Interval Arithmetic

Interval arithmetic is the arithmetic of quantities that lie within specified ranges (i.e., intervals) instead of having definite known values. Interval arithmetic can be especially useful when working with data that is subject to measurement errors or uncertainties. It can be considered a rigorous version of significance arithmetic (a.k.a., automatic precision control).

It is powerful enough to provide rigorous mathematical proofs (de la Llave 1991, Hutchings et al. 2000, Tucker 2002, Gutowski 2003), but rigor comes at a price. In particular, interval arithmetic can be slow, and often gives overly pessimistic results for real-world computations.

For bounded closed intervals and with and , the basic operations give all possible results as and vary independently (Kearfott 1996). In particular,

			(1)
			(2)
			(3)

where and denote the minimum and maximum, respectively. Division is given by

(4)

when . With floating-point arithmetic, lower endpoints are rounded downward and upper endpoints upward so that the computed interval still contains every possible exact result (Kearfott 1996).

Repeated occurrences of the same variable can make an enclosure wider than the actual set of values, since separate interval operations do not retain the dependence between occurrences. For example, if , then

(5)

although for every single (Trott 2006, pp. 54-66). Thus, subtracting two independently varying values from the same interval differs from subtracting one value from itself.

Interval multiplication is not distributive. For bounded closed intervals , , and , one has only the inclusion

(6)

For example,

			(7)
			(8)

where , , and (Trott 2006, pp. 54-66). The expression permits independent choices from in the two products, whereas uses one choice from .