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@ 2593

Last change on this file since 2593 was 2593, checked in by Marek Rychlik, 9 years ago

* empty log message *

File size: 14.2 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(defpackage "POLYNOMIAL"
23 (:use :cl :ring :monom :order :term #| :infix |# )
24 (:export "POLY")
25 (:documentation "Implements polynomials"))
26
27(in-package :polynomial)
28
29(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
30
31(defclass poly ()
32 ((termlist :initarg :termlist :accessor poly-termlist))
33 (:default-initargs :termlist nil))
34
35(defmethod print-object ((self poly) stream)
36 (format stream "#<POLY TERMLIST=~A >" (poly-termlist self)))
37
38(defmethod insert-item ((self poly) (item term))
39 (push item (poly-termlist self))
40 self)
41
42(defmethod append-item ((self poly) (item term))
43 (setf (cdr (last (poly-termlist self))) (list item))
44 self)
45
46;; Leading term
47(defgeneric leading-term (object)
48 (:method ((self poly))
49 (car (poly-termlist self)))
50 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
51
52;; Second term
53(defgeneric second-leading-term (object)
54 (:method ((self poly))
55 (cadar (poly-termlist self)))
56 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
57
58;; Leading coefficient
59(defgeneric leading-coefficient (object)
60 (:method ((self poly))
61 (r-coeff (leading-term self)))
62 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
63
64;; Second coefficient
65(defgeneric second-leading-coefficient (object)
66 (:method ((self poly))
67 (r-coeff (second-leading-term self)))
68 (:documentation "The second leading coefficient of a polynomial. It signals error for a polynomial with at most one term."))
69
70;; Testing for a zero polynomial
71(defmethod r-zerop ((self poly))
72 (null (poly-termlist self)))
73
74;; The number of terms
75(defmethod r-length ((self poly))
76 (length (poly-termlist self)))
77
78(defmethod multiply-by ((self poly) (other monom))
79 (mapc #'(lambda (term) (multiply-by term other))
80 (poly-termlist self))
81 self)
82
83(defmethod multiply-by ((self poly) (other scalar))
84 (mapc #'(lambda (term) (multiply-by term other))
85 (poly-termlist self))
86 self)
87
88(defun fast-add-to (p q order-fn)
89 "Fast destructive addition of termlists
90Note that this assumes the presence of a
91dummy header."
92 (macrolet ((lt (x) `(cadr ,x))
93 (lc (x) `(r-coeff (cadr ,x))))
94 (do ((p p)
95 (q q))
96 ((or (endp (cdr p)) (endp (cdr q)))
97 p)
98 (multiple-value-bind
99 (greater-p equal-p)
100 (funcall order-fn (lt q) (lt p))
101 (cond
102 (greater-p
103 (rotatef (cdr p) (cdr q)))
104 (equal-p
105 (let ((s (add-to (lc p) (lc q))))
106 (if (r-zerop s)
107 (setf (cdr p) (cddr p))
108 (setf (lc p) s
109 q (cdr q)))))))
110 (setf p (cdr p)))))
111
112(defmethod add-to ((self poly) (other poly))
113 "Adds to polynomial SELF another polynomial OTHER.
114This operation destructively modifies both polynomials.
115The result is stored in SELF. This implementation does
116no consing, entirely reusing the sells of SELF and OTHER."
117 (with-slots ((termlist1 termlist) (order1 order))
118 self
119 (with-slots ((termlist2 termlist) (order2 order))
120 other
121 (unless (eq order1 order2)
122 (setf termlist2 (sort order1 termlist2)))
123 ;; Create dummy head
124 (push nil termlist1)
125 (push nil termlist2)
126 (fast-add-to termlist1 termlist2 order1)
127 ;; Remove dummy head
128 (pop termlist1)))
129 self)
130
131(defmethod subtract-from ((self poly) (other poly)))
132
133(defmethod unary-uminus ((self poly)))
134
135#|
136
137(defun poly-standard-extension (plist &aux (k (length plist)))
138 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
139 (declare (list plist) (fixnum k))
140 (labels ((incf-power (g i)
141 (dolist (x (poly-termlist g))
142 (incf (monom-elt (term-monom x) i)))
143 (incf (poly-sugar g))))
144 (setf plist (poly-list-add-variables plist k))
145 (dotimes (i k plist)
146 (incf-power (nth i plist) i))))
147
148(defun saturation-extension (ring f plist
149 &aux
150 (k (length plist))
151 (d (monom-dimension (poly-lm (car plist))))
152 f-x plist-x)
153 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
154 (declare (type ring ring))
155 (setf f-x (poly-list-add-variables f k)
156 plist-x (mapcar #'(lambda (x)
157 (setf (poly-termlist x)
158 (nconc (poly-termlist x)
159 (list (make-term :monom (make-monom :dimension d)
160 :coeff (funcall (ring-uminus ring)
161 (funcall (ring-unit ring)))))))
162 x)
163 (poly-standard-extension plist)))
164 (append f-x plist-x))
165
166
167(defun polysaturation-extension (ring f plist
168 &aux
169 (k (length plist))
170 (d (+ k (monom-dimension (poly-lm (car plist)))))
171 ;; Add k variables to f
172 (f (poly-list-add-variables f k))
173 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
174 (plist (apply #'poly-append (poly-standard-extension plist))))
175 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
176 ;; Add -1 as the last term
177 (declare (type ring ring))
178 (setf (cdr (last (poly-termlist plist)))
179 (list (make-term :monom (make-monom :dimension d)
180 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
181 (append f (list plist)))
182
183(defun saturation-extension-1 (ring f p)
184 "Calculate [F, U*P-1]. It destructively modifies F."
