DC FieldValueLanguage
dc.contributor.authorJeannerod, Claude Pierre-
dc.contributor.authorRump, Siegfried M.-
dc.date.accessioned2019-05-07T10:10:44Z-
dc.date.available2019-05-07T10:10:44Z-
dc.date.issued2018-
dc.identifier.citationMathematics of Computation 310 (87): 803-819 (2018)de_DE
dc.identifier.issn0025-5718de_DE
dc.identifier.urihttp://hdl.handle.net/11420/2654-
dc.description.abstractRounding error analyses of numerical algorithms are most often carried out via repeated applications of the so-called standard models of floating-point arithmetic. Given a round-to-nearest function fl and barring underflow and overflow, such models bound the relative errors E 1 (t) = |t - fl(t)|/|t| and E 2 (t) = |t - fl(t)|/|fl(t)| by the unit roundoff u. This paper investigates the possibility and the usefulness of refining these bounds, both in the case of an arbitrary real t and in the case where t is the exact result of an arithmetic operation on some floating-point numbers. We show that E 1 (t) and E 2 (t) are optimally bounded by u/(1 + u) and u, respectively, when t is real or, under mild assumptions on the base and the precision, when t = x ± y or t = xy with x, y two floating-point numbers. We prove that while this remains true for division in base β > 2, smaller, attainable bounds can be derived for both division in base β = 2 and square root. This set of optimal bounds is then applied to the rounding error analysis of various numerical algorithms: in all cases, we obtain significantly shorter proofs of the best-known error bounds for such algorithms, and/or improvements on these bounds themselves.en
dc.language.isoende_DE
dc.relation.ispartofMathematics of computationde_DE
dc.titleOn relative errors of floating-point operations: Optimal bounds and applicationsde_DE
dc.typeArticlede_DE
dc.type.diniarticle-
dcterms.DCMITypeText-
tuhh.abstract.englishRounding error analyses of numerical algorithms are most often carried out via repeated applications of the so-called standard models of floating-point arithmetic. Given a round-to-nearest function fl and barring underflow and overflow, such models bound the relative errors E 1 (t) = |t - fl(t)|/|t| and E 2 (t) = |t - fl(t)|/|fl(t)| by the unit roundoff u. This paper investigates the possibility and the usefulness of refining these bounds, both in the case of an arbitrary real t and in the case where t is the exact result of an arithmetic operation on some floating-point numbers. We show that E 1 (t) and E 2 (t) are optimally bounded by u/(1 + u) and u, respectively, when t is real or, under mild assumptions on the base and the precision, when t = x ± y or t = xy with x, y two floating-point numbers. We prove that while this remains true for division in base β > 2, smaller, attainable bounds can be derived for both division in base β = 2 and square root. This set of optimal bounds is then applied to the rounding error analysis of various numerical algorithms: in all cases, we obtain significantly shorter proofs of the best-known error bounds for such algorithms, and/or improvements on these bounds themselves.de_DE
tuhh.publisher.doi10.1090/mcom/3234-
tuhh.publication.instituteZuverlässiges Rechnen E-19de_DE
tuhh.type.opus(wissenschaftlicher) Artikel-
tuhh.institute.germanZuverlässiges Rechnen E-19de
tuhh.institute.englishZuverlässiges Rechnen E-19de_DE
tuhh.gvk.hasppnfalse-
dc.type.driverarticle-
dc.type.casraiJournal Article-
tuhh.container.issue310de_DE
tuhh.container.volume87de_DE
tuhh.container.startpage803de_DE
tuhh.container.endpage819de_DE
item.languageiso639-1en-
item.grantfulltextnone-
item.openairetypeArticle-
item.cerifentitytypePublications-
item.creatorOrcidJeannerod, Claude Pierre-
item.creatorOrcidRump, Siegfried M.-
item.fulltextNo Fulltext-
item.creatorGNDJeannerod, Claude Pierre-
item.creatorGNDRump, Siegfried M.-
item.openairecristypehttp://purl.org/coar/resource_type/c_6501-
crisitem.author.deptZuverlässiges Rechnen E-19-
crisitem.author.deptZuverlässiges Rechnen E-19-
crisitem.author.orcid0000-0002-4779-4800-
crisitem.author.parentorgStudiendekanat Elektrotechnik, Informatik und Mathematik-
crisitem.author.parentorgStudiendekanat Elektrotechnik, Informatik und Mathematik-
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