Interval arithmetic

Interval arithmetic is a computer arithmetic for (mathematical) intervals^[1][2][3].

Definition

For real intervals (interval o 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

${\displaystyle {\begin{aligned}{\frac {1}{[y_{1},y_{2}]}}&=\left[{\tfrac {1}{y_{2}}},{\tfrac {1}{y_{1}}}\right]&&0\notin [y_{1},y_{2}]\\{\frac {1}{[y_{1},0]}}&=\left[-\infty ,{\tfrac {1}{y_{1}}}\right]\\{\frac {1}{[0,y_{2}]}}&=\left[{\tfrac {1}{y_{2}}},\infty \right]\\{\frac {1}{[y_{1},y_{2}]}}&=\left[-\infty ,{\tfrac {1}{y_{1}}}\right]\cup \left[{\tfrac {1}{y_{2}}},\infty \right]=[-\infty ,\infty ]&&0\in (y_{1},y_{2})\end{aligned}}}$

Details

Applications

Interval arithmetic is mainly usit i the field o validatit numerics^[4]. It is also usit i other technical areas^[5].

Implementations

Relatit page: Affine arithmetic

Syne the birth o interval arithmetic, many experts have made interval arithmetic programs. The most famous works are INTLAB (made wi MATLAB)^[6], arb^[7], JuliaIntervals^[8][9], an kv^[10].

Community

Thare are several international conferences aboot interval arithmetic. Ane o the most largest meetin is the International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics^[11][12][13]. Thare are also SWIM SWIM (Small Workshop on Interval Methods), PPAM (International Conference on Parallel Processing and Applied Mathematics), an 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
13. 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006), Proceedings of a meeting held 26-29 September 2006, Duisburg, Germany. Published by the Institute of Electrical and Electronics Engineers

External links

• Interval arithmetic (Wolfram Mathworld)
• Interval Methods from Arnold Neumaier Archived 2020-12-02 at the Wayback Machine

Workshops

• SWIM (Summer Workshop on Interval Methods)
• International Conference on Parallel Processing and Applied Mathematics

Libraries

• INTLAB, Institute for Reliable Computing Archived 2020-01-30 at the Wayback Machine
• Ball arithmetic by Joris van der Hoeven
• kv - a C++ Library for Verified Numerical Computation
• Arb - a C library for arbitrary-precision ball arithmetic
• Interval and related software Archived 2020-05-13 at the Wayback Machine