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Main Title: A min-max problem on roots of unity
Author(s): Liesen, Jörg
Tichý, Petr
Type: Research Paper
Abstract: The worst-case residual norms of the GMRES method for linear algebraic systems can, in case of a normal matrix, be characterized by a min-max approximation problem on the matrix eigenvalues. In our paper "The worst-case GMRES for normal matrices" we derive a lower bound on this min-max value (worst-case residual norm) for each step of the GMRES iteration. We conjecture that the lower bound and the min-max value agree up to a factor of $4/\pi$, i.e. that the lower bound multiplied by $4/\pi$ represents an upper bound. In this paper we prove for several different iteration steps that our conjecture is true for a special set of eigenvalues, namely the roots of unity. This case is of interest, since numerical experiments indicate that the ratio of the min-max value and our lower bound is maximal when the eigenvalues are the roots of unity.
Subject(s): min-max problem
polynomial approximation on discrete set
best approximation
best constants
evaluation of convergence
Issue Date: 1-Oct-2003
Date Available: 17-Dec-2021
Language Code: en
DDC Class: 510 Mathematik
MSC 2000: 41A10 Approximation by polynomials
41A44 Best constants
41A50 Best approximation, Chebyshev systems
65F10 Iterative methods for linear systems
Series: Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin
Series Number: 2003, 28
ISSN: 2197-8085
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik
Appears in Collections:Technische Universität Berlin » Publications

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