Global Krylov subspace methods are the most e cient and robust methods to solve Generalized coupled Sylvester matrix equation. In this paper, we propose nested splitting conjugate gradient process for solving this equation. This method has inner and outer iterations, which employs the Generalized conjugate gradient method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergence and symmetric positive defi nite splitting of the coe cient matrices. Convergence properties of this method are investigated. Finally, the e ectiveness of the nested splitting conjugate gradient method is explained by some numerical examples.

The main aim of this paper intends to discuss the solution of fuzzy Sylvester matrix equation AX+XB=C where A= (aij)ÎRn´n, B= (bij) ÎRm´m are crisp M- matrices, C= (cij) ÎRn´m is fuzzy matrix and XÎRn´m is unknown, by applying a special algorithm based on a class of ABS algorithms called Huang algorithm. At first, we transform this system to an (mn)´(mn) fuzzy system of linear equations. Then, we convert this system to three (mn)´(mn) crisp linear systems of equations, and we solve them simultaneously by ABS algorithm.

The global FOM and GMRES algorithms are among the effective methods to solve Sylvester matrix equations. In this paper, we study these algorithms in the case that the coefficient matrices are real symmetric (real symmetric positive definite) and extract two CG-type algorithms for solving Generalized Sylvester matrix equations. The proposed methods are iterative projection methods onto matrix Krylov subspaces. Numerical examples are presented.

Solutions of two problems related to the matrix equation of Sylvester type is given. In the first problem, the procedures for linear matrix inequalities are used to con-struct the solution of this equation. In the second problem, when a matrix is given which is not a solution of this equation, it is required to find such solution of the original equation, which most accurately approximates the given matrix. For this, an algorithm for constructing a general solution of the Sylvester matrix equation is used. The effectiveness of the proposed approaches is illustrated on the examples.

In this study based on the accelerated over relaxation (AOR) method we make an iterative scheme for solving Generalized Lyapunov matrix equation AX + XB + Xm j=1 NjXMj = C, over complex or real matrices. Then we analyze the convergence of the new iterative method in detail. There have been discussions for the calculation of optimal parameters. Finally a numerical example is given to demonstrate the capability of the new method.

The global Generalized minimum residual (Gl-GMRES) method is examined for solving the Generalized Sylvester matrix equation Sqi=1 AiXBi=C. Some new theoretical results are elaborated for the proposed method by employing the Schur complement. These results can be exploited to establish new convergence properties of the Gl-GMRES method for solving general (coupled) linear matrix equations. In addition, the Gl-GMRES method for solving the Generalized Sylvester-transpose matrix equation is briey studied. Finally, some numerical experiments are presented to illustrate the efficiently of the Gl-GMRES method for solving the general linear matrix equations.