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When Is the Product of Intervals Also an Interval?

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Reliable Computing

Abstract

Interval arithmetic is based on the fact that for intervals on the real line, the element-wise product of two intervals is also an interval. This property is not always true: e.g., it is not true if we consider intervals on the set of integers instead of intervals on the set of real numbers. When is an element-wise product or a sum of two intervals always an interval? In this paper, we analyze this problem in a general algebraic setting: we need the corresponding algebraic structures to have (related) addition, multiplication, and order; thus, we consider (consistently) ordered rings. We describe all consistently ordered rings for which the element-wise product and sum of two intervals are always intervals.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Texas at El Paso, El Paso, TX, 79968, USA, e-mail

    Olga Kosheleva

  2. Koenestraat 10, NL 3958 XG, Amerongen, The Netherlands

    Piet G. Vroegindeweij

Authors
  1. Olga Kosheleva
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  2. Piet G. Vroegindeweij
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Kosheleva, O., Vroegindeweij, P.G. When Is the Product of Intervals Also an Interval?. Reliable Computing 4, 179–190 (1998). https://doi.org/10.1023/A:1009937210234

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  • DOI: https://doi.org/10.1023/A:1009937210234

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