Abstract. The Lanczos method is an iterative procedure to compute an orthogonal basis for the Krylov subspace generated by a symmetric matrix A and a starting vector v. An interesting application of this method is the computation of the matrix exponential exp(−τ A)v. This vector plays an important role in the solution of parabolic equations where A results from some form of discretization of an elliptic operator. In the present paper we will argue that for these applications the convergence behavior of this method can be unsatisfactory. We will propose a modified method that resolves this by a simple preconditioned transformation at the cost of an innerouter iteration. A priori error bounds are presented that are independent of the norm of A. This shows that the worst case convergence speed is independent of the mesh width in the spatial discretization of the elliptic operator. We discuss, furthermore, a posteriori error estimation and the tuning of the coupling between the inner and outer iteration. We conclude with several numerical experiments with the proposed method.

There is a class of linear problems for which the computation of the matrix-vector product is very expensive since a time consuming method is necessary to approximate it with some prescribed relative precision. In this paper we investigate the impact of approximately computed matrix-vector products on the convergence and attainable accuracy of several Krylov subspace solvers. We will argue that the sensitivity towards perturbations is mainly determined by the underlying way the Krylov subspace is constructed and does not depend on the optimality properties of the particular method. The obtained insight is used to tune the precision of the matrix-vector product in every iteration step in such a way that an overall efficient process is obtained. Our analysis confirms the empirically found relaxation strategy of Bouras and Frayssé for the GMRES method proposed in [A Relaxation Strategy for Inexact Matrix-Vector Products for Krylov Methods,