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source: branches/f4grobner/polynomial.lisp@ 978

Last change on this file since 978 was 978, checked in by Marek Rychlik, 10 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))
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))
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))
127 (make-poly-from-termlist (monom-times-termlist m (poly-termlist p)) (+ (poly-sugar p) (monom-sugar m))))
128
129(defun term-times-poly (ring term p)
130 (declare (type ring ring))
131 (make-poly-from-termlist (term-times-termlist ring term (poly-termlist p)) (+ (poly-sugar p) (term-sugar term))))
132
133(defun poly-add (ring-and-order p q)
134 (declare (type ring-and-order ring-and-order))
135 (make-poly-from-termlist
136 (termlist-add ring-and-order
137 (poly-termlist p)
138 (poly-termlist q))
139 (max (poly-sugar p) (poly-sugar q))))
140
141(defun poly-sub (ring p q)
142 (make-poly-from-termlist (termlist-sub ring (poly-termlist p) (poly-termlist q)) (max (poly-sugar p) (poly-sugar q))))
143
144(defun poly-uminus (ring p)
145 (make-poly-from-termlist (termlist-uminus ring (poly-termlist p)) (poly-sugar p)))
146
147(defun poly-mul (ring p q)
148 (make-poly-from-termlist (termlist-mul ring (poly-termlist p) (poly-termlist q)) (+ (poly-sugar p) (poly-sugar q))))
149
150(defun poly-expt (ring p n)
151 (make-poly-from-termlist (termlist-expt ring (poly-termlist p) n) (* n (poly-sugar p))))
152
153(defun poly-append (&rest plist)
154 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
155 (apply #'max (mapcar #'poly-sugar plist))))
156
157(defun poly-nreverse (p)
158 (setf (poly-termlist p) (nreverse (poly-termlist p)))
159 p)
160
161(defun poly-contract (p &optional (k 1))
162 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
163 (poly-sugar p)))
164
165(defun poly-extend (p &optional (m (make-monom :dimension 1)))
166 (make-poly-from-termlist
167 (termlist-extend (poly-termlist p) m)
168 (+ (poly-sugar p) (monom-sugar m))))
169
170(defun poly-add-variables (p k)
171 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
172 p)
173
174(defun poly-list-add-variables (plist k)
175 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
176
177(defun poly-standard-extension (plist &aux (k (length plist)))
178 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
179 (declare (list plist) (fixnum k))
180 (labels ((incf-power (g i)
181 (dolist (x (poly-termlist g))
182 (incf (monom-elt (term-monom x) i)))
183 (incf (poly-sugar g))))
184 (setf plist (poly-list-add-variables plist k))
185 (dotimes (i k plist)
186 (incf-power (nth i plist) i))))
187
188(defun saturation-extension (ring f plist &aux (k (length plist)) (d (monom-dimension (poly-lm (car plist)))))
189 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
190 (setf f (poly-list-add-variables f k)
191 plist (mapcar #'(lambda (x)
192 (setf (poly-termlist x) (nconc (poly-termlist x)
193 (list (make-term (make-monom :dimension d)
194 (funcall (ring-uminus ring) (funcall (ring-unit ring)))))))
195 x)
196 (poly-standard-extension plist)))
197 (append f plist))
198
199
200(defun polysaturation-extension (ring f plist &aux (k (length plist))
201 (d (+ k (monom-dimension (poly-lm (car plist))))))
202 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]."
203 (setf f (poly-list-add-variables f k)
204 plist (apply #'poly-append (poly-standard-extension plist))
205 (cdr (last (poly-termlist plist))) (list (make-term (make-monom :dimension d)
206 (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
207 (append f (list plist)))
208
209(defun saturation-extension-1 (ring f p) (polysaturation-extension ring f (list p)))
210
211;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
212;;
213;; Evaluation of polynomial (prefix) expressions
214;;
215;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
216
217(defun coerce-coeff (ring expr vars)
218 "Coerce an element of the coefficient ring to a constant polynomial."
219 ;; Modular arithmetic handler by rat
220 (make-poly-from-termlist (list (make-term (make-monom :dimension (length vars))
221 (funcall (ring-parse ring) expr)))
222 0))
223
224(defun poly-eval (ring expr vars &optional (list-marker '[))
225 (labels ((p-eval (arg) (poly-eval ring arg vars))
226 (p-eval-list (args) (mapcar #'p-eval args))
227 (p-add (x y) (poly-add ring x y)))
228 (cond
229 ((eql expr 0) (make-poly-zero))
230 ((member expr vars :test #'equalp)
231 (let ((pos (position expr vars :test #'equalp)))
232 (make-variable ring (length vars) pos)))
233 ((atom expr)
234 (coerce-coeff ring expr vars))
235 ((eq (car expr) list-marker)
236 (cons list-marker (p-eval-list (cdr expr))))
237 (t
238 (case (car expr)
239 (+ (reduce #'p-add (p-eval-list (cdr expr))))
240 (- (case (length expr)
241 (1 (make-poly-zero))
242 (2 (poly-uminus ring (p-eval (cadr expr))))
243 (3 (poly-sub ring (p-eval (cadr expr)) (p-eval (caddr expr))))
244 (otherwise (poly-sub ring (p-eval (cadr expr))
245 (reduce #'p-add (p-eval-list (cddr expr)))))))
246 (*
247 (if (endp (cddr expr)) ;unary
248 (p-eval (cdr expr))
249 (reduce #'(lambda (p q) (poly-mul ring p q)) (p-eval-list (cdr expr)))))
250 (expt
251 (cond
252 ((member (cadr expr) vars :test #'equalp)
253 ;;Special handling of (expt var pow)
254 (let ((pos (position (cadr expr) vars :test #'equalp)))
255 (make-variable ring (length vars) pos (caddr expr))))
256 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
257 ;; Negative power means division in coefficient ring
258 ;; Non-integer power means non-polynomial coefficient
259 (coerce-coeff ring expr vars))
260 (t (poly-expt ring (p-eval (cadr expr)) (caddr expr)))))
261 (otherwise
262 (coerce-coeff ring expr vars)))))))
263
264(defun spoly (ring f g)
265 "It yields the S-polynomial of polynomials F and G."
266 (declare (type poly f g))
267 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
268 (mf (monom-div lcm (poly-lm f)))
269 (mg (monom-div lcm (poly-lm g))))
270 (declare (type monom mf mg))
271 (multiple-value-bind (c cf cg)
272 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
273 (declare (ignore c))
274 (poly-sub
275 ring
276 (scalar-times-poly ring cg (monom-times-poly mf f))
277 (scalar-times-poly ring cf (monom-times-poly mg g))))))
278
279
280(defun poly-primitive-part (ring p)
281 "Divide polynomial P with integer coefficients by gcd of its
282coefficients and return the result."
283 (declare (type poly p))
284 (if (poly-zerop p)
285 (values p 1)
286 (let ((c (poly-content ring p)))
287 (values (make-poly-from-termlist (mapcar
288 #'(lambda (x)
289 (make-term (term-monom x)
290 (funcall (ring-div ring) (term-coeff x) c)))
291 (poly-termlist p))
292 (poly-sugar p))
293 c))))
294
295(defun poly-content (ring p)
296 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
297to compute the greatest common divisor."
298 (declare (type poly p))
299 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
300
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