Post-Widder Formula Method

The distinct property of the Post-Widder formula

is that it requires sampling of the Laplace space function only on the real line. The difficulty of computing high order derivatives can be mitigated by using finite differences. This gives rise to the well-known Gaver functionals


Unfortunately, this method suffers from very slow convergence
as
so that a series accelerator is demanded for any practical computation. Recent work VALKO2004 has established the utility of the Wynn rho algorithm for the acceleration. This approximates the function

by

where the elements of the matrix

are computed from the rule for an even



This acceleration technique is an extension of the more well-known -algorithm WIMP1981.

Another drawback to a Post-Widder based approach is its sensitivity to roundoff errors. This is due to the potentially large coefficients in the Gaver functionals. Some success using high precision variables to subdue the roundoff error has been reported ABATE2004. Although notable packages exist BAILEY2004, BAILEY2005, the present lack of a standardized library for arbitrary precision variables for low level languages however makes this approach cumbersome for vectors or matrix-valued functions.