# Approximation of Exponential Function of a Matrix by Continued Fraction Expansion

### Masatake Mori

Tokyo Denki University, Japan

## 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.

## Cite this article

Masatake Mori, Approximation of Exponential Function of a Matrix by Continued Fraction Expansion. Publ. Res. Inst. Math. Sci. 10 (1974), no. 1, pp. 257–269

DOI 10.2977/PRIMS/1195192181