Properties of worst-case GMRES
| dc.contributor.author | Faber, Vance | |
| dc.contributor.author | Liesen, Jörg | |
| dc.contributor.author | Tichý, Petr | |
| dc.date.accessioned | 2017-12-14T15:10:30Z | |
| dc.date.available | 2017-12-14T15:10:30Z | |
| dc.date.issued | 2013 | |
| dc.description.abstract | In the convergence analysis of the GMRES method for a given matrix $A$, one quantity of interest is the largest possible residual norm that can be attained, at a given iteration step $k$, over all unit norm initial vectors. This quantity is called the worst-case GMRES residual norm for $A$ and $k$. We show that the worst-case behavior of GMRES for the matrices $A$ and $A^T$ is the same, and we analyze properties of initial vectors for which the worst-case residual norm is attained. In particular, we prove that such vectors satisfy a certain “cross equality.” We show that the worst-case GMRES polynomial may not be uniquely determined, and we consider the relation between the worst-case and the ideal GMRES approximations, giving new examples in which the inequality between the two quantities is strict at all iteration steps $k\geq 3$. Finally, we give a complete characterization of how the values of the approximation problems change in the context of worst-case and ideal GMRES for a real matrix, when one considers complex (rather than real) polynomials and initial vectors. | en |
| dc.identifier.eissn | 1095-7162 | |
| dc.identifier.issn | 0895-4798 | |
| dc.identifier.uri | https://depositonce.tu-berlin.de/handle/11303/7271 | |
| dc.identifier.uri | http://dx.doi.org/10.14279/depositonce-6544 | |
| dc.language.iso | en | en |
| dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | en |
| dc.subject.ddc | 518 Numerische Analysis | de |
| dc.subject.other | GMRES method | en |
| dc.subject.other | worst-case convergence | en |
| dc.subject.other | ideal GMRES | en |
| dc.subject.other | matrix approximation problems | en |
| dc.subject.other | minmax | en |
| dc.title | Properties of worst-case GMRES | en |
| dc.type | Article | en |
| dc.type.version | publishedVersion | en |
| dcterms.bibliographicCitation.doi | 10.1137/13091066X | en |
| dcterms.bibliographicCitation.issue | 4 | en |
| dcterms.bibliographicCitation.journaltitle | SIAM Journal on Matrix Analysis and Applications | en |
| dcterms.bibliographicCitation.originalpublishername | Society for Industrial and Applied Mathematics | en |
| dcterms.bibliographicCitation.originalpublisherplace | Philadelphia, Pa. | en |
| dcterms.bibliographicCitation.pageend | 1519 | en |
| dcterms.bibliographicCitation.pagestart | 1500 | en |
| dcterms.bibliographicCitation.volume | 34 | en |
| tub.accessrights.dnb | free | en |
| tub.affiliation | Fak. 2 Mathematik und Naturwissenschaften::Inst. Mathematik::FG Numerische Lineare Algebra | de |
| tub.affiliation.faculty | Fak. 2 Mathematik und Naturwissenschaften | de |
| tub.affiliation.group | FG Numerische Lineare Algebra | de |
| tub.affiliation.institute | Inst. Mathematik | de |
| tub.publisher.universityorinstitution | Technische Universität Berlin | en |