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source: branches/f4grobner/pol.lisp@ 1980

Last change on this file since 1980 was 1980, checked in by Marek Rychlik, 10 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)
88 (sugar)
89 ))
90
91(defun make-poly-from-termlist (termlist &optional (sugar (termlist-sugar termlist))))
92(defun make-poly-zero (&aux (termlist nil) (sugar -1)))
93(defun make-poly-variable (ring nvars pos &optional (power 1)
94 &aux
95 (termlist (list
96 (make-term-variable ring nvars pos power)))
97 (sugar power)))
98
99(defun poly-unit (ring dimension
100 &aux
101 (termlist (termlist-unit ring dimension))
102 (sugar 0))
103 (make-instance 'poly :termlist termlist :sugar (termlist-sugar termlist)))
104
105
106
107;; Leading term
108(defmacro poly-lt (p) `(car (poly-termlist ,p)))
109
110;; Second term
111(defmacro poly-second-lt (p) `(cadar (poly-termlist ,p)))
112
113;; Leading monomial
114(defun poly-lm (p)
115 (declare (type poly p))
116 (term-monom (poly-lt p)))
117
118;; Second monomial
119(defun poly-second-lm (p)
120 (declare (type poly p))
121 (term-monom (poly-second-lt p)))
122
123;; Leading coefficient
124(defun poly-lc (p)
125 (declare (type poly p))
126 (term-coeff (poly-lt p)))
127
128;; Second coefficient
129(defun poly-second-lc (p)
130 (declare (type poly p))
131 (term-coeff (poly-second-lt p)))
132
133;; Testing for a zero polynomial
134(defun poly-zerop (p)
135 (declare (type poly p))
136 (null (poly-termlist p)))
137
138;; The number of terms
139(defun poly-length (p)
140 (declare (type poly p))
141 (length (poly-termlist p)))
142
143(defun poly-reset-sugar (p)
144 "(Re)sets the sugar of a polynomial P to the sugar of (POLY-TERMLIST P).
145Thus, the sugar is set to the maximum sugar of all monomials of P, or -1
146if P is a zero polynomial."
147 (declare (type poly p))
148 (setf (poly-sugar p) (termlist-sugar (poly-termlist p)))
149 p)
150
151(defun scalar-times-poly (ring c p)
152 "The scalar product of scalar C by a polynomial P. The sugar of the
153original polynomial becomes the sugar of the result."
154 (declare (type ring ring) (type poly p))
155 (make-poly-from-termlist (scalar-times-termlist ring c (poly-termlist p)) (poly-sugar p)))
156
157(defun scalar-times-poly-1 (ring c p)
158 "The scalar product of scalar C by a polynomial P, omitting the head term. The sugar of the
159original polynomial becomes the sugar of the result."
160 (declare (type ring ring) (type poly p))
161 (make-poly-from-termlist (scalar-times-termlist ring c (cdr (poly-termlist p))) (poly-sugar p)))
162
163(defun monom-times-poly (m p)
164 (declare (type monom m) (type poly p))
165 (make-poly-from-termlist
166 (monom-times-termlist m (poly-termlist p))
167 (+ (poly-sugar p) (monom-sugar m))))
168
169(defun term-times-poly (ring term p)
170 (declare (type ring ring) (type term term) (type poly p))
171 (make-poly-from-termlist
172 (term-times-termlist ring term (poly-termlist p))
173 (+ (poly-sugar p) (term-sugar term))))
174
175(defun poly-add (ring-and-order p q)
176 (declare (type ring-and-order ring-and-order) (type poly p q))
177 (make-poly-from-termlist
178 (termlist-add ring-and-order
179 (poly-termlist p)
180 (poly-termlist q))
181 (max (poly-sugar p) (poly-sugar q))))
182
183(defun poly-sub (ring-and-order p q)
184 (declare (type ring-and-order ring-and-order) (type poly p q))
185 (make-poly-from-termlist
186 (termlist-sub ring-and-order (poly-termlist p) (poly-termlist q))
187 (max (poly-sugar p) (poly-sugar q))))
188
