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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 :copy)
33 (:export "POLY"
34 "POLY-DIMENSION"
35 "POLY-TERMLIST"
36 "POLY-TERM-ORDER"
37 "POLY-INSERT-TERM"
38 "LEADING-TERM"
39 "LEADING-COEFFICIENT"
40 "ADD-TO"
41 "ADD"
42 "SUBTRACT-FROM"
43 "SUBTRACT"
44 "CHANGE-TERM-ORDER"
45 "STANDARD-EXTENSION"
46 "STANDARD-EXTENSION-1"
47 "STANDARD-SUM"
48 "SATURATION-EXTENSION"
49 "ALIST->POLY")
50 (:documentation "Implements polynomials. A polynomial is essentially
51a mapping of monomials of the same degree to coefficients. The
52momomials are ordered according to a monomial order."))
53
54(in-package :polynomial)
55
56(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
57
58(defclass poly ()
59 ((dimension :initform nil
60 :initarg :dimension
61 :accessor poly-dimension
62 :documentation "Shared dimension of all terms, the number of variables")
63 (termlist :initform nil :initarg :termlist :accessor poly-termlist
64 :documentation "List of terms.")
65 (order :initform #'lex> :initarg :order :accessor poly-term-order
66 :documentation "Monomial/term order."))
67 (:default-initargs :dimension nil :termlist nil :order #'lex>)
68 (:documentation "A polynomial with a list of terms TERMLIST, ordered
69according to term order ORDER, which defaults to LEX>."))
70
71(defmethod print-object ((self poly) stream)
72 (print-unreadable-object (self stream :type t :identity t)
73 (with-accessors ((dimension poly-dimension)
74 (termlist poly-termlist)
75 (order poly-term-order))
76 self
77 (format stream "DIMENSION=~A TERMLIST=~A ORDER=~A"
78 dimension termlist order))))
79
80(defgeneric change-term-order (self other)
81 (:documentation "Change term order of SELF to the term order of OTHER.")
82 (:method ((self poly) (other poly))
83 (unless (eq (poly-term-order self) (poly-term-order other))
84 (setf (poly-termlist self) (sort (poly-termlist self) (poly-term-order other))
85 (poly-term-order self) (poly-term-order other)))
86 self))
87
88(defgeneric poly-insert-term (self term)
89 (:documentation "Insert a term TERM into SELF before all other
90 terms. Order is not enforced.")
91 (:method ((self poly) (term term))
92 (cond ((null (poly-dimension self))
93 (setf (poly-dimension self) (monom-dimension term)))
94 (t (assert (= (poly-dimension self) (monom-dimension term)))))
95 (push term (poly-termlist self))
96 self))
97
98(defgeneric poly-append-term (self term)
99 (:documentation "Append a term TERM to SELF after all other terms. Order is not enforced.")
100 (:method ((self poly) (term term))
101 (cond ((null (poly-dimension self))
102 (setf (poly-dimension self) (monom-dimension term)))
103 (t (assert (= (poly-dimension self) (monom-dimension term)))))
104 (setf (cdr (last (poly-termlist self))) (list term))
105 self))
106
107(defun alist->poly (alist &aux (poly (make-instance 'poly)))
108 "It reads polynomial from an alist formatted as ( ... (exponents . coeff) ...).
109It can be used to enter simple polynomials by hand, e.g the polynomial
110in two variables, X and Y, given in standard notation as:
111
112 3*X^2*Y^3+2*Y+7
113
114can be entered as
115(ALIST->POLY '(((2 3) . 3) ((0 1) . 2) ((0 0) . 7))).
116
117NOTE: The primary use is for low-level debugging of the package."
118 (dolist (x alist poly)
119 (poly-insert-term poly (make-instance 'term :exponents (car x) :coeff (cdr x)))))
120
121(defmethod update-instance-for-different-class :after ((old monom) (new poly) &key)
122 "Converts OLD of class MONOM to a NEW of class POLY, by making it into a 1-element TERMLIST."
