【作者】 李杨；

【导师】 郭学萍；

【作者基本信息】 华东师范大学 ， 计算数学， 2016， 硕士

【摘要】 近年来,科学和工程计算领域越来越多的出现非线性问题,如何快速有效地数值求解各类非线性问题逐渐受到人们的普遍关注.目前已经有很多求解非线性方程组的数值算法.本文主要讨论一种基于Hermitian和反Hermitian分裂(HSS)的迭代方法用于求解大型、稀疏且带有正定Jacobian矩阵的非线性方程组的多步修正Newton-HSS方法,其中多步修正Newton方法用于求解非线性方程组,HSS方法用于近似求解牛顿方程.当步数m=1时,多步修正Newton-HSS方法就是Newton-HSS方法；当步数m=2时,多步修正Newton-HSS方法就是修正Newton-HSS方法.首先给出了多步修正Newton-HSS方法的算法步骤.然后从以下三个方面对多步修正Newton-HSS算法进行了收敛性分析：1.Lipschitz连续条件下的局部收敛性、半局部收敛性定理；2. Holder连续条件下的半局部收敛性定理；3.全局多步修正Newton-HSS方法的全局收敛定理.实际上, Holder连续弱于Lipschitz连续,在某种程度上,Lipschitz连续是Holder连续的特例.最后通过Lipschitz条件的数值算例及Holder条件的数值算例,以三步修正Newton-HSS算法及四步修正Newton-HSS算法为例,证明了多步修正Newton-HSS方法在运行时间及外迭代次数等方面都优于修正Newton-HSS方法,从而说明了多步修正Newton-HSS算法的可行性及有效性.更多还原

【Abstract】 In recent years, quite a number of nonlinear systems often arise in many scientific and en-gineering computing areas. The numerical solutions for the nonlinear systems are often required. There are a lot of ways to be chosen to solve the nonlinear systems. In this paper, based on the HSS splitting, we establish a class of multi-step modified Newton-HSS (MMN-HSS) methods for solving large sparse system of nonlinear equations with positive definite Jacobian matrices. The MMN-HSS methods use the multi-step modified Newton methods to solve the nonlinear equation-s, and the HSS method to approximately solve the modified Newton equation. When step number m= 1 and m= 2, this method is simplified as the Newton-HSS method and the modified Newton-HSS method, respectively.Firstly, after some straightforward operations, we give the expression of the multi-step modi-fied Newton-HSS algorithm.Secondly, we analysis the convergence of this method from the following aspects. Under the Lipschitz conditions, we not only show the local convergence theorem but also prove the semilocal convergence theorem. We also give the semilocal convergence theorem of the MMN-HSS meth-ods, assuming the nonlinear operator satisfies the Holder continuous condition. Then, we establish the global multi-step modified Newton-HSS method and prove the global convergence theorem. The Holder condition is much milder than the usual Lipschitz condition. To some degree, the Lip-schitz condition is the special example of the Holder condition.Finally, one numerical example with the Lipschitz condition and two examples with the Holder condition are given to confirm the feasibility and effectiveness of our method. The nu-merical results show that the multi-step modified Newton-HSS method outperforms the modified Newton-HSS method in the sense of numbers of iterations and CPU time.更多还原