We establish a near-optimality guarantee for the full orthogonalization method (FOM), showing that the overall convergence of FOM is nearly as good as GMRES. In particular, we prove that at every iteration $k$, there exists an iteration $j\leq k$ for which the FOM residual norm at iteration $j$ is no more than $\sqrt{k+1}$ times larger than the GMRES residual norm at iteration $k$. This bound is sharp, and it has implications for algorithms for approximating the action of a matrix function on a vector.

翻译：我们建立了全正交化方法（FOM）的近最优性保证，证明FOM的整体收敛性几乎与GMRES相当。具体而言，我们证明在每次迭代$k$时，存在一个迭代$j\leq k$，使得第$j$次迭代的FOM残差范数不超过第$k$次迭代的GMRES残差范数的$\sqrt{k+1}$倍。该界是紧的，并且对近似矩阵函数作用于向量的算法具有重要意义。