For the saddle-point problems, we discuss and analyze the updated preconditioned Hermitian and skew-Hermitian splitting (UPHSS) iteration method in detail. On this basis, we introduce a two-stage iteration method for the UPHSS iteration method. Theoretical analysis shows that the UPHSS and two-stage UPHSS iteration methods are convergent to the unique solution of the saddle-point linear system when the parameter is suitably chosen. Numerical examples show the correctness of the theory and the effectiveness of these methods.

中文翻译：

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更新了预处理的Hermitian和Skew-Hermitian分裂型迭代方法，用于解决鞍点问题

对于鞍点问题，我们详细讨论和分析了更新的预处理Hermitian和Skew-Hermitian分裂（UPHSS）迭代方法。在此基础上，针对UPHSS迭代方法引入了两阶段迭代方法。理论分析表明，该UPHSS和两级UPHSS迭代方法收敛到鞍点的线性系统的独特的解决方案，当该参数被适当地选择。数值算例表明了该理论的正确性和这些方法的有效性。