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source: branches/f4grobner/pol.lisp@ 1983

Last change on this file since 1983 was 1983, 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
91(defun make-poly-from-termlist (termlist &optional (sugar (termlist-sugar termlist)))
92 (make-instance 'poly :termlist termlist :sugar sugar))
93
94(defun make-poly-zero (&aux (termlist nil) (sugar -1))
95 (make-instance 'poly :termlist termlist :sugar sugar))
96
97(defun make-poly-variable (ring nvars pos &optional (power 1)
98 &aux
99 (termlist (list
100 (make-term-variable ring nvars pos power)))
101 (sugar power))
102 (make-instance 'poly :termlist termlist :sugar sugar))
103
104(defun poly-unit (ring dimension
105 &aux
106 (termlist (termlist-unit ring dimension))
107 (sugar 0))
108 (make-instance 'poly :termlist termlist :sugar (termlist-sugar termlist)))
109
110
111
112;; Leading term
113(defmacro poly-lt (p) `(car (poly-termlist ,p)))
114
115;; Second term
116(defmacro poly-second-lt (p) `(cadar (poly-termlist ,p)))
117
118;; Leading monomial
119(defun poly-lm (p)
120 (declare (type poly p))
121 (term-monom (poly-lt p)))
122
123;; Second monomial
124(defun poly-second-lm (p)
125 (declare (type poly p))
126 (term-monom (poly-second-lt p)))
127
128;; Leading coefficient
129(defun poly-lc (p)
130 (declare (type poly p))
131 (term-coeff (poly-lt p)))
132
133;; Second coefficient
134(defun poly-second-lc (p)
135 (declare (type poly p))
136 (term-coeff (poly-second-lt p)))
137
138;; Testing for a zero polynomial
139(defun poly-zerop (p)
140 (declare (type poly p))
141 (null (poly-termlist p)))
142
143;; The number of terms
144(defun poly-length (p)
145 (declare (type poly p))
146 (length (poly-termlist p)))
147
148(defun poly-reset-sugar (p)
149 "(Re)sets the sugar of a polynomial P to the sugar of (POLY-TERMLIST P).
150Thus, the sugar is set to the maximum sugar of all monomials of P, or -1
151if P is a zero polynomial."
152 (declare (type poly p))
153 (setf (poly-sugar p) (termlist-sugar (poly-termlist p)))
154 p)
155
156(defun scalar-times-poly (ring c p)
157 "The scalar product of scalar C by a polynomial P. The sugar of the
158original polynomial becomes the sugar of the result."
159 (declare (type ring ring) (type poly p))
160 (make-poly-from-termlist (scalar-times-termlist ring c (poly-termlist p)) (poly-sugar p)))
161
162(defun scalar-times-poly-1 (ring c p)
163 "The scalar product of scalar C by a polynomial P, omitting the head term. The sugar of the
164original polynomial becomes the sugar of the result."
