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