185 (declare (type ring ring))
186 (polysaturation-extension ring f (list p)))
187
188;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
189;;
190;; Evaluation of polynomial (prefix) expressions
191;;
192;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
193
194(defun coerce-coeff (ring expr vars)
195 "Coerce an element of the coefficient ring to a constant polynomial."
196 ;; Modular arithmetic handler by rat
197 (declare (type ring ring))
198 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
199 :coeff (funcall (ring-parse ring) expr)))
200 0))
201
202(defun poly-eval (expr vars
203 &optional
204 (ring +ring-of-integers+)
205 (order #'lex>)
206 (list-marker :[)
207 &aux
208 (ring-and-order (make-ring-and-order :ring ring :order order)))
209 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
210variables VARS. Return the resulting polynomial or list of
211polynomials. Standard arithmetical operators in form EXPR are
212replaced with their analogues in the ring of polynomials, and the
213resulting expression is evaluated, resulting in a polynomial or a list
214of polynomials in internal form. A similar operation in another computer
215algebra system could be called 'expand' or so."
216 (declare (type ring ring))
217 (labels ((p-eval (arg) (poly-eval arg vars ring order))
218 (p-eval-scalar (arg) (poly-eval-scalar arg))
219 (p-eval-list (args) (mapcar #'p-eval args))
220 (p-add (x y) (poly-add ring-and-order x y)))
221 (cond
222 ((null expr) (error "Empty expression"))
223 ((eql expr 0) (make-poly-zero))
224 ((member expr vars :test #'equalp)
225 (let ((pos (position expr vars :test #'equalp)))
226 (make-poly-variable ring (length vars) pos)))
227 ((atom expr)
228 (coerce-coeff ring expr vars))
229 ((eq (car expr) list-marker)
230 (cons list-marker (p-eval-list (cdr expr))))
231 (t
232 (case (car expr)
233 (+ (reduce #'p-add (p-eval-list (cdr expr))))
234 (- (case (length expr)
235 (1 (make-poly-zero))
236 (2 (poly-uminus ring (p-eval (cadr expr))))
237 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
238 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
239 (reduce #'p-add (p-eval-list (cddr expr)))))))
240 (*
241 (if (endp (cddr expr)) ;unary
242 (p-eval (cdr expr))
243 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
244 (/
245 ;; A polynomial can be divided by a scalar
246 (cond
247 ((endp (cddr expr))
248 ;; A special case (/ ?), the inverse
249 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
250 (t
251 (let ((num (p-eval (cadr expr)))
252 (denom-inverse (apply (ring-div ring)
253 (cons (funcall (ring-unit ring))
254 (mapcar #'p-eval-scalar (cddr expr))))))
255 (scalar-times-poly ring denom-inverse num)))))
256 (expt
257 (cond
258 ((member (cadr expr) vars :test #'equalp)
259 ;;Special handling of (expt var pow)
260 (let ((pos (position (cadr expr) vars :test #'equalp)))
261 (make-poly-variable ring (length vars) pos (caddr expr))))
262 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
263 ;; Negative power means division in coefficient ring
264 ;; Non-integer power means non-polynomial coefficient
265 (coerce-coeff ring expr vars))
266 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
267 (otherwise
268 (coerce-coeff ring expr vars)))))))
269
270(defun poly-eval-scalar (expr
271 &optional
272 (ring +ring-of-integers+)
273 &aux
274 (order #'lex>))
275 "Evaluate a scalar expression EXPR in ring RING."
