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