 1 (math-sqr u
)))))))
1542 (math-defintegral calcFunc-arccos
1543 (and (equal u math-integ-var
)
1544 (math-sub (math-mul u
(list 'calcFunc-arccos u
))
1545 (math-from-radians-2
1546 (list 'calcFunc-sqrt
(math-sub 1 (math-sqr u
)))))))
1548 (math-defintegral calcFunc-arctan
1549 (and (equal u math-integ-var
)
1550 (math-sub (math-mul u
(list 'calcFunc-arctan u
))
1551 (math-from-radians-2
1552 (math-div (list 'calcFunc-ln
(math-add 1 (math-sqr u
)))
1555 (math-defintegral calcFunc-sinh
1556 (and (equal u math-integ-var
)
1557 (list 'calcFunc-cosh u
)))
1559 (math-defintegral calcFunc-cosh
1560 (and (equal u math-integ-var
)
1561 (list 'calcFunc-sinh u
)))
1563 (math-defintegral calcFunc-tanh
1564 (and (equal u math-integ-var
)
1565 (list 'calcFunc-ln
(list 'calcFunc-cosh u
))))
1567 (math-defintegral calcFunc-arcsinh
1568 (and (equal u math-integ-var
)
1569 (math-sub (math-mul u
(list 'calcFunc-arcsinh u
))
1570 (list 'calcFunc-sqrt
(math-add (math-sqr u
) 1)))))
1572 (math-defintegral calcFunc-arccosh
1573 (and (equal u math-integ-var
)
1574 (math-sub (math-mul u
(list 'calcFunc-arccosh u
))
1575 (list 'calcFunc-sqrt
(math-sub 1 (math-sqr u
))))))
1577 (math-defintegral calcFunc-arctanh
1578 (and (equal u math-integ-var
)
1579 (math-sub (math-mul u
(list 'calcFunc-arctan u
))
1580 (math-div (list 'calcFunc-ln
1581 (math-add 1 (math-sqr u
)))
1584 ;;; (Ax + B) / (ax^2 + bx + c)^n forms.
1585 (math-defintegral-2 /
1586 (math-integral-rational-funcs u v
))
1588 (defun math-integral-rational-funcs (u v
)
1589 (let ((pu (math-is-polynomial u math-integ-var
1))
1593 (if (and (eq (car-safe v
) '^
) (natnump (nth 2 v
)))
1594 (setq vpow
(nth 2 v
)
1596 (and (setq pv
(math-is-polynomial v math-integ-var
2))
1597 (let ((int (math-mul-thru
1599 (math-integral-q02 (car pv
) (nth 1 pv
)
1600 (nth 2 pv
) v vpow
))))
1602 (setq int
(math-add int
1607 (nth 2 pv
) v vpow
)))))
1610 (defun math-integral-q12 (a b c v vpow
)
1614 (math-sub (math-div math-integ-var b
)
1615 (math-mul (math-div a
(math-sqr b
))
1616 (list 'calcFunc-ln v
))))
1618 (math-div (math-add (list 'calcFunc-ln v
)
1622 (let ((nm1 (math-sub vpow
1))
1623 (nm2 (math-sub vpow
2)))
1625 (math-div a
(math-mul nm1
(math-pow v nm1
)))
1626 (math-div 1 (math-mul nm2
(math-pow v nm2
))))
1629 (setq q
(math-sub (math-mul 4 (math-mul a c
)) (math-sqr b
))))
1630 (let ((part (math-div b
(math-mul 2 c
))))
1631 (math-mul-thru (math-pow c vpow
)
1632 (math-integral-q12 part
1 nil
1633 (math-add math-integ-var part
)
1636 (and (math-ratp q
) (math-negp q
)
1637 (let ((calc-symbolic-mode t
))
1638 (math-ratp (math-sqrt (math-neg q
))))
1639 (throw 'int-rat nil
)) ; should have used calcFunc-apart first
1640 (math-sub (math-div (list 'calcFunc-ln v
) (math-mul 2 c
))
1641 (math-mul-thru (math-div b
(math-mul 2 c
))
1642 (math-integral-q02 a b c v
1))))
1644 (let ((n (1- vpow
)))
1645 (math-sub (math-neg (math-div
1646 (math-add (math-mul b math-integ-var
)
1648 (math-mul n
(math-mul q
(math-pow v n
)))))
1649 (math-mul-thru (math-div (math-mul b
(1- (* 2 n
)))
1651 (math-integral-q02 a b c v n
))))))))
1653 (defun math-integral-q02 (a b c v vpow
)
1657 (math-div (list 'calcFunc-ln v
) b
))
1659 (math-div (math-pow v
(- 1 vpow
))
1660 (math-mul (- 1 vpow
) b
)))))
1662 (setq q
(math-sub (math-mul 4 (math-mul a c
)) (math-sqr b
))))
1663 (let ((part (math-div b
(math-mul 2 c
))))
1664 (math-mul-thru (math-pow c vpow
)
1665 (math-integral-q02 part
1 nil
1666 (math-add math-integ-var part
)
1669 (setq part
(math-add (math-mul 2 (math-mul c math-integ-var
)) b
))
1671 (let ((n (1- vpow
)))
1672 (math-add (math-div part
(math-mul n
(math-mul q
(math-pow v n
))))
1673 (math-mul-thru (math-div (math-mul (- (* 4 n
) 2) c
)
1675 (math-integral-q02 a b c v n
)))))
1676 ((math-guess-if-neg q
)
1677 (setq rq
(list 'calcFunc-sqrt
(math-neg q
)))
1678 ;;(math-div-thru (list 'calcFunc-ln
1679 ;; (math-div (math-sub part rq)
1680 ;; (math-add part rq)))
1682 (math-div (math-mul -
2 (list 'calcFunc-arctanh
1683 (math-div part rq
)))
1686 (setq rq
(list 'calcFunc-sqrt q
))
1687 (math-div (math-mul 2 (math-to-radians-2
1688 (list 'calcFunc-arctan
1689 (math-div part rq
))))
1693 (math-defintegral calcFunc-erf
1694 (and (equal u math-integ-var
)
1695 (math-add (math-mul u
(list 'calcFunc-erf u
))
1696 (math-div 1 (math-mul (list 'calcFunc-exp
(math-sqr u
))
1697 (list 'calcFunc-sqrt
1698 '(var pi var-pi
)))))))
1700 (math-defintegral calcFunc-erfc
1701 (and (equal u math-integ-var
)
1702 (math-sub (math-mul u
(list 'calcFunc-erfc u
))
1703 (math-div 1 (math-mul (list 'calcFunc-exp
(math-sqr u
))
1704 (list 'calcFunc-sqrt
1705 '(var pi var-pi
)))))))
1710 (defvar math-tabulate-initial nil
)
1711 (defvar math-tabulate-function nil
)
1712 (defun calcFunc-table (expr var
&optional low high step
)
1713 (or low
(setq low
'(neg (var inf var-inf
)) high
'(var inf var-inf
)))
1714 (or high
(setq high low low
1))
1715 (and (or (math-infinitep low
) (math-infinitep high
))
1717 (math-scan-for-limits expr
))
1718 (and step
(math-zerop step
) (math-reject-arg step
'nonzerop
))
1719 (let ((known (+ (if (Math-objectp low
) 1 0)
1720 (if (Math-objectp high
) 1 0)
1721 (if (or (null step
) (Math-objectp step
)) 1 0)))
1722 (count '(var inf var-inf
))
1724 (or (= known
2) ; handy optimization
1725 (equal high
'(var inf var-inf
))
1727 (setq count
(math-div (math-sub high low
) (or step
1)))
1728 (or (Math-objectp count
)
1729 (setq count
(math-simplify count
)))
1730 (if (Math-messy-integerp count
)
1731 (setq count
(math-trunc count
)))))
1732 (if (Math-negp count
)
1734 (if (integerp count
)
1735 (let ((var-DUMMY nil
)
1736 (vec math-tabulate-initial
)
1737 (math-working-step-2 (1+ count
))
1738 (math-working-step 0))
1739 (setq expr
(math-evaluate-expr
1740 (math-expr-subst expr var
'(var DUMMY var-DUMMY
))))
1742 (setq math-working-step
(1+ math-working-step
)
1744 vec
(cond ((eq math-tabulate-function
'calcFunc-sum
)
1745 (math-add vec
(math-evaluate-expr expr
)))
1746 ((eq math-tabulate-function
'calcFunc-prod
)
1747 (math-mul vec
(math-evaluate-expr expr
)))
1749 (cons (math-evaluate-expr expr
) vec
)))
1750 low
(math-add low
(or step
1))
1752 (if math-tabulate-function
1754 (cons 'vec
(nreverse vec
))))
1755 (if (Math-integerp count
)
1756 (calc-record-why 'fixnump high
)
1757 (if (Math-num-integerp low
