The Studies Of Relaxed Iterative Methods And Preconditioning Techniques For Equations Of Linear Systems And Saddle Point Problem

Solutions of large-scale linear systems arise widely in various scientific and engi-neering fields. Researches of methods for solving large-scale sparse systems of linear al-gebraic equations have important theoretic significance and practical applications. In this dissertation, we deeply study the iteration solutions of linear algebraic systems, investi-gate the convergence and comparison theorems of matrix splitting methods and discuss the iteration solutions of saddle point problems.An iterative method is studied to search a scaling matrix for diagonal dominance. For any irreducible M-matrix A, an improved iterative method based on the special structures of M-matrices is presented to find a positive diagonal matrix D such that AD is strictly diagonally dominance. Furthermore, from this iterative algorithm, a lower bound for the spectral radius of H-matrix is given.Two-stage iterative method and multisplitting algorithm for waveform relaxation method are investigated. Also the convergence and comparison theorems by choosing different splittings of matrices are obtained. We first present a convergence theorem of two-stage waveform relaxation method when the coefficient matrix is an H-matrix. More-over, comparison theorem for this iterative method of Hermitian positive definite matrix is also obtained, which enriches some existing literatures. Secondly, we consider a non-stationary multisplitting two-stage strategy for waveform relaxation iteration, solving the initial value problem of ordinary differential equations. The convergence theorems and the comparison theorems are discussed in detail when the coefficient matrix has some special properties, which provide theoretical base for the choice of iteration methods.The iterative solutions of saddle point problems are studied. A modified generalized symmetric SOR method which gives three parameters to find the solution of augmented systems is firstly proposed. This modified generalized SSOR method is the extension of the SOR-like method. And the convergence of this new algorithm is discussed under suit-able restrictions on the parameters. Secondly, two generalized SOR methods for solving the saddle point problems are presented by different splittings of the coefficient matrix. And the conditions of two algorithms for their convergence are derived, respectively. Fur-thermore, by choosing different preconditioning matrices, we get different generalized iteration schemes with different convergent rates.An alternating iterative method based on modified Gauss-Seidel method is investi-gated. The spectral radius of iterative matrix of this new method is proved to be smaller than that of the SOR method. It also indicates that the alternating modified Gauss-Seidel method is convergent. Then the convergence theorem for the nonsymmetric positive semidefinite linear system is studied when A=M-N, where M is not always non-singular. The necessary and sufficient conditions for seminorm convergence are given.