The higher-order Levenberg-Marquardt method with Armijo type line search for nonlinear equations

To save more Jacobian calculations and achieve a faster convergence rate, Yang [A higher-order Levenberg-Marquardt method for nonlinear equations, Appl. Math. Comput. 219(22)(2013), pp. 10682-10694, doi:10.1016/j.amc.2013.04.033, 65H10] proposed a higher-order Levenberg-Marquardt (LM) method by computing the LM step and another two approximate LM steps for nonlinear equations. Under the local error bound condition, global and local convergence of this method is proved by using trust region technique. However, it is clear that the last two approximate LM steps may be not necessarily a descent direction, and standard line search technique cannot be used directly to obtain the convergence properties of this higher-order LM method. Hence, in this paper, we employ the nonmonotone second-order Armijo line search proposed by Zhou [On the convergence of the modified Levenberg-Marquardt method with a nonmonotone second order Armijo type line search, J. Comput. Appl. Math. 239 (2013), pp. 152-161] to guarantee the global convergence of this higher-order LM method. Moreover, the local convergence is also preserved under the local error bound condition. Numerical results show that the new method is efficient.