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source: branches/f4grobner/polynomial.lisp@ 983

Last change on this file since 983 was 983, 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 p q)
158 (make-poly-from-termlist (termlist-mul ring (poly-termlist p) (poly-termlist q)) (+ (poly-sugar p) (poly-sugar q))))
159
160(defun poly-expt (ring p n)
161 (make-poly-from-termlist (termlist-expt ring (poly-termlist p) n) (* n (poly-sugar p))))
162
163(defun poly-append (&rest plist)
164 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
165 (apply #'max (mapcar #'poly-sugar plist))))
166
167(defun poly-nreverse (p)
168 (setf (poly-termlist p) (nreverse (poly-termlist p)))
169 p)
170
171(defun poly-contract (p &optional (k 1))
172 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
173 (poly-sugar p)))
174
175(defun poly-extend (p &optional (m (make-monom :dimension 1)))
176 (make-poly-from-termlist
177 (termlist-extend (poly-termlist p) m)
178 (+ (poly-sugar p) (monom-sugar m))))
179
180(defun poly-add-variables (p k)
181 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
182 p)
183
184(defun poly-list-add-variables (plist k)
185 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
186
187(defun poly-standard-extension (plist &aux (k (length plist)))
188 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
189 (declare (list plist) (fixnum k))
190 (labels ((incf-power (g i)
191 (dolist (x (poly-termlist g))
192 (incf (monom-elt (term-monom x) i)))
193 (incf (poly-sugar g))))
194 (setf plist (poly-list-add-variables plist k))
195 (dotimes (i k plist)
196 (incf-power (nth i plist) i))))
197
198(defun saturation-extension (ring f plist &aux (k (length plist)) (d (monom-dimension (poly-lm (car plist)))))
199 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
200 (setf f (poly-list-add-variables f k)
201 plist (mapcar #'(lambda (x)
202 (setf (poly-termlist x) (nconc (poly-termlist x)
203 (list (make-term (make-monom :dimension d)
204 (funcall (ring-uminus ring) (funcall (ring-unit ring)))))))
205 x)
206 (poly-standard-extension plist)))
207 (append f plist))
208
209
210(defun polysaturation-extension (ring f plist &aux (k (length plist))
211 (d (+ k (monom-dimension (poly-lm (car plist))))))
212 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]."
213 (setf f (poly-list-add-variables f k)
214 plist (apply #'poly-append (poly-standard-extension plist))
215 (cdr (last (poly-termlist plist))) (list (make-term (make-monom :dimension d)
216 (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
217 (append f (list plist)))
218
219(defun saturation-extension-1 (ring f p) (polysaturation-extension ring f (list p)))
220
221;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
222;;
223;; Evaluation of polynomial (prefix) expressions
224;;
225;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
226
227(defun coerce-coeff (ring expr vars)
228 "Coerce an element of the coefficient ring to a constant polynomial."
229 ;; Modular arithmetic handler by rat
230 (make-poly-from-termlist (list (make-term (make-monom :dimension (length vars))
231 (funcall (ring-parse ring) expr)))
232 0))
233
234(defun poly-eval (ring expr vars &optional (list-marker '[))
235 (labels ((p-eval (arg) (poly-eval ring arg vars))
236 (p-eval-list (args) (mapcar #'p-eval args))
237 (p-add (x y) (poly-add ring x y)))
238 (cond
239 ((eql expr 0) (make-poly-zero))
240 ((member expr vars :test #'equalp)
241 (let ((pos (position expr vars :test #'equalp)))
242 (make-variable ring (length vars) pos)))
243 ((atom expr)
244 (coerce-coeff ring expr vars))
245 ((eq (car expr) list-marker)
246 (cons list-marker (p-eval-list (cdr expr))))
247 (t
248 (case (car expr)
249 (+ (reduce #'p-add (p-eval-list (cdr expr))))
250 (- (case (length expr)
251 (1 (make-poly-zero))
252 (2 (poly-uminus ring (p-eval (cadr expr))))
253 (3 (poly-sub ring (p-eval (cadr expr)) (p-eval (caddr expr))))
254 (otherwise (poly-sub ring (p-eval (cadr expr))
255 (reduce #'p-add (p-eval-list (cddr expr)))))))
256 (*
257 (if (endp (cddr expr)) ;unary
258 (p-eval (cdr expr))
259 (reduce #'(lambda (p q) (poly-mul ring p q)) (p-eval-list (cdr expr)))))
260 (expt
261 (cond
262 ((member (cadr expr) vars :test #'equalp)
263 ;;Special handling of (expt var pow)
264 (let ((pos (position (cadr expr) vars :test #'equalp)))
265 (make-variable ring (length vars) pos (caddr expr))))
266 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
267 ;; Negative power means division in coefficient ring
268 ;; Non-integer power means non-polynomial coefficient
269 (coerce-coeff ring expr vars))
270 (t (poly-expt ring (p-eval (cadr expr)) (caddr expr)))))
271 (otherwise
272 (coerce-coeff ring expr vars)))))))
273
274(defun spoly (ring f g)
275 "It yields the S-polynomial of polynomials F and G."
276 (declare (type poly f g))
277 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
278 (mf (monom-div lcm (poly-lm f)))
279 (mg (monom-div lcm (poly-lm g))))
280 (declare (type monom mf mg))
281 (multiple-value-bind (c cf cg)
282 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
283 (declare (ignore c))
284 (poly-sub
285 ring
286 (scalar-times-poly ring cg (monom-times-poly mf f))
287 (scalar-times-poly ring cf (monom-times-poly mg g))))))
288
289
290(defun poly-primitive-part (ring p)
291 "Divide polynomial P with integer coefficients by gcd of its
292coefficients and return the result."
293 (declare (type poly p))
294 (if (poly-zerop p)
295 (values p 1)
296 (let ((c (poly-content ring p)))
297 (values (make-poly-from-termlist (mapcar
298 #'(lambda (x)
299 (make-term (term-monom x)
300 (funcall (ring-div ring) (term-coeff x) c)))
301 (poly-termlist p))
302 (poly-sugar p))
303 c))))
304
305(defun poly-content (ring p)
306 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
307to compute the greatest common divisor."
308 (declare (type poly p))
309 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
310
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