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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 :ring)
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(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
71
72(deftype exponent ()
73 "Type of exponent in a monomial."
74 'fixnum)
75
76(defclass monom ()
77 ((dim :initarg :dim)
78 (exponents :initarg :exponents))
79 (:default-initargs :dim 0 :exponents nil))
80
81(defmethod print-object ((m monom) stream)
82 (princ (slot-value m 'exponents)))
83
84;; If a monomial is redefined as structure with slot EXPONENTS, the function
85;; below can be the BOA constructor.
86(defun make-monom (&key
87 (dimension nil dimension-suppied-p)
88 (initial-exponents nil initial-exponents-supplied-p)
89 (initial-exponent nil initial-exponent-supplied-p)
90 &aux
91 (dim (cond (dimension-suppied-p dimension)
92 (initial-exponents-supplied-p (length initial-exponents))
93 (t (error "You must provide DIMENSION or INITIAL-EXPONENTS"))))
94 (exponents (cond
95 ;; when exponents are supplied
96 (initial-exponents-supplied-p
97 (make-array (list dim) :initial-contents initial-exponents
98 :element-type 'exponent))
99 ;; when all exponents are to be identical
100 (initial-exponent-supplied-p
101 (make-array (list dim) :initial-element initial-exponent
102 :element-type 'exponent))
103 ;; otherwise, all exponents are zero
104 (t
105 (make-array (list dim) :element-type 'exponent :initial-element 0)))))
106 "A constructor (factory) of monomials. If DIMENSION is given, a sequence of
107DIMENSION elements of type EXPONENT is constructed, where individual
108elements are the value of INITIAL-EXPONENT, which defaults to 0.
109Alternatively, all elements may be specified as a list
110INITIAL-EXPONENTS."
111 (make-instance 'monom :dim dim :exponents exponents))
112
113;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
114;;
115;; Operations on monomials
116;;
117;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
118
119(defmethod dimension ((m monom))
120 (slot-value m 'dim))
121
122(defmethod ring-elt ((m monom) index)
123 "Return the power in the monomial M of variable number INDEX."
124 (with-slots (exponents)
125 m
126 (elt exponents index)))
127
128(defmethod (setf ring-elt) (new-value (m monom) index)
129 "Return the power in the monomial M of variable number INDEX."
130 (with-slots (exponents)
131 m
132 (setf (elt exponents index) new-value)))
133
134(defmethod ring-total-degree ((m monom) &optional (start 0) (end (dimension m)))
135 "Return the todal degree of a monomoal M. Optinally, a range
136of variables may be specified with arguments START and END."
137 (declare (type fixnum start end))
138 (with-slots (exponents)
139 m
140 (reduce #'+ exponents :start start :end end)))
141
142(defmethod sugar ((m monom) &aux (start 0) (end (dimension m)))
143 "Return the sugar of a monomial M. Optinally, a range
144of variables may be specified with arguments START and END."
145 (declare (type fixnum start end))
146 (monom-total-degree m start end))
147
148(defmethod ring-div ((m1 monom) (m2 monom))
149 "Divide monomial M1 by monomial M2."
150 (with-slots (exponents1 exponents)
151 m1
152 (with-slots (exponents2 exponents2)
153 m2
154 (let* ((exponents (copy-seq exponents1))
155 (dim (reduce #+ exponents)))
156 (map-into result #'- exponents1 exponents2)
157 (make-instance 'monom :dim dim :exponents exponents)))))
158
159#|
160(defun monom-mul (m1 m2 &aux (result (copy-seq m1)))
161 "Multiply monomial M1 by monomial M2."
162 (declare (type monom m1 m2 result))
163 (map-into result #'+ m1 m2))
164
165(defun monom-divides-p (m1 m2)
166 "Returns T if monomial M1 divides monomial M2, NIL otherwise."
