# Interval arithmetic

Interval arithmetic is a specific type of computer arithmetic for (mathematical) intervals.[1][2][3] It is mainly used for the automated detection of errors. There is a value, which isn't known exactly, but which can be given by an interval: the value is known to be inside the interval. There's a function to calculate the error. This function should be given the unknown value, but instead can only be given the interval. The result is a function which will map intervals.

Interval arithmetic can be used to treat rounding errors, or to treat insecurities with measurements: Each measurement has a certain error, which cannot be determined exactly.

## Definition

For real intervals (interval of real numbers), interval arithmetic is defined as follows:[1][2][3]

• Addition: ${\displaystyle [x_{1},x_{2}]+[y_{1},y_{2}]=[x_{1}+y_{1},x_{2}+y_{2}]}$
• Subtraction: ${\displaystyle [x_{1},x_{2}]-[y_{1},y_{2}]=[x_{1}-y_{2},x_{2}-y_{1}]}$
• Multiplication: ${\displaystyle [x_{1},x_{2}]\cdot [y_{1},y_{2}]=[\min(x_{1}y_{1},x_{1}y_{2},x_{2}y_{1},x_{2}y_{2}),\max(x_{1}y_{1},x_{1}y_{2},x_{2}y_{1},x_{2}y_{2})]}$
• Division:
${\displaystyle {\frac {[x_{1},x_{2}]}{[y_{1},y_{2}]}}=[x_{1},x_{2}]\cdot {\frac {1}{[y_{1},y_{2}]}},}$
where
{\displaystyle {\begin{aligned}{\frac {1}{[y_{1},y_{2}]}}&=\left[{\tfrac {1}{y_{2}}},{\tfrac {1}{y_{1}}}\right]&&0\notin [y_{1},y_{2}]\\{\frac {1}{[y_{1},0]}}&=\left[-\infty ,{\tfrac {1}{y_{1}}}\right]\\{\frac {1}{[0,y_{2}]}}&=\left[{\tfrac {1}{y_{2}}},\infty \right]\\{\frac {1}{[y_{1},y_{2}]}}&=\left[-\infty ,{\tfrac {1}{y_{1}}}\right]\cup \left[{\tfrac {1}{y_{2}}},\infty \right]=[-\infty ,\infty ]&&0\in (y_{1},y_{2})\end{aligned}}}

## Applications

Interval arithmetic is mainly used in the field of validated numerics.[4] It is also used in other technical areas.[5]

## Implementations

Since the birth of interval arithmetic, many experts have made interval arithmetic programs. The most famous works are INTLAB (made with MATLAB),[6] arb,[7] JuliaIntervals,[8][9] and kv.[10]

## Community

There are several international conferences about interval arithmetic. One of the most largest meeting is the International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics[11][12][13] (usually abbreviated as SCAN). There are also SWIM (Small Workshop on Interval Methods), PPAM (International Conference on Parallel Processing and Applied Mathematics), and REC (International Workshop on Reliable Engineering Computing).

## References

1. Mayer, G. (2017). Interval analysis: and automatic result verification. Walter de Gruyter GmbH & Co KG.
2. Moore, R. E., Kearfott, R. B., & Cloud, M. J. (2009). Introduction to Interval Analysis. Society for Industrial and Applied Mathematics.
3. Alefeld, G., & Mayer, G. (2000). Interval analysis: theory and applications. Journal of Computational and Applied Mathematics, 121(1-2), 421-464.
4. Tucker, W. (2011). Validated Numerics: A Short Introduction to Rigorous Computations. Princeton University Press.
5. Jaulin, L. Kieffer, M., Didrit, O. Walter, E. (2001). Applied Interval Analysis. Berlin: Springer.
6. S.M. Rump: INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages 77-104. Kluwer Academic Publishers, Dordrecht, 1999.
7. Johansson, F. (2017). Arb: efficient arbitrary-precision midpoint-radius interval arithmetic. IEEE Transactions on Computers, 66(8), 1281-1292.
8. Sanders, D. P., Benet, L., & Kryukov, N. (2016). The julia package ValidatedNumerics. jl and its application to the rigorous characterization of open billiard models. SCAN 2016, 124.
9. ValidatedNumerics.jl: a new framework in Julia, David P. Sanders and Luis Benet, SCAN 2018.
10. Overview of kv – a C++ library for verified numerical computation, Masahide Kashiwagi, SCAN 2018.
11. Scientific Computing, Computer Arithmetic, and Validated Numerics 16th International Symposium, SCAN 2014, Würzburg, Germany, September 21-26, 2014. Revised Selected Papers. Editors: Marco Nehmeier, Jürgen Wolff von Gudenberg, Warwick Tucker. Published by Springer.
12. 13th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Verified Numerical Computations (SCAN'2008), Proceedings of a meeting held 29 September - 3 October 2008, El Paso, Texas, USA. Special volume devoted to materials presented at SCAN 2012. Published by the Institute of Computational Technologies