The Splitting Iteration Methods And Preconditioning Schemes For The Complex Symmetric Linear Systems

The complex linear systems, especially the complex symmetric linear systems, exist widely in the application fields of scientific and engineering computation, which drew considerable attention of researchers. In the practical applications, the coeffi-cient matrices are usually large and sparse. Therefore, iterative methods are employed to solve these problems. For complex linear systems, there are two commonly used treatment methods. The first one is to solve the original complex system with iterative methods, and the second one is solve its equivalent 2×2 block real linear system. Re-gardless of which approach, in order to get better convergence effect, We need to adopt effective preconditioning technique.In this paper, we studied the splitting methods and the preconditioning techniques for the complex symmetric linear systems. The main contribution are as follows.(1) For a class of complex symmetric linear systems, based on the Hermitian and normal splitting, we proposed a modified Hermitian and normal (MHNS) splitting iter-ative method and a modified and simplified Hermitian and normal (MSHNS) splitting iterative method. We proved that MHNS method is unconditionally convergent. In or-der to speed up the computational efficiency, we proposed an inexact version of MHNS (IMHNS) method by the CG method. Furthermore, we designed the preconditioned variant of MSHNS method, as well as the corresponding preconditioners for the Krylov subspace methods.(2) By transforming the original complex linear system into an equivalent 2×2 block real linear system, we established the generalized accelerated over-relaxation (GAOR) method, and proved that GAOR is convergent under suitable conditions. Mean-while, we established the preconditioned form of the GAOR method, and discussed the corresponding preconditioners. Numerical experiments are carried out to show the per-formance of the proposed preconditioners.