1 ;;; calc-poly.el --- polynomial functions for Calc
3 ;; Copyright (C) 1990, 1991, 1992, 1993, 2001 Free Software Foundation, Inc.
5 ;; Author: David Gillespie <daveg@synaptics.com>
6 ;; Maintainer: Jay Belanger <belanger@truman.edu>
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 calcFunc-pcont (expr &optional var)
35 (cond ((Math-primp expr)
36 (cond ((Math-zerop expr) 1)
37 ((Math-messy-integerp expr) (math-trunc expr))
38 ((Math-objectp expr) expr)
39 ((or (equal expr var) (not var)) 1)
42 (math-mul (calcFunc-pcont (nth 1 expr) var)
43 (calcFunc-pcont (nth 2 expr) var)))
45 (math-div (calcFunc-pcont (nth 1 expr) var)
46 (calcFunc-pcont (nth 2 expr) var)))
47 ((and (eq (car expr) '^) (Math-natnump (nth 2 expr)))
48 (math-pow (calcFunc-pcont (nth 1 expr) var) (nth 2 expr)))
49 ((memq (car expr) '(neg polar))
50 (calcFunc-pcont (nth 1 expr) var))
52 (let ((p (math-is-polynomial expr var)))
54 (let ((lead (nth (1- (length p)) p))
55 (cont (math-poly-gcd-list p)))
56 (if (math-guess-if-neg lead)
60 ((memq (car expr) '(+ - cplx sdev))
61 (let ((cont (calcFunc-pcont (nth 1 expr) var)))
64 (let ((c2 (calcFunc-pcont (nth 2 expr) var)))
65 (if (and (math-negp cont)
66 (if (eq (car expr) '-) (math-posp c2) (math-negp c2)))
67 (math-neg (math-poly-gcd cont c2))
68 (math-poly-gcd cont c2))))))
72 (defun calcFunc-pprim (expr &optional var)
73 (let ((cont (calcFunc-pcont expr var)))
74 (if (math-equal-int cont 1)
76 (math-poly-div-exact expr cont var))))
78 (defun math-div-poly-const (expr c)
79 (cond ((memq (car-safe expr) '(+ -))
81 (math-div-poly-const (nth 1 expr) c)
82 (math-div-poly-const (nth 2 expr) c)))
83 (t (math-div expr c))))
85 (defun calcFunc-pdeg (expr &optional var)
87 '(neg (var inf var-inf))
89 (or (math-polynomial-p expr var)
90 (math-reject-arg expr "Expected a polynomial"))
91 (math-poly-degree expr))))
93 (defun math-poly-degree (expr)
94 (cond ((Math-primp expr)
95 (if (eq (car-safe expr) 'var) 1 0))
97 (math-poly-degree (nth 1 expr)))
99 (+ (math-poly-degree (nth 1 expr))
100 (math-poly-degree (nth 2 expr))))
102 (- (math-poly-degree (nth 1 expr))
103 (math-poly-degree (nth 2 expr))))
104 ((and (eq (car expr) '^) (natnump (nth 2 expr)))
105 (* (math-poly-degree (nth 1 expr)) (nth 2 expr)))
106 ((memq (car expr) '(+ -))
107 (max (math-poly-degree (nth 1 expr))
108 (math-poly-degree (nth 2 expr))))
111 (defun calcFunc-plead (expr var)
112 (cond ((eq (car-safe expr) '*)
113 (math-mul (calcFunc-plead (nth 1 expr) var)
114 (calcFunc-plead (nth 2 expr) var)))
115 ((eq (car-safe expr) '/)
116 (math-div (calcFunc-plead (nth 1 expr) var)
117 (calcFunc-plead (nth 2 expr) var)))
118 ((and (eq (car-safe expr) '^) (math-natnump (nth 2 expr)))
119 (math-pow (calcFunc-plead (nth 1 expr) var) (nth 2 expr)))
125 (let ((p (math-is-polynomial expr var)))
127 (nth (1- (length p)) p)
134 ;;; Polynomial quotient, remainder, and GCD.
135 ;;; Originally by Ove Ewerlid (ewerlid@mizar.DoCS.UU.SE).
136 ;;; Modifications and simplifications by daveg.
138 (defvar math-poly-modulus 1)
140 ;;; Return gcd of two polynomials
141 (defun calcFunc-pgcd (pn pd)
142 (if (math-any-floats pn)
143 (math-reject-arg pn "Coefficients must be rational"))
144 (if (math-any-floats pd)
145 (math-reject-arg pd "Coefficients must be rational"))
146 (let ((calc-prefer-frac t)
147 (math-poly-modulus (math-poly-modulus pn pd)))
148 (math-poly-gcd pn pd)))
150 ;;; Return only quotient to top of stack (nil if zero)
152 ;; calc-poly-div-remainder is a local variable for
153 ;; calc-poly-div (in calc-alg.el), but is used by
154 ;; calcFunc-pdiv, which is called by calc-poly-div.
