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The worst-case GMRES for normal matrices

Liesen, Jörg; Tichý, Petr

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

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.