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

<h2> Papers and Tutorials</h2>

<ol>

<li> T.Koyama,

<li>N.Marumo, T.Oaku, A.Takemura,

Properties of powers of functions satisfying second-order linear differential equations with applications to statistics,

<a href="http://arxiv.org/abs/1405.4451"> arxiv:1405.4451</a>

<li> J.Hayakawa, A.Takemura,

Estimation of exponential-polynomial distribution by holonomic gradient descent

<a href="http://arxiv.org/abs/1403.7852"> arxiv:1403.7852</a>

<li> C.Siriteanu, A.Takemura, S.Kuriki,

MIMO Zero-Forcing Detection Performance Evaluation by Holonomic Gradient Method

<a href="http://arxiv.org/abs/1403.3788"> arxiv:1403.3788</a>

<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

Line 42 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,

to appear in Journal of Pure and Applied Algebra

Journal of Pure and Applied Algebra (online),

<li>

T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,

Line 61 Journal of Multivariate Analysis, 116 (2013), 440--455

Line 79 Journal of Multivariate Analysis, 116 (2013), 440--455

<li>T.Koyama, A Holonomic Ideal which Annihilates the Fisher-Bingham Integral,

Funkcialaj Ekvacioj 56 (2013), 51--61.

<a href="https://www.jstage.jst.go.jp/article/fesi/56/1/56_51/_article">jstage</a>

<li>

Hiromasa Nakayama, Kenta Nishiyama, Masayuki Noro, Katsuyoshi Ohara,

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

Line 91 Advances in Applied Mathematics 47 (2011), 639--658,

</ol>

<h2> Three Steps of HGM </h2>

<ol>

<li> Finding 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> Finding 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> Solving 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>

Most software packages are experimental and temporary documents are found in

"asir-contrib manual" (auto-autogenerated part), or

"Experimental Functions in Asir", or "miscellaneous and other documents"

of the

OpenXM documents</a>

or in <a href="./"> this folder</a>.

The nightly snapshot of the asir-contrib can be found in the asir page below,

or look up our <a href="http://www.math.sci.kobe-u.ac.jp/cgi/cvsweb.cgi/">

cvsweb page</a>.

<ol>

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

<li> Command line interfaces are in the folder OpenXM/src/hgm

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

in the OpenXM source tree. See <a href="http://www.math.kobe-u.ac.jp/OpenXM">

tk_jack (for Jack polynomials) are in the

OpenXM distribution page </a>.

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

<li> <a href="http://www.math.kobe-u.ac.jp/OpenXM/Math/hgm/"> hgm package for R </a> (hgm_*tar.gz, hgm-manual.pdf) for the step 3.

To install this package in R, type in

<pre>

R CMD install hgm_*.tar.gz

</pre>

<li> The following packages are

for the computer algebra system

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

They are in the asir-contrib collection.

<ul>

<li> yang.rr (for Pfaffian systems) ,

nk_restriction.rr (for D-module integrations),

tk_jack.rr (for Jack polynomials),

ko_fb_pfaffian.rr (Pfaffian system for the Fisher-Bingham system),

are for the steps 1 or 2.

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

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

<li> ot_hgm_ahg.rr (HGM for A-distributions, very experimental).

</ul>

</ol>

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

Line 86 tk_jack (for Jack polynomials) are in the

Line 147 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 1.10 2014/05/16 11:30:31 takayama Exp $ </pre>

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