Improvement of accuracy in numerical methods for inverting Laplace transforms based on the Post-Widder formula

The paper proposes a new computational version for numerical inversion of Laplace transforms at a point and on an interval, based on the Post-Widder formula. In this version the original sequence of Post-Widder approximants is calculated using operations on series and the approximate value of the sought function is constructed as a limit of this sequence using polynomial and rational extrapolation to the limit. Some procedures are proposed for improvement of accuracy in the problem of inversion at equidistant points of an interval without computing additional approximants. An extension of the method to the two-dimensional case is discussed. The one-dimensional version is illustrated with some probabilistic examples involving both explicit and implicit functions. The accuracy achieved in these examples with using the standard double precision arithmetic, evaluated in terms of relative errors, is about 10−10–10−14 even in case of out-of-scale parameters. Computational aspects are discussed in comparison with previously known realizations of the Post-Widder method.