A new double-step splitting iteration method for certain block two-by-two linear systems

We consider the iterative solution of certain block two-by-two linear systems and introduce a new double-step splitting (NDSS) iteration method. The proposed method is based on the transformed matrix iteration method proposed recently, and obtained by applying two-step and preconditioning techniques for the original linear system. We prove that the NDSS iteration method is convergent under mild conditions. Upper bounds on the spectral radius of the iteration matrix of the NDSS method are presented and the parameters which minimize these bounds are computed. We also consider the inexact NDSS iteration method. The proposed methods are compared theoretically and numerically with some existing ones, which shows the good performance of the NDSS iteration method and its inexact version.