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1;;----------------------------------------------------------------
2;; File: polynomial.lisp
3;;----------------------------------------------------------------
4;;
5;; Author: Marek Rychlik (rychlik@u.arizona.edu)
6;; Date: Thu Aug 27 09:41:24 2015
7;; Copying: (C) Marek Rychlik, 2010. All rights reserved.
8;;
9;;----------------------------------------------------------------
10;;; -*- Mode: Lisp -*-
11;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
12;;;
13;;; Copyright (C) 1999, 2002, 2009, 2015 Marek Rychlik <rychlik@u.arizona.edu>
14;;;
15;;; This program is free software; you can redistribute it and/or modify
16;;; it under the terms of the GNU General Public License as published by
17;;; the Free Software Foundation; either version 2 of the License, or
18;;; (at your option) any later version.
19;;;
20;;; This program is distributed in the hope that it will be useful,
21;;; but WITHOUT ANY WARRANTY; without even the implied warranty of
22;;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
23;;; GNU General Public License for more details.
24;;;
25;;; You should have received a copy of the GNU General Public License
26;;; along with this program; if not, write to the Free Software
27;;; Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
28;;;
29;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
30
31(defpackage "POLYNOMIAL"
32 (:use :cl :utils :monom)
33 (:export "POLY"
34 "POLY-DIMENSION"
35 "POLY-TERMLIST"
36 "POLY-TERM-ORDER"
37 "POLY-INSERT-TERM"
38 "POLY-LEADING-TERM"
39 "POLY-LEADING-COEFFICIENT"
40 "POLY-LEADING-MONOM"
41 "POLY-ADD-TO"
42 "POLY-SUBTRACT-FROM"
43 "CHANGE-TERM-ORDER"
44 "STANDARD-EXTENSION"
45 "STANDARD-EXTENSION-1"
46 "STANDARD-SUM"
47 "SATURATION-EXTENSION"
48 "ALIST->POLY")
49 (:documentation "Implements polynomials. A polynomial is essentially
50a mapping of monomials of the same degree to coefficients. The
51momomials are ordered according to a monomial order."))
52
53(in-package :polynomial)
54
55(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
56
57(defclass poly ()
58 ((dimension :initform nil
59 :initarg :dimension
60 :accessor poly-dimension
61 :documentation "Shared dimension of all terms, the number of variables")
62 (termlist :initform nil :initarg :termlist :accessor poly-termlist
63 :documentation "List of terms.")
64 (order :initform #'lex> :initarg :order :accessor poly-term-order
65 :documentation "Monomial/term order."))
66 (:default-initargs :dimension nil :termlist nil :order #'lex>)
67 (:documentation "A polynomial with a list of terms TERMLIST, ordered
68according to term order ORDER, which defaults to LEX>."))
69
70(defmethod print-object ((self poly) stream)
71 (print-unreadable-object (self stream :type t :identity t)
72 (with-accessors ((dimension poly-dimension)
73 (termlist poly-termlist)
74 (order poly-term-order))
75 self
76 (format stream "DIMENSION=~A TERMLIST=~A ORDER=~A"
77 dimension termlist order))))
78
79(defgeneric change-term-order (self other)
80 (:documentation "Change term order of SELF to the term order of OTHER.")
81 (:method ((self poly) (other poly))
82 (unless (eq (poly-term-order self) (poly-term-order other))
83 (setf (poly-termlist self) (sort (poly-termlist self) (poly-term-order other))
84 (poly-term-order self) (poly-term-order other)))
85 self))
86
87(defgeneric poly-insert-term (self term)
88 (:documentation "Insert a term TERM into SELF before all other
89 terms. Order is not enforced.")
90 (:method ((self poly) (term term))
91 (cond ((null (poly-dimension self))
92 (setf (poly-dimension self) (monom-dimension term)))
93 (t (assert (= (poly-dimension self) (monom-dimension term)))))
94 (push term (poly-termlist self))
95 self))
96
97(defgeneric poly-append-term (self term)
98 (:documentation "Append a term TERM to SELF after all other terms. Order is not enforced.")
99 (:method ((self poly) (term term))
100 (cond ((null (poly-dimension self))
101 (setf (poly-dimension self) (monom-dimension term)))
102 (t (assert (= (poly-dimension self) (monom-dimension term)))))
103 (setf (cdr (last (poly-termlist self))) (list term))
104 self))
105
106(defun alist->poly (alist &aux (poly (make-instance 'poly)))
107 "It reads polynomial from an alist formatted as ( ... (exponents . coeff) ...).
