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Line 12 the Holonomic Gradient Descent Method (HGD) </h1>

<h2> Papers and Tutorials</h2>

<ol>

<li> T.Koyama,

Holonomic Modules Associated with Multivariate Normal Probabilities of Polyhedra,

<a href="http://arxiv.org/abs/1311.6905"> arxiv:1311.6905 </a>

Line 30 T.Hibi et al, Groebner Bases : Statistics and Software

Introduction to the Holonomic Gradient Method (movie), 2013.

<a href="http://www.youtube.com/watch?v=SgyDDLzWTyI"> movie at youtube </a>

<li> T.Sei, A.Kume,

Calculating the normalising constant of the Bingham distribution on the sphere using the holonomic gradient method,

Calculating the Normalising Constant of the Bingham Distribution on the Sphere using the Holonomic Gradient Method,

Statistics and Computing, 2013,

<li> T.Koyama, A.Takemura,

Calculation of Orthant Probabilities by the Holonomic Gradient Method,

<a href="http://arxiv.org/abs/1211.6822"> arxiv:1211.6822</a>

<li>T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,

Holonomic Rank of the Fisher-Bingham System of Differential Equations,

Line 73 Advances in Applied Mathematics 47 (2011), 639--658,

Line 78 Advances in Applied Mathematics 47 (2011), 639--658,

</ol>

<h2> Three Steps of HGM </h2>

<ol>

<li> Find a holonomic system satisfied by the normalizing constant.

We may use computational or theoretical methods to find it.

Groebner basis and related methods are used.

<li> Find an initial value vector for the holonomic system.

This is equivalent to evaluating the normalizing constant and its derivatives

at a point.

This step is usually performed by a series expansion.

<li> Solve the holonomic system numerically. We use several methods

in numerical analysis such as the Runge-Kutta method of solving

ordinary differential equations and efficient solvers of systems of linear

equations.

</ol>

<h2> Software Packages for HGM</h2>

<ol>

<li> <a href="http://www.math.kobe-u.ac.jp/OpenXM/Math/hgm/"> hgm package for R </a>

<li> <a href="http://www.math.kobe-u.ac.jp/OpenXM/Math/hgm/"> hgm package for R </a> for the step 3.

<li> yang (for Pfaffian systems) , nk_restriction (for D-module integrations),

tk_jack (for Jack polynomials) are in the

tk_jack (for Jack polynomials), ko_fb_pfaffian (Pfaffian system for the Fisher-Bingham system)

are for the steps 1 or 2 and in the

<a href="http://www.math.kobe-u.ac.jp/Asir"> asir-contrib </a>

<li> nk_fb_gen_c is a package to generate a C program to perform

maximal Likehood estimates for the Fisher-Bingham distribution by HGD (holonomic gradient descent)

It is in the

<a href="http://www.math.kobe-u.ac.jp/Asir"> asir-contrib </a>

</ol>

<h2> Programs to try examples of our papers </h2>

Line 86 tk_jack (for Jack polynomials) are in the

Line 111 tk_jack (for Jack polynomials) are in the

<li> <a href="http://www.math.kobe-u.ac.jp/OpenXM/Math/Fisher-Bingham-2"> d-dimensional Fisher-Bingham System </a>

</ol>

<pre> $OpenXM$ </pre>

<pre> $OpenXM: OpenXM/src/hgm/doc/ref-hgm.html,v 1.4 2014/03/24 21:03:55 takayama Exp $ </pre>

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