)
1758 (if (Math-num-integerp high
)
1759 (calc-record-why 'integerp step
)
1760 (calc-record-why 'integerp high
))
1761 (calc-record-why 'integerp low
)))
1762 (append (list (or math-tabulate-function
'calcFunc-table
)
1764 (and (not (and (equal low
'(neg (var inf var-inf
)))
1765 (equal high
'(var inf var-inf
))))
1767 (and step
(list step
))))))
1769 (defun math-scan-for-limits (x)
1770 (cond ((Math-primp x
))
1771 ((and (eq (car x
) 'calcFunc-subscr
)
1772 (Math-vectorp (nth 1 x
))
1773 (math-expr-contains (nth 2 x
) var
))
1774 (let* ((calc-next-why nil
)
1775 (low-val (math-solve-for (nth 2 x
) 1 var nil
))
1776 (high-val (math-solve-for (nth 2 x
) (1- (length (nth 1 x
)))
1779 (and low-val
(math-realp low-val
)
1780 high-val
(math-realp high-val
))
1781 (and (Math-lessp high-val low-val
)
1782 (setq temp low-val low-val high-val high-val temp
))
1783 (setq low
(math-max low
(math-ceiling low-val
))
1784 high
(math-min high
(math-floor high-val
)))))
1786 (while (setq x
(cdr x
))
1787 (math-scan-for-limits (car x
))))))
1790 (defvar math-disable-sums nil
)
1791 (defun calcFunc-sum (expr var
&optional low high step
)
1792 (if math-disable-sums
(math-reject-arg))
1793 (let* ((res (let* ((calc-internal-prec (+ calc-internal-prec
2)))
1794 (math-sum-rec expr var low high step
)))
1795 (math-disable-sums t
))
1796 (math-normalize res
)))
1798 (defun math-sum-rec (expr var
&optional low high step
)
1799 (or low
(setq low
'(neg (var inf var-inf
)) high
'(var inf var-inf
)))
1800 (and low
(not high
) (setq high low low
1))
1804 ((not (math-expr-contains expr var
))
1805 (math-mul expr
(math-add (math-div (math-sub high low
) (or step
1))
1807 ((and step
(not (math-equal-int step
1)))
1808 (if (math-negp step
)
1809 (math-sum-rec expr var high low
(math-neg step
))
1810 (let ((lo (math-simplify (math-div low step
))))
1811 (if (math-known-num-integerp lo
)
1812 (math-sum-rec (math-normalize
1813 (math-expr-subst expr var
1814 (math-mul step var
)))
1815 var lo
(math-simplify (math-div high step
)))
1816 (math-sum-rec (math-normalize
1817 (math-expr-subst expr var
1818 (math-add (math-mul step var
)
1821 (math-simplify (math-div (math-sub high low
)
1823 ((memq (setq t1
(math-compare low high
)) '(0 1))
1825 (math-expr-subst expr var low
)
1827 ((setq t1
(math-is-polynomial expr var
20))
1831 (setq poly
(math-poly-mix poly
1
1832 (math-sum-integer-power n
) (car t1
))
1835 (setq n
(math-build-polynomial-expr poly high
))
1836 (if (memq low
'(0 1))
1838 (math-sub n
(math-build-polynomial-expr poly
1839 (math-sub low
1))))))
1840 ((and (memq (car expr
) '(+ -
))
1841 (setq t1
(math-sum-rec (nth 1 expr
) var low high
)
1842 t2
(math-sum-rec (nth 2 expr
) var low high
))
1843 (not (and (math-expr-calls t1
'(calcFunc-sum))
1844 (math-expr-calls t2
'(calcFunc-sum)))))
1845 (list (car expr
) t1 t2
))
1846 ((and (eq (car expr
) '*)
1847 (setq t1
(math-sum-const-factors expr var
)))
1848 (math-mul (car t1
) (math-sum-rec (cdr t1
) var low high
)))
1849 ((and (eq (car expr
) '*) (memq (car-safe (nth 1 expr
)) '(+ -
)))
1850 (math-sum-rec (math-add-or-sub (math-mul (nth 1 (nth 1 expr
))
1852 (math-mul (nth 2 (nth 1 expr
))
1854 nil
(eq (car (nth 1 expr
)) '-
))
1856 ((and (eq (car expr
) '*) (memq (car-safe (nth 2 expr
)) '(+ -
)))
1857 (math-sum-rec (math-add-or-sub (math-mul (nth 1 expr
)
1858 (nth 1 (nth 2 expr
)))
1859 (math-mul (nth 1 expr
)
1860 (nth 2 (nth 2 expr
)))
1861 nil
(eq (car (nth 2 expr
)) '-
))
1863 ((and (eq (car expr
) '/)
1864 (not (math-primp (nth 1 expr
)))
1865 (setq t1
(math-sum-const-factors (nth 1 expr
) var
)))
1867 (math-sum-rec (math-div (cdr t1
) (nth 2 expr
))
1869 ((and (eq (car expr
) '/)
1870 (setq t1
(math-sum-const-factors (nth 2 expr
) var
)))
1871 (math-div (math-sum-rec (math-div (nth 1 expr
) (cdr t1
))
1874 ((eq (car expr
) 'neg
)
1875 (math-neg (math-sum-rec (nth 1 expr
) var low high
)))
1876 ((and (eq (car expr
) '^
)
1877 (not (math-expr-contains (nth 1 expr
) var
))
1878 (setq t1
(math-is-polynomial (nth 2 expr
) var
1)))
1879 (let ((x (math-pow (nth 1 expr
) (nth 1 t1
))))
1880 (math-div (math-mul (math-sub (math-pow x
(math-add 1 high
))
1882 (math-pow (nth 1 expr
) (car t1
)))
1884 ((and (setq t1
(math-to-exponentials expr
))
1885 (setq t1
(math-sum-rec t1 var low high
))
1886 (not (math-expr-calls t1
'(calcFunc-sum))))
1888 ((memq (car expr
) '(calcFunc-ln calcFunc-log10
))
1889 (list (car expr
) (calcFunc-prod (nth 1 expr
) var low high
)))
1890 ((and (eq (car expr
) 'calcFunc-log
)
1892 (not (math-expr-contains (nth 2 expr
) var
)))
1894 (calcFunc-prod (nth 1 expr
) var low high
)
1896 (if (equal val
'(var nan var-nan
)) (setq val nil
))
1898 (let* ((math-tabulate-initial 0)
1899 (math-tabulate-function 'calcFunc-sum
))
1900 (calcFunc-table expr var low high
)))))
1902 (defun calcFunc-asum (expr var low
&optional high step no-mul-flag
)
1903 (or high
(setq high low low
1))
1904 (if (and step
(not (math-equal-int step
1)))
1905 (if (math-negp step
)
1906 (math-mul (math-pow -
1 low
)
1907 (calcFunc-asum expr var high low
(math-neg step
) t
))
1908 (let ((lo (math-simplify (math-div low step
))))
1909 (if (math-num-integerp lo
)
1910 (calcFunc-asum (math-normalize
1911 (math-expr-subst expr var
1912 (math-mul step var
)))
1913 var lo
(math-simplify (math-div high step
)))
1914 (calcFunc-asum (math-normalize
1915 (math-expr-subst expr var
1916 (math-add (math-mul step var
)
1919 (math-simplify (math-div (math-sub high low
)
1921 (math-mul (if no-mul-flag
1 (math-pow -
1 low
))
1922 (calcFunc-sum (math-mul (math-pow -
1 var
) expr
) var low high
))))
1924 (defun math-sum-const-factors (expr var
)
1928 (while (eq (car-safe p
) '*)
1929 (if (math-expr-contains (nth 1 p
) var
)
1930 (setq not-const
(cons (nth 1 p
) not-const
))
1931 (setq const
(cons (nth 1 p
) const
)))
1933 (if (math-expr-contains p var
)
1934 (setq not-const
(cons p not-const
))
1935 (setq const
(cons p const
)))
1937 (cons (let ((temp (car const
)))
1938 (while (setq const
(cdr const
))
1939 (setq temp
(list '* (car const
) temp
)))
1941 (let ((temp (or (car not-const
) 1)))
1942 (while (setq not-const
(cdr not-const
))
1943 (setq temp
(list '* (car not-const
) temp
)))
1946 (defvar math-sum-int-pow-cache
(list '(0 1)))
1947 ;; Following is from CRC Math Tables, 27th ed, pp. 52-53.
