Please use this identifier to cite or link to this item: http://dx.doi.org/10.14279/depositonce-14263
 Main Title: A min-max problem on roots of unity Author(s): Liesen, JörgTichý, Petr Type: Research Paper URI: https://depositonce.tu-berlin.de/handle/11303/15490http://dx.doi.org/10.14279/depositonce-14263 License: http://rightsstatements.org/vocab/InC/1.0/ 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 problempolynomial approximation on discrete setbest approximationbest constantsGMRESevaluation of convergence Issue Date: 1-Oct-2003 Date Available: 17-Dec-2021 Language Code: en DDC Class: 510 Mathematik MSC 2000: 41A10 Approximation by polynomials41A44 Best constants41A50 Best approximation, Chebyshev systems65F10 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