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source: branches/f4grobner/polynomial.lisp@ 187

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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(in-package :grobner)
24
25;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
26;;
27;; Polynomials
28;;
29;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
30
31(defstruct (poly
32 ;;
33 ;; BOA constructor, by default constructs zero polynomial
34 (:constructor make-poly-from-termlist (termlist &optional (sugar (termlist-sugar termlist))))
35 (:constructor make-poly-zero (&aux (termlist nil) (sugar -1)))
36 ;; Constructor of polynomials representing a variable
37 (:constructor make-variable (ring nvars pos &optional (power 1)
38 &aux
39 (termlist (list
40 (make-term-variable ring nvars pos power)))
41 (sugar power)))
42 (:constructor poly-unit (ring dimension
43 &aux
44 (termlist (termlist-unit ring dimension))
45 (sugar 0))))
46 (termlist nil :type list)
47 (sugar -1 :type fixnum))
48
49;; Leading term
50(defmacro poly-lt (p) `(car (poly-termlist ,p)))
51
52;; Second term
53(defmacro poly-second-lt (p) `(cadar (poly-termlist ,p)))
54
55;; Leading monomial
56(defun poly-lm (p) (term-monom (poly-lt p)))
57
58;; Second monomial
59(defun poly-second-lm (p) (term-monom (poly-second-lt p)))
60
61;; Leading coefficient
62(defun poly-lc (p) (term-coeff (poly-lt p)))
63
64;; Second coefficient
65(defun poly-second-lc (p) (term-coeff (poly-second-lt p)))
66
67;; Testing for a zero polynomial
68(defun poly-zerop (p) (null (poly-termlist p)))
69
70;; The number of terms
71(defun poly-length (p) (length (poly-termlist p)))
72
73(defun scalar-times-poly (ring c p)
74 (make-poly-from-termlist (scalar-times-termlist ring c (poly-termlist p)) (poly-sugar p)))
75
76;; The scalar product omitting the head term
77(defun scalar-times-poly-1 (ring c p)
78 (make-poly-from-termlist (scalar-times-termlist ring c (cdr (poly-termlist p))) (poly-sugar p)))
79
80(defun monom-times-poly (m p)
81 (make-poly-from-termlist (monom-times-termlist m (poly-termlist p)) (+ (poly-sugar p) (monom-sugar m))))
82
83(defun term-times-poly (ring term p)
84 (make-poly-from-termlist (term-times-termlist ring term (poly-termlist p)) (+ (poly-sugar p) (term-sugar term))))
85
86(defun poly-add (ring p q)
87 (make-poly-from-termlist (termlist-add ring (poly-termlist p) (poly-termlist q)) (max (poly-sugar p) (poly-sugar q))))
88
89(defun poly-sub (ring p q)
90 (make-poly-from-termlist (termlist-sub ring (poly-termlist p) (poly-termlist q)) (max (poly-sugar p) (poly-sugar q))))
91
92(defun poly-uminus (ring p)
93 (make-poly-from-termlist (termlist-uminus ring (poly-termlist p)) (poly-sugar p)))
94
95(defun poly-mul (ring p q)
96 (make-poly-from-termlist (termlist-mul ring (poly-termlist p) (poly-termlist q)) (+ (poly-sugar p) (poly-sugar q))))
97
98(defun poly-expt (ring p n)
99 (make-poly-from-termlist (termlist-expt ring (poly-termlist p) n) (* n (poly-sugar p))))
100
101(defun poly-append (&rest plist)
102 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
103 (apply #'max (mapcar #'poly-sugar plist))))
104
105(defun poly-nreverse (p)
106 (setf (poly-termlist p) (nreverse (poly-termlist p)))
107 p)
108
109(defun poly-contract (p &optional (k 1))
110 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
111 (poly-sugar p)))
112
113(defun poly-extend (p &optional (m (make-monom 1 :initial-element 0)))
114 (make-poly-from-termlist
115 (termlist-extend (poly-termlist p) m)
116 (+ (poly-sugar p) (monom-sugar m))))
117
118(defun poly-add-variables (p k)
119 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
120 p)
121
122(defun poly-list-add-variables (plist k)
123 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
124
125(defun poly-standard-extension (plist &aux (k (length plist)))
126 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
127 (declare (list plist) (fixnum k))
128 (labels ((incf-power (g i)
129 (dolist (x (poly-termlist g))
130 (incf (monom-elt (term-monom x) i)))
131 (incf (poly-sugar g))))
132 (setf plist (poly-list-add-variables plist k))
133 (dotimes (i k plist)
134 (incf-power (nth i plist) i))))
135
136(defun saturation-extension (ring f plist &aux (k (length plist)) (d (monom-dimension (poly-lm (car plist)))))
137 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
138 (setf f (poly-list-add-variables f k)
139 plist (mapcar #'(lambda (x)
140 (setf (poly-termlist x) (nconc (poly-termlist x)
141 (list (make-term (make-monom d :initial-element 0)
142 (funcall (ring-uminus ring) (funcall (ring-unit ring)))))))
143 x)
144 (poly-standard-extension plist)))
145 (append f plist))
146
147
148(defun polysaturation-extension (ring f plist &aux (k (length plist))
149 (d (+ k (length (poly-lm (car plist))))))
150 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]."
