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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 #| :infix |# )
24 (:export "POLY"
25 "POLY-TERMLIST"
26 "POLY-TERM-ORDER"
27 "CHANGE-TERM-ORDER"
28 "SATURATION-EXTENSION")
29 (:documentation "Implements polynomials"))
30
31(in-package :polynomial)
32
33(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
34
35(defclass poly ()
36 ((termlist :initarg :termlist :accessor poly-termlist
37 :documentation "List of terms.")
38 (order :initarg :order :accessor poly-term-order
39 :documentation "Monomial/term order."))
40 (:default-initargs :termlist nil :order #'lex>)
41 (:documentation "A polynomial with a list of terms TERMLIST, ordered
42according to term order ORDER, which defaults to LEX>."))
43
44(defmethod print-object ((self poly) stream)
45 (format stream "#<POLY TERMLIST=~A ORDER=~A>"
46 (poly-termlist self)
47 (poly-term-order self)))
48
49(defgeneric change-term-order (self other)
50 (:documentation "Change term order of SELF to the term order of OTHER.")
51 (:method ((self poly) (other poly))
52 (unless (eq (poly-term-order self) (poly-term-order other))
53 (setf (poly-termlist self) (sort (poly-termlist self) (poly-term-order other))
54 (poly-term-order self) (poly-term-order other)))
55 self))
56
57(defmethod r-equalp ((self poly) (other poly))
58 "POLY instances are R-EQUALP if they have the same
59order and if all terms are R-EQUALP."
60 (and (every #'r-equalp (poly-termlist self) (poly-termlist other))
61 (eq (poly-term-order self) (poly-term-order other))))
62
63(defmethod insert-item ((self poly) (item term))
64 (push item (poly-termlist self))
65 self)
66
67(defmethod append-item ((self poly) (item term))
68 (setf (cdr (last (poly-termlist self))) (list item))
69 self)
70
71;; Leading term
72(defgeneric leading-term (object)
73 (:method ((self poly))
74 (car (poly-termlist self)))
75 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
76
77;; Second term
78(defgeneric second-leading-term (object)
79 (:method ((self poly))
80 (cadar (poly-termlist self)))
81 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
82
83;; Leading coefficient
84(defgeneric leading-coefficient (object)
85 (:method ((self poly))
86 (r-coeff (leading-term self)))
87 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
88
89;; Second coefficient
90(defgeneric second-leading-coefficient (object)
91 (:method ((self poly))
92 (r-coeff (second-leading-term self)))
93 (:documentation "The second leading coefficient of a polynomial. It
94 signals error for a polynomial with at most one term."))
95
96;; Testing for a zero polynomial
97(defmethod r-zerop ((self poly))
98 (null (poly-termlist self)))
99
100;; The number of terms
101(defmethod r-length ((self poly))
102 (length (poly-termlist self)))
103
104(defmethod multiply-by ((self poly) (other monom))
105 (mapc #'(lambda (term) (multiply-by term other))
106 (poly-termlist self))
107 self)
108
109(defmethod multiply-by ((self poly) (other scalar))
110 (mapc #'(lambda (term) (multiply-by term other))
111 (poly-termlist self))
112 self)
113
114
115(defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)
116 "Return an expression which will efficiently adds/subtracts two
117polynomials, P and Q. The addition/subtraction of coefficients is
118performed by calling ADD/SUBTRACT-METHOD-NAME. If UMINUS-METHOD-NAME
119is supplied, it is used to negate the coefficients of Q which do not
120have a corresponding coefficient in P. The code implements an
121efficient algorithm to add two polynomials represented as sorted lists
122of terms. The code destroys both arguments, reusing the terms to build
123the result."
124 `(macrolet ((lc (x) `(r-coeff (car ,x))))
125 (do ((p ,p)
126 (q ,q)
127 r)
128 ((or (endp p) (endp q))
129 ;; NOTE: R contains the result in reverse order. Can it
130 ;; be more efficient to produce the terms in correct order?
