Generalization of the interval TOR method for solving interval linear systems

Lingjian Pu; Yan Zhu; Shiliang Wu; Lingjian Pu; Yan Zhu; Shiliang Wu

doi:10.3934/math.2025824

AIMS Mathematics

2025, Volume 10, Issue 8: 18454-18474. doi: 10.3934/math.2025824

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Research article

Generalization of the interval TOR method for solving interval linear systems

1.
School of Mathematics, Kunming University, Kunming 650214, Yunnan, China
2.
School of Mathematics, Yunnan Normal University, Kunming 650500, Yunnan, China
3.
Yunnan Key Laboratory of Modern Analytical Mathematics and Applications, Yunnan Normal University, Kunming 650500, Yunnan, China

Received: 27 May 2025 Revised: 16 July 2025 Accepted: 28 July 2025 Published: 14 August 2025
MSC : 15A30, 65G30, 65G40, 65H10

Iterative methods for solving interval linear systems were presented in this paper. By generalizing interval diagonal matrices to interval band matrices, a generalization of the interval two-parameter overrelaxation method (GITOR) was introduced, and the convergence analysis of the proposed method was discussed. Specifically, if the coefficient matrices of the system are strictly diagonally dominant (SDD) matrices, interval M-matrices, or interval H-matrices, the proposed method converges under any initial approximation. Furthermore, an upper bound on the spectral radius of the iterative matrices was provided for interval strictly diagonally dominant matrices. Finally, numerical examples for each type of mentioned interval matrix were studied, demonstrating the efficiency of the proposed method.
- interval linear systems,
- iterative method,
- interval band matrices,
- generalized interval TOR method,
- convergence analysis
Citation: Lingjian Pu, Yan Zhu, Shiliang Wu. Generalization of the interval TOR method for solving interval linear systems[J]. AIMS Mathematics, 2025, 10(8): 18454-18474. doi: 10.3934/math.2025824

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Abstract

Iterative methods for solving interval linear systems were presented in this paper. By generalizing interval diagonal matrices to interval band matrices, a generalization of the interval two-parameter overrelaxation method (GITOR) was introduced, and the convergence analysis of the proposed method was discussed. Specifically, if the coefficient matrices of the system are strictly diagonally dominant (SDD) matrices, interval M-matrices, or interval H-matrices, the proposed method converges under any initial approximation. Furthermore, an upper bound on the spectral radius of the iterative matrices was provided for interval strictly diagonally dominant matrices. Finally, numerical examples for each type of mentioned interval matrix were studied, demonstrating the efficiency of the proposed method.

References

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