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@ 3262

Last change on this file since 3262 was 3262, checked in by Marek Rychlik, 9 years ago

* empty log message *

File size: 15.1 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 :utils :ring :monom :order :term)
24 (:export "POLY"
25 "POLY-TERMLIST"
26 "POLY-TERM-ORDER"
27 "CHANGE-TERM-ORDER"
28 "STANDARD-EXTENSION"
29 "STANDARD-EXTENSION-1"
30 "STANDARD-SUM"
31 "SATURATION-EXTENSION"
32 "ALIST->POLY")
33 (:documentation "Implements polynomials."))
34
35(in-package :polynomial)
36
37(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
38
39(defclass poly ()
40 ((dimension :initform nil
41 :initarg :dimension
42 :accessor poly-dimension
43 :documentation "Shared dimension of all terms, the number of variables")
44 (termlist :initform nil :initarg :termlist :accessor poly-termlist
45 :documentation "List of terms.")
46 (order :initform #'lex> :initarg :order :accessor poly-term-order
47 :documentation "Monomial/term order."))
48 (:default-initargs :dimension nil :termlist nil :order #'lex>)
49 (:documentation "A polynomial with a list of terms TERMLIST, ordered
50according to term order ORDER, which defaults to LEX>."))
51
52(defmethod print-object ((self poly) stream)
53 (print-unreadable-object (self stream :type t :identity t)
54 (with-accessors ((dimension poly-dimension)
55 (termlist poly-termlist)
56 (order poly-term-order))
57 self
58 (format stream "DIMENSION=~A TERMLIST=~A ORDER=~A"
59 dimension termlist order))))
60
61(defgeneric change-term-order (self other)
62 (:documentation "Change term order of SELF to the term order of OTHER.")
63 (:method ((self poly) (other poly))
64 (unless (eq (poly-term-order self) (poly-term-order other))
65 (setf (poly-termlist self) (sort (poly-termlist self) (poly-term-order other))
66 (poly-term-order self) (poly-term-order other)))
67 self))
68
69(defun alist->poly (alist &aux (poly (make-instance 'poly)))
70 "It reads polynomial from an alist formatted as ( ... (exponents . coeff) ...).
71It can be used to enter simple polynomials by hand, e.g the polynomial
72in two variables, X and Y, given in standard notation as:
73
74 3*X^2*Y^3+2*Y+7
75
76can be entered as
77(ALIST->POLY '(((2 3) . 3) ((0 1) . 2) ((0 0) . 7))).
78
79NOTE: The primary use is for low-level debugging of the package."
80 (dolist (x alist poly)
81 (insert-item poly (make-instance 'term :exponents (car x) :coeff (cdr x)))))
82
83
84(defmethod r-equalp ((self poly) (other poly))
85 "POLY instances are R-EQUALP if they have the same
86order and if all terms are R-EQUALP."
87 (and (every #'r-equalp (poly-termlist self) (poly-termlist other))
88 (eq (poly-term-order self) (poly-term-order other))))
89
90(defmethod insert-item ((self poly) (item term))
91 (cond ((null (poly-dimension self))
92 (setf (poly-dimension self) (monom-dimension item)))
93 (t (assert (= (poly-dimension self) (monom-dimension item)))))
94 (push item (poly-termlist self))
95 self)
96
97(defmethod append-item ((self poly) (item term))
98 (cond ((null (poly-dimension self))
99 (setf (poly-dimension self) (monom-dimension item)))
100 (t (assert (= (poly-dimension self) (monom-dimension item)))))
101 (setf (cdr (last (poly-termlist self))) (list item))
102 self)
103
104;; Leading term
105(defgeneric leading-term (object)
106 (:method ((self poly))
107 (car (poly-termlist self)))
108 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
109
110;; Second term
111(defgeneric second-leading-term (object)
112 (:method ((self poly))
113 (cadar (poly-termlist self)))
114 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
115
116;; Leading coefficient
117(defgeneric leading-coefficient (object)
118 (:method ((self poly))
119 (scalar-coeff (leading-term self)))
120 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
121
122;; Second coefficient
123(defgeneric second-leading-coefficient (object)
124 (:method ((self poly))
125 (scalar-coeff (second-leading-term self)))
126 (:documentation "The second leading coefficient of a polynomial. It
127 signals error for a polynomial with at most one term."))
