Analysis of some Krylov subspace approximations to the matrix exponential operator

In this note a theoretical analysis of some Krylov subspace approximations to the matrix expoential operation exp(A)v is presented, and a priori and a posteriori error estimates are established. Several such approximations are considered. The main idea of these techniques is to approximately project the exponential operator onto a small Krylov subspace and to carry out the resulting small exponential matrix computation accurately. This general approach, which has been used with success in several applications, provides a systematic way of defining high-order explicit-type schemes for solving systems of ordinary differential equations or time-dependent partial differential equations.