Laplace approximation of Lauricella functions F A and F D

Butler, R.W.; Wood, Andrew T.A.

Abstract

The Lauricella functions, which are generalizations of the Gauss hypergeometric function 2 F 1, arise naturally in many areas of mathematics and statistics. So far as we are aware, there is little or nothing in the literature on how to calculate numerical approximations for these functions outside those cases in which a simple one-dimensional integral representation or a one-dimensional series representation is available. In this paper we present first-order and second-order Laplace approximations to the Lauricella functions F(n)A and F(n)D. Our extensive numerical results show that these approximations achieve surprisingly good accuracy in a wide variety of examples, including cases well outside the asymptotic framework within which the approximations were derived. Moreover, it turns out that the second-order Laplace approximations are usually more accurate than their first-order versions. The numerical results are complemented by theoretical investigations which suggest that the approximations have good relative error properties outside the asymptotic regimes within which they were derived, including in certain cases where the dimension n goes to infinity.

Citation

Butler, R., & Wood, A. T. (2015). Laplace approximation of Lauricella functions F A and F D. Advances in Computational Mathematics, 41(6), https://doi.org/10.1007/s10444-014-9397-5

• Journal Article Type: Article
• Online Publication Date: Dec 10, 2014
• Publication Date: Dec 1, 2015
• Deposit Date: Nov 23, 2015
• Publicly Available Date: Nov 23, 2015
• Journal: Advances in Computational Mathematics
• Print ISSN: 1019-7168
• Electronic ISSN: 1572-9044
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 41
• Issue: 6
• DOI: https://doi.org/10.1007/s10444-014-9397-5
• Keywords: Gauss hypergeometric function; Lauricella functions; vector-; argument hypergeometric functions
• Public URL: https://nottingham-repository.worktribe.com/output/980750
• Publisher URL: http://link.springer.com/article/10.1007/s10444-014-9397-5
• Contract Date: Nov 23, 2015