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source: branches/f4grobner/polynomial.lisp@ 1129

Last change on this file since 1129 was 1129, 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 :order :term :termlist :infix)
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 "READ-INFIX-FORM"
65 "READ-POLY"
66 "STRING->POLY"
67 ))
68
69(in-package :polynomial)
70
71;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
72;;
73;; Polynomials
74;;
75;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
76
77(defstruct (poly
78 ;;
79 ;; BOA constructor, by default constructs zero polynomial
80 (:constructor make-poly-from-termlist (termlist &optional (sugar (termlist-sugar termlist))))
81 (:constructor make-poly-zero (&aux (termlist nil) (sugar -1)))
82 ;; Constructor of polynomials representing a variable
83 (:constructor make-variable (ring nvars pos &optional (power 1)
84 &aux
85 (termlist (list
86 (make-term-variable ring nvars pos power)))
87 (sugar power)))
88 (:constructor poly-unit (ring dimension
89 &aux
90 (termlist (termlist-unit ring dimension))
91 (sugar 0))))
92 (termlist nil :type list)
93 (sugar -1 :type fixnum))
94
95;; Leading term
96(defmacro poly-lt (p) `(car (poly-termlist ,p)))
97
98;; Second term
99(defmacro poly-second-lt (p) `(cadar (poly-termlist ,p)))
100
101;; Leading monomial
102(defun poly-lm (p) (term-monom (poly-lt p)))
103
104;; Second monomial
105(defun poly-second-lm (p) (term-monom (poly-second-lt p)))
106
107;; Leading coefficient
108(defun poly-lc (p) (term-coeff (poly-lt p)))
109
110;; Second coefficient
111(defun poly-second-lc (p) (term-coeff (poly-second-lt p)))
112
113;; Testing for a zero polynomial
114(defun poly-zerop (p) (null (poly-termlist p)))
115
116;; The number of terms
117(defun poly-length (p) (length (poly-termlist p)))
118
119(defun scalar-times-poly (ring c p)
120 (declare (type ring ring) (poly p))
121 (make-poly-from-termlist (scalar-times-termlist ring c (poly-termlist p)) (poly-sugar p)))
122
123;; The scalar product omitting the head term
124(defun scalar-times-poly-1 (ring c p)
125 (declare (type ring ring) (poly p))
126 (make-poly-from-termlist (scalar-times-termlist ring c (cdr (poly-termlist p))) (poly-sugar p)))
127
128(defun monom-times-poly (m p)
129 (declare (poly p))
130 (make-poly-from-termlist
131 (monom-times-termlist m (poly-termlist p))
132 (+ (poly-sugar p) (monom-sugar m))))
133
134(defun term-times-poly (ring term p)
135 (declare (type ring ring) (type term term) (type poly p))
136 (make-poly-from-termlist
137 (term-times-termlist ring term (poly-termlist p))
138 (+ (poly-sugar p) (term-sugar term))))
139
140(defun poly-add (ring-and-order p q)
141 (declare (type ring-and-order ring-and-order) (type poly p q))
142 (make-poly-from-termlist
143 (termlist-add ring-and-order
144 (poly-termlist p)
145 (poly-termlist q))
146 (max (poly-sugar p) (poly-sugar q))))
147
148(defun poly-sub (ring-and-order p q)
149 (declare (type ring-and-order ring-and-order) (type poly p q))
150 (make-poly-from-termlist
151 (termlist-sub ring-and-order (poly-termlist p) (poly-termlist q))
152 (max (poly-sugar p) (poly-sugar q))))
153
154(defun poly-uminus (ring p)
155 (declare (type ring ring) (type poly p))
156 (make-poly-from-termlist
157 (termlist-uminus ring (poly-termlist p))
158 (poly-sugar p)))
159
160(defun poly-mul (ring-and-order p q)
