This is a joint work with Ken Jackson

Validated, or interval methods for initial value problems (IVPs) for ordinary differential equations (ODEs) produce bounds that are guaranteed to contain the true solution of a problem. These methods have not been as popular as traditional (point) numerical methods for IVPs for ODEs. One reason is that validated methods are normally more complex and significantly more expensive than traditional methods. In addition, early interval methods for ODEs often produced unacceptably wide bounds because of the wrapping effect [2]. However, more sophisticated modern methods [1,3] cope effectively with this difficulty and produce tight bounds for a wide range of problems. Another reason is the lack of flexible, easy-to-modify, and portable software, which researchers can use to experiment easily with different techniques.

We are implementing a package called VNODE (Validated Numerical ODE) for computing enclosures that are guaranteed to contain the true solution of the IVP

y'(t) = f(y),   y(t₀) = y₀,     (1)

at points t_j : j = 1, ..., N satisfying t₀ < t₁ < ... < t_N for f sufficiently differentiable.

Validated methods use interval arithmetic [2], Taylor series expansions, and various interval techniques [1,2,3,4] to bound rounding and truncation errors in the numerical solution of (Eq.1). Usually, one step of a validated method consists of two phases: (1) validate the existence and uniqueness of the true solution of (Eq.1) on the interval [t_j,t_j+1] (j = 0, ..., N-1) and simultaneously bound it on [t_j,t_j+1], and (2) compute a tight enclosure of the solution at t_j+1.

Our objective is to build a modular, flexible, easy-to-use, and efficient package, suitable for both research and industrial applications. To build such a package, we have chosen the object-oriented approach to designing VNODE and the C++ programming language to implement it. To make the structure flexible, we have designed VNODE as a set of classes, which can be grouped into (a) classes describing the mathematical and numerical problems and the computed numerical solution, (b) classes implementing methods used for computing guaranteed bounds of the solution, and (c) classes implementing automatic generation of interval Taylor coefficients. In VNODE, a general method is implemented as a class, which has pointers to abstract classes that implement stepsize selection, validate existence and uniqueness, and compute tight enclosures. These classes serve as a general interface to particular methods. When adding a new method, as a derived class to an abstract class, all the communication is handled by the base class.

In this talk, we briefly overview validated methods for IVPs for ODEs, available software for computing validated solutions of IVPs for ODEs, and available packages for interval arithmetic and automatic generation of Taylor coefficients. Then, we describe the structure of VNODE and illustrate how it can be used. Finally, we discuss problems that we have encountered during the development of VNODE.

1. R. J. Lohner. Einschliess ung der Lösung gewöhnlicher Anfangs-- und Randwertaufgaben und Anwendungen. PhD thesis, Universität Karlsruhe, 1988.

2. R. E. Moore. Interval Analysis. Prentice-Hall, Englewood Cliffs, N.J., 1966.

3. N. S. Nedialkov. Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation. PhD thesis, Department of Computer Science, University of Toronto, Toronto, Canada, M5S 3G4. Available at http://www.cs.toronto.edu/NA/reports.html.

4. R. Rihm. Interval methods for initial value problems in ODEs. In Jürgen Herzberger, editor, Topics in Validated Computations: Proceedings of the IMACS-GAMM International Workshop on Validated Computations, University of Oldenburg, Elsevier Studies in Computational Mathematics. Elsevier, Amsterdam, New York, 1994.

MINISIMPOSIUM SESSION: 8. Software issues