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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(defpackage "MONOM"
23 (:use :cl :ring)
24 (:export "MONOM"
25 "EXPONENT"
26 "MONOM-DIMENSION"
27 "MONOM-EXPONENTS"
28 "MONOM-EQUALP"
29 "MONOM-CLONE"
30 "MAKE-MONOM-VARIABLE")
31 (:documentation
32 "This package implements basic operations on monomials.
33DATA STRUCTURES: Conceptually, monomials can be represented as lists:
34
35 monom: (n1 n2 ... nk) where ni are non-negative integers
36
37However, lists may be implemented as other sequence types, so the
38flexibility to change the representation should be maintained in the
39code to use general operations on sequences whenever possible. The
40optimization for the actual representation should be left to
41declarations and the compiler.
42
43EXAMPLES: Suppose that variables are x and y. Then
44
45 Monom x*y^2 ---> (1 2) "))
46
47(in-package :monom)
48
49(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
50
51(deftype exponent ()
52 "Type of exponent in a monomial."
53 'fixnum)
54
55(defclass monom ()
56 ((dimension :initarg :dimension :accessor monom-dimension)
57 (exponents :initarg :exponents :accessor monom-exponents))
58 (:default-initargs :dimension nil :exponents nil :exponent nil)
59 (:documentation
60 "Implements a monomial, i.e. a product of powers
61of variables, like X*Y^2."))
62
63(defmethod print-object ((self monom) stream)
64 (format stream "#<MONOM DIMENSION=~A EXPONENTS=~A>"
65 (monom-dimension self)
66 (monom-exponents self)))
67
68(defmethod shared-initialize :after ((self monom) slot-names
69 &key
70 dimension
71 exponents
72 exponent
73 &allow-other-keys
74 )
75 (if (eq slot-names t) (setf slot-names '(dimension exponents)))
76 (dolist (slot-name slot-names)
77 (case slot-name
78 (dimension
79 (cond (dimension
80 (setf (slot-value self 'dimension) dimension))
81 (exponents
82 (setf (slot-value self 'dimension) (length exponents)))
83 (t
84 (error "DIMENSION or EXPONENTS must not be NIL"))))
85 (exponents
86 (cond
87 ;; when exponents are supplied
88 (exponents
89 (let ((dim (length exponents)))
90 (when (and dimension (/= dimension dim))
91 (error "EXPONENTS must have length DIMENSION"))
92 (setf (slot-value self 'dimension) dim
93 (slot-value self 'exponents) (make-array dim :initial-contents exponents))))
94 ;; when all exponents are to be identical
95 (t
96 (let ((dim (slot-value self 'dimension)))
97 (setf (slot-value self 'exponents)
98 (make-array (list dim) :initial-element (or exponent 0)
99 :element-type 'exponent)))))))))
100
101(defun monom-clone (m)
102 (make-instance 'monom
103 :dimension (monom-dimension m)
104 :exponents (copy-seq (monom-exponents m))))
105
106(defun monom-equalp (m1 m2)
107 "Returns T iff monomials M1 and M2 have identical
108EXPONENTS."
109 (declare (type monom m1 m2))
110 (equalp (monom-exponents m1) (monom-exponents m2)))
111
112(defmethod r-coeff ((m monom))
113 "A MONOM can be treated as a special case of TERM,
114where the coefficient is 1."
115 1)
116
117(defmethod r-elt ((m monom) index)
118 "Return the power in the monomial M of variable number INDEX."
119 (with-slots (exponents)
120 m
121 (elt exponents index)))
122
123(defmethod (setf r-elt) (new-value (m monom) index)
124 "Return the power in the monomial M of variable number INDEX."
125 (with-slots (exponents)
126 m
127 (setf (elt exponents index) new-value)))
128
129(defmethod r-total-degree ((m monom) &optional (start 0) (end (monom-dimension m)))
130 "Return the todal degree of a monomoal M. Optinally, a range
131of variables may be specified with arguments START and END."
132 (declare (type fixnum start end))
133 (with-slots (exponents)
134 m
135 (reduce #'+ exponents :start start :end end)))
136
137
138(defmethod r-sugar ((m monom) &aux (start 0) (end (monom-dimension m)))
139 "Return the sugar of a monomial M. Optinally, a range
140of variables may be specified with arguments START and END."
141 (declare (type fixnum start end))
142 (r-total-degree m start end))
143
144(defmethod multiply-by ((self monom) (other monom))
145 (with-slots ((exponents1 exponents) (dimension1 dimension))
146 self
147 (with-slots ((exponents2 exponents) (dimension2 dimension))
148 other
149 (unless (= dimension1 dimension2)
150 (error "Incompatible dimensions: ~A and ~A.~%" dimension1 dimension2))
151 (map-into exponents1 #'+ exponents1 exponents2)))
152 self)
153
154(defmethod r/ ((m1 monom) (m2 monom))
155 "Divide monomial M1 by monomial M2."
156 (with-slots ((exponents1 exponents) (dimension1 dimension))
157 m1
158 (with-slots ((exponents2 exponents))
159 m2
160 (let* ((exponents (copy-seq exponents1))
161 (dimension dimension1))
162 (map-into exponents #'- exponents1 exponents2)
163 (make-instance 'monom :dimension dimension :exponents exponents)))))
164
165(defmethod r-divides-p ((m1 monom) (m2 monom))
166 "Returns T if monomial M1 divides monomial M2, NIL otherwise."
