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.bib.
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 1.3 and up.
How to Install
pkg> add LinearOperators
pkg> test LinearOperatorsHow 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 |
shape |
Return the size of a linear operator |
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 (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] # VectorThis 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] # NumberIn the same spirit, the operator full always returns a matrix.
full(opA[:,1]) # nx1 matrixOther Operators
- LimitedLDLFactorizations 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.
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.