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source: branches/f4grobner/polynomial.lisp@ 2702

Last change on this file since 2702 was 2697, 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 :documentation "List of terms.")
36 (order :initarg :order :accessor poly-term-order
37 :documentation "Monomial/term order."))
38 (:default-initargs :termlist nil :order #'lex>)
39 (:documentation "A polynomial with a list of terms TERMLIST, ordered
40according to term order ORDER, which defaults to LEX>."))
41
42(defmethod print-object ((self poly) stream)
43 (format stream "#<POLY TERMLIST=~A ORDER=~A>"
44 (poly-termlist self)
45 (poly-term-order self)))
46
47(defmethod r-equalp ((self poly) (other poly))
48 "POLY instances are R-EQUALP if they have the same
49order and if all terms are R-EQUALP."
50 (and (every #'r-equalp (poly-termlist self) (poly-termlist other))
51 (eq (poly-term-order self) (poly-term-order other))))
52
53(defmethod insert-item ((self poly) (item term))
54 (push item (poly-termlist self))
55 self)
56
57(defmethod append-item ((self poly) (item term))
58 (setf (cdr (last (poly-termlist self))) (list item))
59 self)
60
61;; Leading term
62(defgeneric leading-term (object)
63 (:method ((self poly))
64 (car (poly-termlist self)))
65 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
66
67;; Second term
68(defgeneric second-leading-term (object)
69 (:method ((self poly))
70 (cadar (poly-termlist self)))
71 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
72
73;; Leading coefficient
74(defgeneric leading-coefficient (object)
75 (:method ((self poly))
76 (r-coeff (leading-term self)))
77 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
78
79;; Second coefficient
80(defgeneric second-leading-coefficient (object)
81 (:method ((self poly))
82 (r-coeff (second-leading-term self)))
83 (:documentation "The second leading coefficient of a polynomial. It signals error for a polynomial with at most one term."))
84
85;; Testing for a zero polynomial
86(defmethod r-zerop ((self poly))
87 (null (poly-termlist self)))
88
89;; The number of terms
90(defmethod r-length ((self poly))
91 (length (poly-termlist self)))
92
93(defmethod multiply-by ((self poly) (other monom))
94 (mapc #'(lambda (term) (multiply-by term other))
95 (poly-termlist self))
96 self)
97
98(defmethod multiply-by ((self poly) (other scalar))
99 (mapc #'(lambda (term) (multiply-by term other))
100 (poly-termlist self))
101 self)
102
103
104(defun fast-addition (p q order-fn add-fun)
105 "Returns the sum of two polynomials. Implements an efficient
106algorithm to add two polynomials represented as sorted lists of
107terms. This function destroys both arguments, reusing the terms to
108build the result."
109 (macrolet ((lc (x) `(r-coeff (car ,x))))
110 (do ((p p)
111 (q q)
112 r)
113 ((or (endp p) (endp q))
114 ;; NOTE: R contains the result in reverse order. Can it
115 ;; be more efficient to produce the terms in correct order?
116 (unless (endp q) (setf r (nreconc r q)))
117 r)
118 (multiple-value-bind
119 (greater-p equal-p)
120 (funcall order-fn (car p) (car q))
121 (cond
122 (greater-p
123 (rotatef (cdr p) r p)
124 )
125 (equal-p
126 (let ((s (funcall add-fun (lc p) (lc q))))
127 (cond
128 ((r-zerop s)
129 (setf p (cdr p))
130 )
131 (t
132 (setf (lc p) s)
133 (rotatef (cdr p) r p))))
134 (setf q (cdr q))
135 )
136 (t
137 (rotatef (cdr q) r q)))))))
138
139
140(defmacro def-additive-operation-method (method-name &optional (doc-string nil doc-string-supplied-p))
141 "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
142 `(defmethod ,method-name ((self poly) (other poly))
143 ,@(when doc-string-supplied-p `(,doc-string))
144 (with-slots ((termlist1 termlist) (order1 order))
145 self
146 (with-slots ((termlist2 termlist) (order2 order))
147 other
148 ;; Ensure orders are compatible
149 (unless (eq order1 order2)
150 (setf termlist2 (sort termlist2 order1)
151 order2 order1))
152 (setf termlist1 (fast-addition termlist1 termlist2 order1 #',method-name))))
153 self))
154
155(def-additive-operation-method add-to
156 "Adds to polynomial SELF another polynomial OTHER.
157This operation destructively modifies both polynomials.
158The result is stored in SELF. This implementation does
159no consing, entirely reusing the sells of SELF and OTHER.")
160
161(def-additive-operation-method subtract-from
162 "Subtracts from polynomial SELF another polynomial OTHER.
163This operation destructively modifies both polynomials.
164The result is stored in SELF. This implementation does
165no consing, entirely reusing the sells of SELF and OTHER.")
166
167(defmethod unary-minus ((self poly))
168 "Destructively modifies the coefficients of the polynomial SELF,
169by changing their sign."
