An accelerated adaptive two-step Levenberg–Marquardt method with the modified Metropolis criterion

In this paper, aiming at the nonlinear equations, a new two-step Levenberg–Marquardt method was proposed. We presented a new Levenberg–Marquardt parameter to obtain the trial step. A new modified Metropolis criterion was used to adjust the upper bound of the approximate step. The convergence of the method was analyzed under the H $\ddot{\mathrm{o}}$lderian local error bound condition and the H $\ddot{}o$lderian continuity of the Jacobian. Numerical experiments showed that the new algorithm is effective and competitive in the numbers of functions, Jacobian evaluations and iterations.