[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
|
---|
[727] | 75 | (:constructor make-monom (dimension
|
---|
[726] | 76 | &key
|
---|
| 77 | (initial-contents #() initial-contents-supplied-p)
|
---|
| 78 | (initial-element #() initial-element-supplied-p)
|
---|
| 79 | (exponents (cond
|
---|
| 80 | (initial-contents-supplied-p
|
---|
[727] | 81 | (make-array (list dimension) :initial-contents initial-contents
|
---|
[726] | 82 | :element-type 'exponent))
|
---|
| 83 | (initial-element-supplied-p
|
---|
[727] | 84 | (make-array (list dimension) :initial-element initial-element
|
---|
[726] | 85 | :element-type 'exponent))
|
---|
[727] | 86 | (t (make-array (list dimension) :element-type 'exponent :initial-element 0)))))))
|
---|
[726] | 87 | (exponents nil :type (vector exponent *)))
|
---|
[717] | 88 |
|
---|
[48] | 89 | ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
---|
| 90 | ;;
|
---|
| 91 | ;; Operations on monomials
|
---|
| 92 | ;;
|
---|
| 93 | ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
---|
| 94 |
|
---|
| 95 | (defmacro monom-elt (m index)
|
---|
| 96 | "Return the power in the monomial M of variable number INDEX."
|
---|
[727] | 97 | `(elt (monom-exponents ,m) ,index))
|
---|
[48] | 98 |
|
---|
[742] | 99 | (defun monom-total-degree (m &optional (start 0) (end (length (monom-exponents m))))
|
---|
[48] | 100 | "Return the todal degree of a monomoal M. Optinally, a range
|
---|
| 101 | of variables may be specified with arguments START and END."
|
---|
| 102 | (declare (type monom m) (fixnum start end))
|
---|
[728] | 103 | (reduce #'+ (monom-exponents m) :start start :end end))
|
---|
[48] | 104 |
|
---|
[743] | 105 | (defun monom-sugar (m &aux (start 0) (end (length (monom-exponents m))))
|
---|
[48] | 106 | "Return the sugar of a monomial M. Optinally, a range
|
---|
| 107 | of variables may be specified with arguments START and END."
|
---|
| 108 | (declare (type monom m) (fixnum start end))
|
---|
[728] | 109 | (monom-total-degree (monom-exponents m) start end))
|
---|
[48] | 110 |
|
---|
[729] | 111 | (defun monom-div (m1 m2 &aux (result (copy-structure m1)))
|
---|
[48] | 112 | "Divide monomial M1 by monomial M2."
|
---|
[728] | 113 | (declare (type monom m1 m2))
|
---|
[729] | 114 | (map-into (monom-exponents result) #'- (monom-exponents m1) (monom-exponents m2))
|
---|
| 115 | result)
|
---|
[48] | 116 |
|
---|
[729] | 117 | (defun monom-mul (m1 m2 &aux (result (copy-structure m1)))
|
---|
[48] | 118 | "Multiply monomial M1 by monomial M2."
|
---|
| 119 | (declare (type monom m1 m2 result))
|
---|
[729] | 120 | (map-into (monom-exponents result) #'+ (monom-exponents m1) (monom-exponents m2))
|
---|
| 121 | result)
|
---|
[48] | 122 |
|
---|
| 123 | (defun monom-divides-p (m1 m2)
|
---|
| 124 | "Returns T if monomial M1 divides monomial M2, NIL otherwise."
|
---|
| 125 | (declare (type monom m1 m2))
|
---|
[730] | 126 | (every #'<= (monom-exponents m1) (monom-exponents m2)))
|
---|
[48] | 127 |
|
---|
| 128 | (defun monom-divides-monom-lcm-p (m1 m2 m3)
|
---|
| 129 | "Returns T if monomial M1 divides MONOM-LCM(M2,M3), NIL otherwise."
|
---|
| 130 | (declare (type monom m1 m2 m3))
|
---|
[731] | 131 | (every #'(lambda (x y z) (declare (type exponent x y z)) (<= x (max y z)))
|
---|
| 132 | (monom-exponents m1)
|
---|
| 133 | (monom-exponents m2)
|
---|
| 134 | (monom-exponents m3)))
|
---|
[48] | 135 |
|
---|
| 136 | (defun monom-lcm-divides-monom-lcm-p (m1 m2 m3 m4)
|
---|
| 137 | "Returns T if monomial MONOM-LCM(M1,M2) divides MONOM-LCM(M3,M4), NIL otherwise."
