Preconditioners

InverseDiagonalMidpoint <: AbstractPrecondition

Preconditioner that preconditions the linear system $Ax=b$ with the diagonal matrix of $A_c^{-1}$, where $A_c$ is the midpoint matrix of $A$.

Notes

• An object of type InverseDiagonalMidpoint is a function with method

    (idmp::InverseDiagonalMidpoint)(A::AbstractMatrix{T},
                                    b::AbstractVector{T}) where {T<:Interval}

Example

julia> A = [2..4 -2..1; -1..2 2..4]
2×2 Matrix{Interval{Float64}}:
  [2, 4]  [-2, 1]
 [-1, 2]   [2, 4]

julia> b = [-2..2, -2..2]
2-element Vector{Interval{Float64}}:
 [-2, 2]
 [-2, 2]

julia> idmp = InverseDiagonalMidpoint()
InverseDiagonalMidpoint()

julia> idmp(A, b)
(Interval{Float64}[[0.666666, 1.33334] [-0.666667, 0.333334]; [-0.333334, 0.666667] [0.666666, 1.33334]], Interval{Float64}[[-0.666667, 0.666667], [-0.666667, 0.666667]])

InverseMidpoint <: AbstractPrecondition

Preconditioner that preconditions the linear system $Ax=b$ with $A_c^{-1}$, where $A_c$ is the midpoint matrix of $A$.

Notes

• An object of type InverseMidpoint is a function with method

    (imp::InverseMidpoint)(A::AbstractMatrix{T},
                           b::AbstractVector{T}) where {T<:Interval}

Examples

julia> A = [2..4 -2..1; -1..2 2..4]
2×2 Matrix{Interval{Float64}}:
  [2, 4]  [-2, 1]
 [-1, 2]   [2, 4]

julia> b = [-2..2, -2..2]
2-element Vector{Interval{Float64}}:
 [-2, 2]
 [-2, 2]

julia> imp = InverseMidpoint()
InverseMidpoint()

julia> imp(A, b)
(Interval{Float64}[[0.594594, 1.40541] [-0.540541, 0.540541]; [-0.540541, 0.540541] [0.594594, 1.40541]], Interval{Float64}[[-0.756757, 0.756757], [-0.756757, 0.756757]])

NoPrecondition <: AbstractPrecondition

Type of the trivial preconditioner which does nothing.

Notes

• An object of type NoPrecondition is a function with method

    (np::NoPrecondition)(A::AbstractMatrix{T},
                         b::AbstractVector{T}) where {T<:Interval}

Example

julia> A = [2..4 -2..1; -1..2 2..4]
2×2 Matrix{Interval{Float64}}:
  [2, 4]  [-2, 1]
 [-1, 2]   [2, 4]

julia> b = [-2..2, -2..2]
2-element Vector{Interval{Float64}}:
 [-2, 2]
 [-2, 2]

julia> np = NoPrecondition()
NoPrecondition()

julia> np(A, b)
(Interval{Float64}[[2, 4] [-2, 1]; [-1, 2] [2, 4]], Interval{Float64}[[-2, 2], [-2, 2]])