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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 "CHANGE-TERM-ORDER"
38 "STANDARD-EXTENSION"
39 "STANDARD-EXTENSION-1"
40 "STANDARD-SUM"
41 "SATURATION-EXTENSION"
42 "ALIST->POLY")
43 (:documentation "Implements polynomials. A polynomial is essentially
44a mapping of monomials of the same degree to coefficients. The
45momomials are ordered according to a monomial order."))
46
47(in-package :polynomial)
48
49(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
50
51(defclass poly ()
52 ((dimension :initform nil
53 :initarg :dimension
54 :accessor poly-dimension
55 :documentation "Shared dimension of all terms, the number of variables")
56 (termlist :initform nil :initarg :termlist :accessor poly-termlist
57 :documentation "List of terms.")
58 (order :initform #'lex> :initarg :order :accessor poly-term-order
59 :documentation "Monomial/term order."))
60 (:default-initargs :dimension nil :termlist nil :order #'lex>)
61 (:documentation "A polynomial with a list of terms TERMLIST, ordered
62according to term order ORDER, which defaults to LEX>."))
63
64(defmethod print-object ((self poly) stream)
65 (print-unreadable-object (self stream :type t :identity t)
66 (with-accessors ((dimension poly-dimension)
67 (termlist poly-termlist)
68 (order poly-term-order))
69 self
70 (format stream "DIMENSION=~A TERMLIST=~A ORDER=~A"
71 dimension termlist order))))
72
73(defgeneric change-term-order (self other)
74 (:documentation "Change term order of SELF to the term order of OTHER.")
75 (:method ((self poly) (other poly))
76 (unless (eq (poly-term-order self) (poly-term-order other))
77 (setf (poly-termlist self) (sort (poly-termlist self) (poly-term-order other))
78 (poly-term-order self) (poly-term-order other)))
79 self))
80
81(defun alist->poly (alist &aux (poly (make-instance 'poly)))
82 "It reads polynomial from an alist formatted as ( ... (exponents . coeff) ...).
83It can be used to enter simple polynomials by hand, e.g the polynomial
84in two variables, X and Y, given in standard notation as:
85
86 3*X^2*Y^3+2*Y+7
87
88can be entered as
89(ALIST->POLY '(((2 3) . 3) ((0 1) . 2) ((0 0) . 7))).
90
91NOTE: The primary use is for low-level debugging of the package."
92 (dolist (x alist poly)
93 (poly-insert-term poly (make-instance 'monom :exponents (car x)) (cdr x))))
94
95
96(defmethod update-instance-for-different-class :after ((old term) (new poly) &key)
97 "Converts OLD of class TERM to a NEW of class POLY, by making it into a 1-element TERMLIST."
98 (reinitialize-instance new
99 :dimension (monom-dimension old)
100 :termlist (list old)))
101
102(defmethod update-instance-for-different-class :after ((old monom) (new poly) &key)
103 "Converts OLD of class MONOM to a NEW of class POLY, by making it into a 1-element TERMLIST."
104 (reinitialize-instance new
105 :dimension (monom-dimension old)
106 :termlist (list (cons monom 1))))
107
108(defmethod r-equalp ((self poly) (other poly))
109 "POLY instances are R-EQUALP if they have the same
110order and if all terms are R-EQUALP."
