Interface for Linear Solvers

The linear solver ls must be a subtype of the abstract type AbstractLinearSolver. It is then called as follows

Required methods		Brief description
`ls(J, rhs; a₀ = 0, a₁ = 1, kwargs...)`		Compute the solution `sol` of the linear problem `(a₀ * I + a₁ * J) * sol = rhs` where `J` is the jacobian. Returns `(sol, success::Bool, itnumber)` where `itnumber` is the number of iterations for solving the problem.

Shifts

Two methods _axpy and _axpy_op are provided to help implementing this shift a₀ * I + a₁ * J

Interface for Eigen Solvers

The linear solver eig must be a subtype of the abstract type AbstractEigenSolver. It is then called as follows

Required methods		Brief description
`eig(J, nev::Int; kwargs...)`		Compute the `nev` eigen-elements with largest real part. Returns `(eigenvalues, eigenvectors, success:Bool, itnumber)` where `itnumber` is the number of iterations for completing the computation. `kwargs` can be used to send information about the algorithm (perform bisection,...).
`geteigenvector(eig, eigenvectors, i::Int)`		Returns the ith eigenvectors from the set of `eigenvectors`.

Interface for Bordered Linear Solvers

The bordered linear solver bls must be a subtype of the abstract type AbstractBorderedLinearSolver (which is itself a subtype of AbstractLinearSolver). It is used to solve

\[\tag{BLS}\left[\begin{array}{ll} {\text{shift}\cdot I+J} & {dR} \\ {(\xi_u \cdot dz_u)^T} & {\xi_p \cdot dz_p} \end{array}\right] \cdot\left[\begin{array}{l} {dX} \\ {dl} \end{array}\right] = \left[\begin{array}{l} R \\ n \end{array}\right]\]

where $\xi_u,\xi_p\in\mathbb C$ and $dR,\xi_u\in\mathbb C^N$.

PALC

To work with PALC, the struct bls must implement update_bls to update the internal linear solver. It must return a AbstractBorderedLinearSolver. If nothing has to be done, just return bls.

Required methods		Brief description
`bls(J, dR, dzu, dzp, R, n, ξu::Number, ξp::Number; shift = nothing, dotp = nothing, applyξu! = nothing)`		Compute the solution `dX, dl` of the linear problem (BLS) where `J` is the jacobian and `dR, dzu` are vectors (not necessarily subtypes of `AbstractVector`). `shift = nothing` is used in place of saying `shift=0`. `dotp` is the dot product used for the vector space. If `dotp(x,y) = dot(x,S,y)` for some matrix S, the function `applyξu!*y,x = mul!(y,S,x)`. Returns `(dX, dl, success::Bool, itnumber)` where `itnumber` is the number of iterations for solving the problem.