A modified two steps Levenberg–Marquardt method for nonlinear equations

The modified Levenberg–Marquardt (MLM) method to solve nonlinear equations was introduced by Fan (2012). This method uses an addition of the Levenberg–Marquardt step and an approximate LM step as the trial step at every iteration. Using a trust region technique, the global and cubic convergence of the MLM method under the local error bound condition is proved (Fan, 2012). Recently, Fan proposed an accelerated MLM algorithm by using a line search strategy to generate a modified LM step and showed that the convergence rate of the algorithm is min{1+2δ,3}min{1+2δ,3} which results the cubic convergence for δ≥1δ≥1 (Fan, 2014). In this paper, by introducing an adaptive LM parameter for AMLM algorithm, we propose an efficient AMLM algorithm. The cubic convergence of the new algorithm is presented while numerical experiments show the new algorithm is promising.