A simple direct numerical procedure is shown for the evaluation of the discrete state transition matrix of linear time-invariant systems with a prescribed accuracy. The procedure uses a power-series expansion and checks the accuracy of the elements of the truncated series in the course of computation by using a relative error criterion. The results demonstrate that this procedure, in general, requires a considerably reduced number of matrix terms in comparision with the conventional procedure based on the matrix norm criterion, even in the presence of severe numerical difficulties.