Discrete-Time Integrable Systems and Numerical Algorithms

1997年给出了以下结果。根据Hirota的离散化方法，首次导出了单摆和非对称振子的离散化形式。这里，这些连续的时间运动方程在相空间中是分离的。由此得到的离散时间系统也具有分离量和守恒量。证明了与分离相对应的守恒量的值与原连续时间系统的守恒量是显著相等的。其次，利用Hankel行列式之比定义了迭代函数，得到了求解单方程的Steffensen迭代法的一种新的推广形式，其收敛速度为k+1阶。采用epsilon算法，降低了计算复杂度。对于开普勒方程的一种特殊情况，映射的次数实际上是减少的.1998年，证明了算术-几何平均的Gauss算法可以看作是一个离散时间可积系统，具有椭圆式的theta函数解和守恒量.从这一观察出发，首席研究员介绍了一种算术调和平均算法，它是一个可积的离散时间可积系统。而无限情况下的算术调和平均算法被证明是一个与伯努利位移共轭的混沌动力学。最后，将算术调和平均算法推广到正定对称矩阵空间--凸黎曼流形。作为应用，设计了一个具有二次收敛速度的计算正定矩阵平方根的算法。

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.