IntervalUnionArithmetic.jl

An extension to IntervalArithmetic.jl with interval unions. Interval unions are sets defined by unions of disjoint intervals.

Conversation in PR#452

This package includes constructors, arithmetic (including with intervals and scalars) and complement functions.

Installation

This is a registered julia package:

julia> ]
pkg> add IntervalUnionArithmetic.jl

or the most up to date version:

julia> ]
pkg> add https://github.com/AnderGray/IntervalUnionArithmetic.jl#master

Example

julia> a = interval(0,2) ∪ interval(3,4)
  [0, 2] ∪ [3, 4]

julia> b = interval(1,2) ∪ interval(4,5) ∪ ∅
  [1, 2] ∪ [4, 5]

julia> c = a * b 
  [0, 10] ∪ [12, 20]

Division and sqrt

julia> x = interval(2,5); 
julia> y = interval(-1,1);
julia> x / y                # Standard interval arithmetic
  [-∞, ∞]
  
julia> x1 = intervalUnion(x);   # Convert to interval union
julia> y1 = intervalUnion(y);
julia> x1 / y1              # Does x1 / y1 for y1\{0} if 0 ∈ y1
  [-∞, -2] ∪ [2, ∞]

julia> sqrt(x)
  [1.41421, 2.23607]

julia> sqrt(x1)
  [-2.23607, -1.41421] ∪ [1.41421, 2.23607]

Set operations

julia> a = interval(0,1) ∪ interval(2,3)
  [0, 1] ∪ [2, 3]

julia> b = interval(-1,0.5) ∪ interval(2.5,5)
  [-1, 0.5] ∪ [2.5, 5]

julia> complement(a)         # complement
  [-∞, 0] ∪ [1, 2] ∪ [3, ∞]

julia> a ∩ b                 # Intersect
  [0, 0.5] ∪ [2.5, 3]
  
julia> a \ b                 # Set difference
  [0.5, 1] ∪ [2, 2.5]
  
julia> bisect(a,0.5)         # Cut at a = 0.5
  [0, 0.5] ∪ [0.5, 1] ∪ [2, 3]
  
julia> a ⊂ interval(0,3)     # Subset
  true

Bibiography

• Schichl, Hermann, et al. "Interval unions." BIT Numerical Mathematics 57.2 (2017): 531-556.]
• Montanher, Tiago, et al. "Using interval unions to solve linear systems of equations with uncertainties." BIT Numerical Mathematics 57.3 (2017): 901-926.
• Domes, Ferenc, et al. "Rigorous global filtering methods with interval unions." Beyond Traditional Probabilistic Data Processing Techniques: Interval, Fuzzy etc. Methods and Their Applications. Springer, Cham, 2020. 249-267