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source: branches/f4grobner/polynomial.lisp@ 1065

Last change on this file since 1065 was 1054, 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)
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 ))
65
66(in-package :polynomial)
67
68;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
69;;
70;; Polynomials
71;;
72;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
73
74(defstruct (poly
75 ;;
76 ;; BOA constructor, by default constructs zero polynomial
77 (:constructor make-poly-from-termlist (termlist &optional (sugar (termlist-sugar termlist))))
78 (:constructor make-poly-zero (&aux (termlist nil) (sugar -1)))
79 ;; Constructor of polynomials representing a variable
80 (:constructor make-variable (ring nvars pos &optional (power 1)
81 &aux
82 (termlist (list
83 (make-term-variable ring nvars pos power)))
84 (sugar power)))
85 (:constructor poly-unit (ring dimension
86 &aux
87 (termlist (termlist-unit ring dimension))
88 (sugar 0))))
89 (termlist nil :type list)
90 (sugar -1 :type fixnum))
91
92;; Leading term
93(defmacro poly-lt (p) `(car (poly-termlist ,p)))
94
95;; Second term
96(defmacro poly-second-lt (p) `(cadar (poly-termlist ,p)))
97
98;; Leading monomial
99(defun poly-lm (p) (term-monom (poly-lt p)))
100
101;; Second monomial
102(defun poly-second-lm (p) (term-monom (poly-second-lt p)))
103
104;; Leading coefficient
105(defun poly-lc (p) (term-coeff (poly-lt p)))
106
107;; Second coefficient
108(defun poly-second-lc (p) (term-coeff (poly-second-lt p)))
109
110;; Testing for a zero polynomial
111(defun poly-zerop (p) (null (poly-termlist p)))
112
113;; The number of terms
114(defun poly-length (p) (length (poly-termlist p)))
115
116(defun scalar-times-poly (ring c p)
117 (declare (type ring ring) (poly p))
118 (make-poly-from-termlist (scalar-times-termlist ring c (poly-termlist p)) (poly-sugar p)))
119
120;; The scalar product omitting the head term
121(defun scalar-times-poly-1 (ring c p)
122 (declare (type ring ring) (poly p))
123 (make-poly-from-termlist (scalar-times-termlist ring c (cdr (poly-termlist p))) (poly-sugar p)))
124
125(defun monom-times-poly (m p)
126 (declare (poly p))
127 (make-poly-from-termlist
128 (monom-times-termlist m (poly-termlist p))
129 (+ (poly-sugar p) (monom-sugar m))))
130
131(defun term-times-poly (ring term p)
132 (declare (type ring ring) (type term term) (type poly p))
133 (make-poly-from-termlist
134 (term-times-termlist ring term (poly-termlist p))
135 (+ (poly-sugar p) (term-sugar term))))
136
137(defun poly-add (ring-and-order p q)
138 (declare (type ring-and-order ring-and-order) (type poly p q))
139 (make-poly-from-termlist
140 (termlist-add ring-and-order
141 (poly-termlist p)
142 (poly-termlist q))
143 (max (poly-sugar p) (poly-sugar q))))
144
145(defun poly-sub (ring-and-order p q)
146 (declare (type ring-and-order ring-and-order) (type poly p q))
147 (make-poly-from-termlist
148 (termlist-sub ring-and-order (poly-termlist p) (poly-termlist q))
149 (max (poly-sugar p) (poly-sugar q))))
150
151(defun poly-uminus (ring p)
152 (declare (type ring ring) (type poly p))
153 (make-poly-from-termlist
154 (termlist-uminus ring (poly-termlist p))
155 (poly-sugar p)))
156
157(defun poly-mul (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-mul ring-and-order (poly-termlist p) (poly-termlist q))
161 (+ (poly-sugar p) (poly-sugar q))))
162
163(defun poly-expt (ring-and-order p n)
164 (declare (type ring-and-order ring-and-order) (type poly p))
165 (make-poly-from-termlist (termlist-expt ring-and-order (poly-termlist p) n) (* n (poly-sugar p))))
166
167(defun poly-append (&rest plist)
168 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
169 (apply #'max (mapcar #'poly-sugar plist))))
170
171(defun poly-nreverse (p)
172 (declare (type poly p))
173 (setf (poly-termlist p) (nreverse (poly-termlist p)))
174 p)
175
176(defun poly-contract (p &optional (k 1))
177 (declare (type poly p))
178 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
179 (poly-sugar p)))
180
181(defun poly-extend (p &optional (m (make-monom :dimension 1)))
182 (declare (type poly p))
183 (make-poly-from-termlist
184 (termlist-extend (poly-termlist p) m)
185 (+ (poly-sugar p) (monom-sugar m))))
186
187(defun poly-add-variables (p k)
188 (declare (type poly p))
189 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
190 p)
191
192(defun poly-list-add-variables (plist k)
193 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
194
195(defun poly-standard-extension (plist &aux (k (length plist)))
196 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
197 (declare (list plist) (fixnum k))
198 (labels ((incf-power (g i)
199 (dolist (x (poly-termlist g))
200 (incf (monom-elt (term-monom x) i)))
201 (incf (poly-sugar g))))
202 (setf plist (poly-list-add-variables plist k))
203 (dotimes (i k plist)
204 (incf-power (nth i plist) i))))
205
206(defun saturation-extension (ring f plist &aux (k (length plist)) (d (monom-dimension (poly-lm (car plist)))))
207 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
208 (setf f (poly-list-add-variables f k)
209 plist (mapcar #'(lambda (x)
210 (setf (poly-termlist x) (nconc (poly-termlist x)
211 (list (make-term (make-monom :dimension d)
212 (funcall (ring-uminus ring) (funcall (ring-unit ring)))))))
213 x)
214 (poly-standard-extension plist)))
215 (append f plist))
216
217
218(defun polysaturation-extension (ring f plist &aux (k (length plist))
219 (d (+ k (monom-dimension (poly-lm (car plist))))))
220 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]."
