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source: branches/f4grobner/polynomial.lisp@ 2668

Last change on this file since 2668 was 2668, 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 "POLY-TERMLIST"
26 "POLY-TERM-ORDER")
27 (:documentation "Implements polynomials"))
28
29(in-package :polynomial)
30
31(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
32
33(defclass poly ()
34 ((termlist :initarg :termlist :accessor poly-termlist)
35 (order :initarg :order :accessor poly-term-order))
36 (:default-initargs :termlist nil :order #'lex>))
37
38(defmethod print-object ((self poly) stream)
39 (format stream "#<POLY TERMLIST=~A ORDER=~A>"
40 (poly-termlist self)
41 (poly-term-order self)))
42
43(defmethod r-equalp ((self poly) (other poly))
44 (and (every #'r-equalp (poly-termlist self) (poly-termlist other))
45 (eq (poly-term-order self) (poly-term-order other))))
46
47(defmethod insert-item ((self poly) (item term))
48 (push item (poly-termlist self))
49 self)
50
51(defmethod append-item ((self poly) (item term))
52 (setf (cdr (last (poly-termlist self))) (list item))
53 self)
54
55;; Leading term
56(defgeneric leading-term (object)
57 (:method ((self poly))
58 (car (poly-termlist self)))
59 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
60
61;; Second term
62(defgeneric second-leading-term (object)
63 (:method ((self poly))
64 (cadar (poly-termlist self)))
65 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
66
67;; Leading coefficient
68(defgeneric leading-coefficient (object)
69 (:method ((self poly))
70 (r-coeff (leading-term self)))
71 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
72
73;; Second coefficient
74(defgeneric second-leading-coefficient (object)
75 (:method ((self poly))
76 (r-coeff (second-leading-term self)))
77 (:documentation "The second leading coefficient of a polynomial. It signals error for a polynomial with at most one term."))
78
79;; Testing for a zero polynomial
80(defmethod r-zerop ((self poly))
81 (null (poly-termlist self)))
82
83;; The number of terms
84(defmethod r-length ((self poly))
85 (length (poly-termlist self)))
86
87(defmethod multiply-by ((self poly) (other monom))
88 (mapc #'(lambda (term) (multiply-by term other))
89 (poly-termlist self))
90 self)
91
92(defmethod multiply-by ((self poly) (other scalar))
93 (mapc #'(lambda (term) (multiply-by term other))
94 (poly-termlist self))
95 self)
96
97
98(defun fast-addition (p q order-fn add-fun)
99 (macrolet ((lc (x) `(r-coeff (car ,x))))
100 (do ((p p)
101 (q q)
102 r)
103 ((or (endp p) (endp q))
104 (unless (endp q) (setf r (nreconc r q)))
105 r)
106 (multiple-value-bind
107 (greater-p equal-p)
108 (funcall order-fn (car p) (car q))
109 (cond
110 (greater-p
111 (rotatef (cdr p) r p)
112 )
113 (equal-p
114 (let ((s (funcall add-fun (lc p) (lc q))))
115 (cond
116 ((r-zerop s)
117 (setf p (cdr p))
118 )
119 (t
120 (setf (lc p) s)
121 (rotatef (cdr p) r p))))
122 (setf q (cdr q))
123 )
124 (t
125 (rotatef (cdr q) r q)))))))
126
127
128
129(defmacro def-additive-operation-method (method-name &optional (doc-string nil doc-string-supplied-p))
130 "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
131 `(defmethod ,method-name ((self poly) (other poly))
132 ,@(when doc-string-supplied-p `(,doc-string))
133 (with-slots ((termlist1 termlist) (order1 order))
134 self
135 (with-slots ((termlist2 termlist) (order2 order))
136 other
137 ;; Ensure orders are compatible
138 (unless (eq order1 order2)
139 (setf termlist2 (sort termlist2 order1)
140 order2 order1))
141 (setf termlist1 (fast-addition termlist1 termlist2 order1 #',method-name))))
142 self))
143
144(def-additive-operation-method add-to
145 "Adds to polynomial SELF another polynomial OTHER.
146This operation destructively modifies both polynomials.
147The result is stored in SELF. This implementation does
148no consing, entirely reusing the sells of SELF and OTHER.")
149
150(def-additive-operation-method subtract-from
151 "Subtracts from polynomial SELF another polynomial OTHER.
152This operation destructively modifies both polynomials.
153The result is stored in SELF. This implementation does
154no consing, entirely reusing the sells of SELF and OTHER.")
