RK-RK and REK-RK methods are the two latest while very effective randomized iteration solvers for factorized linear system $UV \beta = y$ by interlacing randomized Kaczmarz (RK) and randomized extended Kaczmarz (REK) updates. This paper considers two latest randomized iterative methods for solving large-scale linear systems 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 factorized 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.
The work is supported by National Natural Science Foundation (11371243).
"On Relaxed Greedy Randomized Iterative Methods for the Solution of Factorized Linear Systems." Taiwanese J. Math. Advance Publication 1 - 27, 2022. https://doi.org/10.11650/tjm/220305