INTERVAL2

INTERVAL2 is a formalism used in numerical analysis to represent and manipulate two-dimensional intervals. It extends traditional interval arithmetic by handling a pair of bound intervals, typically denoted as I1 = [a, b] and I2 = [c, d], which together bound a vector quantity or two correlated quantities.

A typical INTERVAL2 object is a pair of real intervals (I1, I2) that represents all pairs (x, y) with x in I1 and y in I2. This representation can model independent axis-aligned bounds or serve as a bound for a function of two variables. In practice, INTERVAL2 also supports enhanced forms that incorporate dependencies between the two components to reduce over-approximation.

Arithmetic on INTERVAL2 follows extensions of interval arithmetic. For example, addition is performed componentwise: (I1, I2) + (J1, J2) = (I1 + J1, I2 + J2). Multiplication must consider the Cartesian product of bounds and the induced range, and advanced implementations may use affine forms or other dependency-aware techniques to tighten results and mitigate the wrapping effect.

Applications of INTERVAL2 include validated numerics, robust optimization, and error bounding in simulations where two quantities are uncertain within known ranges and may be correlated. It is useful in areas such as control, computer graphics, and scientific computing, where reliable enclosures of multivariate quantities are needed.

Example: if x ∈ [1, 2] and y ∈ [3, 4], then INTERVAL2(x, y) = ([1, 2], [3, 4]). For a function f(x, y) = x^2 + y, the computed enclosure uses the bounds of x and y to obtain a bound for f over the specified domain.

See also: interval arithmetic, affine arithmetic, validated numerics.