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source: branches/f4grobner/division.lisp@ 1724

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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 :monom :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 "GROBNER-TEST"
33 ))
34
35(in-package :division)
36
37(defvar $poly_top_reduction_only nil
38 "If not FALSE, use top reduction only whenever possible.
39Top reduction means that division algorithm stops after the first reduction.")
40
41;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
42;;
43;; An implementation of the division algorithm
44;;
45;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
46
47(defun grobner-op (ring-and-order c1 c2 m f g
48 &aux
49 (ring (ro-ring ring-and-order)))
50 "Returns C2*F-C1*M*G, where F and G are polynomials M is a monomial.
51Assume that the leading terms will cancel."
52 (declare (type ring-and-order ring-and-order))
53 #+grobner-check(funcall (ring-zerop ring)
54 (funcall (ring-sub ring)
55 (funcall (ring-mul ring) c2 (poly-lc f))
56 (funcall (ring-mul ring) c1 (poly-lc g))))
57 #+grobner-check(monom-equal-p (poly-lm f) (monom-mul m (poly-lm g)))
58 ;; Note that below we can drop the leading terms of f ang g for the
59 ;; purpose of polynomial arithmetic.
60 ;;
61 ;; TODO: Make sure that the sugar calculation is correct if leading
62 ;; terms are dropped.
63 (poly-sub ring-and-order
64 (scalar-times-poly-1 ring c2 f)
65 (scalar-times-poly-1 ring c1 (monom-times-poly m g))))
66
67(defun check-loop-invariant (ring-and-order c f a fl r p
68 &aux
69 (ring (ro-ring ring-and-order))
70 (p-zero (make-poly-zero))
71 (a (mapcar #'poly-reverse a))
72 (r (poly-reverse r)))
73 "Check loop invariant of division algorithms, when we divide a
74polynomial F by the list of polynomials FL. The invariant is the
75identity C*F=SUM AI*FI+R+P, where F0 is the initial value of F, A is
76the list of partial quotients, R is the intermediate value of the
77remainder, and P is the intermediate value which eventually becomes
780. A thing to remember is that the terms of polynomials in A and
79the polynomial R have their terms in reversed order. Hence, before
80the arithmetic is performed, we need to fix the order of terms"
81 #|
82 (format t "~&----------------------------------------------------------------~%")
83 (format t "#### Loop invariant check ####:~%C=~A~%F=~A~%A=~A~%FL=~A~%R=~A~%P=~A~%"
84 c f a fl r p)
85 |#
86 (flet ((p-add (x y) (poly-add ring-and-order x y))
87 (p-sub (x y) (poly-sub ring-and-order x y))
88 (p-mul (x y) (poly-mul ring-and-order x y)))
89 (let* ((prod (inner-product a fl p-add p-mul p-zero))
90 (succeeded-p
91 (poly-zerop
92 (p-sub
93 (scalar-times-poly ring c f)
94 (reduce #'p-add (list prod r p))))))
95 (unless succeeded-p
96 (error "#### Polynomial division Loop invariant failed ####:~%C=~A~%F=~A~%A=~A~%FL=~A~%R=~A~%P=~A~%"
97 c f a fl r p))
98 succeeded-p)))
99
100
101(defun poly-pseudo-divide (ring-and-order f fl
102 &aux
103 (ring (ro-ring ring-and-order)))
104 "Pseudo-divide a polynomial F by the list of polynomials FL. Return
105multiple values. The first value is a list of quotients A. The second
106value is the remainder R. The third argument is a scalar coefficient
107C, such that C*F can be divided by FL within the ring of coefficients,
108which is not necessarily a field. Finally, the fourth value is an
109integer count of the number of reductions performed. The resulting
110objects satisfy the equation: C*F= sum A[i]*FL[i] + R. The sugar of
111the quotients is initialized to default."
112 (declare (type poly f) (list fl))
113 ;; Loop invariant: c*f=sum ai*fi+r+p, where p must eventually become 0
114 (do ((r (make-poly-zero))
115 (c (funcall (ring-unit ring)))
116 (a (make-list (length fl) :initial-element (make-poly-zero)))
117 (division-count 0)
118 (p f))
119 ((poly-zerop p)
120 #+grobner-check(check-loop-invariant ring-and-order c f a fl r p)
121 (debug-cgb "~&~3T~d reduction~:p" division-count)
122 (when (poly-zerop r) (debug-cgb " ---> 0"))
123 ;; We obtained the terms in reverse order, so must fix that
124 (setf a (mapcar #'poly-nreverse a)
125 r (poly-nreverse r))
126 ;; Initialize the sugar of the quotients
127 (mapc #'poly-reset-sugar a)
128 (values a r c division-count))
129 (declare (fixnum division-count))
130 ;; Check the loop invariant here
131 #+grobner-check(check-loop-invariant ring-and-order c f a fl r p)
132 (do ((fl fl (rest fl)) ;scan list of divisors
133 (b a (rest b)))
134 ((cond
135 ((endp fl) ;no division occurred
136 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
137 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
138 (pop (poly-termlist p)) ;remove lt(p) from p
139 t)
140 ((monom-divides-p (poly-lm (car fl)) (poly-lm p)) ;division occurred
141 (incf division-count)
142 (multiple-value-bind (gcd c1 c2)
143 (funcall (ring-ezgcd ring) (poly-lc (car fl)) (poly-lc p))
144 (declare (ignore gcd))
145 (let ((m (monom-div (poly-lm p) (poly-lm (car fl)))))
146 ;; Multiply the equation c*f=sum ai*fi+r+p by c1.
