Interval arithmetic

Interval arithmetic is a computer arithmetic for (mathematical) intervals^[1][2][3].

Definition

For real intervals (interval o real numbers), interval arithmetic is defined as follows:^[1][2][3]

• Addition: ${\displaystyle [x_{1},x_{2}]+[y_{1},y_{2}]=[x_{1}+y_{1},x_{2}+y_{2}]}$
• Subtraction: ${\displaystyle [x_{1},x_{2}]-[y_{1},y_{2}]=[x_{1}-y_{2},x_{2}-y_{1}]}$
• Multiplication: ${\displaystyle [x_{1},x_{2}]\cdot [y_{1},y_{2}]=[\min(x_{1}y_{1},x_{1}y_{2},x_{2}y_{1},x_{2}y_{2}),\max(x_{1}y_{1},x_{1}y_{2},x_{2}y_{1},x_{2}y_{2})]}$
• Division:

${\displaystyle {\frac {[x_{1},x_{2}]}{[y_{1},y_{2}]}}=[x_{1},x_{2}]\cdot {\frac {1}{[y_{1},y_{2}]}},}$

where

${\displaystyle {\begin{aligned}{\frac {1}{[y_{1},y_{2}]}}&=\left[{\tfrac {1}{y_{2}}},{\tfrac {1}{y_{1}}}\right]&&0\notin [y_{1},y_{2}]\\{\frac {1}{[y_{1},0]}}&=\left[-\infty ,{\tfrac {1}{y_{1}}}\right]\\{\frac {1}{[0,y_{2}]}}&=\left[{\tfrac {1}{y_{2}}},\infty \right]\\{\frac {1}{[y_{1},y_{2}]}}&=\left[-\infty ,{\tfrac {1}{y_{1}}}\right]\cup \left[{\tfrac {1}{y_{2}}},\infty \right]=[-\infty ,\infty ]&&0\in (y_{1},y_{2})\end{aligned}}}$

Details

Applications

Interval arithmetic is mainly usit i the field o validatit numerics^[4]. It is also usit i other technical areas^[5].

Implementations

Relatit page: Affine arithmetic

Syne the birth o interval arithmetic, many experts have made interval arithmetic programs. The most famous works are INTLAB (made wi MATLAB)^[6], arb^[7], JuliaIntervals^[8][9], an kv^[10].

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^[11][12][13]. 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).