Arbitrary precision numerical integration with bounded error.
Identifieur interne : 000F71 ( Hal/Curation ); précédent : 000F70; suivant : 000F72Arbitrary precision numerical integration with bounded error.
Auteurs : Laurent Fousse [France]Source :
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Abstract
Numerical integration is commonly available in numerical computation systems. We study in this thesis the error on the result of a numerical quadrature using the Newton-Cotes and Gauss-Legendre methods for a monodimensional real function in the context of arbitrary precision. From the algorithmic point of view we give for each method a computation procedure with a guaranteed bound on the error. For the analysis of the Gauss-Legendre method we studied polynomial root refinement schemes (secante, Newton's iteration, dichotomy) and we gave heuristics to make sure the method converges in practice. The algorithms proposed in this thesis were written in a numerical quadrature library called "Correctly Rounded Quadrature" (CRQ) available at http://komite.net/laurent/soft/crq/. We also give a comparison of CRQ with other numerical integration softwares.
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We also give a comparison of CRQ with other numerical integration softwares.</div></front></TEI><hal api="V3"><titleStmt><title xml:lang="en">Arbitrary precision numerical integration with bounded error.</title><title xml:lang="fr">Intégration numérique avec erreur bornée en précision arbitraire</title><author role="aut"><persName><forename type="first">Laurent</forename><surname>Fousse</surname></persName><email>laurent@komite.net</email><idno type="halauthor">137424</idno><affiliation ref="#struct-160"></affiliation></author><editor role="depositor"><persName><forename>Laurent</forename><surname>Fousse</surname></persName><email>laurent@komite.net</email></editor></titleStmt><editionStmt><edition n="v1" type="current"><date type="whenSubmitted">2010-04-28 14:52:28</date><date type="whenModified">2016-05-18 08:54:27</date><date type="whenReleased">2010-04-28 15:02:21</date><date type="whenProduced">2006-12-04</date><date type="whenEndEmbargoed">2010-04-28</date><ref type="file" target="https://tel.archives-ouvertes.fr/tel-00477243/document"><date notBefore="2010-04-28"></date></ref><ref type="file" n="1" target="https://tel.archives-ouvertes.fr/tel-00477243/file/these-20070228.pdf"><date notBefore="2010-04-28"></date></ref></edition><respStmt><resp>contributor</resp><name key="104103"><persName><forename>Laurent</forename><surname>Fousse</surname></persName><email>laurent@komite.net</email></name></respStmt></editionStmt><publicationStmt><distributor>CCSD</distributor><idno type="halId">tel-00477243</idno><idno type="halUri">https://tel.archives-ouvertes.fr/tel-00477243</idno><idno type="halBibtex">fousse:tel-00477243</idno><idno type="halRefHtml">Modélisation et simulation. Université Henri Poincaré - Nancy I, 2006. Français</idno><idno type="halRef">Modélisation et simulation. Université Henri Poincaré - Nancy I, 2006. Français</idno></publicationStmt><seriesStmt><idno type="stamp" n="CNRS">CNRS - Centre national de la recherche scientifique</idno><idno type="stamp" n="INRIA">INRIA - Institut National de Recherche en Informatique et en Automatique</idno><idno type="stamp" n="INPL">Institut National Polytechnique de Lorraine</idno><idno type="stamp" n="LORIA2">Publications du LORIA</idno><idno type="stamp" n="LABO-LORIA-SET" p="LORIA">LABO-LORIA-SET</idno><idno type="stamp" n="LORIA">LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications</idno><idno type="stamp" n="TDS-MACS">Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes</idno><idno type="stamp" n="UNIV-LORRAINE">Université de Lorraine</idno></seriesStmt><notesStmt></notesStmt><sourceDesc><biblStruct><analytic><title xml:lang="en">Arbitrary precision numerical integration with bounded error.