## Krylov.jl

A Julia Basket of Hand-Picked Krylov Methods
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March 2015

# Krylov.jl: A Julia basket of hand-picked Krylov methods

## How to Cite

If you use Krylov.jl in your work, please cite it using the metadata given in `CITATION.cff`.

#### BibTeX

``````@article{montoison-orban-2023,
author  = {Montoison, Alexis and Orban, Dominique},
title   = {{Krylov.jl: A Julia basket of hand-picked Krylov methods}},
journal = {Journal of Open Source Software},
volume  = {8},
number  = {89},
pages   = {5187},
year    = {2023},
doi     = {10.21105/joss.05187}
}
``````

## Content

This package provides implementations of certain of the most useful Krylov method for a variety of problems:

1. 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.
1. 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 systems 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‖.

1. Linear least-norm problems

minimize ‖x‖   subject to   Ax = b

should 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 systems are less common but also occur.

Ax = b   and   Aᴴy = c

where A can have any shape.

1. Saddle-point and Hermitian quasi-definite systems

[M     A]  [x] = [b]
[Aᴴ   -N]  [y]    [c]

where A can have any shape.

1. Generalized saddle-point and non-Hermitian partitioned systems

[M   A]  [x] = [b]
[B   N]  [y]    [c]

where A can have any shape and B has the shape of Aᴴ. 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.

## Features

All solvers in Krylov.jl have in-place version, are compatible with GPU and work in any floating-point data type.

## How to Install

Krylov can be installed and tested through the Julia package manager:

```julia> ]