On the convergence of restarted Krylov subspace methods

We are interested in the convergence analysis of restarted Krylov subspace iterative methods for the solution of large nonsymmetric linear systems. Several contributions in the literature have associated the convergence to some spectral properties of the coefficient matrix, while little work has been devoted to investigating how the singular values of A may influence the convergence. In this paper we present new relations that can be used to monitor the behavior of the restarted methods, especially GMRES, when the coefficient matrix has small (but not tiny) singular values and the right-hand side has a dominant component onto the corresponding left singular space. We also present some simple but insightful relations that highlight the dependence of the restarted schemes on new matrices; moreover, closed forms of the restarted solutions are used to relate the approximations of the unrestarted and restarted approaches.