DC FieldValueLanguage
dc.contributor.authorBünger, Florian-
dc.date.accessioned2020-11-24T09:23:01Z-
dc.date.available2020-11-24T09:23:01Z-
dc.date.issued2010-06-09-
dc.identifier.citationAdvances in Computational Mathematics 2 (35): 193-215 (2011-11-01)de_DE
dc.identifier.issn1019-7168de_DE
dc.identifier.urihttp://hdl.handle.net/11420/7919-
dc.description.abstractWe consider the problem of minimizing or maximizing the quotient, where p = p0 + p1x + ... + pmxm, q = q0 + q1x + ... + qnxn ∈ K[x], K ∈ R, C, are non-zero real or complex polynomials of maximum degree m, n ∈ ℕ respectively and double pipepdouble pipe := (pipep0pipe2 + ... + pipepmpipe2)1/2 is simply the Euclidean norm of the polynomial coefficients. Clearly fm,n is bounded and assumes its maximum and minimum values min fm,n = fm,n(pmin, qmin) and max fm,n = f(pmax, qmax). We prove that minimizers pmin, qmin for K = ¢ and maximizers pmax, qmax for arbitrary K fulfill deg(pmin) = m = deg(pmax), deg(qmin) = n = deg(qmax) and all roots of pmin, qmin, pmax, qmax have modulus one and are simple. For K = ℝ we can only prove the existence of minimizers pmin, qmin of full degree m and n respectively having roots of modulus one. These results are obtained by transferring the optimization problem to that of determining extremal eigenvalues and corresponding eigenvectors of autocorrelation Toeplitz matrices. By the way we give lower bounds for min fm,n for real polynomials which are slightly better than the known ones and inclusions for max fm,n. © 2010 Springer Science+Business Media, LLC.en
dc.language.isoende_DE
dc.publisherBaltzer Science Publ.de_DE
dc.subjectEigenvalues and eigenvectors of autocorrelation Toeplitz matricesde_DE
dc.subjectInequalities of polynomial productsde_DE
dc.subject.ddc004: Informatikde_DE
dc.subject.ddc510: Mathematikde_DE
dc.titleMinimizing and maximizing the Euclidean norm of the product of two polynomialsde_DE
dc.typeArticlede_DE
dc.type.diniarticle-
dcterms.DCMITypeText-
tuhh.abstract.englishWe consider the problem of minimizing or maximizing the quotient, where p = p0 + p1x + ... + pmxm, q = q0 + q1x + ... + qnxn ∈ K[x], K ∈ R, C, are non-zero real or complex polynomials of maximum degree m, n ∈ ℕ respectively and double pipepdouble pipe := (pipep0pipe2 + ... + pipepmpipe2)1/2 is simply the Euclidean norm of the polynomial coefficients. Clearly fm,n is bounded and assumes its maximum and minimum values min fm,n = fm,n(pmin, qmin) and max fm,n = f(pmax, qmax). We prove that minimizers pmin, qmin for K = ¢ and maximizers pmax, qmax for arbitrary K fulfill deg(pmin) = m = deg(pmax), deg(qmin) = n = deg(qmax) and all roots of pmin, qmin, pmax, qmax have modulus one and are simple. For K = ℝ we can only prove the existence of minimizers pmin, qmin of full degree m and n respectively having roots of modulus one. These results are obtained by transferring the optimization problem to that of determining extremal eigenvalues and corresponding eigenvectors of autocorrelation Toeplitz matrices. By the way we give lower bounds for min fm,n for real polynomials which are slightly better than the known ones and inclusions for max fm,n. © 2010 Springer Science+Business Media, LLC.de_DE
tuhh.publisher.doi10.1007/s10444-010-9158-z-
tuhh.publication.instituteZuverlässiges Rechnen E-19de_DE
tuhh.type.opus(wissenschaftlicher) Artikel-
dc.type.driverarticle-
dc.type.casraiJournal Article-
tuhh.container.issue2de_DE
tuhh.container.volume35de_DE
tuhh.container.startpage193de_DE
tuhh.container.endpage215de_DE
dc.identifier.scopus2-s2.0-80051902298de_DE
local.status.inpressfalsede_DE
datacite.resourceTypeJournal Article-
datacite.resourceTypeGeneralText-
item.grantfulltextnone-
item.cerifentitytypePublications-
item.openairetypeArticle-
item.creatorOrcidBünger, Florian-
item.languageiso639-1en-
item.creatorGNDBünger, Florian-
item.fulltextNo Fulltext-
item.openairecristypehttp://purl.org/coar/resource_type/c_6501-
item.mappedtypeArticle-
crisitem.author.deptMathematik E-10-
crisitem.author.orcid0000-0001-7643-0443-
crisitem.author.parentorgStudiendekanat Elektrotechnik, Informatik und Mathematik (E)-
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