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source: branches/f4grobner/division.lisp@ 1270

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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
[1263]63 (scalar-times-poly-1 ring c2 f)
64 (scalar-times-poly-1 ring c1 (monom-times-poly m g))))
[59]65
[1242]66(defun check-loop-invariant (ring-and-order c f a fl r p
[1237]67 &aux
68 (ring (ro-ring ring-and-order))
[1264]69 (p-zero (make-poly-zero))
70 (a (mapcar #'poly-reverse a))
71 (r (poly-reverse r)))
[1238]72 "Check loop invariant of division algorithms, when we divide a
73polynomial F by the list of polynomials FL. The invariant is the
[1242]74identity C*F=SUM AI*FI+R+P, where F0 is the initial value of F, A is
[1238]75the list of partial quotients, R is the intermediate value of the
[1242]76remainder, and P is the intermediate value which eventually becomes
[1269]770. A thing to remember is that the terms of polynomials in A and
78the polynomial R have their terms in reversed order. Hence, before
79the arithmetic is performed, we need to fix the order of terms"
[1257]80 (format t "C=~A~%F=~A~%A=~A~%FL=~A~%R=~A~%P=~A~%" c f a fl r p)
[1242]81 (flet ((p-add (x y) (poly-add ring-and-order x y))
82 (p-sub (x y) (poly-sub ring-and-order x y))
83 (p-mul (x y) (poly-mul ring-and-order x y)))
[1257]84 (let* ((prod (inner-product a fl p-add p-mul p-zero))
85 (succeeded-p
86 (poly-zerop
87 (p-sub
88 (scalar-times-poly ring c f)
89 (reduce #'p-add (list prod r p))))))
[1270]90 (unless succeeded-p
91 (error "Division failed: C=~A~%F=~A~%A=~A~%FL=~A~%R=~A~%P=~A~%" c f a fl r p))
[1257]92 succeeded-p)))
[1237]93
94
[1179]95(defun poly-pseudo-divide (ring-and-order f fl
96 &aux
97 (ring (ro-ring ring-and-order)))
[59]98 "Pseudo-divide a polynomial F by the list of polynomials FL. Return
99multiple values. The first value is a list of quotients A. The second
100value is the remainder R. The third argument is a scalar coefficient
101C, such that C*F can be divided by FL within the ring of coefficients,
102which is not necessarily a field. Finally, the fourth value is an
103integer count of the number of reductions performed. The resulting
[1220]104objects satisfy the equation: C*F= sum A[i]*FL[i] + R. The sugar of
[1221]105the quotients is initialized to default."
[59]106 (declare (type poly f) (list fl))
[1241]107 ;; Loop invariant: c*f=sum ai*fi+r+p, where p must eventually become 0
[59]108 (do ((r (make-poly-zero))
109 (c (funcall (ring-unit ring)))
110 (a (make-list (length fl) :initial-element (make-poly-zero)))
111 (division-count 0)
112 (p f))
113 ((poly-zerop p)
114 (debug-cgb "~&~3T~d reduction~:p" division-count)
115 (when (poly-zerop r) (debug-cgb " ---> 0"))
[1211]116 ;; We obtained the terms in reverse order, so must fix that
[1210]117 (setf a (mapcar #'poly-nreverse a)
118 r (poly-nreverse r))
[1219]119 ;; Initialize the sugar of the quotients
120 (mapc #'poly-reset-sugar a)
[1210]121 (values a r c division-count))
[59]122 (declare (fixnum division-count))
[1252]123 ;; Check the loop invariant here
[1260]124 #+grobner-check(check-loop-invariant ring-and-order c f a fl r p)
[1207]125 (do ((fl fl (rest fl)) ;scan list of divisors
[59]126 (b a (rest b)))
127 ((cond
[1207]128 ((endp fl) ;no division occurred
129 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
130 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
131 (pop (poly-termlist p)) ;remove lt(p) from p
132 t)
133 ((monom-divides-p (poly-lm (car fl)) (poly-lm p)) ;division occurred
134 (incf division-count)
135 (multiple-value-bind (gcd c1 c2)
136 (funcall (ring-ezgcd ring) (poly-lc (car fl)) (poly-lc p))
137 (declare (ignore gcd))
138 (let ((m (monom-div (poly-lm p) (poly-lm (car fl)))))
139 ;; Multiply the equation c*f=sum ai*fi+r+p by c1.
