A Julia Linear Operator Package

Operators behave like matrices but are defined by their effect when applied to a vector. They can be transposed, conjugated, or combined with other operators cheaply. The costly operation is deferred until multiplied with a vector.

Compatibility

Julia 0.3 and 0.4.

How to Install

Pkg.clone("https://github.com/dpo/LinearOperators.jl.git")

Documentation

See https://dpo.github.io/LinearOperators.jl/LinearOperators.html for preliminary documentation.

Example 1

Operators may be defined from matrices and combined using the usual operations, but the result is deferred until the operator is applied.

julia> A1 = rand(5,7);
julia> A2 = sprand(7,3,.3);
julia> op1 = LinearOperator(A1);
julia> op2 = LinearOperator(A2);
julia> op = op1 * op2;  # Does not form A1 * A2
julia> x = rand(3);
julia> y = op * x;

Example 2

Operators may be defined to represent (approximate) inverses.

julia> A = rand(5,5); A = A' * A;
julia> op = opCholesky(A);  # Use, e.g., as a preconditioner
julia> v = rand(5);
julia> norm(A \ v - op * v) / norm(v)
1.6522645623951567e-14

Example 3

Operators may be defined from functions. In the example below, the transposed isn't defined, but it may be inferred from the conjugate transposed. Missing operations are represented as nullable functions. Nullable types were introduced in Julia 0.4 but are provided in Julia 0.3 by Compat.jl.

julia> using Compat  # only required if you use Julia 0.3.
julia> dft = LinearOperator(10, 10, Float64, false, false,
                            v -> fft(v),
                            Nullable{Function}(),  # this operation is "missing".
                            w -> ifft(w));
julia> x = rand(10);
julia> y = dft * x;
julia> norm(dft' * y - x)  # DFT is an orthogonal operator
2.2929868617541516e-16
julia> dft.' * y
julia> 10-element Array{Complex{Float64},1}:
  0.927514+0.227908im
 0.0472611+0.62094im
 ...

Operators Available

Operator	Description
LinearOperator	Base class. Useful to define operators from functions
opEye	Identity operator
opOnes	All ones operator
opZeros	All zeros operator
opDiagonal	Square (equivalent to diagm()) or rectangular diagonal operator
opInverse	Equivalent to \
opCholesky	More efficient than opInverse for symmetric positive definite matrices
opHouseholder	Apply a Householder transformation I-2hh'
opHermitian	Represent a symmetric/hermitian operator based on the diagonal and strict lower triangle

Utility Functions

Function	Description
full	Convert an abstract operator to a dense array
check_ctranspose	Cheap check that A' is correctly implemented
check_hermitian	Cheap check that A = A'
check_positive_definite	Cheap check that an operator is positive (semi-)definite

Other Operations on Operators

Operators can be transposed (A.'), conjugated (conj(A)) and conjugate-transposed (A').

Other Operators

• LLDL features a limited-memory LDL^T factorization operator that may be used as preconditioner in iterative methods
• MUMPS.jl features a full distributed-memory factorization operator that may be used to represent the preconditioner in, e.g., constraint-preconditioned Krylov methods.

Testing

julia> Pkg.test("LinearOperators")