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source: branches/f4grobner/division.lisp@ 1192

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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"
[1177]23 (:use :cl :utils :ring :monomial :polynomial :grobner-debug :term :ring-and-order)
[470]24 (:export "$POLY_TOP_REDUCTION_ONLY"
25 "POLY-PSEUDO-DIVIDE"
[459]26 "POLY-EXACT-DIVIDE"
[491]27 "NORMAL-FORM-STEP"
[459]28 "NORMAL-FORM"
29 "POLY-NORMALIZE"
[472]30 "POLY-NORMALIZE-LIST"
[473]31 "BUCHBERGER-CRITERION"
[459]32 ))
[148]33
[460]34(in-package :division)
35
[469]36(defvar $poly_top_reduction_only nil
37 "If not FALSE, use top reduction only whenever possible.
38Top reduction means that division algorithm stops after the first reduction.")
39
[59]40;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
41;;
42;; An implementation of the division algorithm
43;;
44;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
45
[1176]46(defun grobner-op (ring-and-order c1 c2 m f g
47 &aux
48 (ring (ro-ring ring-and-order)))
[59]49 "Returns C2*F-C1*M*G, where F and G are polynomials M is a monomial.
50Assume that the leading terms will cancel."
[1178]51 (declare (type ring-and-order ring-and-order))
[59]52 #+grobner-check(funcall (ring-zerop ring)
53 (funcall (ring-sub ring)
54 (funcall (ring-mul ring) c2 (poly-lc f))
55 (funcall (ring-mul ring) c1 (poly-lc g))))
56 #+grobner-check(monom-equal-p (poly-lm f) (monom-mul m (poly-lm g)))
57 ;; Note that we can drop the leading terms of f ang g
[1176]58 (poly-sub ring-and-order
[59]59 (scalar-times-poly-1 ring c2 f)
60 (scalar-times-poly-1 ring c1 (monom-times-poly m g))))
61
[1179]62(defun poly-pseudo-divide (ring-and-order f fl
63 &aux
64 (ring (ro-ring ring-and-order)))
[59]65 "Pseudo-divide a polynomial F by the list of polynomials FL. Return
66multiple values. The first value is a list of quotients A. The second
67value is the remainder R. The third argument is a scalar coefficient
68C, such that C*F can be divided by FL within the ring of coefficients,
69which is not necessarily a field. Finally, the fourth value is an
70integer count of the number of reductions performed. The resulting
71objects satisfy the equation: C*F= sum A[i]*FL[i] + R."
72 (declare (type poly f) (list fl))
73 (do ((r (make-poly-zero))
74 (c (funcall (ring-unit ring)))
75 (a (make-list (length fl) :initial-element (make-poly-zero)))
76 (division-count 0)
77 (p f))
78 ((poly-zerop p)
79 (debug-cgb "~&~3T~d reduction~:p" division-count)
80 (when (poly-zerop r) (debug-cgb " ---> 0"))
81 (values (mapcar #'poly-nreverse a) (poly-nreverse r) c division-count))
82 (declare (fixnum division-count))
83 (do ((fl fl (rest fl)) ;scan list of divisors
84 (b a (rest b)))
85 ((cond
86 ((endp fl) ;no division occurred
87 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
88 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
89 (pop (poly-termlist p)) ;remove lt(p) from p
90 t)
91 ((monom-divides-p (poly-lm (car fl)) (poly-lm p)) ;division occurred
92 (incf division-count)
93 (multiple-value-bind (gcd c1 c2)
94 (funcall (ring-ezgcd ring) (poly-lc (car fl)) (poly-lc p))
95 (declare (ignore gcd))
96 (let ((m (monom-div (poly-lm p) (poly-lm (car fl)))))
97 ;; Multiply the equation c*f=sum ai*fi+r+p by c1.
98 (mapl #'(lambda (x)
99 (setf (car x) (scalar-times-poly ring c1 (car x))))
100 a)
101 (setf r (scalar-times-poly ring c1 r)
102 c (funcall (ring-mul ring) c c1)
[1179]103 p (grobner-op ring-and-order c2 c1 m p (car fl)))
[59]104 (push (make-term m c2) (poly-termlist (car b))))
105 t)))))))
106
107(defun poly-exact-divide (ring f g)
108 "Divide a polynomial F by another polynomial G. Assume that exact division
109with no remainder is possible. Returns the quotient."
