Eigen solvers (Eig)

See also Eigen solvers in the BifurcationKit.jl documentation for more information, for example on how to implement your own eigensolver.

The eigensolver is specified through NewtonPar(eigsolver = ...) and is used to monitor the stability of the solutions during continuation (see Bifurcation detection (codim 1)).

Generic eigen solvers

BifurcationKit.jl provides a list of eigensolvers which are directly usable here:

1. DefaultEig for small problems (dense).
2. EigArpack for sparse problems.
3. EigArnoldiMethod for sparse problems.
4. EigKrylovKit for matrix-free problems.

Generalized / DAE eigen solvers

For a differential-algebraic problem $M\dot z = F(z, p)$, the stability is given by the generalized eigenvalue problem

\[J\,\phi = \lambda\, M\,\phi,\]

see Mass matrix. When GridapBifProblem is used (it is an AbstractDAEBifProblem), BifurcationKit.jl automatically wraps the user eigensolver into EigenDAE and calls it with the mass matrix. One can also build the generalized solver explicitly:

1. EigenMassMatrix(M, eig) solves $J\phi = \lambda M\phi$ using the eigensolver eig. For example

using BifurcationKitM0 = GridapBifurcationKit.get_mass_matrix(prob)eig = EigenMassMatrix(M0, EigArnoldiMethod(sigma = 0.1, which = LM()))λ, vp, conv, iter = eig(J, 10)

1. EigenDAE(eig) is the wrapper used internally for AbstractDAEBifProblem. Passing a plain eigensolver to NewtonPar is enough, the mass matrix is added automatically.