【作者】 王伟；

【导师】 于波；

【作者基本信息】 大连理工大学 ， 计算数学， 2006， 硕士

【摘要】 非线性方程组的数值求解常见于许多科学与工程计算领域,具有十分重要的理论意义和实用价值。在Newton法的基础上发展而得到的不精确Newton法是目前求解大规模稀疏非线性方程组的主要方法之一。不精确Newton法是一个内外迭代过程,其外迭代是Newton迭代,而内迭代则是某个线性迭代。关于解非线性方程组和无约束优化的不精确牛顿型方法的研究很多,但关于不精确秩1、秩2修正拟牛顿法的研究尚未见到,这大概是因为秩1、秩2修正拟牛顿方程易于求解的缘故。但拟牛顿法推广到Banach空间上算子方程或minimax问题上时,因为子问题的精确求解变得困难或不可能,就有必要研究不精确拟牛顿法。本文对有限维非线性方程组建立不精确拟牛顿法的收敛性理论,为研究Bnanch空间算子方程和minimax问题的不精确拟牛顿法做好准备。更多还原

【Abstract】 Numerical methods for nonlinear equations are very important in many areas. In-exact Newton method, which is based on Newton method, is one of the main methods for solving large sparse systems of nonlinear equations. Inexact Newton method is an inner-outer iterative procedure, with Newton iteration as its outer iteration and a linear iteration as its inner iteration. There are lots of research results about inexact Newton-like methods for solving nonlinear equations and unconstrained optimization, however, no result on inexact rank one or rank two updated quasi-Newton methods has been seen, maybe because rank one and rank two updated quasi-Newton equations are easier to solve than Newton equations. When quasi-Newton methods is extended to operator equation in Banach space or minimax problems, it is difficult or impossible to solve subproblems exactly, so it is necessary to investigate inexact quasi-Newton methods. In this paper, convergence of inexact quasi-Newton methods for nonlinear equations in R~n is proven under commonly used conditions on updated matrics. The results serve as preparation for convergence theory of inexact quasi-Newton methods for operator equations in Banach space and minimax problems.更多还原