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diff --git a/src/algebra/cmplxrt.spad.pamphlet b/src/algebra/cmplxrt.spad.pamphlet new file mode 100644 index 00000000..5cd1072e --- /dev/null +++ b/src/algebra/cmplxrt.spad.pamphlet @@ -0,0 +1,117 @@ +\documentclass{article} +\usepackage{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} |