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source: branches/f4grobner/polynomial.lisp@ 2446

Last change on this file since 2446 was 2446, checked in by Marek Rychlik, 9 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;;
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(defmethod scalar-multiply ((self poly) scalar)
137 "The scalar product of a polynomial SELF by a scalar SCALAR."
138 (mapc #'(lambda (term) (scalar-multiply term scalar)) (poly-termlist self))
139 self)
140
141(defun scalar-times-poly-1 (ring c p)
142 "The scalar product of scalar C by a polynomial P, omitting the head term. The sugar of the
143original polynomial becomes the sugar of the result."
144 (declare (type ring ring) (type poly p))
145 (make-poly-from-termlist (scalar-times-termlist ring c (cdr (poly-termlist p))) (poly-sugar p)))
146
147(defun monom-times-poly (m p)
148 (declare (type monom m) (type poly p))
149 (make-poly-from-termlist
150 (monom-times-termlist m (poly-termlist p))
151 (+ (poly-sugar p) (monom-sugar m))))
152
153(defun term-times-poly (ring term p)
154 (declare (type ring ring) (type term term) (type poly p))
155 (make-poly-from-termlist
156 (term-times-termlist ring term (poly-termlist p))
157 (+ (poly-sugar p) (term-sugar term))))
158
159(defun poly-add (ring-and-order p q)
160 (declare (type ring-and-order ring-and-order) (type poly p q))
161 (make-poly-from-termlist
162 (termlist-add ring-and-order
163 (poly-termlist p)
164 (poly-termlist q))
165 (max (poly-sugar p) (poly-sugar q))))
166
167(defun poly-sub (ring-and-order p q)
168 (declare (type ring-and-order ring-and-order) (type poly p q))
169 (make-poly-from-termlist
170 (termlist-sub ring-and-order (poly-termlist p) (poly-termlist q))
171 (max (poly-sugar p) (poly-sugar q))))
172
173(defun poly-uminus (ring p)
174 (declare (type ring ring) (type poly p))
175 (make-poly-from-termlist
176 (termlist-uminus ring (poly-termlist p))
177 (poly-sugar p)))
178
179(defun poly-mul (ring-and-order p q)
180 (declare (type ring-and-order ring-and-order) (type poly p q))
181 (make-poly-from-termlist
182 (termlist-mul ring-and-order (poly-termlist p) (poly-termlist q))
183 (+ (poly-sugar p) (poly-sugar q))))
184
185(defun poly-expt (ring-and-order p n)
186 (declare (type ring-and-order ring-and-order) (type poly p))
187 (make-poly-from-termlist (termlist-expt ring-and-order (poly-termlist p) n) (* n (poly-sugar p))))
188
189(defun poly-append (&rest plist)
190 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
191 (apply #'max (mapcar #'poly-sugar plist))))
192
193(defun poly-nreverse (p)
194 "Destructively reverse the order of terms in polynomial P. Returns P"
195 (declare (type poly p))
196 (setf (poly-termlist p) (nreverse (poly-termlist p)))
197 p)
198
199(defun poly-reverse (p)
200 "Returns a copy of the polynomial P with terms in reverse order."
201 (declare (type poly p))
202 (make-poly-from-termlist (reverse (poly-termlist p))
203 (poly-sugar p)))
204
205
206(defun poly-contract (p &optional (k 1))
207 (declare (type poly p))
208 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
209 (poly-sugar p)))
210
211(defun poly-extend (p &optional (m (make-monom :dimension 1)))
212 (declare (type poly p))
213 (make-poly-from-termlist
214 (termlist-extend (poly-termlist p) m)
215 (+ (poly-sugar p) (monom-sugar m))))
216
217(defun poly-add-variables (p k)
218 (declare (type poly p))
219 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
220 p)
221
222(defun poly-list-add-variables (plist k)
223 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
224
225(defun poly-standard-extension (plist &aux (k (length plist)))
226 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
227 (declare (list plist) (fixnum k))
228 (labels ((incf-power (g i)
229 (dolist (x (poly-termlist g))
230 (incf (monom-elt (term-monom x) i)))
231 (incf (poly-sugar g))))
232 (setf plist (poly-list-add-variables plist k))
233 (dotimes (i k plist)
234 (incf-power (nth i plist) i))))
235
236(defun saturation-extension (ring f plist
237 &aux
238 (k (length plist))
239 (d (monom-dimension (poly-lm (car plist))))
240 f-x plist-x)
241 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
242 (declare (type ring ring))
243 (setf f-x (poly-list-add-variables f k)
244 plist-x (mapcar #'(lambda (x)
245 (setf (poly-termlist x)
246 (nconc (poly-termlist x)
247 (list (make-term :monom (make-monom :dimension d)
248 :coeff (funcall (ring-uminus ring)
249 (funcall (ring-unit ring)))))))
250 x)
251 (poly-standard-extension plist)))
252 (append f-x plist-x))
253
254
255(defun polysaturation-extension (ring f plist
256 &aux
257 (k (length plist))
258 (d (+ k (monom-dimension (poly-lm (car plist)))))
259 ;; Add k variables to f
260 (f (poly-list-add-variables f k))
261 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
262 (plist (apply #'poly-append (poly-standard-extension plist))))
263 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
264 ;; Add -1 as the last term
265 (declare (type ring ring))
266 (setf (cdr (last (poly-termlist plist)))
267 (list (make-term :monom (make-monom :dimension d)
268 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
269 (append f (list plist)))
270
271(defun saturation-extension-1 (ring f p)
272 "Calculate [F, U*P-1]. It destructively modifies F."