128
129;; Testing for a zero polynomial
130(defmethod r-zerop ((self poly))
131 (null (poly-termlist self)))
132
133;; The number of terms
134(defmethod r-length ((self poly))
135 (length (poly-termlist self)))
136
137(defmethod multiply-by ((self poly) (other monom))
138 (mapc #'(lambda (term) (multiply-by term other))
139 (poly-termlist self))
140 self)
141
142(defmethod multiply-by ((self poly) (other term))
143 (mapc #'(lambda (term) (multiply-by term other))
144 (poly-termlist self))
145 self)
146
147(defmethod multiply-by ((self poly) (other scalar))
148 (mapc #'(lambda (term) (multiply-by term other))
149 (poly-termlist self))
150 self)
151
152
153(defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)
154 "Return an expression which will efficiently adds/subtracts two
155polynomials, P and Q. The addition/subtraction of coefficients is
156performed by calling ADD/SUBTRACT-METHOD-NAME. If UMINUS-METHOD-NAME
157is supplied, it is used to negate the coefficients of Q which do not
158have a corresponding coefficient in P. The code implements an
159efficient algorithm to add two polynomials represented as sorted lists
160of terms. The code destroys both arguments, reusing the terms to build
161the result."
162 `(macrolet ((lc (x) `(scalar-coeff (car ,x))))
163 (do ((p ,p)
164 (q ,q)
165 r)
166 ((or (endp p) (endp q))
167 ;; NOTE: R contains the result in reverse order. Can it
168 ;; be more efficient to produce the terms in correct order?
169 (unless (endp q)
170 ;; Upon subtraction, we must change the sign of
171 ;; all coefficients in q
172 ,@(when uminus-fn
173 `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))
174 (setf r (nreconc r q)))
175 r)
176 (multiple-value-bind
177 (greater-p equal-p)
178 (funcall ,order-fn (car p) (car q))
179 (cond
180 (greater-p
181 (rotatef (cdr p) r p)
182 )
183 (equal-p
184 (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))
185 (cond
186 ((r-zerop s)
187 (setf p (cdr p))
188 )
189 (t
190 (setf (lc p) s)
191 (rotatef (cdr p) r p))))
192 (setf q (cdr q))
193 )
194 (t
195 ;;Negate the term of Q if UMINUS provided, signallig
196 ;;that we are doing subtraction
197 ,(when uminus-fn
198 `(setf (lc q) (funcall ,uminus-fn (lc q))))
199 (rotatef (cdr q) r q)))))))
200
201
202(defmacro def-add/subtract-method (add/subtract-method-name
203 uminus-method-name
204 &optional
205 (doc-string nil doc-string-supplied-p))
206 "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
207 `(defmethod ,add/subtract-method-name ((self poly) (other poly))
208 ,@(when doc-string-supplied-p `(,doc-string))
209 ;; Ensure orders are compatible
210 (change-term-order other self)
211 (setf (poly-termlist self) (fast-add/subtract
212 (poly-termlist self) (poly-termlist other)
213 (poly-term-order self)
214 #',add/subtract-method-name
215 ,(when uminus-method-name `(function ,uminus-method-name))))
216 self))
217
218(eval-when (:compile-toplevel :load-toplevel :execute)
219
220 (def-add/subtract-method add-to nil
221 "Adds to polynomial SELF another polynomial OTHER.
222This operation destructively modifies both polynomials.
223The result is stored in SELF. This implementation does
224no consing, entirely reusing the sells of SELF and OTHER.")
