Preconditioned Richardson iteration for augmented linear systems

For solving a class of augmented linear systems, we propose a new efficient iteration method, which is called preconditioned Richardson iteration (PR). Under suitable restrictions on the iteration parameters, we prove that the iterative sequences converge to the unique solution of the augmented linear system. Moreover, the optimal iteration parameters and the corresponding optimal convergence factor are discussed in detail. Numerical results show that the PR iteration method has an advantage over several other iteration methods by computing with the preconditioned GMRES methods from the point of view of iteration steps and CPU times.