How Descriptive are GMRES Convergence Bounds?

GMRES is a popular Krylov subspace method for solving linear systems of equations involving a general non-Hermitian coefficient matrix. The conventional bounds on GMRES convergence involve polynomial approximation problems in the complex plane. Three popular approaches pose this approximation problem on the spectrum, the field of values, or pseudospectra of the coefficient matrix. We analyze and compare these bounds, illustrating with six examples the success and failure of each. When the matrix departs from normality due only to a low-dimensional invariant subspace, we discuss how these bounds can be adapted to exploit this structure. Since the Arnoldi process that underpins GMRES provides approximations to the pseudospectra, one can estimate the GMRES convergence bounds as an iteration proceeds.