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Verified High-Order Inversion of Functional Dependencies and Interval Newton Methods

Abstract

A new method for computing verified enclosures of the inverses of given functions over large domains is presented. The approach is based on Taylor Model methods, and the sharpness of the enclosures scales with a high order of the domain. These methods have applications in the solution of implicit equations and the Taylor Model based integration of Differential Algebraic Equations (DAE) as well as other tasks where obtaining verified high-order models of inverse functions is required.

The accuracy of Taylor model methods has been shown to scale with the (n+1)-st order of the underlying domain, and as a consequence, they are particularly well suited to model functions over relatively large domains. Moreover, since Taylor models can control the cancellation and dependency problems that often affect regular interval techniques, the new method can successfully deal with complicated multidimensional problems. As an application of these new methods, a high-order extension of the standard Interval Newton method that converges approximately with the (n+1)-st order of the underlying domain is developed.

Several examples showing various aspects of the practical behavior of the methods are given.

M. Berz, J. Hoefkens, Reliable Computing 7(5) (2001) 379-398

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