close Warning: Can't synchronize with repository "(default)" (The repository directory has changed, you should resynchronize the repository with: trac-admin $ENV repository resync '(default)'). Look in the Trac log for more information.

source: branches/f4grobner/polynomial.lisp@ 2445

Last change on this file since 2445 was 2445, checked in by Marek Rychlik, 9 years ago

* empty log message *
File size: 17.1 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;;
24;; Polynomials
25;;
26;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
27
28(defpackage "POLYNOMIAL"
29 (:use :cl :ring :ring-and-order :monom :order :term :termlist :infix)
30 (:export "POLY"
31 "POLY-TERMLIST"
32 "POLY-SUGAR"
33 "POLY-RESET-SUGAR"
34 "POLY-LT"
35 "MAKE-POLY-FROM-TERMLIST"
36 "MAKE-POLY-ZERO"
37 "MAKE-POLY-VARIABLE"
38 "POLY-UNIT"
39 "POLY-LM"
40 "POLY-SECOND-LM"
41 "POLY-SECOND-LT"
42 "POLY-LC"
43 "POLY-SECOND-LC"
44 "POLY-ZEROP"
45 "POLY-LENGTH"
46 "SCALAR-TIMES-POLY"
47 "SCALAR-TIMES-POLY-1"
48 "MONOM-TIMES-POLY"
49 "TERM-TIMES-POLY"
50 "POLY-ADD"
51 "POLY-SUB"
52 "POLY-UMINUS"
53 "POLY-MUL"
54 "POLY-EXPT"
55 "POLY-APPEND"
56 "POLY-NREVERSE"
57 "POLY-REVERSE"
58 "POLY-CONTRACT"
59 "POLY-EXTEND"
60 "POLY-ADD-VARIABLES"
61 "POLY-LIST-ADD-VARIABLES"
62 "POLY-STANDARD-EXTENSION"
63 "SATURATION-EXTENSION"
64 "POLYSATURATION-EXTENSION"
65 "SATURATION-EXTENSION-1"
66 "COERCE-COEFF"
67 "POLY-EVAL"
68 "POLY-EVAL-SCALAR"
69 "SPOLY"
70 "POLY-PRIMITIVE-PART"
71 "POLY-CONTENT"
72 "READ-INFIX-FORM"
73 "READ-POLY"
74 "STRING->POLY"
75 "POLY->ALIST"
76 "STRING->ALIST"
77 "POLY-EQUAL-NO-SUGAR-P"
78 "POLY-SET-EQUAL-NO-SUGAR-P"
79 "POLY-LIST-EQUAL-NO-SUGAR-P"
80 ))
81
82(in-package :polynomial)
83
84(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
85
86#|
87 ;;
88 ;; BOA constructor, by default constructs zero polynomial
89 (:constructor make-poly-from-termlist (termlist &optional (sugar (termlist-sugar termlist))))
90 (:constructor make-poly-zero (&aux (termlist nil) (sugar -1)))
91 ;; Constructor of polynomials representing a variable
92 (:constructor make-poly-variable (ring nvars pos &optional (power 1)
93 &aux
94 (termlist (list
95 (make-term-variable ring nvars pos power)))
96 (sugar power)))
97 (:constructor poly-unit (ring dimension
98 &aux
99 (termlist (termlist-unit ring dimension))
100 (sugar 0))))
101
102|#
103
104(defclass poly ()
105 ((termlist :initarg :terms :accessor poly-termlist))
106 (:default-initargs :termlist nil))
107
108;; Leading term
109(defgeneric leading-term (object)
110 (:method ((self poly))
111 (car (poly-termlist self))))
112
113;; Second term
114(defgeneric second-leading-term (object)
115 (:method ((self poly))
116 (cadar (poly-termlist self))))
117
118;; Leading coefficient
119(defgeneric leading-coefficient (object)
120 (:method ((self poly))
121 (r-coeff (leading-term self))))
122
123;; Second coefficient
124(defgeneric second-leading-coefficient (object)
125 (:method ((self poly))
126 (term-coeff (second-leading-term self))))
127
128;; Testing for a zero polynomial
129(defmethod r-zerop ((self poly))
130 (null (poly-termlist self)))
131
132;; The number of terms
133(defmethod r-length ((self poly))
134 (length (poly-termlist self)))
135
136(defun poly-reset-sugar (p)
137 "(Re)sets the sugar of a polynomial P to the sugar of (POLY-TERMLIST P).
138Thus, the sugar is set to the maximum sugar of all monomials of P, or -1
139if P is a zero polynomial."
