On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems

For the singular, non-Hermitian, and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the semi-convergence of the Hermitian and skew-Hermitian splitting (HSS) iteration methods. We then investigate the semi-convergence factor and estimate its upper bound for the HSS iteration method. If the semi-convergence condition is satisfied, it is shown that the semi-convergence rate is the same as that of the HSS iteration method applied to a linear system with the coefficient matrix equal to the compression of the original matrix on the range space of its Hermitian part, that is, the matrix obtained from the original matrix by restricting the domain and projecting the range space to the range space of the Hermitian part. In particular, an upper bound is obtained in terms of the largest and the smallest nonzero eigenvalues of the Hermitian part of the coefficient matrix. In addition, applications of the HSS iteration method as a preconditioner for Krylov subspace methods such as GMRES are investigated in detail, and several examples are used to illustrate the theoretical results and examine the numerical effectiveness of the HSS iteration method served either as a preconditioner for GMRES or as a solver.