  Exact methods for exponentiation of matrices of dimension $N$ can be
computationally expensive in terms of execution time ($N^{3}$) and memory
requirements ($N^{2}$) not to mention numerical precision issues. A type of
matrix often exponentiated in the sciences is the rate matrix. Here we explore
five methods to exponentiate rate matrices some of which apply even more
broadly to other matrix types. Three of the methods leverage a mathematical
analogy between computing matrix elements of a matrix exponential and computing
transition probabilities of a dynamical processes (technically a Markov jump
process, MJP, typically simulated using Gillespie). In doing so, we identify a
novel MJP-based method relying on restricting the number of "trajectory" jumps
based on the magnitude of the matrix elements with favorable computational
scaling. We then discuss this method's downstream implications on mixing
properties of Monte Carlo posterior samplers. We also benchmark two other
methods of matrix exponentiation valid for any matrix (beyond rate matrices
and, more generally, positive definite matrices) related to solving
differential equations: Runge-Kutta integrators and Krylov subspace methods.
Under conditions where both the largest matrix element and the number of
non-vanishing elements scale linearly with $N$ -- reasonable conditions for
rate matrices often exponentiated -- computational time scaling with the most
competitive methods (Krylov and one of the MJP-based methods) reduces to $N^2$
with total memory requirements of $N$.
