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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> F.H.Danufane, K.Ohara, N.Takayama,
   Holonomic Gradient Method for the Distribution Function of the Largest Root of Complex Non-central Wishart Matrices,
   <a href="https://arxiv.org/abs/1707.02564"> arxiv:1707.02564 </a>
   <li> T.Koyama,
   An integral formula for the powered sum of the independent, identically and normally distributed random variables,
   <a href="https://arxiv.org/abs/1706.03989"> arxiv:1706.03989 </a>
   <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,
   <a href="http://dl.acm.org/citation.cfm?doid=2930889.2930905"> ACM DL </a>
   
   <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.
   <a href="http://link.springer.com/chapter/10.1007%2F978-3-662-44199-2_105">
   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>
   
   <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 20  Holonomic Modules Associated with Multivariate Normal 
Line 111  Holonomic Modules Associated with Multivariate Normal 
 Pfaffian Systems of A-Hypergeometric Equations I,  Pfaffian Systems of A-Hypergeometric Equations I,
 Bases of Twisted Cohomology Groups,  Bases of Twisted Cohomology Groups,
 <a href="http://arxiv.org/abs/1212.6103"> arxiv:1212.6103 </a>  <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
   <a href="http://dx.doi.org/10.1016/j.aim.2016.10.021"> DOI </a>
   
 <li> <img src="./wakaba01.png" alt="Intro">  <li> <img src="./wakaba01.png" alt="Intro">
 <a href="http://link.springer.com/book/10.1007/978-4-431-54574-3">  <a href="http://link.springer.com/book/10.1007/978-4-431-54574-3">
Line 43  Calculation of Orthant Probabilities by the Holonomic 
Line 136  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 160  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 76  Holonomic Gradient Descent  and its Application to Fis
Line 170  Holonomic Gradient Descent  and its Application to Fis
 <!-- <a href="http://arxiv.org/abs//1005.5273"> arxiv:1005.5273 </a>  -->  <!-- <a href="http://arxiv.org/abs//1005.5273"> arxiv:1005.5273 </a>  -->
 Advances in Applied Mathematics 47 (2011), 639--658,  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>
   
   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>  <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>
   
   <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
   <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> 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>.
   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>
   
   </ul>
   
 <h2> Programs to try examples of our papers </h2>  <h2> Programs to try examples of our papers </h2>
 <ol>  <ol>
 <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 1.22 2016/11/03 23:19:18 takayama Exp $ </pre>
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