The modified Levenberg-Marquardt method for nonlinear equations with cubic convergence

We propose a modified Levenberg-Marquardt method for nonlinear equations, in which not only a LM step but also an approximate LM step are computed at every iteration.To ensure the global convergence of the new method, a new kind of predicted reduction is introduced for the merit function when using the trust region technique.The cubic convergence of the modified LM method is proved under the local error bound condition which is weaker than nonsingularity.Numerical results show that the new method is very efficient and could save many calculations of the Jacobian.