Properties and Traits

size(A::AbstractOperator, [dom,])

Returns the size of an AbstractOperator. Type size(A,1) for the size of the codomain and size(A,2) for the size of the codomain.

Note that the size is always returned as a Tuple, so for a 2D operator the size of the codomain will be (m,) and the size of the domain will be (n,) for an m x n operator.

julia> size(FiniteDiff((10,20), 1))
((9, 20), (10, 20))

julia> size(FiniteDiff((10,20), 1),1)
(9, 20)

julia> size(FiniteDiff((10,20), 1),2)
(10, 20)

ndims(A::AbstractOperator, [dom,])

Returns a Tuple with the number of dimensions of the codomain and domain of an AbstractOperator. Type ndims(A,1) for the number of dimensions of the codomain and ndims(A,2) for the number of dimensions of the codomain.

julia> V = Variation((2,3,4))
Ʋ  ℝ^(2, 3, 4) -> ℝ^(24, 3)

julia> ndims(V)
(2, 3)

julia> ndims(V,1)
2

julia> ndims(V,2)
3

ndoms(L::AbstractOperator, [dom::Int]) -> (number of codomains, number of domains)

Returns the number of codomains and domains of a AbstractOperator. Optionally you can specify the codomain (with dom = 1) or the domain (with dom = 2)

julia> ndoms(Eye(10,10))
(1, 1)

julia> ndoms(hcat(Eye(10,10),Eye(10,10)))
(1, 2)

julia> ndoms(hcat(Eye(10,10),Eye(10,10)),2)
2

julia> ndoms(DCAT(Eye(10,10),Eye(10,10)))
(2, 2)

domain_type(A::AbstractOperator)

Returns the type of the domain.

julia> domain_type(DiagOp(rand(10)))
Float64

julia> domain_type(hcat(Eye(Complex{Float64},(10,)),DiagOp(rand(ComplexF64, 10))))
(ComplexF64, ComplexF64)

codomain_type(A::AbstractOperator)

Returns the type of the codomain.

julia> codomain_type(DiagOp(rand(ComplexF64, 10)))
ComplexF64 (alias for Complex{Float64})

julia> codomain_type(vcat(Eye(Complex{Float64},(10,)),DiagOp(rand(ComplexF64, 10))))
(ComplexF64, ComplexF64)

domain_storage_type(L::AbstractOperator)

Returns the type of the storage for the domain of the operator.

julia> domain_storage_type(DiagOp(rand(10)))
Array{Float64}

julia> domain_storage_type(hcat(Eye(Complex{Float64},(10,)),DiagOp(rand(ComplexF64, 10))))
RecursiveArrayTools.ArrayPartition{ComplexF64, Tuple{Array{ComplexF64}, Array{ComplexF64}}}

codomain_storage_type(L::AbstractOperator)

Returns the type of the storage of for the codomain of the operator.

julia> codomain_storage_type(DiagOp(rand(ComplexF64,10)))
Array{ComplexF64}

julia> codomain_storage_type(vcat(Eye(Complex{Float64},(10,)),DiagOp(rand(ComplexF64,10))))
RecursiveArrayTools.ArrayPartition{ComplexF64, Tuple{Array{ComplexF64}, Array{ComplexF64}}}

LinearAlgebra.diag(A::AbstractOperator)

Returns the diagonal of A. If A is not diagonal, an error is thrown.

The diagonal is defined as the vector d such that A * x = d .* x for all x in the domain of A, where .* is the element-wise multiplication.

LinearAlgebra.opnorm(A::AbstractOperator)

Returns the operator norm of A. The operator norm is defined as the maximum singular value of A. It is computed using the power method by default, unless the operator has a fast implementation.

The operator norm is defined as: ‖A‖ = sup_{x != 0} ‖A*x‖ / ‖x‖.