123 (reinitialize-instance new
124 :dimension (monom-dimension old)
125 :termlist (list (cons old 1))))
126
127(defmethod universal-equalp ((self poly) (other poly))
128 "Implements equality of polynomials."
129 (and (eql (poly-dimension self) (poly-dimension other))
130 (every #'universal-equalp (poly-termlist self) (poly-termlist other))
131 (eq (poly-term-order self) (poly-term-order other))))
132
133(defgeneric leading-term (object)
134 (:method ((self poly))
135 (car (poly-termlist self)))
136 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
137
138(defgeneric second-leading-term (object)
139 (:method ((self poly))
140 (cadar (poly-termlist self)))
141 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
142
143(defgeneric leading-coefficient (object)
144 (:method ((self poly))
145 (term-coeff (leading-term self)))
146 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
147
148
149(defgeneric second-leading-coefficient (object)
150 (:method ((self poly))
151 (term-coeff (second-leading-term self)))
152 (:documentation "The second leading coefficient of a polynomial. It
153 signals error for a polynomial with at most one term."))
154
155(defmethod universal-zerop ((self poly))
156 "Return T iff SELF is a zero polynomial."
157 (null (poly-termlist self)))
158
159(defgeneric poly-length (self)
160 (:documentation "Return the number of terms.")
161 (:method ((self poly))
162 (length (poly-termlist self))))
163
164(defmethod multiply-by ((self poly) (other monom))
165 "Multiply a polynomial SELF by OTHER."
166 (mapc #'(lambda (term) (multiply-by term other))
167 (poly-termlist self))
168 self)
169
170(defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)
171 "Return an expression which will efficiently adds/subtracts two
172polynomials, P and Q. The addition/subtraction of coefficients is
173performed by calling ADD/SUBTRACT-METHOD-NAME. If UMINUS-METHOD-NAME
174is supplied, it is used to negate the coefficients of Q which do not
175have a corresponding coefficient in P. The code implements an
176efficient algorithm to add two polynomials represented as sorted lists
177of terms. The code destroys both arguments, reusing the terms to build
178the result."
179 `(macrolet ((lc (x) `(term-coeff (car ,x))))
180 (do ((p ,p)
181 (q ,q)
182 r)
183 ((or (endp p) (endp q))
184 ;; NOTE: R contains the result in reverse order. Can it
185 ;; be more efficient to produce the terms in correct order?
186 (unless (endp q)
187 ;; Upon subtraction, we must change the sign of
188 ;; all coefficients in q
189 ,@(when uminus-fn
190 `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))
191 (setf r (nreconc r q)))
192 r)
193 (multiple-value-bind
194 (greater-p equal-p)
195 (funcall ,order-fn (car p) (car q))
196 (cond
197 (greater-p
198 (rotatef (cdr p) r p)
199 )
200 (equal-p
201 (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))
202 (cond
203 ((universal-zerop s)
204 (setf p (cdr p))
205 )
206 (t
207 (setf (lc p) s)
208 (rotatef (cdr p) r p))))
209 (setf q (cdr q))
210 )
211 (t
212 ;;Negate the term of Q if UMINUS provided, signallig
213 ;;that we are doing subtraction
214 ,(when uminus-fn
215 `(setf (lc q) (funcall ,uminus-fn (lc q))))
216 (rotatef (cdr q) r q)))))))
217
218
219(defgeneric add-to (self other)
220 (:documentation "Add OTHER to SELF.")
221 (:method ((self number) (other number))
222 (+ self other)))
223
224(defgeneric subtract-from (self other)
225 (:documentation "Subtract OTHER from SELF.")
