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1;;; -*- Mode: Lisp; Package: Maxima; Syntax: Common-Lisp; Base: 10 -*-
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;;
24;; Load this file into Maxima to bootstrap the Grobner package.
25;; NOTE: This file does use symbols defined by Maxima, so it
26;; will not work when loaded in Common Lisp.
27;;
28;; DETAILS: This file implements an interface between the Grobner
29;; basis package NGROBNER, which is a pure Common Lisp package, and
30;; Maxima. NGROBNER for efficiency uses its own representation of
31;; polynomials. Thus, it is necessary to convert Maxima representation
32;; to the internal representation and back. The facilities to do so
33;; are implemented in this file.
34;;
35;; Also, since the NGROBNER package consists of many Lisp files, it is
36;; necessary to load the files. It is possible and preferrable to use
37;; ASDF for this purpose. The default is ASDF. It is also possible to
38;; simply used LOAD and COMPILE-FILE to accomplish this task.
39;;
40;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
41
42(eval-when
43 #+gcl (load eval)
44 #-gcl (:load-toplevel :execute)
45 (defpackage "MAXIMA" (:use :cl))
46 (setf *features* (remove :maxima *features*)))
47
48(in-package :maxima)
49
50#+maxima(macsyma-module cgb-maxima)
51
52(eval-when
53 #+gcl (load eval)
54 #-gcl (:load-toplevel :execute)
55 (format t "~&Loading maxima-grobner ~a ~a~%"
56 "$Revision: 2.0 $" "$Date: 2015/06/02 0:34:17 $"))
57
58;;FUNCTS is loaded because it contains the definition of LCM
59#+maxima($load "functs")
60
61#+sbcl(progn (require 'asdf) (load "ngrobner.asd")(asdf:load-system :ngrobner))
62
63(use-package :ngrobner)
64
65;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
66;;
67;; Maxima expression ring
68;;
69;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
70;;
71;; This is how we perform operations on coefficients
72;; using Maxima functions.
73;;
74;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
75
76(defparameter *maxima-ring*
77 (make-ring
78 ;;(defun coeff-zerop (expr) (meval1 `(($is) (($equal) ,expr 0))))
79 :parse #'(lambda (expr)
80 (when modulus (setf expr ($rat expr)))
81 expr)
82 :unit #'(lambda () (if modulus ($rat 1) 1))
83 :zerop #'(lambda (expr)
84 ;;When is exactly a maxima expression equal to 0?
85 (cond ((numberp expr)
86 (= expr 0))
87 ((atom expr) nil)
88 (t
89 (case (caar expr)
90 (mrat (eql ($ratdisrep expr) 0))
91 (otherwise (eql ($totaldisrep expr) 0))))))
92 :add #'(lambda (x y) (m+ x y))
93 :sub #'(lambda (x y) (m- x y))
94 :uminus #'(lambda (x) (m- x))
95 :mul #'(lambda (x y) (m* x y))
96 ;;(defun coeff-div (x y) (cadr ($divide x y)))
97 :div #'(lambda (x y) (m// x y))
98 :lcm #'(lambda (x y) (meval1 `((|$LCM|) ,x ,y)))
99 :ezgcd #'(lambda (x y) (apply #'values (cdr ($ezgcd ($totaldisrep x) ($totaldisrep y)))))
100 ;; :gcd #'(lambda (x y) (second ($ezgcd x y)))))
101 :gcd #'(lambda (x y) ($gcd x y))))
102
103;; Rebind some global variables for Maxima environment
104(setf *expression-ring* *maxima-ring* ; Coefficient arithmetic done by Maxima
105 *ratdisrep-fun* '$ratdisrep ; Coefficients are converted to general form
106 )
107
108;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
109;;
110;; Maxima expression parsing
111;;
112;;
113;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
114;;
115;; Functions and macros dealing with internal representation
116;; structure.
117;;
118;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
119
120(defun equal-test-p (expr1 expr2)
121 (alike1 expr1 expr2))
122
123(defun coerce-maxima-list (expr)
124 "Convert a Maxima list to Lisp list."