151 (setf f (poly-list-add-variables f k)
152 plist (apply #'poly-append (poly-standard-extension plist))
153 (cdr (last (poly-termlist plist))) (list (make-term (make-monom d :initial-element 0)
154 (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
155 (append f (list plist)))
156
157(defun saturation-extension-1 (ring f p) (polysaturation-extension ring f (list p)))
158
159
160
161;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
162;;
163;; Evaluation of polynomial (prefix) expressions
164;;
165;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
166
167(defun coerce-coeff (ring expr vars)
168 "Coerce an element of the coefficient ring to a constant polynomial."
169 ;; Modular arithmetic handler by rat
170 (make-poly-from-termlist (list (make-term (make-monom (length vars) :initial-element 0)
171 (funcall (ring-parse ring) expr)))
172 0))
173
174(defun poly-eval (ring expr vars &optional (list-marker '[))
175 (labels ((p-eval (arg) (poly-eval ring arg vars))
176 (p-eval-list (args) (mapcar #'p-eval args))
177 (p-add (x y) (poly-add ring x y)))
178 (cond
179 ((eql expr 0) (make-poly-zero))
180 ((member expr vars :test #'equalp)
181 (let ((pos (position expr vars :test #'equalp)))
182 (make-variable ring (length vars) pos)))
183 ((atom expr)
184 (coerce-coeff ring expr vars))
185 ((eq (car expr) list-marker)
186 (cons list-marker (p-eval-list (cdr expr))))
187 (t
188 (case (car expr)
189 (+ (reduce #'p-add (p-eval-list (cdr expr))))
190 (- (case (length expr)
191 (1 (make-poly-zero))
192 (2 (poly-uminus ring (p-eval (cadr expr))))
193 (3 (poly-sub ring (p-eval (cadr expr)) (p-eval (caddr expr))))
194 (otherwise (poly-sub ring (p-eval (cadr expr))
195 (reduce #'p-add (p-eval-list (cddr expr)))))))
196 (*
197 (if (endp (cddr expr)) ;unary
198 (p-eval (cdr expr))
199 (reduce #'(lambda (p q) (poly-mul ring p q)) (p-eval-list (cdr expr)))))
200 (expt
201 (cond
202 ((member (cadr expr) vars :test #'equalp)
203 ;;Special handling of (expt var pow)
204 (let ((pos (position (cadr expr) vars :test #'equalp)))
205 (make-variable ring (length vars) pos (caddr expr))))
206 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
207 ;; Negative power means division in coefficient ring
208 ;; Non-integer power means non-polynomial coefficient
209 (coerce-coeff ring expr vars))
210 (t (poly-expt ring (p-eval (cadr expr)) (caddr expr)))))
211 (otherwise
212 (coerce-coeff ring expr vars)))))))
213
214(defun spoly (ring f g)
215 "It yields the S-polynomial of polynomials F and G."
216 (declare (type poly f g))
217 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
218 (mf (monom-div lcm (poly-lm f)))
219 (mg (monom-div lcm (poly-lm g))))
220 (declare (type monom mf mg))
221 (multiple-value-bind (c cf cg)
222 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
223 (declare (ignore c))
224 (poly-sub
225 ring
226 (scalar-times-poly ring cg (monom-times-poly mf f))
227 (scalar-times-poly ring cf (monom-times-poly mg g))))))
228
229
230(defun poly-primitive-part (ring p)
231 "Divide polynomial P with integer coefficients by gcd of its
232coefficients and return the result."
233 (declare (type poly p))
234 (if (poly-zerop p)
235 (values p 1)
236 (let ((c (poly-content ring p)))
237 (values (make-poly-from-termlist (mapcar
238 #'(lambda (x)
239 (make-term (term-monom x)
240 (funcall (ring-div ring) (term-coeff x) c)))
241 (poly-termlist p))
242 (poly-sugar p))
243 c))))
244
245(defun poly-content (ring p)
246 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
247to compute the greatest common divisor."
248 (declare (type poly p))
249 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
250
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