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source: branches/f4grobner/division.lisp@ 4071

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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 :monom :polynomial :grobner-debug)
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 (c1 c2 m f g)
48 "Returns C2*F-C1*M*G, where F and G are polynomials M is a monomial.
49Assume that the leading terms will cancel."
50 (declare (type monom m)
51 (type poly f g))
52 #+grobner-check(universal-zerop
53 (subtract
54 (multiply c2 (leading-coefficient f))
55 (multiply c1 (leading-coefficient g))))
56 #+grobner-check(universal-equalp (leading-monomial f) (multiply m (leading-monomial 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 (subtract
63 (multiply f c2)
64 (multiply (multiply m g) c1)))
65
66(defun check-loop-invariant (c f a fl r p
67 &aux
68 (p-zero (make-zero-for f))
69 (a (mapcar #'poly-reverse a))
70 (r (poly-reverse r)))
71 "Check loop invariant of division algorithms, when we divide a
72polynomial F by the list of polynomials FL. The invariant is the
73identity C*F=SUM AI*FI+R+P, where F0 is the initial value of F, A is
74the list of partial quotients, R is the intermediate value of the
75remainder, and P is the intermediate value which eventually becomes
760. A thing to remember is that the terms of polynomials in A and
77the polynomial R have their terms in reversed order. Hence, before
78the arithmetic is performed, we need to fix the order of terms"
79 #|
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 |#
84 (let* ((prod (inner-product a fl add multiply p-zero))
85 (succeeded-p (universal-zerop (subtract (multiply f c) (add prod r p)))))
86 (unless succeeded-p
87 (error "#### Polynomial division Loop invariant failed ####:~%C=~A~%F=~A~%A=~A~%FL=~A~%R=~A~%P=~A~%"
88 c f a fl r p))
89 succeeded-p))
90
91
92(defun poly-pseudo-divide (f fl)
93 "Pseudo-divide a polynomial F by the list of polynomials FL. Return
94multiple values. The first value is a list of quotients A. The second
95value is the remainder R. The third argument is a scalar coefficient
96C, such that C*F can be divided by FL within the ring of coefficients,
97which is not necessarily a field. Finally, the fourth value is an
98integer count of the number of reductions performed. The resulting
99objects satisfy the equation: C*F= sum A[i]*FL[i] + R. The sugar of
100the quotients is initialized to default."
101 (declare (type poly f) (list fl))
102 ;; Loop invariant: c*f=sum ai*fi+r+p, where p must eventually become 0
103 (do ((r (make-zero-for f))
104 (c (make-unit-for f))
105 (a (make-list (length fl) :initial-element (make-zero-for f)))
106 (division-count 0)
107 (p f))
108 ((universal-zerop p)
109 #+grobner-check(check-loop-invariant c f a fl r p)
110 (debug-cgb "~&~3T~d reduction~:p" division-count)
111 (when (universal-zerop r) (debug-cgb " ---> 0"))
112 ;; We obtained the terms in reverse order, so must fix that
113 (setf a (mapcar #'poly-reverse a)
114 r (poly-reverse r))
115 ;; Initialize the sugar of the quotients
116 ;; (mapc #'poly-reset-sugar a) ;; TODO: Sugar is currently unimplemented
117 (values a r c division-count))
118 (declare (fixnum division-count))
119 ;; Check the loop invariant here
120 #+grobner-check(check-loop-invariant c f a fl r p)
121 (do ((fl fl (rest fl)) ;scan list of divisors
122 (b a (rest b)))
123 ((cond
124 ((endp fl) ;no division occurred
125 (push (leading-term p) (poly-termlist r)) ;move lt(p) to remainder
126 ;;(setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
127 (pop (poly-termlist p)) ;remove lt(p) from p
128 t)
129 ((divides-p (leading-monomial (car fl)) (leading-monomial p)) ;division occurred
130 (incf division-count)
131 (multiple-value-bind (gcd c1 c2)
132 (universal-ezgcd (leading-coefficient (car fl)) (leading-coefficient p))
133 (declare (ignore gcd))
134 (let ((m (divide (leading-monomial p) (leading-monomial (car fl)))))
135 ;; Multiply the equation c*f=sum ai*fi+r+p by c1.
136 (mapl #'(lambda (x)
137 (setf (car x) (multiply (car x) c1)))
138 a)
139 (setf r (multiply r c1)
140 c (multiply c c1)
141 p (grobner-op c2 c1 m p (car fl)))
142 (push (change-class m 'term :coeff c2)
143 (poly-termlist (car b))))
144 t))))
145 )))
146
147(defun poly-exact-divide (f g)
148 "Divide a polynomial F by another polynomial G. Assume that exact division
149with no remainder is possible. Returns the quotient."
150 (declare (type poly f g))
151 (multiple-value-bind (quot rem coeff division-count)
152 (poly-pseudo-divide f (list g))
153 (declare (ignore division-count coeff)
154 (list quot)
155 (type poly rem)
156 (type fixnum division-count))
157 (unless (universal-zerop rem) (error "Exact division failed."))