1948 (defun math-sum-integer-power (pow)
1949 (let ((calc-prefer-frac t
)
1950 (n (length math-sum-int-pow-cache
)))
1952 (let* ((new (list 0 0))
1954 (pp (cdr (nth (1- n
) math-sum-int-pow-cache
)))
1959 (setq q
(math-div (car pp
) p
)
1960 new
(cons (math-mul q n
) new
)
1961 sum
(math-add sum q
)
1964 (setcar lin
(math-sub 1 (math-mul n sum
)))
1965 (setq math-sum-int-pow-cache
1966 (nconc math-sum-int-pow-cache
(list (nreverse new
)))
1968 (nth pow math-sum-int-pow-cache
)))
1970 (defun math-to-exponentials (expr)
1973 (let ((x (nth 1 expr
))
1974 (pi (if calc-symbolic-mode
'(var pi var-pi
) (math-pi)))
1975 (i (if calc-symbolic-mode
'(var i var-i
) '(cplx 0 1))))
1976 (cond ((eq (car expr
) 'calcFunc-exp
)
1977 (list '^
'(var e var-e
) x
))
1978 ((eq (car expr
) 'calcFunc-sin
)
1979 (or (eq calc-angle-mode
'rad
)
1980 (setq x
(list '/ (list '* x pi
) 180)))
1982 (list '^
'(var e var-e
) (list '* x i
))
1983 (list '^
'(var e var-e
)
1984 (list 'neg
(list '* x i
))))
1986 ((eq (car expr
) 'calcFunc-cos
)
1987 (or (eq calc-angle-mode
'rad
)
1988 (setq x
(list '/ (list '* x pi
) 180)))
1990 (list '^
'(var e var-e
)
1992 (list '^
'(var e var-e
)
1993 (list 'neg
(list '* x i
))))
1995 ((eq (car expr
) 'calcFunc-sinh
)
1997 (list '^
'(var e var-e
) x
)
1998 (list '^
'(var e var-e
) (list 'neg x
)))
2000 ((eq (car expr
) 'calcFunc-cosh
)
2002 (list '^
'(var e var-e
) x
)
2003 (list '^
'(var e var-e
) (list 'neg x
)))
2007 (defun math-to-exps (expr)
2008 (cond (calc-symbolic-mode expr
)
2010 (if (equal expr
'(var e var-e
)) (math-e) expr
))
2011 ((and (eq (car expr
) '^
)
2012 (equal (nth 1 expr
) '(var e var-e
)))
2013 (list 'calcFunc-exp
(nth 2 expr
)))
2015 (cons (car expr
) (mapcar 'math-to-exps
(cdr expr
))))))
2018 (defvar math-disable-prods nil
)
2019 (defun calcFunc-prod (expr var
&optional low high step
)
2020 (if math-disable-prods
(math-reject-arg))
2021 (let* ((res (let* ((calc-internal-prec (+ calc-internal-prec
2)))
2022 (math-prod-rec expr var low high step
)))
2023 (math-disable-prods t
))
2024 (math-normalize res
)))
2026 (defun math-prod-rec (expr var
&optional low high step
)
2027 (or low
(setq low
'(neg (var inf var-inf
)) high
'(var inf var-inf
)))
2028 (and low
(not high
) (setq high
'(var inf var-inf
)))
2032 ((not (math-expr-contains expr var
))
2033 (math-pow expr
(math-add (math-div (math-sub high low
) (or step
1))
2035 ((and step
(not (math-equal-int step
1)))
2036 (if (math-negp step
)
2037 (math-prod-rec expr var high low
(math-neg step
))
2038 (let ((lo (math-simplify (math-div low step
))))
2039 (if (math-known-num-integerp lo
)
2040 (math-prod-rec (math-normalize
2041 (math-expr-subst expr var
2042 (math-mul step var
)))
2043 var lo
(math-simplify (math-div high step
)))
2044 (math-prod-rec (math-normalize
2045 (math-expr-subst expr var
2046 (math-add (math-mul step
2050 (math-simplify (math-div (math-sub high low
)
2052 ((and (memq (car expr
) '(* /))
2053 (setq t1
(math-prod-rec (nth 1 expr
) var low high
)
2054 t2
(math-prod-rec (nth 2 expr
) var low high
))
2055 (not (and (math-expr-calls t1
'(calcFunc-prod))
2056 (math-expr-calls t2
'(calcFunc-prod)))))
2057 (list (car expr
) t1 t2
))
2058 ((and (eq (car expr
) '^
)
2059 (not (math-expr-contains (nth 2 expr
) var
)))
2060 (math-pow (math-prod-rec (nth 1 expr
) var low high
)
2062 ((and (eq (car expr
) '^
)
2063 (not (math-expr-contains (nth 1 expr
) var
)))
2064 (math-pow (nth 1 expr
)
2065 (calcFunc-sum (nth 2 expr
) var low high
)))
2066 ((eq (car expr
) 'sqrt
)
2067 (math-normalize (list 'calcFunc-sqrt
2068 (list 'calcFunc-prod
(nth 1 expr
)
2070 ((eq (car expr
) 'neg
)
2071 (math-mul (math-pow -
1 (math-add (math-sub high low
) 1))
2072 (math-prod-rec (nth 1 expr
) var low high
)))
2073 ((eq (car expr
) 'calcFunc-exp
)
2074 (list 'calcFunc-exp
(calcFunc-sum (nth 1 expr
) var low high
)))
2075 ((and (setq t1
(math-is-polynomial expr var
1))
2078 ((or (and (math-equal-int (nth 1 t1
) 1)
2079 (setq low
(math-simplify
2080 (math-add low
(car t1
)))
2082 (math-add high
(car t1
)))))
2083 (and (math-equal-int (nth 1 t1
) -
1)
2086 (math-sub (car t1
) high
))
2088 (math-sub (car t1
) t2
)))))
2089 (if (or (math-zerop low
) (math-zerop high
))
2091 (if (and (or (math-negp low
) (math-negp high
))
2092 (or (math-num-integerp low
)
2093 (math-num-integerp high
)))
2094 (if (math-posp high
)
2096 (math-mul (math-pow -
1
2098 (math-add low high
) 1))
2100 (list 'calcFunc-fact
2102 (list 'calcFunc-fact
2103 (math-sub -
1 high
)))))
2105 (list 'calcFunc-fact high
)
2106 (list 'calcFunc-fact
(math-sub low
1))))))
2107 ((and (or (and (math-equal-int (nth 1 t1
) 2)
2108 (setq t2
(math-simplify
2109 (math-add (math-mul low
2)
2112 (math-add (math-mul high
2)
2114 (and (math-equal-int (nth 1 t1
) -
2)
2115 (setq t2
(math-simplify
2122 (or (math-integerp t2
)
2123 (and (math-messy-integerp t2
)
2124 (setq t2
(math-trunc t2
)))
2126 (and (math-messy-integerp t3
)
2127 (setq t3
(math-trunc t3
)))))
2128 (if (or (math-zerop t2
) (math-zerop t3
))
2130 (if (or (math-evenp t2
) (math-evenp t3
))
2131 (if (or (math-negp t2
) (math-negp t3
))
2132 (if (math-posp high
)
2135 (list 'calcFunc-dfact
2137 (list 'calcFunc-dfact
2140 (list 'calcFunc-dfact t3
)
2141 (list 'calcFunc-dfact
2146 (list '/ (list '-
(list '- t2 t3
)
2150 (list 'calcFunc-dfact
2152 (list 'calcFunc-dfact
2156 (list 'calcFunc-dfact t3
)
2157 (list 'calcFunc-dfact
2161 (if (equal val
'(var nan var-nan
)) (setq val nil
))
2163 (let* ((math-tabulate-initial 1)
2164 (math-tabulate-function 'calcFunc-prod
))
2165 (calcFunc-table expr var low high
)))))
2170 (defvar math-solve-ranges nil
)
2171 ;;; Attempt to reduce lhs = rhs to solve-var = rhs', where solve-var appears
2172 ;;; in lhs but not in rhs or rhs'; return rhs'.
2173 ;;; Uses global values: solve-*.
