A recursive algorithm for the G transformation and accurate computation of incomplete Bessel functions

摘要

In the present contribution, we develop an efficient algorithm for the recursive computation of the G_n~(1) transformation for the approximation of infinite-range integrals. Previous to this contribution, the theoretically powerful G_n~(1) transformation was handicapped by the lack of an algorithmic implementation. Our proposed algorithm removes this handicap by introducing a recursive computation of the successive G_n~(1) transformations with respect to the order n. This recursion, however, introduces the (x~2 d/dx) operator applied to the integrand. Consequently, we employ the Slevinsky-Safouhi formula I for the analytical and numerical developments of these required successive derivatives. Incomplete Bessel functions, which pose as a numerical challenge, are computed to high pre-determined accuracies using the developed algorithm. The numerical results obtained show the high efficiency of the new method, which does not resort to any numerical integration in the computation.