155
156(defmethod unary-uminus ((self poly)))
157
158#|
159
160(defun poly-standard-extension (plist &aux (k (length plist)))
161 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
162 (declare (list plist) (fixnum k))
163 (labels ((incf-power (g i)
164 (dolist (x (poly-termlist g))
165 (incf (monom-elt (term-monom x) i)))
166 (incf (poly-sugar g))))
167 (setf plist (poly-list-add-variables plist k))
168 (dotimes (i k plist)
169 (incf-power (nth i plist) i))))
170
171(defun saturation-extension (ring f plist
172 &aux
173 (k (length plist))
174 (d (monom-dimension (poly-lm (car plist))))
175 f-x plist-x)
176 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
177 (declare (type ring ring))
178 (setf f-x (poly-list-add-variables f k)
179 plist-x (mapcar #'(lambda (x)
180 (setf (poly-termlist x)
181 (nconc (poly-termlist x)
182 (list (make-term :monom (make-monom :dimension d)
183 :coeff (funcall (ring-uminus ring)
184 (funcall (ring-unit ring)))))))
185 x)
186 (poly-standard-extension plist)))
187 (append f-x plist-x))
188
189
190(defun polysaturation-extension (ring f plist
191 &aux
192 (k (length plist))
193 (d (+ k (monom-dimension (poly-lm (car plist)))))
194 ;; Add k variables to f
195 (f (poly-list-add-variables f k))
196 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
197 (plist (apply #'poly-append (poly-standard-extension plist))))
198 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
199 ;; Add -1 as the last term
200 (declare (type ring ring))
201 (setf (cdr (last (poly-termlist plist)))
202 (list (make-term :monom (make-monom :dimension d)
203 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
204 (append f (list plist)))
205
206(defun saturation-extension-1 (ring f p)
207 "Calculate [F, U*P-1]. It destructively modifies F."
208 (declare (type ring ring))
209 (polysaturation-extension ring f (list p)))
210
211;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
212;;
213;; Evaluation of polynomial (prefix) expressions
214;;
215;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
216
217(defun coerce-coeff (ring expr vars)
218 "Coerce an element of the coefficient ring to a constant polynomial."
219 ;; Modular arithmetic handler by rat
220 (declare (type ring ring))
221 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
222 :coeff (funcall (ring-parse ring) expr)))
223 0))
224
225(defun poly-eval (expr vars
226 &optional
227 (ring +ring-of-integers+)
228 (order #'lex>)
229 (list-marker :[)
230 &aux
231 (ring-and-order (make-ring-and-order :ring ring :order order)))
232 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
233variables VARS. Return the resulting polynomial or list of
234polynomials. Standard arithmetical operators in form EXPR are
235replaced with their analogues in the ring of polynomials, and the
236resulting expression is evaluated, resulting in a polynomial or a list
237of polynomials in internal form. A similar operation in another computer
238algebra system could be called 'expand' or so."
239 (declare (type ring ring))
240 (labels ((p-eval (arg) (poly-eval arg vars ring order))
241 (p-eval-scalar (arg) (poly-eval-scalar arg))
242 (p-eval-list (args) (mapcar #'p-eval args))
243 (p-add (x y) (poly-add ring-and-order x y)))
244 (cond
245 ((null expr) (error "Empty expression"))
246 ((eql expr 0) (make-poly-zero))
247 ((member expr vars :test #'equalp)
248 (let ((pos (position expr vars :test #'equalp)))
249 (make-poly-variable ring (length vars) pos)))
250 ((atom expr)
251 (coerce-coeff ring expr vars))
252 ((eq (car expr) list-marker)
253 (cons list-marker (p-eval-list (cdr expr))))
254 (t
255 (case (car expr)
256 (+ (reduce #'p-add (p-eval-list (cdr expr))))
257 (- (case (length expr)
258 (1 (make-poly-zero))
259 (2 (poly-uminus ring (p-eval (cadr expr))))
260 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
261 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
262 (reduce #'p-add (p-eval-list (cddr expr)))))))
263 (*
264 (if (endp (cddr expr)) ;unary
265 (p-eval (cdr expr))
266 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
267 (/
268 ;; A polynomial can be divided by a scalar
269 (cond
270 ((endp (cddr expr))
271 ;; A special case (/ ?), the inverse
272 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
273 (t
274 (let ((num (p-eval (cadr expr)))
275 (denom-inverse (apply (ring-div ring)
276 (cons (funcall (ring-unit ring))
277 (mapcar #'p-eval-scalar (cddr expr))))))
278 (scalar-times-poly ring denom-inverse num)))))
279 (expt
280 (cond
281 ((member (cadr expr) vars :test #'equalp)
282 ;;Special handling of (expt var pow)
283 (let ((pos (position (cadr expr) vars :test #'equalp)))
284 (make-poly-variable ring (length vars) pos (caddr expr))))
285 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
286 ;; Negative power means division in coefficient ring
287 ;; Non-integer power means non-polynomial coefficient
288 (coerce-coeff ring expr vars))
289 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
290 (otherwise
291 (coerce-coeff ring expr vars)))))))
292
293(defun poly-eval-scalar (expr
294 &optional
295 (ring +ring-of-integers+)
296 &aux
297 (order #'lex>))
298 "Evaluate a scalar expression EXPR in ring RING."
