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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra suls.spad}
\author{Clifton J. Williamson}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain SULS SparseUnivariateLaurentSeries}
<<domain SULS SparseUnivariateLaurentSeries>>=
)abbrev domain SULS SparseUnivariateLaurentSeries
++ Author: Clifton J. Williamson
++ Date Created: 11 November 1994
++ Date Last Updated: June 18, 2010
++ Basic Operations:
++ Related Domains: InnerSparseUnivariatePowerSeries,
++ SparseUnivariateTaylorSeries, SparseUnivariatePuiseuxSeries
++ Also See:
++ AMS Classifications:
++ Keywords: sparse, series
++ Examples:
++ References:
++ Description: Sparse Laurent series in one variable
++ \spadtype{SparseUnivariateLaurentSeries} is a domain representing Laurent
++ series in one variable with coefficients in an arbitrary ring. The
++ parameters of the type specify the coefficient ring, the power series
++ variable, and the center of the power series expansion. For example,
++ \spad{SparseUnivariateLaurentSeries(Integer,x,3)} represents Laurent
++ series in \spad{(x - 3)} with integer coefficients.
SparseUnivariateLaurentSeries(Coef,var,cen): Exports == Implementation where
Coef : Ring
var : Symbol
cen : Coef
I ==> Integer
NNI ==> NonNegativeInteger
OUT ==> OutputForm
P ==> Polynomial Coef
RF ==> Fraction Polynomial Coef
RN ==> Fraction Integer
S ==> String
SUTS ==> SparseUnivariateTaylorSeries(Coef,var,cen)
EFULS ==> ElementaryFunctionsUnivariateLaurentSeries(Coef,SUTS,%)
Exports ==> Join(UnivariateLaurentSeriesConstructorCategory(Coef,SUTS),_
PartialDifferentialDomain(%,Variable var)) with
coerce: Variable(var) -> %
++ \spad{coerce(var)} converts the series variable \spad{var} into a
++ Laurent series.
if Coef has Algebra Fraction Integer then
integrate: (%,Variable(var)) -> %
++ \spad{integrate(f(x))} returns an anti-derivative of the power
++ series \spad{f(x)} with constant coefficient 0.
++ We may integrate a series when we can divide coefficients
++ by integers.
Implementation ==> InnerSparseUnivariatePowerSeries(Coef) add
Rep := InnerSparseUnivariatePowerSeries(Coef)
variable x == var
center x == cen
coerce(v: Variable(var)) ==
zero? cen => monomial(1,1)
monomial(1,1) + monomial(cen,0)
pole? x == negative? order(x,0)
--% operations with Taylor series
coerce(uts:SUTS) == uts pretend %
taylorIfCan uls ==
pole? uls => "failed"
uls pretend SUTS
taylor uls ==
(uts := taylorIfCan uls) case "failed" =>
error "taylor: Laurent series has a pole"
uts :: SUTS
retractIfCan(x:%):Union(SUTS,"failed") == taylorIfCan x
laurent(n,uts) == monomial(1,n) * (uts :: %)
removeZeroes uls == uls
removeZeroes(n,uls) == uls
taylorRep uls == taylor(monomial(1,-order(uls,0)) * uls)
degree uls == order(uls,0)
numer uls == taylorRep uls
denom uls == monomial(1,(-order(uls,0)) :: NNI)$SUTS
(uts:SUTS) * (uls:%) == (uts :: %) * uls
(uls:%) * (uts:SUTS) == uls * (uts :: %)
if Coef has Field then
(uts1:SUTS) / (uts2:SUTS) == (uts1 :: %) / (uts2 :: %)
recip(uls) == iExquo(1,uls,false)
if Coef has IntegralDomain then
uls1 exquo uls2 == iExquo(uls1,uls2,false)
if Coef has Field then
uls1:% / uls2:% ==
(q := uls1 exquo uls2) case "failed" =>
error "quotient cannot be computed"
q :: %
differentiate(uls:%,v:Variable(var)) == differentiate uls
elt(uls1:%,uls2:%) ==
order(uls2,1) < 1 =>
error "elt: second argument must have positive order"
negative?(ord := order(uls1,0)) =>
(recipr := recip uls2) case "failed" =>
error "elt: second argument not invertible"
uls3 := uls1 * monomial(1,-ord)
iCompose(uls3,uls2) * (recipr :: %) ** ((-ord) :: NNI)
iCompose(uls1,uls2)
if Coef has IntegralDomain then
rationalFunction(uls,n) ==
zero?(e := order(uls,0)) =>
negative? n => 0
polynomial(taylor uls,n :: NNI) :: RF
negative?(m := n - e) => 0
poly := polynomial(taylor(monomial(1,-e) * uls),m :: NNI) :: RF
v := variable(uls) :: RF; c := center(uls) :: P :: RF
poly / (v - c) ** ((-e) :: NNI)
rationalFunction(uls,n1,n2) == rationalFunction(truncate(uls,n1,n2),n2)
if Coef has Algebra Fraction Integer then
integrate uls ==
zero? coefficient(uls,-1) =>
error "integrate: series has term of order -1"
integrate(uls)$Rep
integrate(uls:%,v:Variable(var)) == integrate uls
(uls1:%) ** (uls2:%) == exp(log(uls1) * uls2)
exp uls == exp(uls)$EFULS
log uls == log(uls)$EFULS
sin uls == sin(uls)$EFULS
cos uls == cos(uls)$EFULS
tan uls == tan(uls)$EFULS
cot uls == cot(uls)$EFULS
sec uls == sec(uls)$EFULS
csc uls == csc(uls)$EFULS
asin uls == asin(uls)$EFULS
acos uls == acos(uls)$EFULS
atan uls == atan(uls)$EFULS
acot uls == acot(uls)$EFULS
asec uls == asec(uls)$EFULS
acsc uls == acsc(uls)$EFULS
sinh uls == sinh(uls)$EFULS
cosh uls == cosh(uls)$EFULS
tanh uls == tanh(uls)$EFULS
coth uls == coth(uls)$EFULS
sech uls == sech(uls)$EFULS
csch uls == csch(uls)$EFULS
asinh uls == asinh(uls)$EFULS
acosh uls == acosh(uls)$EFULS
atanh uls == atanh(uls)$EFULS
acoth uls == acoth(uls)$EFULS
asech uls == asech(uls)$EFULS
acsch uls == acsch(uls)$EFULS
if Coef has CommutativeRing then
(uls:%) ** (r:RN) == cRationalPower(uls,r)
else
(uls:%) ** (r:RN) ==
negative?(ord0 := order(uls,0)) =>
order := ord0 :: I
(n := order exquo denom(r)) case "failed" =>
error "**: rational power does not exist"
uts := retract(uls * monomial(1,-order))@SUTS
utsPow := (uts ** r) :: %
monomial(1,(n :: I) * numer(r)) * utsPow
uts := retract(uls)@SUTS
(uts ** r) :: %
--% OutputForms
coerce(uls:%): OUT ==
st := getStream uls
if not(explicitlyEmpty? st or explicitEntries? st) _
and (nx := retractIfCan(deref getRef uls))@Union(I,"failed") case I then
count : NNI := _$streamCount$Lisp
degr := min(count,(nx :: I) + count + 1)
extend(uls,degr)
seriesToOutputForm(st,getRef uls,variable uls,center uls,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>>
<<domain SULS SparseUnivariateLaurentSeries>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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