The modified Levenberg–Marquardt method with nonmonotone technique

In this paper, we propose a modified Levenberg–Marquardt (LM) method with a nonmonotone technique for solving nonlinear equations. Under the $\mathrm{H}\ddot{\mathrm{o}}\mathrm{l}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}$ continuity and the $\mathrm{H}\ddot{\mathrm{o}}\mathrm{l}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}$ local error bounds conditions, which are weaker than the local error bounds and the Lipschitz continuity, the global convergence and local convergence of the algorithm are proved. Numerical experiments also show that the algorithm is effective.