Generalized polar decompositions for the approximation of the matrix exponential

In this paper we describe the use of the theory of generalized polar decompositions to approximate a matrix exponential. The algorithms presented in this paper have the property that, if $Z \in \g$, a Lie algebra of matrices, then the approximation for $\exp(Z)$ resides in $\G$, the matrix Lie group of $\g$. 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 $\O{\kappa n^{3}}$, the constant $\kappa$ is small, depending on the order and the Lie algebra $\g$.

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 4th order.