  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.
