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 H_k(z) of the continued fraction expansion of \exp z are shown to satisfy |H_k(z)| ≤ 1 for \mathrm{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.

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