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source: branches/f4grobner/polynomial.lisp@ 431

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