155 (defvar calc-poly-div-remainder)
157 (defun calcFunc-pdiv (pn pd &optional base)
158 (let* ((calc-prefer-frac t)
159 (math-poly-modulus (math-poly-modulus pn pd))
160 (res (math-poly-div pn pd base)))
161 (setq calc-poly-div-remainder (cdr res))
164 ;;; Return only remainder to top of stack
165 (defun calcFunc-prem (pn pd &optional base)
166 (let ((calc-prefer-frac t)
167 (math-poly-modulus (math-poly-modulus pn pd)))
168 (cdr (math-poly-div pn pd base))))
170 (defun calcFunc-pdivrem (pn pd &optional base)
171 (let* ((calc-prefer-frac t)
172 (math-poly-modulus (math-poly-modulus pn pd))
173 (res (math-poly-div pn pd base)))
174 (list 'vec (car res) (cdr res))))
176 (defun calcFunc-pdivide (pn pd &optional base)
177 (let* ((calc-prefer-frac t)
178 (math-poly-modulus (math-poly-modulus pn pd))
179 (res (math-poly-div pn pd base)))
180 (math-add (car res) (math-div (cdr res) pd))))
183 ;;; Multiply two terms, expanding out products of sums.
184 (defun math-mul-thru (lhs rhs)
185 (if (memq (car-safe lhs) '(+ -))
187 (math-mul-thru (nth 1 lhs) rhs)
188 (math-mul-thru (nth 2 lhs) rhs))
189 (if (memq (car-safe rhs) '(+ -))
191 (math-mul-thru lhs (nth 1 rhs))
192 (math-mul-thru lhs (nth 2 rhs)))
193 (math-mul lhs rhs))))
195 (defun math-div-thru (num den)
196 (if (memq (car-safe num) '(+ -))
198 (math-div-thru (nth 1 num) den)
199 (math-div-thru (nth 2 num) den))
203 ;;; Sort the terms of a sum into canonical order.
204 (defun math-sort-terms (expr)
205 (if (memq (car-safe expr) '(+ -))
207 (sort (math-sum-to-list expr)
208 (function (lambda (a b) (math-beforep (car a) (car b))))))
211 (defun math-list-to-sum (lst)
213 (list (if (cdr (car lst)) '- '+)
214 (math-list-to-sum (cdr lst))
217 (math-neg (car (car lst)))
220 (defun math-sum-to-list (tree &optional neg)
221 (cond ((eq (car-safe tree) '+)
222 (nconc (math-sum-to-list (nth 1 tree) neg)
223 (math-sum-to-list (nth 2 tree) neg)))
224 ((eq (car-safe tree) '-)
225 (nconc (math-sum-to-list (nth 1 tree) neg)
226 (math-sum-to-list (nth 2 tree) (not neg))))
227 (t (list (cons tree neg)))))
229 ;;; Check if the polynomial coefficients are modulo forms.
230 (defun math-poly-modulus (expr &optional expr2)
231 (or (math-poly-modulus-rec expr)
232 (and expr2 (math-poly-modulus-rec expr2))
235 (defun math-poly-modulus-rec (expr)
236 (if (and (eq (car-safe expr) 'mod) (Math-natnump (nth 2 expr)))
237 (list 'mod 1 (nth 2 expr))
238 (and (memq (car-safe expr) '(+ - * /))
239 (or (math-poly-modulus-rec (nth 1 expr))
240 (math-poly-modulus-rec (nth 2 expr))))))
243 ;;; Divide two polynomials. Return (quotient . remainder).
244 (defvar math-poly-div-base nil)
245 (defun math-poly-div (u v &optional math-poly-div-base)
246 (if math-poly-div-base
247 (math-do-poly-div u v)
248 (math-do-poly-div (calcFunc-expand u) (calcFunc-expand v))))
250 (defun math-poly-div-exact (u v &optional base)
251 (let ((res (math-poly-div u v base)))
254 (math-reject-arg (list 'vec u v) "Argument is not a polynomial"))))
256 (defun math-do-poly-div (u v)
257 (cond ((math-constp u)
259 (cons (math-div u v) 0)
264 (if (memq (car-safe u) '(+ -))
265 (math-add-or-sub (math-poly-div-exact (nth 1 u) v)
266 (math-poly-div-exact (nth 2 u) v)
271 (cons math-poly-modulus 0))
272 ((and (math-atomic-factorp u) (math-atomic-factorp v))
273 (cons (math-simplify (math-div u v)) 0))
275 (let ((base (or math-poly-div-base
276 (math-poly-div-base u v)))
279 (null (setq vp (math-is-polynomial v base nil 'gen))))
281 (setq up (math-is-polynomial u base nil 'gen)
282 res (math-poly-div-coefs up vp))
283 (cons (math-build-polynomial-expr (car res) base)
284 (math-build-polynomial-expr (cdr res) base)))))))
286 (defun math-poly-div-rec (u v)
287 (cond ((math-constp u)
292 (if (memq (car-safe u) '(+ -))
293 (math-add-or-sub (math-poly-div-rec (nth 1 u) v)
294 (math-poly-div-rec (nth 2 u) v)
297 ((Math-equal u v) math-poly-modulus)
298 ((and (math-atomic-factorp u) (math-atomic-factorp v))
299 (math-simplify (math-div u v)))
303 (let ((base (math-poly-div-base u v))
306 (null (setq vp (math-is-polynomial v base nil 'gen))))
308 (setq up (math-is-polynomial u base nil 'gen)
309 res (math-poly-div-coefs up vp))
310 (math-add (math-build-polynomial-expr (car res) base)
311 (math-div (math-build-polynomial-expr (cdr res) base)
314 ;;; Divide two polynomials in coefficient-list form. Return (quot . rem).