131 (unless (endp q)
132 ;; Upon subtraction, we must change the sign of
133 ;; all coefficients in q
134 ,@(when uminus-fn
135 `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))
136 (setf r (nreconc r q)))
137 r)
138 (multiple-value-bind
139 (greater-p equal-p)
140 (funcall ,order-fn (car p) (car q))
141 (cond
142 (greater-p
143 (rotatef (cdr p) r p)
144 )
145 (equal-p
146 (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))
147 (cond
148 ((r-zerop s)
149 (setf p (cdr p))
150 )
151 (t
152 (setf (lc p) s)
153 (rotatef (cdr p) r p))))
154 (setf q (cdr q))
155 )
156 (t
157 ;;Negate the term of Q if UMINUS provided, signallig
158 ;;that we are doing subtraction
159 ,(when uminus-fn
160 `(setf (lc q) (funcall ,uminus-fn (lc q))))
161 (rotatef (cdr q) r q)))))))
162
163
164(defmacro def-add/subtract-method (add/subtract-method-name
165 uminus-method-name
166 &optional
167 (doc-string nil doc-string-supplied-p))
168 "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
169 `(defmethod ,add/subtract-method-name ((self poly) (other poly))
170 ,@(when doc-string-supplied-p `(,doc-string))
171 ;; Ensure orders are compatible
172 (change-term-order other self)
173 (setf (poly-termlist self) (fast-add/subtract
174 (poly-termlist self) (poly-termlist other)
175 (poly-term-order self)
176 #',add/subtract-method-name
177 ,(when uminus-method-name `(function ,uminus-method-name))))
178 self))
179
180(eval-when (:compile-toplevel :load-toplevel :execute)
181
182 (def-add/subtract-method add-to nil
183 "Adds to polynomial SELF another polynomial OTHER.
184This operation destructively modifies both polynomials.
185The result is stored in SELF. This implementation does
186no consing, entirely reusing the sells of SELF and OTHER.")
187
188 (def-add/subtract-method subtract-from unary-minus
189 "Subtracts from polynomial SELF another polynomial OTHER.
190This operation destructively modifies both polynomials.
191The result is stored in SELF. This implementation does
192no consing, entirely reusing the sells of SELF and OTHER.")
193
194 )
195
196
197
198(defmethod unary-minus ((self poly))
199 "Destructively modifies the coefficients of the polynomial SELF,
200by changing their sign."
201 (mapc #'unary-minus (poly-termlist self))
202 self)
203
204(defun add-termlists (p q order-fn)
205 "Destructively adds two termlists P and Q ordered according to ORDER-FN."
206 (fast-add/subtract p q order-fn #'add-to nil))
207
208(defmacro multiply-term-by-termlist-dropping-zeros (term termlist
209 &optional (reverse-arg-order-P nil))
210 "Multiplies term TERM by a list of term, TERMLIST.
211Takes into accound divisors of zero in the ring, by
212deleting zero terms. Optionally, if REVERSE-ARG-ORDER-P
213is T, change the order of arguments; this may be important
214if we extend the package to non-commutative rings."
215 `(mapcan #'(lambda (other-term)
216 (let ((prod (r*
217 ,@(cond
218 (reverse-arg-order-p
219 `(other-term ,term))
220 (t
221 `(,term other-term))))))
222 (cond
223 ((r-zerop prod) nil)
224 (t (list prod)))))
225 ,termlist))
226
227(defun multiply-termlists (p q order-fn)
228 (cond
229 ((or (endp p) (endp q))
230 ;;p or q is 0 (represented by NIL)
231 nil)
232 ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
233 ((endp (cdr p))
234 (multiply-term-by-termlist-dropping-zeros (car p) q))
235 ((endp (cdr q))
236 (multiply-term-by-termlist-dropping-zeros (car q) p t))
237 (t
238 (cons (r* (car p) (car q))
239 (add-termlists
240 (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
241 (multiply-termlists (cdr p) q order-fn)
242 order-fn)))))
243
244(defmethod multiply-by ((self poly) (other poly))
245 (change-term-order other self)
246 (setf (poly-termlist self) (multiply-termlists (poly-termlist self)
247 (poly-termlist other)
248 (poly-term-order self)))
249 self)
250
251(defmethod r* ((poly1 poly) (poly2 poly))
252 "Non-destructively multiply POLY1 by POLY2."