221 (setf f (poly-list-add-variables f k)
222 plist (apply #'poly-append (poly-standard-extension plist))
223 (cdr (last (poly-termlist plist))) (list (make-term (make-monom :dimension d)
224 (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
225 (append f (list plist)))
226
227(defun saturation-extension-1 (ring f p) (polysaturation-extension ring f (list p)))
228
229;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
230;;
231;; Evaluation of polynomial (prefix) expressions
232;;
233;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
234
235(defun coerce-coeff (ring expr vars)
236 "Coerce an element of the coefficient ring to a constant polynomial."
237 ;; Modular arithmetic handler by rat
238 (make-poly-from-termlist (list (make-term (make-monom :dimension (length vars))
239 (funcall (ring-parse ring) expr)))
240 0))
241
242(defun poly-eval (expr vars
243 &optional
244 (ring *ring-of-integers*)
245 (order #'lex>)
246 (list-marker '[)
247 &aux
248 (ring-and-order (make-ring-and-order :ring ring :order order)))
249 (labels ((p-eval (arg) (poly-eval arg vars ring order))
250 (p-eval-list (args) (mapcar #'p-eval args))
251 (p-add (x y) (poly-add ring-and-order x y)))
252 (cond
253 ((eql expr 0) (make-poly-zero))
254 ((member expr vars :test #'equalp)
255 (let ((pos (position expr vars :test #'equalp)))
256 (make-variable ring (length vars) pos)))
257 ((atom expr)
258 (coerce-coeff ring expr vars))
259 ((eq (car expr) list-marker)
260 (cons list-marker (p-eval-list (cdr expr))))
261 (t
262 (case (car expr)
263 (+ (reduce #'p-add (p-eval-list (cdr expr))))
264 (- (case (length expr)
265 (1 (make-poly-zero))
266 (2 (poly-uminus ring (p-eval (cadr expr))))
267 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
268 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
269 (reduce #'p-add (p-eval-list (cddr expr)))))))
270 (*
271 (if (endp (cddr expr)) ;unary
272 (p-eval (cdr expr))
273 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
274 (expt
275 (cond
276 ((member (cadr expr) vars :test #'equalp)
277 ;;Special handling of (expt var pow)
278 (let ((pos (position (cadr expr) vars :test #'equalp)))
279 (make-variable ring (length vars) pos (caddr expr))))
280 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
281 ;; Negative power means division in coefficient ring
282 ;; Non-integer power means non-polynomial coefficient
283 (coerce-coeff ring expr vars))
284 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
285 (otherwise
286 (coerce-coeff ring expr vars)))))))
287
288(defun spoly (ring f g)
289 "It yields the S-polynomial of polynomials F and G."
290 (declare (type poly f g))
291 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
292 (mf (monom-div lcm (poly-lm f)))
293 (mg (monom-div lcm (poly-lm g))))
294 (declare (type monom mf mg))
295 (multiple-value-bind (c cf cg)
296 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
297 (declare (ignore c))
298 (poly-sub
299 ring
300 (scalar-times-poly ring cg (monom-times-poly mf f))
301 (scalar-times-poly ring cf (monom-times-poly mg g))))))
302
303
304(defun poly-primitive-part (ring p)
305 "Divide polynomial P with integer coefficients by gcd of its
306coefficients and return the result."
307 (declare (type poly p))
308 (if (poly-zerop p)
309 (values p 1)
310 (let ((c (poly-content ring p)))
311 (values (make-poly-from-termlist (mapcar
312 #'(lambda (x)
313 (make-term (term-monom x)
314 (funcall (ring-div ring) (term-coeff x) c)))
315 (poly-termlist p))
316 (poly-sugar p))
317 c))))
318
319(defun poly-content (ring p)
320 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
321to compute the greatest common divisor."
322 (declare (type poly p))
323 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
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