[1201] | 1 | ;;; -*- Mode: Lisp -*-
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[81] | 2 | ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
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| 3 | ;;;
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| 4 | ;;; Copyright (C) 1999, 2002, 2009, 2015 Marek Rychlik <rychlik@u.arizona.edu>
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| 5 | ;;;
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| 6 | ;;; This program is free software; you can redistribute it and/or modify
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| 7 | ;;; it under the terms of the GNU General Public License as published by
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| 8 | ;;; the Free Software Foundation; either version 2 of the License, or
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| 9 | ;;; (at your option) any later version.
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| 10 | ;;;
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| 11 | ;;; This program is distributed in the hope that it will be useful,
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| 12 | ;;; but WITHOUT ANY WARRANTY; without even the implied warranty of
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| 13 | ;;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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| 14 | ;;; GNU General Public License for more details.
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| 15 | ;;;
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| 16 | ;;; You should have received a copy of the GNU General Public License
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| 17 | ;;; along with this program; if not, write to the Free Software
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| 18 | ;;; Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
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| 19 | ;;;
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| 20 | ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
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| 21 |
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[418] | 22 | ;;----------------------------------------------------------------
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| 23 | ;; This package implements BASIC OPERATIONS ON MONOMIALS
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| 24 | ;;----------------------------------------------------------------
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| 25 | ;; DATA STRUCTURES: Conceptually, monomials can be represented as lists:
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| 26 | ;;
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| 27 | ;; monom: (n1 n2 ... nk) where ni are non-negative integers
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| 28 | ;;
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| 29 | ;; However, lists may be implemented as other sequence types,
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| 30 | ;; so the flexibility to change the representation should be
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| 31 | ;; maintained in the code to use general operations on sequences
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| 32 | ;; whenever possible. The optimization for the actual representation
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| 33 | ;; should be left to declarations and the compiler.
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| 34 | ;;----------------------------------------------------------------
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| 35 | ;; EXAMPLES: Suppose that variables are x and y. Then
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| 36 | ;;
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[714] | 37 | ;; Monom x*y^2 ---> (1 2)
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[418] | 38 | ;;
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| 39 | ;;----------------------------------------------------------------
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| 40 |
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[1610] | 41 | (defpackage "MONOM"
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[395] | 42 | (:use :cl)
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[422] | 43 | (:export "MONOM"
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[423] | 44 | "EXPONENT"
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[422] | 45 | "MAKE-MONOM"
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[887] | 46 | "MAKE-MONOM-VARIABLE"
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[396] | 47 | "MONOM-ELT"
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| 48 | "MONOM-DIMENSION"
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| 49 | "MONOM-TOTAL-DEGREE"
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| 50 | "MONOM-SUGAR"
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| 51 | "MONOM-DIV"
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| 52 | "MONOM-MUL"
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| 53 | "MONOM-DIVIDES-P"
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[395] | 54 | "MONOM-DIVIDES-MONOM-LCM-P"
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| 55 | "MONOM-LCM-DIVIDES-MONOM-LCM-P"
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[497] | 56 | "MONOM-LCM-EQUAL-MONOM-LCM-P"
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[395] | 57 | "MONOM-DIVISIBLE-BY-P"
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| 58 | "MONOM-REL-PRIME-P"
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| 59 | "MONOM-EQUAL-P"
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| 60 | "MONOM-LCM"
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| 61 | "MONOM-GCD"
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[504] | 62 | "MONOM-DEPENDS-P"
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[395] | 63 | "MONOM-MAP"
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| 64 | "MONOM-APPEND"
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| 65 | "MONOM-CONTRACT"
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[1153] | 66 | "MONOM->LIST"))
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[81] | 67 |
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[1610] | 68 | (in-package :monom)
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[48] | 69 |
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| 70 | (deftype exponent ()
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| 71 | "Type of exponent in a monomial."
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| 72 | 'fixnum)
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| 73 |
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[880] | 74 | (deftype monom (&optional dim)
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| 75 | "Type of monomial."
