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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 equation AX+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) matrix X0 is taken. Finally, two numerical examples are given to illustrate that the introduced iterative algorithms are more efficient

mathematical problems in engineering

The Relaxed Gradient Based Iterative Algorithm for the Symmetric (Skew Symmetric) Solution of the Sylvester Equation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>A</mml:mi><mml:mi>X</mml:mi><mml:mo>+</mml:mo><mml:mi>X</mml:mi><mml:mi>B</mml:mi><mml:mo>=</mml:mo><mml:mi>C</mml:mi></mml:math>

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