The worst-case GMRES for normal matrices

dc.contributor.authorLiesen, Jörg
dc.contributor.authorTichý, Petr
dc.date.accessioned2021-12-17T10:05:31Z
dc.date.available2021-12-17T10:05:31Z
dc.date.issued2003-09-15
dc.description.abstractWe study the convergence of GMRES for linear algebraic systems with normal matrices. In particular, we explore the standard bound based on a min-max approximation problem on the discrete set of the matrix eigenvalues. We completely characterize the worst-case GMRES-related quantities in the next-to-last iteration step and evaluate the standard bound in terms of explicit polynomials involving the matrix eigenvalues. For a general iteration step, we develop a computable lower and upper bound on the standard bound. Our bounds allow to study the worst-case GMRES residual norm in dependence of the eigenvalue distribution. For hermitian matrices the lower bound is equal to the worst-case residual norm. In addition, numerical experiments show that the lower bound is generally very tight, and support our conjecture that it is to within a constant factor of the actual worst-case residual norm. Since the worst-case residual norm in each step is to within a factor of the square root of the matrix size to what is considered an ``average'' residual norm, our results are of relevance beyond the worst case.en
dc.identifier.issn2197-8085
dc.identifier.urihttps://depositonce.tu-berlin.de/handle/11303/15491
dc.identifier.urihttp://dx.doi.org/10.14279/depositonce-14264
dc.language.isoenen
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subject.ddc510 Mathematiken
dc.subject.otherGMRESen
dc.subject.otherevaluation of convergenceen
dc.subject.otherideal GMRESen
dc.subject.othernormal matricesen
dc.subject.othermin-max problemen
dc.titleThe worst-case GMRES for normal matricesen
dc.typeResearch Paperen
dc.type.versionsubmittedVersionen
tub.accessrights.dnbfreeen
tub.affiliationFak. 2 Mathematik und Naturwissenschaften::Inst. Mathematikde
tub.affiliation.facultyFak. 2 Mathematik und Naturwissenschaftende
tub.affiliation.instituteInst. Mathematikde
tub.publisher.universityorinstitutionTechnische Universität Berlinen
tub.series.issuenumber2003, 27en
tub.series.namePreprint-Reihe des Instituts für Mathematik, Technische Universität Berlinen
tub.subject.msc200065F10 Iterative methods for linear systemsen
tub.subject.msc200065F15 Eigenvalues, eigenvectorsen
tub.subject.msc200065F20 Overdetermined systems, pseudoinversesen
tub.subject.msc200015A06 Linear equationsen
tub.subject.msc200015A09 Matrix inversion, generalized inversesen
tub.subject.msc200015A18 Eigenvalues, singular values, and eigenvectorsen

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