1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
|
\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra updecomp.spad}
\author{Frederic Lehobey}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package UPDECOMP UnivariatePolynomialDecompositionPackage}
<<package UPDECOMP UnivariatePolynomialDecompositionPackage>>=
)abbrev package UPDECOMP UnivariatePolynomialDecompositionPackage
++ Author: Frederic Lehobey
++ Date Created: 17 June 1996
++ Date Last Updated: 4 June 1997
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keyword:
++ Exemples:
++ References:
++ [1] Peter Henrici, Automatic Computations with Power Series,
++ Journal of the Association for Computing Machinery, Volume 3, No. 1,
++ January 1956, 10-15
++ [2] Dexter Kozen and Susan Landau, Polynomial Decomposition
++ Algorithms, Journal of Symbolic Computation (1989) 7, 445-456
-- Decomposition would be speeded up (O(n log n) instead of O(n^2)) by
-- implementing the algorithm described in [3] based on [4] and [5].
++ [3] Joachim von zur Gathen, Functional Decomposition Polynomials:
++ the Tame Case, Journal of Symbolic Computation (1990) 9, 281-299
++ [4] R. P. Brent and H. T. Kung, Fast Algorithms for Manipulating
++ Formal Power Series, Journal of the Association for Computing
++ Machinery, Vol. 25, No. 4, October 1978, 581-595
++ [5] R. P. Brent, Multiple-Precision Zero-Finding Methods and the
++ Complexity of Elementary Function Evaluation, Analytic
++ Computational Complexity, J. F. Traub, Ed., Academic Press,
++ New York 1975, 151-176
++ Description: UnivariatePolynomialDecompositionPackage implements
++ functional decomposition of univariate polynomial with coefficients
++ in an \spad{IntegralDomain} of \spad{CharacteristicZero}.
UnivariatePolynomialDecompositionPackage(R,UP): Exports == Implementation where
R : Join(IntegralDomain,CharacteristicZero)
UP : UnivariatePolynomialCategory(R)
N ==> NonNegativeInteger
LR ==> Record(left: UP, right: UP)
QR ==> Record(quotient: UP, remainder: UP)
Exports ==> with
monicRightFactorIfCan: (UP,N) -> Union(UP,"failed")
++ monicRightFactorIfCan(f,d) returns a candidate to be the
++ monic right factor (h in f = g o h) of degree d of a
++ functional decomposition of the polynomial f or
++ \spad{"failed"} if no such candidate.
rightFactorIfCan: (UP,N,R) -> Union(UP,"failed")
++ rightFactorIfCan(f,d,c) returns a candidate to be the
++ right factor (h in f = g o h) of degree d with leading
++ coefficient c of a functional decomposition of the
++ polynomial f or \spad{"failed"} if no such candidate.
leftFactorIfCan: (UP,UP) -> Union(UP,"failed")
++ leftFactorIfCan(f,h) returns the left factor (g in f = g o h)
++ of the functional decomposition of the polynomial f with
++ given h or \spad{"failed"} if g does not exist.
monicDecomposeIfCan: UP -> Union(LR,"failed")
++ monicDecomposeIfCan(f) returns a functional decomposition
++ of the monic polynomial f of "failed" if it has not found any.
monicCompleteDecompose: UP -> List UP
++ monicCompleteDecompose(f) returns a list of factors of f for
++ the functional decomposition ([ f1, ..., fn ] means
++ f = f1 o ... o fn).
Implementation ==> add
rightFactorIfCan(p,dq,lcq) ==
dp := degree p
zero? lcq =>
error "rightFactorIfCan: leading coefficient may not be zero"
(zero? dp) or (zero? dq) => "failed"
nc := dp exquo dq
nc case "failed" => "failed"
n := nc::N
s := subtractIfCan(dq,1)::N
lcp := leadingCoefficient p
q: UP := monomial(lcq,dq)
for k in 1..s repeat
c: R := 0
for i in 0..subtractIfCan(k,1)::N repeat
c := c+(k::R-(n::R+1)*(i::R))*
coefficient(q,subtractIfCan(dq,i)::N)*
coefficient(p,subtractIfCan(dp+i,k)::N)
cquo := c exquo ((k*n)::R*lcp)
cquo case "failed" => return "failed"
q := q+monomial(cquo::R,subtractIfCan(dq,k)::N)
q
monicRightFactorIfCan(p,dq) == rightFactorIfCan(p,dq,1$R)
import UnivariatePolynomialDivisionPackage(R,UP)
leftFactorIfCan(f,h) ==
g: UP := 0
zero? degree h => "failed"
for i in 0.. while not zero? f repeat
qrf := divideIfCan(f,h)
qrf case "failed" => return "failed"
qr := qrf :: QR
r := qr.remainder
not ground? r => return "failed"
g := g+monomial(ground(r),i)
f := qr.quotient
g
monicDecomposeIfCan f ==
df := degree f
zero? df => "failed"
for dh in 2..subtractIfCan(df,1)::N | zero?(df rem dh) repeat
h := monicRightFactorIfCan(f,dh)
h case UP =>
g := leftFactorIfCan(f,h::UP)
g case UP => return [g::UP,h::UP]
"failed"
monicCompleteDecompose f ==
cf := monicDecomposeIfCan f
cf case "failed" => [ f ]
lr := cf :: LR
append(monicCompleteDecompose lr.left,[lr.right])
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
<<package UPDECOMP UnivariatePolynomialDecompositionPackage>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
