Improvement On Some Numerical Solutions Of Saddle Point Problems

Let m,n be two given positive integers.Consider the following class of linear equation:(?)where A ? Cm×m is a nonsingular matrix,B ? Cm×n is a rectangular matrix((m ? n)),and f ? Cm,g g?Cn are two given vectors,B*is the conjugate transpose of B.Systems of linear equations with the form(1)are called saddle point problems.If B is a matrix of full column rank,i.e,rank B = n,the system(1)is a nonsingular saddle point problem;rank B<n,the system(1)is a singular saddle point problem.In recent years,the scholars have constantly explored new iteration methods for solving the large sparse nonsingular saddle point problems.In 2016,Bai and Benzi proposed the regularized Hermitian and skew-Hermitian splitting(RHSS)iterative method and they show that RHSS method is unconditional convergence.In order to improve the convergence speed of RHSS iteration method,we present the preconditioned RHSS iterative method in the second chapter of this paper.We emphatically analyzes the convergence of the PRHSS method and the eigenvalues of the iteration matrix of this method.Moreover,the singular saddle point problems are also research hotspots recently.Many breakthroughs and achievements have been obtained for the singular saddle point problems especially for A is a real matrix.The third chapter of this paper is also based on this assump-tion,which study the semi-convergence of the SOR-like method for solving the large sparse singular saddle point problems.Moreover,we present the pseudo-optimal parameters and the corresponding pseudo-optimal semi-convergence factor.