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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-ADD-TO"
39 "CHANGE-TERM-ORDER"
40 "STANDARD-EXTENSION"
41 "STANDARD-EXTENSION-1"
42 "STANDARD-SUM"
43 "SATURATION-EXTENSION"
44 "ALIST->POLY")
45 (:documentation "Implements polynomials. A polynomial is essentially
46a mapping of monomials of the same degree to coefficients. The
47momomials are ordered according to a monomial order."))
48
49(in-package :polynomial)
50
51(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
52
53(defclass poly ()
54 ((dimension :initform nil
55 :initarg :dimension
56 :accessor poly-dimension
57 :documentation "Shared dimension of all terms, the number of variables")
58 (termlist :initform nil :initarg :termlist :accessor poly-termlist
59 :documentation "List of terms. This is an association
60list mapping monomials to coefficients, ordered by this polynomial's
61monomial order.")
62 (order :initform #'lex> :initarg :order :accessor poly-term-order
63 :documentation "Monomial/term order."))
64 (:default-initargs :dimension nil :termlist nil :order #'lex>)
65 (:documentation "A polynomial with a list of terms TERMLIST, ordered
66according to term order ORDER, which defaults to LEX>."))
67
68(defmethod print-object ((self poly) stream)
69 (print-unreadable-object (self stream :type t :identity t)
70 (with-accessors ((dimension poly-dimension)
71 (termlist poly-termlist)
72 (order poly-term-order))
73 self
74 (format stream "DIMENSION=~A TERMLIST=~A ORDER=~A"
75 dimension termlist order))))
76
77(defgeneric change-term-order (self other)
78 (:documentation "Change term order of SELF to the term order of OTHER.")
79 (:method ((self poly) (other poly))
80 (unless (eq (poly-term-order self) (poly-term-order other))
81 (setf (poly-termlist self) (sort (poly-termlist self) (poly-term-order other) :key #'car)
82 (poly-term-order self) (poly-term-order other)))
83 self))
84
85(defgeneric poly-insert-term (self monom coeff)
86 (:documentation "Insert a term with monomial MONOM and coefficient COEFF
87before all other terms. Order is not enforced.")
88 (:method ((self poly) (monom monom) coeff)
89 (cond ((null (poly-dimension self))
90 (setf (poly-dimension self) (monom-dimension monom)))
91 (t (assert (= (poly-dimension self) (monom-dimension monom)))))
92 (push (cons monom coeff) (poly-termlist self))
93 self))
94
95(defgeneric poly-append-term (self monom coeff)
96 (:documentation "Append a term with monomial MONOM and coefficient COEFF
97after all other terms. Order is not enforced.")
98 (:method ((self poly) (monom monom) coeff)
99 (cond ((null (poly-dimension self))
100 (setf (poly-dimension self) (monom-dimension monom)))
101 (t (assert (= (poly-dimension self) (monom-dimension monom)))))
102 (setf (cdr (last (poly-termlist self))) (list (cons monom coeff)))
103 self))
104
105(defun alist->poly (alist &aux (poly (make-instance 'poly)))
106 "It reads polynomial from an alist formatted as ( ... (exponents . coeff) ...).
107It can be used to enter simple polynomials by hand, e.g the polynomial
108in two variables, X and Y, given in standard notation as:
109
110 3*X^2*Y^3+2*Y+7
111
112can be entered as
113(ALIST->POLY '(((2 3) . 3) ((0 1) . 2) ((0 0) . 7))).
114
115NOTE: The primary use is for low-level debugging of the package."
116 (dolist (x alist poly)
117 (poly-insert-term poly (make-instance 'monom :exponents (car x)) (cdr x))))
118
119(defmethod update-instance-for-different-class :after ((old monom) (new poly) &key)
120 "Converts OLD of class MONOM to a NEW of class POLY, by making it into a 1-element TERMLIST."
121 (reinitialize-instance new
122 :dimension (monom-dimension old)
123 :termlist (list (cons old 1))))
124
125(defgeneric poly-equalp (self other)
126 (:documentation "Implements equality of polynomials.")
