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https://github.com/JuliaLinearAlgebra/ArnoldiMethod.jl
The Arnoldi Method with Krylov-Schur restart, natively in Julia.
https://github.com/JuliaLinearAlgebra/ArnoldiMethod.jl
eigenvalues eigenvectors julia linear-algebra
Last synced: 3 months ago
JSON representation
The Arnoldi Method with Krylov-Schur restart, natively in Julia.
- Host: GitHub
- URL: https://github.com/JuliaLinearAlgebra/ArnoldiMethod.jl
- Owner: JuliaLinearAlgebra
- License: mit
- Created: 2018-03-11T20:33:36.000Z (over 6 years ago)
- Default Branch: master
- Last Pushed: 2024-05-06T19:54:56.000Z (6 months ago)
- Last Synced: 2024-07-19T06:29:27.488Z (4 months ago)
- Topics: eigenvalues, eigenvectors, julia, linear-algebra
- Language: Julia
- Homepage: https://julialinearalgebra.github.io/ArnoldiMethod.jl/dev
- Size: 568 KB
- Stars: 94
- Watchers: 5
- Forks: 18
- Open Issues: 5
-
Metadata Files:
- Readme: readme.md
- Contributing: CONTRIBUTING.md
- License: LICENSE
- Citation: CITATION.cff
Awesome Lists containing this project
- awesome-sciml - JuliaLinearAlgebra/ArnoldiMethod.jl: Implicitly Restarted Arnoldi Method, natively in Julia
README
# ArnoldiMethod.jl
[](https://github.com/JuliaLinearAlgebra/ArnoldiMethod.jl/actions/workflows/ci.yml) [](https://codecov.io/github/JuliaLinearAlgebra/ArnoldiMethod.jl?branch=master)
The Arnoldi Method with Krylov-Schur restart, natively in Julia.
## Docs
[](https://julialinearalgebra.github.io/ArnoldiMethod.jl/stable) [](https://julialinearalgebra.github.io/ArnoldiMethod.jl/dev)
## Goal
Make `eigs` an efficient and native Julia function.
## Installation
Open the package manager in the REPL via `]` and run
```
(v1.6) pkg> add ArnoldiMethod
```
## Example
```julia
julia> using ArnoldiMethod, LinearAlgebra, SparseArrays
julia> A = spdiagm(
-1 => fill(-1.0, 99),
0 => fill(2.0, 100),
1 => fill(-1.0, 99)
);
julia> decomp, history = partialschur(A, nev=10, tol=1e-6, which=:SR);
julia> decomp
PartialSchur decomposition (Float64) of dimension 10
eigenvalues:
10-element Array{Complex{Float64},1}:
0.0009674354160236865 + 0.0im
0.003868805732811139 + 0.0im
0.008701304061962657 + 0.0im
0.01546025527344699 + 0.0im
0.024139120518486677 + 0.0im
0.0347295035554728 + 0.0im
0.04722115887278571 + 0.0im
0.06160200160067088 + 0.0im
0.0778581192025522 + 0.0im
0.09597378493453936 + 0.0im
julia> history
Converged: 10 of 10 eigenvalues in 174 matrix-vector products
julia> norm(A * decomp.Q - decomp.Q * decomp.R)
6.39386920955869e-8
julia> λs, X = partialeigen(decomp);
julia> norm(A * X - X * Diagonal(λs))
6.393869211477937e-8
```
## ArnoldiMethod.jl is generic
ArnoldiMethod.jl's Schur decomposition is written in Julia, it does not use LAPACK. This allows
you to use arbitrary number types.
We repeat the above example with [DoubleFloats.jl](https://github.com/JuliaMath/DoubleFloats.jl)
and more accuracy.
```julia
julia> using ArnoldiMethod, DoubleFloats, LinearAlgebra, SparseArrays
julia> A = spdiagm(
-1 => fill(Double64(-1), 99),
0 => fill(Double64(2), 100),
1 => fill(Double64(-1), 99)
);
julia> decomp, history = partialschur(A, nev=10, tol=1e-28, which=:SR);
julia> decomp
PartialSchur decomposition (Double64) of dimension 10
eigenvalues:
10-element Vector{Complex{Double64}}:
9.6743541602387015850892187143202406e-04 + 0.0im
3.86880573281130335530623278634505297e-03 + 0.0im
8.70130406196283903200426213162702754e-03 + 0.0im
1.54602552734469798152574737604660783e-02 + 0.0im
2.41391205184865585041130463401142985e-02 + 0.0im
3.47295035554726251259365854776375027e-02 + 0.0im
4.72211588727859409278578405476287512e-02 + 0.0im
6.16020016006677741124091774018629622e-02 + 0.0im
7.78581192025509024705094505968950069e-02 + 0.0im
9.59737849345402152393882633733121172e-02 + 0.0im
julia> history
Converged: 10 of 10 eigenvalues in 442 matrix-vector products
julia> norm(A * decomp.Q - decomp.Q * decomp.R)
4.53243232681764960018699535610331068e-30
julia> norm(decomp.Q' * decomp.Q - I)
3.53573060252329801278244497021683397e-29
```