226 (:method ((self number) (other number))
227 (- self other)))
228
229(defmacro def-add/subtract-method (add/subtract-method-name
230 uminus-method-name
231 &optional
232 (doc-string nil doc-string-supplied-p))
233 "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
234 `(defmethod ,add/subtract-method-name ((self poly) (other poly))
235 ,@(when doc-string-supplied-p `(,doc-string))
236 ;; Ensure orders are compatible
237 (change-term-order other self)
238 (setf (poly-termlist self) (fast-add/subtract
239 (poly-termlist self) (poly-termlist other)
240 (poly-term-order self)
241 #',add/subtract-method-name
242 ,(when uminus-method-name `(function ,uminus-method-name))))
243 self))
244
245(eval-when (:compile-toplevel :load-toplevel :execute)
246
247 (def-add/subtract-method add-to nil
248 "Adds to polynomial SELF another polynomial OTHER.
249This operation destructively modifies both polynomials.
250The result is stored in SELF. This implementation does
251no consing, entirely reusing the sells of SELF and OTHER.")
252
253 (def-add/subtract-method subtract-from unary-minus
254 "Subtracts from polynomial SELF another polynomial OTHER.
255This operation destructively modifies both polynomials.
256The result is stored in SELF. This implementation does
257no consing, entirely reusing the sells of SELF and OTHER.")
258 )
259
260(defmethod unary-minus ((self poly))
261 "Destructively modifies the coefficients of the polynomial SELF,
262by changing their sign."
263 (mapc #'unary-minus (poly-termlist self))
264 self)
265
266(defun add-termlists (p q order-fn)
267 "Destructively adds two termlists P and Q ordered according to ORDER-FN."
268 (fast-add/subtract p q order-fn #'add-to nil))
269
270(defmacro multiply-term-by-termlist-dropping-zeros (term termlist
271 &optional (reverse-arg-order-P nil))
272 "Multiplies term TERM by a list of term, TERMLIST.
273Takes into accound divisors of zero in the ring, by
274deleting zero terms. Optionally, if REVERSE-ARG-ORDER-P
275is T, change the order of arguments; this may be important
276if we extend the package to non-commutative rings."
277 `(mapcan #'(lambda (other-term)
278 (let ((prod (multiply
279 ,@(cond
280 (reverse-arg-order-p
281 `(other-term ,term))
282 (t
283 `(,term other-term))))))
284 (cond
285 ((universal-zerop prod) nil)
286 (t (list prod)))))
287 ,termlist))
288
289(defun multiply-termlists (p q order-fn)
290 "A version of polynomial multiplication, operating
291directly on termlists."
292 (cond
293 ((or (endp p) (endp q))
294 ;;p or q is 0 (represented by NIL)
295 nil)
296 ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
297 ((endp (cdr p))
298 (multiply-term-by-termlist-dropping-zeros (car p) q))
299 ((endp (cdr q))
300 (multiply-term-by-termlist-dropping-zeros (car q) p t))
301 (t
302 (cons (multiply (car p) (car q))
303 (add-termlists
304 (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
305 (multiply-termlists (cdr p) q order-fn)
306 order-fn)))))
307
308(defmethod multiply-by ((self poly) (other poly))
309 (change-term-order other self)
310 (setf (poly-termlist self) (multiply-termlists (poly-termlist self)
311 (poly-termlist other)
312 (poly-term-order self)))
313 self)
314
315(defun add (object1 object2)
316 "Non-destructively add POLY1 by POLY2."
317 (add-to (copy-instance object1) (copy-instance object2)))
318
319(defun subtract (minuend &rest subtrahends)
320 "Non-destructively subtract MINUEND and SUBTRAHENDS."
321 (subtract-from (copy-instance minuend) (reduce #'add subtrahends)))
322
323(defmethod left-tensor-product-by ((self poly) (other monom))
324 (setf (poly-termlist self)
325 (mapcan #'(lambda (term)
326 (let ((prod (left-tensor-product-by term other)))
327 (cond
328 ((universal-zerop prod) nil)
329 (t (list prod)))))
330 (poly-termlist self)))
331 (incf (poly-dimension self) (monom-dimension other))
332 self)
333
334(defmethod right-tensor-product-by ((self poly) (other monom))
335 (setf (poly-termlist self)
336 (mapcan #'(lambda (term)
337 (let ((prod (right-tensor-product-by term other)))
338 (cond
339 ((universal-zerop prod) nil)
340 (t (list prod)))))
341 (poly-termlist self)))
342 (incf (poly-dimension self) (monom-dimension other))
343 self)
344
345
346(defun standard-extension (plist &aux (k (length plist)) (i 0))
347 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
348is a list of polynomials. Destructively modifies PLIST elements."
