Rigorous error bounds for RK methods in the proof of chaotic behaviour

AbstractComplicated dynamical systems can be rigorously analysed by means of Conley index theory. Due to its partly numerical nature such an analysis necessitates bounds on the truncation and the round-off error. These are provided for explicit RK methods in the form of iteration schemes ready-made for applications. The presentation is aimed to simplify error bounds already available so that different error sources can be clearly overlooked. As an immediate application, a computer-assisted analysis elucidates the intricate dynamics of a simple mechanical system

Rigorous Error Analysis Of Numerical Algorithms Via Symbolic Computations

this paper is to propose rigorous error analysis of both the intrinsic and rounding errors based on symbolic computations. Although our main object of interest are algorithms for ODE's, the presented method is widely applicable in analysis. The method was successfuly used in [6, 7] in a computer assisted proof of chaos in the Lorenz equaitons. Unlike the interval analysis, the presented method is disjoint form the algorithm itself. This means that there is no need to rewrite the existing software and in particular the algorithms do not slow down. What is even more important, error estimates for a prescribed set of inputs may be obtained even before the algorithm itself is run. This is espacially convenient if the numerical algorithm is to be run several times for many similar inputs and the requested error bound must be in an a priori prescribed limit. In such a situation one can experiment with error bounds of various settings of the algorithm and the algorithm is run only after a setting with satisfactory error bounds is found. One should mention that the rounding error bounds obtained this way are in general not as good as those produced by the interval arithmetic. However, this is irrelevant if rounding errors are small when compared to the intrinsic error, which is usually the case. The organisation of the paper is as follows. In Section 2 we introduce representable numbers and in Section 3 we discuss binary orders of magnitude. The following section is devoted to representable arithmetic. An auxiliary function \Gamma is introduced in Section 5. Arithmetic expressions are dealt with in Section 6 and rounding error bounds for arithmetic expressions are discussed in Section 7. Section 8 presents error bounds for Runge-Kutta methods. A program in MATHEMATICA evaluatin..