Computing symmetric solutions of general Sylvester matrix equations via Lanczos version of biconjugate residual algorithm

Many problems in control theory can be studied by obtaining the symmetric solution of linear matrix equations. In this investigation, we deal with the symmetric solutions X, Y and Z of the general Sylvester matrix equations A1XB1+C1YD1+E1ZF1=G1,A2XB2+C2YD2+E2ZF2=G2,⋮⋮⋮⋮AtXBt+CtYDt+EtZFt=Gt.The Lanczos version of biconjugate residual (BCR) algorithm is generalized to compute the symmetric solutions of the general Sylvester matrix equations. The convergence properties of this algorithm are discussed and it is proven that it smoothly converges to the symmