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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 :ring :monom :order :term #| :infix |# )
24 (:export "POLY"
25 "POLY-TERMLIST"
26 "POLY-TERM-ORDER")
27 (:documentation "Implements polynomials"))
28
29(in-package :polynomial)
30
31(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
32
33(defclass poly ()
34 ((termlist :initarg :termlist :accessor poly-termlist
35 :documentation "List of terms.")
36 (order :initarg :order :accessor poly-term-order
37 :documentation "Monomial/term order."))
38 (:default-initargs :termlist nil :order #'lex>)
39 (:documentation "A polynomial with a list of terms TERMLIST, ordered
40according to term order ORDER, which defaults to LEX>."))
41
42(defmethod print-object ((self poly) stream)
43 (format stream "#<POLY TERMLIST=~A ORDER=~A>"
44 (poly-termlist self)
45 (poly-term-order self)))
46
47(defmethod r-equalp ((self poly) (other poly))
48 "POLY instances are R-EQUALP if they have the same
49order and if all terms are R-EQUALP."
50 (and (every #'r-equalp (poly-termlist self) (poly-termlist other))
51 (eq (poly-term-order self) (poly-term-order other))))
52
53(defmethod insert-item ((self poly) (item term))
54 (push item (poly-termlist self))
55 self)
56
57(defmethod append-item ((self poly) (item term))
58 (setf (cdr (last (poly-termlist self))) (list item))
59 self)
60
61;; Leading term
62(defgeneric leading-term (object)
63 (:method ((self poly))
64 (car (poly-termlist self)))
65 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
66
67;; Second term
68(defgeneric second-leading-term (object)
69 (:method ((self poly))
70 (cadar (poly-termlist self)))
71 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
72
73;; Leading coefficient
74(defgeneric leading-coefficient (object)
75 (:method ((self poly))
76 (r-coeff (leading-term self)))
77 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
78
79;; Second coefficient
80(defgeneric second-leading-coefficient (object)
81 (:method ((self poly))
82 (r-coeff (second-leading-term self)))
83 (:documentation "The second leading coefficient of a polynomial. It signals error for a polynomial with at most one term."))
84
85;; Testing for a zero polynomial
86(defmethod r-zerop ((self poly))
87 (null (poly-termlist self)))
88
89;; The number of terms
90(defmethod r-length ((self poly))
91 (length (poly-termlist self)))
92
93(defmethod multiply-by ((self poly) (other monom))
94 (mapc #'(lambda (term) (multiply-by term other))
95 (poly-termlist self))
96 self)
97
98(defmethod multiply-by ((self poly) (other scalar))
99 (mapc #'(lambda (term) (multiply-by term other))
100 (poly-termlist self))
101 self)
102
103
104(defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)
105 "Return an expression which will efficiently adds/subtracts two
106polynomials, P and Q. The addition/subtraction of coefficients is
107performed by calling ADD/SUBTRACT-METHOD-NAME. If UMINUS-METHOD-NAME
108is supplied, it is used to negate the coefficients of Q which do not
109have a corresponding coefficient in P. The code implements an
110efficient algorithm to add two polynomials represented as sorted lists
111of terms. The code destroys both arguments, reusing the terms to build
112the result."
113 `(macrolet ((lc (x) `(r-coeff (car ,x))))
114 (do ((p ,p)
115 (q ,q)
116 r)
117 ((or (endp p) (endp q))
118 ;; NOTE: R contains the result in reverse order. Can it
119 ;; be more efficient to produce the terms in correct order?
