【作者】 吴静；

【导师】 黄廷祝；

【作者基本信息】 电子科技大学 ， 应用数学， 2009， 博士

【摘要】 科学与工程的很多重要领域如计算电磁学，高阶微分方程求解，最优化问题，流体力学和油藏模拟等都离不开大型线性代数方程组的求解．大型稀疏线性方程组的求解方法研究已经成为大规模科学与工程计算的核心问题之一，具有重要的理论意义和实际应用价值．本文对求解大型稀疏线性代数方程组的一些迭代解法进行了深入的研究，特别系统研究了非Hermitian线性系统的收敛特性，反对称三角迭代法，交替迭代法的半收敛性理论及广义鞍点问题Uzawa类型算法等．研究了非Hermitian矩阵线性系统单分裂的收敛性理论．首先，对矩阵的Hermitian和反Hermitian分裂（HSS）添加一个参数α，得到了变形的HSS分裂，利用新的分裂方法建立了非Hermitian正定矩阵单分裂的收敛性理论，给出了取特殊分裂时最优参数的选取方法．同时，将非Hermitian正定矩阵单分裂的收敛定理应用到广义交替迭代法和两步多分裂迭代法中，给出了两种方法的收敛理论．其次，利用与HSS分裂相类似的正规和反Hermitian分裂（NSS）方法，研究了非Hermitian不定矩阵单分裂收敛的等价条件，给出了非Hermitian不定矩阵NSS分裂的相关性质．同时，将所得结论用来判定矩阵是否具有对称占优性．研究了两类特殊迭代方法—矩阵双分裂迭代法和反对称三角迭代方法．首先，建立了系数矩阵为两类特殊矩阵—H-矩阵和Hermitian正定矩阵时，双分裂迭代法的收敛性理论，并且得到了Hermitian正定矩阵双分裂的比较理论．这些理论为迭代法的选择提供了一些理论依据．其次，给出了反对称三角迭代法中迭代矩阵的两种新的选取方法，对文[65]进行了拓展，得到了反对称三角迭代法收敛的充分条件，对于最优参数的选择也做了相应的介绍．另外，给出了新方法中H₀的一些特殊选取，并得到了迭代法无条件收敛的理论结果．研究了交替迭代方法．首先对各类交替迭代法，如经典交替迭代法，广义交替迭代法，并行同步迭代法，并行交替同步迭代法的模型一和模型二进行了简单介绍．其次，研究了当系数矩阵为奇异矩阵时各类交替迭代法的半收敛性理论，同时给出了各类交替迭代法的比较理论．研究了鞍点问题中Uzawa类型的迭代解法．在对各类Uzawa类型算法进行回顾后，提出了三个带松弛因子的非线性Uzawa算法，即非线性Uzawa算法的变形，分析了各算法的收敛性问题，得到三个算法的收敛理论．同时，通过数值实验说明了引入松弛因子的必要性，实验结果表明，前两个带松弛因子的非线性Uzawa算法比原算法所需迭代数少．更多还原

【Abstract】 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.更多还原