Zhou et al. and Huang et al. have proposed the modified shift-splitting (MSS) preconditioner and the generalized modified shift-splitting (GMSS) for non-symmetric saddle point problems, respectively. They have used symmetric positive definite and skew-symmetric splitting of the (1, 1)-block in a saddle point problem. In this paper, we use positive definite and skew-symmetric splitting instead and present new modified shift-splitting (NMSS) method for solving large sparse linear systems in saddle point form with a dominant positive definite part in (1, 1)-block. We investigate the convergence and semi-convergence properties of this method for nonsingular and singular saddle point problems. We also use the NMSS method as a preconditioner for GMRES method. The numerical results show that if the (1, 1)-block has a positive definite dominant part, the NMSS-preconditioned GMRES method can cause better performance results compared to other preconditioned GMRES methods such as GMSS, MSS, Uzawa-HSS and PU-STS. Meanwhile, the NMSS preconditioner is made for non-symmetric saddle point problems with symmetric and non-symmetric (1, 1)-blocks.

中文翻译：

---

用于非对称鞍点问题的新型改进的移位分裂预处理器

周等。和黄等。分别针对非对称鞍点问题提出了改进的移位分裂（MSS）预处理器和广义的改进移位分裂（GMSS）。他们在鞍点问题中使用了（1，1）块的对称正定和斜对称拆分。在本文中，我们使用正定和倾斜对称分裂代替，并提出了一种新的改进的移位分裂（NMSS）方法，以求解鞍点形式的大型稀疏线性系统，该线性稀疏线性系统在（1，1）块中具有占优势的正定部分。我们研究此方法对非奇异和奇异鞍点问题的收敛性和半收敛性。我们还将NMSS方法用作GMRES方法的前提。数值结果表明，如果（1，1）块具有正确定的主导部分，与其他预处理的GMRES方法（例如GMSS，MSS，Uzawa-HSS和PU-STS）相比，NMSS预处理的GMRES方法可产生更好的性能结果。同时，针对具有对称和非对称（1，1）块的非对称鞍点问题制作了NMSS预调节器。