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

Authors:
Richard M. Slevinsky;Hassan Safouhi
Affiliations:
Mathematical Section, Campus Saint-Jean, University of Alberta, 8406, 91 Street, Edmonton (AB) T6C 4G9, Canada;Mathematical Section, Campus Saint-Jean, University of Alberta, 8406, 91 Street, Edmonton (AB) T6C 4G9, Canada
Venue:
Applied Numerical Mathematics
Year:
2010

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Abstract

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^2ddx) 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.