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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 "Non-destructively multiply monomial M1 by M2."
156 (multiply-by (monom-clone m1) (monom-clone m2)))
157
158(defmethod r/ ((m1 monom) (m2 monom))
159 "Divide monomial M1 by monomial M2."
160 (with-slots ((exponents1 exponents) (dimension1 dimension))
161 m1
162 (with-slots ((exponents2 exponents))
163 m2
164 (let* ((exponents (copy-seq exponents1))
165 (dimension dimension1))
166 (map-into exponents #'- exponents1 exponents2)
167 (make-instance 'monom :dimension dimension :exponents exponents)))))
168
169(defmethod r-divides-p ((m1 monom) (m2 monom))
170 "Returns T if monomial M1 divides monomial M2, NIL otherwise."
171 (with-slots ((exponents1 exponents))
172 m1
173 (with-slots ((exponents2 exponents))
174 m2
175 (every #'<= exponents1 exponents2))))
176
177
178(defmethod r-divides-lcm-p ((m1 monom) (m2 monom) (m3 monom))
179 "Returns T if monomial M1 divides LCM(M2,M3), NIL otherwise."
180 (every #'(lambda (x y z) (<= x (max y z)))
181 m1 m2 m3))
182
183
184(defmethod r-lcm-divides-lcm-p ((m1 monom) (m2 monom) (m3 monom) (m4 monom))
185 "Returns T if monomial MONOM-LCM(M1,M2) divides MONOM-LCM(M3,M4), NIL otherwise."
186 (declare (type monom m1 m2 m3 m4))
187 (every #'(lambda (x y z w) (<= (max x y) (max z w)))
188 m1 m2 m3 m4))
189
190(defmethod r-lcm-equal-lcm-p (m1 m2 m3 m4)
191 "Returns T if monomial LCM(M1,M2) equals LCM(M3,M4), NIL otherwise."
192 (with-slots ((exponents1 exponents))
193 m1
194 (with-slots ((exponents2 exponents))
195 m2
196 (with-slots ((exponents3 exponents))
197 m3
198 (with-slots ((exponents4 exponents))
199 m4
200 (every
201 #'(lambda (x y z w) (= (max x y) (max z w)))
202 exponents1 exponents2 exponents3 exponents4))))))
203
204(defmethod r-divisible-by-p ((m1 monom) (m2 monom))
205 "Returns T if monomial M1 is divisible by monomial M2, NIL otherwise."
206 (with-slots ((exponents1 exponents))
207 m1
208 (with-slots ((exponents2 exponents))
209 m2
210 (every #'>= exponents1 exponents2))))
211
212(defmethod r-rel-prime-p ((m1 monom) (m2 monom))
213 "Returns T if two monomials M1 and M2 are relatively prime (disjoint)."
214 (with-slots ((exponents1 exponents))
215 m1
216 (with-slots ((exponents2 exponents))
217 m2
218 (every #'(lambda (x y) (zerop (min x y))) exponents1 exponents2))))
219
220
221(defmethod r-equalp ((m1 monom) (m2 monom))
222 "Returns T if two monomials M1 and M2 are equal."
223 (monom-equalp m1 m2))
224
225(defmethod r-lcm ((m1 monom) (m2 monom))
226 "Returns least common multiple of monomials M1 and M2."
227 (with-slots ((exponents1 exponents) (dimension1 dimension))
228 m1
229 (with-slots ((exponents2 exponents))
230 m2
231 (let* ((exponents (copy-seq exponents1))
232 (dimension dimension1))
233 (map-into exponents #'max exponents1 exponents2)
234 (make-instance 'monom :dimension dimension :exponents exponents)))))
235
236
237(defmethod r-gcd ((m1 monom) (m2 monom))
238 "Returns greatest common divisor of monomials M1 and M2."
239 (with-slots ((exponents1 exponents) (dimension1 dimension))
240 m1
241 (with-slots ((exponents2 exponents))
242 m2
243 (let* ((exponents (copy-seq exponents1))
244 (dimension dimension1))
245 (map-into exponents #'min exponents1 exponents2)
246 (make-instance 'monom :dimension dimension :exponents exponents)))))
247
248(defmethod r-depends-p ((m monom) k)
249 "Return T if the monomial M depends on variable number K."
250 (declare (type fixnum k))
251 (with-slots (exponents)
252 m
253 (plusp (elt exponents k))))
254
255(defmethod r-tensor-product ((m1 monom) (m2 monom))
256 (with-slots ((exponents1 exponents) (dimension1 dimension))
257 m1
258 (with-slots ((exponents2 exponents) (dimension2 dimension))
259 m2
260 (make-instance 'monom
261 :dimension (+ dimension1 dimension2)
262 :exponents (concatenate 'vector exponents1 exponents2)))))
263
264(defmethod r-contract ((m monom) k)
265 "Drop the first K variables in monomial M."
266 (declare (fixnum k))
267 (with-slots (dimension exponents)
268 m
269 (setf dimension (- dimension k)
270 exponents (subseq exponents k))))
271
272(defun make-monom-variable (nvars pos &optional (power 1)
273 &aux (m (make-instance 'monom :dimension nvars)))
274 "Construct a monomial in the polynomial ring
275RING[X[0],X[1],X[2],...X[NVARS-1]] over the (unspecified) ring RING
276which represents a single variable. It assumes number of variables
277NVARS and the variable is at position POS. Optionally, the variable
278may appear raised to power POWER. "
279 (declare (type fixnum nvars pos power) (type monom m))
280 (with-slots (exponents)
281 m
282 (setf (elt exponents pos) power)
283 m))
284
285(defmethod r->list ((m monom))
286 "A human-readable representation of a monomial M as a list of exponents."
287 (coerce (monom-exponents m) 'list))
288
289(defmethod r-dimension ((self monom))
290 (monom-dimension self))
291
292(defmethod r-exponents ((self monom))
293 (monom-exponents self))
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