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source: branches/f4grobner/polynomial.lisp@ 1214

Last change on this file since 1214 was 1214, 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 "The scalar product of scalar C by a polynomial P. The sugar of the
124original polynomial becomes the sugar of the result."
125 (declare (type ring ring) (poly p))
126 (make-poly-from-termlist (scalar-times-termlist ring c (poly-termlist p)) (poly-sugar p)))
127
128(defun scalar-times-poly-1 (ring c p)
129 "The scalar product of scalar C by a polynomial P, omitting the head term. The sugar of the
130original polynomial becomes the sugar of the result."
131 (declare (type ring ring) (poly p))
132 (make-poly-from-termlist (scalar-times-termlist ring c (cdr (poly-termlist p))) (poly-sugar p)))
133
134(defun monom-times-poly (m p)
135 (declare (poly p))
136 (make-poly-from-termlist
137 (monom-times-termlist m (poly-termlist p))
138 (+ (poly-sugar p) (monom-sugar m))))
139
140(defun term-times-poly (ring term p)
141 (declare (type ring ring) (type term term) (type poly p))
142 (make-poly-from-termlist
143 (term-times-termlist ring term (poly-termlist p))
144 (+ (poly-sugar p) (term-sugar term))))
145
146(defun poly-add (ring-and-order p q)
147 (declare (type ring-and-order ring-and-order) (type poly p q))
148 (make-poly-from-termlist
149 (termlist-add ring-and-order
150 (poly-termlist p)
151 (poly-termlist q))
152 (max (poly-sugar p) (poly-sugar q))))
153
154(defun poly-sub (ring-and-order p q)
155 (declare (type ring-and-order ring-and-order) (type poly p q))
156 (make-poly-from-termlist
157 (termlist-sub ring-and-order (poly-termlist p) (poly-termlist q))
158 (max (poly-sugar p) (poly-sugar q))))
159
160(defun poly-uminus (ring p)
161 (declare (type ring ring) (type poly p))
162 (make-poly-from-termlist
163 (termlist-uminus ring (poly-termlist p))
164 (poly-sugar p)))
165
166(defun poly-mul (ring-and-order p q)
167 (declare (type ring-and-order ring-and-order) (type poly p q))
168 (make-poly-from-termlist
169 (termlist-mul ring-and-order (poly-termlist p) (poly-termlist q))
170 (+ (poly-sugar p) (poly-sugar q))))
171
172(defun poly-expt (ring-and-order p n)
173 (declare (type ring-and-order ring-and-order) (type poly p))
174 (make-poly-from-termlist (termlist-expt ring-and-order (poly-termlist p) n) (* n (poly-sugar p))))
175
176(defun poly-append (&rest plist)
177 (make-poly-from-termlist (apply #'append (mapcar #'poly-termlist plist))
178 (apply #'max (mapcar #'poly-sugar plist))))
179
180(defun poly-nreverse (p)
181 (declare (type poly p))
182 (setf (poly-termlist p) (nreverse (poly-termlist p)))
183 p)
184
185(defun poly-contract (p &optional (k 1))
186 (declare (type poly p))
187 (make-poly-from-termlist (termlist-contract (poly-termlist p) k)
188 (poly-sugar p)))
189
190(defun poly-extend (p &optional (m (make-monom :dimension 1)))
191 (declare (type poly p))
192 (make-poly-from-termlist
193 (termlist-extend (poly-termlist p) m)
194 (+ (poly-sugar p) (monom-sugar m))))
195
196(defun poly-add-variables (p k)
197 (declare (type poly p))
198 (setf (poly-termlist p) (termlist-add-variables (poly-termlist p) k))
199 p)
200
201(defun poly-list-add-variables (plist k)
202 (mapcar #'(lambda (p) (poly-add-variables p k)) plist))
203
204(defun poly-standard-extension (plist &aux (k (length plist)))
205 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]."
206 (declare (list plist) (fixnum k))
207 (labels ((incf-power (g i)
208 (dolist (x (poly-termlist g))
209 (incf (monom-elt (term-monom x) i)))
210 (incf (poly-sugar g))))
211 (setf plist (poly-list-add-variables plist k))
212 (dotimes (i k plist)
213 (incf-power (nth i plist) i))))
214
215(defun saturation-extension (ring f plist &aux (k (length plist)) (d (monom-dimension (poly-lm (car plist)))))
216 "Calculate [F, U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK]."
