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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 "CHANGE-TERM-ORDER")
28 (:documentation "Implements polynomials"))
29
30(in-package :polynomial)
31
32(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
33
34(defclass poly ()
35 ((termlist :initarg :termlist :accessor poly-termlist
36 :documentation "List of terms.")
37 (order :initarg :order :accessor poly-term-order
38 :documentation "Monomial/term order."))
39 (:default-initargs :termlist nil :order #'lex>)
40 (:documentation "A polynomial with a list of terms TERMLIST, ordered
41according to term order ORDER, which defaults to LEX>."))
42
43(defmethod print-object ((self poly) stream)
44 (format stream "#<POLY TERMLIST=~A ORDER=~A>"
45 (poly-termlist self)
46 (poly-term-order self)))
47
48(defgeneric change-term-order (self other)
49 (:documentation "Change term order of SELF to the term order of OTHER.")
50 (:method ((self poly) (other poly))
51 (unless (eq (poly-term-order self) (poly-term-order other))
52 (setf (poly-termlist self) (sort (poly-termlist self) (poly-term-order other))
53 (poly-term-order self) (poly-term-order other)))
54 self))
55
56(defmethod r-equalp ((self poly) (other poly))
57 "POLY instances are R-EQUALP if they have the same
58order and if all terms are R-EQUALP."
59 (and (every #'r-equalp (poly-termlist self) (poly-termlist other))
60 (eq (poly-term-order self) (poly-term-order other))))
61
62(defmethod insert-item ((self poly) (item term))
63 (push item (poly-termlist self))
64 self)
65
66(defmethod append-item ((self poly) (item term))
67 (setf (cdr (last (poly-termlist self))) (list item))
68 self)
69
70;; Leading term
71(defgeneric leading-term (object)
72 (:method ((self poly))
73 (car (poly-termlist self)))
74 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
75
76;; Second term
77(defgeneric second-leading-term (object)
78 (:method ((self poly))
79 (cadar (poly-termlist self)))
80 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
81
82;; Leading coefficient
83(defgeneric leading-coefficient (object)
84 (:method ((self poly))
85 (r-coeff (leading-term self)))
86 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
87
88;; Second coefficient
89(defgeneric second-leading-coefficient (object)
90 (:method ((self poly))
91 (r-coeff (second-leading-term self)))
92 (:documentation "The second leading coefficient of a polynomial. It
93 signals error for a polynomial with at most one term."))
94
95;; Testing for a zero polynomial
96(defmethod r-zerop ((self poly))
97 (null (poly-termlist self)))
98
99;; The number of terms
100(defmethod r-length ((self poly))
101 (length (poly-termlist self)))
102
103(defmethod multiply-by ((self poly) (other monom))
104 (mapc #'(lambda (term) (multiply-by term other))
105 (poly-termlist self))
106 self)
107
108(defmethod multiply-by ((self poly) (other scalar))
109 (mapc #'(lambda (term) (multiply-by term other))
110 (poly-termlist self))
111 self)
112
113
114(defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)
115 "Return an expression which will efficiently adds/subtracts two
116polynomials, P and Q. The addition/subtraction of coefficients is
117performed by calling ADD/SUBTRACT-METHOD-NAME. If UMINUS-METHOD-NAME
118is supplied, it is used to negate the coefficients of Q which do not
119have a corresponding coefficient in P. The code implements an
120efficient algorithm to add two polynomials represented as sorted lists
121of terms. The code destroys both arguments, reusing the terms to build
122the result."
123 `(macrolet ((lc (x) `(r-coeff (car ,x))))
124 (do ((p ,p)
125 (q ,q)
126 r)
127 ((or (endp p) (endp q))
128 ;; NOTE: R contains the result in reverse order. Can it
129 ;; be more efficient to produce the terms in correct order?
