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source: branches/f4grobner/pol.lisp@ 1988

Last change on this file since 1988 was 1988, checked in by Marek Rychlik, 9 years ago

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1;;; -*- Mode: Lisp -*-
2;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
3;;;
4;;; Copyright (C) 1999, 2002, 2009, 2015 Marek Rychlik <rychlik@u.arizona.edu>
5;;;
6;;; This program is free software; you can redistribute it and/or modify
7;;; it under the terms of the GNU General Public License as published by
8;;; the Free Software Foundation; either version 2 of the License, or
9;;; (at your option) any later version.
10;;;
11;;; This program is distributed in the hope that it will be useful,
12;;; but WITHOUT ANY WARRANTY; without even the implied warranty of
13;;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
14;;; GNU General Public License for more details.
15;;;
16;;; You should have received a copy of the GNU General Public License
17;;; along with this program; if not, write to the Free Software
18;;; Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
19;;;
20;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
21
22;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
23;;
24;; Polynomials
25;;
26;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
27
28(defpackage "POL"
29 (:use :cl :ring :ring-and-order :monom :order :term :termlist :infix)
30 (:export "POLY"
31 "POLY-TERMLIST"
32 "POLY-SUGAR"
33 "POLY-RESET-SUGAR"
34 "POLY-LT"
35 "MAKE-POLY-FROM-TERMLIST"
36 "MAKE-POLY-ZERO"
37 "MAKE-POLY-VARIABLE"
38 "POLY-UNIT"
39 "POLY-LM"
40 "POLY-SECOND-LM"
41 "POLY-SECOND-LT"
42 "POLY-LC"
43 "POLY-SECOND-LC"
44 "POLY-ZEROP"
45 "POLY-LENGTH"
46 "SCALAR-TIMES-POLY"
47 "SCALAR-TIMES-POLY-1"
48 "MONOM-TIMES-POLY"
49 "TERM-TIMES-POLY"
50 "POLY-ADD"
51 "POLY-SUB"
52 "POLY-UMINUS"
53 "POLY-MUL"
54 "POLY-EXPT"
55 "POLY-APPEND"
56 "POLY-NREVERSE"
57 "POLY-REVERSE"
58 "POLY-CONTRACT"
59 "POLY-EXTEND"
60 "POLY-ADD-VARIABLES"
61 "POLY-LIST-ADD-VARIABLES"
62 "POLY-STANDARD-EXTENSION"
63 "SATURATION-EXTENSION"
64 "POLYSATURATION-EXTENSION"
65 "SATURATION-EXTENSION-1"
66 "COERCE-COEFF"
67 "POLY-EVAL"
68 "POLY-EVAL-SCALAR"
69 "SPOLY"
70 "POLY-PRIMITIVE-PART"
71 "POLY-CONTENT"
72 "READ-INFIX-FORM"
73 "READ-POLY"
74 "STRING->POLY"
75 "POLY->ALIST"
76 "STRING->ALIST"
77 "POLY-EQUAL-NO-SUGAR-P"
78 "POLY-SET-EQUAL-NO-SUGAR-P"
79 "POLY-LIST-EQUAL-NO-SUGAR-P"
80 ))
81
82(in-package :pol)
83
84(proclaim '(optimize (speed 0) (space 0) (safety 3) (debug 3)))
85
86(defclass poly ()
87 ((termlist :initarg :termlist :accessor termlist)
88 (sugar :initarg :sugar :accessor sugar)
89 )
90 (:default-initargs :termlist nil :sugar -1))
91
92(defun make-poly-from-termlist (termlist &optional (sugar (termlist-sugar termlist)))
93 (make-instance 'poly :termlist termlist :sugar sugar))
94
95(defun make-poly-zero (&aux (termlist nil) (sugar -1))
96 (make-instance 'poly :termlist termlist :sugar sugar))
97
98(defun make-poly-variable (ring nvars pos &optional (power 1)
99 &aux
100 (termlist (list
101 (make-term-variable ring nvars pos power)))
102 (sugar power))
103 (make-instance 'poly :termlist termlist :sugar sugar))
104
105(defun poly-unit (ring dimension
106 &aux
107 (termlist (termlist-unit ring dimension))
108 (sugar 0))
109 (make-instance 'poly :termlist termlist :sugar (termlist-sugar termlist)))
110
111
112(defmethod print-object ((poly poly) stream)
