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source: branches/f4grobner/polynomial.lisp@ 2547

Last change on this file since 2547 was 2545, checked in by Marek Rychlik, 9 years ago

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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 (:documentation "Implements polynomials"))
26
27(in-package :polynomial)
28
29(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
30
31#|
32 ;;
33 ;; BOA constructor, by default constructs zero polynomial
34 (:constructor make-poly-from-termlist (termlist &optional (sugar (termlist-sugar termlist))))
35 (:constructor make-poly-zero (&aux (termlist nil) (sugar -1)))
36 ;; Constructor of polynomials representing a variable
37 (:constructor make-poly-variable (ring nvars pos &optional (power 1)
38 &aux
39 (termlist (list
40 (make-term-variable ring nvars pos power)))
41 (sugar power)))
42 (:constructor poly-unit (ring dimension
43 &aux
44 (termlist (termlist-unit ring dimension))
45 (sugar 0))))
46
47|#
48
49(defclass poly ()
50 ((termlist :initarg :termlist :accessor poly-termlist))
51 (:default-initargs :termlist nil))
52
53(defmethod print-object ((self poly) stream)
54 (format stream "#<POLY TERMLIST=~A >" (poly-termlist self)))
55
56(defmethod insert-item ((self poly) (item term))
57 (push item (poly-termlist self))
58 self)
59
60(defmethod append-item ((self poly) (item term))
61 (setf (cdr (last (poly-termlist self))) (list item))
62 self)
63
64;; Leading term
65(defgeneric leading-term (object)
66 (:method ((self poly))
67 (car (poly-termlist self)))
68 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
69
70;; Second term
71(defgeneric second-leading-term (object)
72 (:method ((self poly))
73 (cadar (poly-termlist self)))
74 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
75
76;; Leading coefficient
77(defgeneric leading-coefficient (object)
78 (:method ((self poly))
79 (r-coeff (leading-term self)))
80 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
81
82;; Second coefficient
83(defgeneric second-leading-coefficient (object)
84 (:method ((self poly))
85 (r-coeff (second-leading-term self)))
86 (:documentation "The second leading coefficient of a polynomial. It signals error for a polynomial with at most one term."))
87
88;; Testing for a zero polynomial
89(defmethod r-zerop ((self poly))
90 (null (poly-termlist self)))
91
92;; The number of terms
93(defmethod r-length ((self poly))
94 (length (poly-termlist self)))
95
96(defmethod multiply-by ((self poly) (other monom))
97 (mapc #'(lambda (term) (multiply-by term other))
98 (poly-termlist self))
99 self)
100
101(defmethod multiply-by ((self poly) (other scalar))
102 (mapc #'(lambda (term) (multiply-by term other))
103 (poly-termlist self))
104 self)
105
106(defmethod add-to ((self poly) (other poly))
107 (macrolet ((lt (termlist) `(car ,termlist))
108 (lc (termlist) `(r-coeff (car ,termlist))))
109 (with-slots ((termlist1 termlist))
110 self
111 (with-slots ((termlist2 termlist))
112 other
113 (do ((p termlist1 (cdr p))
114 (q termlist2))
115 ((endp p)
116 ;; Include remaining terms of termlist1
117 (setf termlist1 (nconc termlist1 q)))
118 ;; Copy all initial terms of q greater than (lt p) into p
119 (do ((r q (cdr q)))
120 ((lex> (lt r) (lt p)))
121 (push (lt r) p))
122 ;; Now compare leading terms of p and q
123 (cond
124 ((monom= (lt p) (lt q))
125 ;; Simply add coefficients
126 (setf (lc p) (add-to (lc p) (lc q)))))))))
127 self)
128
129(defmethod subtract-from ((self poly) (other poly)))
130
131(defmethod unary-uminus ((self poly)))
132
133#|
134
135(defun poly-standard-extension (plist &aux (k (length plist)))
136 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
137 (declare (list plist) (fixnum k))
138 (labels ((incf-power (g i)
139 (dolist (x (poly-termlist g))
140 (incf (monom-elt (term-monom x) i)))
141 (incf (poly-sugar g))))
142 (setf plist (poly-list-add-variables plist k))
143 (dotimes (i k plist)
144 (incf-power (nth i plist) i))))
145
146(defun saturation-extension (ring f plist
147 &aux
148 (k (length plist))
149 (d (monom-dimension (poly-lm (car plist))))
150 f-x plist-x)
151 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
152 (declare (type ring ring))
153 (setf f-x (poly-list-add-variables f k)
154 plist-x (mapcar #'(lambda (x)
155 (setf (poly-termlist x)
156 (nconc (poly-termlist x)
157 (list (make-term :monom (make-monom :dimension d)
158 :coeff (funcall (ring-uminus ring)
159 (funcall (ring-unit ring)))))))
160 x)
161 (poly-standard-extension plist)))
162 (append f-x plist-x))
163
164
165(defun polysaturation-extension (ring f plist
166 &aux
167 (k (length plist))
168 (d (+ k (monom-dimension (poly-lm (car plist)))))
169 ;; Add k variables to f
170 (f (poly-list-add-variables f k))
171 ;; Set PLIST to [U1*P1,U2*P2,...,UK*PK]
172 (plist (apply #'poly-append (poly-standard-extension plist))))
173 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]. It destructively modifies F."
