One of the famous methods for system of nonlinear equations as well as nonlinear least-squares problems is the Levenberg-Marquardt method. Various modifications of this method have been given by improved LM parameters. In this paper, we give a new efficient Levenberg-Marquardt method by using a geometric mean to update the LM parameter. According to the rank of the Jacobian matrix, some relevant assumptions are given. Under a new local error bound condition, we consider the local convergence properties of our efficient Levenberg-Marquardt method without requiring zero residues. Numerical experiments show that the new efficient Levenberg-Marquardt method is effective.