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This package provides implementations of certain of the most useful Krylov method for a variety of problems:
Ax = b
should be solved when b lies in the range space of A. This situation occurs when
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
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‖.
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
Overdetermined sytems are less common but also occur.
Ax = b and Aᵀy = c
where A can have any shape.
[M A] [x] = [b]
[Aᵀ -N] [y] [c]
where A can have any shape.
Krylov solvers are particularly appropriate in situations where such problems must be solved but a factorization is not possible, either because:
Iterative methods are recommended in either of the following situations:
All solvers in Krylov.jl 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
If you use Krylov.jl in your work, please cite using the format given in CITATION.bib
.
03/10/2015
5 days ago
509 commits