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source: branches/f4grobner/polynomial.lisp@ 2589

Last change on this file since 2589 was 2589, 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(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;; Fast destructive addition of termlists
89(defun fast-add-to (p q)
90 (macrolet ((lt (x) `(cadr ,x))
91 (lc (x) `(r-coeff (cadr ,x))))
92 (do ((p p)
93 (q q))
94 ((or (endp (cdr p)) (endp (cdr q)))
95 p)
96 (multiple-value-bind
97 (greater-p equal-p)
98 (lex> (lt q) (lt p))
99 (cond
100 (greater-p
101 (rotatef (cdr p) (cdr q)))
102 (equal-p
103 (let ((s (add-to (lc p) (lc q))))
104 (if (r-zerop s)
105 (setf (cdr p) (cddr p))
106 (setf (lc p) s
107 q (cdr q)))))))
108 (setf p (cdr p)))))
109
110(defmethod add-to ((self poly) (other poly))
111 "Adds to polynomial SELF another polynomial OTHER.
112This operation destructively modifies both polynomials.
113The result is stored in SELF. This implementation does
114no consing, entirely reusing the sells of SELF and OTHER."
115 (with-slots ((termlist1 termlist))
116 self
117 (with-slots ((termlist2 termlist))
118 other
119 (push nil termlist1)
120 (push nil termlist2)
121 (fast-add-to termlist1 termlist2)
122 (pop termlist)))
123 self)
124
125(defmethod subtract-from ((self poly) (other poly)))
126
127(defmethod unary-uminus ((self poly)))
128
129#|
130
131(defun poly-standard-extension (plist &aux (k (length plist)))
132 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
133 (declare (list plist) (fixnum k))
134 (labels ((incf-power (g i)
135 (dolist (x (poly-termlist g))
136 (incf (monom-elt (term-monom x) i)))
137 (incf (poly-sugar g))))
138 (setf plist (poly-list-add-variables plist k))
139 (dotimes (i k plist)
140 (incf-power (nth i plist) i))))
141
142(defun saturation-extension (ring f plist
143 &aux
144 (k (length plist))
145 (d (monom-dimension (poly-lm (car plist))))
146 f-x plist-x)
147 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
148 (declare (type ring ring))
149 (setf f-x (poly-list-add-variables f k)
150 plist-x (mapcar #'(lambda (x)
151 (setf (poly-termlist x)
152 (nconc (poly-termlist x)
153 (list (make-term :monom (make-monom :dimension d)
154 :coeff (funcall (ring-uminus ring)
155 (funcall (ring-unit ring)))))))
156 x)
157 (poly-standard-extension plist)))
158 (append f-x plist-x))
159
160
161(defun polysaturation-extension (ring f plist
162 &aux
163 (k (length plist))
164 (d (+ k (monom-dimension (poly-lm (car plist)))))
165 ;; Add k variables to f
166 (f (poly-list-add-variables f k))
167 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
168 (plist (apply #'poly-append (poly-standard-extension plist))))
169 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
170 ;; Add -1 as the last term
171 (declare (type ring ring))
172 (setf (cdr (last (poly-termlist plist)))
173 (list (make-term :monom (make-monom :dimension d)
174 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
175 (append f (list plist)))
176
177(defun saturation-extension-1 (ring f p)
178 "Calculate [F, U*P-1]. It destructively modifies F."
179 (declare (type ring ring))
180 (polysaturation-extension ring f (list p)))
181
182;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
183;;
184;; Evaluation of polynomial (prefix) expressions
185;;
186;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
187
188(defun coerce-coeff (ring expr vars)
189 "Coerce an element of the coefficient ring to a constant polynomial."
190 ;; Modular arithmetic handler by rat
191 (declare (type ring ring))
192 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
193 :coeff (funcall (ring-parse ring) expr)))
194 0))
195
196(defun poly-eval (expr vars
197 &optional
198 (ring +ring-of-integers+)
199 (order #'lex>)
200 (list-marker :[)
201 &aux
202 (ring-and-order (make-ring-and-order :ring ring :order order)))
203 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
204variables VARS. Return the resulting polynomial or list of
205polynomials. Standard arithmetical operators in form EXPR are
206replaced with their analogues in the ring of polynomials, and the
207resulting expression is evaluated, resulting in a polynomial or a list
208of polynomials in internal form. A similar operation in another computer
209algebra system could be called 'expand' or so."
210 (declare (type ring ring))
211 (labels ((p-eval (arg) (poly-eval arg vars ring order))
212 (p-eval-scalar (arg) (poly-eval-scalar arg))
213 (p-eval-list (args) (mapcar #'p-eval args))
214 (p-add (x y) (poly-add ring-and-order x y)))
215 (cond
216 ((null expr) (error "Empty expression"))
217 ((eql expr 0) (make-poly-zero))
218 ((member expr vars :test #'equalp)
219 (let ((pos (position expr vars :test #'equalp)))
220 (make-poly-variable ring (length vars) pos)))
221 ((atom expr)
222 (coerce-coeff ring expr vars))
223 ((eq (car expr) list-marker)
224 (cons list-marker (p-eval-list (cdr expr))))
225 (t
226 (case (car expr)
227 (+ (reduce #'p-add (p-eval-list (cdr expr))))
228 (- (case (length expr)
229 (1 (make-poly-zero))
230 (2 (poly-uminus ring (p-eval (cadr expr))))
231 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
232 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
233 (reduce #'p-add (p-eval-list (cddr expr)))))))
234 (*
235 (if (endp (cddr expr)) ;unary
236 (p-eval (cdr expr))
237 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
238 (/
239 ;; A polynomial can be divided by a scalar
240 (cond
241 ((endp (cddr expr))
242 ;; A special case (/ ?), the inverse
243 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
244 (t
245 (let ((num (p-eval (cadr expr)))
246 (denom-inverse (apply (ring-div ring)
247 (cons (funcall (ring-unit ring))
248 (mapcar #'p-eval-scalar (cddr expr))))))
249 (scalar-times-poly ring denom-inverse num)))))
250 (expt
251 (cond
252 ((member (cadr expr) vars :test #'equalp)
253 ;;Special handling of (expt var pow)
254 (let ((pos (position (cadr expr) vars :test #'equalp)))
255 (make-poly-variable ring (length vars) pos (caddr expr))))
256 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
257 ;; Negative power means division in coefficient ring
258 ;; Non-integer power means non-polynomial coefficient
259 (coerce-coeff ring expr vars))
260 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
261 (otherwise
262 (coerce-coeff ring expr vars)))))))
263
264(defun poly-eval-scalar (expr
265 &optional
266 (ring +ring-of-integers+)
267 &aux
268 (order #'lex>))
269 "Evaluate a scalar expression EXPR in ring RING."
