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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra brill.spad}
\author{Frederic Lehobey, James H. Davenport}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package BRILL BrillhartTests}
<<package BRILL BrillhartTests>>=
)abbrev package BRILL BrillhartTests
++ Author: Frederic Lehobey, James H. Davenport
++ Date Created: 28 June 1994
++ Date Last Updated: 11 July 1997
++ Basic Operations: brillhartIrreducible?
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: factorization
++ Examples:
++ References:
++ [1] John Brillhart, Note on Irreducibility Testing,
++ Mathematics of Computation, vol. 35, num. 35, Oct. 1980, 1379-1381
++ [2] James Davenport, On Brillhart Irreducibility. To appear.
++ [3] John Brillhart, On the Euler and Bernoulli polynomials,
++ J. Reine Angew. Math., v. 234, (1969), pp. 45-64
BrillhartTests(UP): Exports == Implementation where
N ==> NonNegativeInteger
Z ==> Integer
UP: UnivariatePolynomialCategory Z
Exports ==> with
brillhartIrreducible?: UP -> Boolean -- See [1]
++ brillhartIrreducible?(p) returns \spad{true} if p can be shown to be
++ irreducible by a remark of Brillhart, \spad{false} is inconclusive.
brillhartIrreducible?: (UP,Boolean) -> Boolean -- See [1]
++ brillhartIrreducible?(p,noLinears) returns \spad{true} if p can be
++ shown to be irreducible by a remark of Brillhart, \spad{false} else.
++ If noLinears is \spad{true}, we are being told p has no linear factors
++ \spad{false} does not mean that p is reducible.
brillhartTrials: () -> N
++ brillhartTrials() returns the number of tests in
++ \spadfun{brillhartIrreducible?}.
brillhartTrials: N -> N
++ brillhartTrials(n) sets to n the number of tests in
++ \spadfun{brillhartIrreducible?} and returns the previous value.
noLinearFactor?: UP -> Boolean -- See [3] p. 47
++ noLinearFactor?(p) returns \spad{true} if p can be shown to have no
++ linear factor by a theorem of Lehmer, \spad{false} else. I insist on
++ the fact that \spad{false} does not mean that p has a linear factor.
Implementation ==> add
import GaloisGroupFactorizationUtilities(Z,UP,Float)
squaredPolynomial(p:UP):Boolean ==
d := degree p
d = 0 => true
odd? d => false
squaredPolynomial reductum p
primeEnough?(n:Z,b:Z):Boolean ==
-- checks if n is prime, with the possible exception of
-- factors whose product is at most b
import Float
bb: Float := b::Float
for i in 2..b repeat
while (d:= n exquo i) case Integer repeat
n:=d::Integer
bb:=bb / i::Float
bb < 1$Float => return false
--- we over-divided, so it can't be prime
prime? n
brillharttrials: N := 6
brillhartTrials():N == brillharttrials
brillhartTrials(n:N):N ==
(brillharttrials,n) := (n,brillharttrials)
n
brillhartIrreducible?(p:UP):Boolean ==
brillhartIrreducible?(p,noLinearFactor? p)
brillhartIrreducible?(p:UP,noLinears:Boolean):Boolean == -- See [1]
zero? brillharttrials => false
origBound := (largeEnough := rootBound(p)+1)
-- see remarks 2 and 4
even0 := even? coefficient(p,0)
even1 := even? p(1)
polyx2 := squaredPolynomial(p)
prime? p(largeEnough) => true
not polyx2 and prime? p(-largeEnough) => true
one? brillharttrials => false
largeEnough := largeEnough+1
primeEnough?(p(largeEnough),if noLinears then 4 else 2) => true
not polyx2 and
primeEnough?(p(-largeEnough),if noLinears then 4 else 2) => true
if odd? largeEnough then
if even0 then largeEnough := largeEnough+1
else
if even1 then largeEnough := largeEnough+1
count :=(if polyx2 then 2 else 1)*(brillharttrials-2)+largeEnough
for i in (largeEnough+1)..count repeat
small := if noLinears then (i-origBound)**2 else (i-origBound)
primeEnough?(p(i),small) => return true
not polyx2 and primeEnough?(p(-i),small) => return true
false
noLinearFactor?(p:UP):Boolean ==
(odd? leadingCoefficient p) and (odd? coefficient(p,0)) and (odd? p(1))
@
\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 BRILL BrillhartTests>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
