An Interval Hermite-Obreschkoff Method for Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation

Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation Nediako Stoyanov Nedialkov Doctor of Philosophy Graduate Department of Computer Science University of Toronto 1999 Compared to standard numerical methods for initia1 value problems (IVPs) for ordinary differential equations (ODEs), validated (aiso called interval) methods for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced. To date, the only effective approach for computing guaranteed enclosures of the solution of an IVP for an ODE has been interval methods based on Taylor series. This thesis derives a new approach, an interval Hermite-Obreschkoff (IHO) method, for cornputing such enclosures. Compared to interval Taylor series (ITS) methods, for the same order and stepsize, our IHO scheme has a smaller truncation error and better stability. As a result, the IHO method allows Iarger stepsizes than the corresponding ITS methods, thus saving computation time. In addition, since fewer Taylor coefficie~its are required by IHO than ITS methods, the IHO method performs better than the ITS methods when the function for cornputing the right side contains many terms. The stability properties of the ITS and IHO methods are investigated. We show as an important by-product of this analysis that the stability of an interval method is determined not only by the stability funct ion of the underlying formula, as in a standard method for an IVP for an ODE, but aIso by the associated formula for the truncation error. This thesis also proposes a Taylor series rnethod for validating existence and uniqueness of the solution, a simple stepsize control, and a program structure appropriate for a large class of validated ODE solvers.