Validated numerics

Validated numerics (or reliable computation) is a numerical analysis with mathematically strict error evaluation. In order to do so, a technology called interval arithmetic is used. 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.

One of the most known implementation of validated numerics is INTLAB (Interval Laboratory). INTLAB was used to create other numerical libraries, and it was also used to solve the Hundred-dollar, Hundred-digit Challenge problems.

References

↑ Tucker, Warwick (2011). Validated Numerics: A Short Introduction to Rigorous Computations. Princeton University Press.
1 2 Rump, S. M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287-449.
1 2 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 Archived 2021-05-09 at the Wayback Machine
Reliable Computing, An open electronic journal devoted to validated numerics.

This short article about mathematics can be made longer. You can help Wikipedia by adding to it.

[1] Tucker, Warwick (2011). Validated Numerics: A Short Introduction to Rigorous Computations. Princeton University Press.

[acta-2] 1 2 Rump, S. M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287-449.

[moore-3] 1 2 Moore, R. E., Kearfott, R. B., & Cloud, M. J. (2009). Introduction to Interval Analysis. Society for Industrial and Applied Mathematics.

[4] Alefeld, G., & Mayer, G. (2000). Interval analysis: theory and applications. Journal of Computational and Applied Mathematics, 121(1-2), 421-464.

[5] Mayer, G. (2017). Interval analysis: and automatic result verification. Walter de Gruyter GmbH & Co KG.

[6] Meyer, K. R., & Schmidt, D. S. (Eds.). (2012). Computer aided proofs in analysis. Springer Science & Business Media.

[7] M. Nakao, M. Plum, Y. Watanabe (2019) Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations (Springer Series in Computational Mathematics).

[rump-8] S.M. Rump: INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages 77-104. Kluwer Academic Publishers, Dordrecht, 1999.

[har-9] Hargreaves, G. I. (2002). Interval analysis in MATLAB. Numerical Algorithms, (2009.1).

[10] 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.