Intervallaritmetik

Intervallaritmetik is a branch of numerical analysis that aims to carry out computations with intervals in order to enclose all possible values of a quantity when inputs are uncertain or affected by rounding errors. An interval [a,b] represents the set of real numbers x with a ≤ x ≤ b, and interval operations are defined to keep the true result inside the computed bounds.

Given two intervals A=[a1,a2] and B=[b1,b2], the basic arithmetic operations are defined as follows: addition A+B=[a1+b1, a2+b2]; subtraction A−B=[a1−b2, a2−b1]; multiplication A·B equals the interval spanned by the four products a1b1, a1b2, a2b1, a2b2, i.e., [min(...), max(...)]. Division A/B is defined when B does not contain zero; then A/B=[min(a1/b1, a1/b2, a2/b1, a2/b2), max(...)] . If B includes zero, division is undefined or requires splitting B into subintervals to avoid zero in the denominator.

Implementations use careful rounding to preserve the inclusion property: lower endpoints are rounded downward and upper endpoints upward so that the computed interval contains the exact result. The interval approach naturally accounts for rounding and input uncertainty, but it may suffer from the dependency problem, where repeated use of the same interval leads to overestimation. Various techniques address this, including splitting the input domain, applying affine arithmetic, or using constraint propagation.

Variants and extensions include Kaucher interval arithmetic, which generalizes traditional intervals to allow improper intervals, and affine arithmetic, which tracks correlations between quantities to reduce overestimation. Intervallaritmetik is used in verified numerical analysis, global optimization, root finding with guaranteed bounds, differential equations, and reliable simulations.

Historically, interval arithmetic was developed by Ramon E. Moore and colleagues in the 1960s and has since become a standard tool for obtaining mathematically guaranteed bounds in computational mathematics.