217 (setf f (poly-list-add-variables f k)
218 plist (mapcar #'(lambda (x)
219 (setf (poly-termlist x) (nconc (poly-termlist x)
220 (list (make-term (make-monom :dimension d)
221 (funcall (ring-uminus ring) (funcall (ring-unit ring)))))))
222 x)
223 (poly-standard-extension plist)))
224 (append f plist))
225
226
227(defun polysaturation-extension (ring f plist &aux (k (length plist))
228 (d (+ k (monom-dimension (poly-lm (car plist))))))
229 "Calculate [F, U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]."
230 (setf f (poly-list-add-variables f k)
231 plist (apply #'poly-append (poly-standard-extension plist))
232 (cdr (last (poly-termlist plist))) (list (make-term (make-monom :dimension d)
233 (funcall (ring-uminus ring) (funcall (ring-unit ring))))))
234 (append f (list plist)))
235
236(defun saturation-extension-1 (ring f p) (polysaturation-extension ring f (list p)))
237
238;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
239;;
240;; Evaluation of polynomial (prefix) expressions
241;;
242;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
243
244(defun coerce-coeff (ring expr vars)
245 "Coerce an element of the coefficient ring to a constant polynomial."
246 ;; Modular arithmetic handler by rat
247 (make-poly-from-termlist (list (make-term (make-monom :dimension (length vars))
248 (funcall (ring-parse ring) expr)))
249 0))
250
251(defun poly-eval (expr vars
252 &optional
253 (ring *ring-of-integers*)
254 (order #'lex>)
255 (list-marker :[)
256 &aux
257 (ring-and-order (make-ring-and-order :ring ring :order order)))
258 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
259variables VARS. Return the resulting polynomial or list of
260polynomials. Standard arithmetical operators in form EXPR are
261replaced with their analogues in the ring of polynomials, and the
262resulting expression is evaluated, resulting in a polynomial or a list
263of polynomials in internal form. A similar operation in another computer
264algebra system could be called 'expand' or so."
265 (labels ((p-eval (arg) (poly-eval arg vars ring order))
266 (p-eval-scalar (arg) (poly-eval-scalar arg))
267 (p-eval-list (args) (mapcar #'p-eval args))
268 (p-add (x y) (poly-add ring-and-order x y)))
269 (cond
270 ((null expr) (error "Empty expression"))
271 ((eql expr 0) (make-poly-zero))
272 ((member expr vars :test #'equalp)
273 (let ((pos (position expr vars :test #'equalp)))
274 (make-variable ring (length vars) pos)))
275 ((atom expr)
276 (coerce-coeff ring expr vars))
277 ((eq (car expr) list-marker)
278 (cons list-marker (p-eval-list (cdr expr))))
279 (t
280 (case (car expr)
281 (+ (reduce #'p-add (p-eval-list (cdr expr))))
282 (- (case (length expr)
283 (1 (make-poly-zero))
284 (2 (poly-uminus ring (p-eval (cadr expr))))
285 (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
286 (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
287 (reduce #'p-add (p-eval-list (cddr expr)))))))
288 (*
289 (if (endp (cddr expr)) ;unary
290 (p-eval (cdr expr))
291 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
292 (/
293 ;; A polynomial can be divided by a scalar
294 (cond
295 ((endp (cddr expr))
296 ;; A special case (/ ?), the inverse
297 (coerce-coeff ring (apply (ring-div ring) (cdr expr)) vars))
298 (t
299 (let ((num (p-eval (cadr expr)))
300 (denom-inverse (apply (ring-div ring)
301 (cons (funcall (ring-unit ring))
302 (mapcar #'p-eval-scalar (cddr expr))))))
303 (scalar-times-poly ring denom-inverse num)))))
304 (expt
305 (cond
306 ((member (cadr expr) vars :test #'equalp)
307 ;;Special handling of (expt var pow)
308 (let ((pos (position (cadr expr) vars :test #'equalp)))
309 (make-variable ring (length vars) pos (caddr expr))))
310 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
311 ;; Negative power means division in coefficient ring
312 ;; Non-integer power means non-polynomial coefficient
313 (coerce-coeff ring expr vars))
314 (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
315 (otherwise
316 (coerce-coeff ring expr vars)))))))
317
318(defun poly-eval-scalar (expr
319 &optional
320 (ring *ring-of-integers*)
321 &aux
322 (order #'lex>))
323 "Evaluate a scalar expression EXPR in ring RING."
