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source: branches/f4grobner/polynomial.lisp@ 432

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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 :monomial :term :termlist)
25 (:export "POLY"
26 "POLY-TERMLIST"
27 "POLY-SUGAR"
28 "POLY-LT"
29 ))
30
31(in-package :polynomial)
32
33;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
34;;
35;; Polynomials
36;;
37;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
38
39(defstruct (poly
40 ;;
41 ;; BOA constructor, by default constructs zero polynomial
42 (:constructor make-poly-from-termlist (termlist &optional (sugar (termlist-sugar termlist))))
43 (:constructor make-poly-zero (&aux (termlist nil) (sugar -1)))
44 ;; Constructor of polynomials representing a variable
45 (:constructor make-variable (ring nvars pos &optional (power 1)
46 &aux
47 (termlist (list
48 (make-term-variable ring nvars pos power)))
49 (sugar power)))
50 (:constructor poly-unit (ring dimension
51 &aux
52 (termlist (termlist-unit ring dimension))
53 (sugar 0))))
54 (termlist nil :type list)
55 (sugar -1 :type fixnum))
56
57;; Leading term
58(defmacro poly-lt (p) `(car (poly-termlist ,p)))
59
60;; Second term
61(defmacro poly-second-lt (p) `(cadar (poly-termlist ,p)))
62
63;; Leading monomial
64(defun poly-lm (p) (term-monom (poly-lt p)))
65
66;; Second monomial
67(defun poly-second-lm (p) (term-monom (poly-second-lt p)))
68
69;; Leading coefficient
70(defun poly-lc (p) (term-coeff (poly-lt p)))
71
72;; Second coefficient
73(defun poly-second-lc (p) (term-coeff (poly-second-lt p)))
74
75;; Testing for a zero polynomial
76(defun poly-zerop (p) (null (poly-termlist p)))
77
78;; The number of terms
79(defun poly-length (p) (length (poly-termlist p)))
80
81(defun scalar-times-poly (ring c p)
82 (make-poly-from-termlist (scalar-times-termlist ring c (poly-termlist p)) (poly-sugar p)))
83
84;; The scalar product omitting the head term
85(defun scalar-times-poly-1 (ring c p)
86 (make-poly-from-termlist (scalar-times-termlist ring c (cdr (poly-termlist p))) (poly-sugar p)))
87
88(defun monom-times-poly (m p)
89 (make-poly-from-termlist (monom-times-termlist m (poly-termlist p)) (+ (poly-sugar p) (monom-sugar m))))
90
91(defun term-times-poly (ring term p)
92 (make-poly-from-termlist (term-times-termlist ring term (poly-termlist p)) (+ (poly-sugar p) (term-sugar term))))
93
94(defun poly-add (ring p q)
95 (make-poly-from-termlist (termlist-add ring (poly-termlist p) (poly-termlist q)) (max (poly-sugar p) (poly-sugar q))))
96
97(defun poly-sub (ring p q)
98 (make-poly-from-termlist (termlist-sub ring (poly-termlist p) (poly-termlist q)) (max (poly-sugar p) (poly-sugar q))))
99
100(defun poly-uminus (ring p)
101 (make-poly-from-termlist (termlist-uminus ring (poly-termlist p)) (poly-sugar p)))
102
103(defun poly-mul (ring p q)
104 (make-poly-from-termlist (termlist-mul ring (poly-termlist p) (poly-termlist q)) (+ (poly-sugar p) (poly-sugar q))))
105
106(defun poly-expt (ring p n)
107 (make-poly-from-termlist (termlist-expt ring (poly-termlist p) n) (* n (poly-sugar p))))
108
109(defun poly-append (&rest plist)
110 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
111 (apply #'max (mapcar #'poly-sugar plist))))
112
113(defun poly-nreverse (p)
114 (setf (poly-termlist p) (nreverse (poly-termlist p)))
115 p)
116
117(defun poly-contract (p &optional (k 1))
118 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
119 (poly-sugar p)))
120
121(defun poly-extend (p &optional (m (make-monom 1 :initial-element 0)))
122 (make-poly-from-termlist
123 (termlist-extend (poly-termlist p) m)
124 (+ (poly-sugar p) (monom-sugar m))))
125
126(defun poly-add-variables (p k)
127 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
128 p)
129
130(defun poly-list-add-variables (plist k)
131 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
132
133(defun poly-standard-extension (plist &aux (k (length plist)))
134 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
135 (declare (list plist) (fixnum k))
136 (labels ((incf-power (g i)
137 (dolist (x (poly-termlist g))
138 (incf (monom-elt (term-monom x) i)))
139 (incf (poly-sugar g))))
140 (setf plist (poly-list-add-variables plist k))
141 (dotimes (i k plist)
142 (incf-power (nth i plist) i))))
143
144(defun saturation-extension (ring f plist &aux (k (length plist)) (d (monom-dimension (poly-lm (car plist)))))
145 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
146 (setf f (poly-list-add-variables f k)
147 plist (mapcar #'(lambda (x)
148 (setf (poly-termlist x) (nconc (poly-termlist x)
149 (list (make-term (make-monom d :initial-element 0)
150 (funcall (ring-uminus ring) (funcall (ring-unit ring)))))))
151 x)
152 (poly-standard-extension plist)))
153 (append f plist))
154
155
156(defun polysaturation-extension (ring f plist &aux (k (length plist))
157 (d (+ k (length (poly-lm (car plist))))))
158 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]."