2174 (defun math-try-solve-for (lhs rhs
&optional sign no-poly
)
2176 (cond ((equal lhs solve-var
)
2177 (setq math-solve-sign sign
)
2178 (if (eq solve-full
'all
)
2179 (let ((vec (list 'vec
(math-evaluate-expr rhs
)))
2181 (while math-solve-ranges
2182 (setq p
(car math-solve-ranges
)
2185 (while (setq p
(cdr p
))
2186 (setq newvec
(nconc newvec
2187 (cdr (math-expr-subst
2188 vec var
(car p
))))))
2190 math-solve-ranges
(cdr math-solve-ranges
)))
2191 (math-normalize vec
))
2195 ((and (eq (car lhs
) '-
)
2196 (eq (car-safe (nth 1 lhs
)) (car-safe (nth 2 lhs
)))
2198 (= (length (nth 1 lhs
)) 2)
2199 (= (length (nth 2 lhs
)) 2)
2200 (setq t1
(get (car (nth 1 lhs
)) 'math-inverse
))
2201 (setq t2
(funcall t1
'(var SOLVEDUM SOLVEDUM
)))
2202 (eq (math-expr-contains-count t2
'(var SOLVEDUM SOLVEDUM
)) 1)
2203 (setq t3
(math-solve-above-dummy t2
))
2204 (setq t1
(math-try-solve-for (math-sub (nth 1 (nth 1 lhs
))
2207 (nth 1 (nth 2 lhs
))))
2210 ((eq (car lhs
) 'neg
)
2211 (math-try-solve-for (nth 1 lhs
) (math-neg rhs
)
2212 (and sign
(- sign
))))
2213 ((and (not (eq solve-full
't
)) (math-try-solve-prod)))
2215 (setq t2
(math-decompose-poly lhs solve-var
15 rhs
)))
2216 (setq t1
(cdr (nth 1 t2
))
2217 t1
(let ((math-solve-ranges math-solve-ranges
))
2218 (cond ((= (length t1
) 5)
2219 (apply 'math-solve-quartic
(car t2
) t1
))
2221 (apply 'math-solve-cubic
(car t2
) t1
))
2223 (apply 'math-solve-quadratic
(car t2
) t1
))
2225 (apply 'math-solve-linear
(car t2
) sign t1
))
2227 (math-poly-all-roots (car t2
) t1
))
2228 (calc-symbolic-mode nil
)
2232 (math-poly-any-root (reverse t1
) 0 t
)
2235 (if (eq (nth 2 t2
) 1)
2237 (math-solve-prod t1
(math-try-solve-for (nth 2 t2
) 0 nil t
)))
2238 (calc-record-why "*Unable to find a symbolic solution")
2240 ((and (math-solve-find-root-term lhs nil
)
2241 (eq (math-expr-contains-count lhs t1
) 1)) ; just in case
2242 (math-try-solve-for (math-simplify
2243 (math-sub (if (or t3
(math-evenp t2
))
2245 (math-neg (math-pow t1 t2
)))
2247 (math-sub (math-normalize
2254 (cond ((not (math-expr-contains (nth 1 lhs
) solve-var
))
2255 (math-try-solve-for (nth 2 lhs
)
2256 (math-sub rhs
(nth 1 lhs
))
2258 ((not (math-expr-contains (nth 2 lhs
) solve-var
))
2259 (math-try-solve-for (nth 1 lhs
)
2260 (math-sub rhs
(nth 2 lhs
))
2262 ((eq (car lhs
) 'calcFunc-eq
)
2263 (math-try-solve-for (math-sub (nth 1 lhs
) (nth 2 lhs
))
2266 (cond ((or (and (eq (car-safe (nth 1 lhs
)) 'calcFunc-sin
)
2267 (eq (car-safe (nth 2 lhs
)) 'calcFunc-cos
))
2268 (and (eq (car-safe (nth 1 lhs
)) 'calcFunc-cos
)
2269 (eq (car-safe (nth 2 lhs
)) 'calcFunc-sin
)))
2270 (math-try-solve-for (math-sub (nth 1 lhs
)
2271 (list (car (nth 1 lhs
))
2273 (math-quarter-circle t
)
2274 (nth 1 (nth 2 lhs
)))))
2276 ((not (math-expr-contains (nth 1 lhs
) solve-var
))
2277 (math-try-solve-for (nth 2 lhs
)
2278 (math-sub (nth 1 lhs
) rhs
)
2279 (and sign
(- sign
))))
2280 ((not (math-expr-contains (nth 2 lhs
) solve-var
))
2281 (math-try-solve-for (nth 1 lhs
)
2282 (math-add rhs
(nth 2 lhs
))
2284 ((and (eq solve-full
't
) (math-try-solve-prod)))
2285 ((and (eq (car lhs
) '%
)
2286 (not (math-expr-contains (nth 2 lhs
) solve-var
)))
2287 (math-try-solve-for (nth 1 lhs
) (math-add rhs
2290 ((eq (car lhs
) 'calcFunc-log
)
2291 (cond ((not (math-expr-contains (nth 2 lhs
) solve-var
))
2292 (math-try-solve-for (nth 1 lhs
) (math-pow (nth 2 lhs
) rhs
)))
2293 ((not (math-expr-contains (nth 1 lhs
) solve-var
))
2294 (math-try-solve-for (nth 2 lhs
) (math-pow
2296 (math-div 1 rhs
))))))
2297 ((and (= (length lhs
) 2)
2299 (setq t1
(get (car lhs
) 'math-inverse
))
2300 (setq t2
(funcall t1 rhs
)))
2301 (setq t1
(get (car lhs
) 'math-inverse-sign
))
2302 (math-try-solve-for (nth 1 lhs
) (math-normalize t2
)
2306 (funcall t1 lhs sign
)))))
2307 ((and (symbolp (car lhs
))
2308 (setq t1
(get (car lhs
) 'math-inverse-n
))
2309 (setq t2
(funcall t1 lhs rhs
)))
2311 ((setq t1
(math-expand-formula lhs
))
2312 (math-try-solve-for t1 rhs sign
))
2314 (calc-record-why "*No inverse known" lhs
)
2318 (defun math-try-solve-prod ()
2319 (cond ((eq (car lhs
) '*)
2320 (cond ((not (math-expr-contains (nth 1 lhs
) solve-var
))
2321 (math-try-solve-for (nth 2 lhs
)
2322 (math-div rhs
(nth 1 lhs
))
2323 (math-solve-sign sign
(nth 1 lhs
))))
2324 ((not (math-expr-contains (nth 2 lhs
) solve-var
))
2325 (math-try-solve-for (nth 1 lhs
)
2326 (math-div rhs
(nth 2 lhs
))
2327 (math-solve-sign sign
(nth 2 lhs
))))
2329 (math-solve-prod (let ((math-solve-ranges math-solve-ranges
))
2330 (math-try-solve-for (nth 2 lhs
) 0))
2331 (math-try-solve-for (nth 1 lhs
) 0)))))
2333 (cond ((not (math-expr-contains (nth 1 lhs
) solve-var
))
2334 (math-try-solve-for (nth 2 lhs
)
2335 (math-div (nth 1 lhs
) rhs
)
2336 (math-solve-sign sign
(nth 1 lhs
))))
2337 ((not (math-expr-contains (nth 2 lhs
) solve-var
))
2338 (math-try-solve-for (nth 1 lhs
)
2339 (math-mul rhs
(nth 2 lhs
))
2340 (math-solve-sign sign
(nth 2 lhs
))))
2341 ((setq t1
(math-try-solve-for (math-sub (nth 1 lhs
)
2342 (math-mul (nth 2 lhs
)
2347 (cond ((not (math-expr-contains (nth 1 lhs
) solve-var
))
2350 (math-add (math-normalize
2351 (list 'calcFunc-log rhs
(nth 1 lhs
)))
2354 (math-mul '(var pi var-pi
)
2358 (list 'calcFunc-ln
(nth 1 lhs
)))))))
2359 ((not (math-expr-contains (nth 2 lhs
) solve-var
))
2360 (cond ((and (integerp (nth 2 lhs
))
2362 (setq t1
(math-integer-log2 (nth 2 lhs
))))
2364 (if (and (eq solve-full t
)
2365 (math-known-realp (nth 1 lhs
)))
2367 (while (>= (setq t1
(1- t1
)) 0)
2368 (setq t2
(list 'calcFunc-sqrt t2
)))
2369 (setq t2
(math-solve-get-sign t2
)))
2370 (while (>= (setq t1
(1- t1
)) 0)
2371 (setq t2
(math-solve-get-sign
2373 (list 'calcFunc-sqrt t2
))))))
2376 (math-normalize t2
)))
2377 ((math-looks-negp (nth 2 lhs
))
2379 (list '^
(nth 1 lhs
) (math-neg (nth 2 lhs
)))
2381 ((and (eq solve-full t
)
2382 (Math-integerp (nth 2 lhs
))
2383 (math-known-realp (nth 1 lhs
)))
2384 (setq t1
(math-normalize
2385 (list 'calcFunc-nroot rhs
(nth 2 lhs
))))
2386 (if (math-evenp (nth 2 lhs
))
2387 (setq t1
(math-solve-get-sign t1
)))
2391 (math-oddp (nth 2 lhs
))
2392 (math-solve-sign sign
(nth 2 lhs
)))))
2393 (t (math-try-solve-for
2398 (if (Math-realp (nth 2 lhs
))
2403 (and (integerp (nth 2 lhs
))
2406 (math-div (nth 2 lhs
) 2))
2413 (and (integerp (nth 2 lhs
))
2418 (list 'calcFunc-nroot
2422 (math-oddp (nth 2 lhs
))
2423 (math-solve-sign sign
(nth 2 lhs
)))))))))
2426 (defun math-solve-prod (lsoln rsoln
)
2431 ((eq solve-full
'all
)
2432 (cons 'vec
(append (cdr lsoln
) (cdr rsoln
))))
2435 (list 'calcFunc-gt
(math-solve-get-sign 1) 0)
2440 ;;; This deals with negative, fractional, and symbolic powers of "x".
