The Worst-Case GMRES for Normal Matrices

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. This bound is sharp, i.e. it is attainable by the GMRES residual norm. The question is how to evaluate or estimate the standard bound, and if it is possible to characterize the GMRES-related quantities for which this bound is attained (worst-case GMRES). In this paper 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 us to study the worst-case GMRES residual norm as a function 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 factor of 4/π 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.