The Relaxed Gradient Based Iterative Algorithm for the Symmetric (Skew Symmetric) Solution of the Sylvester EquationAX+XB=C

Abstract

In this paper, we present two different relaxed gradient based iterative (RGI) algorithms for solving the symmetric and skew symmetric solution of the Sylvester matrix equationAX+XB=C. By using these two iterative methods, it is proved that the iterative solution converges to its true symmetric (skew symmetric) solution under some appropriate assumptions when any initial symmetric (skew symmetric) matrixX0is taken. Finally, two numerical examples are given to illustrate that the introduced iterative algorithms are more efficient.