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 computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

. For computation,

interval arithmetic Interval arithmetic (also known as interval mathematics, interval analysis, or interval computation) is a mathematical technique used to put bounds on rounding errors and measurement errors in mathematical computation. Numerical methods using ...

is used, and all results are represented by intervals. Validated numerics were used by

Warwick Tucker Warwick Tucker is an Australian mathematician at Monash University (previously deputy Chair and Chair at the Department of Mathematics at Uppsala University 2009–2020) who works on dynamical systems, chaos theory and computational mathematics. H ...

in order to solve the 14th of

Smale's problems Smale's problems are a list of eighteen unsolved problems in mathematics proposed by Steve Smale in 1998 and republished in 1999. Smale composed this list in reply to a request from Vladimir Arnold, then vice-president of the International Mathema ...

, 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 a p ...

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 35-country military coalition in response to the Iraqi invasion of Kuwait. Spearheaded by the United States, the coalition's efforts against Iraq were carried out in two key phases: ...

(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 int ...

rocket (1996) * Mistakes in election result totalization

Main topics

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

Tools

References

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. {{Industrial and applied mathematics Numerical analysis Computational science