  The famous greedy randomized Kaczmarz (GRK) method uses the greedy selection
rule on maximum distance to determine a subset of the indices of working rows.
In this paper, with the greedy selection rule on maximum residual, we propose
the greedy randomized Motzkin-Kaczmarz (GRMK) method for linear systems. The
block version of the new method is also presented. We analyze the convergence
of the two methods and provide the corresponding convergence factors. Extensive
numerical experiments show that the GRMK method has almost the same performance
as the GRK method for dense matrices and the former performs better in
computing time for some sparse matrices, and the block versions of the GRMK and
GRK methods always have almost the same performance.
