Efficient Computation of the Matrix Exponential by Generalized Polar Decompositions

Abstract

In this paper we explore the computation of the matrix exponential in a manner that is consistent with Lie-group structure. Our point of departure is the method of generalized polar decompositions, which we modify and combine with similarity transformations that bring the underlying matrix to a form more amenable to efficient computation. We develop techniques valid for a range of Lie groups: the orthogonal group, the symplectic group, Lorentz, isotropy, and scaling groups. However, the GPD approach is equally promising in a more general context. Even when Lie-group structure is not at issue, our algorithm is more efficient in many settings than classical methods for the computation of the matrix exponential.