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source: branches/f4grobner/monom.lisp@ 1884

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1;;; -*- Mode: Lisp -*-
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 "MONOM"
42 (:use :cl)
43 (:export "MONOM"
44 "EXPONENT"
45 "MAKE-MONOM"
46 "MAKE-MONOM-VARIABLE"
47 "MONOM-ELT"
48 "MONOM-DIMENSION"
49 "MONOM-TOTAL-DEGREE"
50 "MONOM-SUGAR"
51 "MONOM-DIV"
52 "MONOM-MUL"
53 "MONOM-DIVIDES-P"
54 "MONOM-DIVIDES-MONOM-LCM-P"
55 "MONOM-LCM-DIVIDES-MONOM-LCM-P"
56 "MONOM-LCM-EQUAL-MONOM-LCM-P"
57 "MONOM-DIVISIBLE-BY-P"
58 "MONOM-REL-PRIME-P"
59 "MONOM-EQUAL-P"
60 "MONOM-LCM"
61 "MONOM-GCD"
62 "MONOM-DEPENDS-P"
63 "MONOM-MAP"
64 "MONOM-APPEND"
65 "MONOM-CONTRACT"
66 "MONOM->LIST"))
67
68(in-package :monom)
69
70(deftype exponent ()
71 "Type of exponent in a monomial."
72 'fixnum)
73
74(deftype monom (&optional dim)
75 "Type of monomial."
76 `(simple-array exponent (,dim)))
77
78;; If a monomial is redefined as structure with slot EXPONENTS, the function
79;; below can be the BOA constructor.
80(defun make-monom (&key
81 (dimension nil dimension-suppied-p)
82 (initial-exponents nil initial-exponents-supplied-p)
83 (initial-exponent nil initial-exponent-supplied-p)
84 &aux
85 (dim (cond (dimension-suppied-p dimension)
86 (initial-exponents-supplied-p (length initial-exponents))
87 (t (error "You must provide DIMENSION nor INITIAL-EXPONENTS"))))
88 (monom (cond
89 ;; when exponents are supplied
90 (initial-exponents-supplied-p
91 (make-array (list dim) :initial-contents initial-exponents
92 :element-type 'exponent))
93 ;; when all exponents are to be identical
94 (initial-exponent-supplied-p
95 (make-array (list dim) :initial-element initial-exponent
96 :element-type 'exponent))
97 ;; otherwise, all exponents are zero
98 (t
99 (make-array (list dim) :element-type 'exponent :initial-element 0)))))
100 "A constructor (factory) of monomials. If DIMENSION is given, a sequence of
101DIMENSION elements of type EXPONENT is constructed, where individual
102elements are the value of INITIAL-EXPONENT, which defaults to 0.
103Alternatively, all elements may be specified as a list
104INITIAL-EXPONENTS."
105 (assert (typep monom `(monom ,dim)))
106 monom)
107
108;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
109;;
110;; Operations on monomials
111;;
112;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
113
114(defun monom-dimension (m)
115 (length m))
116
117(defmacro monom-elt (m index)
118 "Return the power in the monomial M of variable number INDEX."
119 `(elt ,m ,index))
120
121(defun monom-total-degree (m &optional (start 0) (end (monom-dimension m)))
122 "Return the todal degree of a monomoal M. Optinally, a range
123of variables may be specified with arguments START and END."
124 (reduce #'+ m :start start :end end))
125
126(defun monom-sugar (m &aux (start 0) (end (monom-dimension m)))
127 "Return the sugar of a monomial M. Optinally, a range
128of variables may be specified with arguments START and END."
129 (monom-total-degree m start end))
130
131(defun monom-div (m1 m2 &aux (result (copy-seq m1)))
132 "Divide monomial M1 by monomial M2."
133 (map-into result #'- m1 m2))
134
135(defun monom-mul (m1 m2 &aux (result (copy-seq m1)))
136 "Multiply monomial M1 by monomial M2."
137 (map-into result #'+ m1 m2))
138
139(defun monom-divides-p (m1 m2)
140 "Returns T if monomial M1 divides monomial M2, NIL otherwise."
141 (every #'<= m1 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 (every #'(lambda (x y z) (<= x (max y z)))
146 m1 m2 m3))
147
148(defun monom-lcm-divides-monom-lcm-p (m1 m2 m3 m4)
149 "Returns T if monomial MONOM-LCM(M1,M2) divides MONOM-LCM(M3,M4), NIL otherwise."
150 (every #'(lambda (x y z w) (<= (max x y) (max z w)))
151 m1 m2 m3 m4))
152
153
154(defun monom-lcm-equal-monom-lcm-p (m1 m2 m3 m4)
155 "Returns T if monomial MONOM-LCM(M1,M2) equals MONOM-LCM(M3,M4), NIL otherwise."
156 (every #'(lambda (x y z w) (= (max x y) (max z w)))
157 m1 m2 m3 m4))
158
159
160(defun monom-divisible-by-p (m1 m2)
161 "Returns T if monomial M1 is divisible by monomial M2, NIL otherwise."
162 (every #'>= m1 m2))
163
164(defun monom-rel-prime-p (m1 m2)
165 "Returns T if two monomials M1 and M2 are relatively prime (disjoint)."
166 (every #'(lambda (x y) (zerop (min x y))) m1 m2))
167
168(defun monom-equal-p (m1 m2)
169 "Returns T if two monomials M1 and M2 are equal."
170 (every #'= m1 m2))
171
172(defun monom-lcm (m1 m2 &aux (result (copy-seq m1)))
173 "Returns least common multiple of monomials M1 and M2."
174 (map-into result #'max m1 m2))
175
176(defun monom-gcd (m1 m2 &aux (result (copy-seq m1)))
177 "Returns greatest common divisor of monomials M1 and M2."
178 (map-into result #'min m1 m2))
179
180(defun monom-depends-p (m k)
181 "Return T if the monomial M depends on variable number K."
182 (plusp (monom-elt m k)))
183
184(defmacro monom-map (fun m &rest ml &aux (result `(copy-seq ,m)))
185 `(map-into ,result ,fun ,m ,@ml))
186
187(defun monom-append (m1 m2 &aux (dim (+ (length m1) (length m2))))
188 (concatenate `(monom ,dim) m1 m2))
189
190(defmacro monom-contract (m k)
191 "Drop the first K variables in monomial M."
192 `(setf ,m (subseq ,m ,k)))
193
194(defun make-monom-variable (nvars pos &optional (power 1)
195 &aux (m (make-monom :dimension nvars)))
196 "Construct a monomial in the polynomial ring
197RING[X[0],X[1],X[2],...X[NVARS-1]] over the (unspecified) ring RING
198which represents a single variable. It assumes number of variables
199NVARS and the variable is at position POS. Optionally, the variable
200may appear raised to power POWER. "
201 (setf (monom-elt m pos) power)
202 m)
203
204(defun monom->list (m)
205 "A human-readable representation of a monomial M as a list of exponents."
206 (coerce m 'list))
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