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 ;; Maintainers: D. Goel <deego@gnufans.org>
7 ;; Colin Walters <walters@debian.org>
9 ;; This file is part of GNU Emacs.
11 ;; GNU Emacs is distributed in the hope that it will be useful,
12 ;; but WITHOUT ANY WARRANTY. No author or distributor
13 ;; accepts responsibility to anyone for the consequences of using it
14 ;; or for whether it serves any particular purpose or works at all,
15 ;; unless he says so in writing. Refer to the GNU Emacs General Public
16 ;; License for full details.
18 ;; Everyone is granted permission to copy, modify and redistribute
19 ;; GNU Emacs, but only under the conditions described in the
20 ;; GNU Emacs General Public License. A copy of this license is
21 ;; supposed to have been given to you along with GNU Emacs so you
22 ;; can know your rights and responsibilities. It should be in a
23 ;; file named COPYING. Among other things, the copyright notice
24 ;; and this notice must be preserved on all copies.
30 ;; This file is autoloaded from calc-ext.el.
35 (defun calc-Need-calc-poly () nil)
38 (defun calcFunc-pcont (expr &optional var)
39 (cond ((Math-primp expr)
40 (cond ((Math-zerop expr) 1)
41 ((Math-messy-integerp expr) (math-trunc expr))
42 ((Math-objectp expr) expr)
43 ((or (equal expr var) (not var)) 1)
46 (math-mul (calcFunc-pcont (nth 1 expr) var)
47 (calcFunc-pcont (nth 2 expr) var)))
49 (math-div (calcFunc-pcont (nth 1 expr) var)
50 (calcFunc-pcont (nth 2 expr) var)))
51 ((and (eq (car expr) '^) (Math-natnump (nth 2 expr)))
52 (math-pow (calcFunc-pcont (nth 1 expr) var) (nth 2 expr)))
53 ((memq (car expr) '(neg polar))
54 (calcFunc-pcont (nth 1 expr) var))
56 (let ((p (math-is-polynomial expr var)))
58 (let ((lead (nth (1- (length p)) p))
59 (cont (math-poly-gcd-list p)))
60 (if (math-guess-if-neg lead)
64 ((memq (car expr) '(+ - cplx sdev))
65 (let ((cont (calcFunc-pcont (nth 1 expr) var)))
68 (let ((c2 (calcFunc-pcont (nth 2 expr) var)))
69 (if (and (math-negp cont)
70 (if (eq (car expr) '-) (math-posp c2) (math-negp c2)))
71 (math-neg (math-poly-gcd cont c2))
72 (math-poly-gcd cont c2))))))
76 (defun calcFunc-pprim (expr &optional var)
77 (let ((cont (calcFunc-pcont expr var)))
78 (if (math-equal-int cont 1)
80 (math-poly-div-exact expr cont var))))
82 (defun math-div-poly-const (expr c)
83 (cond ((memq (car-safe expr) '(+ -))
85 (math-div-poly-const (nth 1 expr) c)
86 (math-div-poly-const (nth 2 expr) c)))
87 (t (math-div expr c))))
89 (defun calcFunc-pdeg (expr &optional var)
91 '(neg (var inf var-inf))
93 (or (math-polynomial-p expr var)
94 (math-reject-arg expr "Expected a polynomial"))
95 (math-poly-degree expr))))
97 (defun math-poly-degree (expr)
98 (cond ((Math-primp expr)
99 (if (eq (car-safe expr) 'var) 1 0))
100 ((eq (car expr) 'neg)
101 (math-poly-degree (nth 1 expr)))
103 (+ (math-poly-degree (nth 1 expr))
104 (math-poly-degree (nth 2 expr))))
106 (- (math-poly-degree (nth 1 expr))
107 (math-poly-degree (nth 2 expr))))
108 ((and (eq (car expr) '^) (natnump (nth 2 expr)))
109 (* (math-poly-degree (nth 1 expr)) (nth 2 expr)))
110 ((memq (car expr) '(+ -))
111 (max (math-poly-degree (nth 1 expr))
112 (math-poly-degree (nth 2 expr))))
115 (defun calcFunc-plead (expr var)
116 (cond ((eq (car-safe expr) '*)
117 (math-mul (calcFunc-plead (nth 1 expr) var)
118 (calcFunc-plead (nth 2 expr) var)))
119 ((eq (car-safe expr) '/)
120 (math-div (calcFunc-plead (nth 1 expr) var)
121 (calcFunc-plead (nth 2 expr) var)))
122 ((and (eq (car-safe expr) '^) (math-natnump (nth 2 expr)))
123 (math-pow (calcFunc-plead (nth 1 expr) var) (nth 2 expr)))
129 (let ((p (math-is-polynomial expr var)))
131 (nth (1- (length p)) p)
138 ;;; Polynomial quotient, remainder, and GCD.
139 ;;; Originally by Ove Ewerlid (ewerlid@mizar.DoCS.UU.SE).
140 ;;; Modifications and simplifications by daveg.
