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source: branches/f4grobner/polynomial.lisp@ 2743

Last change on this file since 2743 was 2743, 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(defmacro fast-add/subtract (p q order-fn add/subtract-fun
105 &optional
106 (uminus-fun nil uminus-fun-supplied-p))
107 "Return an expression which will efficiently of two polynomials. Implements an efficient
108algorithm to add two polynomials represented as sorted lists of
109terms. This function destroys both arguments, reusing the terms to
110build the result."
111 `(macrolet ((lc (x) `(r-coeff (car ,x))))
112 (do ((p ,p)
113 (q ,q)
114 r)
115 ((or (endp p) (endp q))
116 ;; NOTE: R contains the result in reverse order. Can it
117 ;; be more efficient to produce the terms in correct order?
118 (unless (endp q) (setf r (nreconc r q)))
119 r)
120 (multiple-value-bind
121 (greater-p equal-p)
122 (funcall ,order-fn (car p) (car q))
123 (cond
124 (greater-p
125 (rotatef (cdr p) r p)
126 )
127 (equal-p
128 (let ((s (funcall ,add/subtract-fun (lc p) (lc q))))
129 (cond
130 ((r-zerop s)
131 (setf p (cdr p))
132 )
133 (t
134 (setf (lc p) s)
135 (rotatef (cdr p) r p))))
136 (setf q (cdr q))
137 )
138 (t
139 ;;Negate the term of Q if UMINUS provided, signallig
140 ;;that we are doing subtraction
141 ,@(when uminus-fun-supplied-p
142 `((setf (lc q) (funcall ,uminus-fun (lc q)))))
143 (rotatef (cdr q) r q)))))))
144
145
146(defmacro def-additive-operation-method (method-name
147 &optional
148 (doc-string nil doc-string-supplied-p))
149 "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
150 `(defmethod ,method-name ((self poly) (other poly))
151 ,@(when doc-string-supplied-p `(,doc-string))
152 (with-slots ((termlist1 termlist) (order1 order))
153 self
154 (with-slots ((termlist2 termlist) (order2 order))
155 other
156 ;; Ensure orders are compatible
157 (unless (eq order1 order2)
158 (setf termlist2 (sort termlist2 order1)
159 order2 order1))
160 (setf termlist1 (fast-add/subtract termlist1 termlist2 order1 #',method-name))))
161 self))
162
163(def-additive-operation-method add-to
164 "Adds to polynomial SELF another polynomial OTHER.
165This operation destructively modifies both polynomials.
166The result is stored in SELF. This implementation does
167no consing, entirely reusing the sells of SELF and OTHER.")
168
169(def-additive-operation-method subtract-from
170 "Subtracts from polynomial SELF another polynomial OTHER.
171This operation destructively modifies both polynomials.
172The result is stored in SELF. This implementation does
173no consing, entirely reusing the sells of SELF and OTHER.")
174
175
176(defmethod unary-minus ((self poly))
177 "Destructively modifies the coefficients of the polynomial SELF,
178by changing their sign."
179 (mapc #'unary-minus (poly-termlist self))
180 self)
181
182#|
183
184(defun poly-standard-extension (plist &aux (k (length plist)))
185 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
186is a list of polynomials."
187 (declare (list plist) (fixnum k))
188 (labels ((incf-power (g i)
189 (dolist (x (poly-termlist g))
190 (incf (monom-elt (term-monom x) i)))
191 (incf (poly-sugar g))))
192 (setf plist (poly-list-add-variables plist k))
193 (dotimes (i k plist)
194 (incf-power (nth i plist) i))))
195
196
197
198(defun saturation-extension (ring f plist
199 &aux
200 (k (length plist))
201 (d (monom-dimension (poly-lm (car plist))))
202 f-x plist-x)
203 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
204 (declare (type ring ring))
205 (setf f-x (poly-list-add-variables f k)
206 plist-x (mapcar #'(lambda (x)
207 (setf (poly-termlist x)
208 (nconc (poly-termlist x)
209 (list (make-term :monom (make-monom :dimension d)
210 :coeff (funcall (ring-uminus ring)
211 (funcall (ring-unit ring)))))))
212 x)
213 (poly-standard-extension plist)))
214 (append f-x plist-x))
215
216
217(defun polysaturation-extension (ring f plist
218 &aux
219 (k (length plist))
220 (d (+ k (monom-dimension (poly-lm (car plist)))))
221 ;; Add k variables to f
222 (f (poly-list-add-variables f k))
223 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
224 (plist (apply #'poly-append (poly-standard-extension plist))))
225 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
226 ;; Add -1 as the last term
227 (declare (type ring ring))
228 (setf (cdr (last (poly-termlist plist)))
229 (list (make-term :monom (make-monom :dimension d)
230 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
231 (append f (list plist)))
232
233(defun saturation-extension-1 (ring f p)
234 "Calculate [F, U*P-1]. It destructively modifies F."