111 (and (every #'r-equalp (poly-termlist self) (poly-termlist other))
112 (eq (poly-term-order self) (poly-term-order other))))
113
114(defgeneric poly-insert-term (self monom coeff)
115 (:method ((self poly) (monom monom) coeff)
116 (cond ((null (poly-dimension self))
117 (setf (poly-dimension self) (monom-dimension monom)))
118 (t (assert (= (poly-dimension self) (monom-dimension monom)))))
119 (push (cons monom coeff) (poly-termlist self))
120 self))
121
122(defgeneric poly-append-term (self monom coeff)
123 (:method ((self poly) (monom monom) coeff)
124 (cond ((null (poly-dimension self))
125 (setf (poly-dimension self) (monom-dimension monom)))
126 (t (assert (= (poly-dimension self) (monom-dimension monom)))))
127 (setf (cdr (last (poly-termlist self))) (list (cons monom coeff)))
128 self))
129
130;; Leading term
131(defgeneric leading-term (object)
132 (:method ((self poly))
133 (car (poly-termlist self)))
134 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
135
136;; Second term
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;; Leading coefficient
143(defgeneric leading-coefficient (object)
144 (:method ((self poly))
145 (scalar-coeff (leading-term self)))
146 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
147
148;; Second coefficient
149(defgeneric second-leading-coefficient (object)
150 (:method ((self poly))
151 (scalar-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;; Testing for a zero polynomial
156(defmethod r-zerop ((self poly))
157 (null (poly-termlist self)))
158
159;; The number of terms
160(defmethod r-length ((self poly))
161 (length (poly-termlist self)))
162
163(defmethod multiply-by ((self poly) (other monom))
164 (mapc #'(lambda (term) (multiply-by term other))
165 (poly-termlist self))
166 self)
167
168(defmethod multiply-by ((self poly) (other term))
169 (mapc #'(lambda (term) (multiply-by term other))
170 (poly-termlist self))
171 self)
172
173(defmethod multiply-by ((self poly) (other scalar))
174 (mapc #'(lambda (term) (multiply-by term other))
175 (poly-termlist self))
176 self)
177
178
179(defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)
180 "Return an expression which will efficiently adds/subtracts two
181polynomials, P and Q. The addition/subtraction of coefficients is
182performed by calling ADD/SUBTRACT-METHOD-NAME. If UMINUS-METHOD-NAME
183is supplied, it is used to negate the coefficients of Q which do not
184have a corresponding coefficient in P. The code implements an
185efficient algorithm to add two polynomials represented as sorted lists
186of terms. The code destroys both arguments, reusing the terms to build
187the result."
188 `(macrolet ((lc (x) `(scalar-coeff (car ,x))))
189 (do ((p ,p)
190 (q ,q)
191 r)
192 ((or (endp p) (endp q))
193 ;; NOTE: R contains the result in reverse order. Can it
194 ;; be more efficient to produce the terms in correct order?
195 (unless (endp q)
196 ;; Upon subtraction, we must change the sign of
197 ;; all coefficients in q
198 ,@(when uminus-fn
199 `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))
200 (setf r (nreconc r q)))
201 r)
202 (multiple-value-bind
203 (greater-p equal-p)
204 (funcall ,order-fn (car p) (car q))
205 (cond
206 (greater-p
207 (rotatef (cdr p) r p)
208 )
209 (equal-p
210 (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))
211 (cond
212 ((r-zerop s)
213 (setf p (cdr p))
214 )
215 (t
216 (setf (lc p) s)
217 (rotatef (cdr p) r p))))
218 (setf q (cdr q))
219 )
220 (t
221 ;;Negate the term of Q if UMINUS provided, signallig
222 ;;that we are doing subtraction
223 ,(when uminus-fn
224 `(setf (lc q) (funcall ,uminus-fn (lc q))))
225 (rotatef (cdr q) r q)))))))
226
227
228(defmacro def-add/subtract-method (add/subtract-method-name
229 uminus-method-name
230 &optional
231 (doc-string nil doc-string-supplied-p))
232 "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
233 `(defmethod ,add/subtract-method-name ((self poly) (other poly))
234 ,@(when doc-string-supplied-p `(,doc-string))
235 ;; Ensure orders are compatible
236 (change-term-order other self)
237 (setf (poly-termlist self) (fast-add/subtract
238 (poly-termlist self) (poly-termlist other)
239 (poly-term-order self)
240 #',add/subtract-method-name
241 ,(when uminus-method-name `(function ,uminus-method-name))))
242 self))
243
244(eval-when (:compile-toplevel :load-toplevel :execute)
245
246 (def-add/subtract-method add-to nil
247 "Adds to polynomial SELF another polynomial OTHER.
248This operation destructively modifies both polynomials.
249The result is stored in SELF. This implementation does
250no consing, entirely reusing the sells of SELF and OTHER.")
251
252 (def-add/subtract-method subtract-from unary-minus
253 "Subtracts from polynomial SELF another polynomial OTHER.
254This operation destructively modifies both polynomials.
255The result is stored in SELF. This implementation does
256no consing, entirely reusing the sells of SELF and OTHER.")
257 )
258
259(defmethod unary-minus ((self poly))
260 "Destructively modifies the coefficients of the polynomial SELF,
261by changing their sign."
262 (mapc #'unary-minus (poly-termlist self))
263 self)
264
265(defun add-termlists (p q order-fn)
266 "Destructively adds two termlists P and Q ordered according to ORDER-FN."
