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<title>References for HGM</title>

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

<h2> Papers and Tutorials</h2>

<ol>

<li> H.Hashiguchi, N.Takayama, A.Takemura,

Distribution of Ratio of two Wishart Matrices and Evaluation of Cumulative Probability

by Holonomic Gradient Method,

<a href="https://arxiv.org/abs/1610.09187"> arxiv:1610.09187 </a>

<li> R.Vidunas, A.Takemura,

Differential relations for the largest root distribution

of complex non-central Wishart matrices,

<a href="http://arxiv.org/abs/1609.01799"> arxiv:1609.01799 </a>

<li> S.Mano,

The A-hypergeometric System Associated with the Rational Normal Curve and

Exchangeable Structures,

<a href="http://arxiv.org/abs/1607.03569"> arxiv:1607.03569 </a>

<li> M.Noro,

System of Partial Differential Equations for the Hypergeometric Function 1F1 of a Matrix Argument on Diagonal Regions,

<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 </a>

<li> T.Koyama,

Holonomic gradient method for the probability content of a simplex

region

with a multivariate normal distribution,

<a href="http://arxiv.org/abs/1512.06564"> arxiv:1512.06564 </a>

<li> N.Takayama, S.Kuriki, A.Takemura,

A-Hpergeometric Distributions and Newton Polytopes,

<a href="http://arxiv.org/abs/1510.02269"> arxiv:1510.02269 </a>

<li> G.Weyenberg, R.Yoshida, D.Howe,

Normalizing Kernels in the Billera-Holmes-Vogtmann Treespace,

<a href="http://arxiv.org/abs/1506.00142"> arxiv:1506.00142 </a>

<li> C.Siriteanu, A.Takemura, C.Koutschan, S.Kuriki, D.St.P.Richards, H.Sin,

Exact ZF Analysis and Computer-Algebra-Aided Evaluation

in Rank-1 LoS Rician Fading,

<a href="http://arxiv.org/abs/1507.07056"> arxiv:1507.07056 </a>

<li> K.Ohara, N.Takayama,

Pfaffian Systems of A-Hypergeometric Systems II ---

Holonomic Gradient Method,

<a href="http://arxiv.org/abs/1505.02947"> arxiv:1505.02947 </a>

<li> T.Koyama,

The Annihilating Ideal of the Fisher Integral,

<a href="http://arxiv.org/abs/1503.05261"> arxiv:1503.05261 </a>

<li> T.Koyama, A.Takemura,

Holonomic gradient method for distribution function of a weighted sum

of noncentral chi-square random variables,

<a href="http://arxiv.org/abs/1503.00378"> arxiv:1503.00378 </a>

<li> Y.Goto,

Contiguity relations of Lauricella's F_D revisited,

<a href="http://arxiv.org/abs/1412.3256"> arxiv:1412.3256 </a>

<li>

T.Koyama, H.Nakayama, K.Ohara, T.Sei, N.Takayama,

Software Packages for Holonomic Gradient Method,

Mathematial Software --- ICMS 2014,

4th International Conference, Proceedings.

Edited by Hoon Hong and Chee Yap,

Springer lecture notes in computer science 8592,

706--712.

DOI

</a>

<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>

Line 28 Holonomic Modules Associated with Multivariate Normal

Line 105 Holonomic Modules Associated with Multivariate Normal

Pfaffian Systems of A-Hypergeometric Equations I,

Bases of Twisted Cohomology Groups,

<a href="http://arxiv.org/abs/1212.6103"> arxiv:1212.6103 </a>

(major revision v2 of arxiv:1212.6103)

(major revision v2 of arxiv:1212.6103).

Accepted version is at

Line 51 Calculation of Orthant Probabilities by the Holonomic

Line 130 Calculation of Orthant Probabilities by the Holonomic

<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 74 Journal of Multivariate Analysis, 116 (2013), 440--455

Line 154 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 84 Holonomic Gradient Descent and its Application to Fis

Line 164 Holonomic Gradient Descent and its Application to Fis

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

</ol>

Early papers related to HGM. <br>

<ol>

<li>

H.Dwinwoodie, L.Matusevich, E. Mosteig,

Transform methods for the hypergeometric distribution,

Statistics and Computing 14 (2004), 287--297.

</ol>

<h2> Three Steps of HGM </h2>

<ol>

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

Line 102 equations.

Line 193 equations.

</ol>

<h2> Software Packages for HGM</h2>

Most software packages are experimental and temporary documents are found in

<ul>

<li>

CRAN package <a href="https://cran.r-project.org/web/packages/hgm/index.html"> hgm </a> (for R).

<li>

Some 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

Line 116 cvsweb page</a>.

Line 213 cvsweb page</a>.

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

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

OpenXM distribution page </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.

<li> Experimental version of <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>.

Line 133 maximal Likehood estimates for the Fisher-Bingham dist

Line 234 maximal Likehood estimates for the Fisher-Bingham dist

</ul>

</ol>

</ul>

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

<ol>

<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 1.9 2014/05/15 07:34:05 takayama Exp $ </pre>

<pre> $OpenXM: OpenXM/src/hgm/doc/ref-hgm.html,v 1.21 2016/11/03 23:05:22 takayama Exp $ </pre>

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