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source: branches/f4grobner/polynomial.lisp@ 3123

Last change on this file since 3123 was 3123, 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 :utils :ring :monom :order :term #| :infix |# )
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 ((termlist :initarg :termlist :accessor poly-termlist
41 :documentation "List of terms.")
42 (order :initarg :order :accessor poly-term-order
43 :documentation "Monomial/term order."))
44 (:default-initargs :termlist nil :order #'lex>)
45 (:documentation "A polynomial with a list of terms TERMLIST, ordered
46according to term order ORDER, which defaults to LEX>."))
47
48(defmethod print-object ((self poly) stream)
49 (format stream "#<POLY TERMLIST=~A ORDER=~A>"
50 (poly-termlist self)
51 (poly-term-order self)))
52
53(defgeneric change-term-order (self other)
54 (:documentation "Change term order of SELF to the term order of OTHER.")
55 (:method ((self poly) (other poly))
56 (unless (eq (poly-term-order self) (poly-term-order other))
57 (setf (poly-termlist self) (sort (poly-termlist self) (poly-term-order other))
58 (poly-term-order self) (poly-term-order other)))
59 self))
60
61(defun alist->poly (alist &aux (poly (make-instance 'poly)))
62 "It reads polynomial from an alist formatted as ( ... (exponents . coeff) ...)."
63 (dolist (x alist poly)
64 (insert-item poly (make-instance 'term :exponents (car x) :coeff (cdr x)))))
65
66
67(defmethod r-equalp ((self poly) (other poly))
68 "POLY instances are R-EQUALP if they have the same
69order and if all terms are R-EQUALP."
70 (and (every #'r-equalp (poly-termlist self) (poly-termlist other))
71 (eq (poly-term-order self) (poly-term-order other))))
72
73(defmethod insert-item ((self poly) (item term))
74 (push item (poly-termlist self))
75 self)
76
77(defmethod append-item ((self poly) (item term))
78 (setf (cdr (last (poly-termlist self))) (list item))
79 self)
80
81;; Leading term
82(defgeneric leading-term (object)
83 (:method ((self poly))
84 (car (poly-termlist self)))
85 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
86
87;; Second term
88(defgeneric second-leading-term (object)
89 (:method ((self poly))
90 (cadar (poly-termlist self)))
91 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
92
93;; Leading coefficient
94(defgeneric leading-coefficient (object)
95 (:method ((self poly))
96 (r-coeff (leading-term self)))
97 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
98
99;; Second coefficient
100(defgeneric second-leading-coefficient (object)
101 (:method ((self poly))
102 (r-coeff (second-leading-term self)))
103 (:documentation "The second leading coefficient of a polynomial. It
104 signals error for a polynomial with at most one term."))
105
106;; Testing for a zero polynomial
107(defmethod r-zerop ((self poly))
108 (null (poly-termlist self)))
109
110;; The number of terms
111(defmethod r-length ((self poly))
112 (length (poly-termlist self)))
113
114(defmethod multiply-by ((self poly) (other monom))
115 (mapc #'(lambda (term) (multiply-by term other))
116 (poly-termlist self))
117 self)
118
119(defmethod multiply-by ((self poly) (other term))
120 (mapc #'(lambda (term) (multiply-by term other))
121 (poly-termlist self))
122 self)
123
124(defmethod multiply-by ((self poly) (other scalar))
125 (mapc #'(lambda (term) (multiply-by term other))
126 (poly-termlist self))
127 self)
128
129
130(defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)
131 "Return an expression which will efficiently adds/subtracts two
132polynomials, P and Q. The addition/subtraction of coefficients is
133performed by calling ADD/SUBTRACT-METHOD-NAME. If UMINUS-METHOD-NAME
134is supplied, it is used to negate the coefficients of Q which do not
135have a corresponding coefficient in P. The code implements an
136efficient algorithm to add two polynomials represented as sorted lists
137of terms. The code destroys both arguments, reusing the terms to build
138the result."
139 `(macrolet ((lc (x) `(r-coeff (car ,x))))
140 (do ((p ,p)
141 (q ,q)
142 r)
143 ((or (endp p) (endp q))
144 ;; NOTE: R contains the result in reverse order. Can it
145 ;; be more efficient to produce the terms in correct order?