140 (declare (type poly p))
141 (setf (poly-sugar p) (termlist-sugar (poly-termlist p)))
142 p)
143
144(defun scalar-times-poly (ring c p)
145 "The scalar product of scalar C by a polynomial P. The sugar of the
146original polynomial becomes the sugar of the result."
147 (declare (type ring ring) (type poly p))
148 (make-poly-from-termlist (scalar-times-termlist ring c (poly-termlist p)) (poly-sugar p)))
149
150(defun scalar-times-poly-1 (ring c p)
151 "The scalar product of scalar C by a polynomial P, omitting the head term. The sugar of the
152original polynomial becomes the sugar of the result."
153 (declare (type ring ring) (type poly p))
154 (make-poly-from-termlist (scalar-times-termlist ring c (cdr (poly-termlist p))) (poly-sugar p)))
155
156(defun monom-times-poly (m p)
157 (declare (type monom m) (type poly p))
158 (make-poly-from-termlist
159 (monom-times-termlist m (poly-termlist p))
160 (+ (poly-sugar p) (monom-sugar m))))
161
162(defun term-times-poly (ring term p)
163 (declare (type ring ring) (type term term) (type poly p))
164 (make-poly-from-termlist
165 (term-times-termlist ring term (poly-termlist p))
166 (+ (poly-sugar p) (term-sugar term))))
167
168(defun poly-add (ring-and-order p q)
169 (declare (type ring-and-order ring-and-order) (type poly p q))
170 (make-poly-from-termlist
171 (termlist-add ring-and-order
172 (poly-termlist p)
173 (poly-termlist q))
174 (max (poly-sugar p) (poly-sugar q))))
175
176(defun poly-sub (ring-and-order p q)
177 (declare (type ring-and-order ring-and-order) (type poly p q))
178 (make-poly-from-termlist
179 (termlist-sub ring-and-order (poly-termlist p) (poly-termlist q))
180 (max (poly-sugar p) (poly-sugar q))))
181
182(defun poly-uminus (ring p)
183 (declare (type ring ring) (type poly p))
184 (make-poly-from-termlist
185 (termlist-uminus ring (poly-termlist p))
186 (poly-sugar p)))
187
188(defun poly-mul (ring-and-order p q)
189 (declare (type ring-and-order ring-and-order) (type poly p q))
190 (make-poly-from-termlist
191 (termlist-mul ring-and-order (poly-termlist p) (poly-termlist q))
192 (+ (poly-sugar p) (poly-sugar q))))
193
194(defun poly-expt (ring-and-order p n)
195 (declare (type ring-and-order ring-and-order) (type poly p))
196 (make-poly-from-termlist (termlist-expt ring-and-order (poly-termlist p) n) (* n (poly-sugar p))))
197
198(defun poly-append (&rest plist)
199 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
200 (apply #'max (mapcar #'poly-sugar plist))))
201
202(defun poly-nreverse (p)
203 "Destructively reverse the order of terms in polynomial P. Returns P"
204 (declare (type poly p))
205 (setf (poly-termlist p) (nreverse (poly-termlist p)))
206 p)
207
208(defun poly-reverse (p)
209 "Returns a copy of the polynomial P with terms in reverse order."
210 (declare (type poly p))
211 (make-poly-from-termlist (reverse (poly-termlist p))
212 (poly-sugar p)))
213
214
215(defun poly-contract (p &optional (k 1))
216 (declare (type poly p))
217 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
218 (poly-sugar p)))
219
220(defun poly-extend (p &optional (m (make-monom :dimension 1)))
221 (declare (type poly p))
222 (make-poly-from-termlist
223 (termlist-extend (poly-termlist p) m)
224 (+ (poly-sugar p) (monom-sugar m))))
225
226(defun poly-add-variables (p k)
227 (declare (type poly p))
228 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
229 p)
230
231(defun poly-list-add-variables (plist k)
232 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
233
234(defun poly-standard-extension (plist &aux (k (length plist)))
235 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
236 (declare (list plist) (fixnum k))
237 (labels ((incf-power (g i)
238 (dolist (x (poly-termlist g))
239 (incf (monom-elt (term-monom x) i)))
240 (incf (poly-sugar g))))
241 (setf plist (poly-list-add-variables plist k))
242 (dotimes (i k plist)
243 (incf-power (nth i plist) i))))
244
245(defun saturation-extension (ring f plist
246 &aux
247 (k (length plist))
248 (d (monom-dimension (poly-lm (car plist))))
249 f-x plist-x)
250 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
251 (declare (type ring ring))
252 (setf f-x (poly-list-add-variables f k)
253 plist-x (mapcar #'(lambda (x)
254 (setf (poly-termlist x)
255 (nconc (poly-termlist x)
256 (list (make-term :monom (make-monom :dimension d)
257 :coeff (funcall (ring-uminus ring)
258 (funcall (ring-unit ring)))))))
259 x)
260 (poly-standard-extension plist)))
261 (append f-x plist-x))
262
263
264(defun polysaturation-extension (ring f plist
265 &aux
266 (k (length plist))
267 (d (+ k (monom-dimension (poly-lm (car plist)))))
268 ;; Add k variables to f
269 (f (poly-list-add-variables f k))
270 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
271 (plist (apply #'poly-append (poly-standard-extension plist))))
272 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
273 ;; Add -1 as the last term
274 (declare (type ring ring))
275 (setf (cdr (last (poly-termlist plist)))
276 (list (make-term :monom (make-monom :dimension d)
277 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
278 (append f (list plist)))
279
280(defun saturation-extension-1 (ring f p)
281 "Calculate [F, U*P-1]. It destructively modifies F."
