Iterative Solutions For Large Sparse Linear Systems And Research In Applications

Solutions of large-scale sparse linear systems arise from the large-scale scientific computing and engineering technique, such as fluid mechanics, computational electro-magnetics, optimization problems and linear elastics.So, research of methods for solving large-scale sparse linear systems has important theoretic significance and practical ap-plications.Iterative solutions of several classes of large-scale sparse linear systems are deeply studied in this thesis. In particular, convergence properties and comparison the-ories of matrix splitting methods have been investigated and preconditioning techniques for saddle point problems have been also studied.The preconditioned AOR, SOR and Gauss-Seidel iterative methods are studied. Firstly, the convergence properties of a modified preconditioner with Gauss-Seidel and AOR(SOR) methods for the L-matrix linear systems are analyzed and some comparison results are presented. Consequently, the optimal structure of the modified preconditioner is given. Secondly, convergence conditions of the preconditioned Gauss-Seidel.for the H-matrix linear systems are discussed. Finally, the convergence properties of the precon-ditioned AOR for the 2×2 block linear systems arising from the least-square problems are discussed.The HSS iterative scheme and HSS preconditioning technique are investigated. Firstly, comparing the lopsided Hermitian/skew-Hermitian splitting (LHSS) method and Hermitian/skew-Hermitian splitting (HSS) method, a new criterion for choosing the LHSS or HSS method is presented. Secondly, a modified Hermitian and skew-Hermitian splitting (MHSS) and inexact MHSS (IMHSS) method to solve non-Hermitian positive definite linear systems are presented. The convergence properties of MHSS and IMHSS method are analyzed. Thirdly, we give some spectral properties of HSS preconditioners for the classical saddle point problems, obtain a new interval containing all the eigenval-ues of the preconditioned matrix and present a new sufficient condition to make all the eigenvalues real. Finally, HSS preconditioners for the generalized saddle point problems with nonzero (2,2) block are investigated. Some spectral properties of the preconditioned matrix are presented, which are the remedies for the defect that the earlier results are only concerned with the saddle point problems with zero (2,2) block. It is shown that under certain conditions, all eigenvalues of the preconditioned matrix with the original system being non-Hermitian will form two tight clusters, one is near (0,0) and the other is near (2,0) as the iteration parameter approaches to zero from above.Iterative methods for the saddle point problems are investigated, which consists of five aspects as follows:1.based on the matrix splitting, a iterative scheme is presented to solve the saddle point problems and its convergence properties are discussed; 2. we present a modified symmetric successive overrelaxation (MSSOR) method for solving the saddle point problems, discuss its convergence conditions and obtain its optimal iter-ation parameter; 3.block triangular preconditioners with a parameter for the generalized saddle point problems are analyzed and a new interval containing all the real and complex eigenvalues of the preconditioned matrix is derived; 4. according to the special structure of the classical saddle point problems arising from the discretization of the mixed time-harmonic Maxwell equations, the optimal block diagonal and triangular preconditioners are obtained, meanwhile, a single column nonzero(1,2) block triangular preconditioner is presented and the original block triangular preconditioners are improved; 5.two classes of the modified augment-free and Schur complement-free preconditioners for the indefi-nite(1,1)block of the saddle point problems arising from the discretization of the mixed time-harmonic Maxwell equations are proposed. The optimal, augment-free and Schur complement-free, block diagonal and triangular preconditioners are obtained by analyz-ing the spectral properties of the preconditioned matrix. Numerical experiments are given to illustrate the efficiency of two classes of the optimal preconditioners.