Analysis of Some Krylov Subspace Approximations to the Matrix Exponential Operator

In this note we present a theoretical analysis of some Krylov subspace approximations to the matrix exponential operation exp(A)v and establish a priori and a posteriori error estimates. Several such approximations are considered. The main idea of these techniques is to approximately project the exponential operator onto a small Krylov subspace and 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. 1. Introduction. The problem of approximating the operation exp(A)v for a given vector v and a matrix A is of considerable importance in many applications. For example, this basic operation is at the core of many methods for solving systems of ordinary differential equations (ODE's) or time-dependent partial differential equations (PDE's). Recently, the use of Krylov subspace techniques in this context has been ac- tively investigated in the literature (2, 3, 4, 9, 10). Friesner et al. (2) and Gallopoulos and Saad (3) introduced a few different ways of applying this approximation to the solution of systems of ordinary differential equations. The paper (3) presents some analysis on the quality of the Krylov approximation and on the ODE integration schemes derived from it. In this note we make the following contributions. 1. We introduce and justify a few new approximation schemes (Section 2); 2. We analyze the Krylov subspace approach from an approximation theory view- point. In particular we establish that the Krylov methods are equivalent to interpolating the exponential function on the associated Ritz values (Section 3);