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[1864] | 76 | `(simple-array exponent ,dim))
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[880] | 77 |
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[884] | 78 | ;; If a monomial is redefined as structure with slot EXPONENTS, the function
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| 79 | ;; below can be the BOA constructor.
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[873] | 80 | (defun make-monom (&key
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| 81 | (dimension nil dimension-suppied-p)
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| 82 | (initial-exponents nil initial-exponents-supplied-p)
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| 83 | (initial-exponent nil initial-exponent-supplied-p)
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| 84 | &aux
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| 85 | (dim (cond (dimension-suppied-p dimension)
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| 86 | (initial-exponents-supplied-p (length initial-exponents))
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| 87 | (t (error "You must provide DIMENSION nor INITIAL-EXPONENTS"))))
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| 88 | (monom (cond
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| 89 | ;; when exponents are supplied
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| 90 | (initial-exponents-supplied-p
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| 91 | (make-array (list dim) :initial-contents initial-exponents
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| 92 | :element-type 'exponent))
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| 93 | ;; when all exponents are to be identical
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| 94 | (initial-exponent-supplied-p
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| 95 | (make-array (list dim) :initial-element initial-exponent
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| 96 | :element-type 'exponent))
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| 97 | ;; otherwise, all exponents are zero
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| 98 | (t
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| 99 | (make-array (list dim) :element-type 'exponent :initial-element 0)))))
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[1600] | 100 | "A constructor (factory) of monomials. If DIMENSION is given, a sequence of
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[1599] | 101 | DIMENSION elements of type EXPONENT is constructed, where individual
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| 102 | elements are the value of INITIAL-EXPONENT, which defaults to 0.
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| 103 | Alternatively, all elements may be specified as a list
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| 104 | INITIAL-EXPONENTS."
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[1866] | 105 | (assert (typep monom 'monom))
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| 106 | monom)
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[717] | 107 |
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[837] | 108 |
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[48] | 109 | ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
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| 110 | ;;
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| 111 | ;; Operations on monomials
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| 112 | ;;
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| 113 | ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
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| 114 |
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[745] | 115 | (defun monom-dimension (m)
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[872] | 116 | (length m))
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[745] | 117 |
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[48] | 118 | (defmacro monom-elt (m index)
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| 119 | "Return the power in the monomial M of variable number INDEX."
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[879] | 120 | `(elt ,m ,index))
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[48] | 121 |
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[747] | 122 | (defun monom-total-degree (m &optional (start 0) (end (monom-dimension m)))
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[48] | 123 | "Return the todal degree of a monomoal M. Optinally, a range
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| 124 | of variables may be specified with arguments START and END."
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[871] | 125 | (reduce #'+ m :start start :end end))
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[48] | 126 |
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[747] | 127 | (defun monom-sugar (m &aux (start 0) (end (monom-dimension m)))
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[48] | 128 | "Return the sugar of a monomial M. Optinally, a range
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| 129 | of variables may be specified with arguments START and END."
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[749] | 130 | (monom-total-degree m start end))
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[48] | 131 |
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[876] | 132 | (defun monom-div (m1 m2 &aux (result (copy-seq m1)))
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[48] | 133 | "Divide monomial M1 by monomial M2."
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[870] | 134 | (map-into result #'- m1 m2))
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[48] | 135 |
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[876] | 136 | (defun monom-mul (m1 m2 &aux (result (copy-seq m1)))
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[48] | 137 | "Multiply monomial M1 by monomial M2."
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[870] | 138 | (map-into result #'+ m1 m2))
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[48] | 139 |
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| 140 | (defun monom-divides-p (m1 m2)
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| 141 | "Returns T if monomial M1 divides monomial M2, NIL otherwise."
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[877] | 142 | (every #'<= m1 m2))
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[48] | 143 |
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| 144 | (defun monom-divides-monom-lcm-p (m1 m2 m3)
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| 145 | "Returns T if monomial M1 divides MONOM-LCM(M2,M3), NIL otherwise."