147 (mapl #'(lambda (x)
148 (setf (car x) (scalar-times-poly ring c1 (car x))))
149 a)
150 (setf r (scalar-times-poly ring c1 r)
151 c (funcall (ring-mul ring) c c1)
152 p (grobner-op ring-and-order c2 c1 m p (car fl)))
153 (push (make-term m c2) (poly-termlist (car b))))
154 t))))
155 )))
156
157(defun poly-exact-divide (ring-and-order f g)
158 "Divide a polynomial F by another polynomial G. Assume that exact division
159with no remainder is possible. Returns the quotient."
160 (declare (type poly f g) (type ring-and-order ring-and-order))
161 (multiple-value-bind (quot rem coeff division-count)
162 (poly-pseudo-divide ring-and-order f (list g))
163 (declare (ignore division-count coeff)
164 (list quot)
165 (type poly rem)
166 (type fixnum division-count))
167 (unless (poly-zerop rem) (error "Exact division failed."))
168 (car quot)))
169
170;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
171;;
172;; An implementation of the normal form
173;;
174;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
175
176(defun normal-form-step (ring-and-order fl p r c division-count
177 &aux
178 (ring (ro-ring ring-and-order))
179 (g (find (poly-lm p) fl
180 :test #'monom-divisible-by-p
181 :key #'poly-lm)))
182 (cond
183 (g ;division possible
184 (incf division-count)
185 (multiple-value-bind (gcd cg cp)
186 (funcall (ring-ezgcd ring) (poly-lc g) (poly-lc p))
187 (declare (ignore gcd))
188 (let ((m (monom-div (poly-lm p) (poly-lm g))))
189 ;; Multiply the equation c*f=sum ai*fi+r+p by cg.
190 (setf r (scalar-times-poly ring cg r)
191 c (funcall (ring-mul ring) c cg)
192 ;; p := cg*p-cp*m*g
193 p (grobner-op ring-and-order cp cg m p g))))
194 (debug-cgb "/"))
195 (t ;no division possible
196 (push (poly-lt p) (poly-termlist r)) ;move lt(p) to remainder
197 (setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
198 (pop (poly-termlist p)) ;remove lt(p) from p
199 (debug-cgb "+")))
200 (values p r c division-count))
201
202;;
203;; Merge NORMAL-FORM someday with POLY-PSEUDO-DIVIDE.
204;;
205;; TODO: It is hard to test normal form as there is no loop invariant,
206;; like for POLY-PSEUDO-DIVIDE. Is there a testing strategy? One
207;; method would be to test NORMAL-FORM using POLY-PSEUDO-DIVIDE.
208;;
209(defun normal-form (ring-and-order f fl
210 &optional
211 (top-reduction-only $poly_top_reduction_only)
212 (ring (ro-ring ring-and-order)))
213 #+grobner-check(when (null fl) (warn "normal-form: empty divisor list."))
214 (do ((r (make-poly-zero))
215 (c (funcall (ring-unit ring)))
216 (division-count 0))
217 ((or (poly-zerop f)
218 ;;(endp fl)
219 (and top-reduction-only (not (poly-zerop r))))
220 (progn
221 (debug-cgb "~&~3T~D reduction~:P" division-count)
222 (when (poly-zerop r)
223 (debug-cgb " ---> 0")))
224 (setf (poly-termlist f) (nreconc (poly-termlist r) (poly-termlist f)))
225 (values f c division-count))
226 (declare (fixnum division-count)
227 (type poly r))
228 (multiple-value-setq (f r c division-count)
229 (normal-form-step ring-and-order fl f r c division-count))))
230
231(defun buchberger-criterion (ring-and-order g)
232 "Returns T if G is a Grobner basis, by using the Buchberger
233criterion: for every two polynomials h1 and h2 in G the S-polynomial
234S(h1,h2) reduces to 0 modulo G."
235 (every #'poly-zerop
236 (makelist (normal-form ring-and-order (spoly ring-and-order (elt g i) (elt g j)) g nil)
237 (i 0 (- (length g) 2))
238 (j (1+ i) (1- (length g))))))
239
240
241(defun poly-normalize (ring p &aux (c (poly-lc p)))
242 "Divide a polynomial by its leading coefficient. It assumes
243that the division is possible, which may not always be the
244case in rings which are not fields. The exact division operator
245is assumed to be provided by the RING structure."
246 (mapc #'(lambda (term)
247 (setf (term-coeff term) (funcall (ring-div ring) (term-coeff term) c)))
248 (poly-termlist p))
249 p)
250
251(defun poly-normalize-list (ring plist)
252 "Divide every polynomial in a list PLIST by its leading coefficient. "
253 (mapcar #'(lambda (x) (poly-normalize ring x)) plist))
254
255;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
256;;
257;; The function GROBNER-CHECK is provided primarily for debugging purposes. To
258;; enable verification of grobner bases with BUCHBERGER-CRITERION, do
259;; (pushnew :grobner-check *features*) and compile/load this file.
260;; With this feature, the calculations will slow down CONSIDERABLY.
261;;
262;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
263
264(defun grobner-test (ring-and-order g f)
265 "Test whether G is a Grobner basis and F is contained in G. Return T
266upon success and NIL otherwise."
267 (debug-cgb "~&GROBNER CHECK: ")
268 (let (($poly_grobner_debug nil)
269 (stat1 (buchberger-criterion ring-and-order g))
270 (stat2
271 (every #'poly-zerop
272 (makelist (normal-form ring-and-order (copy-tree (elt f i)) g nil)
273 (i 0 (1- (length f)))))))
274 (unless stat1 (error "~&Buchberger criterion failed, not a grobner basis: ~A" g))
275 (unless stat2
276 (error "~&Original polynomials not in ideal spanned by Grobner basis: ~A" f)))
277 (debug-cgb "~&GROBNER CHECK END")
278 t)
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