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 × N matrix and _u_0 is an N dimensional vector, based on the continued fraction expansion of exp z is given. The approximants Hk(z) of the continued fraction expansion of exp z are shown to satisfy |Hk(z)| ≤ 1 for Re z ≤ 0, which results in an unconditionally stable method when every eigenvalue of A lies in the left half-plane or on the imaginary axis.