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source: branches/f4grobner/division.lisp@ 480

Last change on this file since 480 was 473, checked in by Marek Rychlik, 9 years ago

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[148]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
[459]22(defpackage "DIVISION"
[471]23 (:use :cl :utils :ring :monomial :polynomial :grobner-debug :term)
[470]24 (:export "$POLY_TOP_REDUCTION_ONLY"
25 "POLY-PSEUDO-DIVIDE"
[459]26 "POLY-EXACT-DIVIDE"
27 "NORMAL-FORM"
28 "POLY-NORMALIZE"
[472]29 "POLY-NORMALIZE-LIST"
[473]30 "BUCHBERGER-CRITERION"
[459]31 ))
[148]32
[460]33(in-package :division)
34
[469]35(defvar $poly_top_reduction_only nil
36 "If not FALSE, use top reduction only whenever possible.
37Top reduction means that division algorithm stops after the first reduction.")
38
[59]39;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
40;;
41;; An implementation of the division algorithm
42;;
43;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
44
45(defun grobner-op (ring c1 c2 m f g)
46 "Returns C2*F-C1*M*G, where F and G are polynomials M is a monomial.
47Assume that the leading terms will cancel."
48 #+grobner-check(funcall (ring-zerop ring)
49 (funcall (ring-sub ring)
50 (funcall (ring-mul ring) c2 (poly-lc f))
51 (funcall (ring-mul ring) c1 (poly-lc g))))
52 #+grobner-check(monom-equal-p (poly-lm f) (monom-mul m (poly-lm g)))
53 ;; Note that we can drop the leading terms of f ang g
54 (poly-sub ring
55 (scalar-times-poly-1 ring c2 f)
56 (scalar-times-poly-1 ring c1 (monom-times-poly m g))))
57
58(defun poly-pseudo-divide (ring f fl)
59 "Pseudo-divide a polynomial F by the list of polynomials FL. Return
60multiple values. The first value is a list of quotients A. The second
61value is the remainder R. The third argument is a scalar coefficient
62C, such that C*F can be divided by FL within the ring of coefficients,
63which is not necessarily a field. Finally, the fourth value is an
64integer count of the number of reductions performed. The resulting
65objects satisfy the equation: C*F= sum A[i]*FL[i] + R."
66 (declare (type poly f) (list fl))
67 (do ((r (make-poly-zero))
68 (c (funcall (ring-unit ring)))
69 (a (make-list (length fl) :initial-element (make-poly-zero)))
70 (division-count 0)
71 (p f))
72 ((poly-zerop p)
73 (debug-cgb "~&~3T~d reduction~:p" division-count)
74 (when (poly-zerop r) (debug-cgb " ---> 0"))
75 (values (mapcar #'poly-nreverse a) (poly-nreverse r) c division-count))
76 (declare (fixnum division-count))
77 (do ((fl fl (rest fl)) ;scan list of divisors
78 (b a (rest b)))
79 ((cond
80 ((endp fl) ;no division occurred
81 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
82 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
83 (pop (poly-termlist p)) ;remove lt(p) from p
84 t)
85 ((monom-divides-p (poly-lm (car fl)) (poly-lm p)) ;division occurred
86 (incf division-count)
87 (multiple-value-bind (gcd c1 c2)
88 (funcall (ring-ezgcd ring) (poly-lc (car fl)) (poly-lc p))
89 (declare (ignore gcd))
90 (let ((m (monom-div (poly-lm p) (poly-lm (car fl)))))
91 ;; Multiply the equation c*f=sum ai*fi+r+p by c1.
92 (mapl #'(lambda (x)
93 (setf (car x) (scalar-times-poly ring c1 (car x))))
94 a)
95 (setf r (scalar-times-poly ring c1 r)
96 c (funcall (ring-mul ring) c c1)
97 p (grobner-op ring c2 c1 m p (car fl)))
98 (push (make-term m c2) (poly-termlist (car b))))
99 t)))))))
100
101(defun poly-exact-divide (ring f g)
102 "Divide a polynomial F by another polynomial G. Assume that exact division
103with no remainder is possible. Returns the quotient."
