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Published on 20 March 2025
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Liu,Y. (2025). Convergence Analysis and Improvement of Iterative Solutions for Linear Equation Systems. Theoretical and Natural Science,101,14-23.
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Convergence Analysis and Improvement of Iterative Solutions for Linear Equation Systems

Yang Liu *,1,
  • 1 Xiangtan University, Xiangtan, China

* Author to whom correspondence should be addressed.

https://doi.org/10.54254/2753-8818/2024.CH21606

Abstract

Many physics and mathematics problems generate linear equation systems, and solving linear equation systems has become an important proposition. Therefore, combined with the characteristics of modern computers, various methods for solutions need to be sought. This article introduces applications of linear equation systems in different fields, as well as representative figures and works with outstanding achievements. It focuses on providing formulas and corresponding Matlab codes for Gauss elimination, Jacobi iteration, and G-S iteration to solve equation systems, and rigorously proves the sufficient and necessary condition for convergence of iterative formula and also proves the convergence of different iteration methods under different types of coefficient matrices. Based on these solving methods, two examples are practiced in Matlab, the running time and iteration times of different methods are comprehensively compared. Thus, the superiority of the G-S iterative method is obtained. Finally, when there are zero elements in the diagonal elements of the coefficient matrix, the article proposes an improved method to solve this problem.

Keywords

system of linear equations, iterative methods, convergence

[1]. David C.​Lay Steven R.​Lay Judi J.​McDonald.​Linear Algebra and Its Applications.​

[2]. Wang G.​X.​Zhou Z.​M.​Zhu S.​M.​Ordinary Differential Equation.​

[3]. Huang Y.​Q.​Shu S.​Yang Y.​.​Numerical Method.​

[4]. Shams M, Kausar N, Agarwal P, et al.​Triangular intuitionistic fuzzy linear system of equations with applications:​ an analytical approach[J].​Applied Mathematics in Science and Engineering, 2024, 32(1):​

[5]. Darvishi T M , Khani F , Godarzi M A , et al.​Symmetric modified AOR method to solve systems of linear equations[J].​Journal of Applied Mathematics and Computing, 2011, 36(1-​2):​41-​59.​

[6]. Zhong-​Zhi B.​On convergence of the matrix splitting iteration paradigm for solving systems of linear equations[J].​Applied Mathematics Letters, 2024, 150108969-​.​

[7]. EisenstatC S, ElmanC H, SchultzH M.​Variational Iterative Methods for Nonsymmetric Systems of Linear Equations[J].​SIAM Journal on Numerical Analysis, 2006, 20(2):​345-​357.​

Cite this article

Liu,Y. (2025). Convergence Analysis and Improvement of Iterative Solutions for Linear Equation Systems. Theoretical and Natural Science,101,14-23.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

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About volume

Volume title: Proceedings of CONF-MPCS 2025 Symposium: Mastering Optimization: Strategies for Maximum Efficiency

ISBN:978-1-80590-017-7(Print) / 978-1-80590-018-4(Online)
Conference date: 21 March 2025
Editor:Anil Fernando, Marwan Omar
Series: Theoretical and Natural Science
Volume number: Vol.101
ISSN:2753-8818(Print) / 2753-8826(Online)

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