273 (declare (type ring ring))
274 (polysaturation-extension ring f (list p)))
275
276;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
277;;
278;; Evaluation of polynomial (prefix) expressions
279;;
280;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
281
282(defun coerce-coeff (ring expr vars)
283 "Coerce an element of the coefficient ring to a constant polynomial."
284 ;; Modular arithmetic handler by rat
285 (declare (type ring ring))
286 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
287 :coeff (funcall (ring-parse ring) expr)))
288 0))
289
290(defun poly-eval (expr vars
291 &optional
292 (ring +ring-of-integers+)
293 (order #'lex>)
294 (list-marker :[)
295 &aux
296 (ring-and-order (make-ring-and-order :ring ring :order order)))
297 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
298variables VARS. Return the resulting polynomial or list of
299polynomials. Standard arithmetical operators in form EXPR are
300replaced with their analogues in the ring of polynomials, and the
301resulting expression is evaluated, resulting in a polynomial or a list
302of polynomials in internal form. A similar operation in another computer
303algebra system could be called 'expand' or so."
304 (declare (type ring ring))
305 (labels ((p-eval (arg) (poly-eval arg vars ring order))
306 (p-eval-scalar (arg) (poly-eval-scalar arg))
307 (p-eval-list (args) (mapcar #'p-eval args))
308 (p-add (x y) (poly-add ring-and-order x y)))
309 (cond
310 ((null expr) (error "Empty expression"))
311 ((eql expr 0) (make-poly-zero))
312 ((member expr vars :test #'equalp)
313 (let ((pos (position expr vars :test #'equalp)))
314 (make-poly-variable ring (length vars) pos)))
315 ((atom expr)
316 (coerce-coeff ring expr vars))
317 ((eq (car expr) list-marker)
318 (cons list-marker (p-eval-list (cdr expr))))
319 (t
320 (case (car expr)
321 (+ (reduce #'p-add (p-eval-list (cdr expr))))
322 (- (case (length expr)
323 (1 (make-poly-zero))
324 (2 (poly-uminus ring (p-eval (cadr expr))))
325 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
326 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
327 (reduce #'p-add (p-eval-list (cddr expr)))))))
328 (*
329 (if (endp (cddr expr)) ;unary
330 (p-eval (cdr expr))
331 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
332 (/
333 ;; A polynomial can be divided by a scalar
334 (cond
335 ((endp (cddr expr))
336 ;; A special case (/ ?), the inverse
337 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
338 (t
339 (let ((num (p-eval (cadr expr)))
340 (denom-inverse (apply (ring-div ring)
341 (cons (funcall (ring-unit ring))
342 (mapcar #'p-eval-scalar (cddr expr))))))
343 (scalar-times-poly ring denom-inverse num)))))
344 (expt
345 (cond
346 ((member (cadr expr) vars :test #'equalp)
347 ;;Special handling of (expt var pow)
348 (let ((pos (position (cadr expr) vars :test #'equalp)))
349 (make-poly-variable ring (length vars) pos (caddr expr))))
350 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
351 ;; Negative power means division in coefficient ring
352 ;; Non-integer power means non-polynomial coefficient
353 (coerce-coeff ring expr vars))
354 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
355 (otherwise
356 (coerce-coeff ring expr vars)))))))
357
358(defun poly-eval-scalar (expr
359 &optional
360 (ring +ring-of-integers+)
361 &aux
362 (order #'lex>))
363 "Evaluate a scalar expression EXPR in ring RING."
