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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> 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,  <li> J.Hayakawa, A.Takemura,
 Estimation of exponential-polynomial distribution by holonomic gradient descent  Estimation of exponential-polynomial distribution by holonomic gradient descent
 <a href="http://arxiv.org/abs/1403.7852"> arxiv:1403.7852</a>  <a href="http://arxiv.org/abs/1403.7852"> arxiv:1403.7852</a>
Line 51  Calculation of Orthant Probabilities by the Holonomic 
Line 59  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 74  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,  <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 88  Advances in Applied Mathematics 47 (2011), 639--658,
Line 97  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.
Line 109  of the
Line 118  of the
 <a href="http://www.math.kobe-u.ac.jp/OpenXM/Current/doc/index-doc.html">  <a href="http://www.math.kobe-u.ac.jp/OpenXM/Current/doc/index-doc.html">
 OpenXM documents</a>  OpenXM documents</a>
 or in <a href="./"> this folder</a>.  or in <a href="./"> this folder</a>.
 The nightly snapshot of the asir-contrib can be found in the Asir-Contrib page below,  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/">  or look up our <a href="http://www.math.sci.kobe-u.ac.jp/cgi/cvsweb.cgi/">
 cvsweb page</a>.  cvsweb page</a>.
 <ol>  <ol>
 <li> Command line interfaces are in the folder OpenXM/src/hgm  <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">  in the OpenXM source tree. See <a href="http://www.math.kobe-u.ac.jp/OpenXM">
 OpenXM distribution page </a>.  OpenXM distribution page </a>.
 <li> <a href="http://www.math.kobe-u.ac.jp/OpenXM/Math/hgm/"> hgm package for R </a> for the step 3.  <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> yang (for Pfaffian systems) , nk_restriction (for D-module integrations),  To install this package in R, type in
 tk_jack  (for Jack polynomials), ko_fb_pfaffian (Pfaffian system for the Fisher-Bingham system)  <pre>
 are for the steps 1 or 2 and in the  R CMD install hgm_*.tar.gz
 <a href="http://www.math.kobe-u.ac.jp/Asir"> asir-contrib</a>.  </pre>
 <li> nk_fb_gen_c is a package to generate a C program to perform  <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).  maximal Likehood estimates for the Fisher-Bingham distribution by HGD (holonomic gradient descent).
 It is in the  <li> ot_hgm_ahg.rr (HGM for A-distributions, very experimental).
 <a href="http://www.math.kobe-u.ac.jp/Asir"> asir-contrib</a>.  </ul>
 </ol>  </ol>
   
 <h2> Programs to try examples of our papers </h2>  <h2> Programs to try examples of our papers </h2>
Line 132  It is in the 
Line 151  It is in the 
 <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 1.8 2014/04/03 07:34:31 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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