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source: branches/f4grobner/polynomial.lisp@ 2448

Last change on this file since 2448 was 2448, 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 "POLYNOMIAL"
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 :polynomial)
83
84(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
85
86#|
87 ;;
88 ;; BOA constructor, by default constructs zero polynomial
89 (:constructor make-poly-from-termlist (termlist &optional (sugar (termlist-sugar termlist))))
90 (:constructor make-poly-zero (&aux (termlist nil) (sugar -1)))
91 ;; Constructor of polynomials representing a variable
92 (:constructor make-poly-variable (ring nvars pos &optional (power 1)
93 &aux
94 (termlist (list
95 (make-term-variable ring nvars pos power)))
96 (sugar power)))
97 (:constructor poly-unit (ring dimension
98 &aux
99 (termlist (termlist-unit ring dimension))
100 (sugar 0))))
101
102|#
103
104(defclass poly ()
105 ((termlist :initarg :terms :accessor poly-termlist))
106 (:default-initargs :termlist nil))
107
108;; Leading term
109(defgeneric leading-term (object)
110 (:method ((self poly))
111 (car (poly-termlist self))))
112
113;; Second term
114(defgeneric second-leading-term (object)
115 (:method ((self poly))
116 (cadar (poly-termlist self))))
117
118;; Leading coefficient
119(defgeneric leading-coefficient (object)
120 (:method ((self poly))
121 (r-coeff (leading-term self))))
122
123;; Second coefficient
124(defgeneric second-leading-coefficient (object)
125 (:method ((self poly))
126 (term-coeff (second-leading-term self))))
127
128;; Testing for a zero polynomial
129(defmethod r-zerop ((self poly))
130 (null (poly-termlist self)))
131
132;; The number of terms
133(defmethod r-length ((self poly))
134 (length (poly-termlist self)))
135
136(defgeneric multiply-by (self other)
137 (:method ((self poly) (other scalar))
138 (mapc #'(lambda (term) (multiply-by term other)) (poly-termlist self))
139 self)
140 (:method ((self poly) (other monom))
141 (mapc #'(lambda (term) (multiply-by term monom)) (poly-termlist self))
142 self))
143
144(defgeneric add-to (self other)
145 (:method ((self poly) (other poly))))
146
147(defgeneric subtract-from (self other)
148 (:method ((self poly) (other poly))))
149
150(defmethod unary-uminus (self))
151
152(defun poly-append (&rest plist)
153 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
154 (apply #'max (mapcar #'poly-sugar plist))))
155
156(defun poly-nreverse (p)
157 "Destructively reverse the order of terms in polynomial P. Returns P"
158 (declare (type poly p))
159 (setf (poly-termlist p) (nreverse (poly-termlist p)))
160 p)
161
162(defun poly-reverse (p)
163 "Returns a copy of the polynomial P with terms in reverse order."
164 (declare (type poly p))
165 (make-poly-from-termlist (reverse (poly-termlist p))
166 (poly-sugar p)))
167
168
169(defun poly-contract (p &optional (k 1))
170 (declare (type poly p))
171 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
172 (poly-sugar p)))
173
174(defun poly-extend (p &optional (m (make-monom :dimension 1)))
175 (declare (type poly p))
176 (make-poly-from-termlist
177 (termlist-extend (poly-termlist p) m)
178 (+ (poly-sugar p) (monom-sugar m))))
179
180(defun poly-add-variables (p k)
181 (declare (type poly p))
182 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
183 p)
184
185(defun poly-list-add-variables (plist k)
186 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
187
188(defun poly-standard-extension (plist &aux (k (length plist)))
189 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
190 (declare (list plist) (fixnum k))
191 (labels ((incf-power (g i)
192 (dolist (x (poly-termlist g))
193 (incf (monom-elt (term-monom x) i)))
194 (incf (poly-sugar g))))
195 (setf plist (poly-list-add-variables plist k))
196 (dotimes (i k plist)
197 (incf-power (nth i plist) i))))
198
199(defun saturation-extension (ring f plist
200 &aux
201 (k (length plist))
202 (d (monom-dimension (poly-lm (car plist))))
203 f-x plist-x)
204 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
205 (declare (type ring ring))
206 (setf f-x (poly-list-add-variables f k)
207 plist-x (mapcar #'(lambda (x)
208 (setf (poly-termlist x)
209 (nconc (poly-termlist x)
210 (list (make-term :monom (make-monom :dimension d)
211 :coeff (funcall (ring-uminus ring)
212 (funcall (ring-unit ring)))))))
213 x)
214 (poly-standard-extension plist)))
215 (append f-x plist-x))
216
217
218(defun polysaturation-extension (ring f plist
219 &aux
220 (k (length plist))
221 (d (+ k (monom-dimension (poly-lm (car plist)))))
222 ;; Add k variables to f
223 (f (poly-list-add-variables f k))
224 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
225 (plist (apply #'poly-append (poly-standard-extension plist))))
226 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
227 ;; Add -1 as the last term
228 (declare (type ring ring))
229 (setf (cdr (last (poly-termlist plist)))
230 (list (make-term :monom (make-monom :dimension d)
231 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
232 (append f (list plist)))
233
234(defun saturation-extension-1 (ring f p)
235 "Calculate [F, U*P-1]. It destructively modifies F."
