Generalized Polar Decompositions for the Approximation of the Matrix Exponential

In this paper we describe the use of the theory of generalized polar decompositions [H. Munthe-Kaas, G. R. W. Quispel, and A. Zanna, Found. Comput. Math., 1 (2001), pp. 297--324] to approximate a matrix exponential. The algorithms presented have the property that, if $Z \in {\frak{g}}$, a Lie algebra of matrices, then the approximation for exp(Z) resides in G, the matrix Lie group of ${\frak{g}}$. This property is very relevant when solving Lie-group ODEs and is not usually fulfilled by standard approximations to the matrix exponential. We propose algorithms based on a splitting of Z into matrices having a very simple structure, usually one row and one column (or a few rows and a few columns), whose exponential is computed very cheaply to machine accuracy. The proposed methods have a complexity of ${\cal O}(\kappa n^{3})$, with constant $\kappa$ small, depending on the order and the Lie algebra ${\frak{g}}$. % The algorithms are recommended in cases where it is of fundamental importance that the approximation for the exponential resides in G, and when the order of approximation needed is not too high. We present in detail algorithms up to fourth order.