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source: branches/f4grobner/polynomial.lisp@ 1201

Last change on this file since 1201 was 1201, checked in by Marek Rychlik, 9 years ago

Changed the first line to eliminate 'unsafe' Emacs variables
File size: 14.9 KB
Line 
1;;; -*- Mode: Lisp -*-
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 "POLY-EVAL-SCALAR"
62 "SPOLY"
63 "POLY-PRIMITIVE-PART"
64 "POLY-CONTENT"
65 "READ-INFIX-FORM"
66 "READ-POLY"
67 "STRING->POLY"
68 "POLY->ALIST"
69 "STRING->ALIST"
70 ))
71
72(in-package :polynomial)
73
74;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
75;;
76;; Polynomials
77;;
78;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
79
80(defstruct (poly
81 ;;
82 ;; BOA constructor, by default constructs zero polynomial
83 (:constructor make-poly-from-termlist (termlist &optional (sugar (termlist-sugar termlist))))
84 (:constructor make-poly-zero (&aux (termlist nil) (sugar -1)))
85 ;; Constructor of polynomials representing a variable
86 (:constructor make-variable (ring nvars pos &optional (power 1)
87 &aux
88 (termlist (list
89 (make-term-variable ring nvars pos power)))
90 (sugar power)))
91 (:constructor poly-unit (ring dimension
92 &aux
93 (termlist (termlist-unit ring dimension))
94 (sugar 0))))
95 (termlist nil :type list)
96 (sugar -1 :type fixnum))
97
98;; Leading term
99(defmacro poly-lt (p) `(car (poly-termlist ,p)))
100
101;; Second term
102(defmacro poly-second-lt (p) `(cadar (poly-termlist ,p)))
103
104;; Leading monomial
105(defun poly-lm (p) (term-monom (poly-lt p)))
106
107;; Second monomial
108(defun poly-second-lm (p) (term-monom (poly-second-lt p)))
109
110;; Leading coefficient
111(defun poly-lc (p) (term-coeff (poly-lt p)))
112
113;; Second coefficient
114(defun poly-second-lc (p) (term-coeff (poly-second-lt p)))
115
116;; Testing for a zero polynomial
117(defun poly-zerop (p) (null (poly-termlist p)))
118
119;; The number of terms
120(defun poly-length (p) (length (poly-termlist p)))
121
122(defun scalar-times-poly (ring c p)
123 (declare (type ring ring) (poly p))
124 (make-poly-from-termlist (scalar-times-termlist ring c (poly-termlist p)) (poly-sugar p)))
125
126;; The scalar product omitting the head term
127(defun scalar-times-poly-1 (ring c p)
128 (declare (type ring ring) (poly p))
129 (make-poly-from-termlist (scalar-times-termlist ring c (cdr (poly-termlist p))) (poly-sugar p)))
130
131(defun monom-times-poly (m p)
132 (declare (poly p))
133 (make-poly-from-termlist
134 (monom-times-termlist m (poly-termlist p))
135 (+ (poly-sugar p) (monom-sugar m))))
136
137(defun term-times-poly (ring term p)
138 (declare (type ring ring) (type term term) (type poly p))
139 (make-poly-from-termlist
140 (term-times-termlist ring term (poly-termlist p))
141 (+ (poly-sugar p) (term-sugar term))))
142
143(defun poly-add (ring-and-order p q)
144 (declare (type ring-and-order ring-and-order) (type poly p q))
145 (make-poly-from-termlist
146 (termlist-add ring-and-order
147 (poly-termlist p)
148 (poly-termlist q))
149 (max (poly-sugar p) (poly-sugar q))))
150
151(defun poly-sub (ring-and-order p q)
152 (declare (type ring-and-order ring-and-order) (type poly p q))
153 (make-poly-from-termlist
154 (termlist-sub ring-and-order (poly-termlist p) (poly-termlist q))
155 (max (poly-sugar p) (poly-sugar q))))
156
157(defun poly-uminus (ring p)
158 (declare (type ring ring) (type poly p))
159 (make-poly-from-termlist
160 (termlist-uminus ring (poly-termlist p))
161 (poly-sugar p)))
162
163(defun poly-mul (ring-and-order p q)
164 (declare (type ring-and-order ring-and-order) (type poly p q))
165 (make-poly-from-termlist
166 (termlist-mul ring-and-order (poly-termlist p) (poly-termlist q))
167 (+ (poly-sugar p) (poly-sugar q))))
168
169(defun poly-expt (ring-and-order p n)
170 (declare (type ring-and-order ring-and-order) (type poly p))
171 (make-poly-from-termlist (termlist-expt ring-and-order (poly-termlist p) n) (* n (poly-sugar p))))
172
173(defun poly-append (&rest plist)
174 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
175 (apply #'max (mapcar #'poly-sugar plist))))
176
177(defun poly-nreverse (p)
178 (declare (type poly p))
179 (setf (poly-termlist p) (nreverse (poly-termlist p)))
180 p)
181
182(defun poly-contract (p &optional (k 1))
183 (declare (type poly p))
184 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
185 (poly-sugar p)))
186
187(defun poly-extend (p &optional (m (make-monom :dimension 1)))
188 (declare (type poly p))
189 (make-poly-from-termlist
190 (termlist-extend (poly-termlist p) m)
191 (+ (poly-sugar p) (monom-sugar m))))
192
193(defun poly-add-variables (p k)
194 (declare (type poly p))
195 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
196 p)
197
198(defun poly-list-add-variables (plist k)
