Computation of the noncentral chi-square distribution

This article gives a formula that allows accurate values of the cumulative noncentral chi-square distribution to be computed. Although this distribution has been recognized for a long time, none of the standard references give formulae that are suitable for computing accurate values over an extensive range of the parameters; approximations in terms of the chi-square distribution are usually recommended. A program written by the author, based on the formula given here, has been successful for computations involving more than 10,000 degrees of freedom. Since many steps are required when the degrees of freedom are as large as this, the program is not “fast” but it is believed to be accurate.