A Limitation for Underestimation Via Twin Arithmetic

A Limitation for Underestimation Via Twin Arithmetic Computing an enclosure for the range of a rational function over an interval is one of the main goals of interval analysis. One way to obtain such an enclosure is to use interval arithmetic evaluation of a formula for the function. Often one would like to check how close the overestimation is to the correct range. Kreinovich, Nesterov, and Zheludeva (Reliable Computing 2(2) (1996)) suggested a new kind of twin arithmetic which produces a twin of intervals at the same time: the usual enclosure, i.e., an interval which is an overestimation for the range, and an interval which is contained in the range, i.e. an interval which is an underestimation for the range. We show in this paper that in certain cases the computed inner interval is much smaller than the correct range. For example, if the function has the same value in the two endpoints of the interval then the inner interval is either empty or contains only one point. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

A Limitation for Underestimation Via Twin Arithmetic

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2001 by Kluwer Academic Publishers
Subject
Mathematics; Numeric Computing; Approximations and Expansions; Computational Mathematics and Numerical Analysis; Mathematical Modeling and Industrial Mathematics
ISSN
1385-3139
eISSN
1573-1340
D.O.I.
10.1023/A:1011474231995
Publisher site
See Article on Publisher Site

Abstract

Computing an enclosure for the range of a rational function over an interval is one of the main goals of interval analysis. One way to obtain such an enclosure is to use interval arithmetic evaluation of a formula for the function. Often one would like to check how close the overestimation is to the correct range. Kreinovich, Nesterov, and Zheludeva (Reliable Computing 2(2) (1996)) suggested a new kind of twin arithmetic which produces a twin of intervals at the same time: the usual enclosure, i.e., an interval which is an overestimation for the range, and an interval which is contained in the range, i.e. an interval which is an underestimation for the range. We show in this paper that in certain cases the computed inner interval is much smaller than the correct range. For example, if the function has the same value in the two endpoints of the interval then the inner interval is either empty or contains only one point.

Journal

Reliable ComputingSpringer Journals

Published: Oct 3, 2004

References

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