125 (cond
126 ((and (consp (car expr)) (eql (caar expr) 'mlist)) (cdr expr))
127 (t expr)))
128
129(defun free-of-vars (expr vars) (apply #'$freeof `(,@vars ,expr)))
130
131(defun parse-poly (expr vars &aux (vars (coerce-maxima-list vars)))
132 "Convert a maxima polynomial expression EXPR in variables VARS to internal form."
133 (labels ((parse (arg) (parse-poly arg vars))
134 (parse-list (args) (mapcar #'parse args)))
135 (cond
136 ((eql expr 0) (make-poly-zero))
137 ((member expr vars :test #'equal-test-p)
138 (let ((pos (position expr vars :test #'equal-test-p)))
139 (make-variable *expression-ring* (length vars) pos)))
140 ((free-of-vars expr vars)
141 ;;This means that variable-free CRE and Poisson forms will be converted
142 ;;to coefficients intact
143 (coerce-coeff *expression-ring* expr vars))
144 (t
145 (case (caar expr)
146 (mplus (reduce #'(lambda (x y) (poly-add *expression-ring* x y)) (parse-list (cdr expr))))
147 (mminus (poly-uminus *expression-ring* (parse (cadr expr))))
148 (mtimes
149 (if (endp (cddr expr)) ;unary
150 (parse (cdr expr))
151 (reduce #'(lambda (p q) (poly-mul *expression-ring* p q)) (parse-list (cdr expr)))))
152 (mexpt
153 (cond
154 ((member (cadr expr) vars :test #'equal-test-p)
155 ;;Special handling of (expt var pow)
156 (let ((pos (position (cadr expr) vars :test #'equal-test-p)))
157 (make-variable *expression-ring* (length vars) pos (caddr expr))))
158 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
159 ;; Negative power means division in coefficient ring
160 ;; Non-integer power means non-polynomial coefficient
161 (mtell "~%Warning: Expression ~%~M~%contains power which is not a positive integer. Parsing as coefficient.~%"
162 expr)
163 (coerce-coeff *expression-ring* expr vars))
164 (t (poly-expt *expression-ring* (parse (cadr expr)) (caddr expr)))))
165 (mrat (parse ($ratdisrep expr)))
166 (mpois (parse ($outofpois expr)))
167 (otherwise
168 (coerce-coeff *expression-ring* expr vars)))))))
169
170(defun parse-poly-list (expr vars)
171 (case (caar expr)
172 (mlist (mapcar #'(lambda (p) (parse-poly p vars)) (cdr expr)))
173 (t (merror "Expression ~M is not a list of polynomials in variables ~M."
174 expr vars))))
175(defun parse-poly-list-list (poly-list-list vars)
176 (mapcar #'(lambda (g) (parse-poly-list g vars)) (coerce-maxima-list poly-list-list)))
177
178
179;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
180;;
181;; Conversion from internal form to Maxima general form
182;;
183;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
184
185(defun maxima-head ()
186 (if $poly_return_term_list
187 '(mlist)
188 '(mplus)))
189
190(defun coerce-to-maxima (poly-type object vars)
191 (case poly-type
192 (:polynomial
193 `(,(maxima-head) ,@(mapcar #'(lambda (term) (coerce-to-maxima :term term vars)) (poly-termlist object))))
194 (:poly-list
195 `((mlist) ,@(mapcar #'(lambda (p) (funcall *ratdisrep-fun* (coerce-to-maxima :polynomial p vars))) object)))
196 (:term
197 `((mtimes) ,(funcall *ratdisrep-fun* (term-coeff object))
198 ,@(mapcar #'(lambda (var power) `((mexpt) ,var ,power))
199 vars (monom-exponents (term-monom object)))))
200 ;; Assumes that Lisp and Maxima logicals coincide
201 (:logical object)
202 (otherwise
203 object)))
204
205
206;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
207;;
208;; Unary and binary operation definition facility
209;;
210;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
211
212(defmacro define-unop (maxima-name fun-name
213 &optional (documentation nil documentation-supplied-p))
214 "Define a MAXIMA-level unary operator MAXIMA-NAME corresponding to unary function FUN-NAME."
