On Relaxed Greedy Randomized Iterative Methods for the Solution of Factorized Linear Systems

RK-RK and REK-RK methods are the two latest while very effective ran-domized iteration solvers for factorized linear system UV beta = y by interlacing random-ized Kaczmarz (RK) and randomized extended Kaczmarz (REK) updates. This paper considers two latest randomized iterative methods for solving large-scale linear sys-tems and linear least-squares problems-greedy randomized Kaczmarz (GRK) method and greedy randomized Gauss-Seidel (GRGS) method. By introducing a relaxation parameter omega into the iterates of GRK and GRGS, we construct relaxed GRK and GRGS methods, respectively. In addition, by interlacing their updates, we propose relaxed GRK-GRK and GRGS-GRK methods to solve consistent and inconsistent fac-torized linear systems, respectively. We prove the exponential convergence of these two interlaced methods and show that relaxed GRK-GRK and GRGS-GRK can be more efficient than RK-RK and REK-RK, respectively, if the relaxation parameters are chosen appropriately.