Generalized Preconditioned MHSS Method for a Class of Complex Symmetric Linear Systems

Cui-Xia Li,Shi-Liang Wu,Yan-Jun Liang

doi:10.1155/2014/206821

Abstract

Based on the modified Hermitian and skew-Hermitian splitting (MHSS) and preconditioned MHSS (PMHSS) methods, a generalized preconditioned MHSS (GPMHSS) method for a class of complex symmetric linear systems is presented. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From a practical point of view, we have analyzed and implemented inexact GPMHSS (IGPMHSS) iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments are reported to confirm the efficiency of the proposed methods.

Highlights

IntroductionAbstract and Applied Analysis linear system (1) is that only two linear subsystems with coefficient matrices αI + W and αI + T, both being real and symmetric positive definite, need to be solved at each step
Consider an iterative solution of the system of linear equations as follows: Ax = b, A ∈ Cn×n, x, b ∈ Cn, (1)where A ∈ Cn×n is a complex symmetric matrix of the following form: A = W + iT, (2)with W ∈ Rn×n being symmetric positive definite and T ∈ Rn×n being symmetric positive semidefinite
In the GPMHSS method, it is required to solve two systems of linear equations whose coefficient matrices are αP1 + W and βP2 + T, respectively. This may be very costly and impractical in actual implementations. To overcome this disadvantage and improve the computational efficiency of the GPMHSS iteration method, we propose to solve the two subproblems iteratively [21, 26], which leads to the inexact GPMHSS (IGPMHSS) iterative scheme

Summary

Introduction

Abstract and Applied Analysis linear system (1) is that only two linear subsystems with coefficient matrices αI + W and αI + T, both being real and symmetric positive definite, need to be solved at each step. The MHSS method successfully avoids solving a shifted skew-Hermitian linear subsystem with coefficient matrix αI + iT. Theoretical analysis in [2] shows that the MHSS method converges unconditionally to the unique solution of the complex symmetric linear system (1) when W ∈ Rn×n is real symmetric positive definite and T ∈ Rn×n is real symmetric positive semidefinite. By introducing Rn×n and P2 two symmetric positive ∈ Rn×n, the GPMHSS definite iterative matrices P1 ∈ scheme works as follows.

Convergence Analysis for the GPMHSS Method

The IGPMHSS Iteration

Numerical Experiments

Conclusion