3-1 Bibliography and Recent Advances.
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3-1 Bibliography and Recent Advances.

General references
1D.Cox, J.Little, D.O’Shea, Ideals, Varieties, and Algorithms: An Introduction toComputational Algebraic Geometry and Commutative Algebra, Undergraduate
Texts in Mathematics, Springer, 2006. The chapters 1 and 2 are prerequisite ofthis tutorial.

2D.Zeilberger, A Holonomic Systems Approache to Speical Function Identities,
Journal of Computational and Applied Mathematics 32 (1990), 321–368. The
original article is exciting!

3T.Oaku, Algorithms for b-functions, restrictions, and algebraic local cohomology
groups of D-modules, Advances in Applied Mathematics 19 (1997), 61–105.
Finding a set of generators of a given integration ideal.

4T.Oaku, Y.Shiraki, N.Takayama, Algebraic Algorithms for D-modules andNumerical Analysis, Z.M.Li, W.Sit (editors), Computer Mathematics, 23–39,
World scientiﬁc, 2003. Holonomic (Schwartz) distribution.

5M.Kauers, P.Paule, The concrete tetrahedron, Symbolic sums, recurrence
equations, generating functions, asymptotic estimation, 2011, Springer. An
introductory book on creative telescoping method.

6T.Hibi edited. Gröbner Bases: Statistics and Software Systems, 2014, Springer.
It includes an introduction to holonomic systems and algorithms for them withproofs.

7F.Chyzak, The ABC of Creative Telescoping, Algorithms, Bounds, Complexity,
2014. A survey on the method of creative telescoping.

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3-1 Bibliography and Recent AdvancesFisher-Bingham distribution and its MLE∫exp(t T Θt + θt)|dt|z(Θ, θ) =(1)Sn|dt|: the Haar measure on the hypersphere S n over which t runs.
Θ: (n + 1) × (n + 1) real symmetric matrix. θ: real vector of thelength n + 1. Th The rank of diﬀerential equations is 2n + 2.

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H.Nakayama, K.Nishiyama, M.Noro, K.Ohara, T.Sei, N.Takayama,A.Takemura, Holonomic Gradient Descent and its Application to theFisher-Bingham Integral, Advances in Applied Mathematics 47 (2011),639–658

2T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama, Holonomic
Gradient Descent for the Fisher-Bingham Distribution on then-Dimensional Sphere, arxiv:1201.3239, Computational Statistics (2013,online)

3T.Sei, A.Kume, Calculating the normalising constant of the Bingham
distribution on the sphere using the holonomic gradient method,
Statistics and Computing (2013, online). [Degenerated cases needindependent discussions.]
3-1 Bibliography and Recent AdvancesFisher distribution and its maximal likelihood estimate(MLE)

X , Θ: 3 × 3 real matrices. Θ> : the transpose of Θ. µ: theinvariant measure on SO(3).
∫z(Θ) =exp(Tr (Θ> X )) dµ(X ).
SO(3)Th The rank of diﬀerential equations is 4.
1T.Sei, H.Shibata, A.Takemura, K.Ohara, N.Takayama, Properties andapplications of Fisher distribution on the rotation group, Journal ofMultivariate Analysis (2013) 440–455.

3-1 Bibliography and Recent Advances

The cumulative distribution function for the ﬁrst
eigenvalue of Wishart matrices
Matrix hypergeometric integral 1 F1
X : m × m real matrix.
∫exp(Tr XY )|X |a−(m+1)/2 |Im − X |c−a−(m+1)/2 dX ,0<X <Im

0 < X < Im means that X∏and Im − X are positive deﬁnite
symmetric matrix. dX = i≤j dxij . Th The rank of diﬀerentialequations is 2m .
1.2by hg10.80.6

0.4

0.20051101520m = 10, n = 12, β = (1, 2, . . . , 10)

H.Hashiguchi, Y.Numata, N.Takayama, A.Takemura, Holonomic gradientmethod for the distribution function of the largest root of a Wishart
matrix, Journal of Multivariate Analysis, 117, (2013) 296-312.
3-1 Bibliography and Recent Advances

Orthant probability∫ ∞∫ ∞...

z(τ, θ) =00mm∑∑
exp θ i xi +xi xj τij  dx
i=1i,j=1dx = dx1 · · · dxm . Th The rank of diﬀerential equations is 2m .
1.

RnL. Schläﬂi, On the multiple integral
dxdy . . . dz whose limits are
p1 = a1 x + b1 y + · · · + h1 z > 0, p2 > 0, . . . , pn > 0 andx 2 + y 2 + · · · + z 2 < 1, Quart. J. Pure Appl. Math. 2, 269–301; 3, 54–68,97–108, 1858.

2R.L. Plackett, A reduction formula for normal multivariate integrals,
Biometrika, 41, No.3/4, 351–360, 1954.

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H.I.Gassmann, Multivariate normal probabilities: implementing an oldidea of Plackett’s, Journal of Computational and Graphical Statistics, 12,No.3, 731–752, 2003.

4T.Koyama, A.Takemura, Calculation of orthant probabilities by theholonomic gradient method, arxiv:1211.682.

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3-1 Bibliography and Recent Advances

A-distributions∫z(θ) =Cnd∑∏
ajexp(θj t )ti−bi −1 dt, dt = dt1 · · · dtd
j=1i=1A = (aij ): n × d integer matrix. bi : real number. aj : the j-th∏acolumn vector of the matrix A. t aj = di=1 ti ij . Th The(holonomic) rank of the N.C. of the A-distribution associated toorder polytopes has polynomial complexity.
1T.Hibi, K.Nishiyama, N.Takayama, Pfaﬃan Systems of A-HypergeometricEquations, arxiv:1212.6103 (v2 will be posted soon).

Mathematical byproduct: construction of a basis of twisted or rapid decay
cohomology groups for complement of the zero set of a generic polynomial isreduced to a question in combinatorial commutative algebra.

More research topics will be in this conference!
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3-1 Bibliography and Recent Advances