On the complexity of real root isolation using continued fractions
Identifieur interne : 004452 ( Main/Exploration ); précédent : 004451; suivant : 004453On the complexity of real root isolation using continued fractions
Auteurs : Elias P. Tsigaridas [France] ; Ioannis Z. Emiris [Grèce]Source :
- Theoretical computer science [ 0304-3975 ] ; 2008.
Descripteurs français
- Pascal (Inist)
- Complexité algorithme, Fraction continue, Algorithmique, Implémentation, Nombre entier, Polynôme, Développement fraction continue, Expansion, Nombre réel, Nombre algébrique, Performance, Multiplicité, Algorithme, Complétude, Efficacité, Logiciel, Informatique théorique, 58A25, Pire cas, 13H15, 68Wxx, 68N01.
- Wicri :
- topic : Logiciel.
English descriptors
- KwdEn :
Abstract
We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method's good performance in practice. We derive an expected complexity bound of ========Otilde;B (d6 + d4τ2), where d is the polynomial degree and τ bounds the coefficient bit size, using a standard bound on the expected bit size of the integers in the continued fraction expansion, thus matching the current worst-case complexity bound for real root isolation by exact methods (Sturm, Descartes and Bernstein subdivision). Moreover, using a homothetic transformation we improve the expected complexity bound to OB (d3τ). We compute the multiplicities within the same complexity and extend the algorithm to non-square-free polynomials. Finally, we present an open-source C++ implementation in the algebraic library SYNAPS, and illustrate its completeness and efficiency as compared to some other available software. For this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000.
Affiliations:
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Emiris</name><affiliation wicri:level="1"><inist:fA14 i1="02"><s1>Department of Informatics and Telecommunications, National Kapodistrian University of Athens</s1><s2>Hellas</s2><s3>GRC</s3><sZ>2 aut.</sZ></inist:fA14><country>Grèce</country><wicri:noRegion>Hellas</wicri:noRegion></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">INIST</idno><idno type="inist">08-0241602</idno><date when="2008">2008</date><idno type="stanalyst">PASCAL 08-0241602 INIST</idno><idno type="RBID">Pascal:08-0241602</idno><idno type="wicri:Area/PascalFrancis/Corpus">000314</idno><idno type="wicri:Area/PascalFrancis/Curation">000714</idno><idno type="wicri:Area/PascalFrancis/Checkpoint">000271</idno><idno type="wicri:explorRef" wicri:stream="PascalFrancis" wicri:step="Checkpoint">000271</idno><idno type="wicri:doubleKey">0304-3975:2008:Tsigaridas E:on:the:complexity</idno><idno type="wicri:Area/Main/Merge">004566</idno><idno type="wicri:Area/Main/Curation">004452</idno><idno type="wicri:Area/Main/Exploration">004452</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a">On the complexity of real root isolation using continued fractions</title><author><name sortKey="Tsigaridas, Elias P" sort="Tsigaridas, Elias P" uniqKey="Tsigaridas E" first="Elias P." last="Tsigaridas">Elias P. Tsigaridas</name><affiliation wicri:level="3"><inist:fA14 i1="01"><s1>INRIA -LORIA Lorraine, project VEGAS, 615, rue du Jardin Botanique, B.P. 101</s1><s2>54602 Villers-des-Nancy</s2><s3>FRA</s3><sZ>1 aut.</sZ></inist:fA14><country>France</country><placeName><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region><settlement type="city">Villers-des-Nancy</settlement></placeName></affiliation></author><author><name sortKey="Emiris, Ioannis Z" sort="Emiris, Ioannis Z" uniqKey="Emiris I" first="Ioannis Z." last="Emiris">Ioannis Z. Emiris</name><affiliation wicri:level="1"><inist:fA14 i1="02"><s1>Department of Informatics and Telecommunications, National Kapodistrian University of Athens</s1><s2>Hellas</s2><s3>GRC</s3><sZ>2 aut.</sZ></inist:fA14><country>Grèce</country><wicri:noRegion>Hellas</wicri:noRegion></affiliation></author></analytic><series><title level="j" type="main">Theoretical computer science</title><title level="j" type="abbreviated">Theor. comput. sci.</title><idno type="ISSN">0304-3975</idno><imprint><date when="2008">2008</date></imprint></series></biblStruct></sourceDesc><seriesStmt><title level="j" type="main">Theoretical computer science</title><title level="j" type="abbreviated">Theor. comput. sci.</title><idno type="ISSN">0304-3975</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Algebraic number</term><term>Algorithm</term><term>Algorithm complexity</term><term>Algorithmics</term><term>Completeness</term><term>Computer theory</term><term>Continued fraction expansion</term><term>Continued fractions</term><term>Efficiency</term><term>Expansion</term><term>Implementation</term><term>Integer</term><term>Multiplicity</term><term>Performance</term><term>Polynomial</term><term>Real number</term><term>Software</term></keywords><keywords scheme="Pascal" xml:lang="fr"><term>Complexité algorithme</term><term>Fraction continue</term><term>Algorithmique</term><term>Implémentation</term><term>Nombre entier</term><term>Polynôme</term><term>Développement fraction continue</term><term>Expansion</term><term>Nombre réel</term><term>Nombre algébrique</term><term>Performance</term><term>Multiplicité</term><term>Algorithme</term><term>Complétude</term><term>Efficacité</term><term>Logiciel</term><term>Informatique théorique</term><term>58A25</term><term>Pire cas</term><term>13H15</term><term>68Wxx</term><term>68N01</term></keywords><keywords scheme="Wicri" type="topic" xml:lang="fr"><term>Logiciel</term></keywords></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method's good performance in practice. We derive an expected complexity bound of ========Otilde;<sub>B</sub> (d<sup>6</sup> + d<sup>4</sup>τ<sup>2</sup>), where d is the polynomial degree and τ bounds the coefficient bit size, using a standard bound on the expected bit size of the integers in the continued fraction expansion, thus matching the current worst-case complexity bound for real root isolation by exact methods (Sturm, Descartes and Bernstein subdivision). Moreover, using a homothetic transformation we improve the expected complexity bound to OB (d<sup>3</sup>τ). We compute the multiplicities within the same complexity and extend the algorithm to non-square-free polynomials. Finally, we present an open-source C++ implementation in the algebraic library SYNAPS, and illustrate its completeness and efficiency as compared to some other available software. For this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000.</div></front></TEI><affiliations><list><country><li>France</li><li>Grèce</li></country><region><li>Grand Est</li><li>Lorraine (région)</li></region><settlement><li>Villers-des-Nancy</li></settlement></list><tree><country name="France"><region name="Grand Est"><name sortKey="Tsigaridas, Elias P" sort="Tsigaridas, Elias P" uniqKey="Tsigaridas E" first="Elias P." last="Tsigaridas">Elias P. Tsigaridas</name></region></country><country name="Grèce"><noRegion><name sortKey="Emiris, Ioannis Z" sort="Emiris, Ioannis Z" uniqKey="Emiris I" first="Ioannis Z." last="Emiris">Ioannis Z. Emiris</name></noRegion></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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