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source: branches/f4grobner/polynomial.lisp@ 1204

Last change on this file since 1204 was 1203, checked in by Marek Rychlik, 10 years ago

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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
348 (mapcar
349 #'(lambda (x)
350 (make-term (term-monom x)
351 (funcall (ring-div ring) (term-coeff x) c)))
352 (poly-termlist p))
353 (poly-sugar p))
354 c))))
355
356(defun poly-content (ring p)
357 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
358to compute the greatest common divisor."
359 (declare (type poly p))
360 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
361
362(defun read-infix-form (&key (stream t))
363 "Parser of infix expressions with integer/rational coefficients
364The parser will recognize two kinds of polynomial expressions:
365
366- polynomials in fully expanded forms with coefficients
367 written in front of symbolic expressions; constants can be optionally
368 enclosed in (); for example, the infix form
369 X^2-Y^2+(-4/3)*U^2*W^3-5
370 parses to
371 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
372
373- lists of polynomials; for example
374 [X-Y, X^2+3*Z]
375 parses to
376 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
377 where the first symbol [ marks a list of polynomials.
378
379-other infix expressions, for example
380 [(X-Y)*(X+Y)/Z,(X+1)^2]
381parses to:
382 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
383Currently this function is implemented using M. Kantrowitz's INFIX package."
384 (read-from-string
385 (concatenate 'string
386 "#I("
387 (with-output-to-string (s)
388 (loop
389 (multiple-value-bind (line eof)
390 (read-line stream t)
391 (format s "~A" line)
392 (when eof (return)))))
393 ")")))
394
395(defun read-poly (vars &key
396 (stream t)
397 (ring *ring-of-integers*)
398 (order #'lex>))
399 "Reads an expression in prefix form from a stream STREAM.
400The expression read from the strem should represent a polynomial or a
401list of polynomials in variables VARS, over the ring RING. The
402polynomial or list of polynomials is returned, with terms in each
403polynomial ordered according to monomial order ORDER."
404 (poly-eval (read-infix-form :stream stream) vars ring order))
405
406(defun string->poly (str vars
407 &optional
408 (ring *ring-of-integers*)
409 (order #'lex>))
410 "Converts a string STR to a polynomial in variables VARS."
411 (with-input-from-string (s str)
412 (read-poly vars :stream s :ring ring :order order)))
413
414(defun poly->alist (p)
415 "Convert a polynomial P to an association list. Thus, the format of the
416returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
417MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
418corresponding coefficient in the ring."
419 (cond
420 ((poly-p p)
421 (mapcar #'term->cons (poly-termlist p)))
422 ((and (consp p) (eq (car p) :[))
423 (cons :[ (mapcar #'poly->alist (cdr p))))))
424
425(defun string->alist (str vars
426 &optional
427 (ring *ring-of-integers*)
428 (order #'lex>))
429 "Convert a string STR representing a polynomial or polynomial list to
430an association list (... (MONOM . COEFF) ...)."
431 (poly->alist (string->poly str vars ring order)))
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