Computing analytic matrix functions for a class of exponential integrators

Different methods for numerically stable computation of functions which are closely related to the exponential function are discussed. Such functions appear in the format of the most often used exponential integrators. A generalization of the method base on the tridiagonal reduction is proposed. The new approach, allows to compute all the functions included in the format of a general exponential integrator, in the case when the arguments are symmetric (Hermitian) matrices. Complexity of the considered methods is derived. Some practical issues regarding variable step size implementations as well as the main advantages and disadvantages are of the different methods are also discussed.