364 (declare (type ring ring))
365 (poly-lc (poly-eval expr nil ring order)))
366
367(defun spoly (ring-and-order f g
368 &aux
369 (ring (ro-ring ring-and-order)))
370 "It yields the S-polynomial of polynomials F and G."
371 (declare (type ring-and-order ring-and-order) (type poly f g))
372 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
373 (mf (monom-div lcm (poly-lm f)))
374 (mg (monom-div lcm (poly-lm g))))
375 (declare (type monom mf mg))
376 (multiple-value-bind (c cf cg)
377 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
378 (declare (ignore c))
379 (poly-sub
380 ring-and-order
381 (scalar-times-poly ring cg (monom-times-poly mf f))
382 (scalar-times-poly ring cf (monom-times-poly mg g))))))
383
384
385(defun poly-primitive-part (ring p)
386 "Divide polynomial P with integer coefficients by gcd of its
387coefficients and return the result."
388 (declare (type ring ring) (type poly p))
389 (if (poly-zerop p)
390 (values p 1)
391 (let ((c (poly-content ring p)))
392 (values (make-poly-from-termlist
393 (mapcar
394 #'(lambda (x)
395 (make-term :monom (term-monom x)
396 :coeff (funcall (ring-div ring) (term-coeff x) c)))
397 (poly-termlist p))
398 (poly-sugar p))
399 c))))
400
401(defun poly-content (ring p)
402 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
403to compute the greatest common divisor."
404 (declare (type ring ring) (type poly p))
405 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
406
407(defun read-infix-form (&key (stream t))
408 "Parser of infix expressions with integer/rational coefficients
409The parser will recognize two kinds of polynomial expressions:
410
411- polynomials in fully expanded forms with coefficients
412 written in front of symbolic expressions; constants can be optionally
413 enclosed in (); for example, the infix form
414 X^2-Y^2+(-4/3)*U^2*W^3-5
415 parses to
416 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
417
418- lists of polynomials; for example
419 [X-Y, X^2+3*Z]
420 parses to
421 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
422 where the first symbol [ marks a list of polynomials.
423
424-other infix expressions, for example
425 [(X-Y)*(X+Y)/Z,(X+1)^2]
426parses to:
427 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
428Currently this function is implemented using M. Kantrowitz's INFIX package."
429 (read-from-string
430 (concatenate 'string
431 "#I("
432 (with-output-to-string (s)
433 (loop
434 (multiple-value-bind (line eof)
435 (read-line stream t)
436 (format s "~A" line)
437 (when eof (return)))))
438 ")")))
439
440(defun read-poly (vars &key
441 (stream t)
442 (ring +ring-of-integers+)
443 (order #'lex>))
444 "Reads an expression in prefix form from a stream STREAM.
445The expression read from the strem should represent a polynomial or a
446list of polynomials in variables VARS, over the ring RING. The
447polynomial or list of polynomials is returned, with terms in each
448polynomial ordered according to monomial order ORDER."
449 (poly-eval (read-infix-form :stream stream) vars ring order))
450
451(defun string->poly (str vars
452 &optional
453 (ring +ring-of-integers+)
454 (order #'lex>))
455 "Converts a string STR to a polynomial in variables VARS."
456 (with-input-from-string (s str)
457 (read-poly vars :stream s :ring ring :order order)))
458
459(defun poly->alist (p)
460 "Convert a polynomial P to an association list. Thus, the format of the
461returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
462MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
463corresponding coefficient in the ring."
464 (cond
465 ((poly-p p)
466 (mapcar #'term->cons (poly-termlist p)))
467 ((and (consp p) (eq (car p) :[))
468 (cons :[ (mapcar #'poly->alist (cdr p))))))
469
470(defun string->alist (str vars
471 &optional
472 (ring +ring-of-integers+)
473 (order #'lex>))
474 "Convert a string STR representing a polynomial or polynomial list to
475an association list (... (MONOM . COEFF) ...)."
476 (poly->alist (string->poly str vars ring order)))
477
478(defun poly-equal-no-sugar-p (p q)
479 "Compare polynomials for equality, ignoring sugar."
480 (declare (type poly p q))
481 (equalp (poly-termlist p) (poly-termlist q)))
482
483(defun poly-set-equal-no-sugar-p (p q)
484 "Compare polynomial sets P and Q for equality, ignoring sugar."
485 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
486
487(defun poly-list-equal-no-sugar-p (p q)
488 "Compare polynomial lists P and Q for equality, ignoring sugar."
489 (every #'poly-equal-no-sugar-p p q))
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