Study On Numerical Solutions To Saddle Point Problems And Coupled Sylvester-type Matrix Equations

Numerical solutions of two kinds of linear systems are mainly studied in this thesis.One kind of the systems belongs to the saddle point problem arising from the scientificcomputing and engineering technology; the other is the coupled Sylvester-type matrixequations frequently appearing in system theory and stability analysis. An intensive studyto the two kinds of linear systems is carried out, and some new iterative methods andpreconditioners are proposed.The thesis does research on the iterative methods and preconditioning techniquesfor the saddle point system. Based on the generalized successive over-relaxation(SOR)method and modified SOR-like method, a modified generalized SOR method(MGSOR)involving three parameters is proposed. The convergence of the proposed method is dis-cussed and the restrictive condition on the parameters is presented. Numerical experi-ments are also given to illustrate the effectiveness of the proposed method. Meanwhile,two block triangular augmentation preconditioners with multi-parameters are proposedfor two different saddle point systems. The eigenvalue distribution of the preconditionedmatrices is investigated in details. Theoretical analysis shows that all the eigenvalues ofthe preconditioned matrices are more tightly clustered when the parameters are properlychosen. The numerical experiments confirm the analysis. Moreover, with different valuesof parameters, different preconditioners of the same effect can be obtained.The thesis also does research on the iterative methods for the coupled sylvester-typematrix equations. Based on the gradient-based iterative method(GI) and Jacobi gradient-based iterative method(JGI), a shift-splitting Jacobi-gradient iterative method(SSJGI) isproposed to solve the Lyapunov matrix equations. Its convergence is discussed in detailsand the restrictive condition on parameters is given. This method has fast convergentspeed and small computational cost by choosing the parameters appropriately. The effi-ciency of the SSJGI method is illustrated by numerical experiments. Meanwhile, a finiteiterative method is proposed for the least-norm least-squares solutions of the generalizedcoupled Sylvester matrix equations, based on a matrix form of CGNE method. The prop-erties and convergence of the proposed method are discussed in details, and an exampleis given to show its performance. This method is simple and does not need to calculate any inverse matrix for solving the least-squares problems. In addition, based on Paige'salgorithms as the framework, two matrix iterative methods are proposed to solve generalcoupled matrix equations. By these new iterative methods, we can get the constraint solu-tions, such as symmetric, generalized bisymmetric and (R, S)-symmetric solutions. Somenumerical experiments are given to show the efficiency.