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}:
0.16020814312432063
0.6514589370303275
0.7233087667435946
0.20105623666790973
0.3789448071142515
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.