Interval arithmetic

Interval arithmetic, interval mathematics, interval analysis, or interval computation, is a method developed by mathematicians since the 1950s and 1960s, as an approach to putting bounds on rounding errors and measurement errors in mathematical computation and thus developing numerical methods that yield reliable results. Very simply put, it represents each value as a range of possibilities. For example, instead of estimating the height of someone using standard arithmetic as 2.0 metres, using interval arithmetic we might be certain that that person is somewhere between 1.97 and 2.03 metres.

This concept is suitable for a variety of purposes. The most common use is to keep track of and handle rounding errors directly during the calculation and of uncertainties in the knowledge of the exact values of physical and technical parameters. The latter often arise from measurement errors and tolerances for components or due to limits on computational accuracy. Interval arithmetic also helps find reliable and guaranteed solutions to equations and optimization problems.

Mathematically, instead of working with an uncertain real ${\displaystyle x}$ we work with the two ends of the interval ${\displaystyle [a,b]}$ that contains ${\displaystyle x}$. In interval arithmetic, any variable ${\displaystyle x}$ lies between ${\displaystyle a}$ and ${\displaystyle b}$, or could be one of them. A function ${\displaystyle f}$ when applied to ${\displaystyle x}$ is also uncertain. In interval arithmetic ${\displaystyle f}$ produces an interval ${\displaystyle [c,d]}$ that is all the possible values for ${\displaystyle f(x)}$ for all ${\displaystyle x\in [a,b]}$.

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