189(defun poly-uminus (ring p)
190 (declare (type ring ring) (type poly p))
191 (make-poly-from-termlist
192 (termlist-uminus ring (poly-termlist p))
193 (poly-sugar p)))
194
195(defun poly-mul (ring-and-order p q)
196 (declare (type ring-and-order ring-and-order) (type poly p q))
197 (make-poly-from-termlist
198 (termlist-mul ring-and-order (poly-termlist p) (poly-termlist q))
199 (+ (poly-sugar p) (poly-sugar q))))
200
201(defun poly-expt (ring-and-order p n)
202 (declare (type ring-and-order ring-and-order) (type poly p))
203 (make-poly-from-termlist (termlist-expt ring-and-order (poly-termlist p) n) (* n (poly-sugar p))))
204
205(defun poly-append (&rest plist)
206 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
207 (apply #'max (mapcar #'poly-sugar plist))))
208
209(defun poly-nreverse (p)
210 "Destructively reverse the order of terms in polynomial P. Returns P"
211 (declare (type poly p))
212 (setf (poly-termlist p) (nreverse (poly-termlist p)))
213 p)
214
215(defun poly-reverse (p)
216 "Returns a copy of the polynomial P with terms in reverse order."
217 (declare (type poly p))
218 (make-poly-from-termlist (reverse (poly-termlist p))
219 (poly-sugar p)))
220
221
222(defun poly-contract (p &optional (k 1))
223 (declare (type poly p))
224 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
225 (poly-sugar p)))
226
227(defun poly-extend (p &optional (m (make-monom :dimension 1)))
228 (declare (type poly p))
229 (make-poly-from-termlist
230 (termlist-extend (poly-termlist p) m)
231 (+ (poly-sugar p) (monom-sugar m))))
232
233(defun poly-add-variables (p k)
234 (declare (type poly p))
235 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
236 p)
237
238(defun poly-list-add-variables (plist k)
239 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
240
241(defun poly-standard-extension (plist &aux (k (length plist)))
242 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
243 (declare (list plist) (fixnum k))
244 (labels ((incf-power (g i)
245 (dolist (x (poly-termlist g))
246 (incf (monom-elt (term-monom x) i)))
247 (incf (poly-sugar g))))
248 (setf plist (poly-list-add-variables plist k))
249 (dotimes (i k plist)
250 (incf-power (nth i plist) i))))
251
252(defun saturation-extension (ring f plist
253 &aux
254 (k (length plist))
255 (d (monom-dimension (poly-lm (car plist))))
256 f-x plist-x)
257 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
258 (declare (type ring ring))
259 (setf f-x (poly-list-add-variables f k)
260 plist-x (mapcar #'(lambda (x)
261 (setf (poly-termlist x)
262 (nconc (poly-termlist x)
263 (list (make-term :monom (make-monom :dimension d)
264 :coeff (funcall (ring-uminus ring)
265 (funcall (ring-unit ring)))))))
266 x)
267 (poly-standard-extension plist)))
268 (append f-x plist-x))
269
270
271(defun polysaturation-extension (ring f plist
272 &aux
273 (k (length plist))
274 (d (+ k (monom-dimension (poly-lm (car plist)))))
275 ;; Add k variables to f
276 (f (poly-list-add-variables f k))
277 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
278 (plist (apply #'poly-append (poly-standard-extension plist))))
279 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
280 ;; Add -1 as the last term
281 (declare (type ring ring))
282 (setf (cdr (last (poly-termlist plist)))
283 (list (make-term :monom (make-monom :dimension d)
284 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
285 (append f (list plist)))
286
287(defun saturation-extension-1 (ring f p)
288 "Calculate [F, U*P-1]. It destructively modifies F."