315 (defun math-poly-div-coefs (u v)
316 (cond ((null v) (math-reject-arg nil "Division by zero"))
317 ((< (length u) (length v)) (cons nil u))
323 (let ((qk (math-poly-div-rec (math-simplify (car urev))
327 (if (or q (not (math-zerop qk)))
328 (setq q (cons qk q)))
329 (while (setq up (cdr up) vp (cdr vp))
330 (setcar up (math-sub (car up) (math-mul-thru qk (car vp)))))
331 (setq urev (cdr urev))
333 (while (and urev (Math-zerop (car urev)))
334 (setq urev (cdr urev)))
335 (cons q (nreverse (mapcar 'math-simplify urev)))))
337 (cons (list (math-poly-div-rec (car u) (car v)))
340 ;;; Perform a pseudo-division of polynomials. (See Knuth section 4.6.1.)
341 ;;; This returns only the remainder from the pseudo-division.
342 (defun math-poly-pseudo-div (u v)
344 ((< (length u) (length v)) u)
345 ((or (cdr u) (cdr v))
346 (let ((urev (reverse u))
352 (while (setq up (cdr up) vp (cdr vp))
353 (setcar up (math-sub (math-mul-thru (car vrev) (car up))
354 (math-mul-thru (car urev) (car vp)))))
355 (setq urev (cdr urev))
358 (setcar up (math-mul-thru (car vrev) (car up)))
360 (while (and urev (Math-zerop (car urev)))
361 (setq urev (cdr urev)))
362 (nreverse (mapcar 'math-simplify urev))))
365 ;;; Compute the GCD of two multivariate polynomials.
366 (defun math-poly-gcd (u v)
367 (cond ((Math-equal u v) u)
371 (calcFunc-gcd u (calcFunc-pcont v))))
375 (calcFunc-gcd v (calcFunc-pcont u))))
377 (let ((base (math-poly-gcd-base u v)))
381 (math-build-polynomial-expr
382 (math-poly-gcd-coefs (math-is-polynomial u base nil 'gen)
383 (math-is-polynomial v base nil 'gen))
385 (calcFunc-gcd (calcFunc-pcont u) (calcFunc-pcont u)))))))
387 (defun math-poly-div-list (lst a)
391 (math-mul-list lst a)
392 (mapcar (function (lambda (x) (math-poly-div-exact x a))) lst))))
394 (defun math-mul-list (lst a)
398 (mapcar 'math-neg lst)
400 (mapcar (function (lambda (x) (math-mul x a))) lst)))))
402 ;;; Run GCD on all elements in a list.
403 (defun math-poly-gcd-list (lst)
404 (if (or (memq 1 lst) (memq -1 lst))
405 (math-poly-gcd-frac-list lst)
406 (let ((gcd (car lst)))
407 (while (and (setq lst (cdr lst)) (not (eq gcd 1)))
409 (setq gcd (math-poly-gcd gcd (car lst)))))
410 (if lst (setq lst (math-poly-gcd-frac-list lst)))
413 (defun math-poly-gcd-frac-list (lst)
414 (while (and lst (not (eq (car-safe (car lst)) 'frac)))
415 (setq lst (cdr lst)))
417 (let ((denom (nth 2 (car lst))))
418 (while (setq lst (cdr lst))
419 (if (eq (car-safe (car lst)) 'frac)
420 (setq denom (calcFunc-lcm denom (nth 2 (car lst))))))
421 (list 'frac 1 denom))
424 ;;; Compute the GCD of two monovariate polynomial lists.
425 ;;; Knuth section 4.6.1, algorithm C.
426 (defun math-poly-gcd-coefs (u v)
427 (let ((d (math-poly-gcd (math-poly-gcd-list u)
428 (math-poly-gcd-list v)))
429 (g 1) (h 1) (z 0) hh r delta ghd)
430 (while (and u v (Math-zerop (car u)) (Math-zerop (car v)))
431 (setq u (cdr u) v (cdr v) z (1+ z)))
433 (setq u (math-poly-div-list u d)
434 v (math-poly-div-list v d)))
436 (setq delta (- (length u) (length v)))
438 (setq r u u v v r delta (- delta)))
439 (setq r (math-poly-pseudo-div u v))
442 v (math-poly-div-list r (math-mul g (math-pow h delta)))
443 g (nth (1- (length u)) u)
445 (math-mul (math-pow g delta) (math-pow h (- 1 delta)))
446 (math-poly-div-exact (math-pow g delta)
447 (math-pow h (1- delta))))))
450 (math-mul-list (math-poly-div-list v (math-poly-gcd-list v)) d)))
451 (if (math-guess-if-neg (nth (1- (length v)) v))
452 (setq v (math-mul-list v -1)))
453 (while (>= (setq z (1- z)) 0)
458 ;;; Return true if is a factor containing no sums or quotients.
