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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> 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 30  T.Hibi et al, Groebner Bases : Statistics and Software
Line 30  T.Hibi et al, Groebner Bases : Statistics and Software
 Introduction to the Holonomic Gradient Method (movie), 2013.  Introduction to the Holonomic Gradient Method (movie), 2013.
 <a href="http://www.youtube.com/watch?v=SgyDDLzWTyI"> movie at youtube </a>  <a href="http://www.youtube.com/watch?v=SgyDDLzWTyI"> movie at youtube </a>
   
   
 <li> T.Sei, A.Kume,  <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,  Statistics and Computing, 2013,
 <a href="http://dx.doi.org/10.1007/s11222-013-9434-0">DOI</a>  <a href="http://dx.doi.org/10.1007/s11222-013-9434-0">DOI</a>
   
   <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,  <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>-->
Line 73  Advances in Applied Mathematics 47 (2011), 639--658,
Line 78  Advances in Applied Mathematics 47 (2011), 639--658,
 <a href="http://dx.doi.org/10.1016/j.aam.2011.03.001"> DOI </a>  <a href="http://dx.doi.org/10.1016/j.aam.2011.03.001"> DOI </a>
 </ol>  </ol>
   
   <h2> Three Steps of HGM </h2>
   <ol>
   <li> Find 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.
   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
   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>  <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>.
   The nightly snapshot of the asir-contrib can be found in the Asir-Contrib 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>  <li> <a href="http://www.math.kobe-u.ac.jp/OpenXM/Math/hgm/"> hgm package for R </a> for the step 3.
 <li> yang (for Pfaffian systems) , nk_restriction (for D-module integrations),  <li> yang (for Pfaffian systems) , nk_restriction (for D-module integrations),
 tk_jack  (for Jack polynomials) are in the  tk_jack  (for Jack polynomials), ko_fb_pfaffian (Pfaffian system for the Fisher-Bingham system)
   are for the steps 1 or 2 and in the
 <a href="http://www.math.kobe-u.ac.jp/Asir"> asir-contrib </a>  <a href="http://www.math.kobe-u.ac.jp/Asir"> asir-contrib </a>
   <li> nk_fb_gen_c is a package to generate a C program to perform
   maximal Likehood estimates for the Fisher-Bingham distribution by HGD (holonomic gradient descent)
   It is in the
   <a href="http://www.math.kobe-u.ac.jp/Asir"> asir-contrib </a>
 </ol>  </ol>
   
 <h2> Programs to try examples of our papers </h2>  <h2> Programs to try examples of our papers </h2>
Line 86  tk_jack  (for Jack polynomials) are in the 
Line 120  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>  <li> <a href="http://www.math.kobe-u.ac.jp/OpenXM/Math/Fisher-Bingham-2"> d-dimensional Fisher-Bingham System </a>
 </ol>  </ol>
   
 <pre> $OpenXM$ </pre>  <pre> $OpenXM: OpenXM/src/hgm/doc/ref-hgm.html,v 1.5 2014/03/26 05:02:18 takayama Exp $ </pre>
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