Types

AbstractIntervalMatrix{IT} <: AbstractMatrix{IT}

Abstract supertype for interval matrix types.

IntervalMatrix{T, IT, MT<:AbstractMatrix{IT}} <: AbstractIntervalMatrix{IT}

An interval matrix i.e. a matrix whose coefficients are intervals. This type is parameterized in the number field, the interval type, and the matrix type.

Fields

• mat – matrix whose entries are intervals

Examples

julia> A = IntervalMatrix([-1 .. -0.8 0 .. 0; 0 .. 0 -1 .. -0.8])
2×2 IntervalMatrix{Float64, Interval{Float64}, Matrix{Interval{Float64}}}:
 [-1.0, -0.7999999]   [0.0, 0.0]
  [0.0, 0.0]         [-1.0, -0.7999999]

An interval matrix proportional to the identity matrix can be built using the UniformScaling operator from the standard library LinearAlgebra. For example,

julia> using LinearAlgebra

julia> IntervalMatrix(interval(1)*I, 2)
2×2 IntervalMatrix{Float64, Interval{Float64}, Matrix{Interval{Float64}}}:
 [1.0, 1.0]  [0.0, 0.0]
 [0.0, 0.0]  [1.0, 1.0]

The number of columns can be specified as a third argument, creating a rectangular $m × n$ matrix such that only the entries in the main diagonal, $(1, 1), (2, 2), …, (k, k)$ are specified, where $k = \min(m, n)$:

julia> IntervalMatrix(interval(-1, 1)*I, 2, 3)
2×3 IntervalMatrix{Float64, Interval{Float64}, Matrix{Interval{Float64}}}:
 [-1.0, 1.0]   [0.0, 0.0]  [0.0, 0.0]
  [0.0, 0.0]  [-1.0, 1.0]  [0.0, 0.0]

julia> IntervalMatrix(interval(-1, 1)*I, 3, 2)
3×2 IntervalMatrix{Float64, Interval{Float64}, Matrix{Interval{Float64}}}:
 [-1.0, 1.0]   [0.0, 0.0]
  [0.0, 0.0]  [-1.0, 1.0]
  [0.0, 0.0]   [0.0, 0.0]

An uninitialized interval matrix can be constructed using undef:

julia> m = IntervalMatrix{Float64}(undef, 2, 2);

julia> typeof(m)
IntervalMatrix{Float64, Interval{Float64}, Matrix{Interval{Float64}}}

Note that this constructor implicitly uses a dense matrix, Matrix{Float64}, as the matrix (mat) field in the new interval matrix.

IntervalMatrixPower{T}

A wrapper for the matrix power that can be incremented.

Fields

• M – the original matrix
• Mᵏ – the current matrix power, i.e., $M^k$
• k – the current power index

Notes

The wrapper should only be accessed using the interface functions. The internal representation (such as the fields) are subject to future changes.

Examples

julia> A = IntervalMatrix([interval(2, 2) interval(2, 3); interval(0, 0) interval(-1, 1)])
2×2 IntervalMatrix{Float64, Interval{Float64}, Matrix{Interval{Float64}}}:
 [2.0, 2.0]   [2.0, 3.0]
 [0.0, 0.0]  [-1.0, 1.0]

julia> pow = IntervalMatrixPower(A);

julia> increment!(pow)
2×2 IntervalMatrix{Float64, Interval{Float64}, Matrix{Interval{Float64}}}:
 [4.0, 4.0]  [2.0, 9.0]
 [0.0, 0.0]  [0.0, 1.0]

julia> increment(pow)
2×2 IntervalMatrix{Float64, Interval{Float64}, Matrix{Interval{Float64}}}:
 [8.0, 8.0]  [-1.0, 21.0]
 [0.0, 0.0]  [-1.0, 1.0]

julia> matrix(pow)
2×2 IntervalMatrix{Float64, Interval{Float64}, Matrix{Interval{Float64}}}:
 [4.0, 4.0]  [2.0, 9.0]
 [0.0, 0.0]  [0.0, 1.0]

julia> index(pow)
2

julia> base(pow)
2×2 IntervalMatrix{Float64, Interval{Float64}, Matrix{Interval{Float64}}}:
 [2.0, 2.0]   [2.0, 3.0]
 [0.0, 0.0]  [-1.0, 1.0]

AffineIntervalMatrix1{T, IT, MT0<:AbstractMatrix{T}, MT1<:AbstractMatrix{T}} <: AbstractIntervalMatrix{IT}

Interval matrix representing the matrix

\[A₀ + λA₁,\]

where $A₀$ and $A₁$ are real (or complex) matrices, and $λ$ is an interval.

Fields

• A0 – matrix
• A1 – matrix
• λ – interval

Examples

The matrix $I + [1 1; -1 1] * interval(0, 1)$ is:

julia> using LinearAlgebra

julia> P = AffineIntervalMatrix1(Matrix(1.0I, 2, 2), [1 1; -1 1.], interval(0, 1));

julia> P
2×2 AffineIntervalMatrix1{Float64, Interval{Float64}, Matrix{Float64}, Matrix{Float64}}:
  [1.0, 2.0]  [0.0, 1.0]
 [-1.0, 0.0]  [1.0, 2.0]

AffineIntervalMatrix{T, IT, MT0<:AbstractMatrix{T}, MT<:AbstractMatrix{T}, MTA<:AbstractVector{MT}} <: AbstractIntervalMatrix{IT}

Interval matrix representing the matrix

\[A₀ + λ₁A₁ + λ₂A₂ + … + λₖAₖ,\]

where $A₀$ and $A₁, …, Aₖ$ are real (or complex) matrices, and $λ₁, …, λₖ$ are intervals.

Fields

• A0 – matrix
• A – vector of matrices
• λ – vector of intervals

Notes

This type is the general case of the AffineIntervalMatrix1, which only contains one matrix proportional to an interval.

Examples

The affine matrix $I + [1 1; -1 1] * interval(0, 1) + [0 1; 1 0] * interval(2, 3)$ is:

julia> using LinearAlgebra

julia> A0 = Matrix(1.0I, 2, 2);

julia> A1 = [1 1; -1 1.]; A2 = [0 1; 1 0];

julia> λ1 = interval(0, 1); λ2 = interval(2, 3);

julia> P = AffineIntervalMatrix(A0, [A1, A2], [λ1, λ2])
2×2 AffineIntervalMatrix{Float64, Interval{Float64}, Matrix{Float64}, Matrix{Float64}, Vector{Matrix{Float64}}, Vector{Interval{Float64}}}:
 [1.0, 2.0]  [2.0, 4.0]
 [1.0, 3.0]  [1.0, 2.0]