Experiments with Range Computations Using Extrapolation

Experiments with Range Computations Using Extrapolation The natural interval extension (NIE) used widely in interval analysis has the first order convergence property, i.e., the excess width of the range enclosures obtained with the NIE goes down at least linearly with the domain width. Here, we show how range approximations of higher convergence orders can be obtained from the sequence of range enclosures generated with the NIE and uniform subdivision. We combine the well-known Richardson Extrapolation Process (Sidi, A., Practical Extrapolation Methods, Cambridge University Press, Cambridge, 2003) with Brezinski’s error control method (Brezinski, C., Error Control in Convergence Acceleration Processes, IMA J. Nunerical Analysis 3 (1983), pp. 65–80) to generate non-validated range approximations to the true range. We demonstrate the proposed method for accelerating the convergence orders on several multidimensional examples, varying from one to six dimensions. These numerical experiments also show that considerable computational savings can be obtained with the proposed procedure. However, the theoretical basis of the proposed method remains to be investigated. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

Experiments with Range Computations Using Extrapolation

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2007 by Springer Science + Business Media B.V.
Subject
Mathematics; Numeric Computing; Mathematical Modeling and Industrial Mathematics; Approximations and Expansions; Computational Mathematics and Numerical Analysis
ISSN
1385-3139
eISSN
1573-1340
D.O.I.
10.1007/s11155-006-9022-5
Publisher site
See Article on Publisher Site

Abstract

The natural interval extension (NIE) used widely in interval analysis has the first order convergence property, i.e., the excess width of the range enclosures obtained with the NIE goes down at least linearly with the domain width. Here, we show how range approximations of higher convergence orders can be obtained from the sequence of range enclosures generated with the NIE and uniform subdivision. We combine the well-known Richardson Extrapolation Process (Sidi, A., Practical Extrapolation Methods, Cambridge University Press, Cambridge, 2003) with Brezinski’s error control method (Brezinski, C., Error Control in Convergence Acceleration Processes, IMA J. Nunerical Analysis 3 (1983), pp. 65–80) to generate non-validated range approximations to the true range. We demonstrate the proposed method for accelerating the convergence orders on several multidimensional examples, varying from one to six dimensions. These numerical experiments also show that considerable computational savings can be obtained with the proposed procedure. However, the theoretical basis of the proposed method remains to be investigated.

Journal

Reliable ComputingSpringer Journals

Published: Nov 22, 2006

References

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