We present an algebraic optimizable Schwarz method for banded matrices. It can be interpreted as a modification of the block Jacobi iterative method for the solution of linear systems of equations whose coefficient matrix is banded. By changing selected entries of the matrix we produce an iterative method which is guaranteed to converge in two iterations. Similarly, the modified block diagonal matrix can be used as a preconditioner for a Krylov subspace iterative method such as GMRES. With this preconditioner, the iterative method is also guaranteed to converge in two iterations (joint work with Martin Gander, University of Geneva).