A Robust Method for Calculating the Matrix Exponential Based on the Time Finite Element Method

Schemes that use ordinary-differential-equation solvers for computing the matrix exponential of a constant matrix have generally been considered inefficient compared to other strategies such as scaling and squaring, since they do not exploit the fact that the matrix in question is constant. In this work, we devise a strategy based on the time-finite element that exploits this fact, and thus results in an extremely efficient strategy that requires the solution of a linear system of equations just once at the beginning of the algorithm. Moreover, since the matrix to be inverted is well-conditioned even when the constant matrix is ill-conditioned, the strategy yields very reliable results as demonstrated by means of several challenging examples. The approach outlined here is easily generalized to the case where the matrix is time dependent and where there is a time-dependent forcing function.