108It can be used to enter simple polynomials by hand, e.g the polynomial
109in two variables, X and Y, given in standard notation as:
110
111 3*X^2*Y^3+2*Y+7
112
113can be entered as
114(ALIST->POLY '(((2 3) . 3) ((0 1) . 2) ((0 0) . 7))).
115
116NOTE: The primary use is for low-level debugging of the package."
117 (dolist (x alist poly)
118 (poly-insert-term poly (make-instance 'term :exponents (car x) :coeff (cdr x)))))
119
120(defmethod update-instance-for-different-class :after ((old monom) (new poly) &key)
121 "Converts OLD of class MONOM to a NEW of class POLY, by making it into a 1-element TERMLIST."
122 (reinitialize-instance new
123 :dimension (monom-dimension old)
124 :termlist (list (cons old 1))))
125
126(defmethod universal-equalp ((self poly) (other poly))
127 "Implements equality of polynomials."
128 (and (eql (poly-dimension self) (poly-dimension other))
129 (every #'universal-equalp (poly-termlist self) (poly-termlist other))
130 (eq (poly-term-order self) (poly-term-order other))))
131
132(defgeneric leading-term (object)
133 (:method ((self poly))
134 (car (poly-termlist self)))
135 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
136
137(defgeneric second-leading-term (object)
138 (:method ((self poly))
139 (cadar (poly-termlist self)))
140 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
141
142(defgeneric leading-coefficient (object)
143 (:method ((self poly))
144 (term-coeff (poly-leading-term self)))
145 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
146
147
148(defgeneric second-leading-coefficient (object)
149 (:method ((self poly))
150 (term-coeff (poly-second-leading-term self)))
151 (:documentation "The second leading coefficient of a polynomial. It
152 signals error for a polynomial with at most one term."))
153
154(defmethod universal-zerop ((self poly))
155 "Return T iff SELF is a zero polynomial."
156 (null (poly-termlist self))))
157
158(defgeneric poly-length (self)
159 (:documentation "Return the number of terms.")
160 (:method ((self poly))
161 (length (poly-termlist self))))
162
163(defmethod multiply-by ((self poly) (other monom))
164 "Multiply a polynomial SELF by OTHER."
165 (mapc #'(lambda (term) (multiply-by term other))
166 (poly-termlist self))
167 self)
168
169(defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)
170 "Return an expression which will efficiently adds/subtracts two
171polynomials, P and Q. The addition/subtraction of coefficients is
172performed by calling ADD/SUBTRACT-METHOD-NAME. If UMINUS-METHOD-NAME
173is supplied, it is used to negate the coefficients of Q which do not
174have a corresponding coefficient in P. The code implements an
175efficient algorithm to add two polynomials represented as sorted lists
176of terms. The code destroys both arguments, reusing the terms to build
177the result."
178 `(macrolet ((lc (x) `(caar ,x)))
179 (do ((p ,p)
180 (q ,q)
181 r)
182 ((or (endp p) (endp q))
183 ;; NOTE: R contains the result in reverse order. Can it
184 ;; be more efficient to produce the terms in correct order?
185 (unless (endp q)
186 ;; Upon subtraction, we must change the sign of
187 ;; all coefficients in q
188 ,@(when uminus-fn
189 `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))
190 (setf r (nreconc r q)))
191 r)
192 (multiple-value-bind
193 (greater-p equal-p)
194 (funcall ,order-fn (caar p) (caar q))
195 (cond
196 (greater-p
197 (rotatef (cdr p) r p)
198 )
199 (equal-p
200 (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))
201 (cond
202 ((r-zerop s)
203 (setf p (cdr p))
204 )
205 (t
206 (setf (lc p) s)
207 (rotatef (cdr p) r p))))
208 (setf q (cdr q))
209 )
210 (t
211 ;;Negate the term of Q if UMINUS provided, signallig
212 ;;that we are doing subtraction
213 ,(when uminus-fn
214 `(setf (lc q) (funcall ,uminus-fn (lc q))))
215 (rotatef (cdr q) r q)))))))
216
217
218(defmacro def-add/subtract-method (add/subtract-method-name
219 uminus-method-name
220 &optional
221 (doc-string nil doc-string-supplied-p))
222 "This macro avoids code duplication for two similar operations: POLY-ADD-TO and POLY-SUBTRACT-FROM."
