【作者】 张娜；

【导师】 杨虎；

【作者基本信息】 重庆大学 ， 概率论与数理统计， 2009， 硕士

【摘要】 矩阵的特征值问题在工程计算,物理,天文和微分方程数值求解等大规模数值计算方面起着极其重要的作用。对于给定的矩阵,要用计算机求出它的特征值,但是由于浮点误差和测量误差的存在,特征值的数值解与准确值之间必存在偏离,因此需要估计计算解的可靠性,这是矩阵扰动分析所要研究的重要课题之一。本文研究了不变子空间上矩阵特征值的绝对加法扰动和相对加法扰动,得出了几个新的扰动界。论文主要系统地研究了特征值传统误差界的三种类型,即:Weyl型,Hoffman-Wielandt型和Bauer-Fike型,并进一步研究了几种特殊的情况,得到了加强的结果。论文第三章研究了特征值的Weyl型扰动。在已有的原矩阵A是Hermite矩阵它的扰动矩阵B为一般矩阵得到的扰动界,以及原矩阵A和它的扰动矩阵B均为Hermite矩阵得到的扰动界的基础上,研究了Hermite在半正定条件下Weyl型的相对扰动界。论文第四章研究了特征值的Hoffman-Wielandt型扰动。在已有的几类常见Hoffman-Wielandt扰动界和可对角矩阵特征值的Hoffman-Wielandt型扰动界的基础上,进一步研究了任意矩阵特征值的Hoffman-Wielandt型扰动界,改进了已有的一些扰动界,得到了更易计算的形式。论文第五章研究了特征值的Bauer-Fike型扰动。在一个已有绝对扰动界的基础上,讨论了它的相对扰动界。论文第六章研究了AQ-AB=R型扰动。通过利用矩阵的分块以及矩阵的扩充等方法,改进了原有的一些扰动界,得到了更方便的结果,并把新扰动界与原有的扰动界进行了比较。更多还原

【Abstract】 In some large-scale numerical calculation problems , such as engineering calculation , physics,astronomy and numerical solution to differential equations, matrix eigenvalue problem plays an extremely important role. For a given matrix, if you want to use the computer to derive its eigenvalues, however, because of the existence of floating-point error and measurement error, it must exist deviation between the exact value and the numerical solution of eigenvalue, therefore , estimating the reliability of the calculation is one of the important issues to do some research to the matrix perturbation analysis .In this paper, the absolute additive perturbation and relative additive perturbation of matrix eigenvalue in invariant subspaces are studied, and some new perturbation bounds are given. The traditional three types of error bounds: Weyl, Hoffman-Wielandt, and Bauer-Fike, are systematically studied in this article and doing some further study to several special cases, then strengthened the corresponding results.In chapter three, some research about the Weyl type are studied. Considering the existing results of the original matrix A is the Hermite matrix and its perturbation matrix B is normal matrix , or the original matrix A and its perturbation matrix B both are Hermite matrix , some study to the relative perturbation bounds of the Weyl type , under the conditions of a positive semi-definite Hermite matrix.,which has the basis of some perturbation bounds under the conditions of a positive definite Hermite matrix.In chapter four, some research of the Hermite-Wielandt type are studied. Considering the exsisting results of the normal matrix and the diagonal matrix , some research about the normal Hoffman-Wielandt type of normal matrix are done, and some exsit perturbation bounds are improved , some perturbation bounds are easy .In chapter five, some research of the Bauer-Fike type are studied. A relative perturbation bound are given , which has the basis of its absolute perturbation bound.In chapter six, some research of the AQ-QB=R type are studied. Through the use of sub-block matrix, as well as the expansion of matrix. a easy result are given ,and at the same time improving the exsit results,. In the end , some comparison are given between the new and the old results.更多还原