# Validated numerics

**Validated numerics**, or **rigorous computation**, **verified computation**, **reliable computation**, **numerical verification** (German: *Zuverlässiges Rechnen*) is numerics including mathematically strict error (rounding error, truncation error, discretization error) evaluation, and it is one field of numerical analysis. For computation, interval arithmetic is used, and all results are represented by intervals. Validated numerics were used by Warwick Tucker in order to solve the 14th of Smale's problems[1], and today it is recognized as a powerful tool for the study of dynamical systems[2].

## Importance

Computation without verification may cause unfortunate results. Below are some examples.

### Rump's example

In the 1980s, Rump made an example[3][4]. He made a complicated function and tried to obtain its value. Single precision, double precision, extended precision results seemed to be correct, but its plus-minus sign was different from the true value.

### Phantom solution

Breuer–Plum–McKenna used the spectrum method to solve the boundary value problem of the Emden equation, and reported that an asymmetric solution was obtained[5]. This result to the study conflicted to the theoretical study by Gidas–Ni–Nirenberg which claimed that there is no asymmetric solution[6]. The solution obtained by Breuer–Plum–McKenna was a phantom solution caused by discretization error. This is a rare case, but it tells us that when we want to strictly discuss differential equations, numerical solutions must be verified.

## Main topics

The study of validated numerics is divided into the following fields.

- Verification in numerical linear algebra
- Verification of special functions
- Verification of numerical quadrature[15][16][17]
- Verification of nonlinear equations (The Kantorovich theorem[18], Krawczyk method, interval Newton method, and the Durand–Kerner–Aberth method are studied)
- Verification for solutions of ODEs, PDEs[19] (for PDEs, knowledge of functional analysis are used[18])
- Verification of linear programming[20]
- Verification of computational geometry
- Verification at high-performance computing environment

## Tools

- INTLAB Library made by MATLAB/GNU Octave
- kv Library made by C++. This library can obtain multiple precision outputs by using GNU MPFR.
- Arb Library made by C. It is capable to rigorously compute various special functions.
- CAPD A collection of flexible C++ modules which are mainly designed to computation of homology of sets, maps and validated numerics for dynamical systems.
- JuliaIntervals on GitHub (Library made by Julia)

## Further reading

- Tucker, W. (2011). Validated Numerics: A Short Introduction to Rigorous Computations. Princeton University Press.
- Moore, R. E., Kearfott, R. B., & Cloud, M. J. (2009). Introduction to Interval Analysis. Society for Industrial and Applied Mathematics.
- Rump, S. M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287-449.

## External links

- Validated Numerics for Pedestrians
- Reliable Computing, An open electronic journal devoted to numerical computations with guaranteed accuracy, bounding of ranges, mathematical proofs based on floating-point arithmetic, and other theory and applications of interval arithmetic and directed rounding.

## See also

- computer-assisted proof
- interval arithmetic
- affine arithmetic
- automatic differentiation
- wikibooks:Numerical calculations and rigorous mathematics

### Researchers

## References

- Tucker, W. (1999). "The Lorenz attractor exists."
*Comptes Rendus de l'Académie des Sciences-Series I-Mathematics*, 328(12), 1197–1202. - ZIN ARAI, HIROSHI KOKUBU, AND PAWEÃL PILARCZYK. RECENT DEVELOPMENT IN RIGOROUS COMPUTATIONAL METHODS IN DYNAMICAL SYSTEMS.
- Rump, S. M. (1988). "Algorithms for verified inclusions: Theory and practice." In
*Reliability in computing*(pp. 109–126). Academic Press. - Loh, E., & Walster, G. W. (2002). Rump's example revisited. Reliable Computing, 8(3), 245-248.
- Breuer, B., Plum, M., & McKenna, P. J. (2001). "Inclusions and existence proofs for solutions of a nonlinear boundary value problem by spectral numerical methods." In
*Topics in Numerical Analysis*(pp. 61–77). Springer, Vienna. - Gidas, B., Ni, W. M., & Nirenberg, L. (1979). "Symmetry and related properties via the maximum principle."
*Communications in Mathematical Physics*, 68(3), 209–243. - http://www-users.math.umn.edu/~arnold//disasters/patriot.html
- ARIANE 5 Flight 501 Failure, http://sunnyday.mit.edu/nasa-class/Ariane5-report.html
- Rounding error changes Parliament makeup
- Rump, S. M. (2014). Verified sharp bounds for the real gamma function over the entire floating-point range. Nonlinear Theory and Its Applications, IEICE, 5(3), 339-348.
- Yamanaka, N., Okayama, T., & Oishi, S. I. (2015, November). Verified Error Bounds for the Real Gamma Function Using Double Exponential Formula over Semi-infinite Interval. In International Conference on Mathematical Aspects of Computer and Information Sciences (pp. 224-228). Springer.
- Johansson, F. (2019). Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms. In Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory (pp. 269-293). Springer, Cham.
- Johansson, F. (2019). Computing Hypergeometric Functions Rigorously. ACM Transactions on Mathematical Software (TOMS), 45(3), 30.
- Johansson, F. (2015). Rigorous high-precision computation of the Hurwitz zeta function and its derivatives. Numerical Algorithms, 69(2), 253-270.
- Johansson, F. (2017). Arb: efficient arbitrary-precision midpoint-radius interval arithmetic. IEEE Transactions on Computers, 66(8), 1281-1292.
- Johansson, F. (2018, July). Numerical integration in arbitrary-precision ball arithmetic. In International Congress on Mathematical Software (pp. 255-263). Springer, Cham.
- Johansson, F., & Mezzarobba, M. (2018). Fast and Rigorous Arbitrary-Precision Computation of Gauss--Legendre Quadrature Nodes and Weights. SIAM Journal on Scientific Computing, 40(6), C726-C747.
- Zeidler, E., Nonlinear Functional Analysis and Its Applications I-V. 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).
- Oishi, S., & Tanabe, K. (2009). Numerical Inclusion of Optimum Point for Linear Programming. JSIAM Letters, 1, 5-8.