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source: branches/f4grobner/polynomial.lisp@ 1531

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