Comparison of methods for evaluating functions of a matrix exponential

We compare six different categories of numerical methods for the evaluation of functions of the matrix exponential. These functions are required for exponential integrators, and are not straightforward to evaluate because they are highly susceptible to rounding errors when the matrix has small eigenvalues. The comparison takes into account both accuracy and computational time. A scaling and squaring algorithm and a diagonalisation algorithm are both found to be efficient.