# Eigen solvers (Eig)

The eigen solvers must be subtypes of AbstractEigenSolver.

They provide a way of computing the eigen elements of the Jacobian J. Such eigen solver eigsolve will be called like ev, evecs, itnumber = eigsolve(J, nev; kwargs...) throughout the package, nev being the number of requested eigen elements of largest real part and kwargs being used to send information about the algorithm (perform bisection,...).

Here is an example of the simplest of them (see src/EigSolver.jl for the true implementation) to give you an idea:

struct DefaultEig <: AbstractEigenSolver end

function (l::DefaultEig)(J, nev; kwargs...)
# I put Array so we can call it on small sparse matrices
F = eigen(Array(J))
I = sortperm(F.values, by = x-> real(x), rev = true)
nev2 = min(nev, length(I))
return F.values[I[1:nev2]], F.vectors[:, I[1:nev2]], true, 1
end
Eigenvalues

The eigenvalues must be ordered by increasing real part for the detection of bifurcations to work properly.

Eigenvectors

You have to implement the method geteigenvector(eigsolver, eigenvectors, i::Int) for newtonHopf to work properly.

## Methods for computing eigenvalues

Like for the linear solvers, computing the spectrum of operators $A$ associated to PDE is a highly non trivial task because of the clustering of eigenvalues. Most methods are based on the so-called power method but this only yields the eigenvalues with largest modulus. In case of the Laplacian operator, this can be disastrous and it is better to apply the power method to $(\sigma I-A)^{-1}$ instead.

This method, called Shift-invert, is readily available for the solvers EigArpack and EigArnoldiMethod, see below. It is mostly used to compute interior eigenvalues. For the solver EigKrylovKit, one must implement its own shift invert operator, using for example GMRESKrylovKit.

In some cases, it may be advantageous to consider the Cayley transform $(\sigma I-A)^{-1}(\tau I+A)$ to focus on a specific part of the spectrum. As it is mathematically equivalent to the Shift-invert method, we did not implement it.

## List of implemented eigen solvers

1. Default eigen Julia eigensolver for matrices. You can create it via eig = DefaultEig(). Note that you can also specify how the eigenvalues are ordered (by decreasing real part by default). You can then compute 3 eigenelements of J like eig(J, 3).
2. Eigensolver from Arpack.jl. You can create one via eigsolver = EigArpack() and pass appropriate options (see Arpack.jl). For example, you can compute eigenvalues using Shift-Invert method with shift σ by using EigArpack(σ, :LR). Note that you can specify how the eigenvalues are ordered (by decreasing real part by default). Finally, this method can be used for (sparse) matrix or Matrix-Free formulation. For a matrix J, you can compute 3 eigen-elements using eig(J, 3). In the case of a Matrix-Free jacobian dx -> J(dx), you need to tell the eigensolver the dimension of the state space by giving an example of vector: eig = EigArpack(v0 = zeros(10)). You can then compute 3 eigen-elements using eig(dx -> J(dx), 3).
3. Eigensolver from KrylovKit.jl. You create one via eig = EigKrylovKit() and pass appropriate options (see KrylovKit.jl). This method can be used for (sparse) matrix or Matrix-Free formulation. In the case of a matrix J, you can create a solver like this eig = EigKrylovKit(). Then, you compute 3 eigen-elements using eig(J, 3). In the case of a Matrix-Free jacobian dx -> J(dx), you need to tell the eigensolver the dimension of the state space by giving an example of vector: eig = EigKrylovKit(x₀ = zeros(10)). You can then compute 3 eigen-elements using eig(dx -> J(dx), 3).
4. Eigensolver from ArnoldiMethod.jl. You can create one via eig = EigArnoldiMethod() and pass appropriate options (see ArnoldiMethod.jl). For example, you can compute eigenvalues using the Shift-Invert method with shift σ by using EigArnoldiMethod(σ, LR()). Note that you can also specify how the eigenvalues are ordered (by decreasing real part by default). In the case of a matrix J, you can create a solver like eig = EigArnoldiMethod(). Then, you compute 3 eigen-elements using eig(J, 3). In the case of a Matrix-Free jacobian dx -> J(dx), you need to tell the eigensolver the dimension of the state space by giving an example of vector: eig = EigArnoldiMethod(x₀ = zeros(10)). You can then compute 3 eigen-elements using eig(dx -> J(dx), 3).