120 (unless (endp q)
121 ;; Upon subtraction, we must change the sign of
122 ;; all coefficients in q
123 ,@(when uminus-fn
124 `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))
125 (setf r (nreconc r q)))
126 r)
127 (multiple-value-bind
128 (greater-p equal-p)
129 (funcall ,order-fn (car p) (car q))
130 (cond
131 (greater-p
132 (rotatef (cdr p) r p)
133 )
134 (equal-p
135 (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))
136 (cond
137 ((r-zerop s)
138 (setf p (cdr p))
139 )
140 (t
141 (setf (lc p) s)
142 (rotatef (cdr p) r p))))
143 (setf q (cdr q))
144 )
145 (t
146 ;;Negate the term of Q if UMINUS provided, signallig
147 ;;that we are doing subtraction
148 ,@(when uminus-fn
149 `((setf (lc q) (funcall ,uminus-fn (lc q)))))
150 (rotatef (cdr q) r q)))))))
151
152
153(defmacro def-add/subtract-method (add/subtract-method-name
154 uminus-method-name
155 &optional
156 (doc-string nil doc-string-supplied-p))
157 "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
158 `(defmethod ,add/subtract-method-name ((self poly) (other poly))
159 ,@(when doc-string-supplied-p `(,doc-string))
160 ;; Ensure orders are compatible
161 (unless (eq (poly-term-order self) (poly-term-order other))
162 (setf (poly-termlist other) (sort (poly-termlist other) (poly-term-order self))
163 (poly-term-order other) (poly-term-order self)))
164 (setf (poly-termlist self) (fast-add/subtract
165 (poly-termlist self) (poly-termlist other)
166 (poly-term-order self)
167 #',add/subtract-method-name
168 ,(when uminus-method-name `(function ,uminus-method-name))))
169 self))
170
171(eval-when (:compile-toplevel :load-toplevel :execute)
172
173 (def-add/subtract-method add-to nil
174 "Adds to polynomial SELF another polynomial OTHER.
175This operation destructively modifies both polynomials.
176The result is stored in SELF. This implementation does
177no consing, entirely reusing the sells of SELF and OTHER.")
178
179 (def-add/subtract-method subtract-from unary-minus
180 "Subtracts from polynomial SELF another polynomial OTHER.
181This operation destructively modifies both polynomials.
182The result is stored in SELF. This implementation does
183no consing, entirely reusing the sells of SELF and OTHER.")
184
185)
186
187(defmethod unary-minus ((self poly))
188 "Destructively modifies the coefficients of the polynomial SELF,
189by changing their sign."
190 (mapc #'unary-minus (poly-termlist self))
191 self)
192
193
194(defun fast-add-termlists (p q order-fn)
195 "Destructively adds two termlists P and Q ordered according to ORDER-FN."
196 (fast-add/subtract p q order-fn add-to nil))
197
198(defun multiply-term-by-termlist-dropping-zeros (term termlist)
199 "A product of a term TERM by a list of term, TERMLIST.
200Takes into accound divisors of zero in the ring, by
201deleting zero terms."
202 (mapcan #'(lambda (other-term)
203 (let ((prod (term-multiply term other-term)))
204 (cond
205 ((r-zerop prod) nil)
206 (t (list prod)))))
207 termlist))
208
209(defun multiply-termlist-by-term-dropping-zeros (termlist term)
210 "A product of a termlist TERMLIST by a term TERM>
211Takes into accound divisors of zero in the ring, by
212deleting zero terms."
213 (mapcan #'(lambda (other-term)
214 (let ((prod (term-multiply other-term term)))
215 (cond
216 ((r-zerop prod) nil)
217 (t (list prod)))))
218 termlist))
219
220
221(defun multiply-termlists (p q)
222 (cond
223 ((or (endp p) (endp q)) nil) ;p or q is 0 (represented by NIL)
224 ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
225 ((endp (cdr p))
226 (multiply-term-by-termlist (car p) q)
227 ((endp (cdr q))
228 (multiply-termlist-by-term p (car q))
229 (t
230 (let ((head (multiply-terms (car p) (car q)))
231 (tail (add-termlists
232 (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
233 (multiply-termlists (cdr p) q))))
234
235 (nconc head tail)))))))
236
237#|
238
239(defun poly-standard-extension (plist &aux (k (length plist)))
240 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
241is a list of polynomials."
242 (declare (list plist) (fixnum k))
243 (labels ((incf-power (g i)
244 (dolist (x (poly-termlist g))
245 (incf (monom-elt (term-monom x) i)))
246 (incf (poly-sugar g))))
247 (setf plist (poly-list-add-variables plist k))
248 (dotimes (i k plist)
249 (incf-power (nth i plist) i))))
250
251
252
253(defun saturation-extension (ring f plist
254 &aux
255 (k (length plist))
256 (d (monom-dimension (poly-lm (car plist))))
257 f-x plist-x)
258 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
259 (declare (type ring ring))
260 (setf f-x (poly-list-add-variables f k)
261 plist-x (mapcar #'(lambda (x)
262 (setf (poly-termlist x)
263 (nconc (poly-termlist x)
264 (list (make-term :monom (make-monom :dimension d)
265 :coeff (funcall (ring-uminus ring)
266 (funcall (ring-unit ring)))))))
267 x)
268 (poly-standard-extension plist)))
269 (append f-x plist-x))
270
271
272(defun polysaturation-extension (ring f plist
273 &aux
274 (k (length plist))
275 (d (+ k (monom-dimension (poly-lm (car plist)))))
276 ;; Add k variables to f
277 (f (poly-list-add-variables f k))
278 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
279 (plist (apply #'poly-append (poly-standard-extension plist))))
280 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
281 ;; Add -1 as the last term
282 (declare (type ring ring))
283 (setf (cdr (last (poly-termlist plist)))
284 (list (make-term :monom (make-monom :dimension d)
285 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
286 (append f (list plist)))
287
288(defun saturation-extension-1 (ring f p)
289 "Calculate [F, U*P-1]. It destructively modifies F."
