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source: branches/f4grobner/polynomial.lisp@ 984

Last change on this file since 984 was 984, checked in by Marek Rychlik, 9 years ago

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1;;; -*- Mode: Lisp; Package: Maxima; Syntax: Common-Lisp; Base: 10 -*-
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
23(defpackage "POLYNOMIAL"
24 (:use :cl :ring :ring-and-order :monomial :term :termlist)
25 (:export "POLY"
26 "POLY-TERMLIST"
27 "POLY-SUGAR"
28 "POLY-LT"
29 "MAKE-POLY-FROM-TERMLIST"
30 "MAKE-POLY-ZERO"
31 "MAKE-VARIABLE"
32 "POLY-UNIT"
33 "POLY-LM"
34 "POLY-SECOND-LM"
35 "POLY-SECOND-LT"
36 "POLY-LC"
37 "POLY-SECOND-LC"
38 "POLY-ZEROP"
39 "POLY-LENGTH"
40 "SCALAR-TIMES-POLY"
41 "SCALAR-TIMES-POLY-1"
42 "MONOM-TIMES-POLY"
43 "TERM-TIMES-POLY"
44 "POLY-ADD"
45 "POLY-SUB"
46 "POLY-UMINUS"
47 "POLY-MUL"
48 "POLY-EXPT"
49 "POLY-APPEND"
50 "POLY-NREVERSE"
51 "POLY-CONTRACT"
52 "POLY-EXTEND"
53 "POLY-ADD-VARIABLES"
54 "POLY-LIST-ADD-VARIABLES"
55 "POLY-STANDARD-EXTENSION"
56 "SATURATION-EXTENSION"
57 "POLYSATURATION-EXTENSION"
58 "SATURATION-EXTENSION-1"
59 "COERCE-COEFF"
60 "POLY-EVAL"
61 "SPOLY"
62 "POLY-PRIMITIVE-PART"
63 "POLY-CONTENT"
64 ))
65
66(in-package :polynomial)
67
68;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
69;;
70;; Polynomials
71;;
72;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
73
74(defstruct (poly
75 ;;
76 ;; BOA constructor, by default constructs zero polynomial
77 (:constructor make-poly-from-termlist (termlist &optional (sugar (termlist-sugar termlist))))
78 (:constructor make-poly-zero (&aux (termlist nil) (sugar -1)))
79 ;; Constructor of polynomials representing a variable
80 (:constructor make-variable (ring nvars pos &optional (power 1)
81 &aux
82 (termlist (list
83 (make-term-variable ring nvars pos power)))
84 (sugar power)))
85 (:constructor poly-unit (ring dimension
86 &aux
87 (termlist (termlist-unit ring dimension))
88 (sugar 0))))
89 (termlist nil :type list)
90 (sugar -1 :type fixnum))
91
92;; Leading term
93(defmacro poly-lt (p) `(car (poly-termlist ,p)))
94
95;; Second term
96(defmacro poly-second-lt (p) `(cadar (poly-termlist ,p)))
97
98;; Leading monomial
99(defun poly-lm (p) (term-monom (poly-lt p)))
100
101;; Second monomial
102(defun poly-second-lm (p) (term-monom (poly-second-lt p)))
103
104;; Leading coefficient
105(defun poly-lc (p) (term-coeff (poly-lt p)))
106
107;; Second coefficient
108(defun poly-second-lc (p) (term-coeff (poly-second-lt p)))
109
110;; Testing for a zero polynomial
111(defun poly-zerop (p) (null (poly-termlist p)))
112
113;; The number of terms
114(defun poly-length (p) (length (poly-termlist p)))
115
116(defun scalar-times-poly (ring c p)
117 (declare (type ring ring) (poly p))
118 (make-poly-from-termlist (scalar-times-termlist ring c (poly-termlist p)) (poly-sugar p)))
119
120;; The scalar product omitting the head term
121(defun scalar-times-poly-1 (ring c p)
122 (declare (type ring ring) (poly p))
123 (make-poly-from-termlist (scalar-times-termlist ring c (cdr (poly-termlist p))) (poly-sugar p)))
124
125(defun monom-times-poly (m p)
126 (declare (type ring ring) (poly p))
127 (make-poly-from-termlist
