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 meters, using interval arithmetic we might be certain that that person is somewhere between 1.97 and 2.03 meters.

Whereas classical arithmetic defines operations on individual numbers, interval arithmetic defines a set of operations on intervals:

T · S = { x | there is some y in T, and some z in S, such that x = y · z }.

The basic operations of interval arithmetic are, for two intervals and that are subsets of the real line (-∞, ∞),

• + = =
• − = =
• × =
• ÷ = when 0 is not in .

Division by an interval containing zero is not defined under the basic interval arithmetic. The addition and multiplication operations are commutative, associative and sub-distributive: the set X ( Y + Z ) is a subset of XY + XZ.

Instead of working with an uncertain real we work with the two ends of the interval which contains : lies between and, or could be one of them. Similarly a function when applied to is also uncertain. Instead, in interval arithmetic produces an interval which is all the possible values for for all .

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.