In this work, an equidistant parameterized Gauss–Seidel (EPGS) iteration method based on a scale-splitting formulation of matrix is proposed for solving a class of block two-by-two real linear systems. Then, we investigate the convergence properties of this method and derive the optimal value of a relaxation parameter as well as the corresponding convergence factor. Furthermore, the spectral properties of EPGS-preconditioner are studied when the EPGS splitting matrix serves as a preconditioner to improve Krylov subspace methods. Finally, some numerical experiments are performed and discussed to demonstrate the performance of our method, and numerical results show that this novel method outperforms than classical iteration methods including PMHSS, E-HS, and GSOR.

在这项工作中，提出了一种基于矩阵比例分解公式的等距参数化高斯-赛德尔（EPGS）迭代方法，用于求解一类块二乘二的实线性系统。然后，我们研究该方法的收敛性，并推导松弛参数的最佳值以及相应的收敛因子。此外，当EPGS分裂矩阵作为改进Krylov子空间方法的预处理器时，研究了EPGS预处理器的光谱特性。最后，进行了一些数值实验并进行了讨论，以证明我们的方法的性能，数值结果表明，该新方法的性能优于经典迭代方法（包括PMHSS，E-HS和GSOR）。