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
HCoop / bpt / emacs.git / blob