158 (car quot)))
159
160;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
161;;
162;; An implementation of the normal form
163;;
164;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
165
166(defun normal-form-step (fl p r c division-count
167 &aux
168 (g (find (leading-monomial p) fl
169 :test #'divisible-by-p
170 :key #'leading-monomial)))
171 (cond
172 (g ;division possible
173 (incf division-count)
174 (multiple-value-bind (gcd cg cp)
175 (universal-ezgcd (leading-coefficient g) (leading-coefficient p))
176 (declare (ignore gcd))
177 (let ((m (divide (leading-monomial p) (leading-monomial g))))
178 ;; Multiply the equation c*f=sum ai*fi+r+p by cg.
179 (setf r (multiply r cg)
180 c (multiply c cg)
181 ;; p := cg*p-cp*m*g
182 p (grobner-op cp cg m p g))))
183 (debug-cgb "/"))
184 (t ;no division possible
185 (push (leading-term p) (poly-termlist r)) ;move lt(p) to remainder
186 ;;(setf (poly-sugar r) (max (poly-sugar r) (term-sugar (poly-lt p))))
187 (pop (poly-termlist p)) ;remove lt(p) from p
188 (debug-cgb "+")))
189 (values p r c division-count))
190
191;;
192;; Merge NORMAL-FORM someday with POLY-PSEUDO-DIVIDE.
193;;
194;; TODO: It is hard to test normal form as there is no loop invariant,
195;; like for POLY-PSEUDO-DIVIDE. Is there a testing strategy? One
196;; method would be to test NORMAL-FORM using POLY-PSEUDO-DIVIDE.
197;;
198(defun normal-form (f fl
199 &optional
200 (top-reduction-only $poly_top_reduction_only))
201 #+grobner-check(when (null fl) (warn "normal-form: empty divisor list."))
202 (do ((r (make-zero-for f))
203 (c (make-zero-for f))
204 (division-count 0))
205 ((or (universal-zerop f)
206 ;;(endp fl)
207 (and top-reduction-only (not (universal-zerop r))))
208 (progn
209 (debug-cgb "~&~3T~D reduction~:P" division-count)
210 (when (universal-zerop r)
211 (debug-cgb " ---> 0")))
212 (setf (poly-termlist f) (nreconc (poly-termlist r) (poly-termlist f)))
213 (values f c division-count))
214 (declare (fixnum division-count)
215 (type poly r))
216 (multiple-value-setq (f r c division-count)
217 (normal-form-step fl f r c division-count))))
218
219(defun spoly (f g)
220 "It yields the S-polynomial of polynomials F and G."
221 (declare (type poly f g))
222 (let* ((lcm (universal-lcm (leading-monomial f) (leading-monomial g)))
223 (mf (divide lcm (leading-monomial f)))
224 (mg (divide lcm (leading-monomial g))))
225 (declare (type monom mf mg))
226 (multiple-value-bind (c cf cg)
227 (universal-ezgcd (leading-coefficient f) (leading-coefficient g))
228 (declare (ignore c))
229 (subtract
230 (multiply (multiply mf f) cg)
231 (multiply (multiply mg g) cf)))))
232
233(defun buchberger-criterion (g)
234 "Returns T if G is a Grobner basis, by using the Buchberger
235criterion: for every two polynomials h1 and h2 in G the S-polynomial
236S(h1,h2) reduces to 0 modulo G."
237 (every #'universal-zerop
238 (makelist (normal-form (spoly (elt g i) (elt g j)) g nil)
239 (i 0 (- (length g) 2))
240 (j (1+ i) (1- (length g))))))
241
242
243(defun poly-normalize (p &aux (c (leading-coefficient p)))
244 "Divide a polynomial by its leading coefficient. It assumes
245that the division is possible, which may not always be the
246case in rings which are not fields. The exact division operator
247is assumed to be provided by the RING structure."
248 (mapc #'(lambda (term)
249 (setf (term-coeff term) (divide (term-coeff term) c)))
250 (poly-termlist p))
251 p)
252
253(defun poly-normalize-list (plist)
254 "Divide every polynomial in a list PLIST by its leading coefficient. "
255 (mapcar #'(lambda (x) (poly-normalize x)) plist))
256
257;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
258;;
259;; The function GROBNER-TEST is provided primarily for debugging purposes. To
260;; enable verification of grobner bases with BUCHBERGER-CRITERION, do
261;; (pushnew :grobner-check *features*) and compile/load this file.
262;; With this feature, the calculations will slow down CONSIDERABLY.
263;;
264;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
265
266(defun grobner-test (g f)
267 "Test whether G is a Grobner basis and F is contained in G. Return T
268upon success and NIL otherwise."
269 (debug-cgb "~&GROBNER CHECK: ")
270 (let (($poly_grobner_debug nil)
271 (stat1 (buchberger-criterion g))
272 (stat2
273 (every #'universal-zerop
274 (makelist (normal-form (copy-tree (elt f i)) g nil)
275 (i 0 (1- (length f)))))))
276 (unless stat1 (error "~&Buchberger criterion failed, not a grobner basis: ~A" g))
277 (unless stat2
278 (error "~&Original polynomials not in ideal spanned by Grobner basis: ~A" f)))
279 (debug-cgb "~&GROBNER CHECK END")
280 t)
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