Linear operators
A LinearOperator is a structure representing a linear operator from a VectorSpace to an other. More precisely, a LinearOperator is comprised of the three fields domain::VectorSpace, codomain::VectorSpace and coefficients::AbsractMatrix with matching dimensions and size.
julia> A = LinearOperator(Taylor(1), Taylor(1), [1 2 ; 3 4])LinearOperator : Taylor(1) → Taylor(1) with coefficients Matrix{Int64}: 1 2 3 4
The three fields domain, codomain and coefficients are accessible via the respective functions of the same name.
julia> domain(A)Taylor(1)julia> codomain(A)Taylor(1)julia> coefficients(A)2×2 Matrix{Int64}: 1 2 3 4
For convenience, the methods zeros, ones, fill and fill! are available:
julia> dom, codom = Taylor(1), Taylor(2)(Taylor(1), Taylor(2))julia> zeros(dom, codom)LinearOperator : Taylor(1) → Taylor(2) with coefficients Matrix{Float64}: 0.0 0.0 0.0 0.0 0.0 0.0julia> ones(dom, codom)LinearOperator : Taylor(1) → Taylor(2) with coefficients Matrix{Float64}: 1.0 1.0 1.0 1.0 1.0 1.0julia> fill(2, dom, codom)LinearOperator : Taylor(1) → Taylor(2) with coefficients Matrix{Int64}: 2 2 2 2 2 2julia> fill!(zeros(dom, codom), 2)LinearOperator : Taylor(1) → Taylor(2) with coefficients Matrix{Float64}: 2.0 2.0 2.0 2.0 2.0 2.0
The coefficients of a LinearOperator are indexed according to the indices of the domain and codomain (as given by indices).
julia> A[0:1,0:1] # indices(domain(A)), indices(codomain(A))2×2 Matrix{Int64}: 1 2 3 4
When the domain and/or the codomain of a LinearOperator is a CartesianSpace, its coefficients can be thought of as a block matrix . The function component extracts a LinearOperator composing the cartesian space.
julia> B = LinearOperator(ParameterSpace() × Taylor(1)^2, ParameterSpace() × Taylor(1)^2, reshape(1:25, 5, 5))LinearOperator : 𝕂 × Taylor(1)² → 𝕂 × Taylor(1)² with coefficients Base.ReshapedArray{Int64, 2, UnitRange{Int64}, Tuple{}}: 1 6 11 16 21 2 7 12 17 22 3 8 13 18 23 4 9 14 19 24 5 10 15 20 25julia> B[1:5,1:5] # indices(domain(B)), indices(codomain(B))5×5 Matrix{Int64}: 1 6 11 16 21 2 7 12 17 22 3 8 13 18 23 4 9 14 19 24 5 10 15 20 25julia> component(B, 1, 1) # extract the linear operator associated with the domain ParameterSpace() and codomain ParameterSpace()LinearOperator : 𝕂 → 𝕂 with coefficients SubArray{Int64, 2, Base.ReshapedArray{Int64, 2, UnitRange{Int64}, Tuple{}}, Tuple{UnitRange{Int64}, UnitRange{Int64}}, false}: 1julia> component(B, 2, 2) # extract the linear operator associated with the domain Taylor(1)^2 and codomain Taylor(1)^2LinearOperator : Taylor(1)² → Taylor(1)² with coefficients SubArray{Int64, 2, Base.ReshapedArray{Int64, 2, UnitRange{Int64}, Tuple{}}, Tuple{UnitRange{Int64}, UnitRange{Int64}}, false}: 7 12 17 22 8 13 18 23 9 14 19 24 10 15 20 25julia> component(component(B, 2, 2), 1, 1)LinearOperator : Taylor(1) → Taylor(1) with coefficients SubArray{Int64, 2, Base.ReshapedArray{Int64, 2, UnitRange{Int64}, Tuple{}}, Tuple{UnitRange{Int64}, UnitRange{Int64}}, false}: 7 12 8 13julia> component(component(B, 2, 2), 2, 2)LinearOperator : Taylor(1) → Taylor(1) with coefficients SubArray{Int64, 2, Base.ReshapedArray{Int64, 2, UnitRange{Int64}, Tuple{}}, Tuple{UnitRange{Int64}, UnitRange{Int64}}, false}: 19 24 20 25
Similarly, the function eachcomponent returns a Generator whose iterates yield each LinearOperator composing the cartesian space.
Arithmetic
The addition and subtraction operations are implemented as the + and - functions respectively.
julia> C = LinearOperator(Taylor(1), Taylor(1), [1 2 ; 3 4])LinearOperator : Taylor(1) → Taylor(1) with coefficients Matrix{Int64}: 1 2 3 4julia> D = LinearOperator(Taylor(1), Taylor(2), [1 2 ; 3 4 ; 5 6])LinearOperator : Taylor(1) → Taylor(2) with coefficients Matrix{Int64}: 1 2 3 4 5 6julia> C + DLinearOperator : Taylor(1) → Taylor(2) with coefficients Matrix{Int64}: 2 4 6 8 5 6julia> C - DLinearOperator : Taylor(1) → Taylor(2) with coefficients Matrix{Int64}: 0 0 0 0 -5 -6julia> C + ILinearOperator : Taylor(1) → Taylor(1) with coefficients Matrix{Int64}: 2 2 3 5julia> C - ILinearOperator : Taylor(1) → Taylor(1) with coefficients Matrix{Int64}: 0 2 3 3
The product between LinearOperator is implemented as the * and ^ functions. The division between LinearOperator is implemented as the \ method.
julia> C * DLinearOperator : Taylor(1) → Taylor(1) with coefficients Matrix{Int64}: 7 10 15 22julia> C ^ 3LinearOperator : Taylor(1) → Taylor(1) with coefficients Matrix{Int64}: 37 54 81 118julia> C \ CLinearOperator : Taylor(1) → Taylor(1) with coefficients Matrix{Float64}: 0.9999999999999998 0.0 8.326672684688673e-17 1.0
The action of a LinearOperator is performed by the right product * of a LinearOperator with a Sequence; alternatively, LinearOperator defines a method on a Sequence representing *. Naturally, the resulting sequence is an element of the codomain of the LinearOperator.
Conversely, the operator \ between a LinearOperator and a Sequence corresponds to the action of the inverse of the LinearOperator; the output sequence is an element of the domain of the LinearOperator.
julia> x = Sequence(Taylor(2), [1, 1, 1])Sequence in Taylor(2) with coefficients Vector{Int64}: 1 1 1julia> C * x # C(x)Sequence in Taylor(1) with coefficients Vector{Int64}: 3 7julia> D \ xSequence in Taylor(1) with coefficients Vector{Float64}: -1.0000000000000016 1.000000000000001