236 (declare (type ring ring))
237 (polysaturation-extension ring f (list p)))
238
239;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
240;;
241;; Evaluation of polynomial (prefix) expressions
242;;
243;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
244
245(defun coerce-coeff (ring expr vars)
246 "Coerce an element of the coefficient ring to a constant polynomial."
247 ;; Modular arithmetic handler by rat
248 (declare (type ring ring))
249 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
250 :coeff (funcall (ring-parse ring) expr)))
251 0))
252
253(defun poly-eval (expr vars
254 &optional
255 (ring +ring-of-integers+)
256 (order #'lex>)
257 (list-marker :[)
258 &aux
259 (ring-and-order (make-ring-and-order :ring ring :order order)))
260 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
261variables VARS. Return the resulting polynomial or list of
262polynomials. Standard arithmetical operators in form EXPR are
263replaced with their analogues in the ring of polynomials, and the
264resulting expression is evaluated, resulting in a polynomial or a list
265of polynomials in internal form. A similar operation in another computer
266algebra system could be called 'expand' or so."
267 (declare (type ring ring))
268 (labels ((p-eval (arg) (poly-eval arg vars ring order))
269 (p-eval-scalar (arg) (poly-eval-scalar arg))
270 (p-eval-list (args) (mapcar #'p-eval args))
271 (p-add (x y) (poly-add ring-and-order x y)))
272 (cond
273 ((null expr) (error "Empty expression"))
274 ((eql expr 0) (make-poly-zero))
275 ((member expr vars :test #'equalp)
276 (let ((pos (position expr vars :test #'equalp)))
277 (make-poly-variable ring (length vars) pos)))
278 ((atom expr)
279 (coerce-coeff ring expr vars))
280 ((eq (car expr) list-marker)
281 (cons list-marker (p-eval-list (cdr expr))))
282 (t
283 (case (car expr)
284 (+ (reduce #'p-add (p-eval-list (cdr expr))))
285 (- (case (length expr)
286 (1 (make-poly-zero))
287 (2 (poly-uminus ring (p-eval (cadr expr))))
288 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
289 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
290 (reduce #'p-add (p-eval-list (cddr expr)))))))
291 (*
292 (if (endp (cddr expr)) ;unary
293 (p-eval (cdr expr))
294 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
295 (/
296 ;; A polynomial can be divided by a scalar
297 (cond
298 ((endp (cddr expr))
299 ;; A special case (/ ?), the inverse
300 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
301 (t
302 (let ((num (p-eval (cadr expr)))
303 (denom-inverse (apply (ring-div ring)
304 (cons (funcall (ring-unit ring))
305 (mapcar #'p-eval-scalar (cddr expr))))))
306 (scalar-times-poly ring denom-inverse num)))))
307 (expt
308 (cond
309 ((member (cadr expr) vars :test #'equalp)
310 ;;Special handling of (expt var pow)
311 (let ((pos (position (cadr expr) vars :test #'equalp)))
312 (make-poly-variable ring (length vars) pos (caddr expr))))
313 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
314 ;; Negative power means division in coefficient ring
315 ;; Non-integer power means non-polynomial coefficient
316 (coerce-coeff ring expr vars))
317 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
318 (otherwise
319 (coerce-coeff ring expr vars)))))))
320
321(defun poly-eval-scalar (expr
322 &optional
323 (ring +ring-of-integers+)
324 &aux
325 (order #'lex>))
326 "Evaluate a scalar expression EXPR in ring RING."