282 (declare (type ring ring))
283 (polysaturation-extension ring f (list p)))
284
285;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
286;;
287;; Evaluation of polynomial (prefix) expressions
288;;
289;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
290
291(defun coerce-coeff (ring expr vars)
292 "Coerce an element of the coefficient ring to a constant polynomial."
293 ;; Modular arithmetic handler by rat
294 (declare (type ring ring))
295 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
296 :coeff (funcall (ring-parse ring) expr)))
297 0))
298
299(defun poly-eval (expr vars
300 &optional
301 (ring +ring-of-integers+)
302 (order #'lex>)
303 (list-marker :[)
304 &aux
305 (ring-and-order (make-ring-and-order :ring ring :order order)))
306 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
307variables VARS. Return the resulting polynomial or list of
308polynomials. Standard arithmetical operators in form EXPR are
309replaced with their analogues in the ring of polynomials, and the
310resulting expression is evaluated, resulting in a polynomial or a list
311of polynomials in internal form. A similar operation in another computer
312algebra system could be called 'expand' or so."
313 (declare (type ring ring))
314 (labels ((p-eval (arg) (poly-eval arg vars ring order))
315 (p-eval-scalar (arg) (poly-eval-scalar arg))
316 (p-eval-list (args) (mapcar #'p-eval args))
317 (p-add (x y) (poly-add ring-and-order x y)))
318 (cond
319 ((null expr) (error "Empty expression"))
320 ((eql expr 0) (make-poly-zero))
321 ((member expr vars :test #'equalp)
322 (let ((pos (position expr vars :test #'equalp)))
323 (make-poly-variable ring (length vars) pos)))
324 ((atom expr)
325 (coerce-coeff ring expr vars))
326 ((eq (car expr) list-marker)
327 (cons list-marker (p-eval-list (cdr expr))))
328 (t
329 (case (car expr)
330 (+ (reduce #'p-add (p-eval-list (cdr expr))))
331 (- (case (length expr)
332 (1 (make-poly-zero))
333 (2 (poly-uminus ring (p-eval (cadr expr))))
334 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
335 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
336 (reduce #'p-add (p-eval-list (cddr expr)))))))
337 (*
338 (if (endp (cddr expr)) ;unary
339 (p-eval (cdr expr))
340 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
341 (/
342 ;; A polynomial can be divided by a scalar
343 (cond
344 ((endp (cddr expr))
345 ;; A special case (/ ?), the inverse
346 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
347 (t
348 (let ((num (p-eval (cadr expr)))
349 (denom-inverse (apply (ring-div ring)
350 (cons (funcall (ring-unit ring))
351 (mapcar #'p-eval-scalar (cddr expr))))))
352 (scalar-times-poly ring denom-inverse num)))))
353 (expt
354 (cond
355 ((member (cadr expr) vars :test #'equalp)
356 ;;Special handling of (expt var pow)
357 (let ((pos (position (cadr expr) vars :test #'equalp)))
358 (make-poly-variable ring (length vars) pos (caddr expr))))
359 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
360 ;; Negative power means division in coefficient ring
361 ;; Non-integer power means non-polynomial coefficient
362 (coerce-coeff ring expr vars))
363 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
364 (otherwise
365 (coerce-coeff ring expr vars)))))))
366
367(defun poly-eval-scalar (expr
368 &optional
369 (ring +ring-of-integers+)
370 &aux
371 (order #'lex>))
372 "Evaluate a scalar expression EXPR in ring RING."
