GRKLib: a Guaranteed Runge Kutta Library

Abstract

In this article, we describe a new library for computing guaranteed bounds of the solutions of Initial Value Problems (IVP). Given an initial value problem and an end point, our library computes a sequence of approximation points together with a sequence of approximation errors such that the distance to the true solution of the IVP is below these error terms at each approximation point. These sequences are computed using a classical Runge-Kutta method for which truncation and roundoff errors may be over-approximated. We also compute the propagation of local errors to obtain an enclosure of the global error at each computation step. These techniques are implemented in a C++ library which provides an easy-to-use framework for the rigorous approximation of IVP. This library implements an error control technique based on step size reduction in order to reach a certain tolerance on local errors.