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source: branches/f4grobner/division.lisp@ 1238

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[1199]1;;; -*- Mode: Lisp -*-
[148]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)))
[1205]57 ;; Note that below we can drop the leading terms of f ang g for the
[1206]58 ;; purpose of polynomial arithmetic.
59 ;;
[1212]60 ;; TODO: Make sure that the sugar calculation is correct if leading
61 ;; terms are dropped.
[1176]62 (poly-sub ring-and-order
[1206]63 (scalar-times-poly-1 ring c2 f)
64 (scalar-times-poly-1 ring c1 (monom-times-poly m g))))
[59]65
[1237]66(defun check-loop-invariant (ring-and-order c f0 a fl r f
67 &aux
68 (ring (ro-ring ring-and-order))
69 (p-zero (make-poly-zero)))
[1238]70 "Check loop invariant of division algorithms, when we divide a
71polynomial F by the list of polynomials FL. The invariant is the
72identity C*F0=SUM AI*FI+R+F, where F0 is the initial value of F, A is
73the list of partial quotients, R is the intermediate value of the
74remainder, and F is the intermediate value which eventually becomes
750."
[1237]76 (flet ((p-add (p q) (poly-add ring-and-order p q))
77 (p-sub (p q) (poly-sub ring-and-order p q))
78 (p-mul (p q) (poly-mul ring-and-order p q)))
79 (poly-zerop
80 (p-sub
81 (scalar-times-poly ring c f0)
82 (reduce #'p-add
83 (list (inner-product a fl p-add p-mul p-zero)
84 r
85 f))))))
86
87
[1179]88(defun poly-pseudo-divide (ring-and-order f fl
89 &aux
90 (ring (ro-ring ring-and-order)))
[59]91 "Pseudo-divide a polynomial F by the list of polynomials FL. Return
92multiple values. The first value is a list of quotients A. The second
93value is the remainder R. The third argument is a scalar coefficient
94C, such that C*F can be divided by FL within the ring of coefficients,
95which is not necessarily a field. Finally, the fourth value is an
96integer count of the number of reductions performed. The resulting
[1220]97objects satisfy the equation: C*F= sum A[i]*FL[i] + R. The sugar of
[1221]98the quotients is initialized to default."
[59]99 (declare (type poly f) (list fl))
[1238]100 ;; Loop invariant: c*f0=sum ai*fi+r+p, where f0 is the initial value of f
[59]101 (do ((r (make-poly-zero))
102 (c (funcall (ring-unit ring)))
103 (a (make-list (length fl) :initial-element (make-poly-zero)))
104 (division-count 0)
105 (p f))
106 ((poly-zerop p)
107 (debug-cgb "~&~3T~d reduction~:p" division-count)
108 (when (poly-zerop r) (debug-cgb " ---> 0"))
[1211]109 ;; We obtained the terms in reverse order, so must fix that
[1210]110 (setf a (mapcar #'poly-nreverse a)
111 r (poly-nreverse r))
[1219]112 ;; Initialize the sugar of the quotients
113 (mapc #'poly-reset-sugar a)
[1210]114 (values a r c division-count))
[59]115 (declare (fixnum division-count))
[1207]116 (do ((fl fl (rest fl)) ;scan list of divisors
[59]117 (b a (rest b)))
118 ((cond
[1207]119 ((endp fl) ;no division occurred
120 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
121 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
122 (pop (poly-termlist p)) ;remove lt(p) from p
123 t)
124 ((monom-divides-p (poly-lm (car fl)) (poly-lm p)) ;division occurred
125 (incf division-count)
126 (multiple-value-bind (gcd c1 c2)
127 (funcall (ring-ezgcd ring) (poly-lc (car fl)) (poly-lc p))
128 (declare (ignore gcd))
129 (let ((m (monom-div (poly-lm p) (poly-lm (car fl)))))
130 ;; Multiply the equation c*f=sum ai*fi+r+p by c1.
131 (mapl #'(lambda (x)
132 (setf (car x) (scalar-times-poly ring c1 (car x))))
133 a)
134 (setf r (scalar-times-poly ring c1 r)
135 c (funcall (ring-mul ring) c c1)
136 p (grobner-op ring-and-order c2 c1 m p (car fl)))
137 (push (make-term m c2) (poly-termlist (car b))))
138 t)))))))
[59]139
140(defun poly-exact-divide (ring f g)
141 "Divide a polynomial F by another polynomial G. Assume that exact division
142with no remainder is possible. Returns the quotient."
