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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. 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.

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. 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.

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.

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