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                      5: <title>References for HGM</title> <!-- Use UTF-8 文字 code-->
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                      9:
                     10: <h1> References for the Holonomic Gradient Method (HGM) and
                     11: the Holonomic Gradient Descent Method  (HGD) </h1>
                     12:
                     13: <h2> Papers  and Tutorials</h2>
                     14: <ol>
1.4       takayama   15: <li> T.Koyama,
1.1       takayama   16: Holonomic Modules Associated with Multivariate Normal Probabilities of Polyhedra,
                     17: <a href="http://arxiv.org/abs/1311.6905"> arxiv:1311.6905 </a>
                     18:
                     19: <li> T.Hibi, K.Nishiyama, N.Takayama,
                     20: Pfaffian Systems of A-Hypergeometric Equations I,
                     21: Bases of Twisted Cohomology Groups,
                     22: <a href="http://arxiv.org/abs/1212.6103"> arxiv:1212.6103 </a>
                     23: (major revision v2 of arxiv:1212.6103)
                     24:
                     25: <li> <img src="./wakaba01.png" alt="Intro">
                     26: <a href="http://link.springer.com/book/10.1007/978-4-431-54574-3">
                     27: T.Hibi et al, Groebner Bases : Statistics and Software Systems </a>, Springer, 2013.
                     28:
                     29: <li> <img src="./wakaba01.png" alt="Intro">
                     30: Introduction to the Holonomic Gradient Method (movie), 2013.
                     31: <a href="http://www.youtube.com/watch?v=SgyDDLzWTyI"> movie at youtube </a>
                     32:
1.2       takayama   33:
1.1       takayama   34: <li> T.Sei, A.Kume,
1.2       takayama   35: Calculating the Normalising Constant of the Bingham Distribution on the Sphere using the Holonomic Gradient Method,
1.1       takayama   36: Statistics and Computing, 2013,
                     37: <a href="http://dx.doi.org/10.1007/s11222-013-9434-0">DOI</a>
                     38:
1.4       takayama   39: <li> T.Koyama, A.Takemura,
1.2       takayama   40: Calculation of Orthant Probabilities by the Holonomic Gradient Method,
                     41: <a href="http://arxiv.org/abs/1211.6822"> arxiv:1211.6822</a>
                     42:
1.1       takayama   43: <li>T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,
                     44: Holonomic Rank of the Fisher-Bingham System of Differential Equations,
                     45: <!-- <a href="http://arxiv.org/abs/1205.6144"> arxiv:1205.6144 </a>-->
                     46: to appear in Journal of Pure and Applied Algebra
                     47:
                     48: <li>
                     49: T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,
                     50: Holonomic Gradient Descent for the Fisher-Bingham Distribution on the d-dimensional Sphere,
                     51: <!-- <a href="http://arxiv.org/abs/1201.3239"> 1201.3239 </a> -->
                     52: Computational Statistics (2013)
                     53: <a href="http://dx.doi.org/10.1007/s00180-013-0456-z"> DOI </a>
                     54:
                     55: <li> Hiroki Hashiguchi, Yasuhide Numata, Nobuki Takayama, Akimichi Takemura,
                     56: Holonomic gradient method for the distribution function of the largest root of a Wishart matrix,
                     57: <!-- <a href="http://arxiv.org/abs/1201.0472"> 1201.0472 </a> -->
                     58: Journal of Multivariate Analysis, 117, (2013) 296-312,
                     59: <a href="http://dx.doi.org/10.1016/j.jmva.2013.03.011"> DOI </a>
                     60:
                     61: <li> Tomonari Sei, Hiroki Shibata, Akimichi Takemura, Katsuyoshi Ohara, Nobuki Takayama,
                     62: Properties and applications of Fisher distribution on the rotation group,
                     63: <!-- <a href="http://arxiv.org/abs/1110.0721"> 1110.0721 </a> -->
                     64: Journal of Multivariate Analysis, 116 (2013), 440--455,
                     65: <a href="http://dx.doi.org/10.1016/j.jmva.2013.01.010">DOI</a>
                     66:
                     67: <li>T.Koyama, A Holonomic Ideal which Annihilates the Fisher-Bingham Integral,
                     68: Funkcialaj Ekvacioj 56 (2013), 51--61.
                     69: <!-- <a href="http://dx.doi.org/10.1619/fesi.56.51">DOI</a> -->
                     70: <a href="https://www.jstage.jst.go.jp/article/fesi/56/1/56_51/_article">jstage</a>
                     71:
                     72: <li>
                     73: Hiromasa Nakayama, Kenta Nishiyama, Masayuki Noro, Katsuyoshi Ohara,
                     74: Tomonari Sei, Nobuki Takayama, Akimichi Takemura ,
                     75: Holonomic Gradient Descent  and its Application to Fisher-Bingham Integral,
                     76: <!-- <a href="http://arxiv.org/abs//1005.5273"> arxiv:1005.5273 </a>  -->
                     77: Advances in Applied Mathematics 47 (2011), 639--658,
                     78: <a href="http://dx.doi.org/10.1016/j.aam.2011.03.001"> DOI </a>
                     79: </ol>
                     80:
1.2       takayama   81: <h2> Three Steps of HGM </h2>
                     82: <ol>
                     83: <li> Find a holonomic system satisfied by the normalizing constant.
                     84: We may use computational or theoretical methods to find it.
                     85: Groebner basis and related methods are used.
                     86: <li> Find an initial value vector for the holonomic system.
                     87: This is equivalent to evaluating the normalizing constant and its derivatives
                     88: at a point.
                     89: This step is usually performed by a series expansion.
                     90: <li> Solve the holonomic system numerically. We use several methods
                     91: in numerical analysis such as the Runge-Kutta method of solving
                     92: ordinary differential equations and efficient solvers of systems of linear
                     93: equations.
                     94: </ol>
                     95:
1.1       takayama   96: <h2> Software Packages for HGM</h2>
                     97: <ol>
1.2       takayama   98: <li> <a href="http://www.math.kobe-u.ac.jp/OpenXM/Math/hgm/"> hgm package for R </a> for the step 3.
1.1       takayama   99: <li> yang (for Pfaffian systems) , nk_restriction (for D-module integrations),
1.5     ! takayama  100: tk_jack  (for Jack polynomials), ko_fb_pfaffian (Pfaffian system for the Fisher-Bingham system)
        !           101: are for the steps 1 or 2 and in the
        !           102: <a href="http://www.math.kobe-u.ac.jp/Asir"> asir-contrib </a>
        !           103: <li> nk_fb_gen_c is a package to generate a C program to perform
        !           104: maximal Likehood estimates for the Fisher-Bingham distribution by HGD (holonomic gradient descent)
        !           105: It is in the
1.1       takayama  106: <a href="http://www.math.kobe-u.ac.jp/Asir"> asir-contrib </a>
                    107: </ol>
                    108:
                    109: <h2> Programs to try examples of our papers </h2>
                    110: <ol>
                    111: <li> <a href="http://www.math.kobe-u.ac.jp/OpenXM/Math/Fisher-Bingham-2"> d-dimensional Fisher-Bingham System </a>
                    112: </ol>
                    113:
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