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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 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(defmacro multiply-term-by-termlist-dropping-zeros (term termlist
199 &optional (reverse-order nil))
200 "Multiplies term TERM by a list of term, TERMLIST.
201Takes into accound divisors of zero in the ring, by
202deleting zero terms."
203 `(mapcan #'(lambda (other-term)
204 (let ((prod (term-multiply
205 ,(cond
206 (reverse-order
207 `(other-term ,term)
208 `(,term other-term))))))
209 (cond
210 ((r-zerop prod) nil)
211 (t (list prod)))))
212 ,termlist))
213
214(defun multiply-termlists (p q order-fn)
215 (cond
216 ((or (endp p) (endp q)) nil) ;p or q is 0 (represented by NIL)
217 ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
218 ((endp (cdr p))
219 (multiply-term-by-termlist-dropping-zeros (car p) q)
220 ((endp (cdr q))
221 (multiply-term-by-termlist-dropping-zeros (car q) p t)
222 (t
223 (nconc (multiply-terms (car p) (car q))
224 (tail (add-termlists
225 (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
226 (multiply-termlists (cdr p) q)
227 order-fn))))))))
228
229
230(defmethod multiply-by ((self poly) (other poly))
231 (setf (poly-termlist self) (multiply-termlists (poly-termlist self)
232 (poly-termlist other)
233 (poly-term-order self)))
234 self)
235
236#|
237
238(defun poly-standard-extension (plist &aux (k (length plist)))
239 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
240is a list of polynomials."
241 (declare (list plist) (fixnum k))
242 (labels ((incf-power (g i)
243 (dolist (x (poly-termlist g))
244 (incf (monom-elt (term-monom x) i)))
245 (incf (poly-sugar g))))
246 (setf plist (poly-list-add-variables plist k))
247 (dotimes (i k plist)
248 (incf-power (nth i plist) i))))
249
250
251
252(defun saturation-extension (ring f plist
253 &aux
254 (k (length plist))
255 (d (monom-dimension (poly-lm (car plist))))
256 f-x plist-x)
257 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
258 (declare (type ring ring))
259 (setf f-x (poly-list-add-variables f k)
260 plist-x (mapcar #'(lambda (x)
261 (setf (poly-termlist x)
262 (nconc (poly-termlist x)
263 (list (make-term :monom (make-monom :dimension d)
264 :coeff (funcall (ring-uminus ring)
265 (funcall (ring-unit ring)))))))
266 x)
267 (poly-standard-extension plist)))
268 (append f-x plist-x))
269
270
271(defun polysaturation-extension (ring f plist
272 &aux
273 (k (length plist))
274 (d (+ k (monom-dimension (poly-lm (car plist)))))
275 ;; Add k variables to f
276 (f (poly-list-add-variables f k))
277 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
278 (plist (apply #'poly-append (poly-standard-extension plist))))
279 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
280 ;; Add -1 as the last term
281 (declare (type ring ring))
282 (setf (cdr (last (poly-termlist plist)))
283 (list (make-term :monom (make-monom :dimension d)
284 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
285 (append f (list plist)))
286
287(defun saturation-extension-1 (ring f p)
288 "Calculate [F, U*P-1]. It destructively modifies F."
289 (declare (type ring ring))
290 (polysaturation-extension ring f (list p)))
291
292;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
293;;
294;; Evaluation of polynomial (prefix) expressions
295;;
296;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
297
298(defun coerce-coeff (ring expr vars)
299 "Coerce an element of the coefficient ring to a constant polynomial."
300 ;; Modular arithmetic handler by rat
301 (declare (type ring ring))
302 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
303 :coeff (funcall (ring-parse ring) expr)))
304 0))
305
306(defun poly-eval (expr vars
307 &optional
308 (ring +ring-of-integers+)
309 (order #'lex>)
310 (list-marker :[)
311 &aux
312 (ring-and-order (make-ring-and-order :ring ring :order order)))
313 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
314variables VARS. Return the resulting polynomial or list of
315polynomials. Standard arithmetical operators in form EXPR are
316replaced with their analogues in the ring of polynomials, and the
317resulting expression is evaluated, resulting in a polynomial or a list
318of polynomials in internal form. A similar operation in another computer
319algebra system could be called 'expand' or so."
320 (declare (type ring ring))
321 (labels ((p-eval (arg) (poly-eval arg vars ring order))
322 (p-eval-scalar (arg) (poly-eval-scalar arg))
323 (p-eval-list (args) (mapcar #'p-eval args))
324 (p-add (x y) (poly-add ring-and-order x y)))
325 (cond
326 ((null expr) (error "Empty expression"))
327 ((eql expr 0) (make-poly-zero))
328 ((member expr vars :test #'equalp)
329 (let ((pos (position expr vars :test #'equalp)))
330 (make-poly-variable ring (length vars) pos)))
331 ((atom expr)
332 (coerce-coeff ring expr vars))
333 ((eq (car expr) list-marker)
334 (cons list-marker (p-eval-list (cdr expr))))
335 (t
336 (case (car expr)
337 (+ (reduce #'p-add (p-eval-list (cdr expr))))
338 (- (case (length expr)
339 (1 (make-poly-zero))
340 (2 (poly-uminus ring (p-eval (cadr expr))))
341 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
342 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
343 (reduce #'p-add (p-eval-list (cddr expr)))))))
344 (*
345 (if (endp (cddr expr)) ;unary
346 (p-eval (cdr expr))
347 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
348 (/
349 ;; A polynomial can be divided by a scalar
350 (cond
351 ((endp (cddr expr))
352 ;; A special case (/ ?), the inverse
353 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
354 (t
355 (let ((num (p-eval (cadr expr)))
356 (denom-inverse (apply (ring-div ring)
357 (cons (funcall (ring-unit ring))
358 (mapcar #'p-eval-scalar (cddr expr))))))
359 (scalar-times-poly ring denom-inverse num)))))
360 (expt
361 (cond
362 ((member (cadr expr) vars :test #'equalp)
363 ;;Special handling of (expt var pow)
364 (let ((pos (position (cadr expr) vars :test #'equalp)))
365 (make-poly-variable ring (length vars) pos (caddr expr))))
366 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
367 ;; Negative power means division in coefficient ring
368 ;; Non-integer power means non-polynomial coefficient
369 (coerce-coeff ring expr vars))
370 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
371 (otherwise
372 (coerce-coeff ring expr vars)))))))
373
374(defun poly-eval-scalar (expr
375 &optional
376 (ring +ring-of-integers+)
377 &aux
378 (order #'lex>))
379 "Evaluate a scalar expression EXPR in ring RING."