130 (unless (endp q)
131 ;; Upon subtraction, we must change the sign of
132 ;; all coefficients in q
133 ,@(when uminus-fn
134 `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))
135 (setf r (nreconc r q)))
136 r)
137 (multiple-value-bind
138 (greater-p equal-p)
139 (funcall ,order-fn (car p) (car q))
140 (cond
141 (greater-p
142 (rotatef (cdr p) r p)
143 )
144 (equal-p
145 (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))
146 (cond
147 ((r-zerop s)
148 (setf p (cdr p))
149 )
150 (t
151 (setf (lc p) s)
152 (rotatef (cdr p) r p))))
153 (setf q (cdr q))
154 )
155 (t
156 ;;Negate the term of Q if UMINUS provided, signallig
157 ;;that we are doing subtraction
158 ,(when uminus-fn
159 `(setf (lc q) (funcall ,uminus-fn (lc q))))
160 (rotatef (cdr q) r q)))))))
161
162
163(defmacro def-add/subtract-method (add/subtract-method-name
164 uminus-method-name
165 &optional
166 (doc-string nil doc-string-supplied-p))
167 "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
168 `(defmethod ,add/subtract-method-name ((self poly) (other poly))
169 ,@(when doc-string-supplied-p `(,doc-string))
170 ;; Ensure orders are compatible
171 (change-term-order other self)
172 (setf (poly-termlist self) (fast-add/subtract
173 (poly-termlist self) (poly-termlist other)
174 (poly-term-order self)
175 #',add/subtract-method-name
176 ,(when uminus-method-name `(function ,uminus-method-name))))
177 self))
178
179(eval-when (:compile-toplevel :load-toplevel :execute)
180
181 (def-add/subtract-method add-to nil
182 "Adds to polynomial SELF another polynomial OTHER.
183This operation destructively modifies both polynomials.
184The result is stored in SELF. This implementation does
185no consing, entirely reusing the sells of SELF and OTHER.")
186
187 (def-add/subtract-method subtract-from unary-minus
188 "Subtracts from polynomial SELF another polynomial OTHER.
189This operation destructively modifies both polynomials.
190The result is stored in SELF. This implementation does
191no consing, entirely reusing the sells of SELF and OTHER.")
192
193 )
194
195
196
197(defmethod unary-minus ((self poly))
198 "Destructively modifies the coefficients of the polynomial SELF,
199by changing their sign."
200 (mapc #'unary-minus (poly-termlist self))
201 self)
202
203(defun add-termlists (p q order-fn)
204 "Destructively adds two termlists P and Q ordered according to ORDER-FN."
205 (fast-add/subtract p q order-fn #'add-to nil))
206
207(defmacro multiply-term-by-termlist-dropping-zeros (term termlist
208 &optional (reverse-arg-order-P nil))
209 "Multiplies term TERM by a list of term, TERMLIST.
210Takes into accound divisors of zero in the ring, by
211deleting zero terms. Optionally, if REVERSE-ARG-ORDER-P
212is T, change the order of arguments; this may be important
213if we extend the package to non-commutative rings."
214 `(mapcan #'(lambda (other-term)
215 (let ((prod (r*
216 ,@(cond
217 (reverse-arg-order-p
218 `(other-term ,term))
219 (t
220 `(,term other-term))))))
221 (cond
222 ((r-zerop prod) nil)
223 (t (list prod)))))
224 ,termlist))
225
226(defun multiply-termlists (p q order-fn)
227 (cond
228 ((or (endp p) (endp q))
229 ;;p or q is 0 (represented by NIL)
230 nil)
231 ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
232 ((endp (cdr p))
233 (multiply-term-by-termlist-dropping-zeros (car p) q))
234 ((endp (cdr q))
235 (multiply-term-by-termlist-dropping-zeros (car q) p t))
236 (t
237 (cons (r* (car p) (car q))
238 (add-termlists
239 (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
240 (multiply-termlists (cdr p) q order-fn)
241 order-fn)))))
242
243(defmethod multiply-by ((self poly) (other poly))
244 (change-term-order other self)
245 (setf (poly-termlist self) (multiply-termlists (poly-termlist self)
246 (poly-termlist other)
247 (poly-term-order self)))
248 self)
249
250(defmethod r* ((poly1 poly) (poly2 poly))
251 "Non-destructively multiply POLY1 by POLY2."
252 (multiply-by (copy-instance POLY1) (copy-instance POLY2)))
253
254#|
255
256
257(defun poly-standard-extension (plist &aux (k (length plist)))
258 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
259is a list of polynomials."
