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[81]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
[418]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;;
[714]37;; Monom x*y^2 ---> (1 2)
[418]38;;
39;;----------------------------------------------------------------
40
[394]41(defpackage "MONOMIAL"
[395]42 (:use :cl)
[422]43 (:export "MONOM"
[423]44 "EXPONENT"
[422]45 "MAKE-MONOM"
[396]46 "MONOM-ELT"
47 "MONOM-DIMENSION"
48 "MONOM-TOTAL-DEGREE"
49 "MONOM-SUGAR"
50 "MONOM-DIV"
51 "MONOM-MUL"
52 "MONOM-DIVIDES-P"
[395]53 "MONOM-DIVIDES-MONOM-LCM-P"
54 "MONOM-LCM-DIVIDES-MONOM-LCM-P"
[497]55 "MONOM-LCM-EQUAL-MONOM-LCM-P"
[395]56 "MONOM-DIVISIBLE-BY-P"
57 "MONOM-REL-PRIME-P"
58 "MONOM-EQUAL-P"
59 "MONOM-LCM"
60 "MONOM-GCD"
[504]61 "MONOM-DEPENDS-P"
[395]62 "MONOM-MAP"
63 "MONOM-APPEND"
64 "MONOM-CONTRACT"
65 "MONOM-EXPONENTS"))
[81]66
[419]67(in-package :monomial)
[48]68
69(deftype exponent ()
70 "Type of exponent in a monomial."
71 'fixnum)
72
[723]73(defstruct (monom
[725]74 ;; BOA constructor
[833]75 (:constructor construct-monom (dimension
[726]76 &key
[753]77 (initial-exponents #() initial-exponents-supplied-p)
[754]78 (initial-exponent #() initial-exponent-supplied-p)
[726]79 (exponents (cond
[763]80 ;; when exponents are supplied
[753]81 (initial-exponents-supplied-p
82 (make-array (list dimension) :initial-contents initial-exponents
[726]83 :element-type 'exponent))
[763]84 ;; when all exponents are to be identical
[754]85 (initial-exponent-supplied-p
86 (make-array (list dimension) :initial-element initial-exponent
[726]87 :element-type 'exponent))
[763]88 ;; otherwise, all exponents are zero
[727]89 (t (make-array (list dimension) :element-type 'exponent :initial-element 0)))))))
[726]90 (exponents nil :type (vector exponent *)))
[717]91
[48]92;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
93;;
94;; Operations on monomials
95;;
96;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
97
[745]98(defun monom-dimension (m)
99 (declare (type monom m))
[746]100 (length (monom-exponents m)))
[745]101
[48]102(defmacro monom-elt (m index)
103 "Return the power in the monomial M of variable number INDEX."
[727]104 `(elt (monom-exponents ,m) ,index))
[48]105
[747]106(defun monom-total-degree (m &optional (start 0) (end (monom-dimension m)))
[48]107 "Return the todal degree of a monomoal M. Optinally, a range
108of variables may be specified with arguments START and END."
109 (declare (type monom m) (fixnum start end))
[728]110 (reduce #'+ (monom-exponents m) :start start :end end))
[48]111
[747]112(defun monom-sugar (m &aux (start 0) (end (monom-dimension m)))
[48]113 "Return the sugar of a monomial M. Optinally, a range
114of variables may be specified with arguments START and END."
115 (declare (type monom m) (fixnum start end))
[749]116 (monom-total-degree m start end))
[48]117
[729]118(defun monom-div (m1 m2 &aux (result (copy-structure m1)))
[48]119 "Divide monomial M1 by monomial M2."
[728]120 (declare (type monom m1 m2))
[729]121 (map-into (monom-exponents result) #'- (monom-exponents m1) (monom-exponents m2))
122 result)
[48]123
[729]124(defun monom-mul (m1 m2 &aux (result (copy-structure m1)))
[48]125 "Multiply monomial M1 by monomial M2."
126 (declare (type monom m1 m2 result))
[729]127 (map-into (monom-exponents result) #'+ (monom-exponents m1) (monom-exponents m2))
128 result)
[48]129
130(defun monom-divides-p (m1 m2)
131 "Returns T if monomial M1 divides monomial M2, NIL otherwise."
