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

<h2> Papers and Tutorials</h2>

<ol>

<li> T.Koyama,

<li> Y.Goto, K.Matsumoto,

Pfaffian equations and contiguity relations of the hypergeometric function of type (k+1,k+n+2) and their applications,

<a href="http://arxiv.org/abs/1602.01637"> arxiv:1602.01637

<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 36 Calculating the Normalising Constant of the Bingham Di

Line 52 Calculating the Normalising Constant of the Bingham Di

Statistics and Computing, 2013,

<li> T.Koyama,

<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 66 Journal of Multivariate Analysis, 116 (2013), 440--455

Line 83 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 80 Advances in Applied Mathematics 47 (2011), 639--658,

Line 97 Advances in Applied Mathematics 47 (2011), 639--658,

<h2> Three Steps of HGM </h2>

<ol>

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

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

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

<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> for the step 3.

<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 for the steps 1 or 2 and 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 106 tk_jack (for Jack polynomials) are for the steps 1 or

Line 151 tk_jack (for Jack polynomials) are for the steps 1 or

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

</ol>

<pre> $OpenXM: OpenXM/src/hgm/doc/ref-hgm.html,v 1.1 2014/03/24 06:43:55 takayama Exp $ </pre>

<pre> $OpenXM: OpenXM/src/hgm/doc/ref-hgm.html,v 1.11 2014/05/20 02:12:18 takayama Exp $ </pre>

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