113 (princ (slot-value poly 'termlist)))
114
115(defmethod poly-termlist ((poly poly))
116 (slot-value poly 'termlist))
117
118(defmethod (setf poly-termlist) (new-value (poly poly))
119 (setf (slot-value poly 'termlist) new-value))
120
121;; Leading term
122(defmacro poly-lt (p) `(car (poly-termlist ,p)))
123
124;; Second term
125(defmacro poly-second-lt (p) `(cadr (poly-termlist ,p)))
126
127;; Leading monomial
128(defun poly-lm (p)
129 (declare (type poly p))
130 (term-monom (poly-lt p)))
131
132;; Second monomial
133(defun poly-second-lm (p)
134 (declare (type poly p))
135 (term-monom (poly-second-lt p)))
136
137;; Leading coefficient
138(defun poly-lc (p)
139 (declare (type poly p))
140 (term-coeff (poly-lt p)))
141
142;; Second coefficient
143(defun poly-second-lc (p)
144 (declare (type poly p))
145 (term-coeff (poly-second-lt p)))
146
147;; Testing for a zero polynomial
148(defun poly-zerop (p)
149 (declare (type poly p))
150 (null (poly-termlist p)))
151
152;; The number of terms
153(defun poly-length (p)
154 (declare (type poly p))
155 (length (poly-termlist p)))
156
157(defun poly-reset-sugar (p)
158 "(Re)sets the sugar of a polynomial P to the sugar of (POLY-TERMLIST P).
159Thus, the sugar is set to the maximum sugar of all monomials of P, or -1
160if P is a zero polynomial."
161 (declare (type poly p))
162 (setf (poly-sugar p) (termlist-sugar (poly-termlist p)))
163 p)
164
165(defun scalar-times-poly (ring c p)
166 "The scalar product of scalar C by a polynomial P. The sugar of the
167original polynomial becomes the sugar of the result."
168 (declare (type ring ring) (type poly p))
169 (make-poly-from-termlist (scalar-times-termlist ring c (poly-termlist p)) (poly-sugar p)))
170
171(defun scalar-times-poly-1 (ring c p)
172 "The scalar product of scalar C by a polynomial P, omitting the head term. The sugar of the
173original polynomial becomes the sugar of the result."
174 (declare (type ring ring) (type poly p))
175 (make-poly-from-termlist (scalar-times-termlist ring c (cdr (poly-termlist p))) (poly-sugar p)))
176
177(defun monom-times-poly (m p)
178 (declare (type monom m) (type poly p))
179 (make-poly-from-termlist
180 (monom-times-termlist m (poly-termlist p))
181 (+ (poly-sugar p) (monom-sugar m))))
182
183(defun term-times-poly (ring term p)
184 (declare (type ring ring) (type term term) (type poly p))
185 (make-poly-from-termlist
186 (term-times-termlist ring term (poly-termlist p))
187 (+ (poly-sugar p) (term-sugar term))))
188
189(defun poly-add (ring-and-order p q)
190 (declare (type ring-and-order ring-and-order) (type poly p q))
191 (make-poly-from-termlist
192 (termlist-add ring-and-order
193 (poly-termlist p)
194 (poly-termlist q))
195 (max (poly-sugar p) (poly-sugar q))))
196
197(defun poly-sub (ring-and-order p q)
198 (declare (type ring-and-order ring-and-order) (type poly p q))
199 (make-poly-from-termlist
200 (termlist-sub ring-and-order (poly-termlist p) (poly-termlist q))
201 (max (poly-sugar p) (poly-sugar q))))
202
203(defun poly-uminus (ring p)
204 (declare (type ring ring) (type poly p))
205 (make-poly-from-termlist
206 (termlist-uminus ring (poly-termlist p))
207 (poly-sugar p)))
208
209(defun poly-mul (ring-and-order p q)
210 (declare (type ring-and-order ring-and-order) (type poly p q))
211 (make-poly-from-termlist
212 (termlist-mul ring-and-order (poly-termlist p) (poly-termlist q))
213 (+ (poly-sugar p) (poly-sugar q))))
214
215(defun poly-expt (ring-and-order p n)