199 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
200
201(defun poly-standard-extension (plist &aux (k (length plist)))
202 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
203 (declare (list plist) (fixnum k))
204 (labels ((incf-power (g i)
205 (dolist (x (poly-termlist g))
206 (incf (monom-elt (term-monom x) i)))
207 (incf (poly-sugar g))))
208 (setf plist (poly-list-add-variables plist k))
209 (dotimes (i k plist)
210 (incf-power (nth i plist) i))))
211
212(defun saturation-extension (ring f plist &aux (k (length plist)) (d (monom-dimension (poly-lm (car plist)))))
213 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
214 (setf f (poly-list-add-variables f k)
215 plist (mapcar #'(lambda (x)
216 (setf (poly-termlist x) (nconc (poly-termlist x)
217 (list (make-term (make-monom :dimension d)
218 (funcall (ring-uminus ring) (funcall (ring-unit ring)))))))
219 x)
220 (poly-standard-extension plist)))
221 (append f plist))
222
223
224(defun polysaturation-extension (ring f plist &aux (k (length plist))
225 (d (+ k (monom-dimension (poly-lm (car plist))))))
226 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]."
227 (setf f (poly-list-add-variables f k)
228 plist (apply #'poly-append (poly-standard-extension plist))
229 (cdr (last (poly-termlist plist))) (list (make-term (make-monom :dimension d)
230 (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
231 (append f (list plist)))
232
233(defun saturation-extension-1 (ring f p) (polysaturation-extension ring f (list p)))
234
235;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
236;;
237;; Evaluation of polynomial (prefix) expressions
238;;
239;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
240
241(defun coerce-coeff (ring expr vars)
242 "Coerce an element of the coefficient ring to a constant polynomial."
243 ;; Modular arithmetic handler by rat
244 (make-poly-from-termlist (list (make-term (make-monom :dimension (length vars))
245 (funcall (ring-parse ring) expr)))
246 0))
247
248(defun poly-eval (expr vars
249 &optional
250 (ring *ring-of-integers*)
251 (order #'lex>)
252 (list-marker :[)
253 &aux
254 (ring-and-order (make-ring-and-order :ring ring :order order)))
255 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
256variables VARS. Return the resulting polynomial or list of polynomials.
257Standard arithmetical operators in form EXPR are replaced with their analogues
258in the ring of polynomials, and the resulting expression is evaluated, resulting
259in a polynomial or a list of polynomials in internal form. A similar operation
260in computer algebra system is called 'expand' or so."
261 (labels ((p-eval (arg) (poly-eval arg vars ring order))
262 (p-eval-scalar (arg) (poly-eval-scalar arg))
263 (p-eval-list (args) (mapcar #'p-eval args))
264 (p-add (x y) (poly-add ring-and-order x y)))
265 (cond
266 ((null expr) (error "Empty expression"))
267 ((eql expr 0) (make-poly-zero))
268 ((member expr vars :test #'equalp)
269 (let ((pos (position expr vars :test #'equalp)))
270 (make-variable ring (length vars) pos)))
271 ((atom expr)
272 (coerce-coeff ring expr vars))
273 ((eq (car expr) list-marker)
274 (cons list-marker (p-eval-list (cdr expr))))
275 (t
276 (case (car expr)
277 (+ (reduce #'p-add (p-eval-list (cdr expr))))
278 (- (case (length expr)
279 (1 (make-poly-zero))
280 (2 (poly-uminus ring (p-eval (cadr expr))))
281 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
282 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
283 (reduce #'p-add (p-eval-list (cddr expr)))))))
284 (*
285 (if (endp (cddr expr)) ;unary
286 (p-eval (cdr expr))
287 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
288 (/
289 ;; A polynomial can be divided by a scalar
290 (cond
291 ((endp (cddr expr))
292 ;; A special case (/ ?), the inverse
293 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
294 (t
295 (let ((num (p-eval (cadr expr)))
296 (denom-inverse (apply (ring-div ring)
297 (cons (funcall (ring-unit ring))
298 (mapcar #'p-eval-scalar (cddr expr))))))
299 (scalar-times-poly ring denom-inverse num)))))
300 (expt
301 (cond
302 ((member (cadr expr) vars :test #'equalp)
303 ;;Special handling of (expt var pow)
304 (let ((pos (position (cadr expr) vars :test #'equalp)))
305 (make-variable ring (length vars) pos (caddr expr))))
306 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
307 ;; Negative power means division in coefficient ring
308 ;; Non-integer power means non-polynomial coefficient
309 (coerce-coeff ring expr vars))
310 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
311 (otherwise
312 (coerce-coeff ring expr vars)))))))
313
314(defun poly-eval-scalar (expr
315 &optional
316 (ring *ring-of-integers*)
317 &aux
318 (order #'lex>))
319 "Evaluate a scalar expression EXPR in ring RING."
