A practical formula for computing optimal parameters in the HSS iteration methods

In the HSS iteration methods proposed by Bai, Golub and Ng [Z.-Z. Bai, G.H. Golub, M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM. J. Matrix Anal. Appl. 24 (2003) 603-626], the determination of the optimal parameter is a tough task when solving a non-Hermitian positive definite linear system. In this paper, a new and simple strategy for obtaining the optimal parameter is proposed, which computes the optimal parameter by solving a cubic polynomial equation. The coefficients of this polynomial are determined by several traces of some matrices related to the symmetric and skew-symmetric parts of the coefficient matrix of the real linear system. Numerical experiments show that our new strategy is very effective for approximating the optimal parameter in the HSS iteration methods as it leads to fast convergence of the method.