170 (mapc #'unary-minus (poly-termlist self))
171 self)
172
173#|
174
175(defun poly-standard-extension (plist &aux (k (length plist)))
176 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
177 (declare (list plist) (fixnum k))
178 (labels ((incf-power (g i)
179 (dolist (x (poly-termlist g))
180 (incf (monom-elt (term-monom x) i)))
181 (incf (poly-sugar g))))
182 (setf plist (poly-list-add-variables plist k))
183 (dotimes (i k plist)
184 (incf-power (nth i plist) i))))
185
186(defun saturation-extension (ring f plist
187 &aux
188 (k (length plist))
189 (d (monom-dimension (poly-lm (car plist))))
190 f-x plist-x)
191 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
192 (declare (type ring ring))
193 (setf f-x (poly-list-add-variables f k)
194 plist-x (mapcar #'(lambda (x)
195 (setf (poly-termlist x)
196 (nconc (poly-termlist x)
197 (list (make-term :monom (make-monom :dimension d)
198 :coeff (funcall (ring-uminus ring)
199 (funcall (ring-unit ring)))))))
200 x)
201 (poly-standard-extension plist)))
202 (append f-x plist-x))
203
204
205(defun polysaturation-extension (ring f plist
206 &aux
207 (k (length plist))
208 (d (+ k (monom-dimension (poly-lm (car plist)))))
209 ;; Add k variables to f
210 (f (poly-list-add-variables f k))
211 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
212 (plist (apply #'poly-append (poly-standard-extension plist))))
213 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
214 ;; Add -1 as the last term
215 (declare (type ring ring))
216 (setf (cdr (last (poly-termlist plist)))
217 (list (make-term :monom (make-monom :dimension d)
218 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
219 (append f (list plist)))
220
221(defun saturation-extension-1 (ring f p)
222 "Calculate [F, U*P-1]. It destructively modifies F."
223 (declare (type ring ring))
224 (polysaturation-extension ring f (list p)))
225
226;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
227;;
228;; Evaluation of polynomial (prefix) expressions
229;;
230;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
231
232(defun coerce-coeff (ring expr vars)
233 "Coerce an element of the coefficient ring to a constant polynomial."
234 ;; Modular arithmetic handler by rat
235 (declare (type ring ring))
236 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
237 :coeff (funcall (ring-parse ring) expr)))
238 0))
239
240(defun poly-eval (expr vars
241 &optional
242 (ring +ring-of-integers+)
243 (order #'lex>)
244 (list-marker :[)
245 &aux
246 (ring-and-order (make-ring-and-order :ring ring :order order)))
247 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
248variables VARS. Return the resulting polynomial or list of
249polynomials. Standard arithmetical operators in form EXPR are
250replaced with their analogues in the ring of polynomials, and the
251resulting expression is evaluated, resulting in a polynomial or a list
252of polynomials in internal form. A similar operation in another computer
253algebra system could be called 'expand' or so."
254 (declare (type ring ring))
255 (labels ((p-eval (arg) (poly-eval arg vars ring order))
256 (p-eval-scalar (arg) (poly-eval-scalar arg))
257 (p-eval-list (args) (mapcar #'p-eval args))
258 (p-add (x y) (poly-add ring-and-order x y)))
259 (cond
260 ((null expr) (error "Empty expression"))
261 ((eql expr 0) (make-poly-zero))
262 ((member expr vars :test #'equalp)
263 (let ((pos (position expr vars :test #'equalp)))
264 (make-poly-variable ring (length vars) pos)))
265 ((atom expr)
266 (coerce-coeff ring expr vars))
267 ((eq (car expr) list-marker)
268 (cons list-marker (p-eval-list (cdr expr))))
269 (t
270 (case (car expr)
271 (+ (reduce #'p-add (p-eval-list (cdr expr))))
272 (- (case (length expr)
273 (1 (make-poly-zero))
274 (2 (poly-uminus ring (p-eval (cadr expr))))
275 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
276 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
277 (reduce #'p-add (p-eval-list (cddr expr)))))))
278 (*
279 (if (endp (cddr expr)) ;unary
280 (p-eval (cdr expr))
281 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
282 (/
283 ;; A polynomial can be divided by a scalar
284 (cond
285 ((endp (cddr expr))
286 ;; A special case (/ ?), the inverse
287 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
288 (t
289 (let ((num (p-eval (cadr expr)))
290 (denom-inverse (apply (ring-div ring)
291 (cons (funcall (ring-unit ring))
292 (mapcar #'p-eval-scalar (cddr expr))))))
293 (scalar-times-poly ring denom-inverse num)))))
294 (expt
295 (cond
296 ((member (cadr expr) vars :test #'equalp)
297 ;;Special handling of (expt var pow)
298 (let ((pos (position (cadr expr) vars :test #'equalp)))
299 (make-poly-variable ring (length vars) pos (caddr expr))))
300 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
301 ;; Negative power means division in coefficient ring
302 ;; Non-integer power means non-polynomial coefficient
303 (coerce-coeff ring expr vars))
304 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
305 (otherwise
306 (coerce-coeff ring expr vars)))))))
307
308(defun poly-eval-scalar (expr
309 &optional
310 (ring +ring-of-integers+)
311 &aux
312 (order #'lex>))
313 "Evaluate a scalar expression EXPR in ring RING."
