Add EigenvalueProblem support

[BacAmorim]

As discussed here it would be nice to support eigenvalues problems in the SciML ecosystem.

Some of the things that should be included:

• dense eigenproblems via the Julia's built in eigen function from LinearAlgebra
• Arnoldi-Lanczos methods for sparse eigenproblems (including shift-and-invert): some packages that implement Arnoldi-Lanczos are Arpack.jl, ArnoldiMethod.jl, KrylovKit.jl
• Jacobi-Davidson method: there is JacobiDavidson.jl

Nice to have:

• FEAST and IFEAST method: I did a prototype implementation of the FEAST solver in julia. It needs more work and the really interesting part iterative-FEAST or IFEAST (which does not require matrix factorizations, but instead only needs matrix-vector multiplication) is not implemented yet. I can give it a try

[BacAmorim]

For reference, there are other packages that also provide wrappers for multiple eigensolvers:

• https://github.com/rveltz/BifurcationKit.jl/blob/master/src/EigSolver.jl
• https://github.com/pablosanjose/Quantica.jl/blob/master/src/diagonalizer.jl

But I think it would be beneficial to have separate package just devoted to eigensolvers.

[BacAmorim]

@rveltz @pablosanjose

[pablosanjose]

In the #greatrefactor branch of Quantica.jl I've coalesced a bunch of eigensolvers into a very small submodule dedicated generically to this task.

https://github.com/pablosanjose/Quantica.jl/blob/ad6c3657ff2170dc7bfb696bc654a547c052aa62/src/solvers/eigensolvers.jl#L1

It currently supports ArnoldiMethod.jl, KrylovKit.jl, Arpack.jl and shift-invert through SparseSuite and LinearMaps.jl. I find this design very simple to extend. Having this as a standalone package would be really nice.

[vpuri3]

thanks for bringing this up @BacAmorim. Developing an EigenvalueProblem interface has been on my radar for quite some time, and I'd love to help. If you start iterating on the API or start putting stuff together in a PR, I can surely help! LMK.

Maybe just start a PR to LinearSolve with stuff like

src/Eigen/Eigen.jl

abstract type AbstractEigenvalueProblem <: SciMLBase.DEProblem end
abstract type AbstractEigenvalueAlgorithm <: SciMLBase.DEAlgorithm end
abstract type AbstractEigenCache <: SciMLBase.DECache end
abstract type AbstractEigenvalueSolution <: SciMLBase.AbstractDAESolution end

"""
A u = B (\lambda u)
"""
struct EigenvalueProblem
    A
    B
    u0
end

function Base.summary(io::IO, prob::EigenvalueProblem) end
function Base.show(io::IO, mime::MIME"text/plain", prob::EigenvalueProblem) end

struct EigenCache <: AbstractEigenCache
    A
    B
    alg
    cache
    abstol
    reltol
    maxiters
    verbose
end

struct ArnoldiMethodJL_PartialScur <: AbstractEigenvalueAlgorithm
end

struct EigenvalueSolution <: AbstractEigenvalueSolution
    u
    \lambda
    retcode
    iters
    cache
end

function SciMLBase.solve(cache::Eigencache, alg::ArnoldiMethodJL_PartialSchur)
end

function SciMLBase.init(prob::AbstractEigenvalueProblem, alg::Union{Nothing,AbstractEigenvalueAlgorithm},
                        alg::AbstractBVPDEAlgorithm = nothing;
                        abstol=default_tol(eltype(prob.op)),
                        reltol=default_tol(eltype(prob.op)),
                        maxiters=length(prob.f),
                        verbose=false,
                        kwargs...
                       )

     return EigenCache()
end


function SciMLBase.solve(prob::AbstractEigenvalueProblem, args...; kwargs...)
    solve(init(prob, nothing, args...; kwargs...))
end

function SciMLBase.solve(prob::EigenvalueProblem, alg::Union{AbstractBVPDEAlgorithm,Nothing}, args...; kwargs...)
    solve(init(prob, alg, args...; kwargs...), alg; kwargs...)
end

the user should be able to specify how many eigenvalues they want and pass in vectors/matrix if they want it in place

[vpuri3]

adding GenericArpack here for reference: https://github.com/dgleich/GenericArpack.jl

[rveltz]

TMO it is missing NonlinearEigenProblems.jl