260 (declare (list plist) (fixnum k))
261 (labels ((incf-power (g i)
262 (dolist (x (poly-termlist g))
263 (incf (monom-elt (term-monom x) i)))
264 (incf (poly-sugar g))))
265 (setf plist (poly-list-add-variables plist k))
266 (dotimes (i k plist)
267 (incf-power (nth i plist) i))))
268
269
270
271(defun saturation-extension (ring f plist
272 &aux
273 (k (length plist))
274 (d (monom-dimension (poly-lm (car plist))))
275 f-x plist-x)
276 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
277 (declare (type ring ring))
278 (setf f-x (poly-list-add-variables f k)
279 plist-x (mapcar #'(lambda (x)
280 (setf (poly-termlist x)
281 (nconc (poly-termlist x)
282 (list (make-term :monom (make-monom :dimension d)
283 :coeff (funcall (ring-uminus ring)
284 (funcall (ring-unit ring)))))))
285 x)
286 (poly-standard-extension plist)))
287 (append f-x plist-x))
288
289
290(defun polysaturation-extension (ring f plist
291 &aux
292 (k (length plist))
293 (d (+ k (monom-dimension (poly-lm (car plist)))))
294 ;; Add k variables to f
295 (f (poly-list-add-variables f k))
296 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
297 (plist (apply #'poly-append (poly-standard-extension plist))))
298 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
299 ;; Add -1 as the last term
300 (declare (type ring ring))
301 (setf (cdr (last (poly-termlist plist)))
302 (list (make-term :monom (make-monom :dimension d)
303 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
304 (append f (list plist)))
305
306(defun saturation-extension-1 (ring f p)
307 "Calculate [F, U*P-1]. It destructively modifies F."
308 (declare (type ring ring))
309 (polysaturation-extension ring f (list p)))
310
311;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
312;;
313;; Evaluation of polynomial (prefix) expressions
314;;
315;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
316
317(defun coerce-coeff (ring expr vars)
318 "Coerce an element of the coefficient ring to a constant polynomial."
319 ;; Modular arithmetic handler by rat
320 (declare (type ring ring))
321 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
322 :coeff (funcall (ring-parse ring) expr)))
323 0))
324
325(defun poly-eval (expr vars
326 &optional
327 (ring +ring-of-integers+)
328 (order #'lex>)
329 (list-marker :[)
330 &aux
331 (ring-and-order (make-ring-and-order :ring ring :order order)))
332 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
333variables VARS. Return the resulting polynomial or list of
334polynomials. Standard arithmetical operators in form EXPR are
335replaced with their analogues in the ring of polynomials, and the
336resulting expression is evaluated, resulting in a polynomial or a list
337of polynomials in internal form. A similar operation in another computer
338algebra system could be called 'expand' or so."
339 (declare (type ring ring))
340 (labels ((p-eval (arg) (poly-eval arg vars ring order))
341 (p-eval-scalar (arg) (poly-eval-scalar arg))
342 (p-eval-list (args) (mapcar #'p-eval args))
343 (p-add (x y) (poly-add ring-and-order x y)))
344 (cond
345 ((null expr) (error "Empty expression"))
346 ((eql expr 0) (make-poly-zero))
347 ((member expr vars :test #'equalp)
348 (let ((pos (position expr vars :test #'equalp)))
349 (make-poly-variable ring (length vars) pos)))
350 ((atom expr)
351 (coerce-coeff ring expr vars))
352 ((eq (car expr) list-marker)
353 (cons list-marker (p-eval-list (cdr expr))))
354 (t
355 (case (car expr)
356 (+ (reduce #'p-add (p-eval-list (cdr expr))))
357 (- (case (length expr)
358 (1 (make-poly-zero))
359 (2 (poly-uminus ring (p-eval (cadr expr))))
360 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
361 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
362 (reduce #'p-add (p-eval-list (cddr expr)))))))
363 (*
364 (if (endp (cddr expr)) ;unary
365 (p-eval (cdr expr))
366 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
367 (/
368 ;; A polynomial can be divided by a scalar
369 (cond
370 ((endp (cddr expr))
371 ;; A special case (/ ?), the inverse
372 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
373 (t
374 (let ((num (p-eval (cadr expr)))
375 (denom-inverse (apply (ring-div ring)
376 (cons (funcall (ring-unit ring))
377 (mapcar #'p-eval-scalar (cddr expr))))))
378 (scalar-times-poly ring denom-inverse num)))))
379 (expt
380 (cond
381 ((member (cadr expr) vars :test #'equalp)
382 ;;Special handling of (expt var pow)
383 (let ((pos (position (cadr expr) vars :test #'equalp)))
384 (make-poly-variable ring (length vars) pos (caddr expr))))
385 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
386 ;; Negative power means division in coefficient ring
387 ;; Non-integer power means non-polynomial coefficient
388 (coerce-coeff ring expr vars))
389 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
390 (otherwise
391 (coerce-coeff ring expr vars)))))))
392
393(defun poly-eval-scalar (expr
394 &optional
395 (ring +ring-of-integers+)
396 &aux
397 (order #'lex>))
398 "Evaluate a scalar expression EXPR in ring RING."