373 (declare (type ring ring))
374 (poly-lc (poly-eval expr nil ring order)))
375
376(defun spoly (ring-and-order f g
377 &aux
378 (ring (ro-ring ring-and-order)))
379 "It yields the S-polynomial of polynomials F and G."
380 (declare (type ring-and-order ring-and-order) (type poly f g))
381 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
382 (mf (monom-div lcm (poly-lm f)))
383 (mg (monom-div lcm (poly-lm g))))
384 (declare (type monom mf mg))
385 (multiple-value-bind (c cf cg)
386 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
387 (declare (ignore c))
388 (poly-sub
389 ring-and-order
390 (scalar-times-poly ring cg (monom-times-poly mf f))
391 (scalar-times-poly ring cf (monom-times-poly mg g))))))
392
393
394(defun poly-primitive-part (ring p)
395 "Divide polynomial P with integer coefficients by gcd of its
396coefficients and return the result."
397 (declare (type ring ring) (type poly p))
398 (if (poly-zerop p)
399 (values p 1)
400 (let ((c (poly-content ring p)))
401 (values (make-poly-from-termlist
402 (mapcar
403 #'(lambda (x)
404 (make-term :monom (term-monom x)
405 :coeff (funcall (ring-div ring) (term-coeff x) c)))
406 (poly-termlist p))
407 (poly-sugar p))
408 c))))
409
410(defun poly-content (ring p)
411 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
412to compute the greatest common divisor."
413 (declare (type ring ring) (type poly p))
414 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
415
416(defun read-infix-form (&key (stream t))
417 "Parser of infix expressions with integer/rational coefficients
418The parser will recognize two kinds of polynomial expressions:
419
420- polynomials in fully expanded forms with coefficients
421 written in front of symbolic expressions; constants can be optionally
422 enclosed in (); for example, the infix form
423 X^2-Y^2+(-4/3)*U^2*W^3-5
424 parses to
425 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
426
427- lists of polynomials; for example
428 [X-Y, X^2+3*Z]
429 parses to
430 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
431 where the first symbol [ marks a list of polynomials.
432
433-other infix expressions, for example
434 [(X-Y)*(X+Y)/Z,(X+1)^2]
435parses to:
436 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
437Currently this function is implemented using M. Kantrowitz's INFIX package."
438 (read-from-string
439 (concatenate 'string
440 "#I("
441 (with-output-to-string (s)
442 (loop
443 (multiple-value-bind (line eof)
444 (read-line stream t)
445 (format s "~A" line)
446 (when eof (return)))))
447 ")")))
448
449(defun read-poly (vars &key
450 (stream t)
451 (ring +ring-of-integers+)
452 (order #'lex>))
453 "Reads an expression in prefix form from a stream STREAM.
454The expression read from the strem should represent a polynomial or a
455list of polynomials in variables VARS, over the ring RING. The
456polynomial or list of polynomials is returned, with terms in each
457polynomial ordered according to monomial order ORDER."
458 (poly-eval (read-infix-form :stream stream) vars ring order))
459
460(defun string->poly (str vars
461 &optional
462 (ring +ring-of-integers+)
463 (order #'lex>))
464 "Converts a string STR to a polynomial in variables VARS."
465 (with-input-from-string (s str)
466 (read-poly vars :stream s :ring ring :order order)))
467
468(defun poly->alist (p)
469 "Convert a polynomial P to an association list. Thus, the format of the
470returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
471MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
472corresponding coefficient in the ring."
473 (cond
474 ((poly-p p)
475 (mapcar #'term->cons (poly-termlist p)))
476 ((and (consp p) (eq (car p) :[))
477 (cons :[ (mapcar #'poly->alist (cdr p))))))
478
479(defun string->alist (str vars
480 &optional
481 (ring +ring-of-integers+)
482 (order #'lex>))
483 "Convert a string STR representing a polynomial or polynomial list to
484an association list (... (MONOM . COEFF) ...)."
485 (poly->alist (string->poly str vars ring order)))
486
487(defun poly-equal-no-sugar-p (p q)
488 "Compare polynomials for equality, ignoring sugar."
489 (declare (type poly p q))
490 (equalp (poly-termlist p) (poly-termlist q)))
491
492(defun poly-set-equal-no-sugar-p (p q)
493 "Compare polynomial sets P and Q for equality, ignoring sugar."
494 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
495
496(defun poly-list-equal-no-sugar-p (p q)
497 "Compare polynomial lists P and Q for equality, ignoring sugar."
498 (every #'poly-equal-no-sugar-p p q))
Note: See TracBrowser for help on using the repository browser.