2441 (defun math-solve-poly-funny-powers (sub-rhs) ; uses "t1", "t2"
2443 (let ((pp math-poly-neg-powers
)
2446 (setq fac
(math-pow (car pp
) (or math-poly-mult-powers
1))
2447 t1
(math-mul t1 fac
)
2448 rhs
(math-mul rhs fac
)
2450 (if sub-rhs
(setq t1
(math-sub t1 rhs
)))
2451 (let ((math-poly-neg-powers nil
))
2452 (setq t2
(math-mul (or math-poly-mult-powers
1)
2453 (let ((calc-prefer-frac t
))
2454 (math-div 1 math-poly-frac-powers
)))
2455 t1
(math-is-polynomial (math-simplify (calcFunc-expand t1
)) b
50))))
2457 ;;; This converts "a x^8 + b x^5 + c x^2" to "(a (x^3)^2 + b (x^3) + c) * x^2".
2458 (defun math-solve-crunch-poly (max-degree) ; uses "t1", "t3"
2460 (while (and t1
(Math-zerop (car t1
)))
2464 (let* ((degree (1- (length t1
)))
2466 (while (and (> scale
1) (= (car t3
) 1))
2467 (and (= (% degree scale
) 0)
2473 (if (= (% n scale
) 0)
2474 (setq new-t1
(nconc new-t1
(list (car p
))))
2475 (or (Math-zerop (car p
))
2480 (setq t3
(cons scale
(cdr t3
))
2482 (setq scale
(1- scale
)))
2483 (setq t3
(list (math-mul (car t3
) t2
) (math-mul count t2
)))
2484 (<= (1- (length t1
)) max-degree
)))))
2486 (defun calcFunc-poly (expr var
&optional degree
)
2488 (or (natnump degree
) (math-reject-arg degree
'fixnatnump
))
2490 (let ((p (math-is-polynomial expr var degree
'gen
)))
2495 (math-reject-arg expr
"Expected a polynomial"))))
2497 (defun calcFunc-gpoly (expr var
&optional degree
)
2499 (or (natnump degree
) (math-reject-arg degree
'fixnatnump
))
2501 (let* ((math-poly-base-variable var
)
2502 (d (math-decompose-poly expr var degree nil
)))
2505 (math-reject-arg expr
"Expected a polynomial"))))
2507 (defun math-decompose-poly (lhs solve-var degree sub-rhs
)
2508 (let ((rhs (or sub-rhs
1))
2510 (setq t2
(math-polynomial-base
2514 (let ((math-poly-neg-powers '(1))
2515 (math-poly-mult-powers nil
)
2516 (math-poly-frac-powers 1)
2517 (math-poly-exp-base t
))
2518 (and (not (equal b lhs
))
2519 (or (not (memq (car-safe b
) '(+ -
))) sub-rhs
)
2520 (setq t3
'(1 0) t2
1
2521 t1
(math-is-polynomial lhs b
50))
2522 (if (and (equal math-poly-neg-powers
'(1))
2523 (memq math-poly-mult-powers
'(nil 1))
2524 (eq math-poly-frac-powers
1)
2526 (setq t1
(cons (math-sub (car t1
) rhs
)
2528 (math-solve-poly-funny-powers sub-rhs
))
2529 (math-solve-crunch-poly degree
)
2530 (or (math-expr-contains b solve-var
)
2531 (math-expr-contains (car t3
) solve-var
))))))))
2533 (list (math-pow t2
(car t3
))
2536 (math-pow t2
(nth 1 t3
))
2537 (math-div (math-pow t2
(nth 1 t3
)) rhs
))))))
2539 (defun math-solve-linear (var sign b a
)
2540 (math-try-solve-for var
2541 (math-div (math-neg b
) a
)
2542 (math-solve-sign sign a
)
2545 (defun math-solve-quadratic (var c b a
)
2548 (if (math-looks-evenp b
)
2549 (let ((halfb (math-div b
2)))
2553 (math-solve-get-sign
2555 (list 'calcFunc-sqrt
2556 (math-add (math-sqr halfb
)
2557 (math-mul (math-neg c
) a
))))))
2562 (math-solve-get-sign
2564 (list 'calcFunc-sqrt
2565 (math-add (math-sqr b
)
2566 (math-mul 4 (math-mul (math-neg c
) a
)))))))
2570 (defun math-solve-cubic (var d c b a
)
2571 (let* ((p (math-div b a
))
2575 (aa (math-sub q
(math-div psqr
3)))
2577 (math-div (math-sub (math-mul 2 (math-mul psqr p
))
2578 (math-mul 9 (math-mul p q
)))
2582 (math-try-solve-for (math-pow (math-add var
(math-div p
3)) 3)
2583 (math-neg bb
) nil t
)
2586 (math-mul (math-add var
(math-div p
3))
2587 (math-add (math-sqr (math-add var
(math-div p
3)))
2590 (setq m
(math-mul 2 (list 'calcFunc-sqrt
(math-div aa -
3))))
2599 (math-sub (list 'calcFunc-arccos
2600 (math-div (math-mul 3 bb
)
2604 (math-add 1 (math-solve-get-int
2607 calc-symbolic-mode
))))
2612 (defun math-solve-quartic (var d c b a aa
)
2613 (setq a
(math-div a aa
))
2614 (setq b
(math-div b aa
))
2615 (setq c
(math-div c aa
))
2616 (setq d
(math-div d aa
))
2619 (let* ((asqr (math-sqr a
))
2620 (asqr4 (math-div asqr
4))
2621 (y (let ((solve-full nil
)
2623 (math-solve-cubic solve-var
2625 (math-mul 4 (math-mul b d
))
2628 (math-sub (math-mul a c
)
2632 (rsqr (math-add (math-sub asqr4 b
) y
))
2633 (r (list 'calcFunc-sqrt rsqr
))
2634 (sign1 (math-solve-get-sign 1))
2635 (de (list 'calcFunc-sqrt
2637 (math-sub (math-mul 3 asqr4
)
2639 (if (Math-zerop rsqr
)
2643 (list 'calcFunc-sqrt
2644 (math-sub (math-sqr y
)
2650 (math-mul 4 (math-mul a b
))
2656 (math-sub (math-add (math-mul sign1
(math-div r
2))
2657 (math-solve-get-sign (math-div de
2)))
2661 (defvar math-symbolic-solve nil
)
2662 (defvar math-int-coefs nil
)
2663 (defun math-poly-all-roots (var p
&optional math-factoring
)
2665 (let* ((math-symbolic-solve calc-symbolic-mode
)
2667 (deg (1- (length p
)))
2668 (orig-p (reverse p
))
2669 (math-int-coefs nil
)
2670 (math-int-scale nil
)
2671 (math-double-roots nil
)
2672 (math-int-factors nil
)
2673 (math-int-threshold nil
)
2675 ;; If rational coefficients, look for exact rational factors.