142 (defvar math-poly-modulus 1)
144 ;;; Return gcd of two polynomials
145 (defun calcFunc-pgcd (pn pd)
146 (if (math-any-floats pn)
147 (math-reject-arg pn "Coefficients must be rational"))
148 (if (math-any-floats pd)
149 (math-reject-arg pd "Coefficients must be rational"))
150 (let ((calc-prefer-frac t)
151 (math-poly-modulus (math-poly-modulus pn pd)))
152 (math-poly-gcd pn pd)))
154 ;;; Return only quotient to top of stack (nil if zero)
155 (defun calcFunc-pdiv (pn pd &optional base)
156 (let* ((calc-prefer-frac t)
157 (math-poly-modulus (math-poly-modulus pn pd))
158 (res (math-poly-div pn pd base)))
159 (setq calc-poly-div-remainder (cdr res))
162 ;;; Return only remainder to top of stack
163 (defun calcFunc-prem (pn pd &optional base)
164 (let ((calc-prefer-frac t)
165 (math-poly-modulus (math-poly-modulus pn pd)))
166 (cdr (math-poly-div pn pd base))))
168 (defun calcFunc-pdivrem (pn pd &optional base)
169 (let* ((calc-prefer-frac t)
170 (math-poly-modulus (math-poly-modulus pn pd))
171 (res (math-poly-div pn pd base)))
172 (list 'vec (car res) (cdr res))))
174 (defun calcFunc-pdivide (pn pd &optional base)
175 (let* ((calc-prefer-frac t)
176 (math-poly-modulus (math-poly-modulus pn pd))
177 (res (math-poly-div pn pd base)))
178 (math-add (car res) (math-div (cdr res) pd))))
181 ;;; Multiply two terms, expanding out products of sums.
182 (defun math-mul-thru (lhs rhs)
183 (if (memq (car-safe lhs) '(+ -))
185 (math-mul-thru (nth 1 lhs) rhs)
186 (math-mul-thru (nth 2 lhs) rhs))
187 (if (memq (car-safe rhs) '(+ -))
189 (math-mul-thru lhs (nth 1 rhs))
190 (math-mul-thru lhs (nth 2 rhs)))
191 (math-mul lhs rhs))))
193 (defun math-div-thru (num den)
194 (if (memq (car-safe num) '(+ -))
196 (math-div-thru (nth 1 num) den)
197 (math-div-thru (nth 2 num) den))
201 ;;; Sort the terms of a sum into canonical order.
202 (defun math-sort-terms (expr)
203 (if (memq (car-safe expr) '(+ -))
205 (sort (math-sum-to-list expr)
206 (function (lambda (a b) (math-beforep (car a) (car b))))))
209 (defun math-list-to-sum (lst)
211 (list (if (cdr (car lst)) '- '+)
212 (math-list-to-sum (cdr lst))
215 (math-neg (car (car lst)))
218 (defun math-sum-to-list (tree &optional neg)
219 (cond ((eq (car-safe tree) '+)
220 (nconc (math-sum-to-list (nth 1 tree) neg)
221 (math-sum-to-list (nth 2 tree) neg)))
222 ((eq (car-safe tree) '-)
223 (nconc (math-sum-to-list (nth 1 tree) neg)
224 (math-sum-to-list (nth 2 tree) (not neg))))
225 (t (list (cons tree neg)))))
227 ;;; Check if the polynomial coefficients are modulo forms.
228 (defun math-poly-modulus (expr &optional expr2)
229 (or (math-poly-modulus-rec expr)
230 (and expr2 (math-poly-modulus-rec expr2))
233 (defun math-poly-modulus-rec (expr)
234 (if (and (eq (car-safe expr) 'mod) (Math-natnump (nth 2 expr)))
235 (list 'mod 1 (nth 2 expr))
236 (and (memq (car-safe expr) '(+ - * /))
237 (or (math-poly-modulus-rec (nth 1 expr))
238 (math-poly-modulus-rec (nth 2 expr))))))
241 ;;; Divide two polynomials. Return (quotient . remainder).
