The Faber–Manteuffel theorem for linear operators

dc.contributor.authorFaber, Vance
dc.contributor.authorLiesen, Jörg
dc.contributor.authorTichý, Petr
dc.date.accessioned2017-12-19T15:34:32Z
dc.date.available2017-12-19T15:34:32Z
dc.date.issued2008
dc.description.abstractA short recurrence for orthogonalizing Krylov subspace bases for a matrix A exists if and only if the adjoint of A is a low-degree polynomial in A (i.e., A is normal of low degree). In the area of iterative methods, this result is known as the Faber–Manteuffel theorem [V. Faber and T. Manteuffel, SIAM J. Numer. Anal., 21 (1984), pp. 352–362]. Motivated by the description by J. Liesen and Z. Strakoš, we formulate here this theorem in terms of linear operators on finite dimensional Hilbert spaces and give two new proofs of the necessity part. We have chosen the linear operator rather than the matrix formulation because we found that a matrix-free proof is less technical. Of course, the linear operator result contains the Faber–Manteuffel theorem for matrices.en
dc.identifier.eissn1095-7170
dc.identifier.issn0036-1429
dc.identifier.urihttps://depositonce.tu-berlin.de//handle/11303/7289
dc.identifier.urihttp://dx.doi.org/10.14279/depositonce-6562
dc.language.isoenen
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subject.ddc518 Numerische Analysisde
dc.subject.othercyclic subspacesen
dc.subject.otherKrylov subspacesen
dc.subject.otherorthogonal basesen
dc.subject.otherorthogonalizationen
dc.subject.othershort recurrencesen
dc.subject.othernormal matricesen
dc.titleThe Faber–Manteuffel theorem for linear operatorsen
dc.typeArticleen
dc.type.versionpublishedVersionen
dcterms.bibliographicCitation.doi10.1137/060678087en
dcterms.bibliographicCitation.issue3en
dcterms.bibliographicCitation.journaltitleSIAM Journal on Numerical Analysisen
dcterms.bibliographicCitation.originalpublishernameSociety for Industrial and Applied Mathematicsen
dcterms.bibliographicCitation.originalpublisherplacePhiladelphia, Paen
dcterms.bibliographicCitation.pageend1337en
dcterms.bibliographicCitation.pagestart1323en
dcterms.bibliographicCitation.volume46en
tub.accessrights.dnbdomainen
tub.affiliationFak. 2 Mathematik und Naturwissenschaften>Inst. Mathematik>FG Numerische Lineare Algebrade
tub.affiliation.facultyFak. 2 Mathematik und Naturwissenschaftende
tub.affiliation.groupFG Numerische Lineare Algebrade
tub.affiliation.instituteInst. Mathematikde
tub.publisher.universityorinstitutionTechnische Universität Berlinen
Files
Original bundle
Now showing 1 - 1 of 1
Loading…
Thumbnail Image
Name:
2008_Liesen_et-al.pdf
Size:
187.52 KB
Format:
Adobe Portable Document Format
Description:
Collections