  An algorithm for numerically computing the exponential of a matrix is
presented. We have derived a polynomial expansion of $e^x$ by computing it as
an initial value problem using a symbolic programming language. This algorithm
is shown to be comparable in operation count and convergence with the
state--of--the--art method which is based on a Pade approximation of the
exponential matrix function. The present polynomial form, however, is more
reliable because the evaluation requires only linear combinations of the input
matrix. We also show that the technique used to solve the differential
equation, when implemented symbolically, leads to a rational as well as a
polynomial form of the solution function. The rational form is the well-known
diagonal Pade approximation of $e^x$. The polynomial form, after some
rearranging to minimize operation count, will be used to evaluate the
exponential of a matrix so as to illustrate its advantages as compared with the
Pade form.
