A Class Of Numerical Solution For Saddle Point Problem

Solution of saddle point problems arise in a wide variety of applications throughout computational science and engineering. For example, fluid dynamics, elasticity, electromagnetism, constrained optimization and least square problems.In this paper, two new iterative methods with uncertain parameters are considered for the large and sparse symmetric saddle point problems (successively, SMPSOR-Like method and MGSAOR method). First, a new iterative method is proposed that be based on modified SOR-Like method with parameters, i.e. symmetric modified SOR-Like method or SMPSOR-Like method. SMPSOR-Like method is based on the splitting form of the saddle point matrix, and then the functional equation among the eigenvalues of the iteration matrix of this method and the precondition matrix and also uncertain parameters is established. This method is convergent by the selection of appropriate preprocessing matrix and uncertain parameters. Thus, when the sufficient and necessary conditions of convergence of SMPSOR-Like method are discussed, and an experimental result of this iteration method is given numerically.Second, another new iterative method is proposed that be based on generalized SAOR method, i.e. modified GSAOR method or MGSAOR method. MGSAOR method is also based on the splitting form of the saddle point matrix, and the functional equation among the eigenvalues of the iteration matrix of this method and the precondition matrix and also uncertain parameters is established. This method is also convergent by the selection of appropriate preprocessing matrix and uncertain parameters. Thus, when the sufficient and necessary conditions of convergence of MGSAOR method are discussed, and an experimental result of this iteration method is given numerically.