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
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
. 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
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
.
Importance
Computation without verification may cause unfortunate results. Below are some examples.
Rump's example
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.
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.
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
The Gulf War was a 1990–1991 armed campaign waged by a Coalition of the Gulf War, 35-country military coalition in response to the Iraqi invasion of Kuwait. Spearheaded by the United States, the coalition's efforts against Ba'athist Iraq, ...
(1991)
* Failure of the
Ariane 5
Ariane 5 is a European heavy-lift space launch vehicle developed and operated by Arianespace for the European Space Agency (ESA). It is launched from the Centre Spatial Guyanais (CSG) in French Guiana. It has been used to deliver payloads in ...
rocket (1996)
* Mistakes in election result totalization
Main topics
The study of validated numerics is divided into the following fields:
Tools
See also
References
Further reading
*
Tucker, Warwick (2011). Validated Numerics: A Short Introduction to Rigorous Computations.
Princeton University Press
Princeton University Press is an independent Academic publishing, publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.
The press was founded by Whitney Darrow, ...
.
*
Moore, Ramon Edgar, Kearfott, R. Baker., Cloud, Michael J. (2009). Introduction to Interval Analysis.
Society for Industrial and Applied Mathematics
Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to applied mathematics, computational science, and data science through research, publications, and community. SIAM is the world's largest scientific soci ...
.
* Rump, Siegfried M. (2010). Verification methods: Rigorous results using floating-point arithmetic.
Acta Numerica, 19, 287-449.
External links
Validated Numerics for PedestriansReliable 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.
{{Industrial and applied mathematics
Numerical analysis
Computational science