267 (fast-add/subtract p q order-fn #'add-to nil))
268
269(defmacro multiply-term-by-termlist-dropping-zeros (term termlist
270 &optional (reverse-arg-order-P nil))
271 "Multiplies term TERM by a list of term, TERMLIST.
272Takes into accound divisors of zero in the ring, by
273deleting zero terms. Optionally, if REVERSE-ARG-ORDER-P
274is T, change the order of arguments; this may be important
275if we extend the package to non-commutative rings."
276 `(mapcan #'(lambda (other-term)
277 (let ((prod (r*
278 ,@(cond
279 (reverse-arg-order-p
280 `(other-term ,term))
281 (t
282 `(,term other-term))))))
283 (cond
284 ((r-zerop prod) nil)
285 (t (list prod)))))
286 ,termlist))
287
288(defun multiply-termlists (p q order-fn)
289 "A version of polynomial multiplication, operating
290directly on termlists."
291 (cond
292 ((or (endp p) (endp q))
293 ;;p or q is 0 (represented by NIL)
294 nil)
295 ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
296 ((endp (cdr p))
297 (multiply-term-by-termlist-dropping-zeros (car p) q))
298 ((endp (cdr q))
299 (multiply-term-by-termlist-dropping-zeros (car q) p t))
300 (t
301 (cons (r* (car p) (car q))
302 (add-termlists
303 (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
304 (multiply-termlists (cdr p) q order-fn)
305 order-fn)))))
306
307(defmethod multiply-by ((self poly) (other poly))
308 (change-term-order other self)
309 (setf (poly-termlist self) (multiply-termlists (poly-termlist self)
310 (poly-termlist other)
311 (poly-term-order self)))
312 self)
313
314(defmethod r+ ((poly1 poly) poly2)
315 "Non-destructively add POLY1 by POLY2."
316 (add-to (copy-instance POLY1) (change-class (copy-instance POLY2) 'poly)))
317
318(defmethod r- ((minuend poly) &rest subtrahends)
319 "Non-destructively subtract MINUEND and SUBTRAHENDS."
320 (subtract-from (copy-instance minuend)
321 (change-class (reduce #'r+ subtrahends) 'poly)))
322
323(defmethod r+ ((poly1 term) poly2)
324 "Non-destructively add POLY1 by POLY2."
325 (add-to (change-class (copy-instance poly1) 'poly)
326 (change-class (copy-instance poly2) 'poly)))
327
328(defmethod r- ((minuend term) &rest subtrahends)
329 "Non-destructively subtract MINUEND and SUBTRAHENDS."
330 (subtract-from (change-class (copy-instance minuend) 'poly)
331 (change-class (reduce #'r+ subtrahends) 'poly)))
332
333(defmethod r+ ((poly1 monom) poly2)
334 "Non-destructively add POLY1 by POLY2."
335 (add-to (change-class (copy-instance poly1) 'poly)
336 (change-class (copy-instance poly2) 'poly)))
337
338(defmethod r- ((minuend monom) &rest subtrahends)
339 "Non-destructively subtract MINUEND and SUBTRAHENDS."
340 (subtract-from (change-class (copy-instance minuend) 'poly)
341 (change-class (reduce #'r+ subtrahends) 'poly)))
342
343(defmethod r* ((poly1 poly) (poly2 poly))
344 "Non-destructively multiply POLY1 by POLY2."
345 (multiply-by (copy-instance poly1) (copy-instance poly2)))
346
347(defmethod left-tensor-product-by ((self poly) (other term))
348 (setf (poly-termlist self)
349 (mapcan #'(lambda (term)
350 (let ((prod (left-tensor-product-by term other)))
351 (cond
352 ((r-zerop prod) nil)
353 (t (list prod)))))
354 (poly-termlist self)))
355 self)
356
357(defmethod right-tensor-product-by ((self poly) (other term))
358 (setf (poly-termlist self)
359 (mapcan #'(lambda (term)
360 (let ((prod (right-tensor-product-by term other)))
361 (cond
362 ((r-zerop prod) nil)
363 (t (list prod)))))
364 (poly-termlist self)))
365 self)
366
367(defmethod left-tensor-product-by ((self poly) (other monom))
368 (setf (poly-termlist self)
369 (mapcan #'(lambda (term)
370 (let ((prod (left-tensor-product-by term other)))
371 (cond
372 ((r-zerop prod) nil)
373 (t (list prod)))))
374 (poly-termlist self)))
375 (incf (poly-dimension self) (monom-dimension other))
376 self)
377
378(defmethod right-tensor-product-by ((self poly) (other monom))
379 (setf (poly-termlist self)
380 (mapcan #'(lambda (term)
381 (let ((prod (right-tensor-product-by term other)))
382 (cond
383 ((r-zerop prod) nil)
384 (t (list prod)))))
385 (poly-termlist self)))
386 (incf (poly-dimension self) (monom-dimension other))
387 self)
388
389
390(defun standard-extension (plist &aux (k (length plist)) (i 0))
391 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
392is a list of polynomials. Destructively modifies PLIST elements."