270 (declare (type ring ring))
271 (poly-lc (poly-eval expr nil ring order)))
272
273(defun spoly (ring-and-order f g
274 &aux
275 (ring (ro-ring ring-and-order)))
276 "It yields the S-polynomial of polynomials F and G."
277 (declare (type ring-and-order ring-and-order) (type poly f g))
278 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
279 (mf (monom-div lcm (poly-lm f)))
280 (mg (monom-div lcm (poly-lm g))))
281 (declare (type monom mf mg))
282 (multiple-value-bind (c cf cg)
283 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
284 (declare (ignore c))
285 (poly-sub
286 ring-and-order
287 (scalar-times-poly ring cg (monom-times-poly mf f))
288 (scalar-times-poly ring cf (monom-times-poly mg g))))))
289
290
291(defun poly-primitive-part (ring p)
292 "Divide polynomial P with integer coefficients by gcd of its
293coefficients and return the result."
294 (declare (type ring ring) (type poly p))
295 (if (poly-zerop p)
296 (values p 1)
297 (let ((c (poly-content ring p)))
298 (values (make-poly-from-termlist
299 (mapcar
300 #'(lambda (x)
301 (make-term :monom (term-monom x)
302 :coeff (funcall (ring-div ring) (term-coeff x) c)))
303 (poly-termlist p))
304 (poly-sugar p))
305 c))))
306
307(defun poly-content (ring p)
308 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
309to compute the greatest common divisor."
310 (declare (type ring ring) (type poly p))
311 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
312
313(defun read-infix-form (&key (stream t))
314 "Parser of infix expressions with integer/rational coefficients
315The parser will recognize two kinds of polynomial expressions:
316
317- polynomials in fully expanded forms with coefficients
318 written in front of symbolic expressions; constants can be optionally
319 enclosed in (); for example, the infix form
320 X^2-Y^2+(-4/3)*U^2*W^3-5
321 parses to
322 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
323
324- lists of polynomials; for example
325 [X-Y, X^2+3*Z]
326 parses to
327 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
328 where the first symbol [ marks a list of polynomials.
329
330-other infix expressions, for example
331 [(X-Y)*(X+Y)/Z,(X+1)^2]
332parses to:
333 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
334Currently this function is implemented using M. Kantrowitz's INFIX package."
335 (read-from-string
336 (concatenate 'string
337 "#I("
338 (with-output-to-string (s)
339 (loop
340 (multiple-value-bind (line eof)
341 (read-line stream t)
342 (format s "~A" line)
343 (when eof (return)))))
344 ")")))
345
346(defun read-poly (vars &key
347 (stream t)
348 (ring +ring-of-integers+)
349 (order #'lex>))
350 "Reads an expression in prefix form from a stream STREAM.
351The expression read from the strem should represent a polynomial or a
352list of polynomials in variables VARS, over the ring RING. The
353polynomial or list of polynomials is returned, with terms in each
354polynomial ordered according to monomial order ORDER."
355 (poly-eval (read-infix-form :stream stream) vars ring order))
356
357(defun string->poly (str vars
358 &optional
359 (ring +ring-of-integers+)
360 (order #'lex>))
361 "Converts a string STR to a polynomial in variables VARS."
362 (with-input-from-string (s str)
363 (read-poly vars :stream s :ring ring :order order)))
364
365(defun poly->alist (p)
366 "Convert a polynomial P to an association list. Thus, the format of the
367returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
368MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
369corresponding coefficient in the ring."
370 (cond
371 ((poly-p p)
372 (mapcar #'term->cons (poly-termlist p)))
373 ((and (consp p) (eq (car p) :[))
374 (cons :[ (mapcar #'poly->alist (cdr p))))))
375
376(defun string->alist (str vars
377 &optional
378 (ring +ring-of-integers+)
379 (order #'lex>))
380 "Convert a string STR representing a polynomial or polynomial list to
381an association list (... (MONOM . COEFF) ...)."
382 (poly->alist (string->poly str vars ring order)))
383
384(defun poly-equal-no-sugar-p (p q)
385 "Compare polynomials for equality, ignoring sugar."
386 (declare (type poly p q))
387 (equalp (poly-termlist p) (poly-termlist q)))
388
389(defun poly-set-equal-no-sugar-p (p q)
390 "Compare polynomial sets P and Q for equality, ignoring sugar."
391 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
392
393(defun poly-list-equal-no-sugar-p (p q)
394 "Compare polynomial lists P and Q for equality, ignoring sugar."
395 (every #'poly-equal-no-sugar-p p q))
396|#
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