On multi-step randomized extended Kaczmarz method for solving large sparse inconsistent linear systems

In order to improve the convergence property and computational behavior of the randomized extended Kaczmarz method, we propose a multi-step randomized extended Kaczmarz method, in which we repeatedly update the iterate several times at each iteration step, obtaining a nonstationary inner-outer iteration scheme for solving large-scale, sparse, and inconsistent system of linear equations. For this multi-step randomized extended Kaczmarz method, we prove its convergence, derive an upper bound for its convergence rate, and demonstrate that this upper bound can be smaller than that of the randomized extended Kaczmarz method for several typical choices of the numbers of inner iteration steps. Numerical experiments also show that the multi-step randomized extended Kaczmarz method can perform better than the randomized extended Kaczmarz method if we choose the numbers of inner iteration steps appropriately.