Computing hypergeometric function of matrix argument In the context of the Bingham probability distribution the ${ }_1F_1$ hypergeometric function of matrix argument naturally arises as a normalization constant of the probability distribution function. Thus, it is of interest to evaluate this function effectively. For the more general class of hypergeometric functions $ _pF_q$ of Matrix argument, an algorithm was given by Koev. However, this algorithm still has some open problems discussed in Koevs paper. My question is, whether there is a way to compute the normalization constant of a bingham distribution for arbitrary dimensions effectively.
 A: It would be useful, if you'd point out the precise "open questions" regarding these evaluations that are of interest to you.
But since you mentioned Plamen Koev's work, I am sure you have tried out his matlab code for computing Hypergeometric functions of matrix argument. For the particular case of the normalization constant of the Bingham distribution, I am sure methods for approximating high-dimensional integrals numerically will prove to be effective, because, although scary looking, the normalization constant still has a very nice form:
\begin{equation*}
  {}_1F_1(\frac12,\frac p2, A) := \int_{S^{p-1}} e^{x^TAx}dx,
\end{equation*}
where the integration is wrt to the uniform distribution on the unit hypersphere $S^{p-1}$.
(PS: This integral has previously been discussed on MO, e.g., here in this question of L. Nicolaescu)
Edit: Regarding numerical approximation. Here is what seems to be the latest in this direction. Have a look at the geodesic monte carlo sampling method of Byrne and Girolami; (they discuss sampling from the Bingham distribution). Once you have that, then it should be "easy" to estimate the normalization constant. But I guess the bad thing might again be a lack of guarantees on how long it takes to get a given accuracy approximation---but for now, seems like the abovecited approach might be the most promising.
A: I have used Kume's method of Fisher Bingham sampling in my master's thesis. Using 100-500 samples were enough. Please have a look on the paper; I might have sampling code of my implementation in Matlab somewhere on my machine 
http://link.springer.com/chapter/10.1007/978-3-642-24085-0_69
