Accelerating the modified Levenberg-Marquardt method for nonlinear equations

In this paper we propose an accelerated version of the modified Levenberg-Marquardt method for nonlinear equations (see Jinyan Fan, Mathematics of Computation 81 (2012), no. 277, 447–466). The original version uses the addition of the LM step and the approximate LM step as the trial step at every iteration, and achieves the cubic convergence under the local error bound condition which is weaker than nonsingularity. The notable differences of the accelerated modified LM method from the modified LM method are that we introduce the line search for the approximate LM step and extend the LM parameter to more general cases. The convergence order of the new method is shown to be a continuous function with respect to the LM parameter. We compare it with both the LM method and the modified LM method; on the benchmark problems we observe competitive performance.