摘要
计算科学与工程中的许多问题最后都归结为对线性代数方程组和矩阵特征值问题的求解．因此，对这两个问题求解方法的研究是科学与工程计算的核心问题之一，具有极其重要的理论意义和实际应用价值．
本文深入地研究了与稀疏线性代数系统迭代求解有关的几类特殊矩阵特征值的相互关系和不同的迭代法．特别地，给出了一些迭代法的比较理论，p-循环情况下不同迭代矩阵的特征值关系，定常迭代法预条件技术及其比较理论．
讨论了著名的Stein-Rosengberg定理及其推广形式，给出MPSD迭代法中几类迭代矩阵与Jacobi迭代矩阵谱半径的比较，指出在特定条件下Jacobi迭代矩阵的谱半径更小，丰富了Stein-Rosengberg定理的内涵，完善了Stein-Rosengberg定理的内容．
研究了p-循环矩阵，给出p-循环情况下MPSD迭代矩阵，GMPSD迭代矩阵及GUSAOR迭代矩阵与Jacobi迭代矩阵的特征值比较关系，优于相关文献的结果，特别是R．S．Varga在名著<>中的特征值关系．
研究了几种定常迭代法的预条件技术：
(1)提出Upper Jacobi与Upper Gauss-Seidel型迭代法；在特定的预条件矩阵下，给出预条件Upper Jacobi与Upper Gauss-Seidel型迭代法与对应的初始迭代法的比较关系及Upper Jacobi与Upper Gauss-Seidel型迭代法的比较关系；
(2)给出AOR迭代法在文献[106，107]提出的预条件矩阵下的比较关系；
(3)给出Gauss-Seidel迭代法在文献[107，111]提出的预条件矩阵下的比较关系．
The solution of linear systems of algebraic equations and computing matrices eigenvaluesarises in many problems of computational Science and Engineering computingproblem. So, the research of solutions for the two questions becomes one of the keyissues of scientific and engineering computing and has important theoretic significanceand valuable practical applications. This doctoral dissertation has a comprehensive studyon eigenvalues of some special matrices as well as iterative methods of linear algebraicsystems. In particular, comparison theories of some iterative methods are presented, andrelationships of eigenvalues among some iterative methods with Jacobi iterative methodhave also been studied and preconditioning techniques for the stationary iterative methodsare also investigated and some comparison theories are obtained.
Stein-Rosenberg Theorem and its generalized forms are studied. Comparison relationshipsof the spectral radius between some iterative matrices of MPSD iterative methodand the Jacobi iterative matrices are obtained. We show that the spectral radius of theJacobi iterative method is smaller than others under some assumptions. So, the SteinRosenbergTheorem and other results in references are perfected.
p-cyclic matrices are investigated. Relationships of eigenvalues between MPSD iterativematrices and Jacobi iterative matrices are obtained at first. The relationships ofeigenvalues between GMPSD iterative matrices and Jacobi iterative matrices are establishedlater and the relationships of eigenvalue between GUSAOR iterative matrices andJacobi iterative matrices are investigated in the end, The results in corresponding referencesare improved and perfected.
Preconditioning techniques for stationary iterative methods are studied.
(1) The Upper Jacobi and Upper Gauss-Seidel iterative methods are proposed. In thespecial preconditioner, the Upper Jacobi and Upper Gauss-Seidel iterative methods arestudied, the comparison relationships of preconditioned Upper Jacobi and Upper GaussSeideliterative methods with the origional ones, as well as comparison relationships ofpreconditioned Upper Jacobi method with preconditioned Upper Gauss-Seidel iterativemethod are obtained.
(2) The precodtioned AOR iterative method under the percondtioners in [106, 107]is investigated and some comparison relationships are obtained.
(3) The precodtioned Gauss-Seidel iterative method under the percondtioners in[107, 111] is studied and some comparison relationships are obtained.
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