276 (declare (type ring ring))
277 (poly-lc (poly-eval expr nil ring order)))
278
279(defun spoly (ring-and-order f g
280 &aux
281 (ring (ro-ring ring-and-order)))
282 "It yields the S-polynomial of polynomials F and G."
283 (declare (type ring-and-order ring-and-order) (type poly f g))
284 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
285 (mf (monom-div lcm (poly-lm f)))
286 (mg (monom-div lcm (poly-lm g))))
287 (declare (type monom mf mg))
288 (multiple-value-bind (c cf cg)
289 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
290 (declare (ignore c))
291 (poly-sub
292 ring-and-order
293 (scalar-times-poly ring cg (monom-times-poly mf f))
294 (scalar-times-poly ring cf (monom-times-poly mg g))))))
295
296
297(defun poly-primitive-part (ring p)
298 "Divide polynomial P with integer coefficients by gcd of its
299coefficients and return the result."
300 (declare (type ring ring) (type poly p))
301 (if (poly-zerop p)
302 (values p 1)
303 (let ((c (poly-content ring p)))
304 (values (make-poly-from-termlist
305 (mapcar
306 #'(lambda (x)
307 (make-term :monom (term-monom x)
308 :coeff (funcall (ring-div ring) (term-coeff x) c)))
309 (poly-termlist p))
310 (poly-sugar p))
311 c))))
312
313(defun poly-content (ring p)
314 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
315to compute the greatest common divisor."
316 (declare (type ring ring) (type poly p))
317 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
318
319(defun read-infix-form (&key (stream t))
320 "Parser of infix expressions with integer/rational coefficients
321The parser will recognize two kinds of polynomial expressions:
322
323- polynomials in fully expanded forms with coefficients
324 written in front of symbolic expressions; constants can be optionally
325 enclosed in (); for example, the infix form
326 X^2-Y^2+(-4/3)*U^2*W^3-5
327 parses to
328 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
329
330- lists of polynomials; for example
331 [X-Y, X^2+3*Z]
332 parses to
333 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
334 where the first symbol [ marks a list of polynomials.
335
336-other infix expressions, for example
337 [(X-Y)*(X+Y)/Z,(X+1)^2]
338parses to:
339 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
340Currently this function is implemented using M. Kantrowitz's INFIX package."
341 (read-from-string
342 (concatenate 'string
343 "#I("
344 (with-output-to-string (s)
345 (loop
346 (multiple-value-bind (line eof)
347 (read-line stream t)
348 (format s "~A" line)
349 (when eof (return)))))
350 ")")))
351
352(defun read-poly (vars &key
353 (stream t)
354 (ring +ring-of-integers+)
355 (order #'lex>))
356 "Reads an expression in prefix form from a stream STREAM.
357The expression read from the strem should represent a polynomial or a
358list of polynomials in variables VARS, over the ring RING. The
359polynomial or list of polynomials is returned, with terms in each
360polynomial ordered according to monomial order ORDER."
361 (poly-eval (read-infix-form :stream stream) vars ring order))
362
363(defun string->poly (str vars
364 &optional
365 (ring +ring-of-integers+)
366 (order #'lex>))
367 "Converts a string STR to a polynomial in variables VARS."
368 (with-input-from-string (s str)
369 (read-poly vars :stream s :ring ring :order order)))
370
371(defun poly->alist (p)
372 "Convert a polynomial P to an association list. Thus, the format of the
373returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
374MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
375corresponding coefficient in the ring."
376 (cond
377 ((poly-p p)
378 (mapcar #'term->cons (poly-termlist p)))
379 ((and (consp p) (eq (car p) :[))
380 (cons :[ (mapcar #'poly->alist (cdr p))))))
381
382(defun string->alist (str vars
383 &optional
384 (ring +ring-of-integers+)
385 (order #'lex>))
386 "Convert a string STR representing a polynomial or polynomial list to
387an association list (... (MONOM . COEFF) ...)."
388 (poly->alist (string->poly str vars ring order)))
389
390(defun poly-equal-no-sugar-p (p q)
391 "Compare polynomials for equality, ignoring sugar."
392 (declare (type poly p q))
393 (equalp (poly-termlist p) (poly-termlist q)))
394
395(defun poly-set-equal-no-sugar-p (p q)
396 "Compare polynomial sets P and Q for equality, ignoring sugar."
397 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
398
399(defun poly-list-equal-no-sugar-p (p q)
400 "Compare polynomial lists P and Q for equality, ignoring sugar."
401 (every #'poly-equal-no-sugar-p p q))
402|#
Note: See TracBrowser for help on using the repository browser.