314 (declare (type ring ring))
315 (poly-lc (poly-eval expr nil ring order)))
316
317(defun spoly (ring-and-order f g
318 &aux
319 (ring (ro-ring ring-and-order)))
320 "It yields the S-polynomial of polynomials F and G."
321 (declare (type ring-and-order ring-and-order) (type poly f g))
322 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
323 (mf (monom-div lcm (poly-lm f)))
324 (mg (monom-div lcm (poly-lm g))))
325 (declare (type monom mf mg))
326 (multiple-value-bind (c cf cg)
327 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
328 (declare (ignore c))
329 (poly-sub
330 ring-and-order
331 (scalar-times-poly ring cg (monom-times-poly mf f))
332 (scalar-times-poly ring cf (monom-times-poly mg g))))))
333
334
335(defun poly-primitive-part (ring p)
336 "Divide polynomial P with integer coefficients by gcd of its
337coefficients and return the result."
338 (declare (type ring ring) (type poly p))
339 (if (poly-zerop p)
340 (values p 1)
341 (let ((c (poly-content ring p)))
342 (values (make-poly-from-termlist
343 (mapcar
344 #'(lambda (x)
345 (make-term :monom (term-monom x)
346 :coeff (funcall (ring-div ring) (term-coeff x) c)))
347 (poly-termlist p))
348 (poly-sugar p))
349 c))))
350
351(defun poly-content (ring p)
352 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
353to compute the greatest common divisor."
354 (declare (type ring ring) (type poly p))
355 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
356
357(defun read-infix-form (&key (stream t))
358 "Parser of infix expressions with integer/rational coefficients
359The parser will recognize two kinds of polynomial expressions:
360
361- polynomials in fully expanded forms with coefficients
362 written in front of symbolic expressions; constants can be optionally
363 enclosed in (); for example, the infix form
364 X^2-Y^2+(-4/3)*U^2*W^3-5
365 parses to
366 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
367
368- lists of polynomials; for example
369 [X-Y, X^2+3*Z]
370 parses to
371 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
372 where the first symbol [ marks a list of polynomials.
373
374-other infix expressions, for example
375 [(X-Y)*(X+Y)/Z,(X+1)^2]
376parses to:
377 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
378Currently this function is implemented using M. Kantrowitz's INFIX package."
379 (read-from-string
380 (concatenate 'string
381 "#I("
382 (with-output-to-string (s)
383 (loop
384 (multiple-value-bind (line eof)
385 (read-line stream t)
386 (format s "~A" line)
387 (when eof (return)))))
388 ")")))
389
390(defun read-poly (vars &key
391 (stream t)
392 (ring +ring-of-integers+)
393 (order #'lex>))
394 "Reads an expression in prefix form from a stream STREAM.
395The expression read from the strem should represent a polynomial or a
396list of polynomials in variables VARS, over the ring RING. The
397polynomial or list of polynomials is returned, with terms in each
398polynomial ordered according to monomial order ORDER."
399 (poly-eval (read-infix-form :stream stream) vars ring order))
400
401(defun string->poly (str vars
402 &optional
403 (ring +ring-of-integers+)
404 (order #'lex>))
405 "Converts a string STR to a polynomial in variables VARS."
406 (with-input-from-string (s str)
407 (read-poly vars :stream s :ring ring :order order)))
408
409(defun poly->alist (p)
410 "Convert a polynomial P to an association list. Thus, the format of the
411returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
412MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
413corresponding coefficient in the ring."
414 (cond
415 ((poly-p p)
416 (mapcar #'term->cons (poly-termlist p)))
417 ((and (consp p) (eq (car p) :[))
418 (cons :[ (mapcar #'poly->alist (cdr p))))))
419
420(defun string->alist (str vars
421 &optional
422 (ring +ring-of-integers+)
423 (order #'lex>))
424 "Convert a string STR representing a polynomial or polynomial list to
425an association list (... (MONOM . COEFF) ...)."
426 (poly->alist (string->poly str vars ring order)))
427
428(defun poly-equal-no-sugar-p (p q)
429 "Compare polynomials for equality, ignoring sugar."
430 (declare (type poly p q))
431 (equalp (poly-termlist p) (poly-termlist q)))
432
433(defun poly-set-equal-no-sugar-p (p q)
434 "Compare polynomial sets P and Q for equality, ignoring sugar."
435 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
436
437(defun poly-list-equal-no-sugar-p (p q)
438 "Compare polynomial lists P and Q for equality, ignoring sugar."
439 (every #'poly-equal-no-sugar-p p q))
440|#
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