A method based on analytic continuation, which is well suited for fast digital computer application, has been applied to the point reactor kinetics equations. The most important characteristic of the method is that it yields an analytic criterion for the magnitude of the time step. This criterion is such that the time step automatically expands or contracts, depending on the behavior of the function within each interval. The use of this criterion to determine the time step guarantees that the fractional error in the results increases, at most, linearly with the number of time steps. Furthermore, the magnitude of the time step determined from this criterion can be much larger than the prompt-neutron generation time. Approximate solutions by this method were compared with some analytic solutions to the reactor kinetics equations, and the error accumulation was found, in all cases, to be within the limits predicted by the theory. Comparisons were also made with experimental transients in the Godiva and SPERT I reactors. The approximate results were found to agree well with experiment in the range of reactivity inputs where the feedback model used is valid. In a comparison with another numerical method (RTS code), analytic continuation was found to be 25 times faster.