140 (mapl #'(lambda (x)
141 (setf (car x) (scalar-times-poly ring c1 (car x))))
142 a)
143 (setf r (scalar-times-poly ring c1 r)
144 c (funcall (ring-mul ring) c c1)
145 p (grobner-op ring-and-order c2 c1 m p (car fl)))
146 (push (make-term m c2) (poly-termlist (car b))))
[1248]147 t))))
148 )))
[59]149
150(defun poly-exact-divide (ring f g)
151 "Divide a polynomial F by another polynomial G. Assume that exact division
152with no remainder is possible. Returns the quotient."
153 (declare (type poly f g))
154 (multiple-value-bind (quot rem coeff division-count)
155 (poly-pseudo-divide ring f (list g))
156 (declare (ignore division-count coeff)
157 (list quot)
158 (type poly rem)
159 (type fixnum division-count))
160 (unless (poly-zerop rem) (error "Exact division failed."))
161 (car quot)))
162
163;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
164;;
165;; An implementation of the normal form
166;;
167;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
168
[1180]169(defun normal-form-step (ring-and-order fl p r c division-count
170 &aux
171 (ring (ro-ring ring-and-order))
172 (g (find (poly-lm p) fl
173 :test #'monom-divisible-by-p
174 :key #'poly-lm)))
[59]175 (cond
176 (g ;division possible
177 (incf division-count)
178 (multiple-value-bind (gcd cg cp)
179 (funcall (ring-ezgcd ring) (poly-lc g) (poly-lc p))
180 (declare (ignore gcd))
181 (let ((m (monom-div (poly-lm p) (poly-lm g))))
182 ;; Multiply the equation c*f=sum ai*fi+r+p by cg.
183 (setf r (scalar-times-poly ring cg r)
184 c (funcall (ring-mul ring) c cg)
185 ;; p := cg*p-cp*m*g
[1181]186 p (grobner-op ring-and-order cp cg m p g))))
[59]187 (debug-cgb "/"))
188 (t ;no division possible
189 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
190 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
191 (pop (poly-termlist p)) ;remove lt(p) from p
192 (debug-cgb "+")))
193 (values p r c division-count))
194
195;; Merge it sometime with poly-pseudo-divide
[1182]196(defun normal-form (ring-and-order f fl
197 &optional
198 (top-reduction-only $poly_top_reduction_only)
199 (ring (ro-ring ring-and-order)))
[59]200 #+grobner-check(when (null fl) (warn "normal-form: empty divisor list."))
201 (do ((r (make-poly-zero))
202 (c (funcall (ring-unit ring)))
[1254]203 (division-count 0))
[59]204 ((or (poly-zerop f)
205 ;;(endp fl)
206 (and top-reduction-only (not (poly-zerop r))))
207 (progn
[1239]208 (debug-cgb "~&~3T~D reduction~:P" division-count)
[59]209 (when (poly-zerop r)
210 (debug-cgb " ---> 0")))
211 (setf (poly-termlist f) (nreconc (poly-termlist r) (poly-termlist f)))
212 (values f c division-count))
213 (declare (fixnum division-count)
214 (type poly r))
215 (multiple-value-setq (f r c division-count)
[1182]216 (normal-form-step ring-and-order fl f r c division-count))))
[59]217
[1187]218(defun buchberger-criterion (ring-and-order g)
[59]219 "Returns T if G is a Grobner basis, by using the Buchberger
220criterion: for every two polynomials h1 and h2 in G the S-polynomial
221S(h1,h2) reduces to 0 modulo G."
[1222]222 (every #'poly-zerop
223 (makelist (normal-form ring-and-order (spoly ring-and-order (elt g i) (elt g j)) g nil)
224 (i 0 (- (length g) 2))
225 (j (1+ i) (1- (length g))))))
[59]226
[64]227
228(defun poly-normalize (ring p &aux (c (poly-lc p)))
229 "Divide a polynomial by its leading coefficient. It assumes
230that the division is possible, which may not always be the
231case in rings which are not fields. The exact division operator
[1197]232is assumed to be provided by the RING structure."
[64]233 (mapc #'(lambda (term)
234 (setf (term-coeff term) (funcall (ring-div ring) (term-coeff term) c)))
235 (poly-termlist p))
236 p)
237
238(defun poly-normalize-list (ring plist)
239 "Divide every polynomial in a list PLIST by its leading coefficient. "
240 (mapcar #'(lambda (x) (poly-normalize ring x)) plist))
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