</title><title xml:lang="fr">Intégration numérique avec erreur bornée en précision arbitraire</title><author role="aut"><persName><forename type="first">Laurent</forename><surname>Fousse</surname></persName><email>laurent@komite.net</email><idno type="halAuthorId">137424</idno><affiliation ref="#struct-160"></affiliation></author></analytic><monogr><imprint><date type="dateDefended">2006-12-04</date></imprint><authority type="institution">Université Henri Poincaré - Nancy I</authority><authority type="school">École doctorale IAEM Lorraine</authority><authority type="supervisor">Paul Zimmermann(Paul.Zimmermann@loria.fr)</authority><authority type="jury">Jean-Claude Bajard (président)</authority><authority type="jury">Fabienne Jézéquel (rapporteur)</authority><authority type="jury">Bruno Salvy (rapporteur)</authority><authority type="jury">Didier Galmiche (examinateur)</authority><authority type="jury">Norbert Müller (examinateur)</authority><authority type="jury">Paul Zimmermann (directeur)</authority></monogr><ref type="seeAlso">http://komite.net/laurent/data/pro/publis/these-20070228.pdf</ref></biblStruct></sourceDesc><profileDesc><langUsage><language ident="fr">French</language></langUsage><textClass><keywords scheme="author"><term xml:lang="en">numerical integration</term><term xml:lang="en">arbitrary precision</term><term xml:lang="en">floating-point arithmetics</term><term xml:lang="en">bounded error</term><term xml:lang="fr">intégration numérique</term><term xml:lang="fr">précision arbitraire</term><term xml:lang="fr">arithmétique flottante</term><term xml:lang="fr">erreur bornée</term></keywords><classCode scheme="halDomain" n="info.info-mo">Computer Science [cs]/Modeling and Simulation</classCode><classCode scheme="halDomain" n="info.info-se">Computer Science [cs]/Software Engineering [cs.SE]</classCode><classCode scheme="halTypology" n="THESE">Theses</classCode></textClass><abstract xml:lang="en">Numerical integration is commonly available in numerical computation systems. We study in this thesis the error on the result of a numerical quadrature using the Newton-Cotes and Gauss-Legendre methods for a monodimensional real function in the context of arbitrary precision. From the algorithmic point of view we give for each method a computation procedure with a guaranteed bound on the error. For the analysis of the Gauss-Legendre method we studied polynomial root refinement schemes (secante, Newton's iteration, dichotomy) and we gave heuristics to make sure the method converges in practice. The algorithms proposed in this thesis were written in a numerical quadrature library called "Correctly Rounded Quadrature" (CRQ) available at http://komite.net/laurent/soft/crq/. We also give a comparison of CRQ with other numerical integration softwares.</abstract><abstract xml:lang="fr">L'intégration numérique est une opération fréquemment disponible et utilisée dans les systèmes de calcul numérique. Nous nous intéressons dans ce mémoire à la maîtrise des erreurs commises lors d'un calcul numérique d'intégrale réelle à une dimension dans le contexte de la précision arbitraire pour les deux méthodes d'intégration que sont Newton-Cotes et Gauss-Legendre. Du point de vue algorithmique nous proposons pour chacune des méthodes une procédure de calcul avec une borne effective sur l'erreur totale commise. Dans le cadre de l'étude de la méthode de Gauss-Legendre nous avons étudié les algorithmes connus de raffinement de racines réelles d'un polynôme (la méthode de la sécante, l'itération de Newton, la dichotomie), et nous en avons proposé des heuristiques explicites permettant de s'assurer en pratique de la convergence. Les algorithmes proposés ont été implémentés dans une bibliothèque d'intégration numérique baptisée «Correctly Rounded Quadrature» (CRQ) disponible à l'adresse http://komite.net/laurent/soft/crq/. Nous comparons CRQ avec d'autres logiciels d'intégration dans ce mémoire.</abstract></profileDesc></hal></record>Pour manipuler ce document sous Unix (Dilib)
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