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source: branches/f4grobner/monom.lisp@ 2126

Last change on this file since 2126 was 2126, 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 (:shadowing-import-from :ring "ZEROP" "LCM" "GCD" "+" "-" "*" "/" "EXPT")
44 (:export "MONOM"
45 "EXPONENT"
46 "MAKE-MONOM"
47 "MONOM-DIMENSION"
48 "MONOM-EXPONENTS"
49 "MAKE-MONOM-VARIABLE"))
50
51(in-package :monom)
52
53(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
54
55(deftype exponent ()
56 "Type of exponent in a monomial."
57 'fixnum)
58
59(defclass monom ()
60 ((dim :initarg :dim :accessor monom-dimension)
61 (exponents :initarg :exponents :accessor monom-exponents))
62 (:default-initargs :dim 0 :exponents nil))
63
64(defmethod print-object ((m monom) stream)
65 (princ (slot-value m 'exponents) stream))
66
67;; If a monomial is redefined as structure with slot EXPONENTS, the function
68;; below can be the BOA constructor.
69(defun make-monom (&key
70 (dimension nil dimension-suppied-p)
71 (initial-exponents nil initial-exponents-supplied-p)
72 (initial-exponent nil initial-exponent-supplied-p)
73 &aux
74 (dim (cond (dimension-suppied-p dimension)
75 (initial-exponents-supplied-p (length initial-exponents))
76 (t (error "You must provide DIMENSION or INITIAL-EXPONENTS"))))
77 (exponents (cond
78 ;; when exponents are supplied
79 (initial-exponents-supplied-p
80 (make-array (list dim) :initial-contents initial-exponents
81 :element-type 'exponent))
82 ;; when all exponents are to be identical
83 (initial-exponent-supplied-p
84 (make-array (list dim) :initial-element initial-exponent
85 :element-type 'exponent))
86 ;; otherwise, all exponents are zero
87 (t
88 (make-array (list dim) :element-type 'exponent :initial-element 0)))))
89 "A constructor (factory) of monomials. If DIMENSION is given, a sequence of
90DIMENSION elements of type EXPONENT is constructed, where individual
91elements are the value of INITIAL-EXPONENT, which defaults to 0.
92Alternatively, all elements may be specified as a list
93INITIAL-EXPONENTS."
94 (make-instance 'monom :dim dim :exponents exponents))
95
96;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
97;;
98;; Operations on monomials
99;;
100;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
101
102(defmethod dimension ((m monom))
103 (monom-dimension m))
104
105(defmethod ring-elt ((m monom) index)
106 "Return the power in the monomial M of variable number INDEX."
107 (with-slots (exponents)
108 m
109 (elt exponents index)))
110
111(defmethod (setf ring-elt) (new-value (m monom) index)
112 "Return the power in the monomial M of variable number INDEX."
113 (with-slots (exponents)
114 m
115 (setf (elt exponents index) new-value)))
116
117(defmethod total-degree ((m monom) &optional (start 0) (end (dimension m)))
118 "Return the todal degree of a monomoal M. Optinally, a range
119of variables may be specified with arguments START and END."
120 (declare (type fixnum start end))
121 (with-slots (exponents)
122 m
123 (reduce #'cl:+ exponents :start start :end end)))
124
125
126(defmethod sugar ((m monom) &aux (start 0) (end (dimension m)))
127 "Return the sugar of a monomial M. Optinally, a range
128of variables may be specified with arguments START and END."
129 (declare (type fixnum start end))
130 (with-slots (exponents)
131 m
132 (total-degree exponents start end)))
133
134(defmethod + ((m1 monom) (m2 monom))
135 "Multiply monomial M1 by monomial M2."
136 (with-slots ((exponents1 exponents))
137 m1
138 (with-slots ((exponents2 exponents))
139 m2
140 (let* ((exponents (copy-seq exponents1))
141 (dim (reduce #'cl:+ exponents)))
142 (map-into exponents #'cl:+ exponents1 exponents2)
143 (make-instance 'monom :dim dim :exponents exponents)))))
144
145
146
147(defmethod / ((m1 monom) (m2 monom))
148 "Divide monomial M1 by monomial M2."
149 (with-slots ((exponents1 exponents))
150 m1
151 (with-slots ((exponents2 exponents))
152 m2
153 (let* ((exponents (copy-seq exponents1))
154 (dim (reduce #'cl:+ exponents)))
155 (map-into exponents #'cl:- exponents1 exponents2)
156 (make-instance 'monom :dim dim :exponents exponents)))))
157
158(defmethod divides-p ((m1 monom) (m2 monom))
159 "Returns T if monomial M1 divides monomial M2, NIL otherwise."