459 (defun math-atomic-factorp (expr)
460 (cond ((eq (car-safe expr) '*)
461 (and (math-atomic-factorp (nth 1 expr))
462 (math-atomic-factorp (nth 2 expr))))
463 ((memq (car-safe expr) '(+ - /))
465 ((memq (car-safe expr) '(^ neg))
466 (math-atomic-factorp (nth 1 expr)))
469 ;;; Find a suitable base for dividing a by b.
470 ;;; The base must exist in both expressions.
471 ;;; The degree in the numerator must be higher or equal than the
472 ;;; degree in the denominator.
473 ;;; If the above conditions are not met the quotient is just a remainder.
474 ;;; Return nil if this is the case.
476 (defun math-poly-div-base (a b)
478 (and (setq a-base (math-total-polynomial-base a))
479 (setq b-base (math-total-polynomial-base b))
482 (let ((maybe (assoc (car (car a-base)) b-base)))
484 (if (>= (nth 1 (car a-base)) (nth 1 maybe))
485 (throw 'return (car (car a-base))))))
486 (setq a-base (cdr a-base)))))))
488 ;;; Same as above but for gcd algorithm.
489 ;;; Here there is no requirement that degree(a) > degree(b).
490 ;;; Take the base that has the highest degree considering both a and b.
491 ;;; ("a^20+b^21+x^3+a+b", "a+b^2+x^5+a^22+b^10") --> (a 22)
493 (defun math-poly-gcd-base (a b)
495 (and (setq a-base (math-total-polynomial-base a))
496 (setq b-base (math-total-polynomial-base b))
498 (while (and a-base b-base)
499 (if (> (nth 1 (car a-base)) (nth 1 (car b-base)))
500 (if (assoc (car (car a-base)) b-base)
501 (throw 'return (car (car a-base)))
502 (setq a-base (cdr a-base)))
503 (if (assoc (car (car b-base)) a-base)
504 (throw 'return (car (car b-base)))
505 (setq b-base (cdr b-base)))))))))
507 ;;; Sort a list of polynomial bases.
508 (defun math-sort-poly-base-list (lst)
509 (sort lst (function (lambda (a b)
510 (or (> (nth 1 a) (nth 1 b))
511 (and (= (nth 1 a) (nth 1 b))
512 (math-beforep (car a) (car b))))))))
514 ;;; Given an expression find all variables that are polynomial bases.
515 ;;; Return list in the form '( (var1 degree1) (var2 degree2) ... ).
517 ;; The variable math-poly-base-total-base is local to
518 ;; math-total-polynomial-base, but is used by math-polynomial-p1,
519 ;; which is called by math-total-polynomial-base.
520 (defvar math-poly-base-total-base)
522 (defun math-total-polynomial-base (expr)
523 (let ((math-poly-base-total-base nil))
524 (math-polynomial-base expr 'math-polynomial-p1)
525 (math-sort-poly-base-list math-poly-base-total-base)))
527 ;; The variable math-poly-base-top-expr is local to math-polynomial-base
528 ;; in calc-alg.el, but is used by math-polynomial-p1 which is called
529 ;; by math-polynomial-base.
530 (defvar math-poly-base-top-expr)
532 (defun math-polynomial-p1 (subexpr)
533 (or (assoc subexpr math-poly-base-total-base)
534 (memq (car subexpr) '(+ - * / neg))
535 (and (eq (car subexpr) '^) (natnump (nth 2 subexpr)))
536 (let* ((math-poly-base-variable subexpr)
537 (exponent (math-polynomial-p math-poly-base-top-expr subexpr)))
539 (setq math-poly-base-total-base (cons (list subexpr exponent)
540 math-poly-base-total-base)))))
543 ;; The variable math-factored-vars is local to calcFunc-factors and
544 ;; calcFunc-factor, but is used by math-factor-expr and
545 ;; math-factor-expr-part, which are called (directly and indirectly) by
546 ;; calcFunc-factor and calcFunc-factors.
547 (defvar math-factored-vars)
549 ;; The variable math-fact-expr is local to calcFunc-factors,
550 ;; calcFunc-factor and math-factor-expr, but is used by math-factor-expr-try
551 ;; and math-factor-expr-part, which are called (directly and indirectly) by
552 ;; calcFunc-factor, calcFunc-factors and math-factor-expr.
553 (defvar math-fact-expr)
555 ;; The variable math-to-list is local to calcFunc-factors and
556 ;; calcFunc-factor, but is used by math-accum-factors, which is
557 ;; called (indirectly) by calcFunc-factors and calcFunc-factor.
