For small values of $\Vert \mathbf{T}\Vert $ the zonal polynomial expansion given by (35.8.1) can be summed numerically. For large $\Vert \mathbf{T}\Vert $ the asymptotic approximations referred to in §35.7(iv) are available.

Other methods include numerical quadrature applied to double and multiple integral representations. See Yan (1992) for the ${}_{1}{}^{}F_{1}^{}$ and ${}_{2}{}^{}F_{1}^{}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $\mathbf{O}(m)$ applied to a generalization of the integral (35.5.8).