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Comparative analysis of Jacobi and Gauss-Seidel iterative methods

Pavel Khrapov, Nikita Volkov

TL;DR

This work provides a comprehensive comparative analysis of the Jacobi and Gauss-Seidel iterative methods for solving SLAEs with real and complex matrices, detailing convergence ranges for 2–3 unknowns and introducing a Python-based convergence check based on the complex Hurwitz criterion (Postnikov). For small systems, explicit boundary conditions and convergence regions are derived, while for larger or complex systems a general stability-testing framework is outlined. A statistical study on real random SLAEs demonstrates that Gauss-Seidel generally converges more often than Jacobi for 3–5 unknowns, though there are cases where Jacobi converges while Gauss-Seidel does not. The results guide practical method selection and provide robust convergence testing tools, particularly when diagonal dominance does not hold.

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.

Comparative analysis of Jacobi and Gauss-Seidel iterative methods

TL;DR

This work provides a comprehensive comparative analysis of the Jacobi and Gauss-Seidel iterative methods for solving SLAEs with real and complex matrices, detailing convergence ranges for 2–3 unknowns and introducing a Python-based convergence check based on the complex Hurwitz criterion (Postnikov). For small systems, explicit boundary conditions and convergence regions are derived, while for larger or complex systems a general stability-testing framework is outlined. A statistical study on real random SLAEs demonstrates that Gauss-Seidel generally converges more often than Jacobi for 3–5 unknowns, though there are cases where Jacobi converges while Gauss-Seidel does not. The results guide practical method selection and provide robust convergence testing tools, particularly when diagonal dominance does not hold.

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.
Paper Structure (11 sections, 125 equations, 10 figures, 1 table)

This paper contains 11 sections, 125 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Convergence conditions of the Jacobi and Gauss-Seidel methods
  3. System of linear algebraic equations in two unknowns
  4. Jacobi method
  5. Gauss-Seidel method
  6. System of linear algebraic equations with three unknowns
  7. Jacobi method
  8. Gauss-Seidel method
  9. The general case of systems of linear algebraic equations with complex matrices
  10. Statistical comparison of convergence of Jacobi and Gauss-Seidel methods
  11. Conclusion

Figures (10)

  • Figure 1: Geometric interpretation of the first equation of the system (\ref{['13']})
  • Figure 2: Dependence of the absolute value $|p|$ at the boundary (\ref{['17']}) at $\varphi_{1} = 0$ ($\operatorname{Re} (q) \leq 0$) and $\varphi_{1} = \pi$ ($\operatorname{Re} (q) \geq 0$)
  • Figure 3: Location of roots of the equation (\ref{['11']}) on the complex plane before rotation
  • Figure 4: Location of roots of the equation (\ref{['11']}) on the complex plane after rotation
  • Figure 5: Dependence of the absolute value $|p|$ on $q$
  • ...and 5 more figures