References for the Holonomic Gradient Method (HGM) and
the Holonomic Gradient Descent Method (HGD)
Papers and Tutorials
 F.H.Danufane, K.Ohara, N.Takayama,
Holonomic Gradient Method for the Distribution Function of the Largest Root of Complex Noncentral Wishart Matrices,
arxiv:1707.02564
 T.Koyama,
An integral formula for the powered sum of the independent, identically and normally distributed random variables,
arxiv:1706.03989
 H.Hashiguchi, N.Takayama, A.Takemura,
Distribution of Ratio of two Wishart Matrices and Evaluation of Cumulative Probability
by Holonomic Gradient Method,
arxiv:1610.09187
 R.Vidunas, A.Takemura,
Differential relations for the largest root distribution
of complex noncentral Wishart matrices,
arxiv:1609.01799
 S.Mano,
The Ahypergeometric System Associated with the Rational Normal Curve and
Exchangeable Structures,
arxiv:1607.03569
 M.Noro,
System of Partial Differential Equations for the Hypergeometric Function 1F1 of a Matrix Argument on Diagonal Regions,
ACM DL
 Y.Goto, K.Matsumoto,
Pfaffian equations and contiguity relations of the hypergeometric function of type (k+1,k+n+2) and their applications,
arxiv:1602.01637
 T.Koyama,
Holonomic gradient method for the probability content of a simplex
region
with a multivariate normal distribution,
arxiv:1512.06564
 N.Takayama, S.Kuriki, A.Takemura,
AHpergeometric Distributions and Newton Polytopes,
arxiv:1510.02269
 G.Weyenberg, R.Yoshida, D.Howe,
Normalizing Kernels in the BilleraHolmesVogtmann Treespace,
arxiv:1506.00142
 C.Siriteanu, A.Takemura, C.Koutschan, S.Kuriki, D.St.P.Richards, H.Sin,
Exact ZF Analysis and ComputerAlgebraAided Evaluation
in Rank1 LoS Rician Fading,
arxiv:1507.07056
 K.Ohara, N.Takayama,
Pfaffian Systems of AHypergeometric Systems II 
Holonomic Gradient Method,
arxiv:1505.02947
 T.Koyama,
The Annihilating Ideal of the Fisher Integral,
arxiv:1503.05261
 T.Koyama, A.Takemura,
Holonomic gradient method for distribution function of a weighted sum
of noncentral chisquare random variables,
arxiv:1503.00378
 Y.Goto,
Contiguity relations of Lauricella's F_D revisited,
arxiv:1412.3256

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,
706712.
DOI
 N.Marumo, T.Oaku, A.Takemura,
Properties of powers of functions satisfying secondorder linear differential equations with applications to statistics,
arxiv:1405.4451
 J.Hayakawa, A.Takemura,
Estimation of exponentialpolynomial distribution by holonomic gradient descent
arxiv:1403.7852
 C.Siriteanu, A.Takemura, S.Kuriki,
MIMO ZeroForcing Detection Performance Evaluation by Holonomic Gradient Method
arxiv:1403.3788
 T.Koyama,
Holonomic Modules Associated with Multivariate Normal Probabilities of Polyhedra,
arxiv:1311.6905
 T.Hibi, K.Nishiyama, N.Takayama,
Pfaffian Systems of AHypergeometric Equations I,
Bases of Twisted Cohomology Groups,
arxiv:1212.6103
(major revision v2 of arxiv:1212.6103).
Accepted version is at
DOI

T.Hibi et al, Groebner Bases : Statistics and Software Systems , Springer, 2013.

Introduction to the Holonomic Gradient Method (movie), 2013.
movie at youtube
 T.Sei, A.Kume,
Calculating the Normalising Constant of the Bingham Distribution on the Sphere using the Holonomic Gradient Method,
Statistics and Computing, 2013,
DOI
 T.Koyama, A.Takemura,
Calculation of Orthant Probabilities by the Holonomic Gradient Method,
arxiv:1211.6822
 T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,
Holonomic Rank of the FisherBingham System of Differential Equations,
Journal of Pure and Applied Algebra (online),
DOI

T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,
Holonomic Gradient Descent for the FisherBingham Distribution on the ddimensional Sphere,
Computational Statistics (2013)
DOI
 Hiroki Hashiguchi, Yasuhide Numata, Nobuki Takayama, Akimichi Takemura,
Holonomic gradient method for the distribution function of the largest root of a Wishart matrix,
Journal of Multivariate Analysis, 117, (2013) 296312,
DOI
 Tomonari Sei, Hiroki Shibata, Akimichi Takemura, Katsuyoshi Ohara, Nobuki Takayama,
Properties and applications of Fisher distribution on the rotation group,
Journal of Multivariate Analysis, 116 (2013), 440455,
DOI
 T.Koyama, A Holonomic Ideal which Annihilates the FisherBingham Integral,
Funkcialaj Ekvacioj 56 (2013), 5161.
DOI

Hiromasa Nakayama, Kenta Nishiyama, Masayuki Noro, Katsuyoshi Ohara,
Tomonari Sei, Nobuki Takayama, Akimichi Takemura ,
Holonomic Gradient Descent and its Application to FisherBingham Integral,
Advances in Applied Mathematics 47 (2011), 639658,
DOI
Early papers related to HGM.

H.Dwinwoodie, L.Matusevich, E. Mosteig,
Transform methods for the hypergeometric distribution,
Statistics and Computing 14 (2004), 287297.
Three Steps of HGM
 Finding 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.
 Finding 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.
 Solving the holonomic system numerically. We use several methods
in numerical analysis such as the RungeKutta method of solving
ordinary differential equations and efficient solvers of systems of linear
equations.
Software Packages for HGM

CRAN package hgm (for R).

Some software packages are experimental and temporary documents are found in
"asircontrib manual" (autoautogenerated part), or
"Experimental Functions in Asir", or "miscellaneous and other documents"
of the
OpenXM documents
or in this folder.
The nightly snapshot of the asircontrib can be found in the asir page below,
or look up our
cvsweb page.
 Command line interfaces are in the folder OpenXM/src/hgm
in the OpenXM source tree. See
OpenXM distribution page .
 Experimental version of hgm package for R (hgm_*tar.gz, hgmmanual.pdf) for the step 3.
To install this package in R, type in
R CMD install hgm_*.tar.gz
 The following packages are
for the computer algebra system
Risa/Asir.
They are in the asircontrib collection.
 yang.rr (for Pfaffian systems) ,
nk_restriction.rr (for Dmodule integrations),
tk_jack.rr (for Jack polynomials),
ko_fb_pfaffian.rr (Pfaffian system for the FisherBingham system),
are for the steps 1 or 2.
 nk_fb_gen_c.rr is a package to generate a C program to perform
maximal Likehood estimates for the FisherBingham distribution by HGD (holonomic gradient descent).
 ot_hgm_ahg.rr (HGM for Adistributions, very experimental).
Programs to try examples of our papers
 ddimensional FisherBingham System
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