Awesome

A Julia Linear Operator Package

Documentation	Linux/macOS/Windows/FreeBSD	Coverage	DOI

How to Cite

If you use LinearOperators.jl in your work, please cite using the format given in CITATION.cff.

Philosophy

Operators behave like matrices (with some exceptions - see below) 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

How to use

Check the tutorial.

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
`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

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
`solve_shifted_system!`	Solves linear system $(B + \sigma I) x = b$, where $B$ is a forward L-BFGS operator and $\sigma \geq 0$.

Other Operations on Operators

Operators can be transposed (transpose(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 below).

Differences

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

A = rand(5,5)
opA = LinearOperator(A)
A[:,1] * 3 # Vector
opA[:,1] * 3 # LinearOperator
A[:,1] * [3] # ERROR
opA[:,1] * [3] # Vector

This is also true for A[i,J], which returns vectors on 0.5, and for the scalar A[i,j]. Similarly, opA[1,1] is an operator of size (1,1):"

opA[1,1] # LinearOperator
A[1,1] # Number

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

full(opA[:,1]) # nx1 matrix

Other Operators

LimitedLDLFactorizations features a limited-memory LDL<sup>T</sup> 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.

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.