. Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated 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. The authors survey Taylor series methods for validated solutions of IVPs for ODEs, describe several such methods in a common framework, and identify areas for future research. Key words. initial value problems, ordinary differential equations, interval arithmetic, Taylor series methods. AMS subject classifications. 65L05, 65G10, 65L60. 1. Introduction. We consider validated numerical methods for the solution of the autonomous initial-value problems (IVPs) y 0 (t) = f(y); y(t 0 ) = y 0 ; (1.1) where t 2 [t 0 ; T ] for some T ? t 0 . Here t 0 ; T 2 R,f 2 C k\Gamma1 (D), D ` R n is an open set, f : D ! R n , and y 0 2 D. For expositional c...

Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated (also 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 computing 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...

Hamming once said, "The purpose of computing is insight, not numbers." If that is so, then the speed of our computers should be measured in insights per year, not operations per second. One key insight we wish from nearly all computing in engineering and scientific applications is, "How accurate is the answer?" Standard numerical analysis has developed techniques of forward and backward error analysis to help provide this insight, but even the best codes for computing approximate answers can be fooled. In contrast, validated computation ffl checks that the hypotheses of appropriate existence and uniqueness theorems are satisfied, ffl uses interval arithmetic with directed rounding to capture truncation and rounding errors in computation, and ffl organizes the computations to obtain as tight an enclosure of the answer as possible. These notes for a series of lectures at the VI-th SERC Numerical Analysis Summer School, Leicester University, apply the principles of validated computatio...