Error Bounds for the Krylov Subspace Methods for Computations of Matrix Exponentials

In this paper, we present new a posteriori and a priori error bounds for the Krylov subspace methods for computing e−τAv for a given τ > 0 and v ∈ Cn, where A is a large sparse nonHermitian matrix. The a priori error bounds relate the convergence to λmin( A+A∗ 2 ), λmax( A+A∗ 2 ) (the smallest and the largest eigenvalue of the Hermitian part of A), and |λmax(A−A 2 )| (the largest eigenvalue in absolute value of the skew-Hermitian part of A), which define a rectangular region enclosing the field of values of A. In particular, our bounds explain an observed convergence behavior where the error may first stagnate for a certain number of iterations before it starts to converge. The special case that A is skew-Hermitian is also considered. Numerical examples are given to demonstrate the theoretical bounds.