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source: branches/f4grobner/polynomial.lisp@ 1213

Last change on this file since 1213 was 1213, 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
23(defpackage "POLYNOMIAL"
24 (:use :cl :ring :ring-and-order :monomial :order :term :termlist :infix)
25 (:export "POLY"
26 "POLY-TERMLIST"
27 "POLY-SUGAR"
28 "POLY-LT"
29 "MAKE-POLY-FROM-TERMLIST"
30 "MAKE-POLY-ZERO"
31 "MAKE-VARIABLE"
32 "POLY-UNIT"
33 "POLY-LM"
34 "POLY-SECOND-LM"
35 "POLY-SECOND-LT"
36 "POLY-LC"
37 "POLY-SECOND-LC"
38 "POLY-ZEROP"
39 "POLY-LENGTH"
40 "SCALAR-TIMES-POLY"
41 "SCALAR-TIMES-POLY-1"
42 "MONOM-TIMES-POLY"
43 "TERM-TIMES-POLY"
44 "POLY-ADD"
45 "POLY-SUB"
46 "POLY-UMINUS"
47 "POLY-MUL"
48 "POLY-EXPT"
49 "POLY-APPEND"
50 "POLY-NREVERSE"
51 "POLY-CONTRACT"
52 "POLY-EXTEND"
53 "POLY-ADD-VARIABLES"
54 "POLY-LIST-ADD-VARIABLES"
55 "POLY-STANDARD-EXTENSION"
56 "SATURATION-EXTENSION"
57 "POLYSATURATION-EXTENSION"
58 "SATURATION-EXTENSION-1"
59 "COERCE-COEFF"
60 "POLY-EVAL"
61 "POLY-EVAL-SCALAR"
62 "SPOLY"
63 "POLY-PRIMITIVE-PART"
64 "POLY-CONTENT"
65 "READ-INFIX-FORM"
66 "READ-POLY"
67 "STRING->POLY"
68 "POLY->ALIST"
69 "STRING->ALIST"
70 ))
71
72(in-package :polynomial)
73
74;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
75;;
76;; Polynomials
77;;
78;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
79
80(defstruct (poly
81 ;;
82 ;; BOA constructor, by default constructs zero polynomial
83 (:constructor make-poly-from-termlist (termlist &optional (sugar (termlist-sugar termlist))))
84 (:constructor make-poly-zero (&aux (termlist nil) (sugar -1)))
85 ;; Constructor of polynomials representing a variable
86 (:constructor make-variable (ring nvars pos &optional (power 1)
87 &aux
88 (termlist (list
89 (make-term-variable ring nvars pos power)))
90 (sugar power)))
91 (:constructor poly-unit (ring dimension
92 &aux
93 (termlist (termlist-unit ring dimension))
94 (sugar 0))))
95 (termlist nil :type list)
96 (sugar -1 :type fixnum))
97
98;; Leading term
99(defmacro poly-lt (p) `(car (poly-termlist ,p)))
100
101;; Second term
102(defmacro poly-second-lt (p) `(cadar (poly-termlist ,p)))
103
104;; Leading monomial
105(defun poly-lm (p) (term-monom (poly-lt p)))
106
107;; Second monomial
108(defun poly-second-lm (p) (term-monom (poly-second-lt p)))
109
110;; Leading coefficient
111(defun poly-lc (p) (term-coeff (poly-lt p)))
112
113;; Second coefficient
114(defun poly-second-lc (p) (term-coeff (poly-second-lt p)))
115
116;; Testing for a zero polynomial
117(defun poly-zerop (p) (null (poly-termlist p)))
118
119;; The number of terms
120(defun poly-length (p) (length (poly-termlist p)))
121
122(defun scalar-times-poly (ring c p)
123 (declare (type ring ring) (poly p))
124 (make-poly-from-termlist (scalar-times-termlist ring c (poly-termlist p)) (poly-sugar p)))
125
126(defun scalar-times-poly-1 (ring c p)
127 "The scalar product of scalar C by a polynomial P, omitting the head term. The sugar of the
128original polynomial becomes the sugar of the result."
