Interval arithmetic

Interval arithmetic is a specific type of computer arithmetic for (mathematical) intervals.Mayer, G. (2017). Interval analysis: and automatic result verification. Walter de Gruyter GmbH & Co KG.Moore, R. E., Kearfott, R. B., & Cloud, M. J. (2009). Introduction to Interval Analysis. Society for Industrial and Applied Mathematics.Alefeld, G., & Mayer, G. (2000). Interval analysis: theory and applications. Journal of Computational and Applied Mathematics, 121(1-2), 421-464. 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:

 Addition: 
 Subtraction: 
 Multiplication: 
 Division:

where

Applications
Interval arithmetic is mainly used in the field of validated numerics.Tucker, W. (2011). Validated Numerics: A Short Introduction to Rigorous Computations. Princeton University Press. It is also used in other technical areas.Jaulin, L. Kieffer, M., Didrit, O. Walter, E. (2001). Applied Interval Analysis. Berlin: Springer.

Implementations

Since the birth of interval arithmetic, many experts have made interval arithmetic programs. The most famous works are INTLAB (made with MATLAB),S.M. Rump: INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages 77-104. Kluwer Academic Publishers, Dordrecht, 1999. arb,Johansson, F. (2017). Arb: efficient arbitrary-precision midpoint-radius interval arithmetic. IEEE Transactions on Computers, 66(8), 1281-1292. JuliaIntervals,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.ValidatedNumerics.jl: a new framework in Julia, David P. Sanders and Luis Benet, SCAN 2018. and kv.Overview of kv – a C++ library for verified numerical computation, Masahide Kashiwagi, SCAN 2018.

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 NumericsScientific 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.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 Technologies12th 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 (IEEE) (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

Other websites
 Interval arithmetic (Wolfram Mathworld)
 Interval Methods from Arnold Neumaier , University of Vienna

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

Libraries
 INTLAB, Institute for Reliable Computing 
 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 

Category:Numerical analysis
Category:Computer science
Category:Arithmetics