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source: branches/f4grobner/division.lisp@ 1182

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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(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 we can drop the leading terms of f ang g
58 (poly-sub ring-and-order
59 (scalar-times-poly-1 ring c2 f)
60 (scalar-times-poly-1 ring c1 (monom-times-poly m g))))
61
62(defun poly-pseudo-divide (ring-and-order f fl
63 &aux
64 (ring (ro-ring ring-and-order)))
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)
103 p (grobner-op ring-and-order c2 c1 m p (car fl)))
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;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
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
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)
144 ;; p := cg*p-cp*m*g
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
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)
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))
175 (multiple-value-setq (f r c division-count)
176 (normal-form-step ring-and-order fl f r c division-count))))
177
178(defun buchberger-criterion (ring 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
183 #'poly-zerop
184 (makelist (normal-form ring (spoly ring (elt g i) (elt g j)) g nil)
185 (i 0 (- (length g) 2))
186 (j (1+ i) (1- (length g))))))
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
193is assumed to be provided by the RING structure of the
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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