For solving large consistent linear systems by iteration methods, inspired by the maximum residual Kaczmarz method and the randomized block Kaczmarz method, we propose the maximum residual block Kaczmarz method, which is designed to preferentially eliminate the largest block in the residual vector $r_{k}$ at each iteration. At the same time, in order to further improve the convergence rate, we construct the maximum residual average block Kaczmarz method to avoid the calculation of pseudo-inverse in block iteration, which completes the iteration by projecting the iteration vector $x_{k}$ to each row of the constrained subset of $A$ and applying different extrapolation step sizes to average them. We prove the convergence of these two methods and give the upper bounds on their convergence rates, respectively. Numerical experiments validate our theory and show that our proposed methods are superior to some other block Kaczmarz methods.

翻译：针对大型相容线性系统的迭代求解问题，受最大残差卡茨马兹方法和随机分块卡茨马兹方法的启发，本文提出了一种最大残差分块卡茨马兹方法。该方法旨在每次迭代时优先消除残差向量$r_{k}$中的最大分块。同时，为进一步提升收敛速度，我们构建了最大残差平均分块卡茨马兹方法，以避免分块迭代中伪逆的计算——该方法通过将迭代向量$x_{k}$投影到$A$的约束子集的每一行，并应用不同的外推步长进行平均来完成迭代。我们证明了这两种方法的收敛性，并分别给出了其收敛速度的上界。数值实验验证了理论分析，并表明所提方法优于其他分块卡茨马兹方法。