216 (declare (type ring-and-order ring-and-order) (type poly p))
217 (make-poly-from-termlist (termlist-expt ring-and-order (poly-termlist p) n) (* n (poly-sugar p))))
218
219(defun poly-append (&rest plist)
220 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
221 (apply #'max (mapcar #'poly-sugar plist))))
222
223(defun poly-nreverse (p)
224 "Destructively reverse the order of terms in polynomial P. Returns P"
225 (declare (type poly p))
226 (setf (poly-termlist p) (nreverse (poly-termlist p)))
227 p)
228
229(defun poly-reverse (p)
230 "Returns a copy of the polynomial P with terms in reverse order."
231 (declare (type poly p))
232 (make-poly-from-termlist (reverse (poly-termlist p))
233 (poly-sugar p)))
234
235
236(defun poly-contract (p &optional (k 1))
237 (declare (type poly p))
238 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
239 (poly-sugar p)))
240
241(defun poly-extend (p &optional (m (make-monom :dimension 1)))
242 (declare (type poly p))
243 (make-poly-from-termlist
244 (termlist-extend (poly-termlist p) m)
245 (+ (poly-sugar p) (monom-sugar m))))
246
247(defun poly-add-variables (p k)
248 (declare (type poly p))
249 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
250 p)
251
252(defun poly-list-add-variables (plist k)
253 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
254
255(defun poly-standard-extension (plist &aux (k (length plist)))
256 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
257 (declare (list plist) (fixnum k))
258 (labels ((incf-power (g i)
259 (dolist (x (poly-termlist g))
260 (incf (monom-elt (term-monom x) i)))
261 (incf (poly-sugar g))))
262 (setf plist (poly-list-add-variables plist k))
263 (dotimes (i k plist)
264 (incf-power (nth i plist) i))))
265
266(defun saturation-extension (ring f plist
267 &aux
268 (k (length plist))
269 (d (monom-dimension (poly-lm (car plist))))
270 f-x plist-x)
271 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
272 (declare (type ring ring))
273 (setf f-x (poly-list-add-variables f k)
274 plist-x (mapcar #'(lambda (x)
275 (setf (poly-termlist x)
276 (nconc (poly-termlist x)
277 (list (make-term :monom (make-monom :dimension d)
278 :coeff (funcall (ring-uminus ring)
279 (funcall (ring-unit ring)))))))
280 x)
281 (poly-standard-extension plist)))
282 (append f-x plist-x))
283
284
285(defun polysaturation-extension (ring f plist
286 &aux
287 (k (length plist))
288 (d (+ k (monom-dimension (poly-lm (car plist)))))
289 ;; Add k variables to f
290 (f (poly-list-add-variables f k))
291 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
292 (plist (apply #'poly-append (poly-standard-extension plist))))
293 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
294 ;; Add -1 as the last term
295 (declare (type ring ring))
296 (setf (cdr (last (poly-termlist plist)))
297 (list (make-term :monom (make-monom :dimension d)
298 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
299 (append f (list plist)))
300
301(defun saturation-extension-1 (ring f p)
302 "Calculate [F, U*P-1]. It destructively modifies F."
303 (declare (type ring ring))
304 (polysaturation-extension ring f (list p)))
305
306;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
307;;
308;; Evaluation of polynomial (prefix) expressions
309;;
310;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
311
312(defun coerce-coeff (ring expr vars)
313 "Coerce an element of the coefficient ring to a constant polynomial."