128 (monom-times-termlist m (poly-termlist p))
129 (+ (poly-sugar p) (monom-sugar m))))
130
131(defun term-times-poly (ring term p)
132 (declare (type ring ring) (type term term) (type poly p))
133 (make-poly-from-termlist
134 (term-times-termlist ring term (poly-termlist p))
135 (+ (poly-sugar p) (term-sugar term))))
136
137(defun poly-add (ring-and-order p q)
138 (declare (type ring-and-order ring-and-order) (type poly p q))
139 (make-poly-from-termlist
140 (termlist-add ring-and-order
141 (poly-termlist p)
142 (poly-termlist q))
143 (max (poly-sugar p) (poly-sugar q))))
144
145(defun poly-sub (ring-and-order p q)
146 (declare (type ring-and-order ring-and-order) (type poly p q))
147 (make-poly-from-termlist
148 (termlist-sub ring (poly-termlist p) (poly-termlist q))
149 (max (poly-sugar p) (poly-sugar q))))
150
151(defun poly-uminus (ring p)
152 (declare (type ring ring) (type poly p))
153 (make-poly-from-termlist
154 (termlist-uminus ring (poly-termlist p))
155 (poly-sugar p)))
156
157(defun poly-mul (ring-and-order p q)
158 (declare (type ring-and-order ring-and-order) (type poly p q))
159 (make-poly-from-termlist
160 (termlist-mul ring (poly-termlist p) (poly-termlist q))
161 (+ (poly-sugar p) (poly-sugar q))))
162
163(defun poly-expt (ring p n)
164 (make-poly-from-termlist (termlist-expt ring (poly-termlist p) n) (* n (poly-sugar p))))
165
166(defun poly-append (&rest plist)
167 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
168 (apply #'max (mapcar #'poly-sugar plist))))
169
170(defun poly-nreverse (p)
171 (setf (poly-termlist p) (nreverse (poly-termlist p)))
172 p)
173
174(defun poly-contract (p &optional (k 1))
175 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
176 (poly-sugar p)))
177
178(defun poly-extend (p &optional (m (make-monom :dimension 1)))
179 (make-poly-from-termlist
180 (termlist-extend (poly-termlist p) m)
181 (+ (poly-sugar p) (monom-sugar m))))
182
183(defun poly-add-variables (p k)
184 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
185 p)
186
187(defun poly-list-add-variables (plist k)
188 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
189
190(defun poly-standard-extension (plist &aux (k (length plist)))
191 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
192 (declare (list plist) (fixnum k))
193 (labels ((incf-power (g i)
194 (dolist (x (poly-termlist g))
195 (incf (monom-elt (term-monom x) i)))
196 (incf (poly-sugar g))))
197 (setf plist (poly-list-add-variables plist k))
198 (dotimes (i k plist)
199 (incf-power (nth i plist) i))))
200
201(defun saturation-extension (ring f plist &aux (k (length plist)) (d (monom-dimension (poly-lm (car plist)))))
202 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
203 (setf f (poly-list-add-variables f k)
204 plist (mapcar #'(lambda (x)
205 (setf (poly-termlist x) (nconc (poly-termlist x)
206 (list (make-term (make-monom :dimension d)
207 (funcall (ring-uminus ring) (funcall (ring-unit ring)))))))
208 x)
209 (poly-standard-extension plist)))
210 (append f plist))
211
212
213(defun polysaturation-extension (ring f plist &aux (k (length plist))
214 (d (+ k (monom-dimension (poly-lm (car plist))))))
215 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]."
