 TOPICS # Interval Arithmetic

Interval arithmetic is the arithmetic of quantities that lie within specified ranges (i.e., intervals) instead of having definite known values. Interval arithmetic can be especially useful when working with data that is subject to measurement errors or uncertainties. It can be considered a rigorous version of significance arithmetic (a.k.a., automatic precision control).

It is powerful enough to provide rigorous mathematical proofs (de la Llave 1991, Hutchings et al. 2000, Tucker 2002, Gutowski 2003), but rigor comes at a price. In particular, interval arithmetic can be slow, and often gives overly pessimistic results for real-world computations.

Floating-Point Arithmetic, Interval, Projectively Extended Real Numbers

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de la Llave, R. In Computer Aided Proofs in Analysis (Ed. K. Meyer and D. Schmidt). New York: Springer-Verlag, 1991.Marlov, S. M. In Scientific Computing and Validated Numerics (Ed. G. Alefeld; A. Frommer, and B. Lang). Berlin: Akademie Verlag, 1996.Gutowski, M. W. "Power and Beauty of Interval Methods." 20 Feb 2003. http://arxiv.org/abs/physics/0302034.Hutchings, M.; Morgan, F.; Ritoré; M.; and Ros, A. Electron. Res. Announc. Amer. Math. Soc. 6, 45, 2000.Jaulin, L.; Kieffer, M.; Didrit, O.; and Walter, É. Applied Interval Analysis. London: Springer-Verlag, 2003.Kearfott, B. R. Euromath Bull. 2, 95, 1996.Petkovič M. S.; and Petkovič, L. D. Complex Interval Arithmetic and Its Applications. Berlin: Wiley, 1998.Popova, E. D. and Ullrich, C. P. "Simplication of Symbolic-Numerical Interval Expressions." In Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation (Ed. O. Gloor). New York: ACM Press, pp. 207-214, 1998.Schenkel, A.; Wehr, J.; and Wittwer, P. Math. Phys. Electr. J. 6, 2000.Shokin, Y. I. In Scientific Computing and Validated Numerics (Ed. G. Alefeld; A. Frommer, and B. Lang). Berlin: Akademie Verlag, 1996.Trott, M. "Interval Arithmetic." §1.1.2 in The Mathematica GuideBook for Numerics. New York: Springer-Verlag, pp. 54-66, 2006. http://www.mathematicaguidebooks.org/.Tucker, W. Found. Comput. Math. 2, 53, 2002.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.

## Referenced on Wolfram|Alpha

Interval Arithmetic

## Cite this as:

Weisstein, Eric W. "Interval Arithmetic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IntervalArithmetic.html