Discrete-Time Integrable Systems and Numerical Algorithms

离散时间可积系统和数值算法

基本信息

批准号：
09440077
负责人：
NAKAMURA Yoshimasa
金额：
$ 3.26万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

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.

在1997年给出了以下结果。得出了简单的摆和不对称振荡器的首次离散，就海洛塔离散程序而言。在这里，这些连续的运动方程在相空间中具有分离器。由此产生的离散时间系统还具有隔板和保守数量。事实证明，与saparatorix相对应的保守数量的值非常等于原始连续时间系统的值。其次，制定了用于求解单个非线性方程的Steffensen迭代方法的新扩展，其收敛速率为k+ 1阶。迭代函数是通过使用Hankel确定因素比例来定义的。 EPSILON -ALGORITHM的使用会降低计算复杂性。对于开普勒方程的特殊情况，映射的数量实际上减少了。在1998年，这表明高斯的算术算法可以将其视为具有椭圆形theta函数函数解决方案和保守数量的离散时间可集成系统。从这个观察开始，首席研究员引入了一种算术谐波平均算法，该算法是一种可集成的离散时间可集成系统。虽然无限情况下的算术谐波平均算法被证明是一种混乱的动力学，它与伯努利的转移相结合。最后，建立了算术谐波平均值算法的扩展，以确定的确定对称矩阵的空间，即凸riemannian歧管。作为应用程序，设计了用于计算正定矩阵的平方根的算法，该算法的设计具有二次收敛速率。