Validated numerics
Validated numerics (or reliable computation) is a numerical analysis with mathematically strict error evaluation.[1][2] In order to do so, a technology called interval arithmetic is used.[3][4][5] Validated numerics is needed for the following reasons.
- It is difficult to avoid numerical errors in numerical computation, and computation without error evaluation may cause unfortunate results.
- It can be applied to computer-assisted proofs for mathematical problems.[6][7]
One of the most known implementation of validated numerics is INTLAB[2][3][8][9] (Interval Laboratory). INTLAB was used to create other numerical libraries, and it was also used to solve the Hundred-dollar, Hundred-digit Challenge problems.[10]
References
- ↑ Tucker, Warwick (2011). Validated Numerics: A Short Introduction to Rigorous Computations. Princeton University Press.
- ↑ 2.0 2.1 Rump, S. M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287-449.
- ↑ 3.0 3.1 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.
- ↑ Mayer, G. (2017). Interval analysis: and automatic result verification. Walter de Gruyter GmbH & Co KG.
- ↑ Meyer, K. R., & Schmidt, D. S. (Eds.). (2012). Computer aided proofs in analysis. Springer Science & Business Media.
- ↑ M. Nakao, M. Plum, Y. Watanabe (2019) Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations (Springer Series in Computational Mathematics).
- ↑ S.M. Rump: INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages 77-104. Kluwer Academic Publishers, Dordrecht, 1999.
- ↑ Hargreaves, G. I. (2002). Interval analysis in MATLAB. Numerical Algorithms, (2009.1).
- ↑ Bornemann, F., Laurie, D., & Wagon, S. (2004). The SIAM 100-digit challenge: a study in high-accuracy numerical computing. Society for Industrial and Applied Mathematics.
Related pages
- Affine arithmetic
- Kantorovich theorem (A theorem frequently used in the context of validated numerics)
- Numerical digit
- Numerical integration
- Numerical linear algebra
- Numerical methods for ordinary differential equations
- Numerical methods for partial differential equations
- International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics
Other websites
- Validated Numerics for Pedestrians
- Reliable Computing, An open electronic journal devoted to validated numerics.