167 (declare (type monom m1 m2))
168 (every #'<= m1 m2))
169
170(defun monom-divides-monom-lcm-p (m1 m2 m3)
171 "Returns T if monomial M1 divides MONOM-LCM(M2,M3), NIL otherwise."
172 (declare (type monom m1 m2 m3))
173 (every #'(lambda (x y z) (<= x (max y z)))
174 m1 m2 m3))
175
176(defun monom-lcm-divides-monom-lcm-p (m1 m2 m3 m4)
177 "Returns T if monomial MONOM-LCM(M1,M2) divides MONOM-LCM(M3,M4), NIL otherwise."
178 (declare (type monom m1 m2 m3 m4))
179 (every #'(lambda (x y z w) (<= (max x y) (max z w)))
180 m1 m2 m3 m4))
181
182
183(defun monom-lcm-equal-monom-lcm-p (m1 m2 m3 m4)
184 "Returns T if monomial MONOM-LCM(M1,M2) equals MONOM-LCM(M3,M4), NIL otherwise."
185 (declare (type monom m1 m2 m3 m4))
186 (every #'(lambda (x y z w) (= (max x y) (max z w)))
187 m1 m2 m3 m4))
188
189
190(defun monom-divisible-by-p (m1 m2)
191 "Returns T if monomial M1 is divisible by monomial M2, NIL otherwise."
192 (declare (type monom m1 m2))
193 (every #'>= m1 m2))
194
195(defun monom-rel-prime-p (m1 m2)
196 "Returns T if two monomials M1 and M2 are relatively prime (disjoint)."
197 (declare (type monom m1 m2))
198 (every #'(lambda (x y) (zerop (min x y))) m1 m2))
199
200(defun monom-equal-p (m1 m2)
201 "Returns T if two monomials M1 and M2 are equal."
202 (declare (type monom m1 m2))
203 (every #'= m1 m2))
204
205(defun monom-lcm (m1 m2 &aux (result (copy-seq m1)))
206 "Returns least common multiple of monomials M1 and M2."
207 (declare (type monom m1 m2 result))
208 (map-into result #'max m1 m2))
209
210(defun monom-gcd (m1 m2 &aux (result (copy-seq m1)))
211 "Returns greatest common divisor of monomials M1 and M2."
212 (declare (type monom m1 m2 result))
213 (map-into result #'min m1 m2))
214
215(defun monom-depends-p (m k)
216 "Return T if the monomial M depends on variable number K."
217 (declare (type monom m) (type fixnum k))
218 (plusp (monom-elt m k)))
219
220(defmacro monom-map (fun m &rest ml &aux (result `(copy-seq ,m)))
221 "Map function FUN of one argument over the powers of a monomial M.
222Fun should map a single FIXNUM argument to FIXNUM. Return a sequence
223of results."
224 `(map-into ,result ,fun ,m ,@ml))
225
226(defun monom-append (m1 m2 &aux (dim (+ (length m1) (length m2))))
227 (declare (type monom m1 m2) (fixnum dim))
228 (concatenate `(monom ,dim) m1 m2))
229
230(defun monom-contract (m k)
231 "Drop the first K variables in monomial M."
232 (declare (type monom m) (fixnum k))
233 (subseq m k))
234
235(defun make-monom-variable (nvars pos &optional (power 1)
236 &aux (m (make-monom :dimension nvars)))
237 "Construct a monomial in the polynomial ring
238RING[X[0],X[1],X[2],...X[NVARS-1]] over the (unspecified) ring RING
239which represents a single variable. It assumes number of variables
240NVARS and the variable is at position POS. Optionally, the variable
241may appear raised to power POWER. "
242 (declare (type fixnum nvars pos power) (type monom m))
243 (setf (monom-elt m pos) power)
244 m)
245
246(defun monom->list (m)
247 "A human-readable representation of a monomial M as a list of exponents."
248 (declare (type monom m))
249 (coerce m 'list))
250|#
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