223 `(defmethod ,add/subtract-method-name ((self poly) (other poly))
224 ,@(when doc-string-supplied-p `(,doc-string))
225 ;; Ensure orders are compatible
226 (change-term-order other self)
227 (setf (poly-termlist self) (fast-add/subtract
228 (poly-termlist self) (poly-termlist other)
229 (poly-term-order self)
230 #',add/subtract-method-name
231 ,(when uminus-method-name `(function ,uminus-method-name))))
232 self))
233
234(eval-when (:compile-toplevel :load-toplevel :execute)
235
236 (def-add/subtract-method poly-add-to nil
237 "Adds to polynomial SELF another polynomial OTHER.
238This operation destructively modifies both polynomials.
239The result is stored in SELF. This implementation does
240no consing, entirely reusing the sells of SELF and OTHER.")
241
242 (def-add/subtract-method poly-subtract-from unary-minus
243 "Subtracts from polynomial SELF another polynomial OTHER.
244This operation destructively modifies both polynomials.
245The result is stored in SELF. This implementation does
246no consing, entirely reusing the sells of SELF and OTHER.")
247 )
248
249(defmethod unary-minus ((self poly))
250 "Destructively modifies the coefficients of the polynomial SELF,
251by changing their sign."
252 (mapc #'unary-minus (poly-termlist self))
253 self)
254
255(defun add-termlists (p q order-fn)
256 "Destructively adds two termlists P and Q ordered according to ORDER-FN."
257 (fast-add/subtract p q order-fn #'poly-add-to nil))
258
259(defmacro multiply-term-by-termlist-dropping-zeros (term termlist
260 &optional (reverse-arg-order-P nil))
261 "Multiplies term TERM by a list of term, TERMLIST.
262Takes into accound divisors of zero in the ring, by
263deleting zero terms. Optionally, if REVERSE-ARG-ORDER-P
264is T, change the order of arguments; this may be important
265if we extend the package to non-commutative rings."
266 `(mapcan #'(lambda (other-term)
267 (let ((prod (r*
268 ,@(cond
269 (reverse-arg-order-p
270 `(other-term ,term))
271 (t
272 `(,term other-term))))))
273 (cond
274 ((r-zerop prod) nil)
275 (t (list prod)))))
276 ,termlist))
277
278(defun multiply-termlists (p q order-fn)
279 "A version of polynomial multiplication, operating
280directly on termlists."
281 (cond
282 ((or (endp p) (endp q))
283 ;;p or q is 0 (represented by NIL)
284 nil)
285 ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
286 ((endp (cdr p))
287 (multiply-term-by-termlist-dropping-zeros (car p) q))
288 ((endp (cdr q))
289 (multiply-term-by-termlist-dropping-zeros (car q) p t))
290 (t
291 (cons (r* (car p) (car q))
292 (add-termlists
293 (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
294 (multiply-termlists (cdr p) q order-fn)
295 order-fn)))))
296
297(defmethod multiply-by ((self poly) (other poly))
298 (change-term-order other self)
299 (setf (poly-termlist self) (multiply-termlists (poly-termlist self)
300 (poly-termlist other)
301 (poly-term-order self)))
302 self)
303
304(defmethod r+ ((poly1 poly) poly2)
305 "Non-destructively add POLY1 by POLY2."
306 (poly-add-to (copy-instance POLY1) (change-class (copy-instance POLY2) 'poly)))
307
308(defmethod r- ((minuend poly) &rest subtrahends)
309 "Non-destructively subtract MINUEND and SUBTRAHENDS."
310 (poly-subtract-from (copy-instance minuend)
311 (change-class (reduce #'r+ subtrahends) 'poly)))
312
313(defmethod r+ ((poly1 monom) poly2)
314 "Non-destructively add POLY1 by POLY2."
315 (poly-add-to (change-class (copy-instance poly1) 'poly)
316 (change-class (copy-instance poly2) 'poly)))
317
318(defmethod r- ((minuend monom) &rest subtrahends)
319 "Non-destructively subtract MINUEND and SUBTRAHENDS."
320 (poly-subtract-from (change-class (copy-instance minuend) 'poly)
321 (change-class (reduce #'r+ subtrahends) 'poly)))
322
323(defmethod r* ((poly1 poly) (poly2 poly))
324 "Non-destructively multiply POLY1 by POLY2."
