Polynomials by Conformal Mapping for the Richardson Iteration Method for Complex Linear Systems

Some methods using conformal mappings for determining polynomials for the Richardson iteration scheme for complex linear systems $A{\bf x} = {\bf b}$ are described. The polynomials can be computed fairly rapidly and give rise to iterative schemes with a rate of convergence very close to optimal. The methods described are applicable when the eigenvalues of the matrix A lie in a known bounded simply connected region $\Omega $ in the complex plane. $\Omega $ is assumed to have a smooth boundary.