The set of N equations, X ̊ (t) = AX(t), X(0) = X 0 , where X( t) is an N-vector and A is a constant N × N matrix, may be solved by a recursive method, X( t + T) = exp( AT) X( t). We discuss rational function approximations for the matrix exponential, exp( AT). Calahan has suggested using the Padb ( N, N) approximant, but when the eigenvalues of A are widely spaced in magnitude, the Padé ( N, N) approximant is inaccurate unless T is very small. A new family of rational approximations to the matrix exponential is presented; the best member of the family to use depends upon the distribution of eigenvalues of A. Contour plots of the absolute square error and of the phase error are given for several approximations. A numerical solution of the heat equation, discretized in space, is given as a numerical example.