Numerical Algebra, Control and Optimization (NACO)

Recent advances in numerical methods for nonlinear equations and nonlinear least squares

Pages: 15 - 34, Volume 1, Issue 1, March 2011      doi:10.3934/naco.2011.1.15

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Ya-Xiang Yuan - State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing,Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Zhong Guan Cun Donglu 55, Beijing, 100190, China (email)

Abstract: Nonlinear equations and nonlinear least squares problems have many applications in physics, chemistry, engineering, biology, economics, finance and many other fields. In this paper, we will review some recent results on numerical methods for these two special problems, particularly on Levenberg-Marquardt type methods, quasi-Newton type methods, and trust region algorithms. Discussions on variable projection methods and subspace methods are also given. Some theoretical results about local convergence results of the Levenberg-Marquardt type methods without non-singularity assumption are presented. A few model algorithms based on line searches and trust regions are also given.

Keywords:  Nonlinear equations, nonlinear least squares, Levenberg-Marquardt, quasi-Newton, trust region, variable projection, subspace, local error bound conditions, convergence.
Mathematics Subject Classification:  Primary: 65K10; Secondary: 90C05.

Received: June 2010;      Revised: September 2010;      Available Online: February 2011.