A high-order modified Levenberg-Marquardt method for systems of nonlinear equations with fourth-order convergence

Fan (2014) presented an accelerated modified Levenberg-Marquardt method for nonlinear equations. At every iteration, the accelerated modified LM method computed not only a LM trial step, but also an additional approximate LM step which employed a line search. In this paper, based on the accelerated modified LM method, we compute the approximate LM step one more time at every iteration, and obtain a high-order accelerating modified Levenberg-Marquardt method. Under the local error bound condition which is weaker than nonsingularity, the convergence order of this new method is shown to be fourth. A globally convergence is also given by the trust region technique. Numerical results show that the new method is efficient and could save many calculations of the Jacobian. (C) 2016 Elsevier Inc. All rights reserved.