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source: branches/f4grobner/division.lisp@ 1261

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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 check-loop-invariant (ring-and-order c f a fl r p
67 &aux
68 (ring (ro-ring ring-and-order))
69 (p-zero (make-poly-zero)))
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*F=SUM AI*FI+R+P, where F0 is the initial value of F, A is
73the list of partial quotients, R is the intermediate value of the
74remainder, and P is the intermediate value which eventually becomes
750."
76 (format t "Running loop-invariant check now.~%")
77 (format t "C=~A~%F=~A~%A=~A~%FL=~A~%R=~A~%P=~A~%" c f a fl r p)
78 (flet ((p-add (x y) (poly-add ring-and-order x y))
79 (p-sub (x y) (poly-sub ring-and-order x y))
80 (p-mul (x y) (poly-mul ring-and-order x y)))
81 (let* ((prod (inner-product a fl p-add p-mul p-zero))
82 (succeeded-p
83 (poly-zerop
84 (p-sub
85 (scalar-times-poly ring c f)
86 (reduce #'p-add (list prod r p))))))
87 (if succeeded-p
88 (format t "### Success ###~%")
89 (error "####### Check failed #####"))
90 succeeded-p)))
91
92
93(defun poly-pseudo-divide (ring-and-order f fl
94 &aux
95 (ring (ro-ring ring-and-order)))
96 "Pseudo-divide a polynomial F by the list of polynomials FL. Return
97multiple values. The first value is a list of quotients A. The second
98value is the remainder R. The third argument is a scalar coefficient
99C, such that C*F can be divided by FL within the ring of coefficients,
100which is not necessarily a field. Finally, the fourth value is an
101integer count of the number of reductions performed. The resulting
102objects satisfy the equation: C*F= sum A[i]*FL[i] + R. The sugar of
103the quotients is initialized to default."
104 (declare (type poly f) (list fl))
105 ;; Loop invariant: c*f=sum ai*fi+r+p, where p must eventually become 0
106 (do ((r (make-poly-zero))
107 (c (funcall (ring-unit ring)))
108 (a (make-list (length fl) :initial-element (make-poly-zero)))
109 (division-count 0)
110 (p f))
111 ((poly-zerop p)
112 (debug-cgb "~&~3T~d reduction~:p" division-count)
113 (when (poly-zerop r) (debug-cgb " ---> 0"))
114 ;; We obtained the terms in reverse order, so must fix that
115 (setf a (mapcar #'poly-nreverse a)
116 r (poly-nreverse r))
117 ;; Initialize the sugar of the quotients
118 (mapc #'poly-reset-sugar a)
119 (values a r c division-count))
120 (declare (fixnum division-count))
121 ;; Check the loop invariant here
122 #+grobner-check(check-loop-invariant ring-and-order c f a fl r p)
123 (do ((fl fl (rest fl)) ;scan list of divisors
124 (b a (rest b)))
125 ((cond
126 ((endp fl) ;no division occurred
127 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
128 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
129 (pop (poly-termlist p)) ;remove lt(p) from p
130 t)
131 ((monom-divides-p (poly-lm (car fl)) (poly-lm p)) ;division occurred
132 (incf division-count)
133 (multiple-value-bind (gcd c1 c2)
134 (funcall (ring-ezgcd ring) (poly-lc (car fl)) (poly-lc p))
135 (declare (ignore gcd))
136 (let ((m (monom-div (poly-lm p) (poly-lm (car fl)))))
137 ;; Multiply the equation c*f=sum ai*fi+r+p by c1.
138 (mapl #'(lambda (x)
139 (setf (car x) (scalar-times-poly ring c1 (car x))))
140 a)
141 (setf r (scalar-times-poly ring c1 r)
142 c (funcall (ring-mul ring) c c1)
143 p (grobner-op ring-and-order c2 c1 m p (car fl)))
144 (push (make-term m c2) (poly-termlist (car b))))
145 t))))
146 )))
147
148(defun poly-exact-divide (ring f g)
149 "Divide a polynomial F by another polynomial G. Assume that exact division
150with no remainder is possible. Returns the quotient."
