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source: branches/f4grobner/division.lisp@ 1221

Last change on this file since 1221 was 1221, checked in by Marek Rychlik, 9 years ago

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1;;; -*- Mode: Lisp -*-
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(defpackage "DIVISION"
23 (:use :cl :utils :ring :monomial :polynomial :grobner-debug :term :ring-and-order)
24 (:export "$POLY_TOP_REDUCTION_ONLY"
25 "POLY-PSEUDO-DIVIDE"
26 "POLY-EXACT-DIVIDE"
27 "NORMAL-FORM-STEP"
28 "NORMAL-FORM"
29 "POLY-NORMALIZE"
30 "POLY-NORMALIZE-LIST"
31 "BUCHBERGER-CRITERION"
32 ))
33
34(in-package :division)
35
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
40;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
41;;
42;; An implementation of the division algorithm
43;;
44;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
45
46(defun grobner-op (ring-and-order c1 c2 m f g
47 &aux
48 (ring (ro-ring ring-and-order)))
49 "Returns C2*F-C1*M*G, where F and G are polynomials M is a monomial.
50Assume that the leading terms will cancel."
51 (declare (type ring-and-order ring-and-order))
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 below we can drop the leading terms of f ang g for the
58 ;; purpose of polynomial arithmetic.
59 ;;
60 ;; TODO: Make sure that the sugar calculation is correct if leading
61 ;; terms are dropped.
62 (poly-sub ring-and-order
63 (scalar-times-poly-1 ring c2 f)
64 (scalar-times-poly-1 ring c1 (monom-times-poly m g))))
65
66(defun poly-pseudo-divide (ring-and-order f fl
67 &aux
68 (ring (ro-ring ring-and-order)))
69 "Pseudo-divide a polynomial F by the list of polynomials FL. Return
70multiple values. The first value is a list of quotients A. The second
71value is the remainder R. The third argument is a scalar coefficient
72C, such that C*F can be divided by FL within the ring of coefficients,
73which is not necessarily a field. Finally, the fourth value is an
74integer count of the number of reductions performed. The resulting
75objects satisfy the equation: C*F= sum A[i]*FL[i] + R. The sugar of
76the quotients is initialized to default."
77 (declare (type poly f) (list fl))
78 (do ((r (make-poly-zero))
79 (c (funcall (ring-unit ring)))
80 (a (make-list (length fl) :initial-element (make-poly-zero)))
81 (division-count 0)
82 (p f))
83 ((poly-zerop p)
84 (debug-cgb "~&~3T~d reduction~:p" division-count)
85 (when (poly-zerop r) (debug-cgb " ---> 0"))
86 ;; We obtained the terms in reverse order, so must fix that
87 (setf a (mapcar #'poly-nreverse a)
88 r (poly-nreverse r))
89 ;; Initialize the sugar of the quotients
90 (mapc #'poly-reset-sugar a)
91 (values a r c division-count))
92 (declare (fixnum division-count))
93 (do ((fl fl (rest fl)) ;scan list of divisors
94 (b a (rest b)))
95 ((cond
96 ((endp fl) ;no division occurred
97 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
98 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
99 (pop (poly-termlist p)) ;remove lt(p) from p
100 t)
101 ((monom-divides-p (poly-lm (car fl)) (poly-lm p)) ;division occurred
102 (incf division-count)
103 (multiple-value-bind (gcd c1 c2)
104 (funcall (ring-ezgcd ring) (poly-lc (car fl)) (poly-lc p))
105 (declare (ignore gcd))
106 (let ((m (monom-div (poly-lm p) (poly-lm (car fl)))))
107 ;; Multiply the equation c*f=sum ai*fi+r+p by c1.
108 (mapl #'(lambda (x)
109 (setf (car x) (scalar-times-poly ring c1 (car x))))
110 a)
111 (setf r (scalar-times-poly ring c1 r)
112 c (funcall (ring-mul ring) c c1)
113 p (grobner-op ring-and-order c2 c1 m p (car fl)))
114 (push (make-term m c2) (poly-termlist (car b))))
115 t)))))))
116
117(defun poly-exact-divide (ring f g)
118 "Divide a polynomial F by another polynomial G. Assume that exact division
119with no remainder is possible. Returns the quotient."