253 (multiply-by (copy-instance POLY1) (copy-instance POLY2)))
254
255(defmethod left-tensor-product-by ((self poly) (other term))
256 (setf (poly-termlist self)
257 (mapcan #'(lambda (term)
258 (let ((prod (left-tensor-product-by term other)))
259 (cond
260 ((r-zerop prod) nil)
261 (t (list prod)))))
262 (poly-termlist self)))
263 self)
264
265(defmethod right-tensor-product-by ((self poly) (other term))
266 (setf (poly-termlist self)
267 (mapcan #'(lambda (term)
268 (let ((prod (right-tensor-product-by term other)))
269 (cond
270 ((r-zerop prod) nil)
271 (t (list prod)))))
272 (poly-termlist self)))
273 self)
274
275(defmethod left-tensor-product-by ((self poly) (other monom))
276 (setf (poly-termlist self)
277 (mapcan #'(lambda (term)
278 (let ((prod (left-tensor-product-by term other)))
279 (cond
280 ((r-zerop prod) nil)
281 (t (list prod)))))
282 (poly-termlist self)))
283 self)
284
285(defmethod right-tensor-product-by ((self poly) (other monom))
286 (setf (poly-termlist self)
287 (mapcan #'(lambda (term)
288 (let ((prod (right-tensor-product-by term other)))
289 (cond
290 ((r-zerop prod) nil)
291 (t (list prod)))))
292 (poly-termlist self)))
293 self)
294
295
296(defun standard-extension (plist &aux (k (length plist)) (i 0))
297 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
298is a list of polynomials. Destructively modifies PLIST elements."
299 (mapc #'(lambda (poly)
300 (left-tensor-product-by
301 poly
302 (prog1
303 (make-monom-variable k i)
304 (incf i))))
305 plist))
306
307(defmethod poly-dimension ((poly poly))
308 (cond ((r-zerop poly) -1)
309 (t (monom-dimension (leading-term poly)))))
310
311(defun standard-extension-1 (plist
312 &aux
313 (k (length plist))
314 (plist (poly-standard-extension plist))
315 (nvars (poly-dimension (car plist))))
316 "Calculate [U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK].
317Firstly, new K variables U1, U2, ..., UK, are inserted into each
318polynomial. Subsequently, P1, P2, ..., PK are destructively modified
319tantamount to replacing PI with UI*PI-1."
320 ;; Implementation note: we use STANDARD-EXTENSION and then subtract
321 ;; 1 from each polynomial; since UI*PI has no constant term,
322 ;; we just need to append the constant term at the end
323 ;; of each termlist.
324 (flet ((subtract-1 (p)
325 (append-item p (make-instance 'term :coeff -1 :dimension (+ k nvars)))))
326 (setf plist (mapc #'subtract-1 plist)))
327 plist)
328
329
330(defun standard-sum (F plist
331 &aux
332 (k (length plist))
333 (d (+ k (monom-dimension (poly-lt (car plist)))))
334 ;; Add k variables to f
335 (f (poly-list-add-variables f k))
336 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
337 (plist (apply #'nconc (poly-standard-extension plist))))
338 "Calculate the polynomial U1*P1+U2*P2+...+UK*PK-1, where PLIST=[P1,P2,...,PK].
339Firstly, new K variables, U1, U2, ..., UK, are inserted into each
340polynomial. Subsequently, P1, P2, ..., PK are destructively modified
341tantamount to replacing PI with UI*PI, and the resulting polynomials
342are added. It should be noted that the term order is not modified,
343which is equivalent to using a lexicographic order on the first K
344variables."
345 (setf (cdr (last (poly-termlist plist)))
346 ;; Add -1 as the last term
347 (list (make-term :monom (make-monom :dimension d)
348 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
349 (append f (list plist)))
350
351#|
352
353
354(defun saturation-extension-1 (ring f p)
355 "Calculate [F, U*P-1]. It destructively modifies F."