215 `(defun ,maxima-name (p vars
216 &aux
217 (vars (coerce-maxima-list vars))
218 (p (parse-poly p vars)))
219 ,@(when documentation-supplied-p (list documentation))
220 (coerce-to-maxima :polynomial (,fun-name *expression-ring* p) vars)))
221
222(defmacro define-binop (maxima-name fun-name
223 &optional (documentation nil documentation-supplied-p))
224 "Define a MAXIMA-level binary operator MAXIMA-NAME corresponding to binary function FUN-NAME."
225 `(defmfun ,maxima-name (p q vars
226 &aux
227 (vars (coerce-maxima-list vars))
228 (p (parse-poly p vars))
229 (q (parse-poly q vars)))
230 ,@(when documentation-supplied-p (list documentation))
231 (coerce-to-maxima :polynomial (,fun-name *expression-ring* p q) vars)))
232
233
234;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
235;;
236;; Facilities for evaluating Grobner package expressions
237;; within a prepared environment
238;;
239;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
240
241(defmacro with-monomial-order ((order) &body body)
242 "Evaluate BODY with monomial order set to ORDER."
243 `(let ((*monomial-order* (or (find-order ,order) *monomial-order*)))
244 . ,body))
245
246(defmacro with-coefficient-ring ((ring) &body body)
247 "Evaluate BODY with coefficient ring set to RING."
248 `(let ((*expression-ring* (or (find-ring ,ring) *expression-ring*)))
249 . ,body))
250
251(defmacro with-elimination-orders ((primary secondary elimination-order)
252 &body body)
253 "Evaluate BODY with primary and secondary elimination orders set to PRIMARY and SECONDARY."
254 `(let ((*primary-elimination-order* (or (find-order ,primary) *primary-elimination-order*))
255 (*secondary-elimination-order* (or (find-order ,secondary) *secondary-elimination-order*))
256 (*elimination-order* (or (find-order ,elimination-order) *elimination-order*)))
257 . ,body))
258
259
260;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
261;;
262;; Maxima-level interface functions
263;;
264;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
265
266;; Auxillary function for removing zero polynomial
267(defun remzero (plist) (remove #'poly-zerop plist))
268
269;;Simple operators
270
271(define-binop $poly_add poly-add
272 "Adds two polynomials P and Q")
273
274(define-binop $poly_subtract poly-sub
275 "Subtracts a polynomial Q from P.")
276
277(define-binop $poly_multiply poly-mul
278 "Returns the product of polynomials P and Q.")
279
280(define-binop $poly_s_polynomial spoly
281 "Returns the syzygy polynomial (S-polynomial) of two polynomials P and Q.")
282
283(define-unop $poly_primitive_part poly-primitive-part
284 "Returns the polynomial P divided by GCD of its coefficients.")
285
286(define-unop $poly_normalize poly-normalize
287 "Returns the polynomial P divided by the leading coefficient.")