120 (declare (type poly f g))
121 (multiple-value-bind (quot rem coeff division-count)
122 (poly-pseudo-divide ring f (list g))
123 (declare (ignore division-count coeff)
124 (list quot)
125 (type poly rem)
126 (type fixnum division-count))
127 (unless (poly-zerop rem) (error "Exact division failed."))
128 (car quot)))
129
130
131
132;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
133;;
134;; An implementation of the normal form
135;;
136;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
137
138(defun normal-form-step (ring-and-order fl p r c division-count
139 &aux
140 (ring (ro-ring ring-and-order))
141 (g (find (poly-lm p) fl
142 :test #'monom-divisible-by-p
143 :key #'poly-lm)))
144 (cond
145 (g ;division possible
146 (incf division-count)
147 (multiple-value-bind (gcd cg cp)
148 (funcall (ring-ezgcd ring) (poly-lc g) (poly-lc p))
149 (declare (ignore gcd))
150 (let ((m (monom-div (poly-lm p) (poly-lm g))))
151 ;; Multiply the equation c*f=sum ai*fi+r+p by cg.
152 (setf r (scalar-times-poly ring cg r)
153 c (funcall (ring-mul ring) c cg)
154 ;; p := cg*p-cp*m*g
155 p (grobner-op ring-and-order cp cg m p g))))
156 (debug-cgb "/"))
157 (t ;no division possible
158 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
159 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
160 (pop (poly-termlist p)) ;remove lt(p) from p
161 (debug-cgb "+")))
162 (values p r c division-count))
163
164;; Merge it sometime with poly-pseudo-divide
165(defun normal-form (ring-and-order f fl
166 &optional
167 (top-reduction-only $poly_top_reduction_only)
168 (ring (ro-ring ring-and-order)))
169 ;; Loop invariant: c*f0=sum ai*fi+r+f, where f0 is the initial value of f
170 #+grobner-check(when (null fl) (warn "normal-form: empty divisor list."))
171 (do ((r (make-poly-zero))
172 (c (funcall (ring-unit ring)))
173 (division-count 0))
174 ((or (poly-zerop f)
175 ;;(endp fl)
176 (and top-reduction-only (not (poly-zerop r))))
177 (progn
178 (debug-cgb "~&~3T~d reduction~:p" division-count)
179 (when (poly-zerop r)
180 (debug-cgb " ---> 0")))
181 (setf (poly-termlist f) (nreconc (poly-termlist r) (poly-termlist f)))
182 (values f c division-count))
183 (declare (fixnum division-count)
184 (type poly r))
185 (multiple-value-setq (f r c division-count)
186 (normal-form-step ring-and-order fl f r c division-count))))
187
188(defun buchberger-criterion (ring-and-order g)
189 "Returns T if G is a Grobner basis, by using the Buchberger
190criterion: for every two polynomials h1 and h2 in G the S-polynomial
191S(h1,h2) reduces to 0 modulo G."
192 (every
193 #'poly-zerop
194 (makelist (normal-form ring-and-order (spoly ring-and-order (elt g i) (elt g j)) g nil)
195 (i 0 (- (length g) 2))
196 (j (1+ i) (1- (length g))))))
197
198
199(defun poly-normalize (ring p &aux (c (poly-lc p)))
200 "Divide a polynomial by its leading coefficient. It assumes
201that the division is possible, which may not always be the
202case in rings which are not fields. The exact division operator
203is assumed to be provided by the RING structure."
204 (mapc #'(lambda (term)
205 (setf (term-coeff term) (funcall (ring-div ring) (term-coeff term) c)))
206 (poly-termlist p))
207 p)
208
209(defun poly-normalize-list (ring plist)
210 "Divide every polynomial in a list PLIST by its leading coefficient. "
211 (mapcar #'(lambda (x) (poly-normalize ring x)) plist))
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