A Step Size Control for Lohner's Enclosure Algorithm for Ordinary Differential Equations with Initial Conditions

Abstract

Lohner's enclosure algorithm for ordinary differential equations with initial conditions is supplemented by an automatic control of the step size. In this chapter, a control has been developed mainly in view of the computability of the upper and lower bounds of an enclosure in a close neighborhood of a pole in a restricted three-body problem. Applications to other problems are investigated in the chapter. For any system of equations, the practical determination of the value(s) of a true solution rests on the execution of a suitable algorithm. Rounding errors are then unavoidable, and there are additional procedural errors if the algorithm is chosen as a truncation of an infinite sequence of arithmetic operations. If there are numerical errors of these kinds, an algorithm delivers only an approximation of the value(s) of the unknown true solution. Applications of Lohner's enclosure algorithm are particularly important in the case of perturbation-sensitive neighborhoods in phase spaces. Controls of the kind presented in the chapter are instrumental for the practical computability of enclosures.