Katsuyoshi Ohara

Holonomic gradient descent and its application to the Fisher-Bingham integral

We give a new algorithm to find local maximum and minimum of a holonomic function and apply it for the Fisher-Bingham integral on the sphere

Properties and applications of Fisher distribution on the rotation group

S^n

Comprehensive Gröbner Systems in Rings of Differential Operators, Holonomic D-modules and B-functions

, which is used in the directional statistics. The method utilizes the theory and algorithms of holonomic systems.

Software Packages for Holonomic Gradient Method

We study properties of Fisher distribution (von Mises-Fisher distribution, matrix Langevin distribution) on the rotation group SO(3). In particular we apply the holonomic gradient descent, introduced by Nakayama et al. (2011) [16], and a method of series expansion for evaluating the normalizing constant of the distribution and for computing the maximum likelihood estimate. The rotation group can be identified with the Stiefel manifold of two orthonormal vectors. Therefore from the viewpoint of statistical modeling, it is of interest to compare Fisher distributions on these manifolds. We illustrate the difference with an example of near-earth objects data.

Developing Linear Algebra Packages on Risa/Asir for Eigenproblems

An algorithm for computing comprehensive Gröbner systems (CGS) is introduced in rings of linear partial differential operators. Their applications to b-functions are considered. The resulting algorithm designed for a wide use of computing comprehensive Gröbner systems can be used to compute all the roots of b-functions and relevant holonomic D-modules. Furthermore, with our implementation, effective methods are illustrated for computing holonomic D-modules associated with hypersurface singularities. It is shown that the proposed algorithm is full of versatility.

Comprehensive Gröbner systems in PBW algebras, Bernstein–Sato ideals and holonomic D -modules

We present software packages for the holonomic gradient method (HGM). These packages compute normalizing constants and the probabilities of some regions. While many algorithms which compute in- tegrals over high-dimensional regions utilize the Monte-Carlo method, our HGM utilizes algorithms for solving ordinary differential equations such as the Runge-Kutta-Fehlberg method. As a result, our HGM can evaluate many integrals with a high degree of accuracy and moderate computational time. The source code of our packages is distributed on our web page (12).

A system of hypergeometric differential equations in two variables of rank 9

We are developing linear algebra packages on Risa/Asir, a computer algebra system. The aim is to provide programs for efficiently and exactly solving eigenproblems on the computer algebra system for large scale square matrices over integers or rational numbers. The software package consists of some programs. The followings are currently prepared for solving eigenproblems: computing eigenspaces, the spectral decomposition, Jordan chains and minimal annihilating polynomials.

An Extension and Efficient Calculation of the Horner’s Rule for Matrices

Abstract A computation method of comprehensive Grobner systems is introduced in Poincare–Birkhoff–Witt (PBW) algebras and applications to Bernstein–Sato ideals, holonomic D-modules for parametric cases are considered. The proposed method provides in particular holonomic D-modules associated with primary components of Bernstein–Sato ideals. Furthermore, with our implementation, effective methods are illustrated for computing b-functions of μ-constant deformations of hypersurface isolated singularities by utilizing holonomic D-modules. High versatility of the proposed method is also illustrated by several examples.

THE DESIGN AND IMPLEMENTATION OF OpenXM-RFC 100 AND 101

We study a hypergeometric function in two variables and a system of hypergeometric differential equations associated with this function. This is a regular holonomic system of rank

Holonomic rank of A-hypergeometric differential-difference equations

9