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source: branches/f4grobner/polynomial.lisp@ 542

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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 :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 (make-poly-from-termlist (scalar-times-termlist ring c (poly-termlist p)) (poly-sugar p)))
118
119;; The scalar product omitting the head term
120(defun scalar-times-poly-1 (ring c p)
121 (make-poly-from-termlist (scalar-times-termlist ring c (cdr (poly-termlist p))) (poly-sugar p)))
122
123(defun monom-times-poly (m p)
124 (make-poly-from-termlist (monom-times-termlist m (poly-termlist p)) (+ (poly-sugar p) (monom-sugar m))))
125
126(defun term-times-poly (ring term p)
127 (make-poly-from-termlist (term-times-termlist ring term (poly-termlist p)) (+ (poly-sugar p) (term-sugar term))))
128
129(defun poly-add (ring p q)
130 (make-poly-from-termlist (termlist-add ring (poly-termlist p) (poly-termlist q)) (max (poly-sugar p) (poly-sugar q))))
131
132(defun poly-sub (ring p q)
133 (make-poly-from-termlist (termlist-sub ring (poly-termlist p) (poly-termlist q)) (max (poly-sugar p) (poly-sugar q))))
134
135(defun poly-uminus (ring p)
136 (make-poly-from-termlist (termlist-uminus ring (poly-termlist p)) (poly-sugar p)))
137
138(defun poly-mul (ring p q)
139 (make-poly-from-termlist (termlist-mul ring (poly-termlist p) (poly-termlist q)) (+ (poly-sugar p) (poly-sugar q))))
140
141(defun poly-expt (ring p n)
142 (make-poly-from-termlist (termlist-expt ring (poly-termlist p) n) (* n (poly-sugar p))))
143
144(defun poly-append (&rest plist)
145 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
146 (apply #'max (mapcar #'poly-sugar plist))))
147
148(defun poly-nreverse (p)
149 (setf (poly-termlist p) (nreverse (poly-termlist p)))
150 p)
151
152(defun poly-contract (p &optional (k 1))
153 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
154 (poly-sugar p)))
155
156(defun poly-extend (p &optional (m (make-monom 1 :initial-element 0)))
157 (make-poly-from-termlist
158 (termlist-extend (poly-termlist p) m)
159 (+ (poly-sugar p) (monom-sugar m))))
160
161(defun poly-add-variables (p k)
162 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
163 p)
164
165(defun poly-list-add-variables (plist k)
166 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
167
168(defun poly-standard-extension (plist &aux (k (length plist)))
169 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
170 (declare (list plist) (fixnum k))
171 (labels ((incf-power (g i)
172 (dolist (x (poly-termlist g))
173 (incf (monom-elt (term-monom x) i)))
174 (incf (poly-sugar g))))
175 (setf plist (poly-list-add-variables plist k))
176 (dotimes (i k plist)
177 (incf-power (nth i plist) i))))
178
179(defun saturation-extension (ring f plist &aux (k (length plist)) (d (monom-dimension (poly-lm (car plist)))))
180 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
181 (setf f (poly-list-add-variables f k)
182 plist (mapcar #'(lambda (x)
183 (setf (poly-termlist x) (nconc (poly-termlist x)
184 (list (make-term (make-monom d :initial-element 0)
185 (funcall (ring-uminus ring) (funcall (ring-unit ring)))))))
186 x)
187 (poly-standard-extension plist)))
188 (append f plist))
189
190
191(defun polysaturation-extension (ring f plist &aux (k (length plist))
192 (d (+ k (length (poly-lm (car plist))))))
193 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]."
194 (setf f (poly-list-add-variables f k)
195 plist (apply #'poly-append (poly-standard-extension plist))
196 (cdr (last (poly-termlist plist))) (list (make-term (make-monom d :initial-element 0)
197 (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
198 (append f (list plist)))
199
200(defun saturation-extension-1 (ring f p) (polysaturation-extension ring f (list p)))
201
202
203
204;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
205;;
206;; Evaluation of polynomial (prefix) expressions
207;;
208;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
209
210(defun coerce-coeff (ring expr vars)
211 "Coerce an element of the coefficient ring to a constant polynomial."
212 ;; Modular arithmetic handler by rat
213 (make-poly-from-termlist (list (make-term (make-monom (length vars) :initial-element 0)
214 (funcall (ring-parse ring) expr)))
215 0))
216
217(defun poly-eval (ring expr vars &optional (list-marker '[))
218 (labels ((p-eval (arg) (poly-eval ring arg vars))
219 (p-eval-list (args) (mapcar #'p-eval args))
220 (p-add (x y) (poly-add ring x y)))
221 (cond
222 ((eql expr 0) (make-poly-zero))
223 ((member expr vars :test #'equalp)
224 (let ((pos (position expr vars :test #'equalp)))
225 (make-variable ring (length vars) pos)))
226 ((atom expr)
227 (coerce-coeff ring expr vars))
228 ((eq (car expr) list-marker)
229 (cons list-marker (p-eval-list (cdr expr))))
230 (t
231 (case (car expr)
232 (+ (reduce #'p-add (p-eval-list (cdr expr))))
233 (- (case (length expr)
234 (1 (make-poly-zero))
235 (2 (poly-uminus ring (p-eval (cadr expr))))
236 (3 (poly-sub ring (p-eval (cadr expr)) (p-eval (caddr expr))))
237 (otherwise (poly-sub ring (p-eval (cadr expr))
238 (reduce #'p-add (p-eval-list (cddr expr)))))))
239 (*
240 (if (endp (cddr expr)) ;unary
241 (p-eval (cdr expr))
242 (reduce #'(lambda (p q) (poly-mul ring p q)) (p-eval-list (cdr expr)))))
243 (expt
244 (cond
245 ((member (cadr expr) vars :test #'equalp)
246 ;;Special handling of (expt var pow)
247 (let ((pos (position (cadr expr) vars :test #'equalp)))
248 (make-variable ring (length vars) pos (caddr expr))))
249 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
250 ;; Negative power means division in coefficient ring
251 ;; Non-integer power means non-polynomial coefficient
252 (coerce-coeff ring expr vars))
253 (t (poly-expt ring (p-eval (cadr expr)) (caddr expr)))))
254 (otherwise
255 (coerce-coeff ring expr vars)))))))
256
257(defun spoly (ring f g)
258 "It yields the S-polynomial of polynomials F and G."
259 (declare (type poly f g))
260 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
261 (mf (monom-div lcm (poly-lm f)))
262 (mg (monom-div lcm (poly-lm g))))
263 (declare (type monom mf mg))
264 (multiple-value-bind (c cf cg)
265 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
266 (declare (ignore c))
267 (poly-sub
268 ring
269 (scalar-times-poly ring cg (monom-times-poly mf f))
270 (scalar-times-poly ring cf (monom-times-poly mg g))))))
271
272
273(defun poly-primitive-part (ring p)
274 "Divide polynomial P with integer coefficients by gcd of its
275coefficients and return the result."
276 (declare (type poly p))
277 (if (poly-zerop p)
278 (values p 1)
279 (let ((c (poly-content ring p)))
280 (values (make-poly-from-termlist (mapcar
281 #'(lambda (x)
282 (make-term (term-monom x)
283 (funcall (ring-div ring) (term-coeff x) c)))
284 (poly-termlist p))
285 (poly-sugar p))
286 c))))
287
288(defun poly-content (ring p)
289 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
290to compute the greatest common divisor."
291 (declare (type poly p))
292 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
293
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