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.