A min-max problem on roots of unity

 Source file is available as : Postscript Document

Author(s) : Jörg Liesen , Petr Tichý

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 28-2003

MSC 2000

41A10 Approximation by polynomials
41A44 Best constants
41A50 Best approximation, Chebyshev systems
65F10 Iterative methods for linear systems

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.

Keywords : min-max problem, polynomial approximation on discrete set, best approximation, best constants, GMRES, evaluation of convergence