The improvements of the generalized shift-splitting preconditioners for non-singular and singular saddle point problems

To solve the saddle point problems with symmetric positive definite (1,1) parts, the improved generalized shift-splitting (IGSS) preconditioner is established in this paper, which yields the IGSS iteration method. Theoretical analysis shows that the IGSS iteration method is convergent and semi-convergent unconditionally. The choices of the iteration parameters are discussed. Moreover, some spectral properties, including the eigenvalue and eigenvector distributions of the preconditioned matrix are also investigated. Finally, numerical results are presented to verify the robustness and the efficiency of the proposed iteration method and the corresponding preconditioner for solving the non-singular and singular saddle point problems.