242 (defvar math-poly-div-base nil)
243 (defun math-poly-div (u v &optional math-poly-div-base)
244 (if math-poly-div-base
245 (math-do-poly-div u v)
246 (math-do-poly-div (calcFunc-expand u) (calcFunc-expand v))))
248 (defun math-poly-div-exact (u v &optional base)
249 (let ((res (math-poly-div u v base)))
252 (math-reject-arg (list 'vec u v) "Argument is not a polynomial"))))
254 (defun math-do-poly-div (u v)
255 (cond ((math-constp u)
257 (cons (math-div u v) 0)
262 (if (memq (car-safe u) '(+ -))
263 (math-add-or-sub (math-poly-div-exact (nth 1 u) v)
264 (math-poly-div-exact (nth 2 u) v)
269 (cons math-poly-modulus 0))
270 ((and (math-atomic-factorp u) (math-atomic-factorp v))
271 (cons (math-simplify (math-div u v)) 0))
273 (let ((base (or math-poly-div-base
274 (math-poly-div-base u v)))
277 (null (setq vp (math-is-polynomial v base nil 'gen))))
279 (setq up (math-is-polynomial u base nil 'gen)
280 res (math-poly-div-coefs up vp))
281 (cons (math-build-polynomial-expr (car res) base)
282 (math-build-polynomial-expr (cdr res) base)))))))
284 (defun math-poly-div-rec (u v)
285 (cond ((math-constp u)
290 (if (memq (car-safe u) '(+ -))
291 (math-add-or-sub (math-poly-div-rec (nth 1 u) v)
292 (math-poly-div-rec (nth 2 u) v)
295 ((Math-equal u v) math-poly-modulus)
296 ((and (math-atomic-factorp u) (math-atomic-factorp v))
297 (math-simplify (math-div u v)))
301 (let ((base (math-poly-div-base u v))
304 (null (setq vp (math-is-polynomial v base nil 'gen))))
306 (setq up (math-is-polynomial u base nil 'gen)
307 res (math-poly-div-coefs up vp))
308 (math-add (math-build-polynomial-expr (car res) base)
309 (math-div (math-build-polynomial-expr (cdr res) base)
312 ;;; Divide two polynomials in coefficient-list form. Return (quot . rem).
313 (defun math-poly-div-coefs (u v)
314 (cond ((null v) (math-reject-arg nil "Division by zero"))
315 ((< (length u) (length v)) (cons nil u))
321 (let ((qk (math-poly-div-rec (math-simplify (car urev))
325 (if (or q (not (math-zerop qk)))
326 (setq q (cons qk q)))
327 (while (setq up (cdr up) vp (cdr vp))
328 (setcar up (math-sub (car up) (math-mul-thru qk (car vp)))))
329 (setq urev (cdr urev))
331 (while (and urev (Math-zerop (car urev)))
332 (setq urev (cdr urev)))
333 (cons q (nreverse (mapcar 'math-simplify urev)))))
335 (cons (list (math-poly-div-rec (car u) (car v)))
338 ;;; Perform a pseudo-division of polynomials. (See Knuth section 4.6.1.)
339 ;;; This returns only the remainder from the pseudo-division.
340 (defun math-poly-pseudo-div (u v)
342 ((< (length u) (length v)) u)
343 ((or (cdr u) (cdr v))
344 (let ((urev (reverse u))
350 (while (setq up (cdr up) vp (cdr vp))
351 (setcar up (math-sub (math-mul-thru (car vrev) (car up))
352 (math-mul-thru (car urev) (car vp)))))
353 (setq urev (cdr urev))
356 (setcar up (math-mul-thru (car vrev) (car up)))
358 (while (and urev (Math-zerop (car urev)))
359 (setq urev (cdr urev)))
360 (nreverse (mapcar 'math-simplify urev))))
363 ;;; Compute the GCD of two multivariate polynomials.
364 (defun math-poly-gcd (u v)
365 (cond ((Math-equal u v) u)
369 (calcFunc-gcd u (calcFunc-pcont v))))
373 (calcFunc-gcd v (calcFunc-pcont u))))
375 (let ((base (math-poly-gcd-base u v)))
379 (math-build-polynomial-expr
380 (math-poly-gcd-coefs (math-is-polynomial u base nil 'gen)
381 (math-is-polynomial v base nil 'gen))
383 (calcFunc-gcd (calcFunc-pcont u) (calcFunc-pcont u)))))))
385 (defun math-poly-div-list (lst a)
389 (math-mul-list lst a)
390 (mapcar (function (lambda (x) (math-poly-div-exact x a))) lst))))
392 (defun math-mul-list (lst a)
396 (mapcar 'math-neg lst)
398 (mapcar (function (lambda (x) (math-mul x a))) lst)))))
400 ;;; Run GCD on all elements in a list.
401 (defun math-poly-gcd-list (lst)
402 (if (or (memq 1 lst) (memq -1 lst))
403 (math-poly-gcd-frac-list lst)
404 (let ((gcd (car lst)))
405 (while (and (setq lst (cdr lst)) (not (eq gcd 1)))
407 (setq gcd (math-poly-gcd gcd (car lst)))))
408 (if lst (setq lst (math-poly-gcd-frac-list lst)))
411 (defun math-poly-gcd-frac-list (lst)
412 (while (and lst (not (eq (car-safe (car lst)) 'frac)))
413 (setq lst (cdr lst)))
415 (let ((denom (nth 2 (car lst))))
416 (while (setq lst (cdr lst))
417 (if (eq (car-safe (car lst)) 'frac)
418 (setq denom (calcFunc-lcm denom (nth 2 (car lst))))))
419 (list 'frac 1 denom))
422 ;;; Compute the GCD of two monovariate polynomial lists.
423 ;;; Knuth section 4.6.1, algorithm C.
