Relative Error Analysis of Matrix Exponential Approximations for Numerical Integration

In this paper, we study the relative error in the numerical solution of a linear ordinary differential equation y'(t) = Ay(t), t >= 0, where A is a normal matrix. The numerical solution is obtained by using at any step an approximation of the matrix exponential, e.g., a polynomial or a rational approximation. The error of the numerical solution with respect to the exact solution is due to this approximation as well as to a possible perturbation in the initial value. For an unperturbed initial value, we have found: (1) unlike the absolute error, the relative error always grows linearly in time; (2) in the long-time, the contributions to the relative error relevant to non-rightmost eigenvalues of A disappear.