349 (mapc #'(lambda (poly)
350 (left-tensor-product-by
351 poly
352 (prog1
353 (make-monom-variable k i)
354 (incf i))))
355 plist))
356
357(defun standard-extension-1 (plist
358 &aux
359 (plist (standard-extension plist))
360 (nvars (poly-dimension (car plist))))
361 "Calculate [U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK].
362Firstly, new K variables U1, U2, ..., UK, are inserted into each
363polynomial. Subsequently, P1, P2, ..., PK are destructively modified
364tantamount to replacing PI with UI*PI-1. It assumes that all
365polynomials have the same dimension, and only the first polynomial
366is examined to determine this dimension."
367 ;; Implementation note: we use STANDARD-EXTENSION and then subtract
368 ;; 1 from each polynomial; since UI*PI has no constant term,
369 ;; we just need to append the constant term at the end
370 ;; of each termlist.
371 (flet ((subtract-1 (p)
372 (poly-append-term p (make-instance 'term :dimension nvars :coeff -1))))
373 (setf plist (mapc #'subtract-1 plist)))
374 plist)
375
376
377(defun standard-sum (plist
378 &aux
379 (plist (standard-extension plist))
380 (nvars (poly-dimension (car plist))))
381 "Calculate the polynomial U1*P1+U2*P2+...+UK*PK-1, where PLIST=[P1,P2,...,PK].
382Firstly, new K variables, U1, U2, ..., UK, are inserted into each
383polynomial. Subsequently, P1, P2, ..., PK are destructively modified
384tantamount to replacing PI with UI*PI, and the resulting polynomials
385are added. Finally, 1 is subtracted. It should be noted that the term
386order is not modified, which is equivalent to using a lexicographic
387order on the first K variables."
388 (flet ((subtract-1 (p)
389 (poly-append-term p (make-instance 'term :dimension nvars :coeff -1))))
390 (subtract-1
391 (make-instance
392 'poly
393 :termlist (apply #'nconc (mapcar #'poly-termlist plist))))))
394
395#|
396
397(defun saturation-extension-1 (ring f p)
398 "Calculate [F, U*P-1]. It destructively modifies F."
399 (declare (type ring ring))
400 (polysaturation-extension ring f (list p)))
401
402
403
404
405(defun spoly (ring-and-order f g
406 &aux
407 (ring (ro-ring ring-and-order)))
408 "It yields the S-polynomial of polynomials F and G."
409 (declare (type ring-and-order ring-and-order) (type poly f g))
410 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
411 (mf (monom-div lcm (poly-lm f)))
412 (mg (monom-div lcm (poly-lm g))))
413 (declare (type monom mf mg))
414 (multiple-value-bind (c cf cg)
415 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
416 (declare (ignore c))
417 (poly-sub
418 ring-and-order
419 (scalar-times-poly ring cg (monom-times-poly mf f))
420 (scalar-times-poly ring cf (monom-times-poly mg g))))))
421
422
423(defun poly-primitive-part (ring p)
424 "Divide polynomial P with integer coefficients by gcd of its
425coefficients and return the result."
426 (declare (type ring ring) (type poly p))
427 (if (poly-zerop p)
428 (values p 1)
429 (let ((c (poly-content ring p)))
430 (values (make-poly-from-termlist
431 (mapcar
432 #'(lambda (x)
433 (make-term :monom (term-monom x)
434 :coeff (funcall (ring-div ring) (term-coeff x) c)))
435 (poly-termlist p))
436 (poly-sugar p))
437 c))))
438
439(defun poly-content (ring p)
440 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
441to compute the greatest common divisor."
442 (declare (type ring ring) (type poly p))
443 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
444
445|#
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