143 (declare (type poly f g))
144 (multiple-value-bind (quot rem coeff division-count)
145 (poly-pseudo-divide ring f (list g))
146 (declare (ignore division-count coeff)
147 (list quot)
148 (type poly rem)
149 (type fixnum division-count))
150 (unless (poly-zerop rem) (error "Exact division failed."))
151 (car quot)))
152
153;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
154;;
155;; An implementation of the normal form
156;;
157;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
158
[1180]159(defun normal-form-step (ring-and-order fl p r c division-count
160 &aux
161 (ring (ro-ring ring-and-order))
162 (g (find (poly-lm p) fl
163 :test #'monom-divisible-by-p
164 :key #'poly-lm)))
[59]165 (cond
166 (g ;division possible
167 (incf division-count)
168 (multiple-value-bind (gcd cg cp)
169 (funcall (ring-ezgcd ring) (poly-lc g) (poly-lc p))
170 (declare (ignore gcd))
171 (let ((m (monom-div (poly-lm p) (poly-lm g))))
172 ;; Multiply the equation c*f=sum ai*fi+r+p by cg.
173 (setf r (scalar-times-poly ring cg r)
174 c (funcall (ring-mul ring) c cg)
175 ;; p := cg*p-cp*m*g
[1181]176 p (grobner-op ring-and-order cp cg m p g))))
[59]177 (debug-cgb "/"))
178 (t ;no division possible
179 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
180 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
181 (pop (poly-termlist p)) ;remove lt(p) from p
182 (debug-cgb "+")))
183 (values p r c division-count))
184
185;; Merge it sometime with poly-pseudo-divide
[1182]186(defun normal-form (ring-and-order f fl
187 &optional
188 (top-reduction-only $poly_top_reduction_only)
189 (ring (ro-ring ring-and-order)))
[59]190 ;; Loop invariant: c*f0=sum ai*fi+r+f, where f0 is the initial value of f
191 #+grobner-check(when (null fl) (warn "normal-form: empty divisor list."))
192 (do ((r (make-poly-zero))
193 (c (funcall (ring-unit ring)))
194 (division-count 0))
195 ((or (poly-zerop f)
196 ;;(endp fl)
197 (and top-reduction-only (not (poly-zerop r))))
198 (progn
199 (debug-cgb "~&~3T~d reduction~:p" division-count)
200 (when (poly-zerop r)
201 (debug-cgb " ---> 0")))
202 (setf (poly-termlist f) (nreconc (poly-termlist r) (poly-termlist f)))
203 (values f c division-count))
204 (declare (fixnum division-count)
205 (type poly r))
206 (multiple-value-setq (f r c division-count)
[1182]207 (normal-form-step ring-and-order fl f r c division-count))))
[59]208
[1187]209(defun buchberger-criterion (ring-and-order g)
[59]210 "Returns T if G is a Grobner basis, by using the Buchberger
211criterion: for every two polynomials h1 and h2 in G the S-polynomial
212S(h1,h2) reduces to 0 modulo G."
[1222]213 (every #'poly-zerop
214 (makelist (normal-form ring-and-order (spoly ring-and-order (elt g i) (elt g j)) g nil)
215 (i 0 (- (length g) 2))
216 (j (1+ i) (1- (length g))))))
[59]217
[64]218
219(defun poly-normalize (ring p &aux (c (poly-lc p)))
220 "Divide a polynomial by its leading coefficient. It assumes
221that the division is possible, which may not always be the
222case in rings which are not fields. The exact division operator
[1197]223is assumed to be provided by the RING structure."
[64]224 (mapc #'(lambda (term)
225 (setf (term-coeff term) (funcall (ring-div ring) (term-coeff term) c)))
226 (poly-termlist p))
227 p)
228
229(defun poly-normalize-list (ring plist)
230 "Divide every polynomial in a list PLIST by its leading coefficient. "
231 (mapcar #'(lambda (x) (poly-normalize ring x)) plist))
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