424 (defun math-poly-gcd-coefs (u v)
425 (let ((d (math-poly-gcd (math-poly-gcd-list u)
426 (math-poly-gcd-list v)))
427 (g 1) (h 1) (z 0) hh r delta ghd)
428 (while (and u v (Math-zerop (car u)) (Math-zerop (car v)))
429 (setq u (cdr u) v (cdr v) z (1+ z)))
431 (setq u (math-poly-div-list u d)
432 v (math-poly-div-list v d)))
434 (setq delta (- (length u) (length v)))
436 (setq r u u v v r delta (- delta)))
437 (setq r (math-poly-pseudo-div u v))
440 v (math-poly-div-list r (math-mul g (math-pow h delta)))
441 g (nth (1- (length u)) u)
443 (math-mul (math-pow g delta) (math-pow h (- 1 delta)))
444 (math-poly-div-exact (math-pow g delta)
445 (math-pow h (1- delta))))))
448 (math-mul-list (math-poly-div-list v (math-poly-gcd-list v)) d)))
449 (if (math-guess-if-neg (nth (1- (length v)) v))
450 (setq v (math-mul-list v -1)))
451 (while (>= (setq z (1- z)) 0)
456 ;;; Return true if is a factor containing no sums or quotients.
457 (defun math-atomic-factorp (expr)
458 (cond ((eq (car-safe expr) '*)
459 (and (math-atomic-factorp (nth 1 expr))
460 (math-atomic-factorp (nth 2 expr))))
461 ((memq (car-safe expr) '(+ - /))
463 ((memq (car-safe expr) '(^ neg))
464 (math-atomic-factorp (nth 1 expr)))
467 ;;; Find a suitable base for dividing a by b.
468 ;;; The base must exist in both expressions.
469 ;;; The degree in the numerator must be higher or equal than the
470 ;;; degree in the denominator.
471 ;;; If the above conditions are not met the quotient is just a remainder.
472 ;;; Return nil if this is the case.
474 (defun math-poly-div-base (a b)
476 (and (setq a-base (math-total-polynomial-base a))
477 (setq b-base (math-total-polynomial-base b))
480 (let ((maybe (assoc (car (car a-base)) b-base)))
482 (if (>= (nth 1 (car a-base)) (nth 1 maybe))
483 (throw 'return (car (car a-base))))))
484 (setq a-base (cdr a-base)))))))
486 ;;; Same as above but for gcd algorithm.
487 ;;; Here there is no requirement that degree(a) > degree(b).
488 ;;; Take the base that has the highest degree considering both a and b.
489 ;;; ("a^20+b^21+x^3+a+b", "a+b^2+x^5+a^22+b^10") --> (a 22)
491 (defun math-poly-gcd-base (a b)
493 (and (setq a-base (math-total-polynomial-base a))
494 (setq b-base (math-total-polynomial-base b))
496 (while (and a-base b-base)
497 (if (> (nth 1 (car a-base)) (nth 1 (car b-base)))
498 (if (assoc (car (car a-base)) b-base)
499 (throw 'return (car (car a-base)))
500 (setq a-base (cdr a-base)))
501 (if (assoc (car (car b-base)) a-base)
502 (throw 'return (car (car b-base)))
503 (setq b-base (cdr b-base)))))))))
505 ;;; Sort a list of polynomial bases.
506 (defun math-sort-poly-base-list (lst)
507 (sort lst (function (lambda (a b)
508 (or (> (nth 1 a) (nth 1 b))
509 (and (= (nth 1 a) (nth 1 b))
510 (math-beforep (car a) (car b))))))))
512 ;;; Given an expression find all variables that are polynomial bases.
513 ;;; Return list in the form '( (var1 degree1) (var2 degree2) ... ).
514 ;;; Note dynamic scope of mpb-total-base.
