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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra genups.spad}
\author{Clifton J. Williamson}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package GENUPS GenerateUnivariatePowerSeries}
<<package GENUPS GenerateUnivariatePowerSeries>>=
)abbrev package GENUPS GenerateUnivariatePowerSeries
++ Author: Clifton J. Williamson
++ Date Created: 29 April 1990
++ Date Last Updated: 31 May 1990
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: series, Taylor, Laurent, Puiseux
++ Examples:
++ References:
++ Description:
++ \spadtype{GenerateUnivariatePowerSeries} provides functions that create
++ power series from explicit formulas for their \spad{n}th coefficient.
GenerateUnivariatePowerSeries(R,FE): Exports == Implementation where
R : Join(IntegralDomain,RetractableTo Integer,_
LinearlyExplicitRingOver Integer)
FE : Join(AlgebraicallyClosedField,TranscendentalFunctionCategory,_
FunctionSpace R)
ANY1 ==> AnyFunctions1
EQ ==> Equation
I ==> Integer
NNI ==> NonNegativeInteger
RN ==> Fraction Integer
SEG ==> UniversalSegment
ST ==> Stream
SY ==> Symbol
UTS ==> UnivariateTaylorSeries
ULS ==> UnivariateLaurentSeries
UPXS ==> UnivariatePuiseuxSeries
Exports ==> with
taylor: (I -> FE,EQ FE) -> Any
++ \spad{taylor(n +-> a(n),x = a)} returns
++ \spad{sum(n = 0..,a(n)*(x-a)**n)}.
taylor: (FE,SY,EQ FE) -> Any
++ \spad{taylor(a(n),n,x = a)} returns \spad{sum(n = 0..,a(n)*(x-a)**n)}.
taylor: (I -> FE,EQ FE,SEG NNI) -> Any
++ \spad{taylor(n +-> a(n),x = a,n0..)} returns
++ \spad{sum(n=n0..,a(n)*(x-a)**n)};
++ \spad{taylor(n +-> a(n),x = a,n0..n1)} returns
++ \spad{sum(n = n0..,a(n)*(x-a)**n)}.
taylor: (FE,SY,EQ FE,SEG NNI) -> Any
++ \spad{taylor(a(n),n,x = a,n0..)} returns
++ \spad{sum(n = n0..,a(n)*(x-a)**n)};
++ \spad{taylor(a(n),n,x = a,n0..n1)} returns
++ \spad{sum(n = n0..,a(n)*(x-a)**n)}.
laurent: (I -> FE,EQ FE,SEG I) -> Any
++ \spad{laurent(n +-> a(n),x = a,n0..)} returns
++ \spad{sum(n = n0..,a(n) * (x - a)**n)};
++ \spad{laurent(n +-> a(n),x = a,n0..n1)} returns
++ \spad{sum(n = n0..n1,a(n) * (x - a)**n)}.
laurent: (FE,SY,EQ FE,SEG I) -> Any
++ \spad{laurent(a(n),n,x=a,n0..)} returns
++ \spad{sum(n = n0..,a(n) * (x - a)**n)};
++ \spad{laurent(a(n),n,x=a,n0..n1)} returns
++ \spad{sum(n = n0..n1,a(n) * (x - a)**n)}.
puiseux: (RN -> FE,EQ FE,SEG RN,RN) -> Any
++ \spad{puiseux(n +-> a(n),x = a,r0..,r)} returns
++ \spad{sum(n = r0,r0 + r,r0 + 2*r..., a(n) * (x - a)**n)};
++ \spad{puiseux(n +-> a(n),x = a,r0..r1,r)} returns
++ \spad{sum(n = r0 + k*r while n <= r1, a(n) * (x - a)**n)}.
