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source: branches/f4grobner/polynomial.lisp@ 1111

Last change on this file since 1111 was 1111, checked in by Marek Rychlik, 9 years ago

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1;;; -*- Mode: Lisp; Package: Maxima; Syntax: Common-Lisp; Base: 10 -*-
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-LT"
29 "MAKE-POLY-FROM-TERMLIST"
30 "MAKE-POLY-ZERO"
31 "MAKE-VARIABLE"
32 "POLY-UNIT"
33 "POLY-LM"
34 "POLY-SECOND-LM"
35 "POLY-SECOND-LT"
36 "POLY-LC"
37 "POLY-SECOND-LC"
38 "POLY-ZEROP"
39 "POLY-LENGTH"
40 "SCALAR-TIMES-POLY"
41 "SCALAR-TIMES-POLY-1"
42 "MONOM-TIMES-POLY"
43 "TERM-TIMES-POLY"
44 "POLY-ADD"
45 "POLY-SUB"
46 "POLY-UMINUS"
47 "POLY-MUL"
48 "POLY-EXPT"
49 "POLY-APPEND"
50 "POLY-NREVERSE"
51 "POLY-CONTRACT"
52 "POLY-EXTEND"
53 "POLY-ADD-VARIABLES"
54 "POLY-LIST-ADD-VARIABLES"
55 "POLY-STANDARD-EXTENSION"
56 "SATURATION-EXTENSION"
57 "POLYSATURATION-EXTENSION"
58 "SATURATION-EXTENSION-1"
59 "COERCE-COEFF"
60 "POLY-EVAL"
61 "SPOLY"
62 "POLY-PRIMITIVE-PART"
63 "POLY-CONTENT"
64 "READ-INFIX-FORM"
65 "READ-POLY"
66 "STRING->POLY"
67 ))
68
69(in-package :polynomial)
70
71;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
72;;
73;; Polynomials
74;;
75;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
76
77(defstruct (poly
78 ;;
79 ;; BOA constructor, by default constructs zero polynomial
80 (:constructor make-poly-from-termlist (termlist &optional (sugar (termlist-sugar termlist))))
81 (:constructor make-poly-zero (&aux (termlist nil) (sugar -1)))
82 ;; Constructor of polynomials representing a variable
83 (:constructor make-variable (ring nvars pos &optional (power 1)
84 &aux
85 (termlist (list
86 (make-term-variable ring nvars pos power)))
87 (sugar power)))
88 (:constructor poly-unit (ring dimension
89 &aux
90 (termlist (termlist-unit ring dimension))
91 (sugar 0))))
92 (termlist nil :type list)
93 (sugar -1 :type fixnum))
94
95;; Leading term
96(defmacro poly-lt (p) `(car (poly-termlist ,p)))
97
98;; Second term
99(defmacro poly-second-lt (p) `(cadar (poly-termlist ,p)))
100
101;; Leading monomial
102(defun poly-lm (p) (term-monom (poly-lt p)))
103
104;; Second monomial
105(defun poly-second-lm (p) (term-monom (poly-second-lt p)))
106
107;; Leading coefficient
108(defun poly-lc (p) (term-coeff (poly-lt p)))
109
110;; Second coefficient
111(defun poly-second-lc (p) (term-coeff (poly-second-lt p)))
112
113;; Testing for a zero polynomial
114(defun poly-zerop (p) (null (poly-termlist p)))
115
116;; The number of terms
117(defun poly-length (p) (length (poly-termlist p)))
118
119(defun scalar-times-poly (ring c p)
120 (declare (type ring ring) (poly p))
121 (make-poly-from-termlist (scalar-times-termlist ring c (poly-termlist p)) (poly-sugar p)))
122
123;; The scalar product omitting the head term
124(defun scalar-times-poly-1 (ring c p)
125 (declare (type ring ring) (poly p))
126 (make-poly-from-termlist (scalar-times-termlist ring c (cdr (poly-termlist p))) (poly-sugar p)))
127
128(defun monom-times-poly (m p)
129 (declare (poly p))
130 (make-poly-from-termlist
131 (monom-times-termlist m (poly-termlist p))
132 (+ (poly-sugar p) (monom-sugar m))))
133
134(defun term-times-poly (ring term p)
135 (declare (type ring ring) (type term term) (type poly p))
136 (make-poly-from-termlist
137 (term-times-termlist ring term (poly-termlist p))
138 (+ (poly-sugar p) (term-sugar term))))
139
140(defun poly-add (ring-and-order p q)
141 (declare (type ring-and-order ring-and-order) (type poly p q))
142 (make-poly-from-termlist
143 (termlist-add ring-and-order