515 (defun math-total-polynomial-base (expr)
516 (let ((mpb-total-base nil))
517 (math-polynomial-base expr 'math-polynomial-p1)
518 (math-sort-poly-base-list mpb-total-base)))
520 (defun math-polynomial-p1 (subexpr)
521 (or (assoc subexpr mpb-total-base)
522 (memq (car subexpr) '(+ - * / neg))
523 (and (eq (car subexpr) '^) (natnump (nth 2 subexpr)))
524 (let* ((math-poly-base-variable subexpr)
525 (exponent (math-polynomial-p mpb-top-expr subexpr)))
527 (setq mpb-total-base (cons (list subexpr exponent)
534 (defun calcFunc-factors (expr &optional var)
535 (let ((math-factored-vars (if var t nil))
537 (calc-prefer-frac t))
539 (setq var (math-polynomial-base expr)))
540 (let ((res (math-factor-finish
541 (or (catch 'factor (math-factor-expr-try var))
543 (math-simplify (if (math-vectorp res)
545 (list 'vec (list 'vec res 1)))))))
547 (defun calcFunc-factor (expr &optional var)
548 (let ((math-factored-vars nil)
550 (calc-prefer-frac t))
551 (math-simplify (math-factor-finish
553 (let ((math-factored-vars t))
554 (or (catch 'factor (math-factor-expr-try var)) expr))
555 (math-factor-expr expr))))))
557 (defun math-factor-finish (x)
560 (if (eq (car x) 'calcFunc-Fac-Prot)
561 (math-factor-finish (nth 1 x))
562 (cons (car x) (mapcar 'math-factor-finish (cdr x))))))
564 (defun math-factor-protect (x)
565 (if (memq (car-safe x) '(+ -))
566 (list 'calcFunc-Fac-Prot x)
569 (defun math-factor-expr (expr)
570 (cond ((eq math-factored-vars t) expr)
571 ((or (memq (car-safe expr) '(* / ^ neg))
572 (assq (car-safe expr) calc-tweak-eqn-table))
573 (cons (car expr) (mapcar 'math-factor-expr (cdr expr))))
574 ((memq (car-safe expr) '(+ -))
575 (let* ((math-factored-vars math-factored-vars)
576 (y (catch 'factor (math-factor-expr-part expr))))
582 (defun math-factor-expr-part (x) ; uses "expr"
583 (if (memq (car-safe x) '(+ - * / ^ neg))
584 (while (setq x (cdr x))
585 (math-factor-expr-part (car x)))
586 (and (not (Math-objvecp x))
587 (not (assoc x math-factored-vars))
588 (> (math-factor-contains expr x) 1)
589 (setq math-factored-vars (cons (list x) math-factored-vars))
590 (math-factor-expr-try x))))
592 (defun math-factor-expr-try (x)
593 (if (eq (car-safe expr) '*)
594 (let ((res1 (catch 'factor (let ((expr (nth 1 expr)))
595 (math-factor-expr-try x))))
596 (res2 (catch 'factor (let ((expr (nth 2 expr)))
597 (math-factor-expr-try x)))))
599 (throw 'factor (math-accum-factors (or res1 (nth 1 expr)) 1
600 (or res2 (nth 2 expr))))))
601 (let* ((p (math-is-polynomial expr x 30 'gen))
602 (math-poly-modulus (math-poly-modulus expr))
605 (setq res (math-factor-poly-coefs p))
606 (throw 'factor res)))))
608 (defun math-accum-factors (fac pow facs)
610 (if (math-vectorp fac)
612 (while (setq fac (cdr fac))
613 (setq facs (math-accum-factors (nth 1 (car fac))
614 (* pow (nth 2 (car fac)))
617 (if (and (eq (car-safe fac) '^) (natnump (nth 2 fac)))
618 (setq pow (* pow (nth 2 fac))
622 (or (math-vectorp facs)
623 (setq facs (if (eq facs 1) '(vec)
624 (list 'vec (list 'vec facs 1)))))
626 (while (and (setq found (cdr found))
627 (not (equal fac (nth 1 (car found))))))
630 (setcar (cdr (cdr (car found))) (+ pow (nth 2 (car found))))
632 ;; Put constant term first.
633 (if (and (cdr facs) (Math-ratp (nth 1 (nth 1 facs))))
634 (cons 'vec (cons (nth 1 facs) (cons (list 'vec fac pow)
636 (cons 'vec (cons (list 'vec fac pow) (cdr facs))))))))
637 (math-mul (math-pow fac pow) facs)))
639 (defun math-factor-poly-coefs (p &optional square-free) ; uses "x"
644 ;; Strip off multiples of x.
645 ((Math-zerop (car p))
647 (while (and p (Math-zerop (car p)))
648 (setq z (1+ z) p (cdr p)))
650 (setq p (math-factor-poly-coefs p square-free))
651 (setq p (math-sort-terms (math-factor-expr (car p)))))
652 (math-accum-factors x z (math-factor-protect p))))
654 ;; Factor out content.
655 ((and (not square-free)
656 (not (eq 1 (setq t1 (math-mul (math-poly-gcd-list p)
657 (if (math-guess-if-neg
658 (nth (1- (length p)) p))
660 (math-accum-factors t1 1 (math-factor-poly-coefs
661 (math-poly-div-list p t1) 'cont)))
663 ;; Check if linear in x.
665 (math-add (math-factor-protect
667 (math-factor-expr (car p))))
668 (math-mul x (math-factor-protect
670 (math-factor-expr (nth 1 p)))))))
672 ;; If symbolic coefficients, use FactorRules.
674 (while (and pp (or (Math-ratp (car pp))
675 (and (eq (car (car pp)) 'mod)
676 (Math-integerp (nth 1 (car pp)))
677 (Math-integerp (nth 2 (car pp))))))
680 (let ((res (math-rewrite
681 (list 'calcFunc-thecoefs x (cons 'vec p))
682 '(var FactorRules var-FactorRules))))
683 (or (and (eq (car-safe res) 'calcFunc-thefactors)
685 (math-vectorp (nth 2 res))
688 (while (setq vec (cdr vec))
689 (setq facs (math-accum-factors (car vec) 1 facs)))
691 (math-build-polynomial-expr p x))))
693 ;; Check if rational coefficients (i.e., not modulo a prime).