127 (:method ((self poly) (other poly))
128 (and (eql (poly-dimension self) (poly-dimension other))
129 (every #'r-equalp (poly-termlist self) (poly-termlist other))
130 (eq (poly-term-order self) (poly-term-order other)))))
131
132;; Leading term
133(defgeneric poly-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;; Second term
139(defgeneric poly-second-leading-term (object)
140 (:method ((self poly))
141 (cadar (poly-termlist self)))
142 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
143
144;; Leading coefficient
145(defgeneric poly-leading-coefficient (object)
146 (:method ((self poly))
147 (cdr (poly-leading-term self)))
148 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
149
150;; Second coefficient
151(defgeneric second-leading-coefficient (object)
152 (:method ((self poly))
153 (cdr (second-leading-term self)))
154 (:documentation "The second leading coefficient of a polynomial. It
155 signals error for a polynomial with at most one term."))
156
157;; Testing for a zero polynomial
158(defgeneric poly-zerop (self)
159 (:method ((self poly))
160 (null (poly-termlist self))))
161
162;; The number of terms
163(defgeneric poly-length (self)
164 (:method ((self poly))
165 (length (poly-termlist self))))
166
167(defgeneric poly-multiply-by (self other)
168 (:documentation "Multiply a polynomial SELF by OTHER.")
169 (:method ((self poly) (other monom))
170 "Multiply a polynomial SELF by monomial OTHER"
171 (mapc #'(lambda (term) (cons (monom-multiply-by (car term) other) (cdr other)))
172 (poly-termlist self))
173 self))
174
175(defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)
176 "Return an expression which will efficiently adds/subtracts two
177polynomials, P and Q. The addition/subtraction of coefficients is
178performed by calling ADD/SUBTRACT-METHOD-NAME. If UMINUS-METHOD-NAME
179is supplied, it is used to negate the coefficients of Q which do not
180have a corresponding coefficient in P. The code implements an
181efficient algorithm to add two polynomials represented as sorted lists
182of terms. The code destroys both arguments, reusing the terms to build
183the result."
184 `(macrolet ((lc (x) `(caar ,x)))
185 (do ((p ,p)
186 (q ,q)
187 r)
188 ((or (endp p) (endp q))
189 ;; NOTE: R contains the result in reverse order. Can it
190 ;; be more efficient to produce the terms in correct order?
191 (unless (endp q)
192 ;; Upon subtraction, we must change the sign of
193 ;; all coefficients in q
194 ,@(when uminus-fn
195 `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))
196 (setf r (nreconc r q)))
197 r)
198 (multiple-value-bind
199 (greater-p equal-p)
200 (funcall ,order-fn (caar p) (caar q))
201 (cond
202 (greater-p
203 (rotatef (cdr p) r p)
204 )
205 (equal-p
206 (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))
207 (cond
208 ((r-zerop s)
209 (setf p (cdr p))
210 )
211 (t
212 (setf (lc p) s)
213 (rotatef (cdr p) r p))))
214 (setf q (cdr q))
215 )
216 (t
217 ;;Negate the term of Q if UMINUS provided, signallig
218 ;;that we are doing subtraction
219 ,(when uminus-fn
220 `(setf (lc q) (funcall ,uminus-fn (lc q))))
221 (rotatef (cdr q) r q)))))))
222
223
224(defmacro def-add/subtract-method (add/subtract-method-name
225 uminus-method-name
226 &optional
227 (doc-string nil doc-string-supplied-p))
228 "This macro avoids code duplication for two similar operations: POLY-ADD-TO and POLY-SUBTRACT-FROM."
229 `(defmethod ,add/subtract-method-name ((self poly) (other poly))
230 ,@(when doc-string-supplied-p `(,doc-string))
231 ;; Ensure orders are compatible
232 (change-term-order other self)
233 (setf (poly-termlist self) (fast-add/subtract
234 (poly-termlist self) (poly-termlist other)
235 (poly-term-order self)
236 #',add/subtract-method-name
237 ,(when uminus-method-name `(function ,uminus-method-name))))
238 self))
239
240(eval-when (:compile-toplevel :load-toplevel :execute)
241
242 (def-add/subtract-method poly-add-to nil
243 "Adds to 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 (def-add/subtract-method poly-subtract-from unary-minus
249 "Subtracts from polynomial SELF another polynomial OTHER.
