Software Packages for Holonomic Gradient Method

  • Tamio Koyama
  • Hiromasa Nakayama
  • Katsuyoshi Ohara
  • Tomonari Sei
  • Nobuki Takayama
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8592)


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 integrals 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].


holonomic gradient method normalizing constant region probability Bingham prior R project 


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  1. 1.
    Bishop, C.M.: Pattern Recognition and Machine Learning. Springer (2006)Google Scholar
  2. 2.
    GNU Scientific Library,
  3. 3.
    Koyama, T., Takemura, A.: Calculation of Orthant Probabilities by the Holonomic Gradient Method, arxiv:1211.6822Google Scholar
  4. 4.
    Kume, A., Wood, A.T.A.: On the Derivatives of the Normalizing Constant of the Bingham Distribution. Statistics and Probability Letters 77, 832–837 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Miwa, T., Hayter, H.J., Kuriki, S.: The evaluation of general non-centered orthant probabilities. Journal of the Royal Statistical Society: Series B 65, 223–234 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Nakayama, H., Nishiyama, K., Noro, M., Ohara, K., Sei, T., Takayama, N., Takemura, A.: Holonomic Gradient Descent and its Application to the Fisher-Bingham Integral. Advances in Applied Mathematics 47, 639–658 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    The R: Project for Statistical Computing,
  8. 8.
    Risa/Asir, a computer algebra system,
  9. 9.
    Sei, T., Kume, A.: Calculating the Normalizing Constant of the Bingham Distribution on the Sphere using the Holonomic Gradient Method. Statistics and Computing (2013)Google Scholar
  10. 10.
  11. 11.
    Zeilberger, D.: A Holonomic Systems Approach to Special Function Identities. Journal of Computational and Applied Mathematics 32, 321–368 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Tamio Koyama
    • 1
  • Hiromasa Nakayama
    • 2
  • Katsuyoshi Ohara
    • 3
  • Tomonari Sei
    • 4
  • Nobuki Takayama
    • 5
  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyUniversity of Tokyo, JSPS Research FellowJapan
  2. 2.Department of MathematicsTokai UniversityJapan
  3. 3.Faculty of Mathematics and PhysicsKanazawa UniversityJapan
  4. 4.Department of MathematicsKeio UniversityJapan
  5. 5.Department of MathematicsKobe UniversityJapan

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