Approximation of Exponential Function of a Matrix by Continued Fraction Expansion

Abstract

A numerical method for high order approximation of $u(t)=\exp (tA{)}_{u_0}$, where $A$ is an $N\times N$ matrix and $u_0$ is an $N$ dimensional vector, based on the continued fraction expansion of $\exp z$ is given. The approximants $H_k(z)$ of the continued fraction expansion of $\exp z$ are shown to satisfy $|H_k(z)|\le 1$ for $\mathrm{Re}z\le 0$, which results in an unconditionally stable method when every eigenvalue of $A$ lies in the left half-plane or on the imaginary axis.