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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra galutil.spad}
\author{Frederic Lehobey}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package GALUTIL GaloisGroupUtilities}
<<package GALUTIL GaloisGroupUtilities>>=
)abbrev package GALUTIL GaloisGroupUtilities
++ Author: Frederic Lehobey
++ Date Created: 29 June 1994
++ Date Last Updated: 30 June 1994
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ \spadtype{GaloisGroupUtilities} provides several useful functions.
GaloisGroupUtilities(R): Exports == Implementation where
N ==> NonNegativeInteger
Z ==> Integer
R : Ring
Exports ==> with
pascalTriangle: (N,Z) -> R
++ pascalTriangle(n,r) returns the binomial coefficient
++ \spad{C(n,r)=n!/(r! (n-r)!)}
++ and stores it in a table to prevent recomputation.
rangePascalTriangle: N -> N
++ rangePascalTriangle(n) sets the maximal number of lines which
++ are stored and returns the previous value.
rangePascalTriangle: () -> N
++ rangePascalTriangle() returns the maximal number of lines stored.
sizePascalTriangle: () -> N
++ sizePascalTriangle() returns the number of entries currently stored
++ in the table.
fillPascalTriangle: () -> Void
++ fillPascalTriangle() fills the stored table.
if R has FloatingPointSystem then
safeCeiling: R -> Z
++ safeCeiling(x) returns the integer which is greater than any integer
++ with the same floating point number representation.
safeFloor: R -> Z
++ safeFloor(x) returns the integer which is lower or equal to the
++ largest integer which has the same floating point number
++ representation.
safetyMargin: N -> N
++ safetyMargin(n) sets to n the number of low weight digits we do not
++ trust in the floating point representation and returns the previous
++ value (for use by \spadfun{safeCeiling}).
safetyMargin: () -> N
++ safetyMargin() returns the number of low weight digits we do not
++ trust in the floating point representation (used by
++ \spadfun{safeCeiling}).
Implementation ==> add
if R has FloatingPointSystem then
safetymargin : N := 6
safeFloor(x:R):Z ==
if (shift := order(x)-precision()$R+safetymargin) >= 0 then
x := x+float(1,shift)
retract(floor(x))@Z
safeCeiling(x:R):Z ==
if (shift := order(x)-precision()$R+safetymargin) >= 0 then
x := x+float(1,shift)
retract(ceiling(x))@Z
safetyMargin(n:N):N ==
(safetymargin,n) := (n,safetymargin)
n
safetyMargin():N == safetymargin
pascaltriangle : FlexibleArray(R) := empty()
ncomputed : N := 3
rangepascaltriangle : N := 216
pascalTriangle(n:N, r:Z):R ==
negative? r => 0
(d := n-r) < r => pascalTriangle(n,d)
zero? r => 1$R
one? r => n :: R
n > rangepascaltriangle =>
binomial(n,r)$IntegerCombinatoricFunctions(Z) :: R
n <= ncomputed =>
m := divide(n-4,2)
mq := m.quotient
pascaltriangle((mq+1)*(mq+m.remainder)+r-1)
-- compute the missing lines
for i in (ncomputed+1)..n repeat
for j in 2..(i quo 2) repeat
pascaltriangle := concat!(pascaltriangle,pascalTriangle((i-1)
:: N, j-1)+pascalTriangle((i-1) :: N,j))
ncomputed := i
pascalTriangle(n,r)
rangePascalTriangle(n:N):N ==
if n<ncomputed then
if n<3 then
pascaltriangle := delete!(pascaltriangle,1..#pascaltriangle)
ncomputed := 3
else
d := divide(n-3,2)
dq := d.quotient
pascaltriangle := delete!(pascaltriangle,((dq+1)*(dq+d.remainder)
+1)..#pascaltriangle)
ncomputed := n
(rangepascaltriangle,n) := (n,rangepascaltriangle)
n
rangePascalTriangle():N == rangepascaltriangle
sizePascalTriangle():N == #pascaltriangle
fillPascalTriangle():Void == pascalTriangle(rangepascaltriangle,2)
@
\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 GALUTIL GaloisGroupUtilities>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
