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| Description
| - When solving a linear algebraic system Ax = b with GMRES, the relative residual norm at each step is bounded from above by the so-called ideal GMRES approximation. This worst-case bound is sharp (i.e. it is attainable by the relative GMRES residual norm) in case of a normal matrix A , but it need not characterize the worst-case GMRES behavior if A is nonnormal. Characterizing the tightness of this bound for nonnormal matrices A represents an important and largely open problem in the convergence analysis of Krylov subspace methods. In this paper we address this problem in case A is a single Jordan block. We study the relation between ideal and worst-case GMRES as well as the problem of estimating the ideal GMRES approximation. Furthermore, we prove new results about the radii of the polynomial numerical hulls of Jordan blocks. Using these, we discuss the closeness of the lower bound on the ideal GMRES approximation that is derived from the radius of the polynomial numerical hull.
- When solving a linear algebraic system Ax = b with GMRES, the relative residual norm at each step is bounded from above by the so-called ideal GMRES approximation. This worst-case bound is sharp (i.e. it is attainable by the relative GMRES residual norm) in case of a normal matrix A , but it need not characterize the worst-case GMRES behavior if A is nonnormal. Characterizing the tightness of this bound for nonnormal matrices A represents an important and largely open problem in the convergence analysis of Krylov subspace methods. In this paper we address this problem in case A is a single Jordan block. We study the relation between ideal and worst-case GMRES as well as the problem of estimating the ideal GMRES approximation. Furthermore, we prove new results about the radii of the polynomial numerical hulls of Jordan blocks. Using these, we discuss the closeness of the lower bound on the ideal GMRES approximation that is derived from the radius of the polynomial numerical hull. (en)
- Řešíme-li soustavu lineárních algebraických rovnic $Ax=b$ metodou GMRES, lze relativní normu residua omezit shora pomocí tzv. ideální GMRES. Pro normální matice charakterizuje ideální GMRES nejhorší možné chování metody GMRES (worst-case GMRES). Ideální a worst-case GMRES se však mohou lišit pro matice, které nejsou normální. Charakterizace vztahu mezi ideální a worst-case GMRES pro nenormální matice představuje důležitý otevřený problém v oblasti konvergenční analýzy krylovovských metod. V práci se zabýváme tímto problémem pro případ, kdy $A$ je Jordanův blok. Studujeme vztah mezi ideální a worst-case GMRES jakož i odhad hodnoty ideální GMRES v závislosti na iteračním kroku a na vlastním čísle Jordanova bloku. Prezentujeme nové výsledky týkající se určování velikostí poloměrů polynomiálních numerických obalů Jordanova bloku. Na základě těchto výsledků diskutujeme kvalitu známého odhadu hodnoty ideální GMRES, založeného na poloměru polynomiálního numerického obalu. (cs)
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| Title
| - On Worst-Case GMRES, ideal GMRES, and the Polynomial Numerical Hull of a Jordan block
- O worst-case GMRES, ideální GMRES a o polynomiálním numerickém obalu Jordanova bloku (cs)
- On Worst-Case GMRES, ideal GMRES, and the Polynomial Numerical Hull of a Jordan block (en)
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| skos:prefLabel
| - On Worst-Case GMRES, ideal GMRES, and the Polynomial Numerical Hull of a Jordan block
- O worst-case GMRES, ideální GMRES a o polynomiálním numerickém obalu Jordanova bloku (cs)
- On Worst-Case GMRES, ideal GMRES, and the Polynomial Numerical Hull of a Jordan block (en)
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| skos:notation
| - RIV/67985807:_____/07:00092720!RIV08-AV0-67985807
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| http://linked.open.../vavai/riv/strany
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| http://linked.open...avai/riv/aktivita
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| http://linked.open...avai/riv/aktivity
| - P(1ET400300415), Z(AV0Z10300504)
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| http://linked.open...iv/cisloPeriodika
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| http://linked.open...vai/riv/dodaniDat
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| http://linked.open...aciTvurceVysledku
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| http://linked.open.../riv/druhVysledku
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| http://linked.open...iv/duvernostUdaju
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| http://linked.open...titaPredkladatele
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| http://linked.open...dnocenehoVysledku
| |
| http://linked.open...ai/riv/idVysledku
| - RIV/67985807:_____/07:00092720
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| http://linked.open...riv/jazykVysledku
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| http://linked.open.../riv/klicovaSlova
| - GMRES convergence; ideal GMRES; polynomial numerical hull; Jordan block (en)
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| http://linked.open.../riv/klicoveSlovo
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| http://linked.open...odStatuVydavatele
| - US - Spojené státy americké
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| http://linked.open...ontrolniKodProRIV
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| http://linked.open...i/riv/nazevZdroje
| - Electronic Transactions on Numerical Analysis
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| http://linked.open...in/vavai/riv/obor
| |
| http://linked.open...ichTvurcuVysledku
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| http://linked.open...cetTvurcuVysledku
| |
| http://linked.open...vavai/riv/projekt
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| http://linked.open...UplatneniVysledku
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| http://linked.open...v/svazekPeriodika
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| http://linked.open...iv/tvurceVysledku
| - Tichý, Petr
- Faber, V.
- Liesen, J.
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| http://linked.open...n/vavai/riv/zamer
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| issn
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| number of pages
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