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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra cmplxrt.spad}
\author{Patrizia Gianni}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package CMPLXRT ComplexRootPackage}
<<package CMPLXRT ComplexRootPackage>>=
)abbrev package CMPLXRT ComplexRootPackage
++ Author: P. Gianni
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors: Complex, Float, Fraction, UnivariatePolynomial
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package provides functions complexZeros
++ for finding the complex zeros
++ of univariate polynomials with complex rational number coefficients.
++ The results are to any user specified precision and are returned
++ as either complex rational number or complex floating point numbers
++ depending on the type of the second argument which specifies the
++ precision.
-- Packages for the computation of complex roots of
-- univariate polynomials with rational or gaussian coefficients.
-- Simplified version, the old original based on Gebauer's solver is
-- in ocmplxrt spad
RN ==> Fraction Integer
I ==> Integer
NF ==> Float
ComplexRootPackage(UP,Par) : T == C where
UP : UnivariatePolynomialCategory Complex Integer
Par : Join(Field, OrderedRing) -- will be Float or RN
CP ==> Complex Par
PCI ==> Polynomial Complex Integer
T == with
complexZeros:(UP,Par) -> List CP
++ complexZeros(poly, eps) finds the complex zeros of the
++ univariate polynomial poly to precision eps with
++ solutions returned as complex floats or rationals
++ depending on the type of eps.
C == add
complexZeros(p:UP,eps:Par):List CP ==
x1:Symbol():=new()
x2:Symbol():=new()
vv:Symbol():=new()
lpf:=factors factor(p)$ComplexFactorization(I,UP)
ris:List CP:=empty()
for pf in lpf repeat
pp:=pf.factor pretend SparseUnivariatePolynomial Complex Integer
q:PCI :=multivariate(pp,vv)
q:=eval(q,vv,x1::PCI+complex(0,1)*(x2::PCI))
p1:=map(real,q)$PolynomialFunctions2(Complex I,I)
p2:=map(imag,q)$PolynomialFunctions2(Complex I,I)
lz:=innerSolve([p1,p2],[],[x1,x2],
eps)$InnerNumericFloatSolvePackage(I,Par,Par)
ris:=append([complex(first z,second z) for z in lz],ris)
ris
@
\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 CMPLXRT ComplexRootPackage>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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