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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra fourier.spad}
\author{James Davenport}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain FCOMP FourierComponent}
<<domain FCOMP FourierComponent>>=
)abbrev domain FCOMP FourierComponent
++ Author: James Davenport
++ Date Created: 17 April 1992
++ Date Last Updated: 12 June 1992
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
FourierComponent(E:OrderedSet):
OrderedSet with
sin: E -> $
++ sin(x) makes a sin kernel for use in Fourier series
cos: E -> $
++ cos(x) makes a cos kernel for use in Fourier series
sin?: $ -> Boolean
++ sin?(x) returns true if term is a sin, otherwise false
argument: $ -> E
++ argument(x) returns the argument of a given sin/cos expressions
==
add
--representations
Rep:=Record(SinIfTrue:Boolean, arg:E)
e:E
x,y:$
sin e == [true,e]
cos e == [false,e]
sin? x == x.SinIfTrue
argument x == x.arg
coerce(x):OutputForm ==
hconcat((if x.SinIfTrue then "sin" else "cos")::OutputForm,
bracket((x.arg)::OutputForm))
x<y ==
x.arg < y.arg => true
y.arg < x.arg => false
x.SinIfTrue => false
y.SinIfTrue
@
\section{domain FSERIES FourierSeries}
<<domain FSERIES FourierSeries>>=
)abbrev domain FSERIES FourierSeries
++ Author: James Davenport
++ Date Created: 17 April 1992
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
FourierSeries(R:Join(CommutativeRing,Algebra(Fraction Integer)),
E:Join(OrderedSet,AbelianGroup)):
Algebra(R) with
if E has canonical and R has canonical then canonical
coerce: R -> $
++ coerce(r) converts coefficients into Fourier Series
coerce: FourierComponent(E) -> $
++ coerce(c) converts sin/cos terms into Fourier Series
makeSin: (E,R) -> $
++ makeSin(e,r) makes a sin expression with given argument and coefficient
makeCos: (E,R) -> $
++ makeCos(e,r) makes a sin expression with given argument and coefficient
== FreeModule(R,FourierComponent(E))
add
--representations
Term := Record(k:FourierComponent(E),c:R)
Rep := List Term
multiply : (Term,Term) -> $
w,x1,x2:$
n:NonNegativeInteger
z:Integer
e:FourierComponent(E)
a:E
r:R
1 == [[cos 0,1]]
coerce e ==
sin? e and zero? argument e => 0
if argument e < 0 then
not sin? e => e:=cos(- argument e)
return [[sin(- argument e),-1]]
[[e,1]]
multiply(t1,t2) ==
r:=(t1.c*t2.c)*(1/2)
s1:=argument t1.k
s2:=argument t2.k
sum:=s1+s2
diff:=s1-s2
sin? t1.k =>
sin? t2.k =>
makeCos(diff,r) + makeCos(sum,-r)
makeSin(sum,r) + makeSin(diff,r)
sin? t2.k =>
makeSin(sum,r) + makeSin(diff,r)
makeCos(diff,r) + makeCos(sum,r)
x1*x2 ==
null x1 => 0
null x2 => 0
+/[+/[multiply(t1,t2) for t2 in x2] for t1 in x1]
makeCos(a,r) ==
a<0 => [[cos(-a),r]]
[[cos a,r]]
makeSin(a,r) ==
zero? a => []
a<0 => [[sin(-a),-r]]
[[sin a,r]]
@
\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>>
<<domain FCOMP FourierComponent>>
<<domain FSERIES FourierSeries>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