299 (declare (type ring ring))
300 (poly-lc (poly-eval expr nil ring order)))
301
302(defun spoly (ring-and-order f g
303 &aux
304 (ring (ro-ring ring-and-order)))
305 "It yields the S-polynomial of polynomials F and G."
306 (declare (type ring-and-order ring-and-order) (type poly f g))
307 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
308 (mf (monom-div lcm (poly-lm f)))
309 (mg (monom-div lcm (poly-lm g))))
310 (declare (type monom mf mg))
311 (multiple-value-bind (c cf cg)
312 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
313 (declare (ignore c))
314 (poly-sub
315 ring-and-order
316 (scalar-times-poly ring cg (monom-times-poly mf f))
317 (scalar-times-poly ring cf (monom-times-poly mg g))))))
318
319
320(defun poly-primitive-part (ring p)
321 "Divide polynomial P with integer coefficients by gcd of its
322coefficients and return the result."
323 (declare (type ring ring) (type poly p))
324 (if (poly-zerop p)
325 (values p 1)
326 (let ((c (poly-content ring p)))
327 (values (make-poly-from-termlist
328 (mapcar
329 #'(lambda (x)
330 (make-term :monom (term-monom x)
331 :coeff (funcall (ring-div ring) (term-coeff x) c)))
332 (poly-termlist p))
333 (poly-sugar p))
334 c))))
335
336(defun poly-content (ring p)
337 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
338to compute the greatest common divisor."
339 (declare (type ring ring) (type poly p))
340 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
341
342(defun read-infix-form (&key (stream t))
343 "Parser of infix expressions with integer/rational coefficients
344The parser will recognize two kinds of polynomial expressions:
345
346- polynomials in fully expanded forms with coefficients
347 written in front of symbolic expressions; constants can be optionally
348 enclosed in (); for example, the infix form
349 X^2-Y^2+(-4/3)*U^2*W^3-5
350 parses to
351 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
352
353- lists of polynomials; for example
354 [X-Y, X^2+3*Z]
355 parses to
356 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
357 where the first symbol [ marks a list of polynomials.
358
359-other infix expressions, for example
360 [(X-Y)*(X+Y)/Z,(X+1)^2]
361parses to:
362 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
363Currently this function is implemented using M. Kantrowitz's INFIX package."
364 (read-from-string
365 (concatenate 'string
366 "#I("
367 (with-output-to-string (s)
368 (loop
369 (multiple-value-bind (line eof)
370 (read-line stream t)
371 (format s "~A" line)
372 (when eof (return)))))
373 ")")))
374
375(defun read-poly (vars &key
376 (stream t)
377 (ring +ring-of-integers+)
378 (order #'lex>))
379 "Reads an expression in prefix form from a stream STREAM.
380The expression read from the strem should represent a polynomial or a
381list of polynomials in variables VARS, over the ring RING. The
382polynomial or list of polynomials is returned, with terms in each
383polynomial ordered according to monomial order ORDER."
384 (poly-eval (read-infix-form :stream stream) vars ring order))
385
386(defun string->poly (str vars
387 &optional
388 (ring +ring-of-integers+)
389 (order #'lex>))
390 "Converts a string STR to a polynomial in variables VARS."
391 (with-input-from-string (s str)
392 (read-poly vars :stream s :ring ring :order order)))
393
394(defun poly->alist (p)
395 "Convert a polynomial P to an association list. Thus, the format of the
396returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
397MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
398corresponding coefficient in the ring."
399 (cond
400 ((poly-p p)
401 (mapcar #'term->cons (poly-termlist p)))
402 ((and (consp p) (eq (car p) :[))
403 (cons :[ (mapcar #'poly->alist (cdr p))))))
404
405(defun string->alist (str vars
406 &optional
407 (ring +ring-of-integers+)
408 (order #'lex>))
409 "Convert a string STR representing a polynomial or polynomial list to
410an association list (... (MONOM . COEFF) ...)."
411 (poly->alist (string->poly str vars ring order)))
412
413(defun poly-equal-no-sugar-p (p q)
414 "Compare polynomials for equality, ignoring sugar."
415 (declare (type poly p q))
416 (equalp (poly-termlist p) (poly-termlist q)))
417
418(defun poly-set-equal-no-sugar-p (p q)
419 "Compare polynomial sets P and Q for equality, ignoring sugar."
420 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
421
422(defun poly-list-equal-no-sugar-p (p q)
423 "Compare polynomial lists P and Q for equality, ignoring sugar."
424 (every #'poly-equal-no-sugar-p p q))
425|#
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