558 (defvar math-to-list)
560 (defun calcFunc-factors (math-fact-expr &optional var)
561 (let ((math-factored-vars (if var t nil))
563 (calc-prefer-frac t))
565 (setq var (math-polynomial-base math-fact-expr)))
566 (let ((res (math-factor-finish
567 (or (catch 'factor (math-factor-expr-try var))
569 (math-simplify (if (math-vectorp res)
571 (list 'vec (list 'vec res 1)))))))
573 (defun calcFunc-factor (math-fact-expr &optional var)
574 (let ((math-factored-vars nil)
576 (calc-prefer-frac t))
577 (math-simplify (math-factor-finish
579 (let ((math-factored-vars t))
580 (or (catch 'factor (math-factor-expr-try var)) math-fact-expr))
581 (math-factor-expr math-fact-expr))))))
583 (defun math-factor-finish (x)
586 (if (eq (car x) 'calcFunc-Fac-Prot)
587 (math-factor-finish (nth 1 x))
588 (cons (car x) (mapcar 'math-factor-finish (cdr x))))))
590 (defun math-factor-protect (x)
591 (if (memq (car-safe x) '(+ -))
592 (list 'calcFunc-Fac-Prot x)
595 (defun math-factor-expr (math-fact-expr)
596 (cond ((eq math-factored-vars t) math-fact-expr)
597 ((or (memq (car-safe math-fact-expr) '(* / ^ neg))
598 (assq (car-safe math-fact-expr) calc-tweak-eqn-table))
599 (cons (car math-fact-expr) (mapcar 'math-factor-expr (cdr math-fact-expr))))
600 ((memq (car-safe math-fact-expr) '(+ -))
601 (let* ((math-factored-vars math-factored-vars)
602 (y (catch 'factor (math-factor-expr-part math-fact-expr))))
608 (defun math-factor-expr-part (x) ; uses "expr"
609 (if (memq (car-safe x) '(+ - * / ^ neg))
610 (while (setq x (cdr x))
611 (math-factor-expr-part (car x)))
612 (and (not (Math-objvecp x))
613 (not (assoc x math-factored-vars))
614 (> (math-factor-contains math-fact-expr x) 1)
615 (setq math-factored-vars (cons (list x) math-factored-vars))
616 (math-factor-expr-try x))))
618 ;; The variable math-fet-x is local to math-factor-expr-try, but is
619 ;; used by math-factor-poly-coefs, which is called by math-factor-expr-try.
622 (defun math-factor-expr-try (math-fet-x)
623 (if (eq (car-safe math-fact-expr) '*)
624 (let ((res1 (catch 'factor (let ((math-fact-expr (nth 1 math-fact-expr)))
625 (math-factor-expr-try math-fet-x))))
626 (res2 (catch 'factor (let ((math-fact-expr (nth 2 math-fact-expr)))
627 (math-factor-expr-try math-fet-x)))))
629 (throw 'factor (math-accum-factors (or res1 (nth 1 math-fact-expr)) 1
630 (or res2 (nth 2 math-fact-expr))))))
631 (let* ((p (math-is-polynomial math-fact-expr math-fet-x 30 'gen))
632 (math-poly-modulus (math-poly-modulus math-fact-expr))
635 (setq res (math-factor-poly-coefs p))
636 (throw 'factor res)))))
638 (defun math-accum-factors (fac pow facs)
640 (if (math-vectorp fac)
642 (while (setq fac (cdr fac))
643 (setq facs (math-accum-factors (nth 1 (car fac))
644 (* pow (nth 2 (car fac)))
647 (if (and (eq (car-safe fac) '^) (natnump (nth 2 fac)))
648 (setq pow (* pow (nth 2 fac))
652 (or (math-vectorp facs)
653 (setq facs (if (eq facs 1) '(vec)
654 (list 'vec (list 'vec facs 1)))))
656 (while (and (setq found (cdr found))
657 (not (equal fac (nth 1 (car found))))))
660 (setcar (cdr (cdr (car found))) (+ pow (nth 2 (car found))))
662 ;; Put constant term first.
663 (if (and (cdr facs) (Math-ratp (nth 1 (nth 1 facs))))
664 (cons 'vec (cons (nth 1 facs) (cons (list 'vec fac pow)
666 (cons 'vec (cons (list 'vec fac pow) (cdr facs))))))))
667 (math-mul (math-pow fac pow) facs)))
669 (defun math-factor-poly-coefs (p &optional square-free) ; uses "x"
674 ;; Strip off multiples of math-fet-x.
675 ((Math-zerop (car p))
677 (while (and p (Math-zerop (car p)))
678 (setq z (1+ z) p (cdr p)))
680 (setq p (math-factor-poly-coefs p square-free))
681 (setq p (math-sort-terms (math-factor-expr (car p)))))
682 (math-accum-factors math-fet-x z (math-factor-protect p))))
684 ;; Factor out content.
685 ((and (not square-free)
686 (not (eq 1 (setq t1 (math-mul (math-poly-gcd-list p)
687 (if (math-guess-if-neg
688 (nth (1- (length p)) p))
690 (math-accum-factors t1 1 (math-factor-poly-coefs
691 (math-poly-div-list p t1) 'cont)))
693 ;; Check if linear in math-fet-x.
695 (math-add (math-factor-protect
697 (math-factor-expr (car p))))
698 (math-mul math-fet-x (math-factor-protect
700 (math-factor-expr (nth 1 p)))))))
702 ;; If symbolic coefficients, use FactorRules.