|
---|
| 138 | (declare (type monom m1 m2 m3 m4))
|
---|
[732] | 139 | (every #'(lambda (x y z w) (declare (type exponent x y z w)) (<= (max x y) (max z w)))
|
---|
| 140 | (monom-exponents m1)
|
---|
| 141 | (monom-exponents m2)
|
---|
| 142 | (monom-exponents m3)
|
---|
| 143 | (monom-exponents m4)))
|
---|
[48] | 144 |
|
---|
| 145 | (defun monom-lcm-equal-monom-lcm-p (m1 m2 m3 m4)
|
---|
| 146 | "Returns T if monomial MONOM-LCM(M1,M2) equals MONOM-LCM(M3,M4), NIL otherwise."
|
---|
| 147 | (declare (type monom m1 m2 m3 m4))
|
---|
[733] | 148 | (every #'(lambda (x y z w) (declare (type exponent x y z w)) (= (max x y) (max z w)))
|
---|
| 149 | (monom-exponents m1)
|
---|
| 150 | (monom-exponents m2)
|
---|
| 151 | (monom-exponents m3)
|
---|
| 152 | (monom-exponents m4)))
|
---|
[48] | 153 |
|
---|
| 154 | (defun monom-divisible-by-p (m1 m2)
|
---|
| 155 | "Returns T if monomial M1 is divisible by monomial M2, NIL otherwise."
|
---|
| 156 | (declare (type monom m1 m2))
|
---|
[733] | 157 | (every #'>= (monom-exponents m1) (monom-exponents m2)))
|
---|
[48] | 158 |
|
---|
| 159 | (defun monom-rel-prime-p (m1 m2)
|
---|
| 160 | "Returns T if two monomials M1 and M2 are relatively prime (disjoint)."
|
---|
| 161 | (declare (type monom m1 m2))
|
---|
[734] | 162 | (every #'(lambda (x y) (declare (type exponent x y)) (zerop (min x y)))
|
---|
| 163 | (monom-exponents m1)
|
---|
| 164 | (monom-exponents m2)))
|
---|
[48] | 165 |
|
---|
| 166 | (defun monom-equal-p (m1 m2)
|
---|
| 167 | "Returns T if two monomials M1 and M2 are equal."
|
---|
| 168 | (declare (type monom m1 m2))
|
---|
[735] | 169 | (every #'= (monom-exponents m1) (monom-exponents m2)))
|
---|
[48] | 170 |
|
---|
[736] | 171 | (defun monom-lcm (m1 m2 &aux (result (copy-structure m1)))
|
---|
[48] | 172 | "Returns least common multiple of monomials M1 and M2."
|
---|
| 173 | (declare (type monom m1 m2))
|
---|
[736] | 174 | (map-into (monom-exponents result) #'max
|
---|
| 175 | (monom-exponents m1)
|
---|
[737] | 176 | (monom-exponents m2))
|
---|
| 177 | result)
|
---|
[48] | 178 |
|
---|
[737] | 179 | (defun monom-gcd (m1 m2 &aux (result (copy-structure m1)))
|
---|
[48] | 180 | "Returns greatest common divisor of monomials M1 and M2."
|
---|
| 181 | (declare (type monom m1 m2))
|
---|
[737] | 182 | (map-into (monom-exponents result) #'min (monom-exponents m1) (monom-exponents m2))
|
---|
| 183 | result)
|
---|
[48] | 184 |
|
---|
| 185 | (defun monom-depends-p (m k)
|
---|
| 186 | "Return T if the monomial M depends on variable number K."
|
---|
| 187 | (declare (type monom m) (fixnum k))
|
---|
[738] | 188 | (plusp (monom-elt m k)))
|
---|
[48] | 189 |
|
---|
[738] | 190 | (defmacro monom-map (fun m &rest ml &aux (result `(copy-structure ,m)))
|
---|
| 191 | `(map-into (monom-exponents ,result) ,fun ,m ,@ml))
|
---|
[48] | 192 |
|
---|
| 193 | (defmacro monom-append (m1 m2)
|
---|
[739] | 194 | `(make-monom (list (+ (monom-dimension ,m1)
|
---|
| 195 | (monom-dimension ,m2)))
|
---|
| 196 | :initial-contents (concatenate 'monom (monom-exponents ,m1) (monom-exponents ,m2))))
|
---|
[48] | 197 |
|
---|
| 198 | (defmacro monom-contract (k m)
|
---|
[740] | 199 | `(setf (monom-exponents ,m) (subseq (monom-exponents ,m) ,k)))
|
---|