数值线性代数中的若干问题

【作者】 张振宇；

【导师】 曹志浩；

【作者基本信息】 复旦大学 ， 计算数学， 2003， 博士

【摘要】 本文包含两部分。在第一部分中，我们研究了三种不同的预条件矩阵一块三角预条件矩阵、约束预条件矩阵和块对角预条件矩阵作用到2×2块线性系统(其中右下角块为零)以后的快速迭代解法，给出了预条件作用后系统的特征值的谱和形式，比较了三种不同预条件作用后采用CMRES(20) 方法求解的有效性。在第二部分中，我们给出了在三种情形下亏损矩阵的特征灵敏度的直接摄动方法。第一种情形是特征值一阶摄动系数都不相同，在这种情形下，我们给出了计算特征值1到3阶摄动系数以及计算特征向量1到2阶摄动系数的计算公式；第二种情形是特征值一阶摄动系数有相等的情形，在这种情形下，我们给出了计算特征值1到3阶摄动系数以及计算特征向量1阶摄动系数的计算公式；第三种情形是第一种情形的扩展，此时对应于最低阶Jordan块的特征值一阶摄动系数有一个是零。在这种情形下，我们给出了计算特征值和特征向量1到3阶摄动系数的计算公式。文章最后给出的数值算例表明了以上给出的公式是正确的。更多还原

【Abstract】 The contents of the article include two parts. In the first part, we study three kinds of pre-conditioners -- block-triangular preconditioner, constraint preconditioner and block-diagonalpreconditioner respective1y and fast iterative so1vers fOr indefinite 2 by 2 b1ock linear systemswith zero (2,2) block. The spectrum and fOrm of eigenvalues of each preconditioned nlatrix arepresented. All of the three kinds of preconditioned linear syStems are so1ved by GMRES(20)and theiI efficiencies are cornpared. NumericaJ results are presented by a maJss of tables andfigures. In the second part, we propose direct perturbation methods of eigensensitivity ana1ysisa defective nlatrix fOr three different cases. The first case is that all of the first-order pertur-batioxl coefficients of a defective eigenvalue associated with its Jordan bIocks with same orderare distinct and nonzero. In this case, we give the formulas to caIculate first- to third-orderperturbation coefficients of the eigenvalues and first- to second-order perturbation coefficientsof the eigenvectors. 1n second case where the eigenprobIem for the first-order perturbationcoefficients of a defective eigenvalue haJs repeated eigenvalues, we give the fOrmulas to calculatethe first- to third-order perturbation coefficients of the eigenvalues and first order perturbationcoefficients of the eigenvectors. The third case is an extension of the first case, where one of thefirst-order perturbation coefficients of the eigenvalues associated with the lowest-order Jordanblocks is zero. In this case, we give the formulaJs to calculate first- to third--order perturbationcoefficients of the eigenva1ues and eigenvectors. The nun1erical examples show the validity oftllese methods.更多还原