165 (declare (type ring ring) (type poly p))
166 (make-poly-from-termlist (scalar-times-termlist ring c (cdr (poly-termlist p))) (poly-sugar p)))
167
168(defun monom-times-poly (m p)
169 (declare (type monom m) (type poly p))
170 (make-poly-from-termlist
171 (monom-times-termlist m (poly-termlist p))
172 (+ (poly-sugar p) (monom-sugar m))))
173
174(defun term-times-poly (ring term p)
175 (declare (type ring ring) (type term term) (type poly p))
176 (make-poly-from-termlist
177 (term-times-termlist ring term (poly-termlist p))
178 (+ (poly-sugar p) (term-sugar term))))
179
180(defun poly-add (ring-and-order p q)
181 (declare (type ring-and-order ring-and-order) (type poly p q))
182 (make-poly-from-termlist
183 (termlist-add ring-and-order
184 (poly-termlist p)
185 (poly-termlist q))
186 (max (poly-sugar p) (poly-sugar q))))
187
188(defun poly-sub (ring-and-order p q)
189 (declare (type ring-and-order ring-and-order) (type poly p q))
190 (make-poly-from-termlist
191 (termlist-sub ring-and-order (poly-termlist p) (poly-termlist q))
192 (max (poly-sugar p) (poly-sugar q))))
193
194(defun poly-uminus (ring p)
195 (declare (type ring ring) (type poly p))
196 (make-poly-from-termlist
197 (termlist-uminus ring (poly-termlist p))
198 (poly-sugar p)))
199
200(defun poly-mul (ring-and-order p q)
201 (declare (type ring-and-order ring-and-order) (type poly p q))
202 (make-poly-from-termlist
203 (termlist-mul ring-and-order (poly-termlist p) (poly-termlist q))
204 (+ (poly-sugar p) (poly-sugar q))))
205
206(defun poly-expt (ring-and-order p n)
207 (declare (type ring-and-order ring-and-order) (type poly p))
208 (make-poly-from-termlist (termlist-expt ring-and-order (poly-termlist p) n) (* n (poly-sugar p))))
209
210(defun poly-append (&rest plist)
211 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
212 (apply #'max (mapcar #'poly-sugar plist))))
213
214(defun poly-nreverse (p)
215 "Destructively reverse the order of terms in polynomial P. Returns P"
216 (declare (type poly p))
217 (setf (poly-termlist p) (nreverse (poly-termlist p)))
218 p)
219
220(defun poly-reverse (p)
221 "Returns a copy of the polynomial P with terms in reverse order."
222 (declare (type poly p))
223 (make-poly-from-termlist (reverse (poly-termlist p))
224 (poly-sugar p)))
225
226
227(defun poly-contract (p &optional (k 1))
228 (declare (type poly p))
229 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
230 (poly-sugar p)))
231
232(defun poly-extend (p &optional (m (make-monom :dimension 1)))
233 (declare (type poly p))
234 (make-poly-from-termlist
235 (termlist-extend (poly-termlist p) m)
236 (+ (poly-sugar p) (monom-sugar m))))
237
238(defun poly-add-variables (p k)
239 (declare (type poly p))
240 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
241 p)
242
243(defun poly-list-add-variables (plist k)
244 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
245
246(defun poly-standard-extension (plist &aux (k (length plist)))
247 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
248 (declare (list plist) (fixnum k))
249 (labels ((incf-power (g i)
250 (dolist (x (poly-termlist g))
251 (incf (monom-elt (term-monom x) i)))
252 (incf (poly-sugar g))))
253 (setf plist (poly-list-add-variables plist k))
254 (dotimes (i k plist)
255 (incf-power (nth i plist) i))))
256
257(defun saturation-extension (ring f plist
258 &aux
259 (k (length plist))
260 (d (monom-dimension (poly-lm (car plist))))
261 f-x plist-x)
262 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
263 (declare (type ring ring))
264 (setf f-x (poly-list-add-variables f k)
265 plist-x (mapcar #'(lambda (x)
266 (setf (poly-termlist x)
267 (nconc (poly-termlist x)
268 (list (make-term :monom (make-monom :dimension d)
269 :coeff (funcall (ring-uminus ring)
270 (funcall (ring-unit ring)))))))
271 x)
272 (poly-standard-extension plist)))
273 (append f-x plist-x))
274
275
276(defun polysaturation-extension (ring f plist
277 &aux
278 (k (length plist))
279 (d (+ k (monom-dimension (poly-lm (car plist)))))
280 ;; Add k variables to f
281 (f (poly-list-add-variables f k))
282 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
283 (plist (apply #'poly-append (poly-standard-extension plist))))
284 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
285 ;; Add -1 as the last term
286 (declare (type ring ring))
287 (setf (cdr (last (poly-termlist plist)))
288 (list (make-term :monom (make-monom :dimension d)
289 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
290 (append f (list plist)))
291
292(defun saturation-extension-1 (ring f p)
293 "Calculate [F, U*P-1]. It destructively modifies F."
