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source: branches/f4grobner/division.lisp@ 1298

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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 (a (mapcar #'poly-reverse a))
71 (r (poly-reverse r)))
72 "Check loop invariant of division algorithms, when we divide a
73polynomial F by the list of polynomials FL. The invariant is the
74identity C*F=SUM AI*FI+R+P, where F0 is the initial value of F, A is
75the list of partial quotients, R is the intermediate value of the
76remainder, and P is the intermediate value which eventually becomes
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"
80 (format t "----------------------------------------------------------------~%")
81 (format t "#### Loop invariant check ####:~%C=~A~%F=~A~%A=~A~%FL=~A~%R=~A~%P=~A~%"
82 c f a fl r p)
83 (flet ((p-add (x y) (poly-add ring-and-order x y))
84 (p-sub (x y) (poly-sub ring-and-order x y))
85 (p-mul (x y) (poly-mul ring-and-order x y)))
86 (let* ((prod (inner-product a fl p-add p-mul p-zero))
87 (succeeded-p
88 (poly-zerop
89 (p-sub
90 (scalar-times-poly ring c f)
91 (reduce #'p-add (list prod r p))))))
92 (unless succeeded-p
93 (error "#### Polynomial division Loop invariant failed ####:~%C=~A~%F=~A~%A=~A~%FL=~A~%R=~A~%P=~A~%"
94 c f a fl r p))
95 succeeded-p)))
96
97
98(defun poly-pseudo-divide (ring-and-order f fl
99 &aux
100 (ring (ro-ring ring-and-order)))
101 "Pseudo-divide a polynomial F by the list of polynomials FL. Return
102multiple values. The first value is a list of quotients A. The second
103value is the remainder R. The third argument is a scalar coefficient
104C, such that C*F can be divided by FL within the ring of coefficients,
105which is not necessarily a field. Finally, the fourth value is an
106integer count of the number of reductions performed. The resulting
107objects satisfy the equation: C*F= sum A[i]*FL[i] + R. The sugar of
108the quotients is initialized to default."
109 (declare (type poly f) (list fl))
110 ;; Loop invariant: c*f=sum ai*fi+r+p, where p must eventually become 0
111 (do ((r (make-poly-zero))
112 (c (funcall (ring-unit ring)))
113 (a (make-list (length fl) :initial-element (make-poly-zero)))
114 (division-count 0)
115 (p f))
116 ((poly-zerop p)
117 #+grobner-check(check-loop-invariant ring-and-order c f a fl r p)
118 (debug-cgb "~&~3T~d reduction~:p" division-count)
119 (when (poly-zerop r) (debug-cgb " ---> 0"))
120 ;; We obtained the terms in reverse order, so must fix that
121 (setf a (mapcar #'poly-nreverse a)
122 r (poly-nreverse r))
123 ;; Initialize the sugar of the quotients
124 (mapc #'poly-reset-sugar a)
125 (values a r c division-count))
126 (declare (fixnum division-count))
127 ;; Check the loop invariant here
128 #+grobner-check(check-loop-invariant ring-and-order c f a fl r p)
129 (do ((fl fl (rest fl)) ;scan list of divisors
130 (b a (rest b)))
131 ((cond
132 ((endp fl) ;no division occurred
133 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
134 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
135 (pop (poly-termlist p)) ;remove lt(p) from p
136 t)
137 ((monom-divides-p (poly-lm (car fl)) (poly-lm p)) ;division occurred
138 (incf division-count)
139 (multiple-value-bind (gcd c1 c2)
140 (funcall (ring-ezgcd ring) (poly-lc (car fl)) (poly-lc p))
141 (declare (ignore gcd))
142 (let ((m (monom-div (poly-lm p) (poly-lm (car fl)))))
143 ;; Multiply the equation c*f=sum ai*fi+r+p by c1.
144 (mapl #'(lambda (x)
145 (setf (car x) (scalar-times-poly ring c1 (car x))))
146 a)
147 (setf r (scalar-times-poly ring c1 r)
148 c (funcall (ring-mul ring) c c1)
149 p (grobner-op ring-and-order c2 c1 m p (car fl)))
150 (push (make-term m c2) (poly-termlist (car b))))
151 t))))
152 )))
153
154(defun poly-exact-divide (ring-and-order f g)
155 "Divide a polynomial F by another polynomial G. Assume that exact division
156with no remainder is possible. Returns the quotient."
157 (declare (type poly f g) (type ring-and-order ring-and-order))
158 (multiple-value-bind (quot rem coeff division-count)
159 (poly-pseudo-divide ring-and-order f (list g))
160 (declare (ignore division-count coeff)
161 (list quot)
162 (type poly rem)
163 (type fixnum division-count))
164 (unless (poly-zerop rem) (error "Exact division failed."))