289 (declare (type ring ring))
290 (polysaturation-extension ring f (list p)))
291
292;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
293;;
294;; Evaluation of polynomial (prefix) expressions
295;;
296;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
297
298(defun coerce-coeff (ring expr vars)
299 "Coerce an element of the coefficient ring to a constant polynomial."
300 ;; Modular arithmetic handler by rat
301 (declare (type ring ring))
302 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
303 :coeff (funcall (ring-parse ring) expr)))
304 0))
305
306(defun poly-eval (expr vars
307 &optional
308 (ring +ring-of-integers+)
309 (order #'lex>)
310 (list-marker :[)
311 &aux
312 (ring-and-order (make-ring-and-order :ring ring :order order)))
313 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
314variables VARS. Return the resulting polynomial or list of
315polynomials. Standard arithmetical operators in form EXPR are
316replaced with their analogues in the ring of polynomials, and the
317resulting expression is evaluated, resulting in a polynomial or a list
318of polynomials in internal form. A similar operation in another computer
319algebra system could be called 'expand' or so."
320 (declare (type ring ring))
321 (labels ((p-eval (arg) (poly-eval arg vars ring order))
322 (p-eval-scalar (arg) (poly-eval-scalar arg))
323 (p-eval-list (args) (mapcar #'p-eval args))
324 (p-add (x y) (poly-add ring-and-order x y)))
325 (cond
326 ((null expr) (error "Empty expression"))
327 ((eql expr 0) (make-poly-zero))
328 ((member expr vars :test #'equalp)
329 (let ((pos (position expr vars :test #'equalp)))
330 (make-poly-variable ring (length vars) pos)))
331 ((atom expr)
332 (coerce-coeff ring expr vars))
333 ((eq (car expr) list-marker)
334 (cons list-marker (p-eval-list (cdr expr))))
335 (t
336 (case (car expr)
337 (+ (reduce #'p-add (p-eval-list (cdr expr))))
338 (- (case (length expr)
339 (1 (make-poly-zero))
340 (2 (poly-uminus ring (p-eval (cadr expr))))
341 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
342 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
343 (reduce #'p-add (p-eval-list (cddr expr)))))))
344 (*
345 (if (endp (cddr expr)) ;unary
346 (p-eval (cdr expr))
347 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
348 (/
349 ;; A polynomial can be divided by a scalar
350 (cond
351 ((endp (cddr expr))
352 ;; A special case (/ ?), the inverse
353 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
354 (t
355 (let ((num (p-eval (cadr expr)))
356 (denom-inverse (apply (ring-div ring)
357 (cons (funcall (ring-unit ring))
358 (mapcar #'p-eval-scalar (cddr expr))))))
359 (scalar-times-poly ring denom-inverse num)))))
360 (expt
361 (cond
362 ((member (cadr expr) vars :test #'equalp)
363 ;;Special handling of (expt var pow)
364 (let ((pos (position (cadr expr) vars :test #'equalp)))
365 (make-poly-variable ring (length vars) pos (caddr expr))))
366 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
367 ;; Negative power means division in coefficient ring
368 ;; Non-integer power means non-polynomial coefficient
369 (coerce-coeff ring expr vars))
370 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
371 (otherwise
372 (coerce-coeff ring expr vars)))))))
373
374(defun poly-eval-scalar (expr
375 &optional
376 (ring +ring-of-integers+)
377 &aux
378 (order #'lex>))
379 "Evaluate a scalar expression EXPR in ring RING."
380 (declare (type ring ring))
381 (poly-lc (poly-eval expr nil ring order)))
382
383(defun spoly (ring-and-order f g
384 &aux
385 (ring (ro-ring ring-and-order)))
386 "It yields the S-polynomial of polynomials F and G."
387 (declare (type ring-and-order ring-and-order) (type poly f g))
388 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
389 (mf (monom-div lcm (poly-lm f)))
390 (mg (monom-div lcm (poly-lm g))))
391 (declare (type monom mf mg))
392 (multiple-value-bind (c cf cg)
393 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
394 (declare (ignore c))
395 (poly-sub
396 ring-and-order
397 (scalar-times-poly ring cg (monom-times-poly mf f))
398 (scalar-times-poly ring cf (monom-times-poly mg g))))))
399
400
401(defun poly-primitive-part (ring p)
402 "Divide polynomial P with integer coefficients by gcd of its
403coefficients and return the result."