294 (declare (type ring ring))
295 (polysaturation-extension ring f (list p)))
296
297;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
298;;
299;; Evaluation of polynomial (prefix) expressions
300;;
301;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
302
303(defun coerce-coeff (ring expr vars)
304 "Coerce an element of the coefficient ring to a constant polynomial."
305 ;; Modular arithmetic handler by rat
306 (declare (type ring ring))
307 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
308 :coeff (funcall (ring-parse ring) expr)))
309 0))
310
311(defun poly-eval (expr vars
312 &optional
313 (ring +ring-of-integers+)
314 (order #'lex>)
315 (list-marker :[)
316 &aux
317 (ring-and-order (make-ring-and-order :ring ring :order order)))
318 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
319variables VARS. Return the resulting polynomial or list of
320polynomials. Standard arithmetical operators in form EXPR are
321replaced with their analogues in the ring of polynomials, and the
322resulting expression is evaluated, resulting in a polynomial or a list
323of polynomials in internal form. A similar operation in another computer
324algebra system could be called 'expand' or so."
325 (declare (type ring ring))
326 (labels ((p-eval (arg) (poly-eval arg vars ring order))
327 (p-eval-scalar (arg) (poly-eval-scalar arg))
328 (p-eval-list (args) (mapcar #'p-eval args))
329 (p-add (x y) (poly-add ring-and-order x y)))
330 (cond
331 ((null expr) (error "Empty expression"))
332 ((eql expr 0) (make-poly-zero))
333 ((member expr vars :test #'equalp)
334 (let ((pos (position expr vars :test #'equalp)))
335 (make-poly-variable ring (length vars) pos)))
336 ((atom expr)
337 (coerce-coeff ring expr vars))
338 ((eq (car expr) list-marker)
339 (cons list-marker (p-eval-list (cdr expr))))
340 (t
341 (case (car expr)
342 (+ (reduce #'p-add (p-eval-list (cdr expr))))
343 (- (case (length expr)
344 (1 (make-poly-zero))
345 (2 (poly-uminus ring (p-eval (cadr expr))))
346 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
347 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
348 (reduce #'p-add (p-eval-list (cddr expr)))))))
349 (*
350 (if (endp (cddr expr)) ;unary
351 (p-eval (cdr expr))
352 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
353 (/
354 ;; A polynomial can be divided by a scalar
355 (cond
356 ((endp (cddr expr))
357 ;; A special case (/ ?), the inverse
358 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
359 (t
360 (let ((num (p-eval (cadr expr)))
361 (denom-inverse (apply (ring-div ring)
362 (cons (funcall (ring-unit ring))
363 (mapcar #'p-eval-scalar (cddr expr))))))
364 (scalar-times-poly ring denom-inverse num)))))
365 (expt
366 (cond
367 ((member (cadr expr) vars :test #'equalp)
368 ;;Special handling of (expt var pow)
369 (let ((pos (position (cadr expr) vars :test #'equalp)))
370 (make-poly-variable ring (length vars) pos (caddr expr))))
371 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
372 ;; Negative power means division in coefficient ring
373 ;; Non-integer power means non-polynomial coefficient
374 (coerce-coeff ring expr vars))
375 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
376 (otherwise
377 (coerce-coeff ring expr vars)))))))
378
379(defun poly-eval-scalar (expr
380 &optional
381 (ring +ring-of-integers+)
382 &aux
383 (order #'lex>))
384 "Evaluate a scalar expression EXPR in ring RING."
385 (declare (type ring ring))
386 (poly-lc (poly-eval expr nil ring order)))
387
388(defun spoly (ring-and-order f g
389 &aux
390 (ring (ro-ring ring-and-order)))
391 "It yields the S-polynomial of polynomials F and G."
