 Validated Numerics
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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, and today it is recognized as a powerful tool for the study of dynamical systems.

## 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 (1991)
• Failure of the Ariane 5 rocket (1996)
• Mistakes in election result totalization

## Main topics

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