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Variants of the general interval power function

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Abstract

The article gives an overview of power function variants and compares three differently extensive versions for use in interval arithmetics. Aiming at the general power function for inclusion in interval libraries, which is defined for as many pairs of base and exponent as possible, we observe that the definition of a power for each such pair depends on the context. This problem eventually comes up at the point where reasonable doubts about the definition \(0^0\) and powers with negative base in correlation with a non-integral exponent arise. We come up with several variants serving distinct purposes, yet also recommend a general-purpose power function, which is unique for being restricted to integral exponents on the domain of negative bases. Three different variants of interval power functions satisfy all meaningful general exponentiation needs. We provide a unified treatment to handle the variants and present valuable implementation techniques for the computation of these interval functions along with a mathematic foundation.

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Notes

  1. 1.

    Private communication with Dan Zuras.

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Acknowledgments

We thank the anonymous referees for their detailed and helpful comments.

Author information

Correspondence to Marco Nehmeier.

Additional information

Communicated by V. Kreinovich.

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Heimlich, O., Nehmeier, M. & Wolff von Gudenberg, J. Variants of the general interval power function. Soft Comput 17, 1357–1366 (2013). https://doi.org/10.1007/s00500-013-1008-8

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Keywords

  • Power function
  • Interval arithmetic
  • Interval functions