129 (declare (type ring ring) (poly p))
130 (make-poly-from-termlist (scalar-times-termlist ring c (cdr (poly-termlist p))) (poly-sugar p)))
131
132(defun monom-times-poly (m p)
133 (declare (poly p))
134 (make-poly-from-termlist
135 (monom-times-termlist m (poly-termlist p))
136 (+ (poly-sugar p) (monom-sugar m))))
137
138(defun term-times-poly (ring term p)
139 (declare (type ring ring) (type term term) (type poly p))
140 (make-poly-from-termlist
141 (term-times-termlist ring term (poly-termlist p))
142 (+ (poly-sugar p) (term-sugar term))))
143
144(defun poly-add (ring-and-order p q)
145 (declare (type ring-and-order ring-and-order) (type poly p q))
146 (make-poly-from-termlist
147 (termlist-add ring-and-order
148 (poly-termlist p)
149 (poly-termlist q))
150 (max (poly-sugar p) (poly-sugar q))))
151
152(defun poly-sub (ring-and-order p q)
153 (declare (type ring-and-order ring-and-order) (type poly p q))
154 (make-poly-from-termlist
155 (termlist-sub ring-and-order (poly-termlist p) (poly-termlist q))
156 (max (poly-sugar p) (poly-sugar q))))
157
158(defun poly-uminus (ring p)
159 (declare (type ring ring) (type poly p))
160 (make-poly-from-termlist
161 (termlist-uminus ring (poly-termlist p))
162 (poly-sugar p)))
163
164(defun poly-mul (ring-and-order p q)
165 (declare (type ring-and-order ring-and-order) (type poly p q))
166 (make-poly-from-termlist
167 (termlist-mul ring-and-order (poly-termlist p) (poly-termlist q))
168 (+ (poly-sugar p) (poly-sugar q))))
169
170(defun poly-expt (ring-and-order p n)
171 (declare (type ring-and-order ring-and-order) (type poly p))
172 (make-poly-from-termlist (termlist-expt ring-and-order (poly-termlist p) n) (* n (poly-sugar p))))
173
174(defun poly-append (&rest plist)
175 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
176 (apply #'max (mapcar #'poly-sugar plist))))
177
178(defun poly-nreverse (p)
179 (declare (type poly p))
180 (setf (poly-termlist p) (nreverse (poly-termlist p)))
181 p)
182
183(defun poly-contract (p &optional (k 1))
184 (declare (type poly p))
185 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
186 (poly-sugar p)))
187
188(defun poly-extend (p &optional (m (make-monom :dimension 1)))
189 (declare (type poly p))
190 (make-poly-from-termlist
191 (termlist-extend (poly-termlist p) m)
192 (+ (poly-sugar p) (monom-sugar m))))
193
194(defun poly-add-variables (p k)
195 (declare (type poly p))
196 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
197 p)
198
199(defun poly-list-add-variables (plist k)
200 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
201
202(defun poly-standard-extension (plist &aux (k (length plist)))
203 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
204 (declare (list plist) (fixnum k))
205 (labels ((incf-power (g i)
206 (dolist (x (poly-termlist g))
207 (incf (monom-elt (term-monom x) i)))
208 (incf (poly-sugar g))))
209 (setf plist (poly-list-add-variables plist k))
210 (dotimes (i k plist)
211 (incf-power (nth i plist) i))))
212
213(defun saturation-extension (ring f plist &aux (k (length plist)) (d (monom-dimension (poly-lm (car plist)))))
214 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
215 (setf f (poly-list-add-variables f k)
216 plist (mapcar #'(lambda (x)
217 (setf (poly-termlist x) (nconc (poly-termlist x)
218 (list (make-term (make-monom :dimension d)
219 (funcall (ring-uminus ring) (funcall (ring-unit ring)))))))
220 x)
221 (poly-standard-extension plist)))
222 (append f plist))
223
224
225(defun polysaturation-extension (ring f plist &aux (k (length plist))
226 (d (+ k (monom-dimension (poly-lm (car plist))))))
227 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]."
