Two Efficient Lopsided Double-Step Methods for Solving Complex Symmetric Linear Systems

Abstract

In this paper, an efficient lopsided double-step (LDS) iteration scheme is proposed to quickly solve complex symmetric linear systems, by using the real part and imaginary part of the coefficient matrix. We give detailed analysis of the spectral radius of the iteration matrix and the quasi-optimal parameter for the LDS method. In addition, a modified version of the LDS (MLDS) method is developed by using only one matrix inversion in each iteration, and the convergence properties of the MLDS method are discussed. Particularly, under suitable conditions, the convergence factors of the LDS and the MLDS methods are no more than 0.1768, and this number is less than that of many exiting methods. Numerical experiments are implemented and the results support the contention that the LDS and the MLDS methods are more efficient than several classical methods. Furthermore, we also explore the fixed parameters for the LDS and the MLDS methods in practice, and the numerical results are very satisfactory.