Comparative analysis of Jacobi and Gauss-Seidel iterative methods

TL;DR

This paper compares the convergence properties of Jacobi and Gauss-Seidel iterative methods for solving linear systems with real and complex matrices, providing new algorithms and statistical analysis for various sizes.

Contribution

It introduces a convergence determination algorithm using the Hurwitz criterion and offers a Python implementation for three unknowns.

Findings

Ranges of convergence for both methods in 2D and 3D systems

Interrelationships of convergence ranges

Statistical comparison for systems with 2 to 5 unknowns

Abstract

The paper presents a comparative analysis of iterative numerical methods of Jacobi and Gauss-Seidel for solving systems of linear algebraic equations (SLAEs) with complex and real matrices. The ranges of convergence for both methods for SLAEs in two and three unknowns, as well as the interrelationships of these ranges are obtained. An algorithm for determining the convergence of methods for SLAEs using the complex analog of the Hurwitz criterion is constructed, the realization of this algorithm in Python in the case of SLAEs in three unknowns is given. A statistical comparison of the convergence of both methods for SLAEs with a real matrices and the number of unknowns from two to five is carried out.