228 (setf f (poly-list-add-variables f k)
229 plist (apply #'poly-append (poly-standard-extension plist))
230 (cdr (last (poly-termlist plist))) (list (make-term (make-monom :dimension d)
231 (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
232 (append f (list plist)))
233
234(defun saturation-extension-1 (ring f p) (polysaturation-extension ring f (list p)))
235
236;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
237;;
238;; Evaluation of polynomial (prefix) expressions
239;;
240;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
241
242(defun coerce-coeff (ring expr vars)
243 "Coerce an element of the coefficient ring to a constant polynomial."
244 ;; Modular arithmetic handler by rat
245 (make-poly-from-termlist (list (make-term (make-monom :dimension (length vars))
246 (funcall (ring-parse ring) expr)))
247 0))
248
249(defun poly-eval (expr vars
250 &optional
251 (ring *ring-of-integers*)
252 (order #'lex>)
253 (list-marker :[)
254 &aux
255 (ring-and-order (make-ring-and-order :ring ring :order order)))
256 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
257variables VARS. Return the resulting polynomial or list of
258polynomials. Standard arithmetical operators in form EXPR are
259replaced with their analogues in the ring of polynomials, and the
260resulting expression is evaluated, resulting in a polynomial or a list
261of polynomials in internal form. A similar operation in another computer
262algebra system could be called 'expand' or so."
263 (labels ((p-eval (arg) (poly-eval arg vars ring order))
264 (p-eval-scalar (arg) (poly-eval-scalar arg))
265 (p-eval-list (args) (mapcar #'p-eval args))
266 (p-add (x y) (poly-add ring-and-order x y)))
267 (cond
268 ((null expr) (error "Empty expression"))
269 ((eql expr 0) (make-poly-zero))
270 ((member expr vars :test #'equalp)
271 (let ((pos (position expr vars :test #'equalp)))
272 (make-variable ring (length vars) pos)))
273 ((atom expr)
274 (coerce-coeff ring expr vars))
275 ((eq (car expr) list-marker)
276 (cons list-marker (p-eval-list (cdr expr))))
277 (t
278 (case (car expr)
279 (+ (reduce #'p-add (p-eval-list (cdr expr))))
280 (- (case (length expr)
281 (1 (make-poly-zero))
282 (2 (poly-uminus ring (p-eval (cadr expr))))
283 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
284 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
285 (reduce #'p-add (p-eval-list (cddr expr)))))))
286 (*
287 (if (endp (cddr expr)) ;unary
288 (p-eval (cdr expr))
289 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
290 (/
291 ;; A polynomial can be divided by a scalar
292 (cond
293 ((endp (cddr expr))
294 ;; A special case (/ ?), the inverse
295 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
296 (t
297 (let ((num (p-eval (cadr expr)))
298 (denom-inverse (apply (ring-div ring)
299 (cons (funcall (ring-unit ring))
300 (mapcar #'p-eval-scalar (cddr expr))))))
301 (scalar-times-poly ring denom-inverse num)))))
302 (expt
303 (cond
304 ((member (cadr expr) vars :test #'equalp)
305 ;;Special handling of (expt var pow)
306 (let ((pos (position (cadr expr) vars :test #'equalp)))
307 (make-variable ring (length vars) pos (caddr expr))))
308 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
309 ;; Negative power means division in coefficient ring
310 ;; Non-integer power means non-polynomial coefficient
311 (coerce-coeff ring expr vars))
312 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
313 (otherwise
314 (coerce-coeff ring expr vars)))))))
315
316(defun poly-eval-scalar (expr
317 &optional
318 (ring *ring-of-integers*)
319 &aux
320 (order #'lex>))
321 "Evaluate a scalar expression EXPR in ring RING."
