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