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https://github.com/Jutho/KrylovKit.jl

Krylov methods for linear problems, eigenvalues, singular values and matrix functions
https://github.com/Jutho/KrylovKit.jl

eigenvalues julia krylov linear-algebra linear-maps singular-values vector

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Krylov methods for linear problems, eigenvalues, singular values and matrix functions

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# KrylovKit.jl



A Julia package collecting a number of Krylov-based algorithms for linear problems, singular

value and eigenvalue problems and the application of functions of linear maps or operators

to vectors.



| **Documentation** | **Build Status** | **Digital Object Idenitifier** | **License** |

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

| [![][docs-stable-img]][docs-stable-url] [![][docs-dev-img]][docs-dev-url] | [![][aqua-img]][aqua-url] [![CI][github-img]][github-url] [![][codecov-img]][codecov-url] | [![DOI][doi-img]][doi-url] | [![license][license-img]][license-url] |



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

[docs-dev-url]: https://jutho.github.io/KrylovKit.jl/latest



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

[docs-stable-url]: https://jutho.github.io/KrylovKit.jl/stable



[github-img]: https://github.com/Jutho/KrylovKit.jl/workflows/CI/badge.svg

[github-url]: https://github.com/Jutho/KrylovKit.jl/actions?query=workflow%3ACI



[aqua-img]: https://raw.githubusercontent.com/JuliaTesting/Aqua.jl/master/badge.svg

[aqua-url]: https://github.com/JuliaTesting/Aqua.jl



[codecov-img]: https://codecov.io/gh/Jutho/KrylovKit.jl/branch/master/graph/badge.svg

[codecov-url]: https://codecov.io/gh/Jutho/KrylovKit.jl



[license-img]: http://img.shields.io/badge/license-MIT-brightgreen.svg?style=flat

[license-url]: LICENSE.md



[doi-img]: https://zenodo.org/badge/DOI/10.5281/zenodo.10622234.svg

[doi-url]: https://doi.org/10.5281/zenodo.10622234



## Release notes for the latest version



### v0.7

This version now depends on and uses [VectorInterface.jl](https://github.com/Jutho/VectorInterface.jl)

to define the vector-like behavior of the input vectors, rather than some minimal set of

methods from `Base` and `LinearAlgebra`. The advantage is that many more types from standard

Julia are now supported out of the box, such as nested vectors or immutable objects such as

tuples. For custom user types for which the old set of required methods was implemented, there

are fallback definitions of the methods in VectorInferace.jl such that these types should still

be supported, but this might result in warnings being printed. It is recommend to implement full

support for at least the methods in VectorInterface without bang or with double bang, where the

latter set of methods can use in-place mutation if your type supports this behavior.



In particular, tuples are now supported:



```julia

julia> values, vectors, info = eigsolve(t -> cumsum(t) .+ 0.5 .* reverse(t), (1,0,0,0));



julia> values

4-element Vector{ComplexF64}:

  2.5298897746721303 + 0.0im

  0.7181879189193713 + 0.4653321688070444im

  0.7181879189193713 - 0.4653321688070444im

  0.03373438748912972 + 0.0im



julia> vectors

4-element Vector{NTuple{4, ComplexF64}}:

 (0.25302539267845964 + 0.0im, 0.322913174072047 + 0.0im, 0.48199234088257203 + 0.0im, 0.774201921982351 + 0.0im)

 (0.08084058845575778 + 0.46550907490257704im, 0.16361072959559492 - 0.20526827902633993im, -0.06286027036719286 - 0.6630573167350086im, -0.47879640378455346 - 0.18713670961291684im)

 (0.08084058845575778 - 0.46550907490257704im, 0.16361072959559492 + 0.20526827902633993im, -0.06286027036719286 + 0.6630573167350086im, -0.47879640378455346 + 0.18713670961291684im)

 (0.22573986355213632 + 0.0im, -0.5730667760748933 + 0.0im, 0.655989711683001 + 0.0im, -0.4362493350466509 + 0.0im)

```



## Overview

KrylovKit.jl accepts general functions or callable objects as linear maps, and general Julia

objects with vector like behavior (as defined in the docs) as vectors.



The high level interface of KrylovKit is provided by the following functions:

*   `linsolve`: solve linear systems

*   `eigsolve`: find a few eigenvalues and corresponding eigenvectors

*   `geneigsolve`: find a few generalized eigenvalues and corresponding vectors

*   `svdsolve`: find a few singular values and corresponding left and right singular vectors

*   `exponentiate`: apply the exponential of a linear map to a vector

*   `expintegrator`: [exponential integrator](https://en.wikipedia.org/wiki/Exponential_integrator)

    for a linear non-homogeneous ODE, computes a linear combination of the `ϕⱼ` functions which generalize `ϕ₀(z) = exp(z)`.



## Installation

`KrylovKit.jl` can be installed with the Julia package manager.

From the Julia REPL, type `]` to enter the Pkg REPL mode and run:

```

pkg> add KrylovKit

```



Or, equivalently, via the `Pkg` API:

```julia

julia> import Pkg; Pkg.add("KrylovKit.jl")

```



## Documentation



-   [**STABLE**][docs-stable-url] - **documentation of the most recently tagged version.**

-   [**DEVEL**][docs-dev-url] - *documentation of the in-development version.*



## Project Status



The package is tested against Julia `1.0`, the current stable and the nightly builds of the Julia `master` branch on Linux, macOS, and Windows, 32- and 64-bit architecture and with `1` and `4` threads.



## Questions and Contributions



Contributions are very welcome, as are feature requests and suggestions. Please open an [issue][issues-url] if you encounter any problems.



[issues-url]: https://github.com/Jutho/KrylovKit.jl/issues