Minimal Residual Modified HSS Iteration Method For A Class Of Complex Symmetric Linear Systems

Large sparse linear systems of complex symmetric structure are of important practical significance which arise from many scientific researches and engineering applications. For solving these systems, many effective iteration methods and corresponding preconditioning techniques have been proposed. However, most of these methods are practicing by using the ordinary matrix splitting iteration schemes.Based on these researches, we in this paper study the minimal-residual-like MHSS method for this class of complex symmetric linear equations.In 2010, Bai et al. proposed MHSS iteration method for solving a class of complex symmetric linear systems. In this paper, we first propose two new iteration methods start with the residual updating schemes of the original MHSS method,i.e., one-step minimal residual MHSS iteration method(abbreviated as MRMHSSI) and two-step alternative minimal residual MHSS iteration method(abbreviated as MRMHSS-II). Here in these two methods, we significantly modify the practice of traditional MHSS iteration method and introduce associated iteration parameters such that the residual produced by each iterative step minimizes locally. Then, we analyze the convergence properties of the two methods and formulate the detailed implementation pseudo-codes of these two methods.Finally, numerical examples are presented to demonstrate the feasibility and the efficiency of these two iteration methods. Indeed, from numerical results we can see that they are very effective for solving this class of complex symmetric linear systems, the performances are much better than the traditional MHSS iteration method.