694 ((eq math-poly-modulus 1)
696 ;; Check if there are any squared terms, or a content not = 1.
697 (if (or (eq square-free t)
698 (equal (setq t1 (math-poly-gcd-coefs
699 p (setq t2 (math-poly-deriv-coefs p))))
702 ;; We now have a square-free polynomial with integer coefs.
703 ;; For now, we use a kludgey method that finds linear and
704 ;; quadratic terms using floating-point root-finding.
705 (if (setq t1 (let ((calc-symbolic-mode nil))
706 (math-poly-all-roots nil p t)))
707 (let ((roots (car t1))
708 (csign (if (math-negp (nth (1- (length p)) p)) -1 1))
713 (let ((coef0 (car (car roots)))
714 (coef1 (cdr (car roots))))
715 (setq expr (math-accum-factors
717 (let ((den (math-lcm-denoms
719 (setq scale (math-div scale den))
722 (math-mul den (math-pow x 2))
723 (math-mul (math-mul coef1 den) x))
724 (math-mul coef0 den)))
725 (let ((den (math-lcm-denoms coef0)))
726 (setq scale (math-div scale den))
727 (math-add (math-mul den x)
728 (math-mul coef0 den))))
731 (setq expr (math-accum-factors
734 (math-build-polynomial-expr
735 (math-mul-list (nth 1 t1) scale)
737 (math-build-polynomial-expr p x)) ; can't factor it.
739 ;; Separate out the squared terms (Knuth exercise 4.6.2-34).
740 ;; This step also divides out the content of the polynomial.
741 (let* ((cabs (math-poly-gcd-list p))
742 (csign (if (math-negp (nth (1- (length p)) p)) -1 1))
743 (t1s (math-mul-list t1 csign))
745 (v (car (math-poly-div-coefs p t1s)))
746 (w (car (math-poly-div-coefs t2 t1s))))
748 (not (math-poly-zerop
749 (setq t2 (math-poly-simplify
751 w 1 (math-poly-deriv-coefs v) -1)))))
752 (setq t1 (math-poly-gcd-coefs v t2)
754 v (car (math-poly-div-coefs v t1))
755 w (car (math-poly-div-coefs t2 t1))))
757 t2 (math-accum-factors (math-factor-poly-coefs v t)
760 (setq t2 (math-accum-factors (math-factor-poly-coefs
765 (math-accum-factors (math-mul cabs csign) 1 t2))))
767 ;; Factoring modulo a prime.
768 ((and (= (length (setq temp (math-poly-gcd-coefs
769 p (math-poly-deriv-coefs p))))
773 (setq temp (nthcdr (nth 2 math-poly-modulus) temp)
774 p (cons (car temp) p)))
775 (and (setq temp (math-factor-poly-coefs p))
776 (math-pow temp (nth 2 math-poly-modulus))))
778 (math-reject-arg nil "*Modulo factorization not yet implemented")))))
780 (defun math-poly-deriv-coefs (p)
783 (while (setq p (cdr p))
784 (setq dp (cons (math-mul (car p) n) dp)
788 (defun math-factor-contains (x a)
791 (if (memq (car-safe x) '(+ - * / neg))
793 (while (setq x (cdr x))
794 (setq sum (+ sum (math-factor-contains (car x) a))))
796 (if (and (eq (car-safe x) '^)
798 (* (math-factor-contains (nth 1 x) a) (nth 2 x))
805 ;;; Merge all quotients and expand/simplify the numerator
806 (defun calcFunc-nrat (expr)
807 (if (math-any-floats expr)
808 (setq expr (calcFunc-pfrac expr)))
809 (if (or (math-vectorp expr)
810 (assq (car-safe expr) calc-tweak-eqn-table))
811 (cons (car expr) (mapcar 'calcFunc-nrat (cdr expr)))
812 (let* ((calc-prefer-frac t)
813 (res (math-to-ratpoly expr))
814 (num (math-simplify (math-sort-terms (calcFunc-expand (car res)))))
815 (den (math-simplify (math-sort-terms (calcFunc-expand (cdr res)))))
816 (g (math-poly-gcd num den)))
818 (let ((num2 (math-poly-div num g))
819 (den2 (math-poly-div den g)))
820 (and (eq (cdr num2) 0) (eq (cdr den2) 0)
821 (setq num (car num2) den (car den2)))))
822 (math-simplify (math-div num den)))))
824 ;;; Returns expressions (num . denom).
