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We apply the holonomic gradient method introduced by Nakayama et al.(2011) tothe evaluation of the exact distribution function of the largest root of aWishart matrix, which involves a hypergeometric function 1F1 of a matrixargument. Numerical evaluation of the hypergeometric function has been one ofthe longstanding problems in multivariate distribution theory. The holonomicgradient method offers a totally new approach, which is complementary to theinfinite series expansion around the origin in terms of zonal polynomials. Itallows us to move away from the origin by the use of partial differentialequations satisfied by the hypergeometric function. From numerical viewpoint weshow that the method works well up to dimension 10. From theoretical viewpointthe method offers many challenging problems both to statistics and D-moduletheory.

Holonomic gradient method for the distribution function of the largest root of a Wishart matrix