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source: branches/f4grobner/polynomial.lisp@ 2583

Last change on this file since 2583 was 2583, 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(defpackage "POLYNOMIAL"
23 (:use :cl :ring :monom :order :term #| :infix |# )
24 (:export "POLY")
25 (:documentation "Implements polynomials"))
26
27(in-package :polynomial)
28
29(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
30
31(defclass poly ()
32 ((termlist :initarg :termlist :accessor poly-termlist))
33 (:default-initargs :termlist nil))
34
35(defmethod print-object ((self poly) stream)
36 (format stream "#<POLY TERMLIST=~A >" (poly-termlist self)))
37
38(defmethod insert-item ((self poly) (item term))
39 (push item (poly-termlist self))
40 self)
41
42(defmethod append-item ((self poly) (item term))
43 (setf (cdr (last (poly-termlist self))) (list item))
44 self)
45
46;; Leading term
47(defgeneric leading-term (object)
48 (:method ((self poly))
49 (car (poly-termlist self)))
50 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
51
52;; Second term
53(defgeneric second-leading-term (object)
54 (:method ((self poly))
55 (cadar (poly-termlist self)))
56 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
57
58;; Leading coefficient
59(defgeneric leading-coefficient (object)
60 (:method ((self poly))
61 (r-coeff (leading-term self)))
62 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
63
64;; Second coefficient
65(defgeneric second-leading-coefficient (object)
66 (:method ((self poly))
67 (r-coeff (second-leading-term self)))
68 (:documentation "The second leading coefficient of a polynomial. It signals error for a polynomial with at most one term."))
69
70;; Testing for a zero polynomial
71(defmethod r-zerop ((self poly))
72 (null (poly-termlist self)))
73
74;; The number of terms
75(defmethod r-length ((self poly))
76 (length (poly-termlist self)))
77
78(defmethod multiply-by ((self poly) (other monom))
79 (mapc #'(lambda (term) (multiply-by term other))
80 (poly-termlist self))
81 self)
82
83(defmethod multiply-by ((self poly) (other scalar))
84 (mapc #'(lambda (term) (multiply-by term other))
85 (poly-termlist self))
86 self)
87
88(defmethod add-to ((self poly) (other poly))
89 "Adds to polynomial SELF another polynomial OTHER.
90This operation destructively modifies both polynomials.
91The result is stored in SELF. This implementation does
92no consing, entirely reusing the sells of SELF and OTHER."
93 (macrolet ((lt (termlist) `(car ,termlist))
94 (lc (termlist) `(r-coeff (car ,termlist))))
95 (with-slots ((termlist1 termlist))
96 self
97 (with-slots ((termlist2 termlist))
98 other
99 (do ((p termlist1)
100 (q termlist2))
101 ((or (endp p) (endp q))
102 (setf termlist1 p))
103 (multiple-value-bind
104 (greater-p equal-p)
105 (lex> (lt q) (lt p))
106 (cond
107 (greater-p
108 ;; P' <- Q
109 ;; (CDR P') <- P
110 ;; Q' <- (CDR Q)
111 (rotatef p q (cdr q))
112 (print p) (print q))
113 (equal-p
114 (setf (lc p) (add-to (lc p) (lc q))
115 p (cdr p)
116 q (cdr q))
117 (print p) (print q))))))))
118
119 self)
120
121(defmethod subtract-from ((self poly) (other poly)))
122
123(defmethod unary-uminus ((self poly)))
124
125#|
126
127(defun poly-standard-extension (plist &aux (k (length plist)))
128 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
129 (declare (list plist) (fixnum k))
130 (labels ((incf-power (g i)
131 (dolist (x (poly-termlist g))
132 (incf (monom-elt (term-monom x) i)))
133 (incf (poly-sugar g))))
134 (setf plist (poly-list-add-variables plist k))
135 (dotimes (i k plist)
136 (incf-power (nth i plist) i))))
137
138(defun saturation-extension (ring f plist
139 &aux
140 (k (length plist))
141 (d (monom-dimension (poly-lm (car plist))))
142 f-x plist-x)
143 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
144 (declare (type ring ring))
145 (setf f-x (poly-list-add-variables f k)
146 plist-x (mapcar #'(lambda (x)
147 (setf (poly-termlist x)
148 (nconc (poly-termlist x)
149 (list (make-term :monom (make-monom :dimension d)
150 :coeff (funcall (ring-uminus ring)
151 (funcall (ring-unit ring)))))))
152 x)
153 (poly-standard-extension plist)))
154 (append f-x plist-x))
155
156
157(defun polysaturation-extension (ring f plist
158 &aux
159 (k (length plist))
160 (d (+ k (monom-dimension (poly-lm (car plist)))))
161 ;; Add k variables to f
162 (f (poly-list-add-variables f k))
163 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
164 (plist (apply #'poly-append (poly-standard-extension plist))))
165 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
166 ;; Add -1 as the last term
167 (declare (type ring ring))
168 (setf (cdr (last (poly-termlist plist)))
169 (list (make-term :monom (make-monom :dimension d)
170 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
171 (append f (list plist)))
172
173(defun saturation-extension-1 (ring f p)
174 "Calculate [F, U*P-1]. It destructively modifies F."
