Choosing the best approach to matrix exponentiation

There is no ideal single approach to matrix exponentiation; an application may have some characteristic that enables or precludes a specific approach. Even methods that theoretically yield precise answers can produce extremely large errors when implemented in floating point arithmetic, and simply utilizing double or quadruple precision representations may not ensure accuracy. An empirical method is employed here to examine the efficacy of selected methods of matrix exponentiation for a particular application. The method centers around parametrizing a sample matrix in order to determine the effects of specific characteristics. The matrices to be exponentiated are upper triangular and stochastic. They may have nearly confluent eigenvalues, as well as widely divergent eigenvalues. Such problems are common in queueing applications using phase-type distributions.