144 (poly-termlist p)
145 (poly-termlist q))
146 (max (poly-sugar p) (poly-sugar q))))
147
148(defun poly-sub (ring-and-order p q)
149 (declare (type ring-and-order ring-and-order) (type poly p q))
150 (make-poly-from-termlist
151 (termlist-sub ring-and-order (poly-termlist p) (poly-termlist q))
152 (max (poly-sugar p) (poly-sugar q))))
153
154(defun poly-uminus (ring p)
155 (declare (type ring ring) (type poly p))
156 (make-poly-from-termlist
157 (termlist-uminus ring (poly-termlist p))
158 (poly-sugar p)))
159
160(defun poly-mul (ring-and-order p q)
161 (declare (type ring-and-order ring-and-order) (type poly p q))
162 (make-poly-from-termlist
163 (termlist-mul ring-and-order (poly-termlist p) (poly-termlist q))
164 (+ (poly-sugar p) (poly-sugar q))))
165
166(defun poly-expt (ring-and-order p n)
167 (declare (type ring-and-order ring-and-order) (type poly p))
168 (make-poly-from-termlist (termlist-expt ring-and-order (poly-termlist p) n) (* n (poly-sugar p))))
169
170(defun poly-append (&rest plist)
171 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
172 (apply #'max (mapcar #'poly-sugar plist))))
173
174(defun poly-nreverse (p)
175 (declare (type poly p))
176 (setf (poly-termlist p) (nreverse (poly-termlist p)))
177 p)
178
179(defun poly-contract (p &optional (k 1))
180 (declare (type poly p))
181 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
182 (poly-sugar p)))
183
184(defun poly-extend (p &optional (m (make-monom :dimension 1)))
185 (declare (type poly p))
186 (make-poly-from-termlist
187 (termlist-extend (poly-termlist p) m)
188 (+ (poly-sugar p) (monom-sugar m))))
189
190(defun poly-add-variables (p k)
191 (declare (type poly p))
192 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
193 p)
194
195(defun poly-list-add-variables (plist k)
196 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
197
198(defun poly-standard-extension (plist &aux (k (length plist)))
199 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
200 (declare (list plist) (fixnum k))
201 (labels ((incf-power (g i)
202 (dolist (x (poly-termlist g))
203 (incf (monom-elt (term-monom x) i)))
204 (incf (poly-sugar g))))
205 (setf plist (poly-list-add-variables plist k))
206 (dotimes (i k plist)
207 (incf-power (nth i plist) i))))
208
209(defun saturation-extension (ring f plist &aux (k (length plist)) (d (monom-dimension (poly-lm (car plist)))))
210 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
211 (setf f (poly-list-add-variables f k)
212 plist (mapcar #'(lambda (x)
213 (setf (poly-termlist x) (nconc (poly-termlist x)
214 (list (make-term (make-monom :dimension d)
215 (funcall (ring-uminus ring) (funcall (ring-unit ring)))))))
216 x)
217 (poly-standard-extension plist)))
218 (append f plist))
219
220
221(defun polysaturation-extension (ring f plist &aux (k (length plist))
222 (d (+ k (monom-dimension (poly-lm (car plist))))))
223 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]."
224 (setf f (poly-list-add-variables f k)
225 plist (apply #'poly-append (poly-standard-extension plist))
226 (cdr (last (poly-termlist plist))) (list (make-term (make-monom :dimension d)
227 (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
228 (append f (list plist)))
229
230(defun saturation-extension-1 (ring f p) (polysaturation-extension ring f (list p)))
231
232;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
233;;
234;; Evaluation of polynomial (prefix) expressions
235;;
236;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
237
238(defun coerce-coeff (ring expr vars)
239 "Coerce an element of the coefficient ring to a constant polynomial."