825 (defun math-to-ratpoly (expr)
826 (let ((res (math-to-ratpoly-rec expr)))
827 (cons (math-simplify (car res)) (math-simplify (cdr res)))))
829 (defun math-to-ratpoly-rec (expr)
830 (cond ((Math-primp expr)
832 ((memq (car expr) '(+ -))
833 (let ((r1 (math-to-ratpoly-rec (nth 1 expr)))
834 (r2 (math-to-ratpoly-rec (nth 2 expr))))
835 (if (equal (cdr r1) (cdr r2))
836 (cons (list (car expr) (car r1) (car r2)) (cdr r1))
838 (cons (list (car expr)
839 (math-mul (car r1) (cdr r2))
843 (cons (list (car expr)
845 (math-mul (car r2) (cdr r1)))
847 (let ((g (math-poly-gcd (cdr r1) (cdr r2))))
848 (let ((d1 (and (not (eq g 1)) (math-poly-div (cdr r1) g)))
849 (d2 (and (not (eq g 1)) (math-poly-div
850 (math-mul (car r1) (cdr r2))
852 (if (and (eq (cdr d1) 0) (eq (cdr d2) 0))
853 (cons (list (car expr) (car d2)
854 (math-mul (car r2) (car d1)))
855 (math-mul (car d1) (cdr r2)))
856 (cons (list (car expr)
857 (math-mul (car r1) (cdr r2))
858 (math-mul (car r2) (cdr r1)))
859 (math-mul (cdr r1) (cdr r2)))))))))))
861 (let* ((r1 (math-to-ratpoly-rec (nth 1 expr)))
862 (r2 (math-to-ratpoly-rec (nth 2 expr)))
863 (g (math-mul (math-poly-gcd (car r1) (cdr r2))
864 (math-poly-gcd (cdr r1) (car r2)))))
866 (cons (math-mul (car r1) (car r2))
867 (math-mul (cdr r1) (cdr r2)))
868 (cons (math-poly-div-exact (math-mul (car r1) (car r2)) g)
869 (math-poly-div-exact (math-mul (cdr r1) (cdr r2)) g)))))
871 (let* ((r1 (math-to-ratpoly-rec (nth 1 expr)))
872 (r2 (math-to-ratpoly-rec (nth 2 expr))))
873 (if (and (eq (cdr r1) 1) (eq (cdr r2) 1))
874 (cons (car r1) (car r2))
875 (let ((g (math-mul (math-poly-gcd (car r1) (car r2))
876 (math-poly-gcd (cdr r1) (cdr r2)))))
878 (cons (math-mul (car r1) (cdr r2))
879 (math-mul (cdr r1) (car r2)))
880 (cons (math-poly-div-exact (math-mul (car r1) (cdr r2)) g)
881 (math-poly-div-exact (math-mul (cdr r1) (car r2))
883 ((and (eq (car expr) '^) (integerp (nth 2 expr)))
884 (let ((r1 (math-to-ratpoly-rec (nth 1 expr))))
885 (if (> (nth 2 expr) 0)
886 (cons (math-pow (car r1) (nth 2 expr))
887 (math-pow (cdr r1) (nth 2 expr)))
888 (cons (math-pow (cdr r1) (- (nth 2 expr)))
889 (math-pow (car r1) (- (nth 2 expr)))))))
890 ((eq (car expr) 'neg)
891 (let ((r1 (math-to-ratpoly-rec (nth 1 expr))))
892 (cons (math-neg (car r1)) (cdr r1))))
896 (defun math-ratpoly-p (expr &optional var)
897 (cond ((equal expr var) 1)
898 ((Math-primp expr) 0)
899 ((memq (car expr) '(+ -))
900 (let ((p1 (math-ratpoly-p (nth 1 expr) var))
902 (and p1 (setq p2 (math-ratpoly-p (nth 2 expr) var))
905 (let ((p1 (math-ratpoly-p (nth 1 expr) var))
907 (and p1 (setq p2 (math-ratpoly-p (nth 2 expr) var))
909 ((eq (car expr) 'neg)
910 (math-ratpoly-p (nth 1 expr) var))
912 (let ((p1 (math-ratpoly-p (nth 1 expr) var))
914 (and p1 (setq p2 (math-ratpoly-p (nth 2 expr) var))
916 ((and (eq (car expr) '^)
917 (integerp (nth 2 expr)))
918 (let ((p1 (math-ratpoly-p (nth 1 expr) var)))
919 (and p1 (* p1 (nth 2 expr)))))
921 ((math-poly-depends expr var) nil)
925 (defun calcFunc-apart (expr &optional var)
926 (cond ((Math-primp expr) expr)
928 (math-add (calcFunc-apart (nth 1 expr) var)
929 (calcFunc-apart (nth 2 expr) var)))
931 (math-sub (calcFunc-apart (nth 1 expr) var)
932 (calcFunc-apart (nth 2 expr) var)))
933 ((not (math-ratpoly-p expr var))
934 (math-reject-arg expr "Expected a rational function"))
936 (let* ((calc-prefer-frac t)
937 (rat (math-to-ratpoly expr))
940 (qr (math-poly-div num den))
944 (setq var (math-polynomial-base den)))
945 (math-add q (or (and var
946 (math-expr-contains den var)
947 (math-partial-fractions r den var))