322 (poly-lc (poly-eval expr nil ring order)))
323
324(defun spoly (ring-and-order f g
325 &aux
326 (ring (ro-ring ring-and-order)))
327 "It yields the S-polynomial of polynomials F and G."
328 (declare (type poly f g))
329 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
330 (mf (monom-div lcm (poly-lm f)))
331 (mg (monom-div lcm (poly-lm g))))
332 (declare (type monom mf mg))
333 (multiple-value-bind (c cf cg)
334 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
335 (declare (ignore c))
336 (poly-sub
337 ring-and-order
338 (scalar-times-poly ring cg (monom-times-poly mf f))
339 (scalar-times-poly ring cf (monom-times-poly mg g))))))
340
341
342(defun poly-primitive-part (ring p)
343 "Divide polynomial P with integer coefficients by gcd of its
344coefficients and return the result."
345 (declare (type poly p))
346 (if (poly-zerop p)
347 (values p 1)
348 (let ((c (poly-content ring p)))
349 (values (make-poly-from-termlist
350 (mapcar
351 #'(lambda (x)
352 (make-term (term-monom x)
353 (funcall (ring-div ring) (term-coeff x) c)))
354 (poly-termlist p))
355 (poly-sugar p))
356 c))))
357
358(defun poly-content (ring p)
359 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
360to compute the greatest common divisor."
361 (declare (type poly p))
362 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
363
364(defun read-infix-form (&key (stream t))
365 "Parser of infix expressions with integer/rational coefficients
366The parser will recognize two kinds of polynomial expressions:
367
368- polynomials in fully expanded forms with coefficients
369 written in front of symbolic expressions; constants can be optionally
370 enclosed in (); for example, the infix form
371 X^2-Y^2+(-4/3)*U^2*W^3-5
372 parses to
373 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
374
375- lists of polynomials; for example
376 [X-Y, X^2+3*Z]
377 parses to
378 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
379 where the first symbol [ marks a list of polynomials.
380
381-other infix expressions, for example
382 [(X-Y)*(X+Y)/Z,(X+1)^2]
383parses to:
384 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
385Currently this function is implemented using M. Kantrowitz's INFIX package."
386 (read-from-string
387 (concatenate 'string
388 "#I("
389 (with-output-to-string (s)
390 (loop
391 (multiple-value-bind (line eof)
392 (read-line stream t)
393 (format s "~A" line)
394 (when eof (return)))))
395 ")")))
396
397(defun read-poly (vars &key
398 (stream t)
399 (ring *ring-of-integers*)
400 (order #'lex>))
401 "Reads an expression in prefix form from a stream STREAM.
402The expression read from the strem should represent a polynomial or a
403list of polynomials in variables VARS, over the ring RING. The
404polynomial or list of polynomials is returned, with terms in each
405polynomial ordered according to monomial order ORDER."
406 (poly-eval (read-infix-form :stream stream) vars ring order))
407
408(defun string->poly (str vars
409 &optional
410 (ring *ring-of-integers*)
411 (order #'lex>))
412 "Converts a string STR to a polynomial in variables VARS."
413 (with-input-from-string (s str)
414 (read-poly vars :stream s :ring ring :order order)))
415
416(defun poly->alist (p)
417 "Convert a polynomial P to an association list. Thus, the format of the
418returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
419MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
420corresponding coefficient in the ring."
421 (cond
422 ((poly-p p)
423 (mapcar #'term->cons (poly-termlist p)))
424 ((and (consp p) (eq (car p) :[))
425 (cons :[ (mapcar #'poly->alist (cdr p))))))
426
427(defun string->alist (str vars
428 &optional
429 (ring *ring-of-integers*)
430 (order #'lex>))
431 "Convert a string STR representing a polynomial or polynomial list to
432an association list (... (MONOM . COEFF) ...)."
433 (poly->alist (string->poly str vars ring order)))
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