110 (declare (type poly f g))
111 (multiple-value-bind (quot rem coeff division-count)
112 (poly-pseudo-divide ring f (list g))
113 (declare (ignore division-count coeff)
114 (list quot)
115 (type poly rem)
116 (type fixnum division-count))
117 (unless (poly-zerop rem) (error "Exact division failed."))
118 (car quot)))
119
120
121
122;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
123;;
124;; An implementation of the normal form
125;;
126;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
[1180]127
128(defun normal-form-step (ring-and-order fl p r c division-count
129 &aux
130 (ring (ro-ring ring-and-order))
131 (g (find (poly-lm p) fl
132 :test #'monom-divisible-by-p
[59]133 :key #'poly-lm)))
134 (cond
135 (g ;division possible
136 (incf division-count)
137 (multiple-value-bind (gcd cg cp)
138 (funcall (ring-ezgcd ring) (poly-lc g) (poly-lc p))
139 (declare (ignore gcd))
140 (let ((m (monom-div (poly-lm p) (poly-lm g))))
141 ;; Multiply the equation c*f=sum ai*fi+r+p by cg.
142 (setf r (scalar-times-poly ring cg r)
143 c (funcall (ring-mul ring) c cg)
[1181]144 ;; p := cg*p-cp*m*g
[59]145 p (grobner-op ring-and-order cp cg m p g))))
146 (debug-cgb "/"))
147 (t ;no division possible
148 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
149 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
150 (pop (poly-termlist p)) ;remove lt(p) from p
151 (debug-cgb "+")))
152 (values p r c division-count))
153
[1182]154;; Merge it sometime with poly-pseudo-divide
155(defun normal-form (ring-and-order f fl
156 &optional
157 (top-reduction-only $poly_top_reduction_only)
[59]158 (ring (ro-ring ring-and-order)))
159 ;; Loop invariant: c*f0=sum ai*fi+r+f, where f0 is the initial value of f
160 #+grobner-check(when (null fl) (warn "normal-form: empty divisor list."))
161 (do ((r (make-poly-zero))
162 (c (funcall (ring-unit ring)))
163 (division-count 0))
164 ((or (poly-zerop f)
165 ;;(endp fl)
166 (and top-reduction-only (not (poly-zerop r))))
167 (progn
168 (debug-cgb "~&~3T~d reduction~:p" division-count)
169 (when (poly-zerop r)
170 (debug-cgb " ---> 0")))
171 (setf (poly-termlist f) (nreconc (poly-termlist r) (poly-termlist f)))
172 (values f c division-count))
173 (declare (fixnum division-count)
174 (type poly r))
[1182]175 (multiple-value-setq (f r c division-count)
[59]176 (normal-form-step ring-and-order fl f r c division-count))))
[1187]177
[59]178(defun buchberger-criterion (ring-and-order g)
179 "Returns T if G is a Grobner basis, by using the Buchberger
180criterion: for every two polynomials h1 and h2 in G the S-polynomial
181S(h1,h2) reduces to 0 modulo G."
182 (every
[1190]183 #'poly-zerop
[59]184 (makelist (normal-form ring-and-order (spoly ring-and-order (elt g i) (elt g j)) g nil)
185 (i 0 (- (length g) 2))
186 (j (1+ i) (1- (length g))))))
[64]187
188
189(defun poly-normalize (ring p &aux (c (poly-lc p)))
190 "Divide a polynomial by its leading coefficient. It assumes
191that the division is possible, which may not always be the
192case in rings which are not fields. The exact division operator
[1176]193is assumed to be provided by the RING structure of the
[64]194lCOEFFICIENT-RING package."
195 (mapc #'(lambda (term)
196 (setf (term-coeff term) (funcall (ring-div ring) (term-coeff term) c)))
197 (poly-termlist p))
198 p)
199
200(defun poly-normalize-list (ring plist)
201 "Divide every polynomial in a list PLIST by its leading coefficient. "
202 (mapcar #'(lambda (x) (poly-normalize ring x)) plist))
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