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