Three-steps modified Levenberg–Marquardt method with a new line search for systems of nonlinear equations

Three steps modified Levenberg–Marquardt method for nonlinear equations was introduced by Yang (2013). This method uses the addition of the Levenberg–Marquardt (LM) step and two approximate LM steps as the trial step at every iteration. Using trust region technique, the global and biquadratic convergence of the method is proved by Yang. The main aim of this paper is to introduce a new line search strategy while investigating the convergence properties of the method with this line search technique. Since the search direction of Yang method may be not a descent direction, standard line searches cannot be used directly. In this paper we propose a new nonmonotone third order Armijo type line search technique which guarantees the global convergence of this method while we use an adaptive LM parameter. It is proved that the convergence order of the new method is biquadratic. Numerical results show the new algorithm is efficient and promising.