puiseux: (FE,SY,EQ FE,SEG RN,RN) -> Any
++ \spad{puiseux(a(n),n,x = a,r0..,r)} returns
++ \spad{sum(n = r0,r0 + r,r0 + 2*r..., a(n) * (x - a)**n)};
++ \spad{puiseux(a(n),n,x = a,r0..r1,r)} returns
++ \spad{sum(n = r0 + k*r while n <= r1, a(n) * (x - a)**n)}.
series: (I -> FE,EQ FE) -> Any
++ \spad{series(n +-> a(n),x = a)} returns
++ \spad{sum(n = 0..,a(n)*(x-a)**n)}.
series: (FE,SY,EQ FE) -> Any
++ \spad{series(a(n),n,x = a)} returns
++ \spad{sum(n = 0..,a(n)*(x-a)**n)}.
series: (I -> FE,EQ FE,SEG I) -> Any
++ \spad{series(n +-> a(n),x = a,n0..)} returns
++ \spad{sum(n = n0..,a(n) * (x - a)**n)};
++ \spad{series(n +-> a(n),x = a,n0..n1)} returns
++ \spad{sum(n = n0..n1,a(n) * (x - a)**n)}.
series: (FE,SY,EQ FE,SEG I) -> Any
++ \spad{series(a(n),n,x=a,n0..)} returns
++ \spad{sum(n = n0..,a(n) * (x - a)**n)};
++ \spad{series(a(n),n,x=a,n0..n1)} returns
++ \spad{sum(n = n0..n1,a(n) * (x - a)**n)}.
series: (RN -> FE,EQ FE,SEG RN,RN) -> Any
++ \spad{series(n +-> a(n),x = a,r0..,r)} returns
++ \spad{sum(n = r0,r0 + r,r0 + 2*r..., a(n) * (x - a)**n)};
++ \spad{series(n +-> a(n),x = a,r0..r1,r)} returns
++ \spad{sum(n = r0 + k*r while n <= r1, a(n) * (x - a)**n)}.
series: (FE,SY,EQ FE,SEG RN,RN) -> Any
++ \spad{series(a(n),n,x = a,r0..,r)} returns
++ \spad{sum(n = r0,r0 + r,r0 + 2*r..., a(n) * (x - a)**n)};
++ \spad{series(a(n),n,x = a,r0..r1,r)} returns
++ \spad{sum(n = r0 + k*r while n <= r1, a(n) * (x - a)**n)}.
Implementation ==> add
genStream: (I -> FE,I) -> ST FE
genStream(f,n) == delay concat(f(n),genStream(f,n + 1))
genFiniteStream: (I -> FE,I,I) -> ST FE
genFiniteStream(f,n,m) == delay
n > m => empty()
concat(f(n),genFiniteStream(f,n + 1,m))
taylor(f,eq) ==
(xx := retractIfCan(lhs eq)@Union(SY,"failed")) case "failed" =>
error "taylor: left hand side must be a variable"
x := xx :: SY; a := rhs eq
coerce(series(genStream(f,0))$UTS(FE,x,a))$ANY1(UTS(FE,x,a))
taylor(an:FE,n:SY,eq:EQ FE) ==
taylor(eval(an,(n :: FE) = (#1 :: FE)),eq)
taylor(f:I -> FE,eq:EQ FE,seg:SEG NNI) ==
(xx := retractIfCan(lhs eq)@Union(SY,"failed")) case "failed" =>
error "taylor: left hand side must be a variable"
x := xx :: SY; a := rhs eq
hasHi seg =>
n0 := lo seg; n1 := hi seg
if n1 < n0 then (n0,n1) := (n1,n0)
uts := series(genFiniteStream(f,n0,n1))$UTS(FE,x,a)
uts := uts * monomial(1,n0)$UTS(FE,x,a)
coerce(uts)$ANY1(UTS(FE,x,a))
n0 := lo seg
uts := series(genStream(f,n0))$UTS(FE,x,a)
uts := uts * monomial(1,n0)$UTS(FE,x,a)
