Numerical solutions for generalized trapezoidal fully fuzzy Sylvester matrix equation with sufficient conditions to have a positive solution
Ahmed Abdel Aziz Elsayed, N. Ahmad, Gh. Malkawi, B. Saassouh, O. Adeyeye · Journal of Mathematics and Computer Science · 2023
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.