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In this paper, based on the generalized parameterized inexact Uzawa method (GPIU) presented by Zhang and Wang [Applied Mathematics and Computation, 219(2013) 4225-4231], we introduce and study an improved generalized parameterized inexact Uzawa method (IGPIU) for singular saddle point problems. Moreover, theoretical analysis shows that the semi-convergence of the IGPIU method can be guaranteed by suitable choices of the iteration parameters. Finally, numerical experiments are carried out, which show that the improved generalized parameterized inexact Uzawa method (IGPIU) with appropriate parameters improve the convergence of iteration method efficiently when solving singular saddle point problems from the classic incompressible steady state Stokes problems. [ABSTRACT FROM AUTHOR]

In this paper, based on the generalized parameterized inexact Uzawa method (GPIU) presented by Zhang and Wang [Applied Mathematics and Computation, 219(2013) 4225-4231], we introduce and study an improved generalized parameterized inexact Uzawa method (IGPIU) for singular saddle point problems. Moreover, theoretical analysis shows that the semi-convergence of the IGPIU method can be guaranteed by suitable choices of the iteration parameters. Finally, numerical experiments are carried out, which show that the improved generalized parameterized inexact Uzawa method (IGPIU) with appropriate parameters improve the convergence of iteration method efficiently when solving singular saddle point problems from the classic incompressible steady state Stokes problems. [ABSTRACT FROM AUTHOR]

Recently, Cao, Du and Niu [Shift-splitting preconditioners for saddle point problems, Journal of Computational and Applied Mathematics, 272 (2014) 239-250] introduced a shiftsplitting preconditioner for saddle point problems. In this paper, we establish a modified shift-splitting preconditioner for solving the large sparse augmented systems of linear equations. Furthermore, the preconditioner is based on a modified shift-splitting of the saddle point matrix, resulting in an unconditional convergent fixed-point iteration, which is a generalization of shift-splitting preconditioners. Finally, numerical examples show the spectrum of the new preconditioned matrix for the different parameters. [ABSTRACT FROM AUTHOR]

The non-symmetric algebraic Riccati equation arising in transport theory can be rewritten as a vector equation and the minimal positive solution of the non-symmetric algebraic Riccati equation can be obtained by solving the vector equation. In this paper, based on the Newton-Shamanskii method, we propose a new iterative method called modified Newton-Shamanskii method for solving the vector equation. Some convergence results are presented. The convergence analy- sis shows that sequence of vectors generated by the modified Newton-Shamanskii method is monotonically increasing and converges to the minimal positive solu- tion of the vector equation. Finally, numerical experiments are presented to illustrate the performance of the modified Newton-Shamanskii method. [ABSTRACT FROM AUTHOR]

The non-symmetric algebraic Riccati equation arising in transport theory can be rewritten as a vector equation and the minimal positive solution of the non-symmetric algebraic Riccati equation can be obtained by solving the vector equation. In this paper, based on the Newton-Shamanskii method, we propose a new iterative method called modified Newton-Shamanskii method for solving the vector equation. Some convergence results are presented. The convergence analysis shows that sequence of vectors generated by the modified Newton-Shamanskii method is monotonically increasing and converges to the minimal positive solution of the vector equation. Finally, numerical experiments are presented to illustrate the performance of the modified Newton-Shamanskii method. [ABSTRACT FROM AUTHOR]

In this paper, a modified nonlinear Uzawa algorithm for solving symmetric saddle point problems is proposed, and also the convergence rate is analyzed. The results of numerical experiments are presented when we apply the algorithm to Stokes equations discretized by mixed finite elements. [ABSTRACT FROM AUTHOR]

In this paper, a modified nonlinear Uzawa algorithm for solving symmetric saddle point problems is proposed, and also the convergence rate is analyzed. The results of numerical experiments are presented when we apply the algorithm to Stokes equations discretized by mixed finite elements. [ABSTRACT FROM AUTHOR]