174 ;; Add -1 as the last term
175 (declare (type ring ring))
176 (setf (cdr (last (poly-termlist plist)))
177 (list (make-term :monom (make-monom :dimension d)
178 :coeff (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
179 (append f (list plist)))
180
181(defun saturation-extension-1 (ring f p)
182 "Calculate [F, U*P-1]. It destructively modifies F."
183 (declare (type ring ring))
184 (polysaturation-extension ring f (list p)))
185
186;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
187;;
188;; Evaluation of polynomial (prefix) expressions
189;;
190;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
191
192(defun coerce-coeff (ring expr vars)
193 "Coerce an element of the coefficient ring to a constant polynomial."
194 ;; Modular arithmetic handler by rat
195 (declare (type ring ring))
196 (make-poly-from-termlist (list (make-term :monom (make-monom :dimension (length vars))
197 :coeff (funcall (ring-parse ring) expr)))
198 0))
199
200(defun poly-eval (expr vars
201 &optional
202 (ring +ring-of-integers+)
203 (order #'lex>)
204 (list-marker :[)
205 &aux
206 (ring-and-order (make-ring-and-order :ring ring :order order)))
207 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
208variables VARS. Return the resulting polynomial or list of
209polynomials. Standard arithmetical operators in form EXPR are
210replaced with their analogues in the ring of polynomials, and the
211resulting expression is evaluated, resulting in a polynomial or a list
212of polynomials in internal form. A similar operation in another computer
213algebra system could be called 'expand' or so."
214 (declare (type ring ring))
215 (labels ((p-eval (arg) (poly-eval arg vars ring order))
216 (p-eval-scalar (arg) (poly-eval-scalar arg))
217 (p-eval-list (args) (mapcar #'p-eval args))
218 (p-add (x y) (poly-add ring-and-order x y)))
219 (cond
220 ((null expr) (error "Empty expression"))
221 ((eql expr 0) (make-poly-zero))
222 ((member expr vars :test #'equalp)
223 (let ((pos (position expr vars :test #'equalp)))
224 (make-poly-variable ring (length vars) pos)))
225 ((atom expr)
226 (coerce-coeff ring expr vars))
227 ((eq (car expr) list-marker)
228 (cons list-marker (p-eval-list (cdr expr))))
229 (t
230 (case (car expr)
231 (+ (reduce #'p-add (p-eval-list (cdr expr))))
232 (- (case (length expr)
233 (1 (make-poly-zero))
234 (2 (poly-uminus ring (p-eval (cadr expr))))
235 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
236 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
237 (reduce #'p-add (p-eval-list (cddr expr)))))))
238 (*
239 (if (endp (cddr expr)) ;unary
240 (p-eval (cdr expr))
241 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
242 (/
243 ;; A polynomial can be divided by a scalar
244 (cond
245 ((endp (cddr expr))
246 ;; A special case (/ ?), the inverse
247 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
248 (t
249 (let ((num (p-eval (cadr expr)))
250 (denom-inverse (apply (ring-div ring)
251 (cons (funcall (ring-unit ring))
252 (mapcar #'p-eval-scalar (cddr expr))))))
253 (scalar-times-poly ring denom-inverse num)))))
254 (expt
255 (cond
256 ((member (cadr expr) vars :test #'equalp)
257 ;;Special handling of (expt var pow)
258 (let ((pos (position (cadr expr) vars :test #'equalp)))
259 (make-poly-variable ring (length vars) pos (caddr expr))))
260 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
261 ;; Negative power means division in coefficient ring
262 ;; Non-integer power means non-polynomial coefficient
263 (coerce-coeff ring expr vars))
264 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
265 (otherwise
266 (coerce-coeff ring expr vars)))))))
267
268(defun poly-eval-scalar (expr
269 &optional
270 (ring +ring-of-integers+)
271 &aux
272 (order #'lex>))
273 "Evaluate a scalar expression EXPR in ring RING."
