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Based on the relaxed alternating positive semi-definite splitting (RAPSS) preconditioner, in this paper, a new preconditioner, called variant of relaxed alternating positive semi-definite splitting (VRAPSS) preconditioner, is presented and discussed. Spectral properties of the VRAPSS preconditioned matrix are analyzed in detail. Numerical experiments are provided to verify the efficiency of the VRAPSS preconditioner.

Recently, Fan and Zheng studied the preconditioned generalized local Hermitian and skew-Hermitian splitting (GLHSS) iteration method for non-Hermitian singular saddle point problem, and given its semi-convergence conditions; see Fan and Zheng (2014). In this note, we prove the semi-convergence of preconditioned GLHSS method by another method. The obtained result shows that the conditions for guaranteeing its semi-convergence are easy to check and more weaker.

We study a preconditioned generalized shift-splitting iteration method for solving saddle point problems. The unconditional convergence theory of the preconditioned generalized shift-splitting iteration method is established. When the splitting matrix is used as a preconditioner, we analyze eigenvalue distribution of the preconditioned saddle point matrix. It is proved that complex eigenvalues having nonzero imaginary parts of the preconditioned matrix are located in an intersection of two circles and the real parts of all eigenvalues of the preconditioned matrix are located in a positive interval. Numerical experiments are used to verify our theoretical results and illustrate effectiveness of the proposed iteration method and the corresponding splitting preconditioner.

Based on the Hermitian and skew-Hermitian splitting of the non-Hermitian positive definite (1,1)-block of the saddle point matrix, a new Uzawa-type iteration method is proposed for solving a class of saddle point problems in this paper. The proposed method can be applied not only to the nonsingular saddle point problems but also to the singular ones. The convergence properties for the nonsingular saddle point problems and the semi-convergence properties for the singular ones of the proposed method are carefully discussed under suitable restrictions. Numerical results verify the effectiveness and robustness of the proposed method.

Recently, a class of generalized shift-splitting iteration methods were proposed for solving nonsingular nonsymmetric saddle point problems (Cao et al., 2015). In this paper, the generalized shift-splitting iteration method is extended to solve singular nonsymmetric saddle point problems. Semi-convergence of the generalized shift-splitting iteration method is carefully analyzed, which shows that the iterative sequence generated by the generalized shift-splitting iteration method converges to a solution of the singular saddle point problem unconditionally. Numerical results verify the robustness and efficiency of the generalized shift-splitting iteration method.

In this paper, a new relaxed splitting preconditioner is proposed for the generalized saddle point problems. The proposed preconditioner is based on a new matrix splitting of the coefficient matrix and is easier to be implemented than the dimensional splitting preconditioner. Eigenvalue distribution of the preconditioned matrix is analyzed. An upper bound of the degree of the minimal polynomial of the preconditioned matrix is discussed. Finally, some numerical experiments are illustrated to show the efficiency of the new preconditioner.

In 2009, Zheng and Miao (B. Zheng and S.-X. Miao, Two new modified Gauss-Seidel methods for linear system with M-matrices, J. Comput. Appl. Math. 233 (2009), 922-930) considered the modified Gauss-Seidel method for solving M-matrix linear system with the pre- conditioner Pmax. In this paper, we consider the modified Gauss-Seidel method for solving the linear system with the generalized preconditioner Pmax(�), and study its convergent properties when the coefficient ma- trix is an H-matrix. Numerical experiments are performed with different examples, and the numerical results verify our theoretical analysis.

In this paper, the generalized stationary iterative (GSI) method is studied for solving the saddle point problems. The convergence of this method is studied under suitable restrictions on iteration parameters. Numerical example for solving Stokes equation is presented to show the superiority of GSI method.