coerce(uts)$ANY1(UTS(FE,x,a))
taylor(an,n,eq,seg) ==
taylor(eval(an,(n :: FE) = (#1 :: FE)),eq,seg)
laurent(f,eq,seg) ==
(xx := retractIfCan(lhs eq)@Union(SY,"failed")) case "failed" =>
error "taylor: left hand side must be a variable"
x := xx :: SY; a := rhs eq
hasHi seg =>
n0 := lo seg; n1 := hi seg
if n1 < n0 then (n0,n1) := (n1,n0)
uts := series(genFiniteStream(f,n0,n1))$UTS(FE,x,a)
coerce(laurent(n0,uts)$ULS(FE,x,a))$ANY1(ULS(FE,x,a))
n0 := lo seg
uts := series(genStream(f,n0))$UTS(FE,x,a)
coerce(laurent(n0,uts)$ULS(FE,x,a))$ANY1(ULS(FE,x,a))
laurent(an,n,eq,seg) ==
laurent(eval(an,(n :: FE) = (#1 :: FE)),eq,seg)
modifyFcn:(RN -> FE,I,I,I,I) -> FE
modifyFcn(f,n0,nn,q,m) == (zero?((m - n0) rem nn) => f(m/q); 0)
puiseux(f,eq,seg,r) ==
(xx := retractIfCan(lhs eq)@Union(SY,"failed")) case "failed" =>
error "puiseux: left hand side must be a variable"
x := xx :: SY; a := rhs eq
not positive? r => error "puiseux: last argument must be positive"
hasHi seg =>
r0 := lo seg; r1 := hi seg
if r1 < r0 then (r0,r1) := (r1,r0)
p0 := numer r0; q0 := denom r0
p1 := numer r1; q1 := denom r1
p2 := numer r; q2 := denom r
q := lcm(lcm(q0,q1),q2)
n0 := p0 * (q quo q0); n1 := p1 * (q quo q1)
nn := p2 * (q quo q2)
ulsUnion := laurent(modifyFcn(f,n0,nn,q,#1),eq,segment(n0,n1))
uls := retract(ulsUnion)$ANY1(ULS(FE,x,a))
coerce(puiseux(1/q,uls)$UPXS(FE,x,a))$ANY1(UPXS(FE,x,a))
p0 := numer(r0 := lo seg); q0 := denom r0
p2 := numer r; q2 := denom r
q := lcm(q0,q2)
n0 := p0 * (q quo q0); nn := p2 * (q quo q2)
ulsUnion := laurent(modifyFcn(f,n0,nn,q,#1),eq,segment n0)
uls := retract(ulsUnion)$ANY1(ULS(FE,x,a))
coerce(puiseux(1/q,uls)$UPXS(FE,x,a))$ANY1(UPXS(FE,x,a))
puiseux(an,n,eq,r0,m) ==
puiseux(eval(an,(n :: FE) = (#1 :: FE)),eq,r0,m)
series(f:I -> FE,eq:EQ FE) == puiseux(f(numer #1),eq,segment 0,1)
series(an:FE,n:SY,eq:EQ FE) == puiseux(an,n,eq,segment 0,1)
series(f:I -> FE,eq:EQ FE,seg:SEG I) ==
ratSeg : SEG RN := map(#1::RN,seg)$UniversalSegmentFunctions2(I,RN)
puiseux(f(numer #1),eq,ratSeg,1)
series(an:FE,n:SY,eq:EQ FE,seg:SEG I) ==
ratSeg : SEG RN := map(#1::RN,seg)$UniversalSegmentFunctions2(I,RN)
puiseux(an,n,eq,ratSeg,1)
series(f:RN -> FE,eq:EQ FE,seg:SEG RN,r:RN) == puiseux(f,eq,seg,r)
series(an:FE,n:SY,eq:EQ FE,seg:SEG RN,r:RN) == puiseux(an,n,eq,seg,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>>
<<package GENUPS GenerateUnivariatePowerSeries>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
