ArnoldiMethod.jl

The Arnoldi Method with Krylov-Schur restart, natively in Julia.

Docs

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> 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 and more accuracy.

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