Convergence properties of BCR method for generalized Sylvester matrix equation over generalized reflexive and anti-reflexive matrices

Solving the well-known Lyapunov and Sylvester matrix equations appears in a wide range of applications such as in control theory and signal processing. This article establishes the matrix form of the biconjugate residual (BCR) algorithm for computing the generalized reflexive solution X and the generalized anti-reflexive solution Y of the generalized Sylvester matrix equation It is proven that the proposed BCR algorithm converges within a finite number of iterations in the absence of round-off errors. At the end, various numerical implementations illustrating the effectiveness and accuracy of the proposed BCR algorithm are presented and discussed.