Efficient Isolation of a Polynomial Real Roots
Identifieur interne : 001E74 ( Hal/Corpus ); précédent : 001E73; suivant : 001E75Efficient Isolation of a Polynomial Real Roots
Auteurs : Fabrice Rouillier ; Paul ZimmermannSource :
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Abstract
This paper gives new results for the isolation of real roots of a univariate polynomial using Descartes' rule of signs, following work of Vincent, Uspensky, Collins and Akritas, Johnson, Krandick. The first contribution is a generic algorithm which enables one to describe all the existing strategies in a unified framework. Using that framework, a new algorithm is presented, which is optimal in terms of memory usage, while doing no more computations than other algorithms based on Descartes' rule of signs. We show that these critical optimizations have important consequences by proposing a full efficient solution for isolating the real roots of zero-dimensional polynomial systems.
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<front><div type="abstract" xml:lang="en">This paper gives new results for the isolation of real roots of a univariate polynomial using Descartes' rule of signs, following work of Vincent, Uspensky, Collins and Akritas, Johnson, Krandick. The first contribution is a generic algorithm which enables one to describe all the existing strategies in a unified framework. Using that framework, a new algorithm is presented, which is optimal in terms of memory usage, while doing no more computations than other algorithms based on Descartes' rule of signs. We show that these critical optimizations have important consequences by proposing a full efficient solution for isolating the real roots of zero-dimensional polynomial systems.</div>
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<idno type="stamp" n="INRIA-LORRAINE">INRIA Nancy - Grand Est</idno>
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<notesStmt><note type="audience" n="0">Not set</note>
<note type="report" n="6">Research Report</note>
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<sourceDesc><biblStruct><analytic><title xml:lang="en">Efficient Isolation of a Polynomial Real Roots</title>
<author role="aut"><persName><forename type="first">Fabrice</forename>
<surname>Rouillier</surname>
</persName>
<email>Fabrice.Rouillier@inria.fr</email>
<idno type="idHal">fabrice-rouillier</idno>
<idno type="halAuthorId">74826</idno>
<idno type="arXiv">http://arxiv.org/a/fabricerouillier</idno>
<affiliation ref="#struct-2364"></affiliation>
</author>
<author role="aut"><persName><forename type="first">Paul</forename>
<surname>Zimmermann</surname>
</persName>
<idno type="idHal">paul-zimmermann</idno>
<idno type="halAuthorId">60829</idno>
<affiliation ref="#struct-2364"></affiliation>
</author>
</analytic>
<monogr><idno type="reportNumber">RR-4113</idno>
<imprint><date type="datePub">2001</date>
</imprint>
<authority type="institution">INRIA</authority>
</monogr>
</biblStruct>
</sourceDesc>
<profileDesc><langUsage><language ident="en">English</language>
</langUsage>
<textClass><keywords scheme="author"><term xml:lang="en">USPENSKY'S ALGORITHM</term>
<term xml:lang="en">POLYNOMIAL SYSTEM</term>
<term xml:lang="en">UNIVARIATE POLYNOMIAL</term>
<term xml:lang="en">REAL ROOT ISOLATION</term>
<term xml:lang="en">DESCARTES' RULE OF SIGNS</term>
</keywords>
<classCode scheme="halDomain" n="info.info-oh">Computer Science [cs]/Other [cs.OH]</classCode>
<classCode scheme="halTypology" n="REPORT">Reports</classCode>
</textClass>
<abstract xml:lang="en">This paper gives new results for the isolation of real roots of a univariate polynomial using Descartes' rule of signs, following work of Vincent, Uspensky, Collins and Akritas, Johnson, Krandick. The first contribution is a generic algorithm which enables one to describe all the existing strategies in a unified framework. Using that framework, a new algorithm is presented, which is optimal in terms of memory usage, while doing no more computations than other algorithms based on Descartes' rule of signs. We show that these critical optimizations have important consequences by proposing a full efficient solution for isolating the real roots of zero-dimensional polynomial systems.</abstract>
</profileDesc>
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