IntervalBoxes.jl

An interval box is a Cartesian product of Intervals (or BareIntervals), as defined in IntervalArithmetic.jl. An interval box of dimension $n$ thus represents an axis-aligned set in Euclidean space $\mathbb{R}^n$.

julia> using IntervalBoxes, IntervalArithmetic, IntervalArithmetic.Symbols

julia> X = IntervalBox(1..3, 4..6)
[1.0, 3.0]_com × [4.0, 6.0]_com

julia> Y = IntervalBox(2..5, 2)
[2.0, 5.0]_com²

julia> (1..3) × (4..6)   # \times<TAB>
[1.0, 3.0]_com × [4.0, 6.0]_com

We have defined × to be the Cartesian cross product operator, acting on Intervals and/or IntervalBoxes.

We treat IntervalBoxes as sets as much as possible, and extend Julia's set operations to act on these objects:

julia> X ⊓ Y
[2.0, 3.0]_trv × [4.0, 5.0]_trv

julia> X ⊔ Y
[1.0, 5.0]_trv × [2.0, 6.0]_trv

julia> using StaticArrays

julia> SVector(2, 5) ∈ X ⊓ Y
true

Note that the ⊔ operator produces the interval hull of the union (i.e. the smallest interval box that contains the union).

One-dimensional intervals can also be treated as sets in the same way, as follows. (These set operations are deliberately not defined in IntervalArithmetic.jl.)

julia> x = IntervalBox(1..3)
[1.0, 3.0]¹

julia> y = IntervalBox(2..4)
[2.0, 4.0]¹

julia> x ⊓ y
[2.0, 3.0]¹

Interval arithmetic allows us to compute an enclosure (in general, an over-estimate) of the range of a multi-dimensional function. E.g.:

julia> f((x, y)) = x + y;

julia> f(X)
[5.0, 9.0]

Note that in order to use numbers in functions like this, the numbers must be wrapped in ExactReal to specify that they are exact representations. Alternatively the complete definition can be annotated with @exact:

julia> g1((x, y)) = ExactReal(2) * x + ExactReal(3) * y;

julia> g1(X)
[14.0, 24.0]_com

julia> @exact g2((x, y)) = 2x + 3y;

julia> g2(X)
[14.0, 24.0]_com

Copyright by JuliaIntervals contributors 2017–2024.

This code was originally in IntervalArithmetic.jl, but was removed. The git history can be found in that repo.