175 (declare (type ring ring))
176 (polysaturation-extension ring f (list p)))
177
178;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
179;;
180;; Evaluation of polynomial (prefix) expressions
181;;
182;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
183
184(defun coerce-coeff (ring expr vars)
185 "Coerce an element of the coefficient ring to a constant polynomial."
186 ;; Modular arithmetic handler by rat
187 (declare (type ring ring))
188 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
189 :coeff (funcall (ring-parse ring) expr)))
190 0))
191
192(defun poly-eval (expr vars
193 &optional
194 (ring +ring-of-integers+)
195 (order #'lex>)
196 (list-marker :[)
197 &aux
198 (ring-and-order (make-ring-and-order :ring ring :order order)))
199 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
200variables VARS. Return the resulting polynomial or list of
201polynomials. Standard arithmetical operators in form EXPR are
202replaced with their analogues in the ring of polynomials, and the
203resulting expression is evaluated, resulting in a polynomial or a list
204of polynomials in internal form. A similar operation in another computer
205algebra system could be called 'expand' or so."
206 (declare (type ring ring))
207 (labels ((p-eval (arg) (poly-eval arg vars ring order))
208 (p-eval-scalar (arg) (poly-eval-scalar arg))
209 (p-eval-list (args) (mapcar #'p-eval args))
210 (p-add (x y) (poly-add ring-and-order x y)))
211 (cond
212 ((null expr) (error "Empty expression"))
213 ((eql expr 0) (make-poly-zero))
214 ((member expr vars :test #'equalp)
215 (let ((pos (position expr vars :test #'equalp)))
216 (make-poly-variable ring (length vars) pos)))
217 ((atom expr)
218 (coerce-coeff ring expr vars))
219 ((eq (car expr) list-marker)
220 (cons list-marker (p-eval-list (cdr expr))))
221 (t
222 (case (car expr)
223 (+ (reduce #'p-add (p-eval-list (cdr expr))))
224 (- (case (length expr)
225 (1 (make-poly-zero))
226 (2 (poly-uminus ring (p-eval (cadr expr))))
227 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
228 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
229 (reduce #'p-add (p-eval-list (cddr expr)))))))
230 (*
231 (if (endp (cddr expr)) ;unary
232 (p-eval (cdr expr))
233 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
234 (/
235 ;; A polynomial can be divided by a scalar
236 (cond
237 ((endp (cddr expr))
238 ;; A special case (/ ?), the inverse
239 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
240 (t
241 (let ((num (p-eval (cadr expr)))
242 (denom-inverse (apply (ring-div ring)
243 (cons (funcall (ring-unit ring))
244 (mapcar #'p-eval-scalar (cddr expr))))))
245 (scalar-times-poly ring denom-inverse num)))))
246 (expt
247 (cond
248 ((member (cadr expr) vars :test #'equalp)
249 ;;Special handling of (expt var pow)
250 (let ((pos (position (cadr expr) vars :test #'equalp)))
251 (make-poly-variable ring (length vars) pos (caddr expr))))
252 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
253 ;; Negative power means division in coefficient ring
254 ;; Non-integer power means non-polynomial coefficient
255 (coerce-coeff ring expr vars))
256 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
257 (otherwise
258 (coerce-coeff ring expr vars)))))))
259
260(defun poly-eval-scalar (expr
261 &optional
262 (ring +ring-of-integers+)
263 &aux
264 (order #'lex>))
265 "Evaluate a scalar expression EXPR in ring RING."
266 (declare (type ring ring))
267 (poly-lc (poly-eval expr nil ring order)))
268
269(defun spoly (ring-and-order f g
270 &aux
271 (ring (ro-ring ring-and-order)))
272 "It yields the S-polynomial of polynomials F and G."
