In this work, we analyze the asymptotic convergence factor of minimal residual iteration (MRI) (or GMRES(1)) for solving linear systems $Ax=b$ based on vector-dependent nonlinear eigenvalue problems. The worst-case root-convergence factor is derived for linear systems with $A$ being symmetric or $I-A$ being skew-symmetric. When $A$ is symmetric, the asymptotic convergence factor highly depends on the initial guess. While $M=I-A$ is skew-symmetric, GMRES(1) converges unconditionally and the worst-case root-convergence factor relies solely on the spectral radius of $M$. We also derive the q-linear convergence factor, which is the same as the worst-case root-convergence factor. Numerical experiments are presented to validate our theoretical results.

翻译：本文基于向量相关的非线性特征值问题，分析了求解线性方程组 $Ax=b$ 的最小残差迭代（MRI）（或称 GMRES(1)）的渐近收敛因子。我们推导了当 $A$ 对称或 $I-A$ 斜对称时线性方程组的最坏情况根收敛因子。当 $A$ 对称时，渐近收敛因子高度依赖于初始猜测。而当 $M=I-A$ 斜对称时，GMRES(1) 无条件收敛，且最坏情况根收敛因子仅依赖于 $M$ 的谱半径。我们还推导了 q-线性收敛因子，其值与最坏情况根收敛因子相同。数值实验验证了我们的理论结果。