ABSTRACT In this work, we develop an alternating nonlinear Generalized Minimum Residual (NGMRES) algorithm with depth and periodicity , denoted by aNGMRES(), applied to linear systems. We provide a theoretical analysis to quantify by how much one‐step NGMRES() using Richardson iterations as initial guesses can improve the convergence speed of the underlying fixed‐point iteration for diagonalizable and symmetric positive definite cases. Our theoretical analysis gives us a better understanding of which factors affect the convergence speed. Moreover, under certain conditions, we prove the periodic equivalence between the proposed aNGMRES applied to Richardson iteration and GMRES. Specifically, aNGMRES() and full GMRES are identical at the iteration index . Therefore, aNGMRES() can be regarded as an alternative to GMRES for solving linear systems. For finite , the iterates of aNGMRES() and restarted GMRES (GMRES()) are the same at the end of each periodic interval of length , that is, at the iteration index . In Addition, we present a convergence analysis of aNGMRES when applied to accelerate Richardson iteration. The advantages of aNGMRES() method are that there is no need to solve a least‐squares problem at each iteration which can reduce the computational cost, and it can enhance the robustness against stagnations, which could occur for NGMRES().