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

TL;DR

This paper provides improved upper bounds on the convergence rates of the randomized extended Kaczmarz and Gauss-Seidel algorithms, which are versatile methods for solving various types of linear systems.

Contribution

It offers a novel interpretation of these algorithms as combinations of standard methods and derives refined convergence bounds.

Findings

Refined upper bounds for convergence rates.

Applicable to all linear system types.

Enhanced understanding of algorithm behavior.

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.