288
289;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
290;;
291;; Macro facility for writing Maxima-level wrappers for
292;; functions operating on internal representation
293;;
294;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
295
296(defmacro with-parsed-polynomials (((maxima-vars &optional (maxima-new-vars nil new-vars-supplied-p))
297 &key (polynomials nil)
298 (poly-lists nil)
299 (poly-list-lists nil)
300 (value-type nil))
301 &body body
302 &aux (vars (gensym))
303 (new-vars (gensym)))
304 `(let ((,vars (coerce-maxima-list ,maxima-vars))
305 ,@(when new-vars-supplied-p
306 (list `(,new-vars (coerce-maxima-list ,maxima-new-vars)))))
307 (coerce-to-maxima
308 ,value-type
309 (with-coefficient-ring ($poly_coefficient_ring)
310 (with-monomial-order ($poly_monomial_order)
311 (with-elimination-orders ($poly_primary_elimination_order
312 $poly_secondary_elimination_order
313 $poly_elimination_order)
314 (let ,(let ((args nil))
315 (dolist (p polynomials args)
316 (setf args (cons `(,p (parse-poly ,p ,vars)) args)))
317 (dolist (p poly-lists args)
318 (setf args (cons `(,p (parse-poly-list ,p ,vars)) args)))
319 (dolist (p poly-list-lists args)
320 (setf args (cons `(,p (parse-poly-list-list ,p ,vars)) args))))
321 . ,body))))
322 ,(if new-vars-supplied-p
323 `(append ,vars ,new-vars)
324 vars))))
325
326
327;;Functions
328
329(defmfun $poly_expand (p vars)
330 "This function is equivalent to EXPAND(P) if P parses correctly to a polynomial.
331If the representation is not compatible with a polynomial in variables VARS,
332the result is an error."
333 (with-parsed-polynomials ((vars) :polynomials (p)
334 :value-type :polynomial)
335 p))
336
337(defmfun $poly_expt (p n vars)
338 (with-parsed-polynomials ((vars) :polynomials (p) :value-type :polynomial)
339 (poly-expt *expression-ring* p n)))
340
341(defmfun $poly_content (p vars)
342 (with-parsed-polynomials ((vars) :polynomials (p))
343 (poly-content *expression-ring* p)))
344
345(defmfun $poly_pseudo_divide (f fl vars
346 &aux (vars (coerce-maxima-list vars))
347 (f (parse-poly f vars))
348 (fl (parse-poly-list fl vars)))
349 (multiple-value-bind (quot rem c division-count)
350 (poly-pseudo-divide *expression-ring* f fl)
351 `((mlist)
352 ,(coerce-to-maxima :poly-list quot vars)
353 ,(coerce-to-maxima :polynomial rem vars)
354 ,c
355 ,division-count)))
356
357(defmfun $poly_exact_divide (f g vars)
358 (with-parsed-polynomials ((vars) :polynomials (f g) :value-type :polynomial)
359 (poly-exact-divide *expression-ring* f g)))
360
361(defmfun $poly_normal_form (f fl vars)
362 (with-parsed-polynomials ((vars) :polynomials (f)
363 :poly-lists (fl)
364 :value-type :polynomial)
365 (normal-form *expression-ring* f (remzero fl) nil)))
366
367(defmfun $poly_buchberger_criterion (g vars)
368 (with-parsed-polynomials ((vars) :poly-lists (g) :value-type :logical)
369 (buchberger-criterion *expression-ring* g)))
370
371(defmfun $poly_buchberger (fl vars)
372 (with-parsed-polynomials ((vars) :poly-lists (fl) :value-type :poly-list)
373 (buchberger *expression-ring* (remzero fl) 0 nil)))
374
375(defmfun $poly_reduction (plist vars)
376 (with-parsed-polynomials ((vars) :poly-lists (plist)
377 :value-type :poly-list)
378 (reduction *expression-ring* plist)))
379
380(defmfun $poly_minimization (plist vars)
381 (with-parsed-polynomials ((vars) :poly-lists (plist)
382 :value-type :poly-list)
383 (minimization plist)))
384
385(defmfun $poly_normalize_list (plist vars)
386 (with-parsed-polynomials ((vars) :poly-lists (plist)
387 :value-type :poly-list)
388 (poly-normalize-list *expression-ring* plist)))
