How Descriptive Are Gmres Convergence Bounds?

Eigenvalues with the eigenvector condition number, the eld of values, and pseu-dospectra have all been suggested as the basis for convergence bounds for minimum residual Krylov subspace methods applied to non-normal coeecient matrices. This paper analyzes and compares these bounds, illustrating with six examples the success and failure of each one. Reened bounds based on eigenvalues and the eld of values are suggested to handle low-dimensional non-normality. It is observed that pseudospectral bounds can capture multiple convergence stages. Unfortunately, computation of pseudospectra can be rather expensive. This motivates an adaptive technique for estimating GMRES convergence based on approximate pseudospectra taken from the Arnoldi process that is the basis for GMRES.