求解非线性方程的Ulm类方法及逆特征值问题

发布时间：2018-03-28 03:36

本文选题：Ulm类方法　切入点：牛顿类方法　出处：《浙江师范大学》2015年硕士论文

【摘要】：随着科技的迅速发展及计算机应用的广泛普及,求解非线性方程在经济、计算机科学、信息科学、物理及生命科学等领域中有着广泛的应用.本文主要研究一般非线性算子方程的求解以及可以转化为非线性方程问题的逆特征值问题的数值方法.具体内容如下：第一章介绍一般非线性算子方程与逆特征值问题求解的发展过程以及与本文相关的预备知识,包括逆特征值问题等相关概念,收敛阶,收敛条件,以及Banach空间的相关结论,同时介绍利用优序列证明半局部收敛的方法及构造优序列的两种常用的方法.最后给出了论文的结构.第二章研究用于求解一般非线性算子方程的Ulm类方法,该方法避免计算Jacobian矩阵和求解Jacobian方程.在一定条件下,我们证明了由该Ulm类方法产生的序列局部收敛到方程的解.第三章研究了用于求解逆特征值问题的牛顿类方法的半局部收敛性问题.利用优序列的技巧,在给定特征值互异和Jacobian矩阵在初始点非奇异的假设下,我们建立了一个仅依赖于初始点信息的Kantorovich型的收敛判据.与其他已有的求解逆特征值问题的数值方法的收敛性结果比较,我们的收敛结果摆脱了对逆特征值问题的解的依赖性问题.
[Abstract]:With the rapid development of science and technology and the wide spread of computer application, solving nonlinear equations in economy, computer science, information science, There are many applications in physics and life sciences. In this paper, we study the solution of general nonlinear operator equations and the numerical methods of inverse eigenvalue problems which can be transformed into nonlinear equations. The main contents are as follows. The first chapter introduces the development process of solving general nonlinear operator equations and inverse eigenvalue problems and the preparatory knowledge related to this paper. Including the inverse eigenvalue problems and other related concepts, convergence order, convergence conditions, as well as the relevant conclusions of Banach space, At the same time, the methods of proving semi-local convergence by using superior sequences and two common methods of constructing superior sequences are introduced. Finally, the structure of the paper is given. In chapter 2, the Ulm class method for solving general nonlinear operator equations is studied. The method avoids calculating Jacobian matrix and solving Jacobian equation. We prove that the sequence generated by the Ulm class method converges locally to the solution of the equation. In chapter 3, we study the semi-local convergence problem of Newtonian methods for solving inverse eigenvalue problems. Under the assumption that the eigenvalue is different and the Jacobian matrix is nonsingular at the initial point, we establish a convergence criterion of Kantorovich type which only depends on the information of the initial point. The convergence results are compared with those obtained by other numerical methods for solving inverse eigenvalue problems. Our convergence results are independent of the solution of the inverse eigenvalue problem.
【学位授予单位】：浙江师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.7

【相似文献】