About: Properties of Worst-Case GMRES

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
Description	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 transposed 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 %22cross equality%22. 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 greater than 3. 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 transposed 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 %22cross equality%22. 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 greater than 3. (en)
Title	Properties of Worst-Case GMRES Properties of Worst-Case GMRES (en)
skos:prefLabel	Properties of Worst-Case GMRES Properties of Worst-Case GMRES (en)
skos:notation	RIV/67985807:_____/13:00421797!RIV14-GA0-67985807
http://linked.open...avai/predkladatel	Ústav informatiky AV ČR, v.v.i.
http://linked.open...avai/riv/aktivita	I P
http://linked.open...avai/riv/aktivity	I, P(GA13-06684S)
http://linked.open...iv/cisloPeriodika	4
http://linked.open...vai/riv/dodaniDat	2014
http://linked.open...aciTvurceVysledku	Tichý, Petr
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Ústav informatiky AV ČR, v. v. i.
http://linked.open...dnocenehoVysledku	100405
http://linked.open...ai/riv/idVysledku	RIV/67985807:_____/13:00421797
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	GMRES method; worst-case convergence; ideal GMRES; matrix approximation problems; minmax (en)
http://linked.open.../riv/klicoveSlovo	ideal GMRES matrix approximation problems minmax worst-case convergence GMRES method
http://linked.open...odStatuVydavatele	US - Spojené státy americké
http://linked.open...ontrolniKodProRIV	[746868875921]
http://linked.open...i/riv/nazevZdroje	SIAM Journal on Matrix Analysis and Applications
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	1 (xsd:int)
http://linked.open...cetTvurcuVysledku	3 (xsd:int)
http://linked.open...vavai/riv/projekt	Iterative Methods in Computational Mathematics: Analysis, Preconditioning, and Applications
http://linked.open...UplatneniVysledku	2013
http://linked.open...v/svazekPeriodika	34
http://linked.open...iv/tvurceVysledku	Tichý, Petr Faber, V. Liesen, J.
http://linked.open...ain/vavai/riv/wos	000328902900004
issn	0895-4798
number of pages	20 (xsd:int)
http://bibframe.org/vocab/doi	10.1137/13091066X

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