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 Finite Iterative (R,S)-conjugate solutions of the generalized complex coupled Sylvester-transpose equations Yajun Xie,Yifen Ke Keywords:Generalized complex coupled Sylvester-transpose equations; (R,S)-conjugate; least Frobenius norm solution; optimal approximation solution; numerical test. Abstract: The iterative method of generalized complex coupled Sylvester-transpose equations $AXB+CY^TD=E,~MX^TN+GYH=F$ over (R,S)-conjugate matrix solution $(X,Y)$ is proposed. Usually, the type of matrix arises from some physical problems with some form of generalized reflexive symmetry. On the condition that the coupled matrix equations are consistent, we show the solution pair $(X^*, Y^*)$ can be obtained by generalization of CG iterative method within finite iterative steps in the absence of roundoff-error for any initial guess chosen by the (R,S)-conjugate matrix. Moreover, the optimal approximation (R,S)-conjugate matrix solutions can be derived by searching the least Frobenius norm solution of the novel generalized complex coupled Sylvester-transpose matrix equations. Finally, some numerical examples are given to illustrate the presented iterative algorithm is efficient.