235 (declare (type ring ring))
236 (polysaturation-extension ring f (list p)))
237
238;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
239;;
240;; Evaluation of polynomial (prefix) expressions
241;;
242;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
243
244(defun coerce-coeff (ring expr vars)
245 "Coerce an element of the coefficient ring to a constant polynomial."
246 ;; Modular arithmetic handler by rat
247 (declare (type ring ring))
248 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
249 :coeff (funcall (ring-parse ring) expr)))
250 0))
251
252(defun poly-eval (expr vars
253 &optional
254 (ring +ring-of-integers+)
255 (order #'lex>)
256 (list-marker :[)
257 &aux
258 (ring-and-order (make-ring-and-order :ring ring :order order)))
259 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
260variables VARS. Return the resulting polynomial or list of
261polynomials. Standard arithmetical operators in form EXPR are
262replaced with their analogues in the ring of polynomials, and the
263resulting expression is evaluated, resulting in a polynomial or a list
264of polynomials in internal form. A similar operation in another computer
265algebra system could be called 'expand' or so."
266 (declare (type ring ring))
267 (labels ((p-eval (arg) (poly-eval arg vars ring order))
268 (p-eval-scalar (arg) (poly-eval-scalar arg))
269 (p-eval-list (args) (mapcar #'p-eval args))
270 (p-add (x y) (poly-add ring-and-order x y)))
271 (cond
272 ((null expr) (error "Empty expression"))
273 ((eql expr 0) (make-poly-zero))
274 ((member expr vars :test #'equalp)
275 (let ((pos (position expr vars :test #'equalp)))
276 (make-poly-variable ring (length vars) pos)))
277 ((atom expr)
278 (coerce-coeff ring expr vars))
279 ((eq (car expr) list-marker)
280 (cons list-marker (p-eval-list (cdr expr))))
281 (t
282 (case (car expr)
283 (+ (reduce #'p-add (p-eval-list (cdr expr))))
284 (- (case (length expr)
285 (1 (make-poly-zero))
286 (2 (poly-uminus ring (p-eval (cadr expr))))
287 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
288 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
289 (reduce #'p-add (p-eval-list (cddr expr)))))))
290 (*
291 (if (endp (cddr expr)) ;unary
292 (p-eval (cdr expr))
293 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
294 (/
295 ;; A polynomial can be divided by a scalar
296 (cond
297 ((endp (cddr expr))
298 ;; A special case (/ ?), the inverse
299 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
300 (t
301 (let ((num (p-eval (cadr expr)))
302 (denom-inverse (apply (ring-div ring)
303 (cons (funcall (ring-unit ring))
304 (mapcar #'p-eval-scalar (cddr expr))))))
305 (scalar-times-poly ring denom-inverse num)))))
306 (expt
307 (cond
308 ((member (cadr expr) vars :test #'equalp)
309 ;;Special handling of (expt var pow)
310 (let ((pos (position (cadr expr) vars :test #'equalp)))
311 (make-poly-variable ring (length vars) pos (caddr expr))))
312 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
313 ;; Negative power means division in coefficient ring
314 ;; Non-integer power means non-polynomial coefficient
315 (coerce-coeff ring expr vars))
316 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
317 (otherwise
318 (coerce-coeff ring expr vars)))))))
319
320(defun poly-eval-scalar (expr
321 &optional
322 (ring +ring-of-integers+)
323 &aux
324 (order #'lex>))
325 "Evaluate a scalar expression EXPR in ring RING."
326 (declare (type ring ring))
327 (poly-lc (poly-eval expr nil ring order)))
328
329(defun spoly (ring-and-order f g
330 &aux
331 (ring (ro-ring ring-and-order)))
332 "It yields the S-polynomial of polynomials F and G."