225
226 (def-add/subtract-method subtract-from unary-minus
227 "Subtracts from polynomial SELF another polynomial OTHER.
228This operation destructively modifies both polynomials.
229The result is stored in SELF. This implementation does
230no consing, entirely reusing the sells of SELF and OTHER.")
231 )
232
233(defmethod unary-minus ((self poly))
234 "Destructively modifies the coefficients of the polynomial SELF,
235by changing their sign."
236 (mapc #'unary-minus (poly-termlist self))
237 self)
238
239(defun add-termlists (p q order-fn)
240 "Destructively adds two termlists P and Q ordered according to ORDER-FN."
241 (fast-add/subtract p q order-fn #'add-to nil))
242
243(defmacro multiply-term-by-termlist-dropping-zeros (term termlist
244 &optional (reverse-arg-order-P nil))
245 "Multiplies term TERM by a list of term, TERMLIST.
246Takes into accound divisors of zero in the ring, by
247deleting zero terms. Optionally, if REVERSE-ARG-ORDER-P
248is T, change the order of arguments; this may be important
249if we extend the package to non-commutative rings."
250 `(mapcan #'(lambda (other-term)
251 (let ((prod (r*
252 ,@(cond
253 (reverse-arg-order-p
254 `(other-term ,term))
255 (t
256 `(,term other-term))))))
257 (cond
258 ((r-zerop prod) nil)
259 (t (list prod)))))
260 ,termlist))
261
262(defun multiply-termlists (p q order-fn)
263 "A version of polynomial multiplication, operating
264directly on termlists."
265 (cond
266 ((or (endp p) (endp q))
267 ;;p or q is 0 (represented by NIL)
268 nil)
269 ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
270 ((endp (cdr p))
271 (multiply-term-by-termlist-dropping-zeros (car p) q))
272 ((endp (cdr q))
273 (multiply-term-by-termlist-dropping-zeros (car q) p t))
274 (t
275 (cons (r* (car p) (car q))
276 (add-termlists
277 (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
278 (multiply-termlists (cdr p) q order-fn)
279 order-fn)))))
280
281(defmethod multiply-by ((self poly) (other poly))
282 (change-term-order other self)
283 (setf (poly-termlist self) (multiply-termlists (poly-termlist self)
284 (poly-termlist other)
285 (poly-term-order self)))
286 self)
287
288(defmethod r* ((poly1 poly) (poly2 poly))
289 "Non-destructively multiply POLY1 by POLY2."
290 (multiply-by (copy-instance POLY1) (copy-instance POLY2)))
291
292(defmethod left-tensor-product-by ((self poly) (other term))
293 (setf (poly-termlist self)
294 (mapcan #'(lambda (term)
295 (let ((prod (left-tensor-product-by term other)))
296 (cond
297 ((r-zerop prod) nil)
298 (t (list prod)))))
299 (poly-termlist self)))
300 self)
301
302(defmethod right-tensor-product-by ((self poly) (other term))
303 (setf (poly-termlist self)
304 (mapcan #'(lambda (term)
305 (let ((prod (right-tensor-product-by term other)))
306 (cond
307 ((r-zerop prod) nil)
308 (t (list prod)))))
309 (poly-termlist self)))
310 self)
311
312(defmethod left-tensor-product-by ((self poly) (other monom))
313 (setf (poly-termlist self)
314 (mapcan #'(lambda (term)
315 (let ((prod (left-tensor-product-by term other)))
316 (cond
317 ((r-zerop prod) nil)
318 (t (list prod)))))
319 (poly-termlist self)))
320 (incf (poly-dimension self) (monom-dimension other))
321 self)
322
323(defmethod right-tensor-product-by ((self poly) (other monom))
324 (setf (poly-termlist self)
325 (mapcan #'(lambda (term)
326 (let ((prod (right-tensor-product-by term other)))
327 (cond
328 ((r-zerop prod) nil)
329 (t (list prod)))))
330 (poly-termlist self)))
331 (incf (poly-dimension self) (monom-dimension other))
332 self)
333
334
335(defun standard-extension (plist &aux (k (length plist)) (i 0))
336 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
337is a list of polynomials. Destructively modifies PLIST elements."
