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Line 12  the Holonomic Gradient Descent Method  (HGD) </h1>
Line 12  the Holonomic Gradient Descent Method  (HGD) </h1>
   
 <h2> Papers  and Tutorials</h2>  <h2> Papers  and Tutorials</h2>
 <ol>  <ol>
   <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,  <li> T.Koyama,
 Holonomic Modules Associated with Multivariate Normal Probabilities of Polyhedra,  Holonomic Modules Associated with Multivariate Normal Probabilities of Polyhedra,
 <a href="http://arxiv.org/abs/1311.6905"> arxiv:1311.6905 </a>  <a href="http://arxiv.org/abs/1311.6905"> arxiv:1311.6905 </a>
Line 43  Calculation of Orthant Probabilities by the Holonomic 
Line 55  Calculation of Orthant Probabilities by the Holonomic 
 <li>T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,  <li>T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,
 Holonomic Rank of the Fisher-Bingham System of Differential Equations,  Holonomic Rank of the Fisher-Bingham System of Differential Equations,
 <!-- <a href="http://arxiv.org/abs/1205.6144"> arxiv:1205.6144 </a>-->  <!-- <a href="http://arxiv.org/abs/1205.6144"> arxiv:1205.6144 </a>-->
 to appear in Journal of Pure and Applied Algebra  Journal of Pure and Applied Algebra  (online),
   <a href="http://dx.doi.org/10.1016/j.jpaa.2014.03.004"> DOI </a>
   
 <li>  <li>
 T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,  T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,
Line 66  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,  <li>T.Koyama, A Holonomic Ideal which Annihilates the Fisher-Bingham Integral,
 Funkcialaj Ekvacioj 56 (2013), 51--61.  Funkcialaj Ekvacioj 56 (2013), 51--61.
 <!-- <a href="http://dx.doi.org/10.1619/fesi.56.51">DOI</a> -->  <a href="http://dx.doi.org/10.1619/fesi.56.51">DOI</a>
 <a href="https://www.jstage.jst.go.jp/article/fesi/56/1/56_51/_article">jstage</a>  <!-- <a href="https://www.jstage.jst.go.jp/article/fesi/56/1/56_51/_article">jstage</a> -->
   
 <li>  <li>
 Hiromasa Nakayama, Kenta Nishiyama, Masayuki Noro, Katsuyoshi Ohara,  Hiromasa Nakayama, Kenta Nishiyama, Masayuki Noro, Katsuyoshi Ohara,
Line 80  Advances in Applied Mathematics 47 (2011), 639--658,
Line 93  Advances in Applied Mathematics 47 (2011), 639--658,
   
 <h2> Three Steps of HGM </h2>  <h2> Three Steps of HGM </h2>
 <ol>  <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.  We may use computational or theoretical methods to find it.
 Groebner basis and related methods are used.  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  This is equivalent to evaluating the normalizing constant and its derivatives
 at a point.  at a point.
 This step is usually performed by a series expansion.  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  in numerical analysis such as the Runge-Kutta method of solving
 ordinary differential equations and efficient solvers of systems of linear  ordinary differential equations and efficient solvers of systems of linear
 equations.  equations.
 </ol>  </ol>
   
 <h2> Software Packages for HGM</h2>  <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
   <a href="http://www.math.kobe-u.ac.jp/OpenXM/Current/doc/index-doc.html">
   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>  <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>  </ol>
   
 <h2> Programs to try examples of our papers </h2>  <h2> Programs to try examples of our papers </h2>
Line 106  tk_jack  (for Jack polynomials) are for the steps 1 or
Line 147  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>  <li> <a href="http://www.math.kobe-u.ac.jp/OpenXM/Math/Fisher-Bingham-2"> d-dimensional Fisher-Bingham System </a>
 </ol>  </ol>
   
 <pre> $OpenXM: OpenXM/src/hgm/doc/ref-hgm.html,v 1.3 2014/03/24 07:54:51 takayama Exp $ </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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