# Interval arithmetic

Interval arithmetic is a computer arithmetic for (mathematical) intervals.

## Definition

For real intervals (interval o real numbers), interval arithmetic is defined as follows:

• Addition: $[x_{1},x_{2}]+[y_{1},y_{2}]=[x_{1}+y_{1},x_{2}+y_{2}]$ • Subtraction: $[x_{1},x_{2}]-[y_{1},y_{2}]=[x_{1}-y_{2},x_{2}-y_{1}]$ • Multiplication: $[x_{1},x_{2}]\cdot [y_{1},y_{2}]=[\min(x_{1}y_{1},x_{1}y_{2},x_{2}y_{1},x_{2}y_{2}),\max(x_{1}y_{1},x_{1}y_{2},x_{2}y_{1},x_{2}y_{2})]$ • Division:
${\frac {[x_{1},x_{2}]}{[y_{1},y_{2}]}}=[x_{1},x_{2}]\cdot {\frac {1}{[y_{1},y_{2}]}},$ where
{\begin{aligned}{\frac {1}{[y_{1},y_{2}]}}&=\left[{\tfrac {1}{y_{2}}},{\tfrac {1}{y_{1}}}\right]&&0\notin [y_{1},y_{2}]\\{\frac {1}{[y_{1},0]}}&=\left[-\infty ,{\tfrac {1}{y_{1}}}\right]\\{\frac {1}{[0,y_{2}]}}&=\left[{\tfrac {1}{y_{2}}},\infty \right]\\{\frac {1}{[y_{1},y_{2}]}}&=\left[-\infty ,{\tfrac {1}{y_{1}}}\right]\cup \left[{\tfrac {1}{y_{2}}},\infty \right]=[-\infty ,\infty ]&&0\in (y_{1},y_{2})\end{aligned}} ## Details

### Applications

Interval arithmetic is mainly usit i the field o validatit numerics. It is also usit i other technical areas.

### Implementations

Syne the birth o interval arithmetic, many experts have made interval arithmetic programs. The most famous works are INTLAB (made wi MATLAB), arb, JuliaIntervals, an kv.

## Community

Thare are several international conferences aboot interval arithmetic. Ane o the most largest meetin is the International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics. Thare are also SWIM SWIM (Small Workshop on Interval Methods), PPAM (International Conference on Parallel Processing and Applied Mathematics), an REC (International Workshop on Reliable Engineering Computing).