333 (declare (type ring-and-order ring-and-order) (type poly f g))
334 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
335 (mf (monom-div lcm (poly-lm f)))
336 (mg (monom-div lcm (poly-lm g))))
337 (declare (type monom mf mg))
338 (multiple-value-bind (c cf cg)
339 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
340 (declare (ignore c))
341 (poly-sub
342 ring-and-order
343 (scalar-times-poly ring cg (monom-times-poly mf f))
344 (scalar-times-poly ring cf (monom-times-poly mg g))))))
345
346
347(defun poly-primitive-part (ring p)
348 "Divide polynomial P with integer coefficients by gcd of its
349coefficients and return the result."
350 (declare (type ring ring) (type poly p))
351 (if (poly-zerop p)
352 (values p 1)
353 (let ((c (poly-content ring p)))
354 (values (make-poly-from-termlist
355 (mapcar
356 #'(lambda (x)
357 (make-term :monom (term-monom x)
358 :coeff (funcall (ring-div ring) (term-coeff x) c)))
359 (poly-termlist p))
360 (poly-sugar p))
361 c))))
362
363(defun poly-content (ring p)
364 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
365to compute the greatest common divisor."
366 (declare (type ring ring) (type poly p))
367 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
368
369(defun read-infix-form (&key (stream t))
370 "Parser of infix expressions with integer/rational coefficients
371The parser will recognize two kinds of polynomial expressions:
372
373- polynomials in fully expanded forms with coefficients
374 written in front of symbolic expressions; constants can be optionally
375 enclosed in (); for example, the infix form
376 X^2-Y^2+(-4/3)*U^2*W^3-5
377 parses to
378 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
379
380- lists of polynomials; for example
381 [X-Y, X^2+3*Z]
382 parses to
383 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
384 where the first symbol [ marks a list of polynomials.
385
386-other infix expressions, for example
387 [(X-Y)*(X+Y)/Z,(X+1)^2]
388parses to:
389 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
390Currently this function is implemented using M. Kantrowitz's INFIX package."
391 (read-from-string
392 (concatenate 'string
393 "#I("
394 (with-output-to-string (s)
395 (loop
396 (multiple-value-bind (line eof)
397 (read-line stream t)
398 (format s "~A" line)
399 (when eof (return)))))
400 ")")))
401
402(defun read-poly (vars &key
403 (stream t)
404 (ring +ring-of-integers+)
405 (order #'lex>))
406 "Reads an expression in prefix form from a stream STREAM.
407The expression read from the strem should represent a polynomial or a
408list of polynomials in variables VARS, over the ring RING. The
409polynomial or list of polynomials is returned, with terms in each
410polynomial ordered according to monomial order ORDER."
411 (poly-eval (read-infix-form :stream stream) vars ring order))
412
413(defun string->poly (str vars
414 &optional
415 (ring +ring-of-integers+)
416 (order #'lex>))
417 "Converts a string STR to a polynomial in variables VARS."
418 (with-input-from-string (s str)
419 (read-poly vars :stream s :ring ring :order order)))
420
421(defun poly->alist (p)
422 "Convert a polynomial P to an association list. Thus, the format of the
423returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
424MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
425corresponding coefficient in the ring."
426 (cond
427 ((poly-p p)
428 (mapcar #'term->cons (poly-termlist p)))
429 ((and (consp p) (eq (car p) :[))
430 (cons :[ (mapcar #'poly->alist (cdr p))))))
431
432(defun string->alist (str vars
433 &optional
434 (ring +ring-of-integers+)
435 (order #'lex>))
436 "Convert a string STR representing a polynomial or polynomial list to
437an association list (... (MONOM . COEFF) ...)."
438 (poly->alist (string->poly str vars ring order)))
439
440(defun poly-equal-no-sugar-p (p q)
441 "Compare polynomials for equality, ignoring sugar."
442 (declare (type poly p q))
443 (equalp (poly-termlist p) (poly-termlist q)))
444
445(defun poly-set-equal-no-sugar-p (p q)
446 "Compare polynomial sets P and Q for equality, ignoring sugar."
447 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
448
449(defun poly-list-equal-no-sugar-p (p q)
450 "Compare polynomial lists P and Q for equality, ignoring sugar."
451 (every #'poly-equal-no-sugar-p p q))
452|#
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