求解非线性方程组的一种新的全局收敛的Levenberg-Marquardt算法

doi:10.12286/jssx.2008.4.388

计算数学 ›› 2008, Vol. 30 ›› Issue (4): 388-396.DOI: 10.12286/jssx.2008.4.388

求解非线性方程组的一种新的全局收敛的Levenberg-Marquardt算法

杨柳¹, 陈艳萍²

1. 湘潭大学数学与计算科学学院, 湖南湘潭, 411105 2. 华南师范大学数学科学学院, 广州, 510631

出版日期:2008-11-14 发布日期:2009-02-05
基金资助:
广东省高等学校珠江学者计划; 国家自然科学基金项目(10671163); 973项目(2005CB321703); 湖南省教育厅资助项目(06A069, 06C824); 国家和湖南省重点学科建设项目资助.

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杨柳, 陈艳萍. 求解非线性方程组的一种新的全局收敛的Levenberg-Marquardt算法[J]. 计算数学, 2008, 30(4): 388-396.

Yang Liu, Chen Yanping. A NEW GLOBALLY CONVERGENT LEVENBERG-MARQUARDT METHOD FOR SOLVING NONLINEAR SYSTEM OF EQUATIONS[J]. Mathematica Numerica Sinica, 2008, 30(4): 388-396.

A NEW GLOBALLY CONVERGENT LEVENBERG-MARQUARDT METHOD FOR SOLVING NONLINEAR SYSTEM OF EQUATIONS

Yang Liu¹, Chen Yanping²

1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China 2.School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Online:2008-11-14 Published:2009-02-05

本文提出了求解非线性方程组的一种新的全局收敛的Levenberg-Marquardt算法,即$\mu_k=\alpha_k(\theta\|F_k\|+(1-\theta)\|J_k^TF_k\|), \theta\in[0,1],$ 其中$\alpha_k$利用信赖域技巧来修正. 在不必假设雅可比矩阵非奇异的局部误差界条件下,证明了该算法是全局收敛和局部二次收敛的. 数值试验表明该算法能有效地求解奇异非线性方程组问题.

关键词： 局部误差界, Levenberg-Marquardt方法, 非线性方程组, 全局收敛性, 局部收敛性

In this paper, we propose a new globally convergent Leveberg-Marquardt method for solving nonlinear systems of equations, i.e. $\mu_k=\alpha_k(\theta\|F_k\| +(1-\theta)\|J_k^TF_k\|),\theta\in[0,1],$ where $\alpha_k$ is updated by trust region techniques. Global
and local convergence of this new method are proved without the nonsingularity assumption of the Jacobian matrix. Numerical results show that this new method performs very well for the singular nonlinear systems of equations.

Key words: local error bound, Levenberg-Marquardt method, nonlinear system of equations, global convergence, local convergence

MR(2010)主题分类:

65K05
90C30

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摘要

求解非线性方程组的一种新的全局收敛的Levenberg-Marquardt算法

A NEW GLOBALLY CONVERGENT LEVENBERG-MARQUARDT METHOD FOR SOLVING NONLINEAR SYSTEM OF EQUATIONS

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