2676 (while (and pp
(Math-ratp (car pp
)))
2679 (if (or math-factoring math-symbolic-solve
)
2681 (let ((lead (car orig-p
))
2682 (calc-prefer-frac t
)
2683 (scale (apply 'math-lcm-denoms p
)))
2684 (setq math-int-scale
(math-abs (math-mul scale lead
))
2685 math-int-threshold
(math-div '(float 5 -
2) math-int-scale
)
2686 math-int-coefs
(cdr (math-div (cons 'vec orig-p
) lead
)))))
2688 (let ((calc-prefer-frac nil
)
2689 (calc-symbolic-mode nil
)
2691 (def-p (copy-sequence orig-p
)))
2693 (if (Math-numberp (car pp
))
2696 (while (> deg
(if math-symbolic-solve
2 4))
2697 (let* ((x (math-poly-any-root def-p
'(float 0 0) nil
))
2699 (if (and (eq (car-safe x
) 'cplx
)
2700 (math-nearly-zerop (nth 2 x
) (nth 1 x
)))
2701 (setq x
(calcFunc-re x
)))
2703 (setq roots
(cons x roots
)))
2704 (or (math-numberp x
)
2705 (setq x
(math-evaluate-expr x
)))
2708 (while (setq pp
(cdr pp
))
2711 (setq b
(math-add (math-mul x b
) c
)))
2712 (setq def-p
(cdr def-p
)
2714 (setq p
(reverse def-p
))))
2716 (let ((solve-var '(var DUMMY var-DUMMY
))
2717 (math-solve-sign nil
)
2718 (math-solve-ranges nil
)
2720 (if (= (length p
) (length math-int-coefs
))
2721 (setq p
(reverse math-int-coefs
)))
2722 (setq roots
(append (cdr (apply (cond ((= deg
2)
2723 'math-solve-quadratic
)
2727 'math-solve-quartic
))
2731 (setq roots
(cons (math-div (math-neg (car p
)) (nth 1 p
))
2736 (math-poly-integer-root (car roots
))
2737 (setq roots
(cdr roots
)))
2738 (list math-int-factors
(nreverse math-int-coefs
) math-int-scale
))
2739 (let ((vec nil
) res
)
2741 (let ((root (car roots
))
2742 (solve-full (and solve-full
'all
)))
2743 (if (math-floatp root
)
2744 (setq root
(math-poly-any-root orig-p root t
)))
2745 (setq vec
(append vec
2746 (cdr (or (math-try-solve-for var root nil t
)
2747 (throw 'ouch nil
))))))
2748 (setq roots
(cdr roots
)))
2749 (setq vec
(cons 'vec
(nreverse vec
)))
2750 (if math-symbolic-solve
2751 (setq vec
(math-normalize vec
)))
2752 (if (eq solve-full t
)
2753 (list 'calcFunc-subscr
2755 (math-solve-get-int 1 (1- (length orig-p
)) 1))
2758 (defun math-lcm-denoms (&rest fracs
)
2761 (if (eq (car-safe (car fracs
)) 'frac
)
2762 (setq den
(calcFunc-lcm den
(nth 2 (car fracs
)))))
2763 (setq fracs
(cdr fracs
)))
2766 (defun math-poly-any-root (p x polish
) ; p is a reverse poly coeff list
2767 (let* ((newt (if (math-zerop x
)
2768 (math-poly-newton-root
2769 p
'(cplx (float 123 -
6) (float 1 -
4)) 4)
2770 (math-poly-newton-root p x
4)))
2771 (res (if (math-zerop (cdr newt
))
2773 (if (and (math-lessp (cdr newt
) '(float 1 -
3)) (not polish
))
2774 (setq newt
(math-poly-newton-root p
(car newt
) 30)))
2775 (if (math-zerop (cdr newt
))
2777 (math-poly-laguerre-root p x polish
)))))
2778 (and math-symbolic-solve
(math-floatp res
)
2782 (defun math-poly-newton-root (p x iters
)
2783 (let* ((calc-prefer-frac nil
)
2784 (calc-symbolic-mode nil
)
2785 (try-integer math-int-coefs
)
2787 (while (and (> (setq iters
(1- iters
)) 0)
2789 (math-working "newton" x
)
2792 (while (setq pp
(cdr pp
))
2793 (setq d
(math-add (math-mul x d
) b
)
2794 b
(math-add (math-mul x b
) (car pp
))))
2795 (not (math-zerop d
)))
2797 (setq dx
(math-div b d
)
2800 (let ((adx (math-abs-approx dx
)))
2801 (and (math-lessp adx math-int-threshold
)
2802 (let ((iroot (math-poly-integer-root x
)))
2805 (setq try-integer nil
))))))
2806 (or (not (or (eq dx
0)
2807 (math-nearly-zerop dx
(math-abs-approx x
))))
2808 (progn (setq dx
0) nil
)))))
2809 (cons x
(if (math-zerop x
)
2810 1 (math-div (math-abs-approx dx
) (math-abs-approx x
))))))
2812 (defun math-poly-integer-root (x)
2813 (and (math-lessp (calcFunc-xpon (math-abs-approx x
)) calc-internal-prec
)
2815 (let* ((calc-prefer-frac t
)
2816 (xre (calcFunc-re x
))
2817 (xim (calcFunc-im x
))
2818 (xresq (math-sqr xre
))
2819 (ximsq (math-sqr xim
)))
2820 (if (math-lessp ximsq
(calcFunc-scf xresq -
1))
2821 ;; Look for linear factor
2822 (let* ((rnd (math-div (math-round (math-mul xre math-int-scale
))
2824 (icp math-int-coefs
)
2827 (while (setq icp
(cdr icp
))
2828 (setq newcoef
(cons rem newcoef
)
2829 rem
(math-add (car icp
)
2830 (math-mul rem rnd
))))
2831 (and (math-zerop rem
)
2833 (setq math-int-coefs
(nreverse newcoef
)
2834 math-int-factors
(cons (list (math-neg rnd
))
2837 ;; Look for irreducible quadratic factor
2838 (let* ((rnd1 (math-div (math-round
2839 (math-mul xre
(math-mul -
2 math-int-scale
)))
2841 (sqscale (math-sqr math-int-scale
))
2842 (rnd0 (math-div (math-round (math-mul (math-add xresq ximsq
)
2845 (rem1 (car math-int-coefs
))
2846 (icp (cdr math-int-coefs
))
2849 (found (assoc (list rnd0 rnd1
(math-posp xim
))
2853 (setq math-double-roots
(delq found math-double-roots
)
2855 (while (setq icp
(cdr icp
))
2857 newcoef
(cons rem1 newcoef
)
2858 rem1
(math-sub rem0
(math-mul this rnd1
))
2859 rem0
(math-sub (car icp
) (math-mul this rnd0
)))))
2860 (and (math-zerop rem0
)
2862 (let ((aa (math-div rnd1 -
2)))
2863 (or found
(setq math-int-coefs
(reverse newcoef
)
2864 math-double-roots
(cons (list
2869 math-int-factors
(cons (cons rnd0 rnd1
)
2872 (let ((calc-symbolic-mode math-symbolic-solve
))
2873 (math-mul (math-sqrt (math-sub (math-sqr aa
)
2875 (if (math-negp xim
) -
1 1)))))))))))
2877 ;;; The following routine is from Numerical Recipes, section 9.5.