324 (poly-lc (poly-eval expr nil ring order)))
325
326(defun spoly (ring-and-order f g
327 &aux
328 (ring (ro-ring ring-and-order)))
329 "It yields the S-polynomial of polynomials F and G."
330 (declare (type poly f g))
331 (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
332 (mf (monom-div lcm (poly-lm f)))
333 (mg (monom-div lcm (poly-lm g))))
334 (declare (type monom mf mg))
335 (multiple-value-bind (c cf cg)
336 (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
337 (declare (ignore c))
338 (poly-sub
339 ring-and-order
340 (scalar-times-poly ring cg (monom-times-poly mf f))
341 (scalar-times-poly ring cf (monom-times-poly mg g))))))
342
343
344(defun poly-primitive-part (ring p)
345 "Divide polynomial P with integer coefficients by gcd of its
346coefficients and return the result."
347 (declare (type poly p))
348 (if (poly-zerop p)
349 (values p 1)
350 (let ((c (poly-content ring p)))
351 (values (make-poly-from-termlist
352 (mapcar
353 #'(lambda (x)
354 (make-term (term-monom x)
355 (funcall (ring-div ring) (term-coeff x) c)))
356 (poly-termlist p))
357 (poly-sugar p))
358 c))))
359
360(defun poly-content (ring p)
361 "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
362to compute the greatest common divisor."
363 (declare (type poly p))
364 (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
365
366(defun read-infix-form (&key (stream t))
367 "Parser of infix expressions with integer/rational coefficients
368The parser will recognize two kinds of polynomial expressions:
369
370- polynomials in fully expanded forms with coefficients
371 written in front of symbolic expressions; constants can be optionally
372 enclosed in (); for example, the infix form
373 X^2-Y^2+(-4/3)*U^2*W^3-5
374 parses to
375 (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5))
376
377- lists of polynomials; for example
378 [X-Y, X^2+3*Z]
379 parses to
380 (:[ (- X Y) (+ (EXPT X 2) (* 3 Z)))
381 where the first symbol [ marks a list of polynomials.
382
383-other infix expressions, for example
384 [(X-Y)*(X+Y)/Z,(X+1)^2]
385parses to:
386 (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2))
387Currently this function is implemented using M. Kantrowitz's INFIX package."
388 (read-from-string
389 (concatenate 'string
390 "#I("
391 (with-output-to-string (s)
392 (loop
393 (multiple-value-bind (line eof)
394 (read-line stream t)
395 (format s "~A" line)
396 (when eof (return)))))
397 ")")))
398
399(defun read-poly (vars &key
400 (stream t)
401 (ring *ring-of-integers*)
402 (order #'lex>))
403 "Reads an expression in prefix form from a stream STREAM.
404The expression read from the strem should represent a polynomial or a
405list of polynomials in variables VARS, over the ring RING. The
406polynomial or list of polynomials is returned, with terms in each
407polynomial ordered according to monomial order ORDER."
408 (poly-eval (read-infix-form :stream stream) vars ring order))
409
410(defun string->poly (str vars
411 &optional
412 (ring *ring-of-integers*)
413 (order #'lex>))
414 "Converts a string STR to a polynomial in variables VARS."
415 (with-input-from-string (s str)
416 (read-poly vars :stream s :ring ring :order order)))
417
418(defun poly->alist (p)
419 "Convert a polynomial P to an association list. Thus, the format of the
420returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where
421MONOM[I] is a list of exponents in the monomial and COEFF[I] is the
422corresponding coefficient in the ring."
423 (cond
424 ((poly-p p)
425 (mapcar #'term->cons (poly-termlist p)))
426 ((and (consp p) (eq (car p) :[))
427 (cons :[ (mapcar #'poly->alist (cdr p))))))
428
429(defun string->alist (str vars
430 &optional
431 (ring *ring-of-integers*)
432 (order #'lex>))
433 "Convert a string STR representing a polynomial or polynomial list to
434an association list (... (MONOM . COEFF) ...)."
435 (poly->alist (string->poly str vars ring order)))
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