On the stability of some algorithms for computing the action of the matrix exponential

This paper deals with the stability of some one-step methods for computing the action of a matrix exponential on a vector, i.e. eAbeAb, where A is an n-by-n matrix and b is a vector of dimension n. The methods are based on certain assumptions made for the case that A is a matrix having sufficiently small norm. If the norm of A is larger, a fixed step size is used to perform a scaling of A  . An analysis of the roundoff error of the computed eAbeAb shows that these methods are backward stable if A is Hermitian, and forward stable if, for example, A is skew-Hermitian or if A is real and essentially nonnegative and b   is real and nonnegative. In addition, an upper bound on the forward error of the computed eAbeAb is obtained for all A and b  . This bound, which is expressed in terms of the condition number, is evaluated for several examples of the data. The results of this paper partly apply to the algorithm proposed by Al-Mohy and Higham (2011) for computing eAbeAb.