338 (mapc #'(lambda (poly)
339 (left-tensor-product-by
340 poly
341 (prog1
342 (make-monom-variable k i)
343 (incf i))))
344 plist))
345
346(defun standard-extension-1 (plist
347 &aux
348 (plist (standard-extension plist))
349 (nvars (poly-dimension (car plist))))
350 "Calculate [U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK].
351Firstly, new K variables U1, U2, ..., UK, are inserted into each
352polynomial. Subsequently, P1, P2, ..., PK are destructively modified
353tantamount to replacing PI with UI*PI-1. It assumes that all
354polynomials have the same dimension, and only the first polynomial
355is examined to determine this dimension."
356 ;; Implementation note: we use STANDARD-EXTENSION and then subtract
357 ;; 1 from each polynomial; since UI*PI has no constant term,
358 ;; we just need to append the constant term at the end
359 ;; of each termlist.
360 (flet ((subtract-1 (p)
361 (append-item p (make-instance 'term :coeff -1 :dimension nvars))))
362 (setf plist (mapc #'subtract-1 plist)))
363 plist)
364
365
366(defun standard-sum (plist
367 &aux
368 (plist (standard-extension plist))
369 (nvars (poly-dimension (car plist))))
370 "Calculate the polynomial U1*P1+U2*P2+...+UK*PK-1, where PLIST=[P1,P2,...,PK].
371Firstly, new K variables, U1, U2, ..., UK, are inserted into each
372polynomial. Subsequently, P1, P2, ..., PK are destructively modified
373tantamount to replacing PI with UI*PI, and the resulting polynomials
374are added. Finally, 1 is subtracted. It should be noted that the term
375order is not modified, which is equivalent to using a lexicographic
376order on the first K variables."
377 (flet ((subtract-1 (p)
378 (append-item p (make-instance 'term :coeff -1 :dimension nvars))))
379 (subtract-1
380 (make-instance
381 'poly
382 :termlist (apply #'nconc (mapcar #'poly-termlist plist))))))
383
384#|
385
386(defun saturation-extension-1 (ring f p)
387 "Calculate [F, U*P-1]. It destructively modifies F."
388 (declare (type ring ring))
389 (polysaturation-extension ring f (list p)))
390
391
392
393
394(defun spoly (ring-and-order f g
395 &aux
396 (ring (ro-ring ring-and-order)))
397 "It yields the S-polynomial of polynomials F and G."
398 (declare (type ring-and-order ring-and-order) (type poly f g))
399 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
400 (mf (monom-div lcm (poly-lm f)))
401 (mg (monom-div lcm (poly-lm g))))
402 (declare (type monom mf mg))
403 (multiple-value-bind (c cf cg)
404 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
405 (declare (ignore c))
406 (poly-sub
407 ring-and-order
408 (scalar-times-poly ring cg (monom-times-poly mf f))
409 (scalar-times-poly ring cf (monom-times-poly mg g))))))
410
411
412(defun poly-primitive-part (ring p)
413 "Divide polynomial P with integer coefficients by gcd of its
414coefficients and return the result."
415 (declare (type ring ring) (type poly p))
416 (if (poly-zerop p)
417 (values p 1)
418 (let ((c (poly-content ring p)))
419 (values (make-poly-from-termlist
420 (mapcar
421 #'(lambda (x)
422 (make-term :monom (term-monom x)
423 :coeff (funcall (ring-div ring) (term-coeff x) c)))
424 (poly-termlist p))
425 (poly-sugar p))
426 c))))
427
428(defun poly-content (ring p)
429 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
430to compute the greatest common divisor."
431 (declare (type ring ring) (type poly p))
432 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
433
434|#
Note: See TracBrowser for help on using the repository browser.