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

Iranian Journal of Numerical Analysis and Optimization

مقاله 4، دوره 15، Issue 3 - شماره پیاپی 34، آذر 2025، صفحه 930-951   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2025.91136.1561

نویسندگان

B. Harika¹؛ D. Rajaiah^{* 2}؛ A. Shivaji¹؛ L.P. Rajkumar¹

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

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

چکیده

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.

کلیدواژه‌ها

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

مراجع

[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.