Refined upper bounds for the convergence of the randomized extended Kaczmarz and Gauss-Seidel algorithms

Abstract: The randomized extended Kaczmarz and Gauss-Seidel algorithms have attracted much attention because of their ability to treat all types of linear systems (consistent or inconsistent, full rank or rank-deficient). In this paper, we interpret the randomized extended Kaczmarz and Gauss-Seidel algorithms as specific combinations of the randomized Kaczmarz and Gauss-Seidel algorithms and present refined upper bounds for their convergence.