A two-step Broyden-like method for nonlinear equations

In this paper, based on a nonmonotone derivative-free line search, we propose a two-step Broyden-like method (denoted by TS-BLM) for solving the nonlinear equations. TS-BLM computes one quasi-Newton step at the beginning iterations. When the iterations are close to the solution set of the nonlinear equations, an additional approximate quasi-Newton step is calculated by solving a linear system formed by using the previous Broyden-like matrix. We prove that TS-BLM converges globally under appropriate conditions. Moreover, we analyze the convergence rate of TS-BLM depending on the approximation of the Broyden-like matrix to the Jacobian. Some numerical results are reported to show the superior numerical performances of TS-BLM compared with the traditional BLM.