159 (setf f (poly-list-add-variables f k)
160 plist (apply #'poly-append (poly-standard-extension plist))
161 (cdr (last (poly-termlist plist))) (list (make-term (make-monom d :initial-element 0)
162 (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
163 (append f (list plist)))
164
165(defun saturation-extension-1 (ring f p) (polysaturation-extension ring f (list p)))
166
167
168
169;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
170;;
171;; Evaluation of polynomial (prefix) expressions
172;;
173;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
174
175(defun coerce-coeff (ring expr vars)
176 "Coerce an element of the coefficient ring to a constant polynomial."
177 ;; Modular arithmetic handler by rat
178 (make-poly-from-termlist (list (make-term (make-monom (length vars) :initial-element 0)
179 (funcall (ring-parse ring) expr)))
180 0))
181
182(defun poly-eval (ring expr vars &optional (list-marker '[))
183 (labels ((p-eval (arg) (poly-eval ring arg vars))
184 (p-eval-list (args) (mapcar #'p-eval args))
185 (p-add (x y) (poly-add ring x y)))
186 (cond
187 ((eql expr 0) (make-poly-zero))
188 ((member expr vars :test #'equalp)
189 (let ((pos (position expr vars :test #'equalp)))
190 (make-variable ring (length vars) pos)))
191 ((atom expr)
192 (coerce-coeff ring expr vars))
193 ((eq (car expr) list-marker)
194 (cons list-marker (p-eval-list (cdr expr))))
195 (t
196 (case (car expr)
197 (+ (reduce #'p-add (p-eval-list (cdr expr))))
198 (- (case (length expr)
199 (1 (make-poly-zero))
200 (2 (poly-uminus ring (p-eval (cadr expr))))
201 (3 (poly-sub ring (p-eval (cadr expr)) (p-eval (caddr expr))))
202 (otherwise (poly-sub ring (p-eval (cadr expr))
203 (reduce #'p-add (p-eval-list (cddr expr)))))))
204 (*
205 (if (endp (cddr expr)) ;unary
206 (p-eval (cdr expr))
207 (reduce #'(lambda (p q) (poly-mul ring p q)) (p-eval-list (cdr expr)))))
208 (expt
209 (cond
210 ((member (cadr expr) vars :test #'equalp)
211 ;;Special handling of (expt var pow)
212 (let ((pos (position (cadr expr) vars :test #'equalp)))
213 (make-variable ring (length vars) pos (caddr expr))))
214 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
215 ;; Negative power means division in coefficient ring
216 ;; Non-integer power means non-polynomial coefficient
217 (coerce-coeff ring expr vars))
218 (t (poly-expt ring (p-eval (cadr expr)) (caddr expr)))))
219 (otherwise
220 (coerce-coeff ring expr vars)))))))
221
222(defun spoly (ring f g)
223 "It yields the S-polynomial of polynomials F and G."
224 (declare (type poly f g))
225 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
226 (mf (monom-div lcm (poly-lm f)))
227 (mg (monom-div lcm (poly-lm g))))
228 (declare (type monom mf mg))
229 (multiple-value-bind (c cf cg)
230 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
231 (declare (ignore c))
232 (poly-sub
233 ring
234 (scalar-times-poly ring cg (monom-times-poly mf f))
235 (scalar-times-poly ring cf (monom-times-poly mg g))))))
236
237
238(defun poly-primitive-part (ring p)
239 "Divide polynomial P with integer coefficients by gcd of its
240coefficients and return the result."
241 (declare (type poly p))
242 (if (poly-zerop p)
243 (values p 1)
244 (let ((c (poly-content ring p)))
245 (values (make-poly-from-termlist (mapcar
246 #'(lambda (x)
247 (make-term (term-monom x)
248 (funcall (ring-div ring) (term-coeff x) c)))
249 (poly-termlist p))
250 (poly-sugar p))
251 c))))
252
253(defun poly-content (ring p)
254 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
255to compute the greatest common divisor."
256 (declare (type poly p))
257 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
258
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