161 (declare (type ring-and-order ring-and-order) (type poly p q))
162 (make-poly-from-termlist
163 (termlist-mul ring-and-order (poly-termlist p) (poly-termlist q))
164 (+ (poly-sugar p) (poly-sugar q))))
165
166(defun poly-expt (ring-and-order p n)
167 (declare (type ring-and-order ring-and-order) (type poly p))
168 (make-poly-from-termlist (termlist-expt ring-and-order (poly-termlist p) n) (* n (poly-sugar p))))
169
170(defun poly-append (&rest plist)
171 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
172 (apply #'max (mapcar #'poly-sugar plist))))
173
174(defun poly-nreverse (p)
175 (declare (type poly p))
176 (setf (poly-termlist p) (nreverse (poly-termlist p)))
177 p)
178
179(defun poly-contract (p &optional (k 1))
180 (declare (type poly p))
181 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
182 (poly-sugar p)))
183
184(defun poly-extend (p &optional (m (make-monom :dimension 1)))
185 (declare (type poly p))
186 (make-poly-from-termlist
187 (termlist-extend (poly-termlist p) m)
188 (+ (poly-sugar p) (monom-sugar m))))
189
190(defun poly-add-variables (p k)
191 (declare (type poly p))
192 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
193 p)
194
195(defun poly-list-add-variables (plist k)
196 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
197
198(defun poly-standard-extension (plist &aux (k (length plist)))
199 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
200 (declare (list plist) (fixnum k))
201 (labels ((incf-power (g i)
202 (dolist (x (poly-termlist g))
203 (incf (monom-elt (term-monom x) i)))
204 (incf (poly-sugar g))))
205 (setf plist (poly-list-add-variables plist k))
206 (dotimes (i k plist)
207 (incf-power (nth i plist) i))))
208
209(defun saturation-extension (ring f plist &aux (k (length plist)) (d (monom-dimension (poly-lm (car plist)))))
210 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
211 (setf f (poly-list-add-variables f k)
212 plist (mapcar #'(lambda (x)
213 (setf (poly-termlist x) (nconc (poly-termlist x)
214 (list (make-term (make-monom :dimension d)
215 (funcall (ring-uminus ring) (funcall (ring-unit ring)))))))
216 x)
217 (poly-standard-extension plist)))
218 (append f plist))
219
220
221(defun polysaturation-extension (ring f plist &aux (k (length plist))
222 (d (+ k (monom-dimension (poly-lm (car plist))))))
223 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]."
224 (setf f (poly-list-add-variables f k)
225 plist (apply #'poly-append (poly-standard-extension plist))
226 (cdr (last (poly-termlist plist))) (list (make-term (make-monom :dimension d)
227 (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
228 (append f (list plist)))
229
230(defun saturation-extension-1 (ring f p) (polysaturation-extension ring f (list p)))
231
232;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
233;;
234;; Evaluation of polynomial (prefix) expressions
235;;
236;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
237
238(defun coerce-coeff (ring expr vars)
239 "Coerce an element of the coefficient ring to a constant polynomial."
240 ;; Modular arithmetic handler by rat
241 (make-poly-from-termlist (list (make-term (make-monom :dimension (length vars))
242 (funcall (ring-parse ring) expr)))
243 0))
244
245(defun poly-eval-scalar (expr
246 &optional
247 (ring *ring-of-integers*)
248 &aux
249 (order #'lex>))
250 "Evaluate a scalar expression EXPR in ring RING."
