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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(defstruct (monom
74 ;; BOA constructor
75 ;; NOTE: having a non-default constructor inhibits reading via #S(...)
76 (:constructor make-monom (&key
77 (dimension nil dimension-suppied-p)
78 (initial-exponents nil initial-exponents-supplied-p)
79 (initial-exponent nil initial-exponent-supplied-p)
80 &aux
81 (dim (cond (dimension-suppied-p dimension)
82 (initial-exponents-supplied-p (length initial-exponents))
83 (t (error "You must provide DIMENSION nor INITIAL-EXPONENTS"))))
84 (exponents (cond
85 ;; when exponents are supplied
86 (initial-exponents-supplied-p
87 (make-array (list dim) :initial-contents initial-exponents
88 :element-type 'exponent))
89 ;; when all exponents are to be identical
90 (initial-exponent-supplied-p
91 (make-array (list dim) :initial-element initial-exponent
92 :element-type 'exponent))
93 ;; otherwise, all exponents are zero
94 (t
95 (make-array (list dim) :element-type 'exponent :initial-element 0))))))
96 )
97 (exponents nil :type (vector exponent *)))
98
99
100;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
101;;
102;; Operations on monomials
103;;
104;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
105
106(defun monom-dimension (m)
107 (declare (type monom m))
108 (length (monom-exponents m)))
109
110(defmacro monom-elt (m index)
111 "Return the power in the monomial M of variable number INDEX."
112 `(elt (monom-exponents ,m) ,index))
113
114(defun monom-total-degree (m &optional (start 0) (end (monom-dimension m)))
115 "Return the todal degree of a monomoal M. Optinally, a range
116of variables may be specified with arguments START and END."
117 (declare (type monom m) (fixnum start end))
118 (reduce #'+ (monom-exponents m) :start start :end end))
119
120(defun monom-sugar (m &aux (start 0) (end (monom-dimension m)))
121 "Return the sugar of a monomial M. Optinally, a range
122of variables may be specified with arguments START and END."
123 (declare (type monom m) (fixnum start end))
124 (monom-total-degree m start end))
125
126(defun monom-div (m1 m2 &aux (result (copy-structure m1)))
127 "Divide monomial M1 by monomial M2."
128 (declare (type monom m1 m2))
129 (map-into (monom-exponents result) #'- (monom-exponents m1) (monom-exponents m2))
130 result)
131
132(defun monom-mul (m1 m2 &aux (result (copy-structure m1)))
133 "Multiply monomial M1 by monomial M2."
134 (declare (type monom m1 m2 result))
135 (map-into (monom-exponents result) #'+ (monom-exponents m1) (monom-exponents m2))
136 result)
137
138(defun monom-divides-p (m1 m2)
139 "Returns T if monomial M1 divides monomial M2, NIL otherwise."
140 (declare (type monom m1 m2))
141 (every #'<= (monom-exponents m1) (monom-exponents m2)))
142
143(defun monom-divides-monom-lcm-p (m1 m2 m3)
144 "Returns T if monomial M1 divides MONOM-LCM(M2,M3), NIL otherwise."
145 (declare (type monom m1 m2 m3))
146 (every #'(lambda (x y z) (declare (type exponent x y z)) (<= x (max y z)))
147 (monom-exponents m1)
148 (monom-exponents m2)
149 (monom-exponents m3)))
150
151(defun monom-lcm-divides-monom-lcm-p (m1 m2 m3 m4)
152 "Returns T if monomial MONOM-LCM(M1,M2) divides MONOM-LCM(M3,M4), NIL otherwise."
153 (declare (type monom m1 m2 m3 m4))
154 (every #'(lambda (x y z w) (declare (type exponent x y z w)) (<= (max x y) (max z w)))
155 (monom-exponents m1)
156 (monom-exponents m2)
157 (monom-exponents m3)
158 (monom-exponents m4)))
159
160(defun monom-lcm-equal-monom-lcm-p (m1 m2 m3 m4)
161 "Returns T if monomial MONOM-LCM(M1,M2) equals MONOM-LCM(M3,M4), NIL otherwise."
162 (declare (type monom m1 m2 m3 m4))
163 (every #'(lambda (x y z w) (declare (type exponent x y z w)) (= (max x y) (max z w)))
164 (monom-exponents m1)
165 (monom-exponents m2)
166 (monom-exponents m3)
167 (monom-exponents m4)))
168
169(defun monom-divisible-by-p (m1 m2)
170 "Returns T if monomial M1 is divisible by monomial M2, NIL otherwise."
171 (declare (type monom m1 m2))
172 (every #'>= (monom-exponents m1) (monom-exponents m2)))
173
174(defun monom-rel-prime-p (m1 m2)
175 "Returns T if two monomials M1 and M2 are relatively prime (disjoint)."
176 (declare (type monom m1 m2))
177 (every #'(lambda (x y) (declare (type exponent x y)) (zerop (min x y)))
178 (monom-exponents m1)
179 (monom-exponents m2)))
180
181(defun monom-equal-p (m1 m2)
182 "Returns T if two monomials M1 and M2 are equal."
183 (declare (type monom m1 m2))
184 (every #'= (monom-exponents m1) (monom-exponents m2)))
185
186(defun monom-lcm (m1 m2 &aux (result (copy-structure m1)))
187 "Returns least common multiple of monomials M1 and M2."
188 (declare (type monom m1 m2))
189 (map-into (monom-exponents result) #'max
190 (monom-exponents m1)
191 (monom-exponents m2))
192 result)
193
194(defun monom-gcd (m1 m2 &aux (result (copy-structure m1)))
195 "Returns greatest common divisor of monomials M1 and M2."
196 (declare (type monom m1 m2))
197 (map-into (monom-exponents result) #'min (monom-exponents m1) (monom-exponents m2))
198 result)
199
200(defun monom-depends-p (m k)
201 "Return T if the monomial M depends on variable number K."
202 (declare (type monom m) (fixnum k))
203 (plusp (monom-elt m k)))
204
205(defmacro monom-map (fun m &rest ml &aux (result `(copy-structure ,m)))
206 `(map-into (monom-exponents ,result) ,fun ,m ,@ml))
207
208(defmacro monom-append (m1 m2)
209 `(make-monom :exponents (concatenate 'vector (monom-exponents ,m1) (monom-exponents ,m2))))
210
211(defmacro monom-contract (k m)
212 `(setf (monom-exponents ,m) (subseq (monom-exponents ,m) ,k)))
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