Analysis of the relaxed deteriorated PSS preconditioner for singular saddle point linear systems

The relaxed deteriorated PSS (RDPSS) preconditioner proposed by Cao et al. [13] is further generalized to solve singular saddle point linear systems. Properties of the RDPSS splitting and convergence of the corresponding iteration method are studied. By using the Moore-Penrose inverse, the RDPSS iteration method and the RDPSS preconditioned GMRES are both proved to converge to the least squares solution of the singular saddle point linear system. The RDPSS preconditioned matrix is also analyzed, results about eigenvalue distributions are derived. Moreover, the RDPSS preconditioner is generalized to a class of more general preconditioners, the corresponding convergence and eigenvalue distribution results are analyzed. Numerical experiments are presented to illustrate the effectiveness of the RDPSS preconditioner and its variants to accelerate GMRES for solving singular saddle point linear systems.