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.

我们考虑某些块二乘二线性系统的迭代解，并引入一种新的双步分裂（NDSS）迭代方法。该方法基于最近提出的变换矩阵迭代方法，并通过对原始线性系统应用两步和预处理技术获得。我们证明了NDSS迭代方法在温和条件下是收敛的。给出了NDSS方法迭代矩阵的谱半径的上限，并计算了使这些界限最小的参数。我们还考虑了不精确的NDSS迭代方法。从理论上和数值上将所提出的方法与现有方法进行了比较，这表明NDSS迭代方法及其不精确版本具有良好的性能。