The Study Of Splitting Iterative Methods For Equations Of Linear Systems And Uzawa-type Algorithms For Generalized Saddle Point Problems

Solutions of large-scale sparse linear algebraic systems are deeply involved in vari-ous scientific and engineering fields,such as computational electromagnetics,numericalsolutions of high-order differential equations,optimization problems,fluid mechanics andreservoir modeling.Moreover,research of methods for solving large-scale sparse systemsof linear algebraic equations becomes one of the key issues of large-scale scientific andengineering computing and such research has important theoretic significance and prac-tical applications.This doctoral dissertation deeply studies iteration methods to iterativesolutions of large-scale sparse linear algebraic equations.In particular,convergence char-acteristic of non-Hermitian linear systems,skew-symmetric triangular splitting iterativemethods,semiconvergence theorems of alternating iterative methods and Uzawa-type al-gorithms for generalized saddle point problems are deeply studied in this thesis.Study convergence theorems of single splittings for linear systems with non-Hermitian matrices.We first present a new iterative methods based on Hermitian andskew-Hermitian splitting(HSS) by appending a parameterαto HSS splitting.Using thisnew method,we establish convergence theorems of single splittings for non-Hermitianpositive definite matrices and give a method for choosing the optimal parameters forthe special splitting.Moreover,we apply convergence theorems of single splittings fornon-Hermitian positive definite to generalized alternating methods and two-stage mul-tisplittings methods and present their convergence theorems.Secondly,with the help ofnormal and skew-Hermitian splitting(NSS),which are similar to HSS splitting,equivalentconditions and related properties for convergence of splitting of non-Hermitian indefinitematrices are presented.Moreover,we use the obtained conclusion to determine whethera matrix has a dominant symmetric part.Study two classes of special iterative methods:double splitting iterative methods andskew-symmetric triangular iterative methods.We first present convergence theorems ofdouble splittings of Hermitian positive definite matrices and H-matrices.Furthermore,comparison theorems for double splittings of Hermitian positive definite matrices are alsoobtained,which provide theoretical base for the choice of iteration methods.Secondly,wecontribute two new methods for selection of iterative matrices of skew-symmetric triangu- lar iterative methods,extending those in [65] and get sufficient conditions for convergenceof skew-symmetric triangular iterative methods.Choices for the optimal parameters arealso introduced.In addition,we give some special selections of Ho for the new methodsand obtained theoretical results for the unconditional convergence of these methods.Study alternating iterative methods.We first briefly introduce various types of alter-nating iterative methods,such as the classice alternating iterative methods,the generalizedalternating iterative methods,parallel synchronous iterative methods,parallel alternatingsynchronous iterative methods of model 1 and model 2.Then,we present semiconver-gence theorems of all kinds of alternating iterative methods with singular coefficient ma-trices.Moreover,comparison theorems for alternating iterative methods are also obtained.Study Uzawa-type iterative methods for generalized saddle point problems,we re-call all types of Uzawa-type algorithms and then present three nonlinear Uzawa-type al-gorithms with relaxation parameters,which the extensions of the original ones.Further-more,convergence of the algorithms is discussed and numerical experiments verify thenecessity of the introduction of the relaxation parameters of these three algorithms,andshow that the nonlinear Uzawa-type algorithms with relaxation parameters requires lessiteration numbers than that of the original algorithms.