151 (declare (type poly f g))
152 (multiple-value-bind (quot rem coeff division-count)
153 (poly-pseudo-divide ring f (list g))
154 (declare (ignore division-count coeff)
155 (list quot)
156 (type poly rem)
157 (type fixnum division-count))
158 (unless (poly-zerop rem) (error "Exact division failed."))
159 (car quot)))
160
161;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
162;;
163;; An implementation of the normal form
164;;
165;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
166
167(defun normal-form-step (ring-and-order fl p r c division-count
168 &aux
169 (ring (ro-ring ring-and-order))
170 (g (find (poly-lm p) fl
171 :test #'monom-divisible-by-p
172 :key #'poly-lm)))
173 (cond
174 (g ;division possible
175 (incf division-count)
176 (multiple-value-bind (gcd cg cp)
177 (funcall (ring-ezgcd ring) (poly-lc g) (poly-lc p))
178 (declare (ignore gcd))
179 (let ((m (monom-div (poly-lm p) (poly-lm g))))
180 ;; Multiply the equation c*f=sum ai*fi+r+p by cg.
181 (setf r (scalar-times-poly ring cg r)
182 c (funcall (ring-mul ring) c cg)
183 ;; p := cg*p-cp*m*g
184 p (grobner-op ring-and-order cp cg m p g))))
185 (debug-cgb "/"))
186 (t ;no division possible
187 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
188 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
189 (pop (poly-termlist p)) ;remove lt(p) from p
190 (debug-cgb "+")))
191 (values p r c division-count))
192
193;; Merge it sometime with poly-pseudo-divide
194(defun normal-form (ring-and-order f fl
195 &optional
196 (top-reduction-only $poly_top_reduction_only)
197 (ring (ro-ring ring-and-order)))
198 #+grobner-check(when (null fl) (warn "normal-form: empty divisor list."))
199 (do ((r (make-poly-zero))
200 (c (funcall (ring-unit ring)))
201 (division-count 0))
202 ((or (poly-zerop f)
203 ;;(endp fl)
204 (and top-reduction-only (not (poly-zerop r))))
205 (progn
206 (debug-cgb "~&~3T~D reduction~:P" division-count)
207 (when (poly-zerop r)
208 (debug-cgb " ---> 0")))
209 (setf (poly-termlist f) (nreconc (poly-termlist r) (poly-termlist f)))
210 (values f c division-count))
211 (declare (fixnum division-count)
212 (type poly r))
213 (multiple-value-setq (f r c division-count)
214 (normal-form-step ring-and-order fl f r c division-count))))
215
216(defun buchberger-criterion (ring-and-order g)
217 "Returns T if G is a Grobner basis, by using the Buchberger
218criterion: for every two polynomials h1 and h2 in G the S-polynomial
219S(h1,h2) reduces to 0 modulo G."
220 (every #'poly-zerop
221 (makelist (normal-form ring-and-order (spoly ring-and-order (elt g i) (elt g j)) g nil)
222 (i 0 (- (length g) 2))
223 (j (1+ i) (1- (length g))))))
224
225
226(defun poly-normalize (ring p &aux (c (poly-lc p)))
227 "Divide a polynomial by its leading coefficient. It assumes
228that the division is possible, which may not always be the
229case in rings which are not fields. The exact division operator
230is assumed to be provided by the RING structure."
231 (mapc #'(lambda (term)
232 (setf (term-coeff term) (funcall (ring-div ring) (term-coeff term) c)))
233 (poly-termlist p))
234 p)
235
236(defun poly-normalize-list (ring plist)
237 "Divide every polynomial in a list PLIST by its leading coefficient. "
238 (mapcar #'(lambda (x) (poly-normalize ring x)) plist))
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