250This operation destructively modifies both polynomials.
251The result is stored in SELF. This implementation does
252no consing, entirely reusing the sells of SELF and OTHER.")
253 )
254
255(defmethod unary-minus ((self poly))
256 "Destructively modifies the coefficients of the polynomial SELF,
257by changing their sign."
258 (mapc #'unary-minus (poly-termlist self))
259 self)
260
261(defun add-termlists (p q order-fn)
262 "Destructively adds two termlists P and Q ordered according to ORDER-FN."
263 (fast-add/subtract p q order-fn #'poly-add-to nil))
264
265(defmacro multiply-term-by-termlist-dropping-zeros (term termlist
266 &optional (reverse-arg-order-P nil))
267 "Multiplies term TERM by a list of term, TERMLIST.
268Takes into accound divisors of zero in the ring, by
269deleting zero terms. Optionally, if REVERSE-ARG-ORDER-P
270is T, change the order of arguments; this may be important
271if we extend the package to non-commutative rings."
272 `(mapcan #'(lambda (other-term)
273 (let ((prod (r*
274 ,@(cond
275 (reverse-arg-order-p
276 `(other-term ,term))
277 (t
278 `(,term other-term))))))
279 (cond
280 ((r-zerop prod) nil)
281 (t (list prod)))))
282 ,termlist))
283
284(defun multiply-termlists (p q order-fn)
285 "A version of polynomial multiplication, operating
286directly on termlists."
287 (cond
288 ((or (endp p) (endp q))
289 ;;p or q is 0 (represented by NIL)
290 nil)
291 ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
292 ((endp (cdr p))
293 (multiply-term-by-termlist-dropping-zeros (car p) q))
294 ((endp (cdr q))
295 (multiply-term-by-termlist-dropping-zeros (car q) p t))
296 (t
297 (cons (r* (car p) (car q))
298 (add-termlists
299 (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
300 (multiply-termlists (cdr p) q order-fn)
301 order-fn)))))
302
303(defmethod multiply-by ((self poly) (other poly))
304 (change-term-order other self)
305 (setf (poly-termlist self) (multiply-termlists (poly-termlist self)
306 (poly-termlist other)
307 (poly-term-order self)))
308 self)
309
310(defmethod r+ ((poly1 poly) poly2)
311 "Non-destructively add POLY1 by POLY2."
312 (poly-add-to (copy-instance POLY1) (change-class (copy-instance POLY2) 'poly)))
313
314(defmethod r- ((minuend poly) &rest subtrahends)
315 "Non-destructively subtract MINUEND and SUBTRAHENDS."
316 (poly-subtract-from (copy-instance minuend)
317 (change-class (reduce #'r+ subtrahends) 'poly)))
318
319(defmethod r+ ((poly1 monom) poly2)
320 "Non-destructively add POLY1 by POLY2."
321 (poly-add-to (change-class (copy-instance poly1) 'poly)
322 (change-class (copy-instance poly2) 'poly)))
323
324(defmethod r- ((minuend monom) &rest subtrahends)
325 "Non-destructively subtract MINUEND and SUBTRAHENDS."
326 (poly-subtract-from (change-class (copy-instance minuend) 'poly)
327 (change-class (reduce #'r+ subtrahends) 'poly)))
328
329(defmethod r* ((poly1 poly) (poly2 poly))
330 "Non-destructively multiply POLY1 by POLY2."
331 (multiply-by (copy-instance poly1) (copy-instance poly2)))
332
333(defmethod left-tensor-product-by ((self poly) (other monom))
334 (setf (poly-termlist self)
335 (mapcan #'(lambda (term)
336 (let ((prod (left-tensor-product-by term other)))
337 (cond
338 ((r-zerop prod) nil)
339 (t (list prod)))))
340 (poly-termlist self)))
341 (incf (poly-dimension self) (monom-dimension other))
342 self)
343
344(defmethod right-tensor-product-by ((self poly) (other monom))
345 (setf (poly-termlist self)
346 (mapcan #'(lambda (term)
347 (let ((prod (right-tensor-product-by term other)))
348 (cond
349 ((r-zerop prod) nil)
350 (t (list prod)))))
351 (poly-termlist self)))
352 (incf (poly-dimension self) (monom-dimension other))
353 self)
354
355
356(defun standard-extension (plist &aux (k (length plist)) (i 0))
357 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
358is a list of polynomials. Destructively modifies PLIST elements."
