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

In: Developments in Reliable Computing

Author

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  • Nedialko S. Nedialkov

    (University of Toronto, Department of Computer Science)

  • Kenneth R. Jackson

    (University of Toronto, Department of Computer Science)

Abstract

To date, the only effective approach for computing guaranteed bounds on the solution of an initial value problem (IVP) for an ordinary differential equation (ODE) has been interval methods based on Taylor series. This paper derives a new approach, an interval Hermite-Obreschkoff (IHO) method, for computing such enclosures. Compared to interval Taylor series (ITS) methods, for the same stepsize and order, our IHO scheme has a smaller truncation error, better stability, and requires fewer Taylor coefficients and high-order Jacobians. 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 function of the underlying formula, as in a standard method for an IVP for an ODE, but also by the associated formula for the truncation error.

Suggested Citation

  • Nedialko S. Nedialkov & Kenneth R. Jackson, 1999. "An Interval Hermite-Obreschkoff Method for Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation," Springer Books, in: Tibor Csendes (ed.), Developments in Reliable Computing, pages 289-310, Springer.
    • Handle: RePEc:spr:sprchp:978-94-017-1247-7_23
    DOI: 10.1007/978-94-017-1247-7_23
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