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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;; 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 "MONOMIAL"
42 (:use :cl)
43 (:export "MONOM"
44 "EXPONENT"
45 "MAKE-MONOM"
46 "MONOM-ELT"
47 "MONOM-DIMENSION"
48 "MONOM-TOTAL-DEGREE"
49 "MONOM-SUGAR"
50 "MONOM-DIV"
51 "MONOM-MUL"
52 "MONOM-DIVIDES-P"
53 "MONOM-DIVIDES-MONOM-LCM-P"
54 "MONOM-LCM-DIVIDES-MONOM-LCM-P"
55 "MONOM-LCM-EQUAL-MONOM-LCM-P"
56 "MONOM-DIVISIBLE-BY-P"
57 "MONOM-REL-PRIME-P"
58 "MONOM-EQUAL-P"
59 "MONOM-LCM"
60 "MONOM-GCD"
61 "MONOM-MAP"
62 "MONOM-APPEND"
63 "MONOM-CONTRACT"
64 "MONOM-EXPONENTS"))
65
66(in-package :monomial)
67
68(deftype exponent ()
69 "Type of exponent in a monomial."
70 'fixnum)
71
72(deftype monom (&optional dim)
73 "Type of monomial."
74 `(simple-array exponent (,dim)))
75
76;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
77;;
78;; Construction of monomials
79;;
80;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
81
82(defmacro make-monom (dim &key (initial-contents nil initial-contents-supplied-p)
83 (initial-element 0 initial-element-supplied-p))
84 "Make a monomial with DIM variables. Additional argument
85INITIAL-CONTENTS specifies the list of powers of the consecutive
86variables. The alternative additional argument INITIAL-ELEMENT
87specifies the common power for all variables."
88 ;;(declare (fixnum dim))
89 `(make-array ,dim
90 :element-type 'exponent
91 ,@(when initial-contents-supplied-p `(:initial-contents ,initial-contents))
92 ,@(when initial-element-supplied-p `(:initial-element ,initial-element))))
93
94
95;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
96;;
97;; Operations on monomials
98;;
99;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
100
101(defmacro monom-elt (m index)
102 "Return the power in the monomial M of variable number INDEX."
103 `(elt ,m ,index))
104
105(defun monom-dimension (m)
106 "Return the number of variables in the monomial M."
107 (length m))
108
109(defun monom-total-degree (m &optional (start 0) (end (length m)))
110 "Return the todal degree of a monomoal M. Optinally, a range
111of variables may be specified with arguments START and END."
112 (declare (type monom m) (fixnum start end))
113 (reduce #'+ m :start start :end end))
114
115(defun monom-sugar (m &aux (start 0) (end (length m)))
116 "Return the sugar of a monomial M. Optinally, a range
117of variables may be specified with arguments START and END."
118 (declare (type monom m) (fixnum start end))
119 (monom-total-degree m start end))
120
121(defun monom-div (m1 m2 &aux (result (copy-seq m1)))
122 "Divide monomial M1 by monomial M2."
123 (declare (type monom m1 m2 result))
124 (map-into result #'- m1 m2))
125
126(defun monom-mul (m1 m2 &aux (result (copy-seq m1)))
127 "Multiply monomial M1 by monomial M2."
128 (declare (type monom m1 m2 result))
129 (map-into result #'+ m1 m2))
130
131(defun monom-divides-p (m1 m2)
132 "Returns T if monomial M1 divides monomial M2, NIL otherwise."
133 (declare (type monom m1 m2))
134 (every #'<= m1 m2))
135
136(defun monom-divides-monom-lcm-p (m1 m2 m3)
137 "Returns T if monomial M1 divides MONOM-LCM(M2,M3), NIL otherwise."
138 (declare (type monom m1 m2 m3))
139 (every #'(lambda (x y z) (declare (type exponent x y z)) (<= x (max y z))) m1 m2 m3))
140
141(defun monom-lcm-divides-monom-lcm-p (m1 m2 m3 m4)
142 "Returns T if monomial MONOM-LCM(M1,M2) divides MONOM-LCM(M3,M4), NIL otherwise."
143 (declare (type monom m1 m2 m3 m4))
144 (every #'(lambda (x y z w) (declare (type exponent x y z w)) (<= (max x y) (max z w))) m1 m2 m3 m4))
145
146(defun monom-lcm-equal-monom-lcm-p (m1 m2 m3 m4)
147 "Returns T if monomial MONOM-LCM(M1,M2) equals MONOM-LCM(M3,M4), NIL otherwise."
148 (declare (type monom m1 m2 m3 m4))
149 (every #'(lambda (x y z w) (declare (type exponent x y z w)) (= (max x y) (max z w))) m1 m2 m3 m4))
150
151(defun monom-divisible-by-p (m1 m2)
152 "Returns T if monomial M1 is divisible by monomial M2, NIL otherwise."
153 (declare (type monom m1 m2))
154 (every #'>= m1 m2))
155
156(defun monom-rel-prime-p (m1 m2)
157 "Returns T if two monomials M1 and M2 are relatively prime (disjoint)."
158 (declare (type monom m1 m2))
159 (every #'(lambda (x y) (declare (type exponent x y)) (zerop (min x y))) m1 m2))
160
161(defun monom-equal-p (m1 m2)
162 "Returns T if two monomials M1 and M2 are equal."
163 (declare (type monom m1 m2))
164 (every #'= m1 m2))
165
166(defun monom-lcm (m1 m2 &aux (result (copy-seq m1)))
167 "Returns least common multiple of monomials M1 and M2."
168 (declare (type monom m1 m2))
169 (map-into result #'max m1 m2))
170
171(defun monom-gcd (m1 m2 &aux (result (copy-seq m1)))
172 "Returns greatest common divisor of monomials M1 and M2."
173 (declare (type monom m1 m2))
174 (map-into result #'min m1 m2))
175
176(defun monom-depends-p (m k)
177 "Return T if the monomial M depends on variable number K."
178 (declare (type monom m) (fixnum k))
179 (plusp (elt m k)))
180
181(defmacro monom-map (fun m &rest ml &aux (result `(copy-seq ,m)))
182 `(map-into ,result ,fun ,m ,@ml))
183
184(defmacro monom-append (m1 m2)
185 `(concatenate 'monom ,m1 ,m2))
186
187(defmacro monom-contract (k m)
188 `(subseq ,m ,k))
189
190(defun monom-exponents (m)
191 (declare (type monom m))
192 (coerce m 'list))
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