Efficient scaling and squaring method for the matrix exponential

This work presents a new algorithm to compute the matrix exponential within a given tolerance. Combined with the scaling and squaring procedure, the algorithm incorporates Taylor, partitioned and classical Padé methods shown to be superior in performance to the approximants used in state-of-the-art software. The algorithm computes matrix–matrix products and also matrix inverses, but it can be implemented to avoid the computation of inverses, making it convenient for some problems. If the matrix A belongs to a Lie algebra, then exp(A) belongs to its associated Lie group, being a property which is preserved by diagonal Padé approximants, and the algorithm has another option to use only these. Numerical experiments show the superior performance with respect to state-of-the-art implementations.