Repository: DepositOnce – institutional repository for research data and publications of TU Berlin https://depositonce.tu-berlin.de
@techreport { 11303_15491,
author = {Liesen, Jörg AND Tichý, Petr},
title = {The worst-case GMRES for normal matrices},
institution = {Technische Universität Berlin},
year = {2003},
doi = {10.14279/depositonce-14264},
url = {http://dx.doi.org/10.14279/depositonce-14264},
keywords = {GMRES, evaluation of convergence, ideal GMRES, normal matrices, min-max problem},
abstract = {We study the convergence of GMRES for linear algebraic systems with normal matrices. In particular, we explore the standard bound based on a min-max approximation problem on the discrete set of the matrix eigenvalues. We completely characterize the worst-case GMRES-related quantities in the next-to-last iteration step and evaluate the standard bound in terms of explicit polynomials involving the matrix eigenvalues. For a general iteration step, we develop a computable lower and upper bound on the standard bound. Our bounds allow to study the worst-case GMRES residual norm in dependence of the eigenvalue distribution. For hermitian matrices the lower bound is equal to the worst-case residual norm. In addition, numerical experiments show that the lower bound is generally very tight, and support our conjecture that it is to within a constant factor of the actual worst-case residual norm. Since the worst-case residual norm in each step is to within a factor of the square root of the matrix size to what is considered an ``average'' residual norm, our results are of relevance beyond the worst case.}
}