Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later

TLDR

The exponential of a matrix can be computed by many methods—including approximation theory, differential equations, eigenvalues, and characteristic polynomials—but practical stability and efficiency considerations make some preferable while none are fully satisfactory. The paper updates the 1978 work by describing a few recent developments. The update includes a separate bibliography.

Abstract

In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed. In practice, consideration of computational stability and efficiency indicates that some of the methods are preferable to others but that none are completely satisfactory. Most of this paper was originally published in 1978. An update, with a separate bibliography, describes a few recent developments.