704 (while (and pp (or (Math-ratp (car pp))
705 (and (eq (car (car pp)) 'mod)
706 (Math-integerp (nth 1 (car pp)))
707 (Math-integerp (nth 2 (car pp))))))
710 (let ((res (math-rewrite
711 (list 'calcFunc-thecoefs math-fet-x (cons 'vec p))
712 '(var FactorRules var-FactorRules))))
713 (or (and (eq (car-safe res) 'calcFunc-thefactors)
715 (math-vectorp (nth 2 res))
718 (while (setq vec (cdr vec))
719 (setq facs (math-accum-factors (car vec) 1 facs)))
721 (math-build-polynomial-expr p math-fet-x))))
723 ;; Check if rational coefficients (i.e., not modulo a prime).
724 ((eq math-poly-modulus 1)
726 ;; Check if there are any squared terms, or a content not = 1.
727 (if (or (eq square-free t)
728 (equal (setq t1 (math-poly-gcd-coefs
729 p (setq t2 (math-poly-deriv-coefs p))))
732 ;; We now have a square-free polynomial with integer coefs.
733 ;; For now, we use a kludgey method that finds linear and
734 ;; quadratic terms using floating-point root-finding.
735 (if (setq t1 (let ((calc-symbolic-mode nil))
736 (math-poly-all-roots nil p t)))
737 (let ((roots (car t1))
738 (csign (if (math-negp (nth (1- (length p)) p)) -1 1))
743 (let ((coef0 (car (car roots)))
744 (coef1 (cdr (car roots))))
745 (setq expr (math-accum-factors
747 (let ((den (math-lcm-denoms
749 (setq scale (math-div scale den))
752 (math-mul den (math-pow math-fet-x 2))
753 (math-mul (math-mul coef1 den)
755 (math-mul coef0 den)))
756 (let ((den (math-lcm-denoms coef0)))
757 (setq scale (math-div scale den))
758 (math-add (math-mul den math-fet-x)
759 (math-mul coef0 den))))
762 (setq expr (math-accum-factors
765 (math-build-polynomial-expr
766 (math-mul-list (nth 1 t1) scale)
768 (math-build-polynomial-expr p math-fet-x)) ; can't factor it.
770 ;; Separate out the squared terms (Knuth exercise 4.6.2-34).
771 ;; This step also divides out the content of the polynomial.
772 (let* ((cabs (math-poly-gcd-list p))
773 (csign (if (math-negp (nth (1- (length p)) p)) -1 1))
774 (t1s (math-mul-list t1 csign))
776 (v (car (math-poly-div-coefs p t1s)))
777 (w (car (math-poly-div-coefs t2 t1s))))
779 (not (math-poly-zerop
780 (setq t2 (math-poly-simplify
782 w 1 (math-poly-deriv-coefs v) -1)))))
783 (setq t1 (math-poly-gcd-coefs v t2)
785 v (car (math-poly-div-coefs v t1))
786 w (car (math-poly-div-coefs t2 t1))))
788 t2 (math-accum-factors (math-factor-poly-coefs v t)
791 (setq t2 (math-accum-factors (math-factor-poly-coefs
796 (math-accum-factors (math-mul cabs csign) 1 t2))))
798 ;; Factoring modulo a prime.
799 ((and (= (length (setq temp (math-poly-gcd-coefs
800 p (math-poly-deriv-coefs p))))
804 (setq temp (nthcdr (nth 2 math-poly-modulus) temp)
805 p (cons (car temp) p)))
806 (and (setq temp (math-factor-poly-coefs p))
807 (math-pow temp (nth 2 math-poly-modulus))))
809 (math-reject-arg nil "*Modulo factorization not yet implemented")))))
811 (defun math-poly-deriv-coefs (p)
814 (while (setq p (cdr p))
815 (setq dp (cons (math-mul (car p) n) dp)
819 (defun math-factor-contains (x a)
822 (if (memq (car-safe x) '(+ - * / neg))
824 (while (setq x (cdr x))
825 (setq sum (+ sum (math-factor-contains (car x) a))))
827 (if (and (eq (car-safe x) '^)
829 (* (math-factor-contains (nth 1 x) a) (nth 2 x))
836 ;;; Merge all quotients and expand/simplify the numerator
837 (defun calcFunc-nrat (expr)
838 (if (math-any-floats expr)
839 (setq expr (calcFunc-pfrac expr)))
840 (if (or (math-vectorp expr)
841 (assq (car-safe expr) calc-tweak-eqn-table))
842 (cons (car expr) (mapcar 'calcFunc-nrat (cdr expr)))
843 (let* ((calc-prefer-frac t)
844 (res (math-to-ratpoly expr))
845 (num (math-simplify (math-sort-terms (calcFunc-expand (car res)))))
846 (den (math-simplify (math-sort-terms (calcFunc-expand (cdr res)))))
847 (g (math-poly-gcd num den)))
849 (let ((num2 (math-poly-div num g))
850 (den2 (math-poly-div den g)))
851 (and (eq (cdr num2) 0) (eq (cdr den2) 0)
852 (setq num (car num2) den (car den2)))))
853 (math-simplify (math-div num den)))))
855 ;;; Returns expressions (num . denom).
