Tight upper bounds for the convergence of the randomized extended Kaczmarz and Gauss–Seidel algorithms

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 present tight upper bounds for the convergence of the randomized extended Kaczmarz and Gauss–Seidel algorithms. Numerical experiments are given to illustrate the theoretical results.