314 ;; Modular arithmetic handler by rat
315 (declare (type ring ring))
316 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
317 :coeff (funcall (ring-parse ring) expr)))
318 0))
319
320(defun poly-eval (expr vars
321 &optional
322 (ring +ring-of-integers+)
323 (order #'lex>)
324 (list-marker :[)
325 &aux
326 (ring-and-order (make-ring-and-order :ring ring :order order)))
327 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
328variables VARS. Return the resulting polynomial or list of
329polynomials. Standard arithmetical operators in form EXPR are
330replaced with their analogues in the ring of polynomials, and the
331resulting expression is evaluated, resulting in a polynomial or a list
332of polynomials in internal form. A similar operation in another computer
333algebra system could be called 'expand' or so."
334 (declare (type ring ring))
335 (labels ((p-eval (arg) (poly-eval arg vars ring order))
336 (p-eval-scalar (arg) (poly-eval-scalar arg))
337 (p-eval-list (args) (mapcar #'p-eval args))
338 (p-add (x y) (poly-add ring-and-order x y)))
339 (cond
340 ((null expr) (error "Empty expression"))
341 ((eql expr 0) (make-poly-zero))
342 ((member expr vars :test #'equalp)
343 (let ((pos (position expr vars :test #'equalp)))
344 (make-poly-variable ring (length vars) pos)))
345 ((atom expr)
346 (coerce-coeff ring expr vars))
347 ((eq (car expr) list-marker)
348 (cons list-marker (p-eval-list (cdr expr))))
349 (t
350 (case (car expr)
351 (+ (reduce #'p-add (p-eval-list (cdr expr))))
352 (- (case (length expr)
353 (1 (make-poly-zero))
354 (2 (poly-uminus ring (p-eval (cadr expr))))
355 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
356 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
357 (reduce #'p-add (p-eval-list (cddr expr)))))))
358 (*
359 (if (endp (cddr expr)) ;unary
360 (p-eval (cdr expr))
361 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
362 (/
363 ;; A polynomial can be divided by a scalar
364 (cond
365 ((endp (cddr expr))
366 ;; A special case (/ ?), the inverse
367 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
368 (t
369 (let ((num (p-eval (cadr expr)))
370 (denom-inverse (apply (ring-div ring)
371 (cons (funcall (ring-unit ring))
372 (mapcar #'p-eval-scalar (cddr expr))))))
373 (scalar-times-poly ring denom-inverse num)))))
374 (expt
375 (cond
376 ((member (cadr expr) vars :test #'equalp)
377 ;;Special handling of (expt var pow)
378 (let ((pos (position (cadr expr) vars :test #'equalp)))
379 (make-poly-variable ring (length vars) pos (caddr expr))))
380 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
381 ;; Negative power means division in coefficient ring
382 ;; Non-integer power means non-polynomial coefficient
383 (coerce-coeff ring expr vars))
384 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
385 (otherwise
386 (coerce-coeff ring expr vars)))))))
387
388(defun poly-eval-scalar (expr
389 &optional
390 (ring +ring-of-integers+)
391 &aux
392 (order #'lex>))
393 "Evaluate a scalar expression EXPR in ring RING."
394 (declare (type ring ring))
395 (poly-lc (poly-eval expr nil ring order)))
396
397(defun spoly (ring-and-order f g
398 &aux
399 (ring (ro-ring ring-and-order)))
400 "It yields the S-polynomial of polynomials F and G."
401 (declare (type ring-and-order ring-and-order) (type poly f g))
402 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
403 (mf (monom-div lcm (poly-lm f)))
404 (mg (monom-div lcm (poly-lm g))))
405 (declare (type monom mf mg))
406 (multiple-value-bind (c cf cg)
407 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
408 (declare (ignore c))
409 (poly-sub
410 ring-and-order
411 (scalar-times-poly ring cg (monom-times-poly mf f))
412 (scalar-times-poly ring cf (monom-times-poly mg g))))))
413
414
415(defun poly-primitive-part (ring p)
416 "Divide polynomial P with integer coefficients by gcd of its
417coefficients and return the result."