856 (defun math-to-ratpoly (expr)
857 (let ((res (math-to-ratpoly-rec expr)))
858 (cons (math-simplify (car res)) (math-simplify (cdr res)))))
860 (defun math-to-ratpoly-rec (expr)
861 (cond ((Math-primp expr)
863 ((memq (car expr) '(+ -))
864 (let ((r1 (math-to-ratpoly-rec (nth 1 expr)))
865 (r2 (math-to-ratpoly-rec (nth 2 expr))))
866 (if (equal (cdr r1) (cdr r2))
867 (cons (list (car expr) (car r1) (car r2)) (cdr r1))
869 (cons (list (car expr)
870 (math-mul (car r1) (cdr r2))
874 (cons (list (car expr)
876 (math-mul (car r2) (cdr r1)))
878 (let ((g (math-poly-gcd (cdr r1) (cdr r2))))
879 (let ((d1 (and (not (eq g 1)) (math-poly-div (cdr r1) g)))
880 (d2 (and (not (eq g 1)) (math-poly-div
881 (math-mul (car r1) (cdr r2))
883 (if (and (eq (cdr d1) 0) (eq (cdr d2) 0))
884 (cons (list (car expr) (car d2)
885 (math-mul (car r2) (car d1)))
886 (math-mul (car d1) (cdr r2)))
887 (cons (list (car expr)
888 (math-mul (car r1) (cdr r2))
889 (math-mul (car r2) (cdr r1)))
890 (math-mul (cdr r1) (cdr r2)))))))))))
892 (let* ((r1 (math-to-ratpoly-rec (nth 1 expr)))
893 (r2 (math-to-ratpoly-rec (nth 2 expr)))
894 (g (math-mul (math-poly-gcd (car r1) (cdr r2))
895 (math-poly-gcd (cdr r1) (car r2)))))
897 (cons (math-mul (car r1) (car r2))
898 (math-mul (cdr r1) (cdr r2)))
899 (cons (math-poly-div-exact (math-mul (car r1) (car r2)) g)
900 (math-poly-div-exact (math-mul (cdr r1) (cdr r2)) g)))))
902 (let* ((r1 (math-to-ratpoly-rec (nth 1 expr)))
903 (r2 (math-to-ratpoly-rec (nth 2 expr))))
904 (if (and (eq (cdr r1) 1) (eq (cdr r2) 1))
905 (cons (car r1) (car r2))
906 (let ((g (math-mul (math-poly-gcd (car r1) (car r2))
907 (math-poly-gcd (cdr r1) (cdr r2)))))
909 (cons (math-mul (car r1) (cdr r2))
910 (math-mul (cdr r1) (car r2)))
911 (cons (math-poly-div-exact (math-mul (car r1) (cdr r2)) g)
912 (math-poly-div-exact (math-mul (cdr r1) (car r2))
914 ((and (eq (car expr) '^) (integerp (nth 2 expr)))
915 (let ((r1 (math-to-ratpoly-rec (nth 1 expr))))
916 (if (> (nth 2 expr) 0)
917 (cons (math-pow (car r1) (nth 2 expr))
918 (math-pow (cdr r1) (nth 2 expr)))
919 (cons (math-pow (cdr r1) (- (nth 2 expr)))
920 (math-pow (car r1) (- (nth 2 expr)))))))
921 ((eq (car expr) 'neg)
922 (let ((r1 (math-to-ratpoly-rec (nth 1 expr))))
923 (cons (math-neg (car r1)) (cdr r1))))
927 (defun math-ratpoly-p (expr &optional var)
928 (cond ((equal expr var) 1)
929 ((Math-primp expr) 0)
930 ((memq (car expr) '(+ -))
931 (let ((p1 (math-ratpoly-p (nth 1 expr) var))
933 (and p1 (setq p2 (math-ratpoly-p (nth 2 expr) var))
936 (let ((p1 (math-ratpoly-p (nth 1 expr) var))
938 (and p1 (setq p2 (math-ratpoly-p (nth 2 expr) var))
940 ((eq (car expr) 'neg)
941 (math-ratpoly-p (nth 1 expr) var))
943 (let ((p1 (math-ratpoly-p (nth 1 expr) var))
945 (and p1 (setq p2 (math-ratpoly-p (nth 2 expr) var))
947 ((and (eq (car expr) '^)
948 (integerp (nth 2 expr)))
949 (let ((p1 (math-ratpoly-p (nth 1 expr) var)))
950 (and p1 (* p1 (nth 2 expr)))))
952 ((math-poly-depends expr var) nil)
956 (defun calcFunc-apart (expr &optional var)
957 (cond ((Math-primp expr) expr)
959 (math-add (calcFunc-apart (nth 1 expr) var)
960 (calcFunc-apart (nth 2 expr) var)))
962 (math-sub (calcFunc-apart (nth 1 expr) var)
963 (calcFunc-apart (nth 2 expr) var)))
964 ((not (math-ratpoly-p expr var))
965 (math-reject-arg expr "Expected a rational function"))
967 (let* ((calc-prefer-frac t)
968 (rat (math-to-ratpoly expr))
971 (qr (math-poly-div num den))
975 (setq var (math-polynomial-base den)))
976 (math-add q (or (and var
977 (math-expr-contains den var)
978 (math-partial-fractions r den var))
979 (math-div r den)))))))
982 (defun math-padded-polynomial (expr var deg)
