Multi-Step projection method for solving nonlinear equations

This paper presents a projection based multi-step conjugate gradient like method for solving large-scale monotone nonlinear equations. The proposed method extends the multi-step conjugate gradient method (MSCGM) originally developed by Ford et al. for unconstrained optimization problems. In this extension, the gradient components in MSCGM are replaced with residuals, and a hyperplane projection technique is incorporated. Under suitable assumptions, we establish the global convergence of the method, showing that the entire sequence of iterates converges to a solution. Numerical experiments on standard test problems demonstrate that the proposed approach is efficient, stable, and competitive with existing methods.