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https://github.com/JuliaSmoothOptimizers/Krylov.jl

A Julia Basket of Hand-Picked Krylov Methods
https://github.com/JuliaSmoothOptimizers/Krylov.jl

julia julia-language krylov linear-algebra linear-systems optimization

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A Julia Basket of Hand-Picked Krylov Methods

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



| **Documentation** | **CI** | **Coverage** | **DOI** | **Downloads** |

|:-----------------:|:------:|:------------:|:-------:|:-------------:|

| [![docs-stable][docs-stable-img]][docs-stable-url] [![docs-dev][docs-dev-img]][docs-dev-url] | [![build-gh][build-gh-img]][build-gh-url] [![build-cirrus][build-cirrus-img]][build-cirrus-url] | [![codecov][codecov-img]][codecov-url] | [![doi][doi-img]][doi-url] | [![downloads][downloads-img]][downloads-url] |



[docs-stable-img]: https://img.shields.io/badge/docs-stable-blue.svg

[docs-stable-url]: https://JuliaSmoothOptimizers.github.io/Krylov.jl/stable

[docs-dev-img]: https://img.shields.io/badge/docs-dev-purple.svg

[docs-dev-url]: https://JuliaSmoothOptimizers.github.io/Krylov.jl/dev

[build-gh-img]: https://github.com/JuliaSmoothOptimizers/Krylov.jl/workflows/CI/badge.svg?branch=main

[build-gh-url]: https://github.com/JuliaSmoothOptimizers/Krylov.jl/actions

[build-cirrus-img]: https://img.shields.io/cirrus/github/JuliaSmoothOptimizers/Krylov.jl?logo=Cirrus%20CI

[build-cirrus-url]: https://cirrus-ci.com/github/JuliaSmoothOptimizers/Krylov.jl

[codecov-img]: https://codecov.io/gh/JuliaSmoothOptimizers/Krylov.jl/branch/main/graph/badge.svg

[codecov-url]: https://app.codecov.io/gh/JuliaSmoothOptimizers/Krylov.jl

[doi-img]: https://joss.theoj.org/papers/10.21105/joss.05187/status.svg

[doi-url]: https://doi.org/10.21105/joss.05187

[downloads-img]: https://img.shields.io/badge/dynamic/json?url=http%3A%2F%2Fjuliapkgstats.com%2Fapi%2Fv1%2Fmonthly_downloads%2FKrylov&query=total_requests&suffix=%2Fmonth&label=Downloads

[downloads-url]: https://juliapkgstats.com/pkg/Krylov



## How to Cite



If you use Krylov.jl in your work, please cite it using the metadata given in [`CITATION.cff`](https://github.com/JuliaSmoothOptimizers/Krylov.jl/blob/main/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_**.



2. 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‖.




3. 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.



4. Adjoint systems





  Ax = b   and   Aᴴy = c




where **_A_** can have any shape.



5. Saddle-point and Hermitian quasi-definite systems





  [M     A]  [x]            =           [b]

  


  [Aᴴ        -N]  [y]    [c]




where **_A_** can have any shape.



6. 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

julia> ]

pkg> add Krylov

pkg> test Krylov

```



## Bug reports and discussions



If you think you found a bug, feel free to open an [issue](https://github.com/JuliaSmoothOptimizers/Krylov.jl/issues).

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](https://github.com/JuliaSmoothOptimizers/Organization/discussions). This forum is for general discussion about this repository and the [JuliaSmoothOptimizers](https://github.com/JuliaSmoothOptimizers) organization, so questions about any of our packages are welcome.