356 (declare (type ring ring))
357 (polysaturation-extension ring f (list p)))
358
359;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
360;;
361;; Evaluation of polynomial (prefix) expressions
362;;
363;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
364
365(defun coerce-coeff (ring expr vars)
366 "Coerce an element of the coefficient ring to a constant polynomial."
367 ;; Modular arithmetic handler by rat
368 (declare (type ring ring))
369 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
370 :coeff (funcall (ring-parse ring) expr)))
371 0))
372
373(defun poly-eval (expr vars
374 &optional
375 (ring +ring-of-integers+)
376 (order #'lex>)
377 (list-marker :[)
378 &aux
379 (ring-and-order (make-ring-and-order :ring ring :order order)))
380 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
381variables VARS. Return the resulting polynomial or list of
382polynomials. Standard arithmetical operators in form EXPR are
383replaced with their analogues in the ring of polynomials, and the
384resulting expression is evaluated, resulting in a polynomial or a list
385of polynomials in internal form. A similar operation in another computer
386algebra system could be called 'expand' or so."
387 (declare (type ring ring))
388 (labels ((p-eval (arg) (poly-eval arg vars ring order))
389 (p-eval-scalar (arg) (poly-eval-scalar arg))
390 (p-eval-list (args) (mapcar #'p-eval args))
391 (p-add (x y) (poly-add ring-and-order x y)))
392 (cond
393 ((null expr) (error "Empty expression"))
394 ((eql expr 0) (make-poly-zero))
395 ((member expr vars :test #'equalp)
396 (let ((pos (position expr vars :test #'equalp)))
397 (make-poly-variable ring (length vars) pos)))
398 ((atom expr)
399 (coerce-coeff ring expr vars))
400 ((eq (car expr) list-marker)
401 (cons list-marker (p-eval-list (cdr expr))))
402 (t
403 (case (car expr)
404 (+ (reduce #'p-add (p-eval-list (cdr expr))))
405 (- (case (length expr)
406 (1 (make-poly-zero))
407 (2 (poly-uminus ring (p-eval (cadr expr))))
408 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
409 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
410 (reduce #'p-add (p-eval-list (cddr expr)))))))
411 (*
412 (if (endp (cddr expr)) ;unary
413 (p-eval (cdr expr))
414 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
415 (/
416 ;; A polynomial can be divided by a scalar
417 (cond
418 ((endp (cddr expr))
419 ;; A special case (/ ?), the inverse
420 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
421 (t
422 (let ((num (p-eval (cadr expr)))
423 (denom-inverse (apply (ring-div ring)
424 (cons (funcall (ring-unit ring))
425 (mapcar #'p-eval-scalar (cddr expr))))))
426 (scalar-times-poly ring denom-inverse num)))))
427 (expt
428 (cond
429 ((member (cadr expr) vars :test #'equalp)
430 ;;Special handling of (expt var pow)
431 (let ((pos (position (cadr expr) vars :test #'equalp)))
432 (make-poly-variable ring (length vars) pos (caddr expr))))
433 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
434 ;; Negative power means division in coefficient ring
435 ;; Non-integer power means non-polynomial coefficient
436 (coerce-coeff ring expr vars))
437 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
438 (otherwise
439 (coerce-coeff ring expr vars)))))))
440
441(defun poly-eval-scalar (expr
442 &optional
443 (ring +ring-of-integers+)
444 &aux
445 (order #'lex>))
446 "Evaluate a scalar expression EXPR in ring RING."
447 (declare (type ring ring))
448 (poly-lc (poly-eval expr nil ring order)))
449
450(defun spoly (ring-and-order f g
451 &aux
452 (ring (ro-ring ring-and-order)))
453 "It yields the S-polynomial of polynomials F and G."
