lauricella {mggd}

Lauricella D-Hypergeometric Function

Description

Computes the Lauricella D-hypergeometric Function function.

Usage

lauricella(a, b, g, x, eps = 1e-06)

Arguments

a

numeric.

b

numeric vector.

g

numeric.

x

numeric vector. x must have the same length as b.

eps

numeric. Precision for the nested sums (default 1e-06).

Details

If n is the length of the b and x vectors, the Lauricella D-hypergeometric Function function is given by:

\displaystyle{F_D^{(n)}\left(a, b_1, ..., b_n, g; x_1, ..., x_n\right) = \sum_{m_1 \geq 0} ... \sum_{m_n \geq 0}{ \frac{ (a)_{m_1+...+m_n}(b_1)_{m_1} ... (b_n)_{m_n} }{ (g)_{m_1+...+m_n} } \frac{x_1^{m_1}}{m_1!} ... \frac{x_n^{m_n}}{m_n!} } }

where (x)_p is the Pochhammer symbol (see pochhammer).

If |x_i| < 1, i = 1, \dots, n, this sum converges. Otherwise there is an error.

The eps argument gives the required precision for its computation. It is the attr(, "epsilon") attribute of the returned value.

Sometimes, the convergence is too slow and the required precision cannot be reached. If this happens, the attr(, "epsilon") attribute is the precision that was really reached.

Value

A numeric value: the value of the Lauricella function, with two attributes attr(, "epsilon") (precision of the result) and attr(, "k") (number of iterations).

Author(s)

Pierre Santagostini, Nizar Bouhlel

References

N. Bouhlel, A. Dziri, Kullback-Leibler Divergence Between Multivariate Generalized Gaussian Distributions. IEEE Signal Processing Letters, vol. 26 no. 7, July 2019. doi:10.1109/LSP.2019.2915000