An accurate restarting for shift-and-invert Krylov subspaces computing matrix exponential actions of nonsymmetric matrices

An accurate residual--time (AccuRT) restarting for computing matrix exponential actions of nonsymmetric matrices by the shift-and-invert (SAI) Krylov subspace method is proposed. The proposed restarting method is an extension of the recently proposed RT (residual--time) restarting and it is designed to avoid a possible accuracy loss in the conventional RT restarting. An expensive part of the SAI Krylov method is solution of linear systems with the shifted matrix. Since the AccuRT algorithm adjusts the shift value, we discuss how the proposed restarting can be implemented with just a single LU~factorization (or a preconditioner setup) of the shifted matrix. Numerical experiments demonstrate an improved accuracy and efficiency of the approach.