104 (declare (type poly f g))
105 (multiple-value-bind (quot rem coeff division-count)
106 (poly-pseudo-divide ring f (list g))
107 (declare (ignore division-count coeff)
108 (list quot)
109 (type poly rem)
110 (type fixnum division-count))
111 (unless (poly-zerop rem) (error "Exact division failed."))
112 (car quot)))
113
114
115
116;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
117;;
118;; An implementation of the normal form
119;;
120;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
121
122(defun normal-form-step (ring fl p r c division-count
123 &aux (g (find (poly-lm p) fl
124 :test #'monom-divisible-by-p
125 :key #'poly-lm)))
126 (cond
127 (g ;division possible
128 (incf division-count)
129 (multiple-value-bind (gcd cg cp)
130 (funcall (ring-ezgcd ring) (poly-lc g) (poly-lc p))
131 (declare (ignore gcd))
132 (let ((m (monom-div (poly-lm p) (poly-lm g))))
133 ;; Multiply the equation c*f=sum ai*fi+r+p by cg.
134 (setf r (scalar-times-poly ring cg r)
135 c (funcall (ring-mul ring) c cg)
136 ;; p := cg*p-cp*m*g
137 p (grobner-op ring cp cg m p g))))
138 (debug-cgb "/"))
139 (t ;no division possible
140 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
141 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
142 (pop (poly-termlist p)) ;remove lt(p) from p
143 (debug-cgb "+")))
144 (values p r c division-count))
145
146;; Merge it sometime with poly-pseudo-divide
147(defun normal-form (ring f fl &optional (top-reduction-only $poly_top_reduction_only))
148 ;; Loop invariant: c*f0=sum ai*fi+r+f, where f0 is the initial value of f
149 #+grobner-check(when (null fl) (warn "normal-form: empty divisor list."))
150 (do ((r (make-poly-zero))
151 (c (funcall (ring-unit ring)))
152 (division-count 0))
153 ((or (poly-zerop f)
154 ;;(endp fl)
155 (and top-reduction-only (not (poly-zerop r))))
156 (progn
157 (debug-cgb "~&~3T~d reduction~:p" division-count)
158 (when (poly-zerop r)
159 (debug-cgb " ---> 0")))
160 (setf (poly-termlist f) (nreconc (poly-termlist r) (poly-termlist f)))
161 (values f c division-count))
162 (declare (fixnum division-count)
163 (type poly r))
164 (multiple-value-setq (f r c division-count)
165 (normal-form-step ring fl f r c division-count))))
166
167(defun buchberger-criterion (ring g)
168 "Returns T if G is a Grobner basis, by using the Buchberger
169criterion: for every two polynomials h1 and h2 in G the S-polynomial
170S(h1,h2) reduces to 0 modulo G."
171 (every
172 #'poly-zerop
173 (makelist (normal-form ring (spoly ring (elt g i) (elt g j)) g nil)
174 (i 0 (- (length g) 2))
175 (j (1+ i) (1- (length g))))))
[64]176
177
178(defun poly-normalize (ring p &aux (c (poly-lc p)))
179 "Divide a polynomial by its leading coefficient. It assumes
180that the division is possible, which may not always be the
181case in rings which are not fields. The exact division operator
182is assumed to be provided by the RING structure of the
183COEFFICIENT-RING package."
184 (mapc #'(lambda (term)
185 (setf (term-coeff term) (funcall (ring-div ring) (term-coeff term) c)))
186 (poly-termlist p))
187 p)
188
189(defun poly-normalize-list (ring plist)
190 "Divide every polynomial in a list PLIST by its leading coefficient. "
191 (mapcar #'(lambda (x) (poly-normalize ring x)) plist))
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