Matrix form of the Bi-CGSTAB method for solving the coupled sylvester matrix equations

The bi-conjugate gradient stabilised (Bi-CGSTAB) method is one of the efficient computational tools to solve the non-Hermitian linear systems Ax = b. By employing Kronecker product and vectorisation operator, this study investigates the matrix form of the Bi-CGSTAB method for solving the coupled Sylvester matrix equations Σ_i=1^k (A_iXB_i + C_iYD_i) = M, Σ_i=1^k(E_iXF_i + G_iYH_i) = N [including (second-order) Sylvester and Lyapunov matrix equations as special cases] encountered in many systems and control applications. Several numerical examples are given to compare the efficiency and performance of the investigated method with some existing algorithms.