359 (mapc #'(lambda (poly)
360 (left-tensor-product-by
361 poly
362 (prog1
363 (make-monom-variable k i)
364 (incf i))))
365 plist))
366
367(defun standard-extension-1 (plist
368 &aux
369 (plist (standard-extension plist))
370 (nvars (poly-dimension (car plist))))
371 "Calculate [U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK].
372Firstly, new K variables U1, U2, ..., UK, are inserted into each
373polynomial. Subsequently, P1, P2, ..., PK are destructively modified
374tantamount to replacing PI with UI*PI-1. It assumes that all
375polynomials have the same dimension, and only the first polynomial
376is examined to determine this dimension."
377 ;; Implementation note: we use STANDARD-EXTENSION and then subtract
378 ;; 1 from each polynomial; since UI*PI has no constant term,
379 ;; we just need to append the constant term at the end
380 ;; of each termlist.
381 (flet ((subtract-1 (p)
382 (poly-append-term p (make-instance 'monom :dimension nvars) -1)))
383 (setf plist (mapc #'subtract-1 plist)))
384 plist)
385
386
387(defun standard-sum (plist
388 &aux
389 (plist (standard-extension plist))
390 (nvars (poly-dimension (car plist))))
391 "Calculate the polynomial U1*P1+U2*P2+...+UK*PK-1, where PLIST=[P1,P2,...,PK].
392Firstly, new K variables, U1, U2, ..., UK, are inserted into each
393polynomial. Subsequently, P1, P2, ..., PK are destructively modified
394tantamount to replacing PI with UI*PI, and the resulting polynomials
395are added. Finally, 1 is subtracted. It should be noted that the term
396order is not modified, which is equivalent to using a lexicographic
397order on the first K variables."
398 (flet ((subtract-1 (p)
399 (poly-append-term p (make-instance 'monom :dimension nvars) -1)))
400 (subtract-1
401 (make-instance
402 'poly
403 :termlist (apply #'nconc (mapcar #'poly-termlist plist))))))
404
405#|
406
407(defun saturation-extension-1 (ring f p)
408 "Calculate [F, U*P-1]. It destructively modifies F."
409 (declare (type ring ring))
410 (polysaturation-extension ring f (list p)))
411
412
413
414
415(defun spoly (ring-and-order f g
416 &aux
417 (ring (ro-ring ring-and-order)))
418 "It yields the S-polynomial of polynomials F and G."
419 (declare (type ring-and-order ring-and-order) (type poly f g))
420 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
421 (mf (monom-div lcm (poly-lm f)))
422 (mg (monom-div lcm (poly-lm g))))
423 (declare (type monom mf mg))
424 (multiple-value-bind (c cf cg)
425 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
426 (declare (ignore c))
427 (poly-sub
428 ring-and-order
429 (scalar-times-poly ring cg (monom-times-poly mf f))
430 (scalar-times-poly ring cf (monom-times-poly mg g))))))
431
432
433(defun poly-primitive-part (ring p)
434 "Divide polynomial P with integer coefficients by gcd of its
435coefficients and return the result."
436 (declare (type ring ring) (type poly p))
437 (if (poly-zerop p)
438 (values p 1)
439 (let ((c (poly-content ring p)))
440 (values (make-poly-from-termlist
441 (mapcar
442 #'(lambda (x)
443 (make-term :monom (term-monom x)
444 :coeff (funcall (ring-div ring) (term-coeff x) c)))
445 (poly-termlist p))
446 (poly-sugar p))
447 c))))
448
449(defun poly-content (ring p)
450 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
451to compute the greatest common divisor."
452 (declare (type ring ring) (type poly p))
453 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
454
455|#
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