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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, and today it is recognized as a powerful tool for the study of dynamical systems.
Computation without verification may cause unfortunate results. Below are some examples.
In the 1980s, Rump made an example. 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.
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. This result to the study conflicted to the theoretical study by Gidas-Ni-Nirenberg which claimed that there is no asymmetric solution. 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.
Accidents caused by numerical errors
The following examples are known as accidents caused by numerical errors:
Failure of intercepting missiles in the Gulf War (1991)
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^Loh, Eugene; Walster, G. William (2002). Rump's example revisited. Reliable Computing, 8(3), 245-248.
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^Ogita, T. (2008). Verified Numerical Computation of Matrix Determinant. SCAN'2008 El Paso, Texas September 29-October 3, 2008, 86.
^Shinya Miyajima, Verified computation for the Hermitian positive definite solution of the conjugate discrete-time algebraic Riccati equation, Journal of Computational and Applied Mathematics, Volume 350, Pages 80-86, April 2019.
^Shinya Miyajima, Fast verified computation for the minimal nonnegative solution of the nonsymmetric algebraic Riccati equation, Computational and Applied Mathematics, Volume 37, Issue 4, Pages 4599-4610, September 2018.
^Shinya Miyajima, Fast verified computation for the solution of the T-congruence Sylvester equation, Japan Journal of Industrial and Applied Mathematics, Volume 35, Issue 2, Pages 541-551, July 2018.
^Shinya Miyajima, Fast verified computation for the solvent of the quadratic matrix equation, The Electronic Journal of Linear Algebra, Volume 34, Pages 137-151, March 2018
^Shinya Miyajima, Fast verified computation for solutions of algebraic Riccati equations arising in transport theory, Numerical Linear Algebra with Applications, Volume 24, Issue 5, Pages 1-12, October 2017.
^Shinya Miyajima, Fast verified computation for stabilizing solutions of discrete-time algebraic Riccati equations, Journal of Computational and Applied Mathematics, Volume 319, Pages 352-364, August 2017.
^Shinya Miyajima, Fast verified computation for solutions of continuous-time algebraic Riccati equations, Japan Journal of Industrial and Applied Mathematics, Volume 32, Issue 2, Pages 529-544, July 2015.
^Rump, Siegfried 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, Naoya; Okayama, Tomoaki; Oishi, Shin'ichi (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, Fredrik (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,
^Johansson, Fredrik (2019). Computing Hypergeometric Functions Rigorously. ACM Transactions on Mathematical Software (TOMS), 45(3), 30.
^Johansson, Fredrik (2015). Rigorous high-precision computation of the Hurwitz zeta function and its derivatives. Numerical Algorithms, 69(2), 253-270.
^Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe (2019) Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations (Springer Series in Computational Mathematics).
^Oishi, Shin'ichi; Tanabe, Kunio (2009). Numerical Inclusion of Optimum Point for Linear Programming. JSIAM Letters, 1, 5-8.
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.