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Main Title: Worst-case and ideal GMRES for a Jordan block
Author(s): Tichý, Petr
Liesen, Jörg
Type: Research Paper
Abstract: We investigate the convergence of GMRES for an $n$ by $n$ Jordan block. For each $k$ that divides $n$ we derive the exact form of the $k$th ideal GMRES polynomial. We show that for a Jordan block, the worst-case and ideal GMRES approximations are the same in these steps. Moreover, we derive lower and upper bounds on the norm of the $k$th ideal GMRES matrix polynomial. For the Jordan block with eigenvalue one, we present an explicit formula for its singular value decomposition and use it to improve the bound on the ideal GMRES residual norm in the considered steps $k$.
Subject(s): Krylov subspace methods
convergence behavior
ideal GMRES
worst-case GMRES
Jordan block
Issue Date: 22-Aug-2005
Date Available: 17-Dec-2021
Language Code: en
DDC Class: 510 Mathematik
MSC 2000: 65F10 Iterative methods for linear systems
Series: Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin
Series Number: 2005, 19
ISSN: 2197-8085
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik
Appears in Collections:Technische Universität Berlin » Publications

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