392 (declare (type ring-and-order ring-and-order) (type poly f g))
393 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
394 (mf (monom-div lcm (poly-lm f)))
395 (mg (monom-div lcm (poly-lm g))))
396 (declare (type monom mf mg))
397 (multiple-value-bind (c cf cg)
398 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
399 (declare (ignore c))
400 (poly-sub
401 ring-and-order
402 (scalar-times-poly ring cg (monom-times-poly mf f))
403 (scalar-times-poly ring cf (monom-times-poly mg g))))))
404
405
406(defun poly-primitive-part (ring p)
407 "Divide polynomial P with integer coefficients by gcd of its
408coefficients and return the result."
409 (declare (type ring ring) (type poly p))
410 (if (poly-zerop p)
411 (values p 1)
412 (let ((c (poly-content ring p)))
413 (values (make-poly-from-termlist
414 (mapcar
415 #'(lambda (x)
416 (make-term :monom (term-monom x)
417 :coeff (funcall (ring-div ring) (term-coeff x) c)))
418 (poly-termlist p))
419 (poly-sugar p))
420 c))))
421
422(defun poly-content (ring p)
423 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
424to compute the greatest common divisor."
425 (declare (type ring ring) (type poly p))
426 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
427
428(defun read-infix-form (&key (stream t))
429 "Parser of infix expressions with integer/rational coefficients
430The parser will recognize two kinds of polynomial expressions:
431
432- polynomials in fully expanded forms with coefficients
433 written in front of symbolic expressions; constants can be optionally
434 enclosed in (); for example, the infix form
435 X^2-Y^2+(-4/3)*U^2*W^3-5
436 parses to
437 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
438
439- lists of polynomials; for example
440 [X-Y, X^2+3*Z]
441 parses to
442 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
443 where the first symbol [ marks a list of polynomials.
444
445-other infix expressions, for example
446 [(X-Y)*(X+Y)/Z,(X+1)^2]
447parses to:
448 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
449Currently this function is implemented using M. Kantrowitz's INFIX package."
450 (read-from-string
451 (concatenate 'string
452 "#I("
453 (with-output-to-string (s)
454 (loop
455 (multiple-value-bind (line eof)
456 (read-line stream t)
457 (format s "~A" line)
458 (when eof (return)))))
459 ")")))
460
461(defun read-poly (vars &key
462 (stream t)
463 (ring +ring-of-integers+)
464 (order #'lex>))
465 "Reads an expression in prefix form from a stream STREAM.
466The expression read from the strem should represent a polynomial or a
467list of polynomials in variables VARS, over the ring RING. The
468polynomial or list of polynomials is returned, with terms in each
469polynomial ordered according to monomial order ORDER."
470 (poly-eval (read-infix-form :stream stream) vars ring order))
471
472(defun string->poly (str vars
473 &optional
474 (ring +ring-of-integers+)
475 (order #'lex>))
476 "Converts a string STR to a polynomial in variables VARS."
477 (with-input-from-string (s str)
478 (read-poly vars :stream s :ring ring :order order)))
479
480(defun poly->alist (p)
481 "Convert a polynomial P to an association list. Thus, the format of the
482returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
483MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
484corresponding coefficient in the ring."
485 (cond
486 ((poly-p p)
487 (mapcar #'term->cons (poly-termlist p)))
488 ((and (consp p) (eq (car p) :[))
489 (cons :[ (mapcar #'poly->alist (cdr p))))))
490
491(defun string->alist (str vars
492 &optional
493 (ring +ring-of-integers+)
494 (order #'lex>))
495 "Convert a string STR representing a polynomial or polynomial list to
496an association list (... (MONOM . COEFF) ...)."
497 (poly->alist (string->poly str vars ring order)))
498
499(defun poly-equal-no-sugar-p (p q)
500 "Compare polynomials for equality, ignoring sugar."
501 (declare (type poly p q))
502 (equalp (poly-termlist p) (poly-termlist q)))
503
504(defun poly-set-equal-no-sugar-p (p q)
505 "Compare polynomial sets P and Q for equality, ignoring sugar."
506 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
507
508(defun poly-list-equal-no-sugar-p (p q)
509 "Compare polynomial lists P and Q for equality, ignoring sugar."
510 (every #'poly-equal-no-sugar-p p q))
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