Continuous random variate generation by fast numerical inversion

The inversion method for generating nonuniform random variates has some advantages compared to other generation methods, since it monotonically transforms uniform random numbers into non-uniform random variates. Hence, it is the method of choice in the simulation literature. However, except for some simple cases where the inverse of the cumulative distribution function is a simple function we need numerical methods. Often inversion by "brute force" is used, applying either very slow iterative methods or linear interpolation of the CDF and huge tables. But then the user has to accept unnecessarily large errors or excessive memory requirements, that slow down the algorithm. In this article, we demonstrate that with Hermite interpolation of the inverse CDF we can obtain very small error bounds close to machine precision. Using our adaptive interval splitting method, this accuracy is reached with moderately sized tables that allow for a fast and simple generation procedure.