  In this paper, we analyze the greedy randomized Kaczmarz (GRK) method
proposed in Bai and Wu (SIAM J. Sci. Comput., 40(1):A592--A606, 2018) for
solving linear systems. We develop more precise greedy probability criteria to
effectively select the working row from the coefficient matrix. Notably, we
prove that the linear convergence of the GRK method is deterministic and
demonstrate that using a tighter threshold parameter can lead to a faster
convergence rate. Our result revises existing convergence analyses, which are
solely based on the expected error by realizing that the iterates of the GRK
method are random variables. Consequently, we obtain an improved iteration
complexity for the GRK method. Moreover, the Polyak's heavy ball momentum
technique is incorporated to improve the performance of the GRK method. We
propose a refined convergence analysis, compared with the technique used in
Loizou and Richt\'{a}rik (Comput. Optim. Appl., 77(3):653--710, 2020), of
momentum variants of randomized iterative methods, which shows that the
proposed GRK method with momentum (mGRK) also enjoys a deterministic linear
convergence. Numerical experiments show that the mGRK method is more efficient
than the GRK method.