216 (setf f (poly-list-add-variables f k)
217 plist (apply #'poly-append (poly-standard-extension plist))
218 (cdr (last (poly-termlist plist))) (list (make-term (make-monom :dimension d)
219 (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
220 (append f (list plist)))
221
222(defun saturation-extension-1 (ring f p) (polysaturation-extension ring f (list p)))
223
224;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
225;;
226;; Evaluation of polynomial (prefix) expressions
227;;
228;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
229
230(defun coerce-coeff (ring expr vars)
231 "Coerce an element of the coefficient ring to a constant polynomial."
232 ;; Modular arithmetic handler by rat
233 (make-poly-from-termlist (list (make-term (make-monom :dimension (length vars))
234 (funcall (ring-parse ring) expr)))
235 0))
236
237(defun poly-eval (ring expr vars &optional (list-marker '[))
238 (labels ((p-eval (arg) (poly-eval ring arg vars))
239 (p-eval-list (args) (mapcar #'p-eval args))
240 (p-add (x y) (poly-add ring x y)))
241 (cond
242 ((eql expr 0) (make-poly-zero))
243 ((member expr vars :test #'equalp)
244 (let ((pos (position expr vars :test #'equalp)))
245 (make-variable ring (length vars) pos)))
246 ((atom expr)
247 (coerce-coeff ring expr vars))
248 ((eq (car expr) list-marker)
249 (cons list-marker (p-eval-list (cdr expr))))
250 (t
251 (case (car expr)
252 (+ (reduce #'p-add (p-eval-list (cdr expr))))
253 (- (case (length expr)
254 (1 (make-poly-zero))
255 (2 (poly-uminus ring (p-eval (cadr expr))))
256 (3 (poly-sub ring (p-eval (cadr expr)) (p-eval (caddr expr))))
257 (otherwise (poly-sub ring (p-eval (cadr expr))
258 (reduce #'p-add (p-eval-list (cddr expr)))))))
259 (*
260 (if (endp (cddr expr)) ;unary
261 (p-eval (cdr expr))
262 (reduce #'(lambda (p q) (poly-mul ring p q)) (p-eval-list (cdr expr)))))
263 (expt
264 (cond
265 ((member (cadr expr) vars :test #'equalp)
266 ;;Special handling of (expt var pow)
267 (let ((pos (position (cadr expr) vars :test #'equalp)))
268 (make-variable ring (length vars) pos (caddr expr))))
269 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
270 ;; Negative power means division in coefficient ring
271 ;; Non-integer power means non-polynomial coefficient
272 (coerce-coeff ring expr vars))
273 (t (poly-expt ring (p-eval (cadr expr)) (caddr expr)))))
274 (otherwise
275 (coerce-coeff ring expr vars)))))))
276
277(defun spoly (ring f g)
278 "It yields the S-polynomial of polynomials F and G."
279 (declare (type poly f g))
280 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
281 (mf (monom-div lcm (poly-lm f)))
282 (mg (monom-div lcm (poly-lm g))))
283 (declare (type monom mf mg))
284 (multiple-value-bind (c cf cg)
285 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
286 (declare (ignore c))
287 (poly-sub
288 ring
289 (scalar-times-poly ring cg (monom-times-poly mf f))
290 (scalar-times-poly ring cf (monom-times-poly mg g))))))
291
292
293(defun poly-primitive-part (ring p)
294 "Divide polynomial P with integer coefficients by gcd of its
295coefficients and return the result."
296 (declare (type poly p))
297 (if (poly-zerop p)
298 (values p 1)
299 (let ((c (poly-content ring p)))
300 (values (make-poly-from-termlist (mapcar
301 #'(lambda (x)
302 (make-term (term-monom x)
303 (funcall (ring-div ring) (term-coeff x) c)))
304 (poly-termlist p))
305 (poly-sugar p))
306 c))))
307
308(defun poly-content (ring p)
309 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
310to compute the greatest common divisor."
311 (declare (type poly p))
312 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
313
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