HSS Method with a Complex Parameter for the Solution of Complex Linear System

Abstract

In this paper, a complex parameter is employed in the Hermitian and skew-Hermitian splitting (HSS) method (Bai, Golub and Ng: SIAM J. Matrix Anal. Appl., 24(2003), 603-626) for solving the complex linear system $Ax=f$. The convergence of the resulting method is proved when the spectrum of the matrix $A$ lie in the right upper (or lower) part of the complex plane. We also derive an upper bound of the spectral radius of the HSS iteration matrix, and an estimated optimal parameter $α$(denoted by $α_{est}$) of this upper bound is presented. Numerical experiments on two modified model problems show that the HSS method with $α_{est}$ has a smaller spectral radius than that with the real parameter which minimizes the corresponding upper bound. In particular, for the 'dominant' imaginary part of the matrix $A$, this improvement is considerable. We also test the GMRES method preconditioned by the HSS preconditioning matrix with our parameter $α_{est}$.