(untitled) (Journal of Computational Mathematics, 2018)
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.