Some research on Levenberg-Marquardt method for the nonlinear equations

Levenberg-Marquardt method is one of the most important methods for solving systems of nonlinear equations. In this paper, we consider the convergence of a new Levenberg-Marquardt method (i.e. lambda(k) = theta vertical bar vertical bar F-k vertical bar vertical bar + (1 - theta)vertical bar vertical bar J(k)(T)F(k)vertical bar vertical bar, where theta is an element of [0, 1] is a real parameter) for solving a system of singular nonlinear equations F(x) = 0, where F is a mapping from R-n into R-m. We will show that if vertical bar vertical bar F(x)vertical bar vertical bar provides a local error bound which is weaker than the condition of nonsingular for the system of nonsingular for the system of nonlinear equations, the sequence generated by the new Levenberg-Marquardt method convergence to a point of the solution set X* quadratically. Numerical experiments and comparisons are reported. (c) 2006 Elsevier Inc. All rights reserved.