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 = real, rev = true)
nev2 = min(nev, length(I))
return F.values[I[1:nev2]], F.vectors[:, I[1:nev2]], true, 1
end
The eigenvalues must be ordered by increasing real part for the detection of bifurcations to work properly.
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
- Default
eigen
Julia eigensolver for matrices. You can create it viaeig = DefaultEig()
. Note that you can also specify how the eigenvalues are ordered (by decreasing real part by default). You can then compute 3 eigenelements ofJ
likeeig(J, 3)
. - Eigensolver from
Arpack.jl
. You can create one viaeigsolver = EigArpack()
and pass appropriate options (see Arpack.jl). For example, you can compute eigenvalues using Shift-Invert method with shiftσ
by usingEigArpack(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 matrixJ
, you can compute 3 eigen-elements usingeig(J, 3)
. In the case of a Matrix-Free jacobiandx -> 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 usingeig(dx -> J(dx), 3)
. - Eigensolver from
KrylovKit.jl
. You create one viaeig = 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 matrixJ
, you can create a solver like thiseig = EigKrylovKit()
. Then, you compute 3 eigen-elements usingeig(J, 3)
. In the case of a Matrix-Free jacobiandx -> 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 usingeig(dx -> J(dx), 3)
. - Eigensolver from
ArnoldiMethod.jl
. You can create one viaeig = EigArnoldiMethod()
and pass appropriate options (see ArnoldiMethod.jl). For example, you can compute eigenvalues using the Shift-Invert method with shiftσ
by usingEigArnoldiMethod(σ, LR())
. Note that you can also specify how the eigenvalues are ordered (by decreasing real part by default). In the case of a matrixJ
, you can create a solver likeeig = EigArnoldiMethod()
. Then, you compute 3 eigen-elements usingeig(J, 3)
. In the case of a Matrix-Free jacobiandx -> 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 usingeig(dx -> J(dx), 3)
.
This is probably due to iterative refinement conducted by SuiteSparse
as explained in this blog post. You can disable this using
using SuiteSparse
SuiteSparse.UMFPACK.umf_ctrl[8] = 0