On greedy randomized augmented kaczmarz method for solving large sparse inconsistent linear systems

For solving large-scale inconsistent systems of linear equations by iteration methods, with the application of the greedy randomized Kaczmarz method to a consistent augmented linear system, we find an appropriate balance between the updates of the two iteration vectors in the randomized extended Kaczmarz method, and, based on this balance, we propose a greedy randomized augmented Kaczmarz method. We prove the convergence of the greedy randomized augmented Kaczmarz method and derive an upper bound for its expected convergence rate. Numerical results show that the greedy randomized augmented Kaczmarz method can be much more effective than the randomized extended Kaczmarz method as well as the partially randomized extended Kaczmarz method.