A generalized preconditioned MHSS method for a class of complex symmetric linear systems (Q2858012)

The authors consider the numerical solution of complex linear systems the matrices of which have symmetric real and imaginary parts (assuming the real part to be positive definite). Instead of the HSS iterative method (see [\textit{Z.-Z. Bai, G. H. Golub} and \textit{M. K. Ng}, SIAM J. Matrix Anal. Appl. 24, No. 3, 603--626 (2003; Zbl 1036.65032)]), they propose another (also Peaceman-Rachford-like) procedure in which the matrices of the systems to be solved are real and contain two iteration parameters. This method is applied to a preconditioned form of the original system. It is proved that the resulting iteration converges for any positive value of the first parameter and an interval of values of the second parameter. Concerning the optimal choice of the parameters, a preliminary answer is given. In their numerical experiments these parameters are determined numerically exhibiting a rather different influence of the two parameters on the speed of convergence. The new method shows to be decisively better by CPU time than the considered other ones, including the HSS-iteration.