A New Theoretical Estimate for the Convergence Rate of the Maximal Weighted Residual Kaczmarz Algorithm

In this note we prove a new theoretical estimate for the convergence rate of the maximal weighted residual Kaczmarz algorithm for solving a consistent linear system. The estimate depends only on quantities that are easy to compute and not on the number of equations in the system. We compare the maximal weighted residual Kaczmarz algorithm and the greedy randomized Kaczmarz algorithm by two sets of examples. Numerical results show that the maximal weighted residual Kaczmarz algorithm requires almost the same number of iterations as that of the greedy randomized Kaczmarz algorithm for underdetermined linear systems and less iterations for overdetermined linear systems. Due to less computational cost in the row index selection strategy, the maximal weighted residual Kaczmarz algorithm is more efficient than the greedy randomized Kaczmarz algorithm.