Stability of the Generalized Polar Decomposition Method for the Approximation of the Matrix Exponential

Abstract

Generalized polar decomposition method (or briefly GPD method) has been introduced by Munthe-Kaas and Zanna to approximate the matrix exponential. In this paper, we investigate the numerical stability of that method with respect to roundoff propagation. The numerical GPD method includes two parts: splitting of a matrix $Z\in \mathfrak{g}$, a Lie algebra of matrices and computing $\exp(Z)\mathbf{v}$ for a vector $\mathbf{v}$. We show that the former is stable provided that $\|Z\|$ is not so large, while the latter is not stable in general except with some restrictions on the entries of the matrix Z and the vector $\mathbf{v}$.