close Warning: Can't synchronize with repository "(default)" (The repository directory has changed, you should resynchronize the repository with: trac-admin $ENV repository resync '(default)'). Look in the Trac log for more information.

source: branches/f4grobner/polynomial.lisp@ 985

Last change on this file since 985 was 985, checked in by Marek Rychlik, 10 years ago

* empty log message *
File size: 10.8 KB
Line 
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-and-order p n)
164 (declare (type ring-and-order ring-and-order) (type poly p))
165 (make-poly-from-termlist (termlist-expt ring (poly-termlist p) n) (* n (poly-sugar p))))
166
167(defun poly-append (&rest plist)
168 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
169 (apply #'max (mapcar #'poly-sugar plist))))
170
171(defun poly-nreverse (p)
172 (setf (poly-termlist p) (nreverse (poly-termlist p)))
173 p)
174
175(defun poly-contract (p &optional (k 1))
176 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
177 (poly-sugar p)))
178
179(defun poly-extend (p &optional (m (make-monom :dimension 1)))
180 (make-poly-from-termlist
181 (termlist-extend (poly-termlist p) m)
182 (+ (poly-sugar p) (monom-sugar m))))
183
184(defun poly-add-variables (p k)
185 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
186 p)
187
188(defun poly-list-add-variables (plist k)
189 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
190
191(defun poly-standard-extension (plist &aux (k (length plist)))
192 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
193 (declare (list plist) (fixnum k))
194 (labels ((incf-power (g i)
195 (dolist (x (poly-termlist g))
196 (incf (monom-elt (term-monom x) i)))
197 (incf (poly-sugar g))))
198 (setf plist (poly-list-add-variables plist k))
199 (dotimes (i k plist)
200 (incf-power (nth i plist) i))))
201
202(defun saturation-extension (ring f plist &aux (k (length plist)) (d (monom-dimension (poly-lm (car plist)))))
203 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
204 (setf f (poly-list-add-variables f k)
205 plist (mapcar #'(lambda (x)
206 (setf (poly-termlist x) (nconc (poly-termlist x)
207 (list (make-term (make-monom :dimension d)
208 (funcall (ring-uminus ring) (funcall (ring-unit ring)))))))
209 x)
210 (poly-standard-extension plist)))
211 (append f plist))
212
213
214(defun polysaturation-extension (ring f plist &aux (k (length plist))
215 (d (+ k (monom-dimension (poly-lm (car plist))))))
216 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]."
217 (setf f (poly-list-add-variables f k)
218 plist (apply #'poly-append (poly-standard-extension plist))
219 (cdr (last (poly-termlist plist))) (list (make-term (make-monom :dimension d)
220 (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
221 (append f (list plist)))
222
223(defun saturation-extension-1 (ring f p) (polysaturation-extension ring f (list p)))
224
225;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
226;;
227;; Evaluation of polynomial (prefix) expressions
228;;
229;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
230
231(defun coerce-coeff (ring expr vars)
232 "Coerce an element of the coefficient ring to a constant polynomial."
233 ;; Modular arithmetic handler by rat
234 (make-poly-from-termlist (list (make-term (make-monom :dimension (length vars))
235 (funcall (ring-parse ring) expr)))
236 0))
237
238(defun poly-eval (ring expr vars &optional (list-marker '[))
239 (labels ((p-eval (arg) (poly-eval ring arg vars))
240 (p-eval-list (args) (mapcar #'p-eval args))
241 (p-add (x y) (poly-add ring x y)))
242 (cond
243 ((eql expr 0) (make-poly-zero))
244 ((member expr vars :test #'equalp)
245 (let ((pos (position expr vars :test #'equalp)))
246 (make-variable ring (length vars) pos)))
247 ((atom expr)
248 (coerce-coeff ring expr vars))
249 ((eq (car expr) list-marker)
250 (cons list-marker (p-eval-list (cdr expr))))
251 (t
252 (case (car expr)
253 (+ (reduce #'p-add (p-eval-list (cdr expr))))
254 (- (case (length expr)
255 (1 (make-poly-zero))
256 (2 (poly-uminus ring (p-eval (cadr expr))))
257 (3 (poly-sub ring (p-eval (cadr expr)) (p-eval (caddr expr))))
258 (otherwise (poly-sub ring (p-eval (cadr expr))
259 (reduce #'p-add (p-eval-list (cddr expr)))))))
260 (*
261 (if (endp (cddr expr)) ;unary
262 (p-eval (cdr expr))
263 (reduce #'(lambda (p q) (poly-mul ring p q)) (p-eval-list (cdr expr)))))
264 (expt
265 (cond
266 ((member (cadr expr) vars :test #'equalp)
267 ;;Special handling of (expt var pow)
268 (let ((pos (position (cadr expr) vars :test #'equalp)))
269 (make-variable ring (length vars) pos (caddr expr))))
270 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
271 ;; Negative power means division in coefficient ring
272 ;; Non-integer power means non-polynomial coefficient
273 (coerce-coeff ring expr vars))
274 (t (poly-expt ring (p-eval (cadr expr)) (caddr expr)))))
275 (otherwise
276 (coerce-coeff ring expr vars)))))))
277
278(defun spoly (ring f g)
279 "It yields the S-polynomial of polynomials F and G."
280 (declare (type poly f g))
281 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
282 (mf (monom-div lcm (poly-lm f)))
283 (mg (monom-div lcm (poly-lm g))))
284 (declare (type monom mf mg))
285 (multiple-value-bind (c cf cg)
286 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
287 (declare (ignore c))
288 (poly-sub
289 ring
290 (scalar-times-poly ring cg (monom-times-poly mf f))
291 (scalar-times-poly ring cf (monom-times-poly mg g))))))
292
293
294(defun poly-primitive-part (ring p)
295 "Divide polynomial P with integer coefficients by gcd of its
296coefficients and return the result."
297 (declare (type poly p))
298 (if (poly-zerop p)
299 (values p 1)
300 (let ((c (poly-content ring p)))
301 (values (make-poly-from-termlist (mapcar
302 #'(lambda (x)
303 (make-term (term-monom x)
304 (funcall (ring-div ring) (term-coeff x) c)))
305 (poly-termlist p))
306 (poly-sugar p))
307 c))))
308
309(defun poly-content (ring p)
310 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
311to compute the greatest common divisor."
312 (declare (type poly p))
313 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
314
Note: See TracBrowser for help on using the repository browser.