解非线性方程组的加速修正Newton-HSS方法

发布时间：2018-12-30 13:57

【摘要】：近年来,非线性问题越来越多地出现在科学和工程计算领域,如何提高其求解效率亦已引起人们的广泛重视.本文主要讨论用于求解大型、稀疏且带有非Hermitian型Jacobian矩阵的非线性方程组的加速修正Newton-HSS方法.它以Hermitian和反Hermitian分裂(HSS)的迭代方法为基础,其中加速修正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.当步长参数a = 1,= 1时,加速修正Newton-HSS方法简化为修正Newton-HSS 方法.本文首先给出了求解带非Hermitian型、正定Jacobian矩阵的加速修正Newton-HSS方法1和带非Hermitian型、奇异Jacobian矩阵的加速修正Newton-HSS方法2的算法步骤.接着从以下方面对加速修正Newton-HSS算法进行了收敛性分析:1.加速修正Newton-HSS方法1在Lipschitz条件下的局部收敛性;2.加速修正Newton-HSS方法1在Lipschitz条件下的半局部收敛性;3.加速修正Newton-HSS方法2在Lipschitz条件下的局部收敛性.最后,通过Lipschitz条件下的数值算例6.1,以a = 1,b取最优参数及b = 1,a取最优参数为例,说明了加速修正Newton-HSS方法1在内迭代次数及运行时间等方面都优于修正Newton-HSS方法.又通过Lipschitz条件下的数值算例6.2,说明了加速修正Newton-HSS方法2的高效性.从而,说明了加速修正Newton-HSS算法的可行性和有效性.
[Abstract]:In recent years, more and more nonlinear problems appear in the fields of science and engineering calculation, and how to improve the efficiency of solving them has been paid more and more attention. In this paper we mainly discuss the accelerated modified Newton-HSS method for solving large sparse nonlinear equations with non Hermitian type Jacobian matrices. It is based on the iterative method of Hermitian and anti-Hermitian splitting (HSS), in which the accelerated modified Newton method is used to solve nonlinear equations, and the HSS method is used to approximate solve Newton equation. When the step size parameter a = 1g = 1, the accelerated modified Newton-HSS method is simplified to the modified Newton-HSS method. In this paper, we first give the steps of the accelerated modified Newton-HSS method 1 with non Hermitian type, positive definite Jacobian matrix and the accelerated modified Newton-HSS method 2 with non Hermitian type and singular Jacobian matrix. Then the convergence of the accelerated modified Newton-HSS algorithm is analyzed from the following aspects: 1. Accelerating the Local Convergence of modified Newton-HSS method 1 under Lipschitz condition; 2. Accelerating the Semi-local Convergence of modified Newton-HSS method 1 under Lipschitz condition; 3. The local convergence of the modified Newton-HSS method 2 under the Lipschitz condition is accelerated. Finally, through the numerical example 6.1 under the Lipschitz condition, taking a = 1nb for the optimal parameter and b = 1a for the optimal parameter as an example, It is shown that the accelerated modified Newton-HSS method 1 is superior to the modified Newton-HSS method in terms of the number of iterations and the running time. The efficiency of the accelerated modified Newton-HSS method 2 is illustrated by a numerical example of 6.2 under the Lipschitz condition. Thus, the feasibility and effectiveness of accelerating the modified Newton-HSS algorithm are illustrated.
【学位授予单位】：华东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.7

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