393 (mapc #'(lambda (poly)
394 (left-tensor-product-by
395 poly
396 (prog1
397 (make-monom-variable k i)
398 (incf i))))
399 plist))
400
401(defun standard-extension-1 (plist
402 &aux
403 (plist (standard-extension plist))
404 (nvars (poly-dimension (car plist))))
405 "Calculate [U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK].
406Firstly, new K variables U1, U2, ..., UK, are inserted into each
407polynomial. Subsequently, P1, P2, ..., PK are destructively modified
408tantamount to replacing PI with UI*PI-1. It assumes that all
409polynomials have the same dimension, and only the first polynomial
410is examined to determine this dimension."
411 ;; Implementation note: we use STANDARD-EXTENSION and then subtract
412 ;; 1 from each polynomial; since UI*PI has no constant term,
413 ;; we just need to append the constant term at the end
414 ;; of each termlist.
415 (flet ((subtract-1 (p)
416 (poly-append-term p (make-instance 'monom :dimension nvars) -1)))
417 (setf plist (mapc #'subtract-1 plist)))
418 plist)
419
420
421(defun standard-sum (plist
422 &aux
423 (plist (standard-extension plist))
424 (nvars (poly-dimension (car plist))))
425 "Calculate the polynomial U1*P1+U2*P2+...+UK*PK-1, where PLIST=[P1,P2,...,PK].
426Firstly, new K variables, U1, U2, ..., UK, are inserted into each
427polynomial. Subsequently, P1, P2, ..., PK are destructively modified
428tantamount to replacing PI with UI*PI, and the resulting polynomials
429are added. Finally, 1 is subtracted. It should be noted that the term
430order is not modified, which is equivalent to using a lexicographic
431order on the first K variables."
432 (flet ((subtract-1 (p)
433 (append-item p (make-instance 'term :coeff -1 :dimension nvars))))
434 (subtract-1
435 (make-instance
436 'poly
437 :termlist (apply #'nconc (mapcar #'poly-termlist plist))))))
438
439#|
440
441(defun saturation-extension-1 (ring f p)
442 "Calculate [F, U*P-1]. It destructively modifies F."
443 (declare (type ring ring))
444 (polysaturation-extension ring f (list p)))
445
446
447
448
449(defun spoly (ring-and-order f g
450 &aux
451 (ring (ro-ring ring-and-order)))
452 "It yields the S-polynomial of polynomials F and G."
453 (declare (type ring-and-order ring-and-order) (type poly f g))
454 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
455 (mf (monom-div lcm (poly-lm f)))
456 (mg (monom-div lcm (poly-lm g))))
457 (declare (type monom mf mg))
458 (multiple-value-bind (c cf cg)
459 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
460 (declare (ignore c))
461 (poly-sub
462 ring-and-order
463 (scalar-times-poly ring cg (monom-times-poly mf f))
464 (scalar-times-poly ring cf (monom-times-poly mg g))))))
465
466
467(defun poly-primitive-part (ring p)
468 "Divide polynomial P with integer coefficients by gcd of its
469coefficients and return the result."
470 (declare (type ring ring) (type poly p))
471 (if (poly-zerop p)
472 (values p 1)
473 (let ((c (poly-content ring p)))
474 (values (make-poly-from-termlist
475 (mapcar
476 #'(lambda (x)
477 (make-term :monom (term-monom x)
478 :coeff (funcall (ring-div ring) (term-coeff x) c)))
479 (poly-termlist p))
480 (poly-sugar p))
481 c))))
482
483(defun poly-content (ring p)
484 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
485to compute the greatest common divisor."
486 (declare (type ring ring) (type poly p))
487 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
488
489|#
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