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 0.0
julia> ones(dom, codom)
LinearOperator : Taylor(1) → Taylor(2) with coefficients Matrix{Float64}: 1.0 1.0 1.0 1.0 1.0 1.0
julia> fill(2, dom, codom)
LinearOperator : Taylor(1) → Taylor(2) with coefficients Matrix{Int64}: 2 2 2 2 2 2
julia> 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 25
julia> 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 25
julia> 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}: 1
julia> component(B, 2, 2) # extract the linear operator associated with the domain Taylor(1)^2 and codomain Taylor(1)^2
LinearOperator : 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 25
julia> 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 13
julia> 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. Their bar counterparts add_bar
(unicode alias +\bar<tab>
) and sub_bar
(unicode alias -\bar<tab>
) give the result projected in the smallest compatible domain and codomain between the operands.
julia> C = LinearOperator(Taylor(1), Taylor(1), [1 2 ; 3 4])
LinearOperator : Taylor(1) → Taylor(1) with coefficients Matrix{Int64}: 1 2 3 4
julia> 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 6
julia> C + D
LinearOperator : Taylor(1) → Taylor(2) with coefficients Matrix{Int64}: 2 4 6 8 5 6
julia> C - D
LinearOperator : Taylor(1) → Taylor(2) with coefficients Matrix{Int64}: 0 0 0 0 -5 -6
julia> add_bar(C, D) # project(C + D, Taylor(1), Taylor(1))
LinearOperator : Taylor(1) → Taylor(1) with coefficients Matrix{Int64}: 2 4 6 8
julia> sub_bar(C, D) # project(C - D, Taylor(1), Taylor(1))
LinearOperator : Taylor(1) → Taylor(1) with coefficients Matrix{Int64}: 0 0 0 0
julia> C + I
LinearOperator : Taylor(1) → Taylor(1) with coefficients Matrix{Int64}: 2 2 3 5
julia> C - I
LinearOperator : 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 * D
LinearOperator : Taylor(1) → Taylor(1) with coefficients Matrix{Int64}: 7 10 15 22
julia> C ^ 3
LinearOperator : Taylor(1) → Taylor(1) with coefficients Matrix{Int64}: 37 54 81 118
julia> C \ C
LinearOperator : Taylor(1) → Taylor(1) with coefficients Matrix{Float64}: 1.0 0.0 0.0 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 1
julia> C * x # C(x)
Sequence in Taylor(1) with coefficients Vector{Int64}: 3 7
julia> D \ x
Sequence in Taylor(1) with coefficients Vector{Float64}: -1.0000000000000018 1.0000000000000013
API
RadiiPolynomial.LinearOperator
— TypeLinearOperator{T<:VectorSpace,S<:VectorSpace,R<:AbstractMatrix}
Compactly supported linear operator with effective domain and codomain.
Fields:
domain :: T
codomain :: S
coefficients :: R
Constructors:
LinearOperator(::VectorSpace, ::VectorSpace, ::AbstractMatrix)
LinearOperator(coefficients::AbstractMatrix)
: equivalent toLinearOperator(ParameterSpace()^size(coefficients, 2), ParameterSpace()^size(coefficients, 1), coefficients)
Examples
julia> LinearOperator(Taylor(1), Taylor(1), [1 2 ; 3 4])
LinearOperator : Taylor(1) → Taylor(1) with coefficients Matrix{Int64}:
1 2
3 4
julia> LinearOperator(Taylor(2), ParameterSpace(), [1.0 0.5 0.25])
LinearOperator : Taylor(2) → 𝕂 with coefficients Matrix{Float64}:
1.0 0.5 0.25
julia> LinearOperator([1 2 3 ; 4 5 6])
LinearOperator : 𝕂³ → 𝕂² with coefficients Matrix{Int64}:
1 2 3
4 5 6