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source: branches/f4grobner/monom.lisp@ 2086

Last change on this file since 2086 was 2086, 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;; This package implements BASIC OPERATIONS ON MONOMIALS
24;;----------------------------------------------------------------
25;; DATA STRUCTURES: Conceptually, monomials can be represented as lists:
26;;
27;; monom: (n1 n2 ... nk) where ni are non-negative integers
28;;
29;; However, lists may be implemented as other sequence types,
30;; so the flexibility to change the representation should be
31;; maintained in the code to use general operations on sequences
32;; whenever possible. The optimization for the actual representation
33;; should be left to declarations and the compiler.
34;;----------------------------------------------------------------
35;; EXAMPLES: Suppose that variables are x and y. Then
36;;
37;; Monom x*y^2 ---> (1 2)
38;;
39;;----------------------------------------------------------------
40
41(defpackage "MONOM"
42 (:use :cl :ring)
43 (:export "MONOM"
44 "EXPONENT"
45 "MAKE-MONOM"
46 "MAKE-MONOM-VARIABLE"
47 "MONOM-ELT"
48 "MONOM-DIMENSION"
49 "MONOM-TOTAL-DEGREE"
50 "MONOM-SUGAR"
51 "MONOM-DIV"
52 "MONOM-MUL"
53 "MONOM-DIVIDES-P"
54 "MONOM-DIVIDES-MONOM-LCM-P"
55 "MONOM-LCM-DIVIDES-MONOM-LCM-P"
56 "MONOM-LCM-EQUAL-MONOM-LCM-P"
57 "MONOM-DIVISIBLE-BY-P"
58 "MONOM-REL-PRIME-P"
59 "MONOM-EQUAL-P"
60 "MONOM-LCM"
61 "MONOM-GCD"
62 "MONOM-DEPENDS-P"
63 "MONOM-MAP"
64 "MONOM-APPEND"
65 "MONOM-CONTRACT"
66 "MONOM->LIST"))
67
68(in-package :monom)
69
70(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
71
72(deftype exponent ()
73 "Type of exponent in a monomial."
74 'fixnum)
75
76(defclass monom ()
77 ((dim :initarg :dim )
78 (exponents :initarg :exponents))
79 (:default-initargs :dim 0 :exponents nil))
80
81(defmethod print-object ((m monom) stream)
82 (princ (slot-value m 'exponents) stream))
83
84;; If a monomial is redefined as structure with slot EXPONENTS, the function
85;; below can be the BOA constructor.
86(defun make-monom (&key
87 (dimension nil dimension-suppied-p)
88 (initial-exponents nil initial-exponents-supplied-p)
89 (initial-exponent nil initial-exponent-supplied-p)
90 &aux
91 (dim (cond (dimension-suppied-p dimension)
92 (initial-exponents-supplied-p (length initial-exponents))
93 (t (error "You must provide DIMENSION or INITIAL-EXPONENTS"))))
94 (exponents (cond
95 ;; when exponents are supplied
96 (initial-exponents-supplied-p
97 (make-array (list dim) :initial-contents initial-exponents
98 :element-type 'exponent))
99 ;; when all exponents are to be identical
100 (initial-exponent-supplied-p
101 (make-array (list dim) :initial-element initial-exponent
102 :element-type 'exponent))
103 ;; otherwise, all exponents are zero
104 (t
105 (make-array (list dim) :element-type 'exponent :initial-element 0)))))
106 "A constructor (factory) of monomials. If DIMENSION is given, a sequence of
107DIMENSION elements of type EXPONENT is constructed, where individual
108elements are the value of INITIAL-EXPONENT, which defaults to 0.
109Alternatively, all elements may be specified as a list
110INITIAL-EXPONENTS."
111 (make-instance 'monom :dim dim :exponents exponents))
112
113;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
114;;
115;; Operations on monomials
116;;
117;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
118
119(defmethod dimension ((m monom))
120 (slot-value m 'dim))
121
122(defmethod ring-elt ((m monom) index)
123 "Return the power in the monomial M of variable number INDEX."
124 (with-slots (exponents)
125 m
126 (elt exponents index)))
127
128(defmethod (setf ring-elt) (new-value (m monom) index)
129 "Return the power in the monomial M of variable number INDEX."
130 (with-slots (exponents)
131 m
132 (setf (elt exponents index) new-value)))
133
134(defmethod total-degree ((m monom) &optional (start 0) (end (dimension m)))
135 "Return the todal degree of a monomoal M. Optinally, a range
136of variables may be specified with arguments START and END."
137 (declare (type fixnum start end))
138 (with-slots (exponents)
139 m
140 (reduce #'+ exponents :start start :end end)))
141
142
143(defmethod sugar ((m monom) &aux (start 0) (end (dimension m)))
144 "Return the sugar of a monomial M. Optinally, a range
145of variables may be specified with arguments START and END."
146 (declare (type fixnum start end))
147 (with-slots (exponents)
148 m
149 (total-degree exponents start end)))
150
151(defmethod ring-mul ((m1 monom) (m2 monom))
152 "Multiply monomial M1 by monomial M2."
153 (with-slots ((exponents1 exponents))
154 m1
155 (with-slots ((exponents2 exponents))
156 m2
157 (let* ((exponents (copy-seq exponents1))
158 (dim (reduce #'+ exponents)))
159 (map-into exponents #'+ exponents1 exponents2)
160 (make-instance 'monom :dim dim :exponents exponents)))))
161
162
163
164(defmethod ring-div ((m1 monom) (m2 monom))
165 "Divide monomial M1 by monomial M2."
