Efficient parameterized HSS iteration methods for complex symmetric linear systems

Abstract

Based on the new HSS (NHSS) iteration method proposed by Pour and Goughery (2015) and the efficient PSHSS iteration method by Zeng and Ma (2016), we introduce an efficient parameterized HSS (PNHSS) and a parameterized single-step HSS (PS*HSS) iteration methods for solving a class of complex symmetric linear systems. Convergence properties of the PNHSS and the PS*HSS iteration methods are studied, which show that the iterative sequences are convergent to the unique solution of the linear system for any initial guess under a loose restriction on the parameter ω. Furthermore, we derive an upper bound for the spectral radius of the PNHSS iteration matrix, and the quasi-optimal parameters α∗ and ω∗ which minimize the above upper bound are also considered. Both theoretical and numerical results show that the PNHSS and the PS*HSS iteration methods outperform the NHSS and the SHSS iteration methods. Little difference about the computational efficiency from the point of view of the CPU times between the PS*HSS, the PNHSS and the PSHSS iteration methods is justified by using the experimental optimal parameters. However, sometimes the PS*HSS and the PNHSS iteration methods are more efficient than the PSHSS iteration method when the experimental optimal parameters are not used.