454 (declare (type ring-and-order ring-and-order) (type poly f g))
455 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
456 (mf (monom-div lcm (poly-lm f)))
457 (mg (monom-div lcm (poly-lm g))))
458 (declare (type monom mf mg))
459 (multiple-value-bind (c cf cg)
460 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
461 (declare (ignore c))
462 (poly-sub
463 ring-and-order
464 (scalar-times-poly ring cg (monom-times-poly mf f))
465 (scalar-times-poly ring cf (monom-times-poly mg g))))))
466
467
468(defun poly-primitive-part (ring p)
469 "Divide polynomial P with integer coefficients by gcd of its
470coefficients and return the result."
471 (declare (type ring ring) (type poly p))
472 (if (poly-zerop p)
473 (values p 1)
474 (let ((c (poly-content ring p)))
475 (values (make-poly-from-termlist
476 (mapcar
477 #'(lambda (x)
478 (make-term :monom (term-monom x)
479 :coeff (funcall (ring-div ring) (term-coeff x) c)))
480 (poly-termlist p))
481 (poly-sugar p))
482 c))))
483
484(defun poly-content (ring p)
485 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
486to compute the greatest common divisor."
487 (declare (type ring ring) (type poly p))
488 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
489
490(defun read-infix-form (&key (stream t))
491 "Parser of infix expressions with integer/rational coefficients
492The parser will recognize two kinds of polynomial expressions:
493
494- polynomials in fully expanded forms with coefficients
495 written in front of symbolic expressions; constants can be optionally
496 enclosed in (); for example, the infix form
497 X^2-Y^2+(-4/3)*U^2*W^3-5
498 parses to
499 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
500
501- lists of polynomials; for example
502 [X-Y, X^2+3*Z]
503 parses to
504 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
505 where the first symbol [ marks a list of polynomials.
506
507-other infix expressions, for example
508 [(X-Y)*(X+Y)/Z,(X+1)^2]
509parses to:
510 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
511Currently this function is implemented using M. Kantrowitz's INFIX package."
512 (read-from-string
513 (concatenate 'string
514 "#I("
515 (with-output-to-string (s)
516 (loop
517 (multiple-value-bind (line eof)
518 (read-line stream t)
519 (format s "~A" line)
520 (when eof (return)))))
521 ")")))
522
523(defun read-poly (vars &key
524 (stream t)
525 (ring +ring-of-integers+)
526 (order #'lex>))
527 "Reads an expression in prefix form from a stream STREAM.
528The expression read from the strem should represent a polynomial or a
529list of polynomials in variables VARS, over the ring RING. The
530polynomial or list of polynomials is returned, with terms in each
531polynomial ordered according to monomial order ORDER."
532 (poly-eval (read-infix-form :stream stream) vars ring order))
533
534(defun string->poly (str vars
535 &optional
536 (ring +ring-of-integers+)
537 (order #'lex>))
538 "Converts a string STR to a polynomial in variables VARS."
539 (with-input-from-string (s str)
540 (read-poly vars :stream s :ring ring :order order)))
541
542(defun poly->alist (p)
543 "Convert a polynomial P to an association list. Thus, the format of the
544returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
545MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
546corresponding coefficient in the ring."
547 (cond
548 ((poly-p p)
549 (mapcar #'term->cons (poly-termlist p)))
550 ((and (consp p) (eq (car p) :[))
551 (cons :[ (mapcar #'poly->alist (cdr p))))))
552
553(defun string->alist (str vars
554 &optional
555 (ring +ring-of-integers+)
556 (order #'lex>))
557 "Convert a string STR representing a polynomial or polynomial list to
558an association list (... (MONOM . COEFF) ...)."
559 (poly->alist (string->poly str vars ring order)))
560
561(defun poly-equal-no-sugar-p (p q)
562 "Compare polynomials for equality, ignoring sugar."
563 (declare (type poly p q))
564 (equalp (poly-termlist p) (poly-termlist q)))
565
566(defun poly-set-equal-no-sugar-p (p q)
567 "Compare polynomial sets P and Q for equality, ignoring sugar."
568 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
569
570(defun poly-list-equal-no-sugar-p (p q)
571 "Compare polynomial lists P and Q for equality, ignoring sugar."
572 (every #'poly-equal-no-sugar-p p q))
573|#
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