273 (declare (type ring-and-order ring-and-order) (type poly f g))
274 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
275 (mf (monom-div lcm (poly-lm f)))
276 (mg (monom-div lcm (poly-lm g))))
277 (declare (type monom mf mg))
278 (multiple-value-bind (c cf cg)
279 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
280 (declare (ignore c))
281 (poly-sub
282 ring-and-order
283 (scalar-times-poly ring cg (monom-times-poly mf f))
284 (scalar-times-poly ring cf (monom-times-poly mg g))))))
285
286
287(defun poly-primitive-part (ring p)
288 "Divide polynomial P with integer coefficients by gcd of its
289coefficients and return the result."
290 (declare (type ring ring) (type poly p))
291 (if (poly-zerop p)
292 (values p 1)
293 (let ((c (poly-content ring p)))
294 (values (make-poly-from-termlist
295 (mapcar
296 #'(lambda (x)
297 (make-term :monom (term-monom x)
298 :coeff (funcall (ring-div ring) (term-coeff x) c)))
299 (poly-termlist p))
300 (poly-sugar p))
301 c))))
302
303(defun poly-content (ring p)
304 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
305to compute the greatest common divisor."
306 (declare (type ring ring) (type poly p))
307 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
308
309(defun read-infix-form (&key (stream t))
310 "Parser of infix expressions with integer/rational coefficients
311The parser will recognize two kinds of polynomial expressions:
312
313- polynomials in fully expanded forms with coefficients
314 written in front of symbolic expressions; constants can be optionally
315 enclosed in (); for example, the infix form
316 X^2-Y^2+(-4/3)*U^2*W^3-5
317 parses to
318 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
319
320- lists of polynomials; for example
321 [X-Y, X^2+3*Z]
322 parses to
323 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
324 where the first symbol [ marks a list of polynomials.
325
326-other infix expressions, for example
327 [(X-Y)*(X+Y)/Z,(X+1)^2]
328parses to:
329 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
330Currently this function is implemented using M. Kantrowitz's INFIX package."
331 (read-from-string
332 (concatenate 'string
333 "#I("
334 (with-output-to-string (s)
335 (loop
336 (multiple-value-bind (line eof)
337 (read-line stream t)
338 (format s "~A" line)
339 (when eof (return)))))
340 ")")))
341
342(defun read-poly (vars &key
343 (stream t)
344 (ring +ring-of-integers+)
345 (order #'lex>))
346 "Reads an expression in prefix form from a stream STREAM.
347The expression read from the strem should represent a polynomial or a
348list of polynomials in variables VARS, over the ring RING. The
349polynomial or list of polynomials is returned, with terms in each
350polynomial ordered according to monomial order ORDER."
351 (poly-eval (read-infix-form :stream stream) vars ring order))
352
353(defun string->poly (str vars
354 &optional
355 (ring +ring-of-integers+)
356 (order #'lex>))
357 "Converts a string STR to a polynomial in variables VARS."
358 (with-input-from-string (s str)
359 (read-poly vars :stream s :ring ring :order order)))
360
361(defun poly->alist (p)
362 "Convert a polynomial P to an association list. Thus, the format of the
363returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
364MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
365corresponding coefficient in the ring."
366 (cond
367 ((poly-p p)
368 (mapcar #'term->cons (poly-termlist p)))
369 ((and (consp p) (eq (car p) :[))
370 (cons :[ (mapcar #'poly->alist (cdr p))))))
371
372(defun string->alist (str vars
373 &optional
374 (ring +ring-of-integers+)
375 (order #'lex>))
376 "Convert a string STR representing a polynomial or polynomial list to
377an association list (... (MONOM . COEFF) ...)."
378 (poly->alist (string->poly str vars ring order)))
379
380(defun poly-equal-no-sugar-p (p q)
381 "Compare polynomials for equality, ignoring sugar."
382 (declare (type poly p q))
383 (equalp (poly-termlist p) (poly-termlist q)))
384
385(defun poly-set-equal-no-sugar-p (p q)
386 "Compare polynomial sets P and Q for equality, ignoring sugar."
387 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
388
389(defun poly-list-equal-no-sugar-p (p q)
390 "Compare polynomial lists P and Q for equality, ignoring sugar."
391 (every #'poly-equal-no-sugar-p p q))
392|#
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