2878 (defun math-poly-laguerre-root (p x polish
)
2879 (let* ((calc-prefer-frac nil
)
2880 (calc-symbolic-mode nil
)
2883 (try-newt (not polish
))
2887 (and (or (< (setq iters
(1+ iters
)) 50)
2888 (math-reject-arg x
"*Laguerre's method failed to converge"))
2889 (let ((err (math-abs-approx (car p
)))
2890 (abx (math-abs-approx x
))
2894 (while (setq pp
(cdr pp
))
2895 (setq f
(math-add (math-mul x f
) d
)
2896 d
(math-add (math-mul x d
) b
)
2897 b
(math-add (math-mul x b
) (car pp
))
2898 err
(math-add (math-abs-approx b
) (math-mul abx err
))))
2899 (math-lessp (calcFunc-scf err
(- -
2 calc-internal-prec
))
2900 (math-abs-approx b
)))
2901 (or (not (math-zerop d
))
2902 (not (math-zerop f
))
2904 (setq x
(math-pow (math-neg b
) (list 'frac
1 m
)))
2906 (let* ((g (math-div d b
))
2908 (h (math-sub g2
(math-mul 2 (math-div f b
))))
2910 (math-mul (1- m
) (math-sub (math-mul m h
) g2
))))
2911 (gp (math-add g sq
))
2912 (gm (math-sub g sq
)))
2913 (if (math-lessp (calcFunc-abssqr gp
) (calcFunc-abssqr gm
))
2915 (setq dx
(math-div m gp
)
2918 (math-lessp (math-abs-approx dx
)
2919 (calcFunc-scf (math-abs-approx x
) -
3)))
2920 (let ((newt (math-poly-newton-root p x1
7)))
2923 (if (math-zerop (cdr newt
))
2924 (setq x
(car newt
) x1 x
)
2925 (if (math-lessp (cdr newt
) '(float 1 -
6))
2926 (let ((newt2 (math-poly-newton-root
2928 (if (math-zerop (cdr newt2
))
2929 (setq x
(car newt2
) x1 x
)
2930 (setq x
(car newt
))))))))
2932 (math-nearly-equal x x1
))))
2933 (let ((cdx (math-abs-approx dx
)))
2938 (math-lessp cdx dxold
)
2941 (let ((digs (calcFunc-xpon
2942 (math-div (math-abs-approx x
) cdx
))))
2944 "*Could not attain full precision")
2946 (let ((calc-internal-prec (max 3 digs
)))
2947 (setq x
(math-normalize x
))))))
2951 (math-lessp (calcFunc-scf (math-abs-approx x
)
2952 (- calc-internal-prec
))
2954 (or (and (math-floatp x
)
2955 (math-poly-integer-root x
))
2958 (defun math-solve-above-dummy (x)
2959 (and (not (Math-primp x
))
2960 (if (and (equal (nth 1 x
) '(var SOLVEDUM SOLVEDUM
))
2964 (while (and (setq x
(cdr x
))
2965 (not (setq res
(math-solve-above-dummy (car x
))))))
2968 (defun math-solve-find-root-term (x neg
) ; sets "t2", "t3"
2969 (if (math-solve-find-root-in-prod x
)
2972 (and (memq (car-safe x
) '(+ -
))
2973 (or (math-solve-find-root-term (nth 1 x
) neg
)
2974 (math-solve-find-root-term (nth 2 x
)
2975 (if (eq (car x
) '-
) (not neg
) neg
))))))
2977 (defun math-solve-find-root-in-prod (x)
2979 (math-expr-contains x solve-var
)
2980 (or (and (eq (car x
) 'calcFunc-sqrt
)
2982 (and (eq (car x
) '^
)
2983 (or (and (memq (math-quarter-integer (nth 2 x
)) '(1 2 3))
2985 (and (eq (car-safe (nth 2 x
)) 'frac
)
2986 (eq (nth 2 (nth 2 x
)) 3)
2988 (and (memq (car x
) '(* /))
2989 (or (and (not (math-expr-contains (nth 1 x
) solve-var
))
2990 (math-solve-find-root-in-prod (nth 2 x
)))
2991 (and (not (math-expr-contains (nth 2 x
) solve-var
))
2992 (math-solve-find-root-in-prod (nth 1 x
))))))))
2995 (defun math-solve-system (exprs solve-vars solve-full
)
2996 (setq exprs
(mapcar 'list
(if (Math-vectorp exprs
)
2999 solve-vars
(if (Math-vectorp solve-vars
)
3002 (or (let ((math-solve-simplifying nil
))
3003 (math-solve-system-rec exprs solve-vars nil
))
3004 (let ((math-solve-simplifying t
))
3005 (math-solve-system-rec exprs solve-vars nil
))))
3007 ;;; The following backtracking solver works by choosing a variable
3008 ;;; and equation, and trying to solve the equation for the variable.
3009 ;;; If it succeeds it calls itself recursively with that variable and
3010 ;;; equation removed from their respective lists, and with the solution
3011 ;;; added to solns as well as being substituted into all existing
3012 ;;; equations. The algorithm terminates when any solution path
3013 ;;; manages to remove all the variables from var-list.
3015 ;;; To support calcFunc-roots, entries in eqn-list and solns are
3016 ;;; actually lists of equations.
3018 (defun math-solve-system-rec (eqn-list var-list solns
)
3023 ;; Try each variable in turn.
3029 (elim (eq (car-safe vv
) 'calcFunc-elim
)))
3031 (setq vv
(nth 1 vv
)))
3033 ;; Try each equation in turn.
3042 ;; Try to solve for vv the list of equations e2.
3044 (setq res2
(or (and (eq (car e2
) eprev
)
3046 (math-solve-for (car e2
) 0 vv
3048 (setq eprev
(car e2
)
3049 res
(cons (if (eq solve-full
'all
)
3057 ;; Found a solution. Now try other variables.
3058 (setq res
(nreverse res
)
3059 res
(math-solve-system-rec
3061 'math-solve-system-subst
3063 (copy-sequence eqn-list
)))
3064 (delq (car v
) (copy-sequence var-list
))
3065 (let ((math-solve-simplifying nil
)
3071 (math-solve-system-subst
3076 (cons (cons vv
(apply 'append res
))
3084 ;; Eliminated all variables, so now put solution into the proper format.
3085 (setq solns
(sort solns
3088 (not (memq (car x
) (memq (car y
) solve-vars
)))))))
3089 (if (eq solve-full
'all
)
3094 (mapcar (function (lambda (x) (cons 'vec
(cdr x
)))) solns
)
3095 (mapcar (function (lambda (x) (cons 'vec x
))) eqn-list
)))))
3099 (mapcar (function (lambda (x) (cons 'calcFunc-eq x
))) solns
)
3100 (mapcar 'car eqn-list
)))))))
3102 (defun math-solve-system-subst (x) ; uses "res" and "v"
3106 (setq accum
(nconc accum
3109 (if math-solve-simplifying
3111 (math-expr-subst (car x
) vv r
))
3112 (math-expr-subst (car x
) vv r
))))
3119 (defun math-get-from-counter (name)
3120 (let ((ctr (assq name calc-command-flags
)))
3122 (setcdr ctr
(1+ (cdr ctr
)))
3123 (setq ctr
(cons name
1)
3124 calc-command-flags
(cons ctr calc-command-flags
)))
3127 (defun math-solve-get-sign (val)
3128 (setq val
(math-simplify val
))
3129 (if (and (eq (car-safe val
) '*)
3130 (Math-numberp (nth 1 val
)))
3131 (list '* (nth 1 val
) (math-solve-get-sign (nth 2 val
)))
3132 (and (eq (car-safe val
) 'calcFunc-sqrt
)
3133 (eq (car-safe (nth 1 val
)) '^
)
3134 (setq val
(math-normalize (list '^
3136 (math-div (nth 2 (nth 1 val
)) 2)))))
3138 (if (and (calc-var-value 'var-GenCount
)
3139 (Math-natnump var-GenCount
)
3140 (not (eq solve-full
'all
)))
3142 (math-mul (list 'calcFunc-as var-GenCount
) val
)
3143 (setq var-GenCount
(math-add var-GenCount
1))
3144 (calc-refresh-evaltos 'var-GenCount
))
3145 (let* ((var (concat "s" (int-to-string (math-get-from-counter 'solve-sign
))))
3146 (var2 (list 'var
(intern var
) (intern (concat "var-" var
)))))
3147 (if (eq solve-full
'all
)
3148 (setq math-solve-ranges
(cons (list var2
1 -
1)
3149 math-solve-ranges
)))
3150 (math-mul var2 val
)))
3151 (calc-record-why "*Choosing positive solution")
3154 (defun math-solve-get-int (val &optional range first
)
3156 (if (and (calc-var-value 'var-GenCount
)
3157 (Math-natnump var-GenCount
)
3158 (not (eq solve-full
'all
)))
3160 (math-mul val
(list 'calcFunc-an var-GenCount
))
3161 (setq var-GenCount
(math-add var-GenCount
1))
3162 (calc-refresh-evaltos 'var-GenCount
))
3163 (let* ((var (concat "n" (int-to-string
3164 (math-get-from-counter 'solve-int
))))
3165 (var2 (list 'var
(intern var
) (intern (concat "var-" var
)))))
3166 (if (and range
(eq solve-full
'all
))
3167 (setq math-solve-ranges
(cons (cons var2
3168 (cdr (calcFunc-index
3169 range
(or first
0))))
3170 math-solve-ranges
)))
3171 (math-mul val var2
)))
3172 (calc-record-why "*Choosing 0 for arbitrary integer in solution")
3175 (defun math-solve-sign (sign expr
