A higher-order Levenberg–Marquardt method for nonlinear equations

In this paper, we present a high-order Levenberg–Marquardt method for nonlinear equations. At every iteration, not only a LM step is computed but also two approximate LM steps are computed which use the previous calculated Jacobian. Under the local error bound condition which is weaker than nonsingularity, the new method has biquadratic convergence. A globally convergent LM algorithm is also given by the trust region technique. Numerical results show that the new fourth-order LM algorithm performs very well and could save many Jacobian calculations especially for large scale problems.