Abstract
This paper gives a brief sketch of the development of interval arithmetic. Early books consider interval arithmetic for closed and bounded real intervals. It was then extended to unbounded real intervals. Considering \(-\infty \) and \(+\infty \) only as bounds but not as elements of unbounded real intervals leads to an exception-free calculus. Formulas for computing the lower and the upper bound of the interval operations including the dot product are independent of each other. On the computer high speed can and should be obtained by computing both bounds in parallel and simultaneously. Another increase of speed and accuracy can be obtained by computing dot products exactly. Arithmetic for closed real intervals even can be extended to open and half-open real intervals, to connected sets of real numbers. Also this leads to a calculus that is free of exceptions.
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Notes
- 1.
The number of atoms in the universe is less than \(10^{80}\).
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Acknowledgements
The author owes thanks to Goetz Alefeld and Gerd Bohlender for useful comments on the paper.
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Kulisch, U.W. (2020). High Speed Exception-Free Interval Arithmetic, from Closed and Bounded Real Intervals to Connected Sets of Real Numbers. In: Kosheleva, O., Shary, S., Xiang, G., Zapatrin, R. (eds) Beyond Traditional Probabilistic Data Processing Techniques: Interval, Fuzzy etc. Methods and Their Applications. Studies in Computational Intelligence, vol 835. Springer, Cham. https://doi.org/10.1007/978-3-030-31041-7_18
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