In this work, we develop an alternating nonlinear Generalized Minimum Residual (NGMRES) algorithm with depth $m$ and periodicity $p$, denoted by aNGMRES($m, p$), applied to linear systems. We provide a theoretical analysis to quantify by how much one-step NGMRES($m$) 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($\infty,p$) and full GMRES are identical at the iteration index $jp$. Therefore, aNGMRES($\infty,p$) can be regarded as an alternative to GMRES for solving linear systems. For finite $m$, the iterates of aNGMRES($m,m+1$) and restarted GMRES (GMRES($m+1$)) are the same at the end of each periodic interval of length $p$, i.e, at the iteration index $jp$. In Addition, we present a convergence analysis of aNGMRES when applied to accelerate Richardson iteration. The advantages of aNGMRES($m,p$) 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($m$).

翻译：本文提出了一种应用于线性系统的交替非线性广义最小残差（NGMRES）算法，其深度为$m$，周期为$p$，记为aNGMRES($m, p$)。我们通过理论分析量化了以Richardson迭代作为初始猜测的单步NGMRES($m$)在可对角化及对称正定情况下对基础定点迭代收敛速度的提升程度。该理论分析有助于深入理解影响收敛速度的关键因素。此外，在特定条件下，我们证明了所提出的aNGMRES应用于Richardson迭代时与GMRES具有周期等价性。具体而言，aNGMRES($\\infty,p$)与完全GMRES在第$jp$次迭代时完全一致，因此aNGMRES($\\infty,p$)可作为求解线性系统的GMRES替代方案。对于有限深度$m$，aNGMRES($m,m+1$)与重启GMRES（GMRES($m+1$)）在长度为$p$的每个周期区间终点（即第$jp$次迭代）具有相同的迭代值。此外，本文给出了aNGMRES加速Richardson迭代时的收敛性分析。aNGMRES($m,p$)方法的优势在于：无需在每次迭代中求解最小二乘问题以降低计算成本，并能增强对NGMRES($m$)可能出现的停滞现象的鲁棒性。