167 (with-slots ((exponents1 exponents))
168 m1
169 (with-slots ((exponents2 exponents))
170 m2
171 (every #'<= exponents1 exponents2))))
172
173
174(defmethod r-divides-lcm-p ((m1 monom) (m2 monom) (m3 monom))
175 "Returns T if monomial M1 divides LCM(M2,M3), NIL otherwise."
176 (every #'(lambda (x y z) (<= x (max y z)))
177 m1 m2 m3))
178
179
180(defmethod r-lcm-divides-lcm-p ((m1 monom) (m2 monom) (m3 monom) (m4 monom))
181 "Returns T if monomial MONOM-LCM(M1,M2) divides MONOM-LCM(M3,M4), NIL otherwise."
182 (declare (type monom m1 m2 m3 m4))
183 (every #'(lambda (x y z w) (<= (max x y) (max z w)))
184 m1 m2 m3 m4))
185
186(defmethod r-lcm-equal-lcm-p (m1 m2 m3 m4)
187 "Returns T if monomial LCM(M1,M2) equals LCM(M3,M4), NIL otherwise."
188 (with-slots ((exponents1 exponents))
189 m1
190 (with-slots ((exponents2 exponents))
191 m2
192 (with-slots ((exponents3 exponents))
193 m3
194 (with-slots ((exponents4 exponents))
195 m4
196 (every
197 #'(lambda (x y z w) (= (max x y) (max z w)))
198 exponents1 exponents2 exponents3 exponents4))))))
199
200(defmethod r-divisible-by-p ((m1 monom) (m2 monom))
201 "Returns T if monomial M1 is divisible by monomial M2, NIL otherwise."
202 (with-slots ((exponents1 exponents))
203 m1
204 (with-slots ((exponents2 exponents))
205 m2
206 (every #'>= exponents1 exponents2))))
207
208(defmethod r-rel-prime-p ((m1 monom) (m2 monom))
209 "Returns T if two monomials M1 and M2 are relatively prime (disjoint)."
210 (with-slots ((exponents1 exponents))
211 m1
212 (with-slots ((exponents2 exponents))
213 m2
214 (every #'(lambda (x y) (zerop (min x y))) exponents1 exponents2))))
215
216
217(defmethod r-equalp ((m1 monom) (m2 monom))
218 "Returns T if two monomials M1 and M2 are equal."
219 (monom-equalp m1 m2))
220
221(defmethod r-lcm ((m1 monom) (m2 monom))
222 "Returns least common multiple of monomials M1 and M2."
223 (with-slots ((exponents1 exponents) (dimension1 dimension))
224 m1
225 (with-slots ((exponents2 exponents))
226 m2
227 (let* ((exponents (copy-seq exponents1))
228 (dimension dimension1))
229 (map-into exponents #'max exponents1 exponents2)
230 (make-instance 'monom :dimension dimension :exponents exponents)))))
231
232
233(defmethod r-gcd ((m1 monom) (m2 monom))
234 "Returns greatest common divisor of monomials M1 and M2."
235 (with-slots ((exponents1 exponents) (dimension1 dimension))
236 m1
237 (with-slots ((exponents2 exponents))
238 m2
239 (let* ((exponents (copy-seq exponents1))
240 (dimension dimension1))
241 (map-into exponents #'min exponents1 exponents2)
242 (make-instance 'monom :dimension dimension :exponents exponents)))))
243
244(defmethod r-depends-p ((m monom) k)
245 "Return T if the monomial M depends on variable number K."
246 (declare (type fixnum k))
247 (with-slots (exponents)
248 m
249 (plusp (elt exponents k))))
250
251(defmethod r-tensor-product ((m1 monom) (m2 monom))
252 (with-slots ((exponents1 exponents) (dimension1 dimension))
253 m1
254 (with-slots ((exponents2 exponents) (dimension2 dimension))
255 m2
256 (make-instance 'monom
257 :dimension (+ dimension1 dimension2)
258 :exponents (concatenate 'vector exponents1 exponents2)))))
259
260(defmethod r-contract ((m monom) k)
261 "Drop the first K variables in monomial M."
262 (declare (fixnum k))
263 (with-slots (dimension exponents)
264 m
265 (setf dimension (- dimension k)
266 exponents (subseq exponents k))))
267
268(defun make-monom-variable (nvars pos &optional (power 1)
269 &aux (m (make-instance 'monom :dimension nvars)))
270 "Construct a monomial in the polynomial ring
271RING[X[0],X[1],X[2],...X[NVARS-1]] over the (unspecified) ring RING
272which represents a single variable. It assumes number of variables
273NVARS and the variable is at position POS. Optionally, the variable
274may appear raised to power POWER. "
275 (declare (type fixnum nvars pos power) (type monom m))
276 (with-slots (exponents)
277 m
278 (setf (elt exponents pos) power)
279 m))
280
281(defmethod r->list ((m monom))
282 "A human-readable representation of a monomial M as a list of exponents."
283 (coerce (monom-exponents m) 'list))
284
285(defmethod r-dimension ((self monom))
286 (monom-dimension self))
287
288(defmethod r-exponents ((self monom))
289 (monom-exponents self))
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