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source: branches/f4grobner/polynomial.lisp@ 987

Last change on this file since 987 was 987, 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-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 (declare (type poly p))
173 (setf (poly-termlist p) (nreverse (poly-termlist p)))
174 p)
175
176(defun poly-contract (p &optional (k 1))
177 (declare (type poly p))
178 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
179 (poly-sugar p)))
180
181(defun poly-extend (p &optional (m (make-monom :dimension 1)))
182 (declare (type poly p))
183 (make-poly-from-termlist
184 (termlist-extend (poly-termlist p) m)
185 (+ (poly-sugar p) (monom-sugar m))))
186
187(defun poly-add-variables (p k)
188 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
189 p)
190
191(defun poly-list-add-variables (plist k)
192 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
193
194(defun poly-standard-extension (plist &aux (k (length plist)))
195 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
196 (declare (list plist) (fixnum k))
197 (labels ((incf-power (g i)
198 (dolist (x (poly-termlist g))
199 (incf (monom-elt (term-monom x) i)))
200 (incf (poly-sugar g))))
201 (setf plist (poly-list-add-variables plist k))
202 (dotimes (i k plist)
203 (incf-power (nth i plist) i))))
204
205(defun saturation-extension (ring f plist &aux (k (length plist)) (d (monom-dimension (poly-lm (car plist)))))
206 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
207 (setf f (poly-list-add-variables f k)
208 plist (mapcar #'(lambda (x)
209 (setf (poly-termlist x) (nconc (poly-termlist x)
210 (list (make-term (make-monom :dimension d)
211 (funcall (ring-uminus ring) (funcall (ring-unit ring)))))))
212 x)
213 (poly-standard-extension plist)))
214 (append f plist))
215
216
217(defun polysaturation-extension (ring f plist &aux (k (length plist))
218 (d (+ k (monom-dimension (poly-lm (car plist))))))
219 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]."
220 (setf f (poly-list-add-variables f k)
221 plist (apply #'poly-append (poly-standard-extension plist))
222 (cdr (last (poly-termlist plist))) (list (make-term (make-monom :dimension d)
223 (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
224 (append f (list plist)))
225
226(defun saturation-extension-1 (ring f p) (polysaturation-extension ring f (list p)))
227
228;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
229;;
230;; Evaluation of polynomial (prefix) expressions
231;;
232;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
233
234(defun coerce-coeff (ring expr vars)
235 "Coerce an element of the coefficient ring to a constant polynomial."
236 ;; Modular arithmetic handler by rat
237 (make-poly-from-termlist (list (make-term (make-monom :dimension (length vars))
238 (funcall (ring-parse ring) expr)))
239 0))
240
241(defun poly-eval (ring expr vars &optional (list-marker '[))
242 (labels ((p-eval (arg) (poly-eval ring arg vars))
243 (p-eval-list (args) (mapcar #'p-eval args))
244 (p-add (x y) (poly-add ring x y)))
245 (cond
246 ((eql expr 0) (make-poly-zero))
247 ((member expr vars :test #'equalp)
248 (let ((pos (position expr vars :test #'equalp)))
249 (make-variable ring (length vars) pos)))
250 ((atom expr)
251 (coerce-coeff ring expr vars))
252 ((eq (car expr) list-marker)
253 (cons list-marker (p-eval-list (cdr expr))))
254 (t
255 (case (car expr)
256 (+ (reduce #'p-add (p-eval-list (cdr expr))))
257 (- (case (length expr)
258 (1 (make-poly-zero))
259 (2 (poly-uminus ring (p-eval (cadr expr))))
260 (3 (poly-sub ring (p-eval (cadr expr)) (p-eval (caddr expr))))
261 (otherwise (poly-sub ring (p-eval (cadr expr))
262 (reduce #'p-add (p-eval-list (cddr expr)))))))
263 (*
264 (if (endp (cddr expr)) ;unary
265 (p-eval (cdr expr))
266 (reduce #'(lambda (p q) (poly-mul ring p q)) (p-eval-list (cdr expr)))))
267 (expt
268 (cond
269 ((member (cadr expr) vars :test #'equalp)
270 ;;Special handling of (expt var pow)
271 (let ((pos (position (cadr expr) vars :test #'equalp)))
272 (make-variable ring (length vars) pos (caddr expr))))
273 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
274 ;; Negative power means division in coefficient ring
275 ;; Non-integer power means non-polynomial coefficient
276 (coerce-coeff ring expr vars))
277 (t (poly-expt ring (p-eval (cadr expr)) (caddr expr)))))
278 (otherwise
279 (coerce-coeff ring expr vars)))))))
280
281(defun spoly (ring f g)
282 "It yields the S-polynomial of polynomials F and G."
283 (declare (type poly f g))
284 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
285 (mf (monom-div lcm (poly-lm f)))
286 (mg (monom-div lcm (poly-lm g))))
287 (declare (type monom mf mg))
288 (multiple-value-bind (c cf cg)
289 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
290 (declare (ignore c))
291 (poly-sub
292 ring
293 (scalar-times-poly ring cg (monom-times-poly mf f))
294 (scalar-times-poly ring cf (monom-times-poly mg g))))))
295
296
297(defun poly-primitive-part (ring p)
298 "Divide polynomial P with integer coefficients by gcd of its
299coefficients and return the result."
300 (declare (type poly p))
301 (if (poly-zerop p)
302 (values p 1)
303 (let ((c (poly-content ring p)))
304 (values (make-poly-from-termlist (mapcar
305 #'(lambda (x)
306 (make-term (term-monom x)
307 (funcall (ring-div ring) (term-coeff x) c)))
308 (poly-termlist p))
309 (poly-sugar p))
310 c))))
311
312(defun poly-content (ring p)
313 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
314to compute the greatest common divisor."
315 (declare (type poly p))
316 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
317
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