Reliable Computing 7: 449–465, 2001.
2001 Kluwer Academic Publishers. Printed in the Netherlands.
An Effective High-Order Interval Method for
Validating Existence and Uniqueness of the
Solution of an IVP for an ODE
NEDIALKO S. NEDIALKOV
Department of Computing and Software, McMaster University, Hamilton, Ontario, L8S 4L7,
Canada, e-mail: firstname.lastname@example.org
KENNETH R. JACKSON
Department of Computer Science, University of Toronto, Toronto, Ontario, M5S 3G4, Canada,
JOHN D. PRYCE
Computer Information Systems Engineering Department, Cranﬁeld University, RMCS Shrivenham,
Swindon SN6 8LA, UK, e-mail: email@example.comﬁeld.ac.uk
(Received: 6 December 1999; accepted: 15 January 2001)
Abstract. Validated methods for initial value problems for ordinary differential equations produce
bounds that are guaranteed to contain the true solution of a problem. When computing such bounds,
these methods verify that a unique solution to the problem exists in the interval of integration and
compute a priori bounds for the solution in this interval. A major difﬁculty in this veriﬁcation phase
is how to take as large a stepsize as possible, subject to some tolerance requirement. We propose a
high-order enclosure method for proving existence and uniqueness of the solution and computing a
We consider the set of autonomous initial value problems (IVPs)
(D), k ≥ 2,
D. The condition (1.2) permits the initial
) to be in an interval, rather than specifying a particular value. We assume
that the representation of
contains only a ﬁnite number of constants, variables,
This work was supported in part by the Natural Sciences and Engineering Research Coun-
cil of Canada, the Information Technology Research Centre of Ontario, and Communications and
Information Technology Ontario.
Most of this work was done when N. S. Nedialkov was at the Department of Computer Science
of the University of Toronto.