Matrix GPBiCG algorithms for solving the general coupled matrix equations

Linear matrix equations have important applications in control and system theory. In the study, we apply Kronecker product and vectorisation operator to extend the generalised product bi-conjugate gradient (GPBiCG) algorithms for solving the general coupled matrix equations Σ^l_j=1(A)_i,1,_jX₁B_i,_1,j+A_i,_2,jX₂B_i,2,j+...+A_i,l,jX_i,l,j) = D_i for i = 1,2,...,l (including the (coupled) Sylvester, the second-order Sylvester and coupled Markovian jump Lyapunov matrix equations). We propose four effective matrix algorithms for finding solutions of the matrix equations. Numerical examples and comparison with other well-known algorithms demonstrate the effectiveness of the proposed matrix algorithms.