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If you use Krylov.jl in your work, please cite using the format given in CITATION.bib.
This package provides implementations of certain of the most useful Krylov method for a variety of problems:
- Square or rectangular full-rank systems
Ax = b
should be solved when b lies in the range space of A. This situation occurs when
- A is square and nonsingular,
- A is tall and has full column rank and b lies in the range of A.
- Linear least-squares problems
minimize ‖b - Ax‖
should be solved when b is not in the range of A (inconsistent systems), regardless of the shape and rank of A. This situation mainly occurs when
- A is square and singular,
- A is tall and thin.
Underdetermined sytems are less common but also occur.
If there are infinitely many such x (because A is column rank-deficient), one with minimum norm is identified
minimize ‖x‖ subject to x ∈ argmin ‖b - Ax‖.
- Linear least-norm problems
minimize ‖x‖ subject to Ax = b
sould be solved when A is column rank-deficient but b is in the range of A (consistent systems), regardless of the shape of A. This situation mainly occurs when
- A is square and singular,
- A is short and wide.
Overdetermined sytems are less common but also occur.
- Adjoint systems
Ax = b and Aᵀy = c
where A can have any shape.
- Saddle-point and symmetric quasi-definite (SQD) systems
[M A] [x] = [b]
[Aᵀ -N] [y] [c]
where A can have any shape.
- Generalized saddle-point and unsymmetric partitioned systems
[M A] [x] = [b]
[B N] [y] [c]
where A can have any shape and B has the shape of Aᵀ. A, B, b and c must be all nonzero.
Krylov solvers are particularly appropriate in situations where such problems must be solved but a factorization is not possible, either because:
- A is not available explicitly,
- A would be dense or would consume an excessive amount of memory if it were materialized,
- factors would consume an excessive amount of memory.
Iterative methods are recommended in either of the following situations:
- the problem is sufficiently large that a factorization is not feasible or would be slow,
- an effective preconditioner is known in cases where the problem has unfavorable spectral structure,
- the operator can be represented efficiently as a sparse matrix,
- the operator is fast, i.e., can be applied with better complexity than if it were materialized as a matrix. Certain fast operators would materialize as dense matrices.
All solvers in Krylov.jl have in-place version, are compatible with GPU and work in any floating-point data type.
Krylov can be installed and tested through the Julia package manager:
julia> ]
pkg> add Krylov
pkg> test Krylov