The Hermitian and skew-Hermitian splitting (HSS) method is a well-known method employed in solving the non-Hermitian positive definite linear systems. However, the performance of the HSS method highly depends on the parameter value. In this paper, we address this issue by introducing two practical selections of this parameter, which are computed based on the Rayleigh quotient of the Hermitian part of the coefficient matrix. On the basis of the new selections, we propose the variable-parameter HSS methods for non-Hermitian positive definite linear systems and provide a convergent analysis under reasonable assumptions. Furthermore, to verify the effectiveness of our selections, we conduct experiments on several large and sparse linear systems. The numerical results show their efficiency over the existing quasi-optimal parameter.

Hermitian 和斜 Hermitian 分裂 (HSS) 方法是用于求解非 Hermitian 正定线性系统的众所周知的方法。然而，HSS 方法的性能在很大程度上取决于参数值。在本文中，我们通过引入该参数的两个实际选择来解决这个问题，这些选择是根据系数矩阵的厄米特部分的瑞利商计算的。在新选择的基础上，我们提出了非厄米正定线性系统的变参数 HSS 方法，并提供了合理假设下的收敛分析。此外，为了验证我们选择的有效性，我们在几个大型稀疏线性系统上进行了实验。数值结果显示了它们在现有准最优参数上的效率。