389
390(defmfun $poly_grobner (f vars)
391 (with-parsed-polynomials ((vars) :poly-lists (f)
392 :value-type :poly-list)
393 (grobner *expression-ring* (remzero f))))
394
395(defmfun $poly_reduced_grobner (f vars)
396 (with-parsed-polynomials ((vars) :poly-lists (f)
397 :value-type :poly-list)
398 (reduced-grobner *expression-ring* (remzero f))))
399
400(defmfun $poly_depends_p (p var mvars
401 &aux (vars (coerce-maxima-list mvars))
402 (pos (position var vars)))
403 (if (null pos)
404 (merror "~%Variable ~M not in the list of variables ~M." var mvars)
405 (poly-depends-p (parse-poly p vars) pos)))
406
407(defmfun $poly_elimination_ideal (flist k vars)
408 (with-parsed-polynomials ((vars) :poly-lists (flist)
409 :value-type :poly-list)
410 (elimination-ideal *expression-ring* flist k nil 0)))
411
412(defmfun $poly_colon_ideal (f g vars)
413 (with-parsed-polynomials ((vars) :poly-lists (f g) :value-type :poly-list)
414 (colon-ideal *expression-ring* f g nil)))
415
416(defmfun $poly_ideal_intersection (f g vars)
417 (with-parsed-polynomials ((vars) :poly-lists (f g) :value-type :poly-list)
418 (ideal-intersection *expression-ring* f g nil)))
419
420(defmfun $poly_lcm (f g vars)
421 (with-parsed-polynomials ((vars) :polynomials (f g) :value-type :polynomial)
422 (poly-lcm *expression-ring* f g)))
423
424(defmfun $poly_gcd (f g vars)
425 ($first ($divide (m* f g) ($poly_lcm f g vars))))
426
427(defmfun $poly_grobner_equal (g1 g2 vars)
428 (with-parsed-polynomials ((vars) :poly-lists (g1 g2))
429 (grobner-equal *expression-ring* g1 g2)))
430
431(defmfun $poly_grobner_subsetp (g1 g2 vars)
432 (with-parsed-polynomials ((vars) :poly-lists (g1 g2))
433 (grobner-subsetp *expression-ring* g1 g2)))
434
435(defmfun $poly_grobner_member (p g vars)
436 (with-parsed-polynomials ((vars) :polynomials (p) :poly-lists (g))
437 (grobner-member *expression-ring* p g)))
438
439(defmfun $poly_ideal_saturation1 (f p vars)
440 (with-parsed-polynomials ((vars) :poly-lists (f) :polynomials (p)
441 :value-type :poly-list)
442 (ideal-saturation-1 *expression-ring* f p 0)))
443
444(defmfun $poly_saturation_extension (f plist vars new-vars)
445 (with-parsed-polynomials ((vars new-vars)
446 :poly-lists (f plist)
447 :value-type :poly-list)
448 (saturation-extension *expression-ring* f plist)))
449
450(defmfun $poly_polysaturation_extension (f plist vars new-vars)
451 (with-parsed-polynomials ((vars new-vars)
452 :poly-lists (f plist)
453 :value-type :poly-list)
454 (polysaturation-extension *expression-ring* f plist)))
455
456(defmfun $poly_ideal_polysaturation1 (f plist vars)
457 (with-parsed-polynomials ((vars) :poly-lists (f plist)
458 :value-type :poly-list)
459 (ideal-polysaturation-1 *expression-ring* f plist 0 nil)))
460
461(defmfun $poly_ideal_saturation (f g vars)
462 (with-parsed-polynomials ((vars) :poly-lists (f g)
463 :value-type :poly-list)
464 (ideal-saturation *expression-ring* f g 0 nil)))
465
466(defmfun $poly_ideal_polysaturation (f ideal-list vars)
467 (with-parsed-polynomials ((vars) :poly-lists (f)
468 :poly-list-lists (ideal-list)
469 :value-type :poly-list)
470 (ideal-polysaturation *expression-ring* f ideal-list 0 nil)))
471
472(defmfun $poly_lt (f vars)
473 (with-parsed-polynomials ((vars) :polynomials (f) :value-type :polynomial)
474 (make-poly-from-termlist (list (poly-lt f)))))
475
476(defmfun $poly_lm (f vars)
477 (with-parsed-polynomials ((vars) :polynomials (f) :value-type :polynomial)
478 (make-poly-from-termlist (list (make-term (poly-lm f) (funcall (ring-unit *expression-ring*)))))))
479
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