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

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. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.