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source: branches/f4grobner/polynomial.lisp@ 979

Last change on this file since 979 was 979, 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
132 (term-times-termlist ring term (poly-termlist p))
133 (+ (poly-sugar p) (term-sugar term))))
134
135(defun poly-add (ring-and-order p q)
136 (declare (type ring-and-order ring-and-order))
137 (make-poly-from-termlist
138 (termlist-add ring-and-order
139 (poly-termlist p)
140 (poly-termlist q))
141 (max (poly-sugar p) (poly-sugar q))))
142
143(defun poly-sub (ring p q)
144 (make-poly-from-termlist (termlist-sub ring (poly-termlist p) (poly-termlist q)) (max (poly-sugar p) (poly-sugar q))))
145
146(defun poly-uminus (ring p)
147 (make-poly-from-termlist (termlist-uminus ring (poly-termlist p)) (poly-sugar p)))
148
149(defun poly-mul (ring p q)
150 (make-poly-from-termlist (termlist-mul ring (poly-termlist p) (poly-termlist q)) (+ (poly-sugar p) (poly-sugar q))))
151
152(defun poly-expt (ring p n)
153 (make-poly-from-termlist (termlist-expt ring (poly-termlist p) n) (* n (poly-sugar p))))
154
155(defun poly-append (&rest plist)
156 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
157 (apply #'max (mapcar #'poly-sugar plist))))
158
159(defun poly-nreverse (p)
160 (setf (poly-termlist p) (nreverse (poly-termlist p)))
161 p)
162
163(defun poly-contract (p &optional (k 1))
164 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
165 (poly-sugar p)))
166
167(defun poly-extend (p &optional (m (make-monom :dimension 1)))
168 (make-poly-from-termlist
169 (termlist-extend (poly-termlist p) m)
170 (+ (poly-sugar p) (monom-sugar m))))
171
172(defun poly-add-variables (p k)
173 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
174 p)
175
176(defun poly-list-add-variables (plist k)
177 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
178
179(defun poly-standard-extension (plist &aux (k (length plist)))
180 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
181 (declare (list plist) (fixnum k))
182 (labels ((incf-power (g i)
183 (dolist (x (poly-termlist g))
184 (incf (monom-elt (term-monom x) i)))
185 (incf (poly-sugar g))))
186 (setf plist (poly-list-add-variables plist k))
187 (dotimes (i k plist)
188 (incf-power (nth i plist) i))))
189
190(defun saturation-extension (ring f plist &aux (k (length plist)) (d (monom-dimension (poly-lm (car plist)))))
191 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
192 (setf f (poly-list-add-variables f k)
193 plist (mapcar #'(lambda (x)
194 (setf (poly-termlist x) (nconc (poly-termlist x)
195 (list (make-term (make-monom :dimension d)
196 (funcall (ring-uminus ring) (funcall (ring-unit ring)))))))
197 x)
198 (poly-standard-extension plist)))
199 (append f plist))
200
201
202(defun polysaturation-extension (ring f plist &aux (k (length plist))
203 (d (+ k (monom-dimension (poly-lm (car plist))))))
204 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]."
205 (setf f (poly-list-add-variables f k)
206 plist (apply #'poly-append (poly-standard-extension plist))
207 (cdr (last (poly-termlist plist))) (list (make-term (make-monom :dimension d)
208 (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
209 (append f (list plist)))
210
211(defun saturation-extension-1 (ring f p) (polysaturation-extension ring f (list p)))
212
213;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
214;;
215;; Evaluation of polynomial (prefix) expressions
216;;
217;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
218
219(defun coerce-coeff (ring expr vars)
220 "Coerce an element of the coefficient ring to a constant polynomial."
221 ;; Modular arithmetic handler by rat
222 (make-poly-from-termlist (list (make-term (make-monom :dimension (length vars))
223 (funcall (ring-parse ring) expr)))
224 0))
225
226(defun poly-eval (ring expr vars &optional (list-marker '[))
227 (labels ((p-eval (arg) (poly-eval ring arg vars))
228 (p-eval-list (args) (mapcar #'p-eval args))
229 (p-add (x y) (poly-add ring x y)))
230 (cond
231 ((eql expr 0) (make-poly-zero))
232 ((member expr vars :test #'equalp)
233 (let ((pos (position expr vars :test #'equalp)))
234 (make-variable ring (length vars) pos)))
235 ((atom expr)
236 (coerce-coeff ring expr vars))
237 ((eq (car expr) list-marker)
238 (cons list-marker (p-eval-list (cdr expr))))
239 (t
240 (case (car expr)
241 (+ (reduce #'p-add (p-eval-list (cdr expr))))
242 (- (case (length expr)
243 (1 (make-poly-zero))
244 (2 (poly-uminus ring (p-eval (cadr expr))))
245 (3 (poly-sub ring (p-eval (cadr expr)) (p-eval (caddr expr))))
246 (otherwise (poly-sub ring (p-eval (cadr expr))
247 (reduce #'p-add (p-eval-list (cddr expr)))))))
248 (*
249 (if (endp (cddr expr)) ;unary
250 (p-eval (cdr expr))
251 (reduce #'(lambda (p q) (poly-mul ring p q)) (p-eval-list (cdr expr)))))
252 (expt
253 (cond
254 ((member (cadr expr) vars :test #'equalp)
255 ;;Special handling of (expt var pow)
256 (let ((pos (position (cadr expr) vars :test #'equalp)))
257 (make-variable ring (length vars) pos (caddr expr))))
258 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
259 ;; Negative power means division in coefficient ring
260 ;; Non-integer power means non-polynomial coefficient
261 (coerce-coeff ring expr vars))
262 (t (poly-expt ring (p-eval (cadr expr)) (caddr expr)))))
263 (otherwise
264 (coerce-coeff ring expr vars)))))))
265
266(defun spoly (ring f g)
267 "It yields the S-polynomial of polynomials F and G."
268 (declare (type poly f g))
269 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
270 (mf (monom-div lcm (poly-lm f)))
271 (mg (monom-div lcm (poly-lm g))))
272 (declare (type monom mf mg))
273 (multiple-value-bind (c cf cg)
274 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
275 (declare (ignore c))
276 (poly-sub
277 ring
278 (scalar-times-poly ring cg (monom-times-poly mf f))
279 (scalar-times-poly ring cf (monom-times-poly mg g))))))
280
281
282(defun poly-primitive-part (ring p)
283 "Divide polynomial P with integer coefficients by gcd of its
284coefficients and return the result."
285 (declare (type poly p))
286 (if (poly-zerop p)
287 (values p 1)
288 (let ((c (poly-content ring p)))
289 (values (make-poly-from-termlist (mapcar
290 #'(lambda (x)
291 (make-term (term-monom x)
292 (funcall (ring-div ring) (term-coeff x) c)))
293 (poly-termlist p))
294 (poly-sugar p))
295 c))))
296
297(defun poly-content (ring p)
298 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
299to compute the greatest common divisor."
300 (declare (type poly p))
301 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
302
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