Numerical solutions for generalized trapezoidal fully fuzzy Sylvester matrix equation with sufficient conditions to have a positive solution

Volume 31, Issue 2, pp 102--136 http://dx.doi.org/10.22436/jmcs.031.02.01

Publication Date: April 26, 2023 Submission Date: November 21, 2021 Revision Date: February 03, 2023 Accteptance Date: March 10, 2023

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Authors

A. A. A. Elsayed - Department of Mathematics, Institute of Applied Technology, Mohamed Bin Zayed City 33884, United Arab Emirates. N. Ahmad - School of Quantitative Sciences, Universiti Utara Malaysia, Sintok 06010, Kedah, Malaysia. Gh. Malkawi - Faculty of Engineering, Math and Natural Science Division, Higher Colleges of Technology (HCT), Al Ain Campus, Abu Dhabi 17155, United Arab Emirates. B. Saassouh - Academic Support Department, Abu Dhabi Polytechnic College, Abu Dhabi 111499, United Arab Emirates. O. Adeyeye - School of Quantitative Sciences, Universiti Utara Malaysia, Sintok 06010, Kehah, Malaysia.

Abstract

This paper proposes three methods for solving a generalized trapezoidal fully fuzzy Sylvester matrix equation (GTrFFSME) and its special cases. The GTrFFSME is converted to an equivalent system of generalized crisp Sylvester matrix equations based on a new constructed fuzzy multiplication operation between three trapezoidal fuzzy numbers. An analytical solution to the GTrFFSME is obtained by developing a fuzzy matrix vectorization method, and the numerical solution is obtained by developing fuzzy gradient and fuzzy least-squares iterative methods. The necessary and sufficient conditions for the GTrFFSME to have a unique positive fuzzy solution are proved in addition to the convergence for the fuzzy gradient and fuzzy least-square methods. The constructed methods can solve other fuzzy equations such as Sylvester, Lyapunov and Stein matrix equations up to size \(\mathrm{100\times 100}\). We illustrate the proposed methods by solving numerical examples with different size systems.

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Keywords

• Generalized fully fuzzy Sylvester matrix equations
• gradient iterative
• numerical fuzzy solution
• least-squares iterative
• trapezoidal fuzzy multiplication

MSC

•  03B52
•  94D05
•  08A72
•  15B15

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