In this article, a parameterized extended shift-splitting (PESS) method and its induced preconditioner are given for solving nonsingular and nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) part. The convergence analysis of the (Formula presented.) iteration method is discussed. The distribution of eigenvalues of the preconditioned matrix is provided. A number of experiments are given to verify the efficiency of the (Formula presented.) method for solving nonsymmetric saddle-point problems.@en

Global Krylov subspace methods are among the most efficient algorithms to solve matrix equation AX=B. Deflation and augmentation techniques are used to accelerate the convergence of Krylov subspace methods. There are two different approaches for deflated and augmented methods: an augmentation space is applied explicitly in every step, or the global method is used for solving a projected problem and then a correction step is applied at the end. In this paper, we present a framework of deflation and augmentation approaches for accelerating the convergence of the global methods for the solution of nonsingular linear matrix equations AX=B. Then, we define deflated and augmented global algorithms. Also, we analyze the deflated and augmented global minimal residual and global orthogonal residual methods. Finally, we present numerical examples to illustrate the effectiveness of different versions of the new algorithms.@en