1998 Fiscal Year Final Research Report Summary
Discrete-Time Integrable Systems and Numerical Algorithms
| Project/Area Number |
09440077
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Osaka University |
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Principal Investigator |
NAKAMURA Yoshimasa Graduate School of Engineering Science, Osaka University, 大学院・基礎工学研究科, 教授 (50172458)
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| Co-Investigator(Kenkyū-buntansha) |
KAJIWARA Kenji Faculty of Engineering, Doshisha University, Associate Professor, 工学部, 助教授 (40268115)
HIROTA Ryogo Faculty of Engineering and Science, Waseda University, Professor, 理工学部, 教授 (00066599)
NAGAI Hideo Graduate School of Engineering Science, Professor, 大学院・基礎工学研究科, 教授 (70110848)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Discrete-Time Integrable Systems / Algorithms / Simple Pendulum / Steffensen Iteration / Arithmetic-Geometric Mean Algorithm |
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| Research Abstract |
In 1997 the following results are given. First time discretizations of the simple pendulum and asymmetric oscillator in terms of Hirota's discretization procedure are derived. Here these continuous time equations of motion have separatorix in the phase space. The resulting discrete time systems also have separatorix and conserved quantities. It is proved that the value of the conserved quantity corresponding to the separatorix is remarkable equal to that of the original continuous time system. Secondly, a new extension of the Steffensen iteration method for solving a single nonlinear equation is formulated whose convergence rate is of order k+ 1. The iteration function is defined by using a ratio of Hankel determinants. The use of epsilon -algorithm diminishes the computational complexity. For a special case of the Kepler equation, the numbers of mappings are actually decreased.
In 1998, it is shown that Gauss' algorithm for arithmetic-geometric mean can be regarded as a discrete-time integrable system having an elliptic theta function solution and a conserved quantity. Starting from this observation the head investigator introduces an arithmetic-harmonic mean algorithm which is an integrable discrete time integrable system. While the arithmetic-harmonic mean algorithm in infinite case is proved to be a chaotic dynamics which is conjugate to the Bernoulli shift. Finally, an extension of the arithmetic-harmonic mean algorithm to the space of positive definite symmetric matrices, a convex Riemannian manifold, is established. As an application an algorithm for computing square root of a positive definite matrix is designed which has a quadratic convergence rate.
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