404 (declare (type ring ring) (type poly p))
405 (if (poly-zerop p)
406 (values p 1)
407 (let ((c (poly-content ring p)))
408 (values (make-poly-from-termlist
409 (mapcar
410 #'(lambda (x)
411 (make-term :monom (term-monom x)
412 :coeff (funcall (ring-div ring) (term-coeff x) c)))
413 (poly-termlist p))
414 (poly-sugar p))
415 c))))
416
417(defun poly-content (ring p)
418 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
419to compute the greatest common divisor."
420 (declare (type ring ring) (type poly p))
421 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
422
423(defun read-infix-form (&key (stream t))
424 "Parser of infix expressions with integer/rational coefficients
425The parser will recognize two kinds of polynomial expressions:
426
427- polynomials in fully expanded forms with coefficients
428 written in front of symbolic expressions; constants can be optionally
429 enclosed in (); for example, the infix form
430 X^2-Y^2+(-4/3)*U^2*W^3-5
431 parses to
432 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
433
434- lists of polynomials; for example
435 [X-Y, X^2+3*Z]
436 parses to
437 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
438 where the first symbol [ marks a list of polynomials.
439
440-other infix expressions, for example
441 [(X-Y)*(X+Y)/Z,(X+1)^2]
442parses to:
443 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
444Currently this function is implemented using M. Kantrowitz's INFIX package."
445 (read-from-string
446 (concatenate 'string
447 "#I("
448 (with-output-to-string (s)
449 (loop
450 (multiple-value-bind (line eof)
451 (read-line stream t)
452 (format s "~A" line)
453 (when eof (return)))))
454 ")")))
455
456(defun read-poly (vars &key
457 (stream t)
458 (ring +ring-of-integers+)
459 (order #'lex>))
460 "Reads an expression in prefix form from a stream STREAM.
461The expression read from the strem should represent a polynomial or a
462list of polynomials in variables VARS, over the ring RING. The
463polynomial or list of polynomials is returned, with terms in each
464polynomial ordered according to monomial order ORDER."
465 (poly-eval (read-infix-form :stream stream) vars ring order))
466
467(defun string->poly (str vars
468 &optional
469 (ring +ring-of-integers+)
470 (order #'lex>))
471 "Converts a string STR to a polynomial in variables VARS."
472 (with-input-from-string (s str)
473 (read-poly vars :stream s :ring ring :order order)))
474
475(defun poly->alist (p)
476 "Convert a polynomial P to an association list. Thus, the format of the
477returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
478MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
479corresponding coefficient in the ring."
480 (cond
481 ((poly-p p)
482 (mapcar #'term->cons (poly-termlist p)))
483 ((and (consp p) (eq (car p) :[))
484 (cons :[ (mapcar #'poly->alist (cdr p))))))
485
486(defun string->alist (str vars
487 &optional
488 (ring +ring-of-integers+)
489 (order #'lex>))
490 "Convert a string STR representing a polynomial or polynomial list to
491an association list (... (MONOM . COEFF) ...)."
492 (poly->alist (string->poly str vars ring order)))
493
494(defun poly-equal-no-sugar-p (p q)
495 "Compare polynomials for equality, ignoring sugar."
496 (declare (type poly p q))
497 (equalp (poly-termlist p) (poly-termlist q)))
498
499(defun poly-set-equal-no-sugar-p (p q)
500 "Compare polynomial sets P and Q for equality, ignoring sugar."
501 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
502
503(defun poly-list-equal-no-sugar-p (p q)
504 "Compare polynomial lists P and Q for equality, ignoring sugar."
505 (every #'poly-equal-no-sugar-p p q))
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