In this paper, a matrix-free method for solving large-scale systems of nonlinear equations is presented. The method is derived via the quasi-Newton approach, where the approximation to Broyden’s update is constructed using a diagonal matrix with an acceleration parameter. A key feature of the method is that it is matrix-free, making it suitable for solving large-scale problems efficiently. Furthermore, the convergence analysis of the new method is discussed under certain standard conditions. Preliminary numerical experiments on selected test problems demonstrate that the proposed method is both effective and promising.