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Compared to standard numerical methods for initial value problems (IVPs) for ordinary diierential 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… (More)

Compared to standard numerical methods for initial value problems (IVPs) for ordinary diierential 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… (More)

This paper is one of a series underpinning the authors' DAETS code for solving DAE initial value problems by Taylor series expansion. First, building on the second author's structural analysis of DAEs (BIT 41 (2001) 364–394), it describes and justifies the method used in DAETS to compute Taylor coefficients (TCs) using automatic differentiation. The DAE may… (More)

We overview the current state of interval methods and software for computing bounds on solutions in initial value problems (IVPs) for ordinary differential equations (ODEs). We introduce the VNODE-LP solver for IVP ODEs, a successor of the author's VNODE package. VNODE-LP is implemented entirely using literate programming. A major goal of the VNODE-LP work… (More)

The numerical solution of initial value problems (IVPs) for ODEs is one of the fundamental problems in computation. Today, there are many well-established algorithms for solving IVPs. However, traditional integration methods usually provide only approximate values for the solution. Precise error bounds are rarely available. The error estimates, which are… (More)

We investigate solution techniques for numerical constraint satisfaction problems and validated numerical set integration methods for computing reachable sets of nonlinear hybrid dynamical systems in presence of uncertainty. To use interval simulation tools with higher dimensional hybrid systems, while assuming large domains for either initial continuous… (More)

In interval computations, the range of each intermediate result r is described by an interval r. To decrease excess interval width, we can keep some information on how r depends on the input x = (x 1 ; : : : ; x n). There are several successful methods of approximating this dependence; in these methods, the dependence is approximated by linear functions… (More)