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 non-Hermitian matrix. The a priori error bounds relate the convergence to λ_min (A+A ∗/2), λ_max(A+A ∗/2) (the smallest and the largest eigenvalue of the Hermitian part of A), and |λ_max(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.