983 (let ((p (math-is-polynomial expr var deg)))
984 (append p (make-list (- deg (length p)) 0))))
986 (defun math-partial-fractions (r den var)
987 (let* ((fden (calcFunc-factors den var))
988 (tdeg (math-polynomial-p den var))
993 (tz (make-list (1- tdeg) 0))
994 (calc-matrix-mode 'scalar))
995 (and (not (and (= (length fden) 2) (eq (nth 2 (nth 1 fden)) 1)))
997 (while (setq fp (cdr fp))
998 (let ((rpt (nth 2 (car fp)))
999 (deg (math-polynomial-p (nth 1 (car fp)) var))
1005 (setq dvar (append '(vec) lz '(1) tz)
1009 dnum (math-add dnum (math-mul dvar
1010 (math-pow var deg2)))
1011 dlist (cons (and (= deg2 (1- deg))
1012 (math-pow (nth 1 (car fp)) rpt))
1016 (while (setq fpp (cdr fpp))
1018 (setq mult (math-mul mult
1019 (math-pow (nth 1 (car fpp))
1020 (nth 2 (car fpp)))))))
1021 (setq dnum (math-mul dnum mult)))
1022 (setq eqns (math-add eqns (math-mul dnum
1028 (setq eqns (math-div (cons 'vec (math-padded-polynomial r var tdeg))
1034 (cons 'vec (math-padded-polynomial
1037 (and (math-vectorp eqns)
1040 (setq eqns (nreverse eqns))
1042 (setq num (cons (car eqns) num)
1045 (setq num (math-build-polynomial-expr
1047 res (math-add res (math-div num (car dlist)))
1049 (setq dlist (cdr dlist)))
1050 (math-normalize res)))))))
1054 (defun math-expand-term (expr)
1055 (cond ((and (eq (car-safe expr) '*)
1056 (memq (car-safe (nth 1 expr)) '(+ -)))
1057 (math-add-or-sub (list '* (nth 1 (nth 1 expr)) (nth 2 expr))
1058 (list '* (nth 2 (nth 1 expr)) (nth 2 expr))
1059 nil (eq (car (nth 1 expr)) '-)))
1060 ((and (eq (car-safe expr) '*)
1061 (memq (car-safe (nth 2 expr)) '(+ -)))
1062 (math-add-or-sub (list '* (nth 1 expr) (nth 1 (nth 2 expr)))
1063 (list '* (nth 1 expr) (nth 2 (nth 2 expr)))
1064 nil (eq (car (nth 2 expr)) '-)))
1065 ((and (eq (car-safe expr) '/)
1066 (memq (car-safe (nth 1 expr)) '(+ -)))
1067 (math-add-or-sub (list '/ (nth 1 (nth 1 expr)) (nth 2 expr))
1068 (list '/ (nth 2 (nth 1 expr)) (nth 2 expr))
1069 nil (eq (car (nth 1 expr)) '-)))
1070 ((and (eq (car-safe expr) '^)
1071 (memq (car-safe (nth 1 expr)) '(+ -))
1072 (integerp (nth 2 expr))
1073 (if (> (nth 2 expr) 0)
1074 (or (and (or (> math-mt-many 500000) (< math-mt-many -500000))
1075 (math-expand-power (nth 1 expr) (nth 2 expr)
1079 (list '^ (nth 1 expr) (1- (nth 2 expr)))))
1080 (if (< (nth 2 expr) 0)
1081 (list '/ 1 (list '^ (nth 1 expr) (- (nth 2 expr))))))))
1084 (defun calcFunc-expand (expr &optional many)
1085 (math-normalize (math-map-tree 'math-expand-term expr many)))
1087 (defun math-expand-power (x n &optional var else-nil)
1088 (or (and (natnump n)
1089 (memq (car-safe x) '(+ -))
1092 (while (memq (car-safe x) '(+ -))
1093 (setq terms (cons (if (eq (car x) '-)
1094 (math-neg (nth 2 x))
1098 (setq terms (cons x terms))
1102 (or (math-expr-contains (car p) var)
1103 (setq terms (delq (car p) terms)
1104 cterms (cons (car p) cterms)))
1107 (setq terms (cons (apply 'calcFunc-add cterms)
1109 (if (= (length terms) 2)
1113 (setq accum (list '+ accum
1114 (list '* (calcFunc-choose n i)
1116 (list '^ (nth 1 terms) i)
1117 (list '^ (car terms)
1126 (setq accum (list '+ accum
1127 (list '^ (car p1) 2))
1129 (while (setq p2 (cdr p2))
1130 (setq accum (list '+ accum
1141 (setq accum (list '+ accum (list '^ (car p1) 3))
1143 (while (setq p2 (cdr p2))
1144 (setq accum (list '+
1150 (list '^ (car p1) 2)
1155 (list '^ (car p2) 2))))
1157 (while (setq p3 (cdr p3))
1158 (setq accum (list '+ accum
1170 (defun calcFunc-expandpow (x n)
1171 (math-normalize (math-expand-power x n)))
1173 (provide 'calc-poly)
1175 ;;; arch-tag: d2566c51-2ccc-45f1-8c50-f3462c2953ff
1176 ;;; calc-poly.el ends here