Discrete-Time Integrable Systems and Numerical Algorithms

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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.