Bordered linear solvers

The bordered linear solvers must be subtypes of AbstractBorderedLinearSolver <: AbstractLinearSolver.

The methods provided here solve bordered linear equations. More precisely, one is interested in the solution $u$ to $J\cdot u = v$ where

Such linear solver bdlsolve will be called like sol, success, itnumber = bdlsolve(A, b, c, d, v1, v2) throughout the package.

Full matrix

This easiest way to solve $(E)$ is by forming the matrix $J$. In case it is sparse, it should be relatively efficient. You can create such bordered linear solver using bls = MatrixBLS(ls) where ls::AbstractLinearSolver is a linear solver (which defaults to \) used to solve invert $J$.

Bordering method

The general solution to $(E)$ when $A$ is non singular is $x_1=A^{-1}v_1, x_2=A^{-1}b$, $u_2 = \frac{1}{d - (c,x_2)}(v_2 - (c,x_1))$ and $u_1=x_1-u_2x_2$. This is the default method used in the package. It is very efficient for large scale problems because it is entirely Matrix-Free and one can use preconditioners. You can create such bordered linear solver using bls = BorderingBLS(ls) where ls::AbstractLinearSolver is a linear solver which defaults to \. The intermediate solutions $x_1=A^{-1}v_1, x_2=A^{-1}b$ are formed using ls.

1. Using such method with ls being a GMRES method is the main way to solve (E) in this package.
2. In the case where ls = DefaultLS(), the factorisation of A is cached so the second linear solve is very fast

Full Matrix-Free

In cases where $A$ is singular but $J$ is not, the bordering method may fail. It can thus be advantageous to form the Matrix-Free version of $J$ and call a generic linear solver to find the solution to $(E)$. You can create such bordered linear solver using bls = MatrixFreeBLS(ls) where ls::AbstractLinearSolver is a (Matrix Free) linear solver which is used to invert J.

For now, this linear solver only works with AbstractArray