948 (math-div r den)))))))
951 (defun math-padded-polynomial (expr var deg)
952 (let ((p (math-is-polynomial expr var deg)))
953 (append p (make-list (- deg (length p)) 0))))
955 (defun math-partial-fractions (r den var)
956 (let* ((fden (calcFunc-factors den var))
957 (tdeg (math-polynomial-p den var))
962 (tz (make-list (1- tdeg) 0))
963 (calc-matrix-mode 'scalar))
964 (and (not (and (= (length fden) 2) (eq (nth 2 (nth 1 fden)) 1)))
966 (while (setq fp (cdr fp))
967 (let ((rpt (nth 2 (car fp)))
968 (deg (math-polynomial-p (nth 1 (car fp)) var))
974 (setq dvar (append '(vec) lz '(1) tz)
978 dnum (math-add dnum (math-mul dvar
979 (math-pow var deg2)))
980 dlist (cons (and (= deg2 (1- deg))
981 (math-pow (nth 1 (car fp)) rpt))
985 (while (setq fpp (cdr fpp))
987 (setq mult (math-mul mult
988 (math-pow (nth 1 (car fpp))
989 (nth 2 (car fpp)))))))
990 (setq dnum (math-mul dnum mult)))
991 (setq eqns (math-add eqns (math-mul dnum
997 (setq eqns (math-div (cons 'vec (math-padded-polynomial r var tdeg))
1003 (cons 'vec (math-padded-polynomial
1006 (and (math-vectorp eqns)
1009 (setq eqns (nreverse eqns))
1011 (setq num (cons (car eqns) num)
1014 (setq num (math-build-polynomial-expr
1016 res (math-add res (math-div num (car dlist)))
1018 (setq dlist (cdr dlist)))
1019 (math-normalize res)))))))
1023 (defun math-expand-term (expr)
1024 (cond ((and (eq (car-safe expr) '*)
1025 (memq (car-safe (nth 1 expr)) '(+ -)))
1026 (math-add-or-sub (list '* (nth 1 (nth 1 expr)) (nth 2 expr))
1027 (list '* (nth 2 (nth 1 expr)) (nth 2 expr))
1028 nil (eq (car (nth 1 expr)) '-)))
1029 ((and (eq (car-safe expr) '*)
1030 (memq (car-safe (nth 2 expr)) '(+ -)))
1031 (math-add-or-sub (list '* (nth 1 expr) (nth 1 (nth 2 expr)))
1032 (list '* (nth 1 expr) (nth 2 (nth 2 expr)))
1033 nil (eq (car (nth 2 expr)) '-)))
1034 ((and (eq (car-safe expr) '/)
1035 (memq (car-safe (nth 1 expr)) '(+ -)))
1036 (math-add-or-sub (list '/ (nth 1 (nth 1 expr)) (nth 2 expr))
1037 (list '/ (nth 2 (nth 1 expr)) (nth 2 expr))
1038 nil (eq (car (nth 1 expr)) '-)))
1039 ((and (eq (car-safe expr) '^)
1040 (memq (car-safe (nth 1 expr)) '(+ -))
1041 (integerp (nth 2 expr))
1042 (if (> (nth 2 expr) 0)
1043 (or (and (or (> mmt-many 500000) (< mmt-many -500000))
1044 (math-expand-power (nth 1 expr) (nth 2 expr)
1048 (list '^ (nth 1 expr) (1- (nth 2 expr)))))
1049 (if (< (nth 2 expr) 0)
1050 (list '/ 1 (list '^ (nth 1 expr) (- (nth 2 expr))))))))
1053 (defun calcFunc-expand (expr &optional many)
1054 (math-normalize (math-map-tree 'math-expand-term expr many)))
1056 (defun math-expand-power (x n &optional var else-nil)
1057 (or (and (natnump n)
1058 (memq (car-safe x) '(+ -))
1061 (while (memq (car-safe x) '(+ -))
1062 (setq terms (cons (if (eq (car x) '-)
1063 (math-neg (nth 2 x))
1067 (setq terms (cons x terms))
1071 (or (math-expr-contains (car p) var)
1072 (setq terms (delq (car p) terms)
1073 cterms (cons (car p) cterms)))
1076 (setq terms (cons (apply 'calcFunc-add cterms)
1078 (if (= (length terms) 2)
1082 (setq accum (list '+ accum
1083 (list '* (calcFunc-choose n i)
1085 (list '^ (nth 1 terms) i)
1086 (list '^ (car terms)
1095 (setq accum (list '+ accum
1096 (list '^ (car p1) 2))
1098 (while (setq p2 (cdr p2))
1099 (setq accum (list '+ accum
1110 (setq accum (list '+ accum (list '^ (car p1) 3))
1112 (while (setq p2 (cdr p2))
1113 (setq accum (list '+
1119 (list '^ (car p1) 2)
1124 (list '^ (car p2) 2))))
1126 (while (setq p3 (cdr p3))
1127 (setq accum (list '+ accum
1139 (defun calcFunc-expandpow (x n)
1140 (math-normalize (math-expand-power x n)))
1142 ;;; calc-poly.el ends here