Parameters of power iteration:

• Maximum number of iterations: 100
• Tolerance for convergence: 1e-6

These parameters can be adjusted in the estimate_opnorm function.

is_linear(A)

Returns true if A is a linear operator. Operator A is linear if A * (x+y) = A * x + A * y and A * (α * x) = α * A * x for all x and y in the domain of A and all scalars α.

is_eye(A)

Returns true if A is an identity operator. Operator A is an identity operator if A * x = x for all x in the domain of A.

is_null(A)

Returns true if A is a null operator. Operator A is null if A * x = 0 for all x in the domain of A, where 0 is the zero element of the codomain of A (e.g. zero vector).

is_symmetric(A)

Returns true if A is a symmetric operator. Operator A is symmetric if A * x = A' * x for all x in the domain of A, where A' is the adjoint of A. In other words, A is symmetric if it is equal to its adjoint.

is_diagonal(A)

Returns true if A is a diagonal operator. Operator A is diagonal if (A * x)[i] = (A * eᵢ)[i] for all x in the domain of A and all i, where eᵢ is the i-th canonical basis vector. In other words, (A * x)[i] depends only on x[i].

is_AcA_diagonal(A)

Returns true if A*A' is diagonal, where A' is the adjoint of A. Compounds A*A' is diagonal if ((A*A') * x)[i] = (A*A') * eᵢ)[i] for all x in the domain of A and all i, where eᵢ is the i-th canonical basis vector. In other words, ((A*A') * x)[i] depends only on x[i].

diag_AcA(A)

Returns the diagonal of A*A', where A' is the adjoint of A. If A*A' is not diagonal, an error is thrown.

is_AAc_diagonal(A)

Returns true if A'*A is diagonal, where A' is the adjoint of A. Compounds A'*A is diagonal if ((A'*A) * x)[i] = (A'*A) * eᵢ)[i] for all x in the domain of A and all i, where eᵢ is the i-th canonical basis vector. In other words, ((A'*A) * x)[i] depends only on x[i].

diag_AAc(A)

Returns the diagonal of A'*A, where A' is the adjoint of A. If A'*A is not diagonal, an error is thrown.

is_orthogonal(A)

Returns true if A is an orthogonal operator. Operator A is orthogonal if A * A' = A' * A = I, where A' is the adjoint of A and I is the identity operator.

is_invertible(A)

Returns true if A is an invertible operator. Operator A is invertible if there exists an operator B such that A * B = B * A = I, where I is the identity operator.

is_full_row_rank(A)

Returns true if A has full row rank. Operator A has full row rank if the rows of A are linearly independent. In other words, the number of linearly independent rows of A is equal to the number of rows of A.

is_full_row_rank(A)

Returns true if A has full column rank. Operator A has full column rank if the columns of A are linearly independent. In other words, the number of linearly independent columns of A is equal to the number of columns of A.

displacement(A::AbstractOperator)

Returns the displacement of the operator.

julia> A = AffineAdd(Eye(4),[1.;2.;3.;4.])
I+d  ℝ^4 -> ℝ^4

julia> displacement(A)
4-element Vector{Float64}:
 1.0
 2.0
 3.0
 4.0

remove_displacement(A::AbstractOperator)

Removes the displacement of the operator.

is_sliced(A)

Returns true if A is a sliced operator. Operator A is sliced if it applies to only a subset of the input values.

julia> is_sliced(DiagOp(rand(10)))
false

julia> is_sliced(DiagOp(rand(10)) * GetIndex((20,), 1:10))
true

remove_slicing(A)

Returns the operator A without slicing. Operator A is sliced if it applies to only a subset of the input values.

is_thread_safe(L::AbstractOperator)

Returns whether the operator is thread safe (i.e. it can be used on multiple arrays simulaneously).

estimate_opnorm(A::AbstractOperator)

Estimates the operator norm of A. The operator norm is defined as the maximum singular value of A. It is computed using the power method with reduced iterations unless the operator has a fast implementation.

The operator norm is defined as: ‖A‖ = sup_{x != 0} ‖A*x‖ / ‖x‖.

Parameters of power iteration:

• Maximum number of iterations: 20
• Tolerance for convergence: 0.01

These parameters can be adjusted by passing maxit and tol keyword arguments. E.g.:

julia> estimate_opnorm(A; maxit=50, tol=1e-6)