Convergence analysis of triangular and symmetric splitting method for fuzzy stochastic linear systems

Document Type : Research Article

Authors

• B. Harika ¹
• D. Rajaiah ²
• A. Shivaji ¹
• L.P. Rajkumar ¹

¹ Department of Mathematics, Kakatiya University, Telangana, India.

² Department of Mathematics, Kakatiya Institute of Technology and Science, Warangal, India.

10.22067/ijnao.2025.91136.1561

Abstract

In this article, the triangular and symmetric splitting iterative method is suggested for solving linear homogeneous systems of equations $πQ = 0$, where $Q$ is the stochastic rate matrix and π is the steady state vector. The homogeneous system is converted to the nonhomogeneous regularized fuzzy linear system $Ax = b$ with the small perturbation parameter $0 < r ≤ 1$. The regularized fuzzy linear system is converted into an embedded linear system. The iterative scheme is established; convergence criteria and its sensitivity analysis are analyzed using the numerical examples and conver-gence theorems. From the numerical results, it is evident to conclude that the proposed method is effective and efficient compared to the theoretical results.

Keywords

• Stochastic rate matrix
• Triangular and symmetric splitting Method
• Fuzzy linear system
• Error analysis

Main Subjects

• Numerical linear algebra and its applications

---

References

[1] Abbasbandy, S., Ezzati, R. and Jafarian, A. LU decomposition method for solving fuzzy system of linear equations, Appl. Math. Comput. 172(1) (2006), 633–43.
[2] Abbasbandy, S. and Jafarian, A. Steepest descent method for system of fuzzy linear equations, Appl. Math. Comput. 175(1) (2006), 823–33.
[3] Allahviranloo, T. Numerical methods for fuzzy system of linear equations, Appl. Math. Comput. 155(2)(2004), 493–502.
[4] Allahviranloo, T. The Adomian decomposition method for fuzzy system of linear equations, Appl. Math. Comput. 163(2)(2005), 553–563.
[5] Chakraverty, S. and Behera, D. Fuzzy system of linear equations with crisp coefficients, J. Intell. Fuzzy Syst. 25(1) (2013), 201–207.
[6] Dehghan, M. and Hashemi, B. Iterative solution of fuzzy linear systems, Appl. Math. Comput. 175(1) (2006), 645–74.

[7] Dehghan, M., Hashemi, B. and Ghatee, M. Solution of the Fully Fuzzy Linear Systems Using Iterative Techniques, Chaos Solitons Fractals. 34(2)(2007), 316–36.
[8] Edalatpanah, S.A. Modified iterative methods for solving fully fuzzy linear systems, Fuzzy Syst. 3 (2017), 55–73.
[9] Ezzati, R. Solving fuzzy linear systems, Soft. Comput. 15(1)(2011),193–197.
[10] Fariborzi Araghi, M.A. and Fallahzadeh, A. Inherited LU factorization for solving fuzzy system of linear equations, Soft. Comput. 17(1) (2014), 159–163.
[11] Friedman, M., Ming, M. and Kandel A. Fuzzy linear systems, Fuzzy Sets Syst. 96(2) (1998), 201–209.
[12] Huang, Z., Chen, Z., Zhang, S., Wang, S. and Wang, K. A new method based on Jacobi iteration for fuzzy linear systems, Thai J. Math. 21(1) (2023), 29–37.
[13] Koam, A.N.A., Akram, M., Muhammad, G. and Hussain, N. LU decomposition scheme for solving m-polar fuzzy system of linear equations, Math. Probl. Eng. 20(1)(2020), 1–19.
[14] Miao, S. X., Zheng, B. and Wang, K. Block SOR methods for fuzzy linear systems, Jour. Appl. Math. Comput., 26(1)(2008), 201–18. 
[15] Nasseri, S. H., Matinfar, M. and Sohrabi, M. QR-decomposition method for solving fuzzy system of linear equations, Int. J. Math. Comput. 4(1)(2009), 129–136.
[16] Rajaiah, D., Rajkumar, L.P. and Malla Reddy, P. Steady state probability vector of positive definite regularized linear systems of circulant stochastic matrices, Linear MultiLinear Algebra, 65(1) (2016), 140–155.
[17] Wang, K. and Wu, Y. Uzawa-SOR method for fuzzy linear system, Int. J. of Inf. and Com. Sci., 1(2)(2012), 30–3.

[18] Wang, K. and Zheng, B. Symmetric successive over relaxation methods for fuzzy linear systems, J. Appl. Math. Comput. 175(2) (2006), 891–901.
[19] Wang, K. and Zheng, B. Block iterative methods for fuzzy linear systems, J. Appl. Math. Comput. 25(1) (2006), 119–36.
[20] Wang. Y.R. and Chen, Y.L. A modified Jacobi iterative method for large-size linear systems, J. Shandong Univ. (Natural Science) 55(6) (2020), 122–6.
[21] Yin, J.F. and Wang, K. Splitting iterative methods for fuzzy system of linear equations, Comput. Math. Model. 20(2)(2009), 326–35.