399 (declare (type ring ring))
400 (poly-lc (poly-eval expr nil ring order)))
401
402(defun spoly (ring-and-order f g
403 &aux
404 (ring (ro-ring ring-and-order)))
405 "It yields the S-polynomial of polynomials F and G."
406 (declare (type ring-and-order ring-and-order) (type poly f g))
407 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
408 (mf (monom-div lcm (poly-lm f)))
409 (mg (monom-div lcm (poly-lm g))))
410 (declare (type monom mf mg))
411 (multiple-value-bind (c cf cg)
412 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
413 (declare (ignore c))
414 (poly-sub
415 ring-and-order
416 (scalar-times-poly ring cg (monom-times-poly mf f))
417 (scalar-times-poly ring cf (monom-times-poly mg g))))))
418
419
420(defun poly-primitive-part (ring p)
421 "Divide polynomial P with integer coefficients by gcd of its
422coefficients and return the result."
423 (declare (type ring ring) (type poly p))
424 (if (poly-zerop p)
425 (values p 1)
426 (let ((c (poly-content ring p)))
427 (values (make-poly-from-termlist
428 (mapcar
429 #'(lambda (x)
430 (make-term :monom (term-monom x)
431 :coeff (funcall (ring-div ring) (term-coeff x) c)))
432 (poly-termlist p))
433 (poly-sugar p))
434 c))))
435
436(defun poly-content (ring p)
437 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
438to compute the greatest common divisor."
439 (declare (type ring ring) (type poly p))
440 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
441
442(defun read-infix-form (&key (stream t))
443 "Parser of infix expressions with integer/rational coefficients
444The parser will recognize two kinds of polynomial expressions:
445
446- polynomials in fully expanded forms with coefficients
447 written in front of symbolic expressions; constants can be optionally
448 enclosed in (); for example, the infix form
449 X^2-Y^2+(-4/3)*U^2*W^3-5
450 parses to
451 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
452
453- lists of polynomials; for example
454 [X-Y, X^2+3*Z]
455 parses to
456 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
457 where the first symbol [ marks a list of polynomials.
458
459-other infix expressions, for example
460 [(X-Y)*(X+Y)/Z,(X+1)^2]
461parses to:
462 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
463Currently this function is implemented using M. Kantrowitz's INFIX package."
464 (read-from-string
465 (concatenate 'string
466 "#I("
467 (with-output-to-string (s)
468 (loop
469 (multiple-value-bind (line eof)
470 (read-line stream t)
471 (format s "~A" line)
472 (when eof (return)))))
473 ")")))
474
475(defun read-poly (vars &key
476 (stream t)
477 (ring +ring-of-integers+)
478 (order #'lex>))
479 "Reads an expression in prefix form from a stream STREAM.
480The expression read from the strem should represent a polynomial or a
481list of polynomials in variables VARS, over the ring RING. The
482polynomial or list of polynomials is returned, with terms in each
483polynomial ordered according to monomial order ORDER."
484 (poly-eval (read-infix-form :stream stream) vars ring order))
485
486(defun string->poly (str vars
487 &optional
488 (ring +ring-of-integers+)
489 (order #'lex>))
490 "Converts a string STR to a polynomial in variables VARS."
491 (with-input-from-string (s str)
492 (read-poly vars :stream s :ring ring :order order)))
493
494(defun poly->alist (p)
495 "Convert a polynomial P to an association list. Thus, the format of the
496returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
497MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
498corresponding coefficient in the ring."
499 (cond
500 ((poly-p p)
501 (mapcar #'term->cons (poly-termlist p)))
502 ((and (consp p) (eq (car p) :[))
503 (cons :[ (mapcar #'poly->alist (cdr p))))))
504
505(defun string->alist (str vars
506 &optional
507 (ring +ring-of-integers+)
508 (order #'lex>))
509 "Convert a string STR representing a polynomial or polynomial list to
510an association list (... (MONOM . COEFF) ...)."
511 (poly->alist (string->poly str vars ring order)))
512
513(defun poly-equal-no-sugar-p (p q)
514 "Compare polynomials for equality, ignoring sugar."
515 (declare (type poly p q))
516 (equalp (poly-termlist p) (poly-termlist q)))
517
518(defun poly-set-equal-no-sugar-p (p q)
519 "Compare polynomial sets P and Q for equality, ignoring sugar."
520 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
521
522(defun poly-list-equal-no-sugar-p (p q)
523 "Compare polynomial lists P and Q for equality, ignoring sugar."
524 (every #'poly-equal-no-sugar-p p q))
525|#
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