418 (declare (type ring ring) (type poly p))
419 (if (poly-zerop p)
420 (values p 1)
421 (let ((c (poly-content ring p)))
422 (values (make-poly-from-termlist
423 (mapcar
424 #'(lambda (x)
425 (make-term :monom (term-monom x)
426 :coeff (funcall (ring-div ring) (term-coeff x) c)))
427 (poly-termlist p))
428 (poly-sugar p))
429 c))))
430
431(defun poly-content (ring p)
432 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
433to compute the greatest common divisor."
434 (declare (type ring ring) (type poly p))
435 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
436
437(defun read-infix-form (&key (stream t))
438 "Parser of infix expressions with integer/rational coefficients
439The parser will recognize two kinds of polynomial expressions:
440
441- polynomials in fully expanded forms with coefficients
442 written in front of symbolic expressions; constants can be optionally
443 enclosed in (); for example, the infix form
444 X^2-Y^2+(-4/3)*U^2*W^3-5
445 parses to
446 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
447
448- lists of polynomials; for example
449 [X-Y, X^2+3*Z]
450 parses to
451 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
452 where the first symbol [ marks a list of polynomials.
453
454-other infix expressions, for example
455 [(X-Y)*(X+Y)/Z,(X+1)^2]
456parses to:
457 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
458Currently this function is implemented using M. Kantrowitz's INFIX package."
459 (read-from-string
460 (concatenate 'string
461 "#I("
462 (with-output-to-string (s)
463 (loop
464 (multiple-value-bind (line eof)
465 (read-line stream t)
466 (format s "~A" line)
467 (when eof (return)))))
468 ")")))
469
470(defun read-poly (vars &key
471 (stream t)
472 (ring +ring-of-integers+)
473 (order #'lex>))
474 "Reads an expression in prefix form from a stream STREAM.
475The expression read from the strem should represent a polynomial or a
476list of polynomials in variables VARS, over the ring RING. The
477polynomial or list of polynomials is returned, with terms in each
478polynomial ordered according to monomial order ORDER."
479 (poly-eval (read-infix-form :stream stream) vars ring order))
480
481(defun string->poly (str vars
482 &optional
483 (ring +ring-of-integers+)
484 (order #'lex>))
485 "Converts a string STR to a polynomial in variables VARS."
486 (with-input-from-string (s str)
487 (read-poly vars :stream s :ring ring :order order)))
488
489(defun poly->alist (p)
490 "Convert a polynomial P to an association list. Thus, the format of the
491returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
492MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
493corresponding coefficient in the ring."
494 (cond
495 ((poly-p p)
496 (mapcar #'term->cons (poly-termlist p)))
497 ((and (consp p) (eq (car p) :[))
498 (cons :[ (mapcar #'poly->alist (cdr p))))))
499
500(defun string->alist (str vars
501 &optional
502 (ring +ring-of-integers+)
503 (order #'lex>))
504 "Convert a string STR representing a polynomial or polynomial list to
505an association list (... (MONOM . COEFF) ...)."
506 (poly->alist (string->poly str vars ring order)))
507
508(defun poly-equal-no-sugar-p (p q)
509 "Compare polynomials for equality, ignoring sugar."
510 (declare (type poly p q))
511 (equalp (poly-termlist p) (poly-termlist q)))
512
513(defun poly-set-equal-no-sugar-p (p q)
514 "Compare polynomial sets P and Q for equality, ignoring sugar."
515 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
516
517(defun poly-list-equal-no-sugar-p (p q)
518 "Compare polynomial lists P and Q for equality, ignoring sugar."
519 (every #'poly-equal-no-sugar-p p q))
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