274 (declare (type ring ring))
275 (poly-lc (poly-eval expr nil ring order)))
276
277(defun spoly (ring-and-order f g
278 &aux
279 (ring (ro-ring ring-and-order)))
280 "It yields the S-polynomial of polynomials F and G."
281 (declare (type ring-and-order ring-and-order) (type poly f g))
282 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
283 (mf (monom-div lcm (poly-lm f)))
284 (mg (monom-div lcm (poly-lm g))))
285 (declare (type monom mf mg))
286 (multiple-value-bind (c cf cg)
287 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
288 (declare (ignore c))
289 (poly-sub
290 ring-and-order
291 (scalar-times-poly ring cg (monom-times-poly mf f))
292 (scalar-times-poly ring cf (monom-times-poly mg g))))))
293
294
295(defun poly-primitive-part (ring p)
296 "Divide polynomial P with integer coefficients by gcd of its
297coefficients and return the result."
298 (declare (type ring ring) (type poly p))
299 (if (poly-zerop p)
300 (values p 1)
301 (let ((c (poly-content ring p)))
302 (values (make-poly-from-termlist
303 (mapcar
304 #'(lambda (x)
305 (make-term :monom (term-monom x)
306 :coeff (funcall (ring-div ring) (term-coeff x) c)))
307 (poly-termlist p))
308 (poly-sugar p))
309 c))))
310
311(defun poly-content (ring p)
312 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
313to compute the greatest common divisor."
314 (declare (type ring ring) (type poly p))
315 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
316
317(defun read-infix-form (&key (stream t))
318 "Parser of infix expressions with integer/rational coefficients
319The parser will recognize two kinds of polynomial expressions:
320
321- polynomials in fully expanded forms with coefficients
322 written in front of symbolic expressions; constants can be optionally
323 enclosed in (); for example, the infix form
324 X^2-Y^2+(-4/3)*U^2*W^3-5
325 parses to
326 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
327
328- lists of polynomials; for example
329 [X-Y, X^2+3*Z]
330 parses to
331 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
332 where the first symbol [ marks a list of polynomials.
333
334-other infix expressions, for example
335 [(X-Y)*(X+Y)/Z,(X+1)^2]
336parses to:
337 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
338Currently this function is implemented using M. Kantrowitz's INFIX package."
339 (read-from-string
340 (concatenate 'string
341 "#I("
342 (with-output-to-string (s)
343 (loop
344 (multiple-value-bind (line eof)
345 (read-line stream t)
346 (format s "~A" line)
347 (when eof (return)))))
348 ")")))
349
350(defun read-poly (vars &key
351 (stream t)
352 (ring +ring-of-integers+)
353 (order #'lex>))
354 "Reads an expression in prefix form from a stream STREAM.
355The expression read from the strem should represent a polynomial or a
356list of polynomials in variables VARS, over the ring RING. The
357polynomial or list of polynomials is returned, with terms in each
358polynomial ordered according to monomial order ORDER."
359 (poly-eval (read-infix-form :stream stream) vars ring order))
360
361(defun string->poly (str vars
362 &optional
363 (ring +ring-of-integers+)
364 (order #'lex>))
365 "Converts a string STR to a polynomial in variables VARS."
366 (with-input-from-string (s str)
367 (read-poly vars :stream s :ring ring :order order)))
368
369(defun poly->alist (p)
370 "Convert a polynomial P to an association list. Thus, the format of the
371returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
372MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
373corresponding coefficient in the ring."
374 (cond
375 ((poly-p p)
376 (mapcar #'term->cons (poly-termlist p)))
377 ((and (consp p) (eq (car p) :[))
378 (cons :[ (mapcar #'poly->alist (cdr p))))))
379
380(defun string->alist (str vars
381 &optional
382 (ring +ring-of-integers+)
383 (order #'lex>))
384 "Convert a string STR representing a polynomial or polynomial list to
385an association list (... (MONOM . COEFF) ...)."
386 (poly->alist (string->poly str vars ring order)))
387
388(defun poly-equal-no-sugar-p (p q)
389 "Compare polynomials for equality, ignoring sugar."
390 (declare (type poly p q))
391 (equalp (poly-termlist p) (poly-termlist q)))
392
393(defun poly-set-equal-no-sugar-p (p q)
394 "Compare polynomial sets P and Q for equality, ignoring sugar."
395 (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p )))
396
397(defun poly-list-equal-no-sugar-p (p q)
398 "Compare polynomial lists P and Q for equality, ignoring sugar."
399 (every #'poly-equal-no-sugar-p p q))
400|#
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