146 (unless (endp q)
147 ;; Upon subtraction, we must change the sign of
148 ;; all coefficients in q
149 ,@(when uminus-fn
150 `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))
151 (setf r (nreconc r q)))
152 r)
153 (multiple-value-bind
154 (greater-p equal-p)
155 (funcall ,order-fn (car p) (car q))
156 (cond
157 (greater-p
158 (rotatef (cdr p) r p)
159 )
160 (equal-p
161 (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))
162 (cond
163 ((r-zerop s)
164 (setf p (cdr p))
165 )
166 (t
167 (setf (lc p) s)
168 (rotatef (cdr p) r p))))
169 (setf q (cdr q))
170 )
171 (t
172 ;;Negate the term of Q if UMINUS provided, signallig
173 ;;that we are doing subtraction
174 ,(when uminus-fn
175 `(setf (lc q) (funcall ,uminus-fn (lc q))))
176 (rotatef (cdr q) r q)))))))
177
178
179(defmacro def-add/subtract-method (add/subtract-method-name
180 uminus-method-name
181 &optional
182 (doc-string nil doc-string-supplied-p))
183 "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
184 `(defmethod ,add/subtract-method-name ((self poly) (other poly))
185 ,@(when doc-string-supplied-p `(,doc-string))
186 ;; Ensure orders are compatible
187 (change-term-order other self)
188 (setf (poly-termlist self) (fast-add/subtract
189 (poly-termlist self) (poly-termlist other)
190 (poly-term-order self)
191 #',add/subtract-method-name
192 ,(when uminus-method-name `(function ,uminus-method-name))))
193 self))
194
195(eval-when (:compile-toplevel :load-toplevel :execute)
196
197 (def-add/subtract-method add-to nil
198 "Adds to polynomial SELF another polynomial OTHER.
199This operation destructively modifies both polynomials.
200The result is stored in SELF. This implementation does
201no consing, entirely reusing the sells of SELF and OTHER.")
202
203 (def-add/subtract-method subtract-from unary-minus
204 "Subtracts from polynomial SELF another polynomial OTHER.
205This operation destructively modifies both polynomials.
206The result is stored in SELF. This implementation does
207no consing, entirely reusing the sells of SELF and OTHER.")
208 )
209
210(defmethod unary-minus ((self poly))
211 "Destructively modifies the coefficients of the polynomial SELF,
212by changing their sign."
213 (mapc #'unary-minus (poly-termlist self))
214 self)
215
216(defun add-termlists (p q order-fn)
217 "Destructively adds two termlists P and Q ordered according to ORDER-FN."
218 (fast-add/subtract p q order-fn #'add-to nil))
219
220(defmacro multiply-term-by-termlist-dropping-zeros (term termlist
221 &optional (reverse-arg-order-P nil))
222 "Multiplies term TERM by a list of term, TERMLIST.
223Takes into accound divisors of zero in the ring, by
224deleting zero terms. Optionally, if REVERSE-ARG-ORDER-P
225is T, change the order of arguments; this may be important
226if we extend the package to non-commutative rings."
227 `(mapcan #'(lambda (other-term)
228 (let ((prod (r*
229 ,@(cond
230 (reverse-arg-order-p
231 `(other-term ,term))
232 (t
233 `(,term other-term))))))
234 (cond
235 ((r-zerop prod) nil)
236 (t (list prod)))))
237 ,termlist))
238
239(defun multiply-termlists (p q order-fn)
240 (cond
241 ((or (endp p) (endp q))
242 ;;p or q is 0 (represented by NIL)
243 nil)
244 ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
245 ((endp (cdr p))
246 (multiply-term-by-termlist-dropping-zeros (car p) q))
247 ((endp (cdr q))
248 (multiply-term-by-termlist-dropping-zeros (car q) p t))
249 (t
250 (cons (r* (car p) (car q))
251 (add-termlists
252 (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
253 (multiply-termlists (cdr p) q order-fn)
254 order-fn)))))
255
256(defmethod multiply-by ((self poly) (other poly))
257 (change-term-order other self)
258 (setf (poly-termlist self) (multiply-termlists (poly-termlist self)
259 (poly-termlist other)
260 (poly-term-order self)))
261 self)
262
263(defmethod r* ((poly1 poly) (poly2 poly))
264 "Non-destructively multiply POLY1 by POLY2."
265 (multiply-by (copy-instance POLY1) (copy-instance POLY2)))
266
267(defmethod left-tensor-product-by ((self poly) (other term))
268 (setf (poly-termlist self)
269 (mapcan #'(lambda (term)
270 (let ((prod (left-tensor-product-by term other)))
271 (cond
272 ((r-zerop prod) nil)
273 (t (list prod)))))
274 (poly-termlist self)))
275 self)
276
277(defmethod right-tensor-product-by ((self poly) (other term))
278 (setf (poly-termlist self)
279 (mapcan #'(lambda (term)
280 (let ((prod (right-tensor-product-by term other)))
281 (cond
282 ((r-zerop prod) nil)
283 (t (list prod)))))
284 (poly-termlist self)))
285 self)
286
287(defmethod left-tensor-product-by ((self poly) (other monom))
288 (setf (poly-termlist self)
289 (mapcan #'(lambda (term)
290 (let ((prod (left-tensor-product-by term other)))
291 (cond
292 ((r-zerop prod) nil)
293 (t (list prod)))))
294 (poly-termlist self)))
295 self)
296
297(defmethod right-tensor-product-by ((self poly) (other monom))
298 (setf (poly-termlist self)
299 (mapcan #'(lambda (term)
300 (let ((prod (right-tensor-product-by term other)))
301 (cond
302 ((r-zerop prod) nil)
303 (t (list prod)))))
304 (poly-termlist self)))
305 self)
306
307
308(defun standard-extension (plist &aux (k (length plist)) (i 0))
309 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
310is a list of polynomials. Destructively modifies PLIST elements."
311 (mapc #'(lambda (poly)
312 (left-tensor-product-by
313 poly
314 (prog1
315 (make-monom-variable k i)
316 (incf i))))
317 plist))
318
319(defmethod poly-dimension ((poly poly))
320 (cond ((r-zerop poly) -1)
321 (t (monom-dimension (leading-term poly)))))
322
323(defun standard-extension-1 (plist
324 &aux
325 (plist (standard-extension plist))
326 (nvars (poly-dimension (car plist))))
327 "Calculate [U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK].
328Firstly, new K variables U1, U2, ..., UK, are inserted into each
329polynomial. Subsequently, P1, P2, ..., PK are destructively modified
330tantamount to replacing PI with UI*PI-1. It assumes that all
331polynomials have the same dimension, and only the first polynomial
332is examined to determine this dimension."
333 ;; Implementation note: we use STANDARD-EXTENSION and then subtract
334 ;; 1 from each polynomial; since UI*PI has no constant term,
335 ;; we just need to append the constant term at the end
336 ;; of each termlist.
337 (flet ((subtract-1 (p)
338 (append-item p (make-instance 'term :coeff -1 :dimension nvars))))
339 (setf plist (mapc #'subtract-1 plist)))
340 plist)
341
342
343(defun standard-sum (plist
344 &aux
345 (plist (standard-extension plist))
346 (nvars (poly-dimension (car plist))))
347 "Calculate the polynomial U1*P1+U2*P2+...+UK*PK-1, where PLIST=[P1,P2,...,PK].
348Firstly, new K variables, U1, U2, ..., UK, are inserted into each
349polynomial. Subsequently, P1, P2, ..., PK are destructively modified
350tantamount to replacing PI with UI*PI, and the resulting polynomials
351are added. Finally, 1 is subtracted. It should be noted that the term
352order is not modified, which is equivalent to using a lexicographic
353order on the first K variables."
354 (flet ((subtract-1 (p)
355 (append-item p (make-instance 'term :coeff -1 :dimension nvars))))
356 (subtract-1
357 (make-instance
358 'poly
359 :termlist (apply #'nconc (mapcar #'poly-termlist plist))))))
360
361#|
362
363(defun saturation-extension-1 (ring f p)
364 "Calculate [F, U*P-1]. It destructively modifies F."
365 (declare (type ring ring))
366 (polysaturation-extension ring f (list p)))
367
368
369
370
371(defun spoly (ring-and-order f g
372 &aux
373 (ring (ro-ring ring-and-order)))
374 "It yields the S-polynomial of polynomials F and G."
375 (declare (type ring-and-order ring-and-order) (type poly f g))
376 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
377 (mf (monom-div lcm (poly-lm f)))
378 (mg (monom-div lcm (poly-lm g))))
379 (declare (type monom mf mg))
380 (multiple-value-bind (c cf cg)
381 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
382 (declare (ignore c))
383 (poly-sub
384 ring-and-order
385 (scalar-times-poly ring cg (monom-times-poly mf f))
386 (scalar-times-poly ring cf (monom-times-poly mg g))))))
387
388
389(defun poly-primitive-part (ring p)
390 "Divide polynomial P with integer coefficients by gcd of its
391coefficients and return the result."
392 (declare (type ring ring) (type poly p))
393 (if (poly-zerop p)
394 (values p 1)
395 (let ((c (poly-content ring p)))
396 (values (make-poly-from-termlist
397 (mapcar
398 #'(lambda (x)
399 (make-term :monom (term-monom x)
400 :coeff (funcall (ring-div ring) (term-coeff x) c)))
401 (poly-termlist p))
402 (poly-sugar p))
403 c))))
404
405(defun poly-content (ring p)
406 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
407to compute the greatest common divisor."
408 (declare (type ring ring) (type poly p))
409 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
410
411|#
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