132 (declare (type monom m1 m2))
[730]133 (every #'<= (monom-exponents m1) (monom-exponents m2)))
[48]134
135(defun monom-divides-monom-lcm-p (m1 m2 m3)
136 "Returns T if monomial M1 divides MONOM-LCM(M2,M3), NIL otherwise."
137 (declare (type monom m1 m2 m3))
[731]138 (every #'(lambda (x y z) (declare (type exponent x y z)) (<= x (max y z)))
139 (monom-exponents m1)
140 (monom-exponents m2)
141 (monom-exponents m3)))
[48]142
143(defun monom-lcm-divides-monom-lcm-p (m1 m2 m3 m4)
144 "Returns T if monomial MONOM-LCM(M1,M2) divides MONOM-LCM(M3,M4), NIL otherwise."
145 (declare (type monom m1 m2 m3 m4))
[732]146 (every #'(lambda (x y z w) (declare (type exponent x y z w)) (<= (max x y) (max z w)))
147 (monom-exponents m1)
148 (monom-exponents m2)
149 (monom-exponents m3)
150 (monom-exponents m4)))
[48]151
152(defun monom-lcm-equal-monom-lcm-p (m1 m2 m3 m4)
153 "Returns T if monomial MONOM-LCM(M1,M2) equals MONOM-LCM(M3,M4), NIL otherwise."
154 (declare (type monom m1 m2 m3 m4))
[733]155 (every #'(lambda (x y z w) (declare (type exponent x y z w)) (= (max x y) (max z w)))
156 (monom-exponents m1)
157 (monom-exponents m2)
158 (monom-exponents m3)
159 (monom-exponents m4)))
[48]160
161(defun monom-divisible-by-p (m1 m2)
162 "Returns T if monomial M1 is divisible by monomial M2, NIL otherwise."
163 (declare (type monom m1 m2))
[733]164 (every #'>= (monom-exponents m1) (monom-exponents m2)))
[48]165
166(defun monom-rel-prime-p (m1 m2)
167 "Returns T if two monomials M1 and M2 are relatively prime (disjoint)."
168 (declare (type monom m1 m2))
[734]169 (every #'(lambda (x y) (declare (type exponent x y)) (zerop (min x y)))
170 (monom-exponents m1)
171 (monom-exponents m2)))
[48]172
173(defun monom-equal-p (m1 m2)
174 "Returns T if two monomials M1 and M2 are equal."
175 (declare (type monom m1 m2))
[735]176 (every #'= (monom-exponents m1) (monom-exponents m2)))
[48]177
[736]178(defun monom-lcm (m1 m2 &aux (result (copy-structure m1)))
[48]179 "Returns least common multiple of monomials M1 and M2."
180 (declare (type monom m1 m2))
[736]181 (map-into (monom-exponents result) #'max
182 (monom-exponents m1)
[737]183 (monom-exponents m2))
184 result)
[48]185
[737]186(defun monom-gcd (m1 m2 &aux (result (copy-structure m1)))
[48]187 "Returns greatest common divisor of monomials M1 and M2."
188 (declare (type monom m1 m2))
[737]189 (map-into (monom-exponents result) #'min (monom-exponents m1) (monom-exponents m2))
190 result)
[48]191
192(defun monom-depends-p (m k)
193 "Return T if the monomial M depends on variable number K."
194 (declare (type monom m) (fixnum k))
[738]195 (plusp (monom-elt m k)))
[48]196
[738]197(defmacro monom-map (fun m &rest ml &aux (result `(copy-structure ,m)))
198 `(map-into (monom-exponents ,result) ,fun ,m ,@ml))
[48]199
200(defmacro monom-append (m1 m2)
[764]201 `(make-monom (+ (monom-dimension ,m1) (monom-dimension ,m2))
[765]202 :initial-exponents (concatenate 'vector (monom-exponents ,m1) (monom-exponents ,m2))))
[48]203
204(defmacro monom-contract (k m)
[740]205 `(setf (monom-exponents ,m) (subseq (monom-exponents ,m) ,k)))
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