166 (with-slots ((exponents1 exponents))
167 m1
168 (with-slots ((exponents2 exponents))
169 m2
170 (let* ((exponents (copy-seq exponents1))
171 (dim (reduce #'+ exponents)))
172 (map-into exponents #'- exponents1 exponents2)
173 (make-instance 'monom :dim dim :exponents exponents)))))
174
175(defmethod divides-p ((m1 monom) (m2 monom))
176 "Returns T if monomial M1 divides monomial M2, NIL otherwise."
177 (with-slots ((exponents1 exponents))
178 m1
179 (with-slots ((exponents2 exponents))
180 m2
181 (every #'<= exponents1 exponents2))))
182
183
184(defmethod divides-lcm-p ((m1 monom) (m2 monom) (m3 monom))
185 "Returns T if monomial M1 divides LCM(M2,M3), NIL otherwise."
186 (every #'(lambda (x y z) (<= x (max y z)))
187 m1 m2 m3))
188
189
190(defmethod lcm-divides-lcm-p ((m1 monom) (m2 monom) (m3 monom) (m4 monom))
191 "Returns T if monomial MONOM-LCM(M1,M2) divides MONOM-LCM(M3,M4), NIL otherwise."
192 (declare (type monom m1 m2 m3 m4))
193 (every #'(lambda (x y z w) (<= (max x y) (max z w)))
194 m1 m2 m3 m4))
195
196(defmethod lcm-equal-lcm-p (m1 m2 m3 m4)
197 "Returns T if monomial LCM(M1,M2) equals LCM(M3,M4), NIL otherwise."
198 (with-slots (exponents1 exponents)
199 m1
200 (with-slots (exponents2 exponents)
201 m2
202 (with-slots (exponents3 exponents)
203 m3
204 (with-slots (exponents4 exponents)
205 m4
206 (every
207 #'(lambda (x y z w) (= (max x y) (max z w)))
208 exponents1 exponents2 exponents3 exponents4))))))
209
210(defmethod divisible-by-p ((m1 monom) (m2 monom))
211 "Returns T if monomial M1 is divisible by monomial M2, NIL otherwise."
212
213 (every #'>= m1 m2))
214
215(defmethod rel-prime-p ((m1 monom) (m2 monom))
216 "Returns T if two monomials M1 and M2 are relatively prime (disjoint)."
217 (with-slots (exponents1 exponents)
218 m1
219 (with-slots (exponents2 exponents)
220 m2
221 (every #'(lambda (x y) (zerop (min x y))) exponents1 exponents2))))
222
223
224(defmethod equal-p ((m1 monom) (m2 monom))
225 "Returns T if two monomials M1 and M2 are equal."
226 (with-slots (exponents1 exponents)
227 m1
228 (with-slots (exponents2 exponents)
229 m2
230 (every #'= exponents1 exponents2))))
231
232(defmethod ring-lcm ((m1 monom) (m2 monom))
233 "Returns least common multiple of monomials M1 and M2."
234 (with-slots (exponents1 exponents)
235 m1
236 (with-slots (exponents2 exponents)
237 m2
238 (let* ((exponents (copy-seq exponents1))
239 (dim (reduce #'+ exponents)))
240 (map-into exponents #'max exponents1 exponents2)
241 (make-instance 'monom :dim dim :exponents exponents)))))
242
243
244(defmethod ring-gcd ((m1 monom) (m2 monom))
245 "Returns greatest common divisor of monomials M1 and M2."
246 (with-slots (exponents1 exponents)
247 m1
248 (with-slots (exponents2 exponents)
249 m2
250 (let* ((exponents (copy-seq exponents1))
251 (dim (reduce #'+ exponents)))
252 (map-into exponents #'min exponents1 exponents2)
253 (make-instance 'monom :dim dim :exponents exponents)))))
254
255(defmethod depends-p ((m monom) k)
256 "Return T if the monomial M depends on variable number K."
257 (declare (type fixnum k))
258 (with-slots (exponents)
259 m
260 (plusp (elt exponents k))))
261
262(defmethod tensor-mul ((m1 monom) (m2 monom)
263 &aux (dim (+ (dimension m1) (dimension m2))))
264 (declare (fixnum dim))
265 (make-instance 'monom :dim dim
266 :exponents (concatenate 'vector exponents1 exponents2)))
267
268(defmethod contract ((m monom) k)
269 "Drop the first K variables in monomial M."
270 (declare (fixnum k))
271 (with-slots (dim exponents)
272 m
273 (setf dim (- dim k)
274 exponents (subseq exponents k))))
275
276(defun make-monom-variable (nvars pos &optional (power 1)
277 &aux (m (make-monom :dimension nvars)))
278 "Construct a monomial in the polynomial ring
279RING[X[0],X[1],X[2],...X[NVARS-1]] over the (unspecified) ring RING
280which represents a single variable. It assumes number of variables
281NVARS and the variable is at position POS. Optionally, the variable
282may appear raised to power POWER. "
283 (declare (type fixnum nvars pos power) (type monom m))
284 (setf (monom-elt m pos) power)
285 m)
286
287(defmethod monom->list ((m monom))
288 "A human-readable representation of a monomial M as a list of exponents."
289 (with-slots (exponents)
290 m
291 (coerce exponents 'list)))
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