Fast and Unconditional Convergent MRMHSS Iteration Method for Solving Complex Symmetric Linear Systems

doi:10.1007/s42967-024-00436-4

Abstract

Abstract: Based on the modified Hermitian and skew-Hermitian splitting (MHSS) iteration scheme and a novel minimum residual technique with the aid of a positive definite matrix, a novel minimum residual MHSS (NMRMHSS) iteration method was proposed for solving complex symmetric systems of linear equations. As is known, the NMRMHSS iteration is unconditional convergent; however, its numerical performance is degraded. In this work, we consider to improve the rate of the convergence of the NMRMHSS iteration method and inherit its theoretical property. To the end, we first combine the minimization technique of NMRMHSS with an accelerating method and obtain a fast and unconditional convergent iteration method. Then, the convergence is demonstrated, which indicates that the contraction factor of our method is smaller than that of NMRMHSS. Besides, the theoretical analysis shows that our method has more widespread application for solving complex symmetric linear systems. Finally, numerical results are reported to illustrate the numerical behavior of the proposed iteration method.

Key words: Complex symmetric matrix, Modified Hermitian and skew-Hermitian splitting (MHSS), Minimum residual technique, Unconditionally convergent, Symmetric positive semi-definite

Wei-Hong Zhang, Yi-Qing Luo, Yu-Jiang Wu. Fast and Unconditional Convergent MRMHSS Iteration Method for Solving Complex Symmetric Linear Systems[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 2043-2060.

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