290 (declare (type ring ring))
291 (polysaturation-extension ring f (list p)))
292
293;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
294;;
295;; Evaluation of polynomial (prefix) expressions
296;;
297;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
298
299(defun coerce-coeff (ring expr vars)
300 "Coerce an element of the coefficient ring to a constant polynomial."
301 ;; Modular arithmetic handler by rat
302 (declare (type ring ring))
303 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
304 :coeff (funcall (ring-parse ring) expr)))
305 0))
306
307(defun poly-eval (expr vars
308 &optional
309 (ring +ring-of-integers+)
310 (order #'lex>)
311 (list-marker :[)
312 &aux
313 (ring-and-order (make-ring-and-order :ring ring :order order)))
314 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
315variables VARS. Return the resulting polynomial or list of
316polynomials. Standard arithmetical operators in form EXPR are
317replaced with their analogues in the ring of polynomials, and the
318resulting expression is evaluated, resulting in a polynomial or a list
319of polynomials in internal form. A similar operation in another computer
320algebra system could be called 'expand' or so."
321 (declare (type ring ring))
322 (labels ((p-eval (arg) (poly-eval arg vars ring order))
323 (p-eval-scalar (arg) (poly-eval-scalar arg))
324 (p-eval-list (args) (mapcar #'p-eval args))
325 (p-add (x y) (poly-add ring-and-order x y)))
326 (cond
327 ((null expr) (error "Empty expression"))
328 ((eql expr 0) (make-poly-zero))
329 ((member expr vars :test #'equalp)
330 (let ((pos (position expr vars :test #'equalp)))
331 (make-poly-variable ring (length vars) pos)))
332 ((atom expr)
333 (coerce-coeff ring expr vars))
334 ((eq (car expr) list-marker)
335 (cons list-marker (p-eval-list (cdr expr))))
336 (t
337 (case (car expr)
338 (+ (reduce #'p-add (p-eval-list (cdr expr))))
339 (- (case (length expr)
340 (1 (make-poly-zero))
341 (2 (poly-uminus ring (p-eval (cadr expr))))
342 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
343 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
344 (reduce #'p-add (p-eval-list (cddr expr)))))))
345 (*
346 (if (endp (cddr expr)) ;unary
347 (p-eval (cdr expr))
348 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
349 (/
350 ;; A polynomial can be divided by a scalar
351 (cond
352 ((endp (cddr expr))
353 ;; A special case (/ ?), the inverse
354 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
355 (t
356 (let ((num (p-eval (cadr expr)))
357 (denom-inverse (apply (ring-div ring)
358 (cons (funcall (ring-unit ring))
359 (mapcar #'p-eval-scalar (cddr expr))))))
360 (scalar-times-poly ring denom-inverse num)))))
361 (expt
362 (cond
363 ((member (cadr expr) vars :test #'equalp)
364 ;;Special handling of (expt var pow)
365 (let ((pos (position (cadr expr) vars :test #'equalp)))
366 (make-poly-variable ring (length vars) pos (caddr expr))))
367 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
368 ;; Negative power means division in coefficient ring
369 ;; Non-integer power means non-polynomial coefficient
370 (coerce-coeff ring expr vars))
371 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
372 (otherwise
373 (coerce-coeff ring expr vars)))))))
374
375(defun poly-eval-scalar (expr
376 &optional
377 (ring +ring-of-integers+)
378 &aux
379 (order #'lex>))
380 "Evaluate a scalar expression EXPR in ring RING."
381 (declare (type ring ring))
382 (poly-lc (poly-eval expr nil ring order)))
383
384(defun spoly (ring-and-order f g
385 &aux
386 (ring (ro-ring ring-and-order)))
387 "It yields the S-polynomial of polynomials F and G."
