Convergence rates of the Kaczmarz-Tanabe method for linear systems

TL;DR

This paper analyzes the convergence rates of the Kaczmarz-Tanabe method for solving linear systems, highlighting key factors affecting its speed and stability, supported by theoretical analysis and numerical tests.

Contribution

It provides a detailed convergence rate analysis of the Kaczmarz-Tanabe method using singular value decomposition, identifying key factors influencing its performance.

Findings

Convergence rate depends on the second maximum singular value of Q.

The minimum non-zero singular value of A influences convergence speed.

Numerical tests confirm theoretical convergence results.

Abstract

In this paper, we investigate the Kaczmarz-Tanabe method for exact and inexact linear systems. The Kaczmarz-Tanabe method is derived from the Kaczmarz method, but is more stable than that. We analyze the convergence and the convergence rate of the Kaczmarz-Tanabe method based on the singular value decomposition theory, and discover two important factors, i.e., the second maximum singular value of $Q$ and the minimum non-zero singular value of $A$, that influence the convergence speed and the amplitude of fluctuation of the Kaczmarz-Tanabe method (even for the Kaczmarz method). Numerical tests verify the theoretical results of the Kaczmarz-Tanabe method.