165 (car quot)))
166
167;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
168;;
169;; An implementation of the normal form
170;;
171;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
172
173(defun normal-form-step (ring-and-order fl p r c division-count
174 &aux
175 (ring (ro-ring ring-and-order))
176 (g (find (poly-lm p) fl
177 :test #'monom-divisible-by-p
178 :key #'poly-lm)))
179 (cond
180 (g ;division possible
181 (incf division-count)
182 (multiple-value-bind (gcd cg cp)
183 (funcall (ring-ezgcd ring) (poly-lc g) (poly-lc p))
184 (declare (ignore gcd))
185 (let ((m (monom-div (poly-lm p) (poly-lm g))))
186 ;; Multiply the equation c*f=sum ai*fi+r+p by cg.
187 (setf r (scalar-times-poly ring cg r)
188 c (funcall (ring-mul ring) c cg)
189 ;; p := cg*p-cp*m*g
190 p (grobner-op ring-and-order cp cg m p g))))
191 (debug-cgb "/"))
192 (t ;no division possible
193 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
194 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
195 (pop (poly-termlist p)) ;remove lt(p) from p
196 (debug-cgb "+")))
197 (values p r c division-count))
198
199;; Merge it sometime with poly-pseudo-divide
200(defun normal-form (ring-and-order f fl
201 &optional
202 (top-reduction-only $poly_top_reduction_only)
203 (ring (ro-ring ring-and-order)))
204 #+grobner-check(when (null fl) (warn "normal-form: empty divisor list."))
205 (do ((r (make-poly-zero))
206 (c (funcall (ring-unit ring)))
207 (division-count 0))
208 ((or (poly-zerop f)
209 ;;(endp fl)
210 (and top-reduction-only (not (poly-zerop r))))
211 (progn
212 (debug-cgb "~&~3T~D reduction~:P" division-count)
213 (when (poly-zerop r)
214 (debug-cgb " ---> 0")))
215 (setf (poly-termlist f) (nreconc (poly-termlist r) (poly-termlist f)))
216 (values f c division-count))
217 (declare (fixnum division-count)
218 (type poly r))
219 (multiple-value-setq (f r c division-count)
220 (normal-form-step ring-and-order fl f r c division-count))))
221
222(defun buchberger-criterion (ring-and-order g)
223 "Returns T if G is a Grobner basis, by using the Buchberger
224criterion: for every two polynomials h1 and h2 in G the S-polynomial
225S(h1,h2) reduces to 0 modulo G."
226 (every #'poly-zerop
227 (makelist (normal-form ring-and-order (spoly ring-and-order (elt g i) (elt g j)) g nil)
228 (i 0 (- (length g) 2))
229 (j (1+ i) (1- (length g))))))
230
231
232(defun poly-normalize (ring p &aux (c (poly-lc p)))
233 "Divide a polynomial by its leading coefficient. It assumes
234that the division is possible, which may not always be the
235case in rings which are not fields. The exact division operator
236is assumed to be provided by the RING structure."
237 (mapc #'(lambda (term)
238 (setf (term-coeff term) (funcall (ring-div ring) (term-coeff term) c)))
239 (poly-termlist p))
240 p)
241
242(defun poly-normalize-list (ring plist)
243 "Divide every polynomial in a list PLIST by its leading coefficient. "
244 (mapcar #'(lambda (x) (poly-normalize ring x)) plist))
245
246;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
247;;
248;; The function GROBNER-CHECK is provided primarily for debugging purposes. To
249;; enable verification of grobner bases with BUCHBERGER-CRITERION, do
250;; (pushnew :grobner-check *features*) and compile/load this file.
251;; With this feature, the calculations will slow down CONSIDERABLY.
252;;
253;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
254
255(defun grobner-test (ring-and-order g f)
256 "Test whether G is a Grobner basis and F is contained in G. Return T
257upon success and NIL otherwise."
258 (debug-cgb "~&GROBNER CHECK: ")
259 (let (($poly_grobner_debug nil)
260 (stat1 (buchberger-criterion ring-and-order g))
261 (stat2
262 (every #'poly-zerop
263 (makelist (normal-form ring-and-order (copy-tree (elt f i)) g nil)
264 (i 0 (1- (length f)))))))
265 (unless stat1 (error "~&Buchberger criterion failed."))
266 (unless stat2
267 (error "~&Original polys not in ideal spanned by Grobner.")))
268 (debug-cgb "~&GROBNER CHECK END")
269 t)
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