388 (declare (type ring-and-order ring-and-order) (type poly f g))
389 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
390 (mf (monom-div lcm (poly-lm f)))
391 (mg (monom-div lcm (poly-lm g))))
392 (declare (type monom mf mg))
393 (multiple-value-bind (c cf cg)
394 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
395 (declare (ignore c))
396 (poly-sub
397 ring-and-order
398 (scalar-times-poly ring cg (monom-times-poly mf f))
399 (scalar-times-poly ring cf (monom-times-poly mg g))))))
400
401
402(defun poly-primitive-part (ring p)
403 "Divide polynomial P with integer coefficients by gcd of its
404coefficients and return the result."
405 (declare (type ring ring) (type poly p))
406 (if (poly-zerop p)
407 (values p 1)
408 (let ((c (poly-content ring p)))
409 (values (make-poly-from-termlist
410 (mapcar
411 #'(lambda (x)
412 (make-term :monom (term-monom x)
413 :coeff (funcall (ring-div ring) (term-coeff x) c)))
414 (poly-termlist p))
415 (poly-sugar p))
416 c))))
417
418(defun poly-content (ring p)
419 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
420to compute the greatest common divisor."
421 (declare (type ring ring) (type poly p))
422 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
423
424(defun read-infix-form (&key (stream t))
425 "Parser of infix expressions with integer/rational coefficients
426The parser will recognize two kinds of polynomial expressions:
427
428- polynomials in fully expanded forms with coefficients
429 written in front of symbolic expressions; constants can be optionally
430 enclosed in (); for example, the infix form
431 X^2-Y^2+(-4/3)*U^2*W^3-5
432 parses to
433 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
434
435- lists of polynomials; for example
436 [X-Y, X^2+3*Z]
437 parses to
438 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
439 where the first symbol [ marks a list of polynomials.
440
441-other infix expressions, for example
442 [(X-Y)*(X+Y)/Z,(X+1)^2]
443parses to:
444 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
445Currently this function is implemented using M. Kantrowitz's INFIX package."
446 (read-from-string
447 (concatenate 'string
448 "#I("
449 (with-output-to-string (s)
450 (loop
451 (multiple-value-bind (line eof)
452 (read-line stream t)
453 (format s "~A" line)
454 (when eof (return)))))
455 ")")))
456
457(defun read-poly (vars &key
458 (stream t)
459 (ring +ring-of-integers+)
460 (order #'lex>))
461 "Reads an expression in prefix form from a stream STREAM.
462The expression read from the strem should represent a polynomial or a
463list of polynomials in variables VARS, over the ring RING. The
464polynomial or list of polynomials is returned, with terms in each
465polynomial ordered according to monomial order ORDER."
466 (poly-eval (read-infix-form :stream stream) vars ring order))
467
468(defun string->poly (str vars
469 &optional
470 (ring +ring-of-integers+)
471 (order #'lex>))
472 "Converts a string STR to a polynomial in variables VARS."
473 (with-input-from-string (s str)
474 (read-poly vars :stream s :ring ring :order order)))
475
476(defun poly->alist (p)
477 "Convert a polynomial P to an association list. Thus, the format of the
478returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
479MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
480corresponding coefficient in the ring."
481 (cond
482 ((poly-p p)
483 (mapcar #'term->cons (poly-termlist p)))
484 ((and (consp p) (eq (car p) :[))
485 (cons :[ (mapcar #'poly->alist (cdr p))))))
486
487(defun string->alist (str vars
488 &optional
489 (ring +ring-of-integers+)
490 (order #'lex>))
491 "Convert a string STR representing a polynomial or polynomial list to
492an association list (... (MONOM . COEFF) ...)."
493 (poly->alist (string->poly str vars ring order)))
494
495(defun poly-equal-no-sugar-p (p q)
496 "Compare polynomials for equality, ignoring sugar."
497 (declare (type poly p q))
498 (equalp (poly-termlist p) (poly-termlist q)))
499
500(defun poly-set-equal-no-sugar-p (p q)
501 "Compare polynomial sets P and Q for equality, ignoring sugar."
502 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
503
504(defun poly-list-equal-no-sugar-p (p q)
505 "Compare polynomial lists P and Q for equality, ignoring sugar."
506 (every #'poly-equal-no-sugar-p p q))
507|#
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