On the exponential convergence of the Kaczmarz algorithm

TL;DR

This paper establishes a new exponential convergence rate for the Kaczmarz algorithm by interpreting its solution path as a dynamical system and applying stability analysis.

Contribution

It introduces a novel dynamical system perspective to analyze the Kaczmarz algorithm, leading to improved convergence bounds.

Findings

Proves exponential convergence of the Kaczmarz algorithm.

Derives a new bound on the convergence rate.

Compares the new bound with existing bounds.

Abstract

The Kaczmarz algorithm (KA) is a popular method for solving a system of linear equations. In this note we derive a new exponential convergence result for the KA. The key allowing us to establish the new result is to rewrite the KA in such a way that its solution path can be interpreted as the output from a particular dynamical system. The asymptotic stability results of the corresponding dynamical system can then be leveraged to prove exponential convergence of the KA. The new bound is also compared to existing bounds.