327 (declare (type ring ring))
328 (poly-lc (poly-eval expr nil ring order)))
329
330(defun spoly (ring-and-order f g
331 &aux
332 (ring (ro-ring ring-and-order)))
333 "It yields the S-polynomial of polynomials F and G."
334 (declare (type ring-and-order ring-and-order) (type poly f g))
335 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
336 (mf (monom-div lcm (poly-lm f)))
337 (mg (monom-div lcm (poly-lm g))))
338 (declare (type monom mf mg))
339 (multiple-value-bind (c cf cg)
340 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
341 (declare (ignore c))
342 (poly-sub
343 ring-and-order
344 (scalar-times-poly ring cg (monom-times-poly mf f))
345 (scalar-times-poly ring cf (monom-times-poly mg g))))))
346
347
348(defun poly-primitive-part (ring p)
349 "Divide polynomial P with integer coefficients by gcd of its
350coefficients and return the result."
351 (declare (type ring ring) (type poly p))
352 (if (poly-zerop p)
353 (values p 1)
354 (let ((c (poly-content ring p)))
355 (values (make-poly-from-termlist
356 (mapcar
357 #'(lambda (x)
358 (make-term :monom (term-monom x)
359 :coeff (funcall (ring-div ring) (term-coeff x) c)))
360 (poly-termlist p))
361 (poly-sugar p))
362 c))))
363
364(defun poly-content (ring p)
365 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
366to compute the greatest common divisor."
367 (declare (type ring ring) (type poly p))
368 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
369
370(defun read-infix-form (&key (stream t))
371 "Parser of infix expressions with integer/rational coefficients
372The parser will recognize two kinds of polynomial expressions:
373
374- polynomials in fully expanded forms with coefficients
375 written in front of symbolic expressions; constants can be optionally
376 enclosed in (); for example, the infix form
377 X^2-Y^2+(-4/3)*U^2*W^3-5
378 parses to
379 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
380
381- lists of polynomials; for example
382 [X-Y, X^2+3*Z]
383 parses to
384 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
385 where the first symbol [ marks a list of polynomials.
386
387-other infix expressions, for example
388 [(X-Y)*(X+Y)/Z,(X+1)^2]
389parses to:
390 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
391Currently this function is implemented using M. Kantrowitz's INFIX package."
392 (read-from-string
393 (concatenate 'string
394 "#I("
395 (with-output-to-string (s)
396 (loop
397 (multiple-value-bind (line eof)
398 (read-line stream t)
399 (format s "~A" line)
400 (when eof (return)))))
401 ")")))
402
403(defun read-poly (vars &key
404 (stream t)
405 (ring +ring-of-integers+)
406 (order #'lex>))
407 "Reads an expression in prefix form from a stream STREAM.
408The expression read from the strem should represent a polynomial or a
409list of polynomials in variables VARS, over the ring RING. The
410polynomial or list of polynomials is returned, with terms in each
411polynomial ordered according to monomial order ORDER."
412 (poly-eval (read-infix-form :stream stream) vars ring order))
413
414(defun string->poly (str vars
415 &optional
416 (ring +ring-of-integers+)
417 (order #'lex>))
418 "Converts a string STR to a polynomial in variables VARS."
419 (with-input-from-string (s str)
420 (read-poly vars :stream s :ring ring :order order)))
421
422(defun poly->alist (p)
423 "Convert a polynomial P to an association list. Thus, the format of the
424returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
425MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
426corresponding coefficient in the ring."
427 (cond
428 ((poly-p p)
429 (mapcar #'term->cons (poly-termlist p)))
430 ((and (consp p) (eq (car p) :[))
431 (cons :[ (mapcar #'poly->alist (cdr p))))))
432
433(defun string->alist (str vars
434 &optional
435 (ring +ring-of-integers+)
436 (order #'lex>))
437 "Convert a string STR representing a polynomial or polynomial list to
438an association list (... (MONOM . COEFF) ...)."
439 (poly->alist (string->poly str vars ring order)))
440
441(defun poly-equal-no-sugar-p (p q)
442 "Compare polynomials for equality, ignoring sugar."
443 (declare (type poly p q))
444 (equalp (poly-termlist p) (poly-termlist q)))
445
446(defun poly-set-equal-no-sugar-p (p q)
447 "Compare polynomial sets P and Q for equality, ignoring sugar."
448 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
449
450(defun poly-list-equal-no-sugar-p (p q)
451 "Compare polynomial lists P and Q for equality, ignoring sugar."
452 (every #'poly-equal-no-sugar-p p q))
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