240 ;; Modular arithmetic handler by rat
241 (make-poly-from-termlist (list (make-term (make-monom :dimension (length vars))
242 (funcall (ring-parse ring) expr)))
243 0))
244
245(defun poly-eval (expr vars
246 &optional
247 (ring *ring-of-integers*)
248 (order #'lex>)
249 (list-marker '[)
250 &aux
251 (ring-and-order (make-ring-and-order :ring ring :order order)))
252 (labels ((p-eval (arg) (poly-eval arg vars ring order))
253 (p-eval-list (args) (mapcar #'p-eval args))
254 (p-add (x y) (poly-add ring-and-order x y)))
255 (cond
256 ((eql expr 0) (make-poly-zero))
257 ((member expr vars :test #'equalp)
258 (let ((pos (position expr vars :test #'equalp)))
259 (make-variable ring (length vars) pos)))
260 ((atom expr)
261 (coerce-coeff ring expr vars))
262 ((eq (car expr) list-marker)
263 (cons list-marker (p-eval-list (cdr expr))))
264 (t
265 (case (car expr)
266 (+ (reduce #'p-add (p-eval-list (cdr expr))))
267 (- (case (length expr)
268 (1 (make-poly-zero))
269 (2 (poly-uminus ring (p-eval (cadr expr))))
270 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
271 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
272 (reduce #'p-add (p-eval-list (cddr expr)))))))
273 (*
274 (if (endp (cddr expr)) ;unary
275 (p-eval (cdr expr))
276 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
277 (/
278 (print expr)
279 ;; A polynomial can be divided by a scalar
280 (cond ((endp (cddr expr))
281 ;; A special case (/ x)
282 (apply (ring-div ring) (cdr expr)))
283 (t
284 (let ((num (p-eval (cons '* (cdr (butlast expr)))))
285 (denom (p-eval-scalar (cons '/ (last expr)))))))))
286 (expt
287 (cond
288 ((member (cadr expr) vars :test #'equalp)
289 ;;Special handling of (expt var pow)
290 (let ((pos (position (cadr expr) vars :test #'equalp)))
291 (make-variable ring (length vars) pos (caddr expr))))
292 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
293 ;; Negative power means division in coefficient ring
294 ;; Non-integer power means non-polynomial coefficient
295 (coerce-coeff ring expr vars))
296 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
297 (otherwise
298 (coerce-coeff ring expr vars)))))))
299
300(defun spoly (ring f g)
301 "It yields the S-polynomial of polynomials F and G."
302 (declare (type poly f g))
303 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
304 (mf (monom-div lcm (poly-lm f)))
305 (mg (monom-div lcm (poly-lm g))))
306 (declare (type monom mf mg))
307 (multiple-value-bind (c cf cg)
308 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
309 (declare (ignore c))
310 (poly-sub
311 ring
312 (scalar-times-poly ring cg (monom-times-poly mf f))
313 (scalar-times-poly ring cf (monom-times-poly mg g))))))
314
315
316(defun poly-primitive-part (ring p)
317 "Divide polynomial P with integer coefficients by gcd of its
318coefficients and return the result."
319 (declare (type poly p))
320 (if (poly-zerop p)
321 (values p 1)
322 (let ((c (poly-content ring p)))
323 (values (make-poly-from-termlist (mapcar
324 #'(lambda (x)
325 (make-term (term-monom x)
326 (funcall (ring-div ring) (term-coeff x) c)))
327 (poly-termlist p))
328 (poly-sugar p))
329 c))))
330
331(defun poly-content (ring p)
332 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
333to compute the greatest common divisor."
334 (declare (type poly p))
335 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
336
337(defun read-infix-form (&key (stream t))
338 "Parser of infix expressions with integer/rational coefficients
339The parser will recognize two kinds of polynomial expressions:
340
341- polynomials in fully expanded forms with coefficients
342 written in front of symbolic expressions; constants can be optionally
343 enclosed in (); for example, the infix form
344 X^2-Y^2+(-4/3)*U^2*W^3-5
345 parses to
346 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
347
348- lists of polynomials; for example
349 [X-Y, X^2+3*Z]
350 parses to
351 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
352 where the first symbol [ marks a list of polynomials.
353
354-other infix expressions, for example
355 [(X-Y)*(X+Y)/Z,(X+1)^2]
356parses to:
357 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
358Currently this function is implemented using M. Kantrowitz's INFIX package."
359 (read-from-string
360 (concatenate 'string
361 "#I("
362 (with-output-to-string (s)
363 (loop
364 (multiple-value-bind (line eof)
365 (read-line stream t)
366 (format s "~A" line)
367 (when eof (return)))))
368 ")")))
369
370(defun read-poly (vars &key (stream t))
371 "Reads an expression in prefix form from a stream STREAM.
372If the expression represents a polynomial or a list of polynomials in variables VARS then
373the polynomial or list of polynomials is returned."
374 (poly-eval (read-infix-form :stream stream) vars))
375
376(defun string->poly (str vars)
377 "Converts a string STR to a polynomial in variables VARS"
378 (with-input-from-string (s str)
379 (read-poly vars :stream s)))
380
381
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