# Linear solvers (LS)

If you provide your own linear solver, it must be a subtype of AbstractLinearSolver otherwise BifurcationKit.jl will not recognize it. See example just below.

The linear solvers provide a way of inverting the Jacobian J or solving J * x = rhs. Such linear solver linsolve will be called like sol, success, itnumber = linsolve(J, rhs) throughout the package.

Here is an example of the simplest one (see src/LinearSolver.jl for the true implementation) to give you an idea, the backslash operator:

struct DefaultLS <: AbstractLinearSolver end

function (l::DefaultLS)(J, rhs)
return J \ rhs, true, 1
end

Note that for newton to work, the linear solver must return 3 arguments. The first one is the result, the second one is whether the computation was successful and the third is the number of iterations required to perform the computation.

You can then call it as follows (and it will be called like this in newton)

ls = DefaultLS()
J = rand(2, 2) # example of linear operator
ls(J, rand(2))

## List of implemented linear solvers

1. Default \ solver based on LU or Cholesky depending on the type of the Jacobian. This works for sparse matrices as well. You can create one via linsolver = DefaultLS().
2. GMRES from IterativeSolvers.jl. You can create one via linsolver = GMRESIterativeSolvers() and pass appropriate options.
3. GMRES from KrylovKit.jl. You can create one via linsolver = GMRESKrylovKit() and pass appropriate options.
Different linear solvers

By tuning the options of GMRESKrylovKit, you can select CG, GMRES... see KrylovKit.jl.

Other solvers

It is very straightforward to implement the Conjugate Gradients from IterativeSolvers.jl by copying the interface done for gmres. Same goes for minres,... Not needing them, I did not implement this.

## Preconditioner

Preconditioners should be considered when using Matrix Free methods such as GMRES. GMRESIterativeSolvers provides a very simple interface for using them. For GMRESKrylovKit, we implemented a left preconditioner. Note that, for GMRESKrylovKit, you are not restricted to use Vectors anymore. Finally, here are some packages to use preconditioners

1. IncompleteLU.jl an ILU like preconditioner
2. AlgebraicMultigrid.jl Algebraic Multigrid (AMG) preconditioners. This works especially well for symmetric positive definite matrices.
3. Preconditioners.jl A convenient interface to conveniently called most of the above preconditioners using a single syntax.
4. We provide a preconditioner based on deflation of eigenvalues (also called preconditioner based on Leading Invariant Subspaces) using a partial Schur decomposition. There are two ways to define one i.e. PrecPartialSchurKrylovKit and PrecPartialSchurArnoldiMethod.
Using Preconditioners

Apart from setting a preconditioner for a linear solver, it can be advantageous to change the preconditioner during computations, e.g. during a call to continuation or newton. This can be achieved by taking advantage of the callbacks to these methods. See the example 2d Ginzburg-Landau equation (finite differences, codim 2, Hopf aBS).