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[875] | 146 | (every #'(lambda (x y z) (<= x (max y z)))
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[869] | 147 | m1 m2 m3))
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[48] | 148 |
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| 149 | (defun monom-lcm-divides-monom-lcm-p (m1 m2 m3 m4)
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| 150 | "Returns T if monomial MONOM-LCM(M1,M2) divides MONOM-LCM(M3,M4), NIL otherwise."
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[869] | 151 | (every #'(lambda (x y z w) (<= (max x y) (max z w)))
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| 152 | m1 m2 m3 m4))
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| 153 |
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[48] | 154 |
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| 155 | (defun monom-lcm-equal-monom-lcm-p (m1 m2 m3 m4)
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| 156 | "Returns T if monomial MONOM-LCM(M1,M2) equals MONOM-LCM(M3,M4), NIL otherwise."
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[869] | 157 | (every #'(lambda (x y z w) (= (max x y) (max z w)))
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| 158 | m1 m2 m3 m4))
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[48] | 159 |
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[869] | 160 |
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[48] | 161 | (defun monom-divisible-by-p (m1 m2)
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| 162 | "Returns T if monomial M1 is divisible by monomial M2, NIL otherwise."
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[869] | 163 | (every #'>= m1 m2))
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[48] | 164 |
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| 165 | (defun monom-rel-prime-p (m1 m2)
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| 166 | "Returns T if two monomials M1 and M2 are relatively prime (disjoint)."
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[875] | 167 | (every #'(lambda (x y) (zerop (min x y))) m1 m2))
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[48] | 168 |
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| 169 | (defun monom-equal-p (m1 m2)
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| 170 | "Returns T if two monomials M1 and M2 are equal."
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[878] | 171 | (every #'= m1 m2))
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[48] | 172 |
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[874] | 173 | (defun monom-lcm (m1 m2 &aux (result (copy-seq m1)))
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[48] | 174 | "Returns least common multiple of monomials M1 and M2."
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[868] | 175 | (map-into result #'max m1 m2))
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[48] | 176 |
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[874] | 177 | (defun monom-gcd (m1 m2 &aux (result (copy-seq m1)))
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[48] | 178 | "Returns greatest common divisor of monomials M1 and M2."
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[868] | 179 | (map-into result #'min m1 m2))
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[48] | 180 |
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| 181 | (defun monom-depends-p (m k)
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| 182 | "Return T if the monomial M depends on variable number K."
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[738] | 183 | (plusp (monom-elt m k)))
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[48] | 184 |
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[874] | 185 | (defmacro monom-map (fun m &rest ml &aux (result `(copy-seq ,m)))
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[868] | 186 | `(map-into ,result ,fun ,m ,@ml))
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[48] | 187 |
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| 188 | (defmacro monom-append (m1 m2)
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[1867] | 189 | `(concatenate 'vector ,m1 ,m2))
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[48] | 190 |
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[1632] | 191 | (defmacro monom-contract (m k)
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[1638] | 192 | "Drop the first K variables in monomial M."
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[868] | 193 | `(setf ,m (subseq ,m ,k)))
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[886] | 194 |
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| 195 | (defun make-monom-variable (nvars pos &optional (power 1)
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| 196 | &aux (m (make-monom :dimension nvars)))
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| 197 | "Construct a monomial in the polynomial ring
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| 198 | RING[X[0],X[1],X[2],...X[NVARS-1]] over the (unspecified) ring RING
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| 199 | which represents a single variable. It assumes number of variables
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| 200 | NVARS and the variable is at position POS. Optionally, the variable
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| 201 | may appear raised to power POWER. "
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| 202 | (setf (monom-elt m pos) power)
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| 203 | m)
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[1151] | 204 |
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| 205 | (defun monom->list (m)
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[1152] | 206 | "A human-readable representation of a monomial M as a list of exponents."
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[1161] | 207 | (coerce m 'list))
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