380 (declare (type ring ring))
381 (poly-lc (poly-eval expr nil ring order)))
382
383(defun spoly (ring-and-order f g
384 &aux
385 (ring (ro-ring ring-and-order)))
386 "It yields the S-polynomial of polynomials F and G."
387 (declare (type ring-and-order ring-and-order) (type poly f g))
388 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
389 (mf (monom-div lcm (poly-lm f)))
390 (mg (monom-div lcm (poly-lm g))))
391 (declare (type monom mf mg))
392 (multiple-value-bind (c cf cg)
393 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
394 (declare (ignore c))
395 (poly-sub
396 ring-and-order
397 (scalar-times-poly ring cg (monom-times-poly mf f))
398 (scalar-times-poly ring cf (monom-times-poly mg g))))))
399
400
401(defun poly-primitive-part (ring p)
402 "Divide polynomial P with integer coefficients by gcd of its
403coefficients and return the result."
404 (declare (type ring ring) (type poly p))
405 (if (poly-zerop p)
406 (values p 1)
407 (let ((c (poly-content ring p)))
408 (values (make-poly-from-termlist
409 (mapcar
410 #'(lambda (x)
411 (make-term :monom (term-monom x)
412 :coeff (funcall (ring-div ring) (term-coeff x) c)))
413 (poly-termlist p))
414 (poly-sugar p))
415 c))))
416
417(defun poly-content (ring p)
418 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
419to compute the greatest common divisor."
420 (declare (type ring ring) (type poly p))
421 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
422
423(defun read-infix-form (&key (stream t))
424 "Parser of infix expressions with integer/rational coefficients
425The parser will recognize two kinds of polynomial expressions:
426
427- polynomials in fully expanded forms with coefficients
428 written in front of symbolic expressions; constants can be optionally
429 enclosed in (); for example, the infix form
430 X^2-Y^2+(-4/3)*U^2*W^3-5
431 parses to
432 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
433
434- lists of polynomials; for example
435 [X-Y, X^2+3*Z]
436 parses to
437 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
438 where the first symbol [ marks a list of polynomials.
439
440-other infix expressions, for example
441 [(X-Y)*(X+Y)/Z,(X+1)^2]
442parses to:
443 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
444Currently this function is implemented using M. Kantrowitz's INFIX package."
445 (read-from-string
446 (concatenate 'string
447 "#I("
448 (with-output-to-string (s)
449 (loop
450 (multiple-value-bind (line eof)
451 (read-line stream t)
452 (format s "~A" line)
453 (when eof (return)))))
454 ")")))
455
456(defun read-poly (vars &key
457 (stream t)
458 (ring +ring-of-integers+)
459 (order #'lex>))
460 "Reads an expression in prefix form from a stream STREAM.
461The expression read from the strem should represent a polynomial or a
462list of polynomials in variables VARS, over the ring RING. The
463polynomial or list of polynomials is returned, with terms in each
464polynomial ordered according to monomial order ORDER."
465 (poly-eval (read-infix-form :stream stream) vars ring order))
466
467(defun string->poly (str vars
468 &optional
469 (ring +ring-of-integers+)
470 (order #'lex>))
471 "Converts a string STR to a polynomial in variables VARS."
472 (with-input-from-string (s str)
473 (read-poly vars :stream s :ring ring :order order)))
474
475(defun poly->alist (p)
476 "Convert a polynomial P to an association list. Thus, the format of the
477returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
478MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
479corresponding coefficient in the ring."
480 (cond
481 ((poly-p p)
482 (mapcar #'term->cons (poly-termlist p)))
483 ((and (consp p) (eq (car p) :[))
484 (cons :[ (mapcar #'poly->alist (cdr p))))))
485
486(defun string->alist (str vars
487 &optional
488 (ring +ring-of-integers+)
489 (order #'lex>))
490 "Convert a string STR representing a polynomial or polynomial list to
491an association list (... (MONOM . COEFF) ...)."
492 (poly->alist (string->poly str vars ring order)))
493
494(defun poly-equal-no-sugar-p (p q)
495 "Compare polynomials for equality, ignoring sugar."
496 (declare (type poly p q))
497 (equalp (poly-termlist p) (poly-termlist q)))
498
499(defun poly-set-equal-no-sugar-p (p q)
500 "Compare polynomial sets P and Q for equality, ignoring sugar."
501 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
502
503(defun poly-list-equal-no-sugar-p (p q)
504 "Compare polynomial lists P and Q for equality, ignoring sugar."
505 (every #'poly-equal-no-sugar-p p q))
506|#
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