On the Convergence of Levenberg-Marquardt Method for Solving Nonlinear Systems

• Published in: Bio-Inspired Computing - Theories and Applications pp. 117 - 122
• Main Authors: Fang, Minglei, Xu, Feng, Zhu, Zhibin, Jiang, Lihua, Geng, Xianya
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg
• Series: Communications in Computer and Information Science
• Subjects: L-M method; Local error bound; Nonlinear equation
• ISBN: 3662450488; 9783662450482
• ISSN: 1865-0929; 1865-0937
• DOI: 10.1007/978-3-662-45049-9_19

Levenberg-Marquardt (L-M forshort) method is one of the most important methods for solving systems of nonlinear equations. In this paper, we consider the convergence under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda_{k}=\min(\|F_{k}\|,\|J_{k}^{T}F_{k}\|)$\end{document} of L-M method. We will show that if  ∥ F(xk) ∥ provides a local error bound, which is weaker than the condition of nonsingularity for the system of nonlinear equations, the sequence generated by the L-M method converges to the point of the solution set quadratically. As well, numerical experiments are reported.