Validated numerics, or rigorous computation, verified computation, reliable computation, numerical verification () 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 mitigate rounding and measurement errors in mathematical computation by computing function bounds. Numeri ...

is most often used, where 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 (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

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 , combatant2 = , commander1 = , commander2 = , strength1 = Over 950,000 soldiers3,113 tanks1,800 aircraft2,200 artillery systems , page = https://www.govinfo.gov/content/pkg/GAOREPORTS-PEMD-96- ...

(1991) * Failure of the

Ariane 5 Ariane 5 is a retired European heavy-lift space launch vehicle operated by Arianespace for the European Space Agency (ESA). It was launched from the Guiana Space Centre (CSG) in French Guiana. It was used to deliver payloads into geostationar ...

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