251 (poly-lc (poly-eval expr nil ring order)))
252
253(defun poly-eval (expr vars
254 &optional
255 (ring *ring-of-integers*)
256 (order #'lex>)
257 (list-marker '[)
258 &aux
259 (ring-and-order (make-ring-and-order :ring ring :order order)))
260 (labels ((p-eval (arg) (poly-eval arg vars ring order))
261 (p-eval-list (args) (mapcar #'p-eval args))
262 (p-add (x y) (poly-add ring-and-order x y)))
263 (cond
264 ((null expr) (error "Empty expression"))
265 ((eql expr 0) (make-poly-zero))
266 ((member expr vars :test #'equalp)
267 (let ((pos (position expr vars :test #'equalp)))
268 (make-variable ring (length vars) pos)))
269 ((atom expr)
270 (coerce-coeff ring expr vars))
271 ((eq (car expr) list-marker)
272 (cons list-marker (p-eval-list (cdr expr))))
273 (t
274 (case (car expr)
275 (+ (reduce #'p-add (p-eval-list (cdr expr))))
276 (- (case (length expr)
277 (1 (make-poly-zero))
278 (2 (poly-uminus ring (p-eval (cadr expr))))
279 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
280 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
281 (reduce #'p-add (p-eval-list (cddr expr)))))))
282 (*
283 (if (endp (cddr expr)) ;unary
284 (p-eval (cdr expr))
285 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
286 (/
287 ;; A polynomial can be divided by a scalar
288 (cond
289 ((endp (cddr expr))
290 ;; A special case (/ ?), the inverse
291 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
292 (t
293 (let ((num (p-eval (cadr expr)))
294 (denom-inverse (poly-eval-scalar '(/ 1 2 3) ring)))
295 (scalar-times-poly ring denom-inverse num)))))
296 (expt
297 (cond
298 ((member (cadr expr) vars :test #'equalp)
299 ;;Special handling of (expt var pow)
300 (let ((pos (position (cadr expr) vars :test #'equalp)))
301 (make-variable ring (length vars) pos (caddr expr))))
302 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
303 ;; Negative power means division in coefficient ring
304 ;; Non-integer power means non-polynomial coefficient
305 (coerce-coeff ring expr vars))
306 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
307 (otherwise
308 (coerce-coeff ring expr vars)))))))
309
310(defun spoly (ring f g)
311 "It yields the S-polynomial of polynomials F and G."
312 (declare (type poly f g))
313 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
314 (mf (monom-div lcm (poly-lm f)))
315 (mg (monom-div lcm (poly-lm g))))
316 (declare (type monom mf mg))
317 (multiple-value-bind (c cf cg)
318 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
319 (declare (ignore c))
320 (poly-sub
321 ring
322 (scalar-times-poly ring cg (monom-times-poly mf f))
323 (scalar-times-poly ring cf (monom-times-poly mg g))))))
324
325
326(defun poly-primitive-part (ring p)
327 "Divide polynomial P with integer coefficients by gcd of its
328coefficients and return the result."
329 (declare (type poly p))
330 (if (poly-zerop p)
331 (values p 1)
332 (let ((c (poly-content ring p)))
333 (values (make-poly-from-termlist (mapcar
334 #'(lambda (x)
335 (make-term (term-monom x)
336 (funcall (ring-div ring) (term-coeff x) c)))
337 (poly-termlist p))
338 (poly-sugar p))
339 c))))
340
341(defun poly-content (ring p)
342 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
343to compute the greatest common divisor."
344 (declare (type poly p))
345 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
346
347(defun read-infix-form (&key (stream t))
348 "Parser of infix expressions with integer/rational coefficients
349The parser will recognize two kinds of polynomial expressions:
350
351- polynomials in fully expanded forms with coefficients
352 written in front of symbolic expressions; constants can be optionally
353 enclosed in (); for example, the infix form
354 X^2-Y^2+(-4/3)*U^2*W^3-5
355 parses to
356 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
357
358- lists of polynomials; for example
359 [X-Y, X^2+3*Z]
360 parses to
361 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
362 where the first symbol [ marks a list of polynomials.
363
364-other infix expressions, for example
365 [(X-Y)*(X+Y)/Z,(X+1)^2]
366parses to:
367 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
368Currently this function is implemented using M. Kantrowitz's INFIX package."
369 (read-from-string
370 (concatenate 'string
371 "#I("
372 (with-output-to-string (s)
373 (loop
374 (multiple-value-bind (line eof)
375 (read-line stream t)
376 (format s "~A" line)
377 (when eof (return)))))
378 ")")))
379
380(defun read-poly (vars &key (stream t))
381 "Reads an expression in prefix form from a stream STREAM.
382If the expression represents a polynomial or a list of polynomials in variables VARS then
383the polynomial or list of polynomials is returned."
384 (poly-eval (read-infix-form :stream stream) vars))
385
386(defun string->poly (str vars)
387 "Converts a string STR to a polynomial in variables VARS"
388 (with-input-from-string (s str)
389 (read-poly vars :stream s)))
390
391
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