Convergence of HS version of BCR algorithm to solve the generalized Sylvester matrix equation over generalized reflexive matrices

In recent years, several linear matrix equations such as Lyapunov and Sylvester matrix equations have received considerable attention due to their important applications in engineering and applied mathematics. In this work, an iterative technique based on the Hestenes-Stiefel (HS) version of biconjugate residual (BCR) algorithm is introduced to solve the generalized Sylvester matrix equation∑i=1f(AiXBi)+∑j=1g(CjYDj)=E,over the generalized reflexive matrices X and Y. We show that the proposed iterative technique converges to the generalized reflexive solutions within a finite number of iterations in the absence of round-off errors. Numerical examples confirm the efficiency and accuracy of the proposed iterative technique and also comparisons with other algorithms are made.