160 (with-slots ((exponents1 exponents))
161 m1
162 (with-slots ((exponents2 exponents))
163 m2
164 (every #'<= exponents1 exponents2))))
165
166
167(defmethod divides-lcm-p ((m1 monom) (m2 monom) (m3 monom))
168 "Returns T if monomial M1 divides LCM(M2,M3), NIL otherwise."
169 (every #'(lambda (x y z) (<= x (max y z)))
170 m1 m2 m3))
171
172
173(defmethod lcm-divides-lcm-p ((m1 monom) (m2 monom) (m3 monom) (m4 monom))
174 "Returns T if monomial MONOM-LCM(M1,M2) divides MONOM-LCM(M3,M4), NIL otherwise."
175 (declare (type monom m1 m2 m3 m4))
176 (every #'(lambda (x y z w) (<= (max x y) (max z w)))
177 m1 m2 m3 m4))
178
179(defmethod lcm-equal-lcm-p (m1 m2 m3 m4)
180 "Returns T if monomial LCM(M1,M2) equals LCM(M3,M4), NIL otherwise."
181 (with-slots (exponents1 exponents)
182 m1
183 (with-slots (exponents2 exponents)
184 m2
185 (with-slots (exponents3 exponents)
186 m3
187 (with-slots (exponents4 exponents)
188 m4
189 (every
190 #'(lambda (x y z w) (= (max x y) (max z w)))
191 exponents1 exponents2 exponents3 exponents4))))))
192
193(defmethod divisible-by-p ((m1 monom) (m2 monom))
194 "Returns T if monomial M1 is divisible by monomial M2, NIL otherwise."
195
196 (every #'>= m1 m2))
197
198(defmethod rel-prime-p ((m1 monom) (m2 monom))
199 "Returns T if two monomials M1 and M2 are relatively prime (disjoint)."
200 (with-slots (exponents1 exponents)
201 m1
202 (with-slots (exponents2 exponents)
203 m2
204 (every #'(lambda (x y) (cl:zerop (min x y))) exponents1 exponents2))))
205
206
207(defmethod equal-p ((m1 monom) (m2 monom))
208 "Returns T if two monomials M1 and M2 are equal."
209 (with-slots (exponents1 exponents)
210 m1
211 (with-slots (exponents2 exponents)
212 m2
213 (every #'= exponents1 exponents2))))
214
215(defmethod lcm ((m1 monom) (m2 monom))
216 "Returns least common multiple of monomials M1 and M2."
217 (with-slots (exponents1 exponents)
218 m1
219 (with-slots (exponents2 exponents)
220 m2
221 (let* ((exponents (copy-seq exponents1))
222 (dim (reduce #'cl:+ exponents)))
223 (map-into exponents #'max exponents1 exponents2)
224 (make-instance 'monom :dim dim :exponents exponents)))))
225
226
227(defmethod gcd ((m1 monom) (m2 monom))
228 "Returns greatest common divisor of monomials M1 and M2."
229 (with-slots (exponents1 exponents)
230 m1
231 (with-slots (exponents2 exponents)
232 m2
233 (let* ((exponents (copy-seq exponents1))
234 (dim (reduce #'cl:+ exponents)))
235 (map-into exponents #'min exponents1 exponents2)
236 (make-instance 'monom :dim dim :exponents exponents)))))
237
238(defmethod depends-p ((m monom) k)
239 "Return T if the monomial M depends on variable number K."
240 (declare (type fixnum k))
241 (with-slots (exponents)
242 m
243 (plusp (elt exponents k))))
244
245(defmethod ring-tensor-mul ((m1 monom) (m2 monom)
246 &aux (dim (cl:+ (dimension m1) (dimension m2))))
247 (declare (fixnum dim))
248 (with-slots (exponents1 exponents)
249 m1
250 (with-slots (exponents2 exponents)
251 m2
252 (make-instance 'monom
253 :dim dim
254 :exponents (concatenate 'vector exponents1 exponents2)))))
255
256(defmethod contract ((m monom) k)
257 "Drop the first K variables in monomial M."
258 (declare (fixnum k))
259 (with-slots (dim exponents)
260 m
261 (setf dim (- dim k)
262 exponents (subseq exponents k))))
263
264(defun make-monom-variable (nvars pos &optional (power 1)
265 &aux (m (make-monom :dimension nvars)))
266 "Construct a monomial in the polynomial ring
267RING[X[0],X[1],X[2],...X[NVARS-1]] over the (unspecified) ring RING
268which represents a single variable. It assumes number of variables
269NVARS and the variable is at position POS. Optionally, the variable
270may appear raised to power POWER. "
271 (declare (type fixnum nvars pos power) (type monom m))
272 (with-slots (exponents)
273 m
274 (setf (elt exponents pos) power)
275 m))
276
277(defmethod monom->list ((m monom))
278 "A human-readable representation of a monomial M as a list of exponents."
279 (with-slots (exponents)
280 m
281 (coerce exponents 'list)))
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