320 (poly-lc (poly-eval expr nil ring order)))
321
322(defun spoly (ring-and-order f g
323 &aux
324 (ring (ro-ring ring-and-order)))
325 "It yields the S-polynomial of polynomials F and G."
326 (declare (type poly f g))
327 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
328 (mf (monom-div lcm (poly-lm f)))
329 (mg (monom-div lcm (poly-lm g))))
330 (declare (type monom mf mg))
331 (multiple-value-bind (c cf cg)
332 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
333 (declare (ignore c))
334 (poly-sub
335 ring-and-order
336 (scalar-times-poly ring cg (monom-times-poly mf f))
337 (scalar-times-poly ring cf (monom-times-poly mg g))))))
338
339
340(defun poly-primitive-part (ring p)
341 "Divide polynomial P with integer coefficients by gcd of its
342coefficients and return the result."
343 (declare (type poly p))
344 (if (poly-zerop p)
345 (values p 1)
346 (let ((c (poly-content ring p)))
347 (values (make-poly-from-termlist (mapcar
348 #'(lambda (x)
349 (make-term (term-monom x)
350 (funcall (ring-div ring) (term-coeff x) c)))
351 (poly-termlist p))
352 (poly-sugar p))
353 c))))
354
355(defun poly-content (ring p)
356 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
357to compute the greatest common divisor."
358 (declare (type poly p))
359 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
360
361(defun read-infix-form (&key (stream t))
362 "Parser of infix expressions with integer/rational coefficients
363The parser will recognize two kinds of polynomial expressions:
364
365- polynomials in fully expanded forms with coefficients
366 written in front of symbolic expressions; constants can be optionally
367 enclosed in (); for example, the infix form
368 X^2-Y^2+(-4/3)*U^2*W^3-5
369 parses to
370 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
371
372- lists of polynomials; for example
373 [X-Y, X^2+3*Z]
374 parses to
375 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
376 where the first symbol [ marks a list of polynomials.
377
378-other infix expressions, for example
379 [(X-Y)*(X+Y)/Z,(X+1)^2]
380parses to:
381 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
382Currently this function is implemented using M. Kantrowitz's INFIX package."
383 (read-from-string
384 (concatenate 'string
385 "#I("
386 (with-output-to-string (s)
387 (loop
388 (multiple-value-bind (line eof)
389 (read-line stream t)
390 (format s "~A" line)
391 (when eof (return)))))
392 ")")))
393
394(defun read-poly (vars &key
395 (stream t)
396 (ring *ring-of-integers*)
397 (order #'lex>))
398 "Reads an expression in prefix form from a stream STREAM.
399The expression read from the strem should represent a polynomial or a
400list of polynomials in variables VARS, over the ring RING. The
401polynomial or list of polynomials is returned, with terms in each
402polynomial ordered according to monomial order ORDER."
403 (poly-eval (read-infix-form :stream stream) vars ring order))
404
405(defun string->poly (str vars
406 &optional
407 (ring *ring-of-integers*)
408 (order #'lex>))
409 "Converts a string STR to a polynomial in variables VARS."
410 (with-input-from-string (s str)
411 (read-poly vars :stream s :ring ring :order order)))
412
413(defun poly->alist (p)
414 "Convert a polynomial P to an association list. Thus, the format of the
415returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
416MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
417corresponding coefficient in the ring."
418 (cond
419 ((poly-p p)
420 (mapcar #'term->cons (poly-termlist p)))
421 ((and (consp p) (eq (car p) :[))
422 (cons :[ (mapcar #'poly->alist (cdr p))))))
423
424(defun string->alist (str vars
425 &optional
426 (ring *ring-of-integers*)
427 (order #'lex>))
428 "Convert a string STR representing a polynomial or polynomial list to
429an association list (... (MONOM . COEFF) ...)."
430 (poly->alist (string->poly str vars ring order)))
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