Single-step PMQHSS and new PMQHSS methods for complex symmetric linear systems with strongly dominant skew-Hermitian parts

In order to solve complex symmetric linear equations more stably and quickly, we design a single-step preconditioned MQHSS (SPMQHSS) iteration method and a new preconditioned MQHSS (NPMQHSS) iteration method. Under suitable conditions, we give the convergence theories of the SPMQHSS and NPMQHSS iteration methods. The upper bounds on the spectral radius of the SPMQHSS and NPMQHSS methods and the quasi-optimal parameters which minimize two upper bounds are given, respectively. Furthermore, we also present the reason why the SPMQHSS and NPMQHSS methods have the same quasi-optimal parameters. In addition, we also analyze the properties of the SPMQHSS method with the parameter matrix being selected as the skew-Hermitian part of the coefficient matrix. Finally, some tested problems are reported to validate the theoretical correct and compare the effectiveness of the proposed methods with several existing ones.