A Julia Linear Operator Package

Operators behave like matrices (with exceptions) 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 1.6 and up.

How to Install

pkg> add LinearOperators
pkg> test LinearOperators

Operators Available

Operator	Description
`LinearOperator`	Base class. Useful to define operators from functions
`TimedLinearOperator`	Linear operator instrumented with timers from `TimerOutputs`
`BlockDiagonalOperator`	Block-diagonal linear operator
`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
`opLDL`	Similar to `opCholesky`, for general sparse symmetric matrices
`opHouseholder`	Apply a Householder transformation `I-2hh'`
`opHermitian`	Represent a symmetric/hermitian operator based on the diagonal and strict lower triangle
`opRestriction`	Represent a selection of "rows" when composed on the left with an existing operator
`opExtension`	Represent a selection of "columns" when composed on the right with an existing operator
`LBFGSOperator`	Limited-memory BFGS approximation in operator form (damped or not)
`InverseLBFGSOperator`	Inverse of a limited-memory BFGS approximation in operator form (damped or not)
`LSR1Operator`	Limited-memory SR1 approximation in operator form
`kron`	Kronecker tensor product in linear operator form

Utility Functions

Function	Description
`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
`diag`	Extract the diagonal of an operator
`Matrix`	Convert an abstract operator to a dense array
`hermitian`	Determine whether the operator is Hermitian
`push!`	For L-BFGS or L-SR1 operators, push a new pair {s,y}
`reset!`	For L-BFGS or L-SR1 operators, reset the data
`show`	Display basic information about an operator
`size`	Return the size of a linear operator
`symmetric`	Determine whether the operator is symmetric
`normest`	Estimate the 2-norm

Other Operations on Operators

Operators can be transposed (A.'), conjugated (conj(A)) and conjugate-transposed (A'). Operators can be sliced (A[:,3], A[2:4,1:5], A[1,1]), but unlike matrices, slices always return operators (see differences).

Differences

Unlike matrices, an operator never reduces to a vector or a number.

using LinearOperators
A = rand(5,5)
opA = LinearOperator(A)
A[:,1] * 3 isa Vector

true

opA[:,1] * 3 isa LinearOperator

true

opA[:,1] * [3] isa Vector

true

However, the following returns an error

A[:,1] * [3]

This is also true for A[i,:], which would return a vector and for the scalar A[i,j]. Similarly, opA[1,1] is an operator of size (1,1):"

(opA[1,1] * [3])[1] - A[1,1] * 3

0.0

In the same spirit, the operator Matrix always returns a matrix.

Matrix(opA[:,1])

5×1 Matrix{Float64}:
 0.2586014478438151
 0.5343056230920415
 0.6894386736882767
 0.00804130308107498
 0.7410212398721603

Other Operators

LLDL features a limited-memory LDLᵀ 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")

Bug reports and discussions

If you think you found a bug, feel free to open an issue. Focused suggestions and requests can also be opened as issues. Before opening a pull request, start an issue or a discussion on the topic, please.

If you want to ask a question not suited for a bug report, feel free to start a discussion here. This forum is for general discussion about this repository and the JuliaSmoothOptimizers organization, so questions about any of our packages are welcome.