)
3177 (let ((s1 (math-possible-signs expr
)))
3178 (cond ((memq s1
'(4 6))
3183 (defun math-looks-evenp (expr)
3184 (if (Math-integerp expr
)
3186 (if (memq (car expr
) '(* /))
3187 (math-looks-evenp (nth 1 expr
)))))
3189 (defun math-solve-for (lhs rhs solve-var solve-full
&optional sign
)
3190 (if (math-expr-contains rhs solve-var
)
3191 (math-solve-for (math-sub lhs rhs
) 0 solve-var solve-full
)
3192 (and (math-expr-contains lhs solve-var
)
3193 (math-with-extra-prec 1
3194 (let* ((math-poly-base-variable solve-var
)
3195 (res (math-try-solve-for lhs rhs sign
)))
3196 (if (and (eq solve-full
'all
)
3197 (math-known-realp solve-var
))
3198 (let ((old-len (length res
))
3203 (and (not (memq (car-safe x
)
3207 new-len
(length res
))
3208 (if (< new-len old-len
)
3209 (calc-record-why (if (= new-len
1)
3210 "*All solutions were complex"
3212 "*Omitted %d complex solutions"
3213 (- old-len new-len
)))))))
3216 (defun math-solve-eqn (expr var full
)
3217 (if (memq (car-safe expr
) '(calcFunc-neq calcFunc-lt calcFunc-gt
3218 calcFunc-leq calcFunc-geq
))
3219 (let ((res (math-solve-for (cons '-
(cdr expr
))
3221 (if (eq (car expr
) 'calcFunc-neq
) nil
1))))
3223 (if (eq math-solve-sign
1)
3224 (list (car expr
) var res
)
3225 (if (eq math-solve-sign -
1)
3226 (list (car expr
) res var
)
3227 (or (eq (car expr
) 'calcFunc-neq
)
3229 "*Can't determine direction of inequality"))
3230 (and (memq (car expr
) '(calcFunc-neq calcFunc-lt calcFunc-gt
))
3231 (list 'calcFunc-neq var res
))))))
3232 (let ((res (math-solve-for expr
0 var full
)))
3234 (list 'calcFunc-eq var res
)))))
3236 (defun math-reject-solution (expr var func
)
3237 (if (math-expr-contains expr var
)
3238 (or (equal (car calc-next-why
) '(* "Unable to find a symbolic solution"))
3239 (calc-record-why "*Unable to find a solution")))
3240 (list func expr var
))
3242 (defun calcFunc-solve (expr var
)
3243 (or (if (or (Math-vectorp expr
) (Math-vectorp var
))
3244 (math-solve-system expr var nil
)
3245 (math-solve-eqn expr var nil
))
3246 (math-reject-solution expr var
'calcFunc-solve
)))
3248 (defun calcFunc-fsolve (expr var
)
3249 (or (if (or (Math-vectorp expr
) (Math-vectorp var
))
3250 (math-solve-system expr var t
)
3251 (math-solve-eqn expr var t
))
3252 (math-reject-solution expr var
'calcFunc-fsolve
)))
3254 (defun calcFunc-roots (expr var
)
3255 (let ((math-solve-ranges nil
))
3256 (or (if (or (Math-vectorp expr
) (Math-vectorp var
))
3257 (math-solve-system expr var
'all
)
3258 (math-solve-for expr
0 var
'all
))
3259 (math-reject-solution expr var
'calcFunc-roots
))))
3261 (defun calcFunc-finv (expr var
)
3262 (let ((res (math-solve-for expr math-integ-var var nil
)))
3264 (math-normalize (math-expr-subst res math-integ-var var
))
3265 (math-reject-solution expr var
'calcFunc-finv
))))
3267 (defun calcFunc-ffinv (expr var
)
3268 (let ((res (math-solve-for expr math-integ-var var t
)))
3270 (math-normalize (math-expr-subst res math-integ-var var
))
3271 (math-reject-solution expr var
'calcFunc-finv
))))
3274 (put 'calcFunc-inv
'math-inverse
3275 (function (lambda (x) (math-div 1 x
))))
3276 (put 'calcFunc-inv
'math-inverse-sign -
1)
3278 (put 'calcFunc-sqrt
'math-inverse
3279 (function (lambda (x) (math-sqr x
))))
3281 (put 'calcFunc-conj
'math-inverse
3282 (function (lambda (x) (list 'calcFunc-conj x
))))
3284 (put 'calcFunc-abs
'math-inverse
3285 (function (lambda (x) (math-solve-get-sign x
))))
3287 (put 'calcFunc-deg
'math-inverse
3288 (function (lambda (x) (list 'calcFunc-rad x
))))
3289 (put 'calcFunc-deg
'math-inverse-sign
1)
3291 (put 'calcFunc-rad
'math-inverse
3292 (function (lambda (x) (list 'calcFunc-deg x
))))
3293 (put 'calcFunc-rad
'math-inverse-sign
1)
3295 (put 'calcFunc-ln
'math-inverse
3296 (function (lambda (x) (list 'calcFunc-exp x
))))
3297 (put 'calcFunc-ln
'math-inverse-sign
1)
3299 (put 'calcFunc-log10
'math-inverse
3300 (function (lambda (x) (list 'calcFunc-exp10 x
))))
3301 (put 'calcFunc-log10
'math-inverse-sign
1)
3303 (put 'calcFunc-lnp1
'math-inverse
3304 (function (lambda (x) (list 'calcFunc-expm1 x
))))
3305 (put 'calcFunc-lnp1
'math-inverse-sign
1)
3307 (put 'calcFunc-exp
'math-inverse
3308 (function (lambda (x) (math-add (math-normalize (list 'calcFunc-ln x
))
3310 (math-mul '(var pi var-pi
)
3312 '(var i var-i
))))))))
3313 (put 'calcFunc-exp
'math-inverse-sign
1)
3315 (put 'calcFunc-expm1
'math-inverse
3316 (function (lambda (x) (math-add (math-normalize (list 'calcFunc-lnp1 x
))
3318 (math-mul '(var pi var-pi
)
3320 '(var i var-i
))))))))
3321 (put 'calcFunc-expm1
'math-inverse-sign
1)
3323 (put 'calcFunc-sin
'math-inverse
3324 (function (lambda (x) (let ((n (math-solve-get-int 1)))
3325 (math-add (math-mul (math-normalize
3326 (list 'calcFunc-arcsin x
))
3328 (math-mul (math-half-circle t
)
3331 (put 'calcFunc-cos
'math-inverse
3332 (function (lambda (x) (math-add (math-solve-get-sign
3334 (list 'calcFunc-arccos x
)))
3336 (math-full-circle t
))))))
3338 (put 'calcFunc-tan
'math-inverse
3339 (function (lambda (x) (math-add (math-normalize (list 'calcFunc-arctan x
))
3341 (math-half-circle t
))))))
3343 (put 'calcFunc-arcsin
'math-inverse
3344 (function (lambda (x) (math-normalize (list 'calcFunc-sin x
)))))
3346 (put 'calcFunc-arccos
'math-inverse
3347 (function (lambda (x) (math-normalize (list 'calcFunc-cos x
)))))
3349 (put 'calcFunc-arctan
'math-inverse
3350 (function (lambda (x) (math-normalize (list 'calcFunc-tan x
)))))
3352 (put 'calcFunc-sinh
'math-inverse
3353 (function (lambda (x) (let ((n (math-solve-get-int 1)))
3354 (math-add (math-mul (math-normalize
3355 (list 'calcFunc-arcsinh x
))
3357 (math-mul (math-half-circle t
)
3361 (put 'calcFunc-sinh
'math-inverse-sign
1)
3363 (put 'calcFunc-cosh
'math-inverse
3364 (function (lambda (x) (math-add (math-solve-get-sign
3366 (list 'calcFunc-arccosh x
)))
3367 (math-mul (math-full-circle t
)
3369 '(var i var-i
)))))))
3371 (put 'calcFunc-tanh
'math-inverse
3372 (function (lambda (x) (math-add (math-normalize
3373 (list 'calcFunc-arctanh x
))
3374 (math-mul (math-half-circle t
)
3376 '(var i var-i
)))))))
3377 (put 'calcFunc-tanh
'math-inverse-sign
1)
3379 (put 'calcFunc-arcsinh
'math-inverse
3380 (function (lambda (x) (math-normalize (list 'calcFunc-sinh x
)))))
3381 (put 'calcFunc-arcsinh
'math-inverse-sign
1)
3383 (put 'calcFunc-arccosh
'math-inverse
3384 (function (lambda (x) (math-normalize (list 'calcFunc-cosh x
)))))
3386 (put 'calcFunc-arctanh
'math-inverse
3387 (function (lambda (x) (math-normalize (list 'calcFunc-tanh x
)))))
3388 (put 'calcFunc-arctanh
'math-inverse-sign
1)
3392 (defun calcFunc-taylor (expr var num
)
3393 (let ((x0 0) (v var
))
3394 (if (memq (car-safe var
) '(+ - calcFunc-eq
))
3395 (setq x0
(if (eq (car var
) '+) (math-neg (nth 2 var
)) (nth 2 var
))
3397 (or (and (eq (car-safe v
) 'var
)
3398 (math-expr-contains expr v
)
3400 (let ((accum (math-expr-subst expr v x0
))
3401 (var2 (if (eq (car var
) 'calcFunc-eq
)
3407 (while (and (<= (setq n
(1+ n
)) num
)
3408 (setq fprime
(calcFunc-deriv fprime v nil t
)))
3409 (setq fprime
(math-simplify fprime
)
3410 nfac
(math-mul nfac n
)
3411 accum
(math-add accum
3412 (math-div (math-mul (math-pow var2 n
)
3417 (math-normalize accum
))))
3418 (list 'calcFunc-taylor expr var num
))))
3420 ;;; calcalg2.el ends here
