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
|
\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra ratfact.spad}
\author{Patrizia Gianni}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package RATFACT RationalFactorize}
<<package RATFACT RationalFactorize>>=
)abbrev package RATFACT RationalFactorize
++ Author: P. Gianni
++ Date created: ??
++ Date last updated: December 1993
++ Factorization of extended polynomials with rational coefficients.
++ This package implements factorization of extended polynomials
++ whose coefficients are rational numbers. It does this by taking the
++ lcm of the coefficients of the polynomial and creating a polynomial
++ with integer coefficients. The algorithm in \spadtype{GaloisGroupFactorizer} is then
++ used to factor the integer polynomial. The result is normalized
++ with respect to the original lcm of the denominators.
++ Keywords: factorization, hensel, rational number
I ==> Integer
RN ==> Fraction Integer
RationalFactorize(RP) : public == private where
BP ==> SparseUnivariatePolynomial(I)
RP : UnivariatePolynomialCategory RN
public ==> with
factor : RP -> Factored RP
++ factor(p) factors an extended polynomial p over the rational numbers.
factorSquareFree : RP -> Factored RP
++ factorSquareFree(p) factors an extended squareFree
++ polynomial p over the rational numbers.
private ==> add
import GaloisGroupFactorizer (BP)
ParFact ==> Record(irr:BP,pow:I)
FinalFact ==> Record(contp:I,factors:List(ParFact))
URNI ==> UnivariatePolynomialCategoryFunctions2(RN,RP,I,BP)
UIRN ==> UnivariatePolynomialCategoryFunctions2(I,BP,RN,RP)
fUnion ==> Union("nil", "sqfr", "irred", "prime")
FFE ==> Record(flg:fUnion, fctr:RP, xpnt:I)
factor(p:RP) : Factored(RP) ==
p = 0 => 0
pden: I := lcm([denom c for c in coefficients p])
pol : RP := pden*p
ipol: BP := map(numer,pol)$URNI
ffact: FinalFact := henselFact(ipol,false)
makeFR(((ffact.contp)/pden)::RP,
[["prime",map(coerce,u.irr)$UIRN,u.pow]$FFE
for u in ffact.factors])
factorSquareFree(p:RP) : Factored(RP) ==
p = 0 => 0
pden: I := lcm([denom c for c in coefficients p])
pol : RP := pden*p
ipol: BP := map(numer,pol)$URNI
ffact: FinalFact := henselFact(ipol,true)
makeFR(((ffact.contp)/pden)::RP,
[["prime",map(coerce,u.irr)$UIRN,u.pow]$FFE
for u in ffact.factors])
@
\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 RATFACT RationalFactorize>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
