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2002 Fiscal Year Final Research Report Summary

Research of Algorithms in terms of Information Geometry Structure and Discrete Time Integrable Systems

Research Project

Project/Area Number 12440025
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKYOTO UNIVERSITY (2001-2002)
Osaka University (2000)

Principal Investigator

NAKAMURA Yoshimasa  Kyoto University, Graduate School of Informatics, Professor, 情報学研究科, 教授 (50172458)

Co-Investigator(Kenkyū-buntansha) EGUCHI Shinto  Ministry of Education, Culture, Sports, Science and Technology, The Institute of Statistical Mathematics, Professor, 統計数理研究所, 教授 (10168776)
TSUJIMOTO Satoshi  Kyoto University, Graduate School of Informatics, Lecturer, 情報学研究科, 講師 (60287977)
OHARA Atsumi  Osaka University, Graduate School of Engineering Science, Associate Professor, 基礎工学研究科, 助教授 (90221168)
OHTA Yasuhiro  Hiroshima University, Graduate School of Engineering, Research Assistant, 工学研究科, 助手 (10213745)
Project Period (FY) 2000 – 2002
Keywordsdiscrete-time integrable systems / numerical algorithms / parallel computing / information geometry / arithmetic harmonic mean algorithm
Research Abstract

Nakamura formulated the arithmetic-harmonic mean (AHM) algorithm which converges to the square of a given positive matrix by using a successive use of arithmetic mean and harmonic mean operations on the space of positive matrices. From the viewpoint of information geometry the AHM algorithm plots the midpoints of mutually dual geodesics which connect 2-points on the space. Ohara generalized the space of positive matrices to that of symmetric cones and made clear the information geometry structure of the generalized space, for example, these mean operations determine midpoints of geodesics on it.
When the positivity is lost, the AHM algorithm does not converge in general. Kondo and Nakamura found that the n-th terms of the recurrence relation takes a determinantal form. The solvable logistic map has a similar property. Based on the determinantal expression and solvability it is shown that the corresponding Lyapunov exponent of the recurrence relation is positive without using invariant measure and computation of integrations.
Nakamura and Tsujimoto started to investigate parallel computing by the discrete-time Toda equation (the qd algorithm), a prototype of discrete-time integrable systems which work as numerical algorithms. They construct a parallel computer system with a dispersive memory and two CPUs. Decomposing the qd table from side to side they computed two pieces by each CPU. Then it is shown that the computation time of a tri-diagonal matrix eigenvalue problem decreases to almost 60 percent of that of one CPU case. Moreover by decomposing the qd table aslant they showed that the parallel computation rate becomes better.
These results will be useful for designing new numerical algorithms in terms of discrete-time integrable systems.

  • Research Products

    (26 results)

All Other

All Publications (26 results)

  • [Publications] A.Mukaihira, Y.Nakamura: "Integrable discretization of the modified KdV equation and applications"Inverse Problems. Vol.16. 413-424 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Tsujimoto, Y.Nakamura, M.Iwasaki: "Discrete Lotka-Volterra system computes singular values"Inverse Problems. Vol.17. 53-58 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Mukaihira, Y.Nakamura: "Schur flow for orthogonal polynomials on the unit circle and its integrable discretization"Journal of Computational and Applied Mathematics. Vol.139. 75-94 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Minesaki, Y.Nakamura: "The discrete relativistic Toda molecule equation and a Pad\'e approximation algorithm"Numerical Algorithms. Vol.27. 219-235 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Nakamura: "Continued fractions and integrable systems"Centre de Recherches Math'ematiques Proceedings & Lecture Notes. Vol.31. 153-163 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Kondo, Y.Nakamura: "Determinantal solutions of solvable chaotic systems"Journal of Computational and Applied Mathematics. Vol.145. 361-372 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Minesaki, Y.Nakamura: "A new discretization of the Kepler motion which conserves the Runge-Lenz vector"Physics Letters A. Vol.306. 127-133 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Iwasaki, Y.Nakamura: "Convergence of solution of the discrete Lotka-Volterra system"Inverse Problems. Vol.18. 1569-1578 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Uohashi, A.Ohara: "Jordan Algebras and Dual Affine Connections on Symmetric Cones"Positivity. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] R.Hirota, M.Iwao, S.Tsujimoto: "Soliton equations exhibiting Pfaffian solutions"Glasgow Mathematical Journal. Vol.43A. 33-41 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] C.R.Gilson, J.J.C.Nimmo, S.Tsujimoto: "Pfaffianization of the discrete KP equation"Journal of Physics A. Vol.34. 10569-10575 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Tsujimoto: "On a discrete analogue of the two-dimensional Toda lattice hierarchy"Publications of RIMS, Kyoto Univ.. Vol.38. 113-133 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 中村 佳正編著: "第5章 可積分系とアルゴリズム"可積分系の応用数理(中村執筆分担)(裳華房). 171-223 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 甘利俊一, 外山敬介編: "9.3節 主成分分析"脳科学大事典(中村執筆分担)(朝倉書店). 821-823 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A. Mukaihira and Y. Nakamura: "Integrable discretization of the modified KdV equation and applications"Inverse Problems. Vol.16. 413-424 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Tsujimoto, Y. Nakamura and M. Iwasaki: "Discrete Lotka-Volterra system computes singular values"Inverse Problems. Vol. 17. 53-58 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A. Mukaihira and Y. Nakamura: "Schur flow for orthogonal polynomials on the unit circle and its integrable discretization"Journal of Computational and Applied Mathematics. Vol.139. 7594 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y, Minesaki and Y. Nakamura: "The discrete relativistic Toda molecule equation and a Pad\' e approximation algorithm"Numerical Algorithms. Vol.27. 219-235 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Nakamura: "Continued fractions and integrable systems"Centre de Recherches Math'ematiques Proceedings & Lecture Notes. Vol. 31. 153-163 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. Kondo and Y. Nakamura: "Determinantal solutions of solvable chaotic systems"Journal of Computational and Applied Mathematics. Vol.145. 361-372 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Minesaki and Y. Nakamura: "A new discretization of the Kepler motion which conserves the Runge-Lenz vector"Physics Letters A.. Vol.306. 127-133 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. Iwasaki and Y. Nakamura: "Convergence of solution of the discrete Lotka-Volterra system"Inverse Problems. Vol.18. 1569-1578 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. Uohashi and A. Ohara: "Jordan algebras and dual affine connections on symmetric cones"Positivity. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] R. Hirota, M. Iwao and S. Tsujimoto: "Soliton equations exhibiting Pfaffian solutions"Glasgow Mathematical Journal. Vol.43A. 33-41 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] C. R. Gilson, J. J. C. Nimmo and S. Tsujimoto: "Pfaffianization of the discrete KP equation"Journal of Physics A. Vol.34. 10569-10575 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Tsujimoto: "On a discrete analogue of the two-dimensional Toda lattice hierarchy"Publications of RIMS, Kyoto Univ.. Vol.38. 113-133 (2002)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2004-04-14  

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