325 (multiply-by (copy-instance poly1) (copy-instance poly2)))
326
327(defmethod left-tensor-product-by ((self poly) (other monom))
328 (setf (poly-termlist self)
329 (mapcan #'(lambda (term)
330 (let ((prod (left-tensor-product-by term other)))
331 (cond
332 ((r-zerop prod) nil)
333 (t (list prod)))))
334 (poly-termlist self)))
335 (incf (poly-dimension self) (monom-dimension other))
336 self)
337
338(defmethod right-tensor-product-by ((self poly) (other monom))
339 (setf (poly-termlist self)
340 (mapcan #'(lambda (term)
341 (let ((prod (right-tensor-product-by term other)))
342 (cond
343 ((r-zerop prod) nil)
344 (t (list prod)))))
345 (poly-termlist self)))
346 (incf (poly-dimension self) (monom-dimension other))
347 self)
348
349
350(defun standard-extension (plist &aux (k (length plist)) (i 0))
351 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
352is a list of polynomials. Destructively modifies PLIST elements."
353 (mapc #'(lambda (poly)
354 (left-tensor-product-by
355 poly
356 (prog1
357 (make-monom-variable k i)
358 (incf i))))
359 plist))
360
361(defun standard-extension-1 (plist
362 &aux
363 (plist (standard-extension plist))
364 (nvars (poly-dimension (car plist))))
365 "Calculate [U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK].
366Firstly, new K variables U1, U2, ..., UK, are inserted into each
367polynomial. Subsequently, P1, P2, ..., PK are destructively modified
368tantamount to replacing PI with UI*PI-1. It assumes that all
369polynomials have the same dimension, and only the first polynomial
370is examined to determine this dimension."
371 ;; Implementation note: we use STANDARD-EXTENSION and then subtract
372 ;; 1 from each polynomial; since UI*PI has no constant term,
373 ;; we just need to append the constant term at the end
374 ;; of each termlist.
375 (flet ((subtract-1 (p)
376 (poly-append-term p (make-instance 'monom :dimension nvars) -1)))
377 (setf plist (mapc #'subtract-1 plist)))
378 plist)
379
380
381(defun standard-sum (plist
382 &aux
383 (plist (standard-extension plist))
384 (nvars (poly-dimension (car plist))))
385 "Calculate the polynomial U1*P1+U2*P2+...+UK*PK-1, where PLIST=[P1,P2,...,PK].
386Firstly, new K variables, U1, U2, ..., UK, are inserted into each
387polynomial. Subsequently, P1, P2, ..., PK are destructively modified
388tantamount to replacing PI with UI*PI, and the resulting polynomials
389are added. Finally, 1 is subtracted. It should be noted that the term
390order is not modified, which is equivalent to using a lexicographic
391order on the first K variables."
392 (flet ((subtract-1 (p)
393 (poly-append-term p (make-instance 'monom :dimension nvars) -1)))
394 (subtract-1
395 (make-instance
396 'poly
397 :termlist (apply #'nconc (mapcar #'poly-termlist plist))))))
398
399#|
400
401(defun saturation-extension-1 (ring f p)
402 "Calculate [F, U*P-1]. It destructively modifies F."
403 (declare (type ring ring))
404 (polysaturation-extension ring f (list p)))
405
406
407
408
409(defun spoly (ring-and-order f g
410 &aux
411 (ring (ro-ring ring-and-order)))
412 "It yields the S-polynomial of polynomials F and G."
413 (declare (type ring-and-order ring-and-order) (type poly f g))
414 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
415 (mf (monom-div lcm (poly-lm f)))
416 (mg (monom-div lcm (poly-lm g))))
417 (declare (type monom mf mg))
418 (multiple-value-bind (c cf cg)
419 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
420 (declare (ignore c))
421 (poly-sub
422 ring-and-order
423 (scalar-times-poly ring cg (monom-times-poly mf f))
424 (scalar-times-poly ring cf (monom-times-poly mg g))))))
425
426
427(defun poly-primitive-part (ring p)
428 "Divide polynomial P with integer coefficients by gcd of its
429coefficients and return the result."
430 (declare (type ring ring) (type poly p))
431 (if (poly-zerop p)
432 (values p 1)
433 (let ((c (poly-content ring p)))
434 (values (make-poly-from-termlist
435 (mapcar
436 #'(lambda (x)
437 (make-term :monom (term-monom x)
438 :coeff (funcall (ring-div ring) (term-coeff x) c)))
439 (poly-termlist p))
440 (poly-sugar p))
441 c))))
442
443(defun poly-content (ring p)
444 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
445to compute the greatest common divisor."
446 (declare (type ring ring) (type poly p))
447 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
448
449|#
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