Laplace approximation of Lauricella functions F_A and F_D
Butler, R.W. and Wood, Andrew T.A. (2014) Laplace approximation of Lauricella functions F_A and F_D. Advances in Computational Mathematics . ISSN 1572-9044 (In Press)
The Lauricella functions, which are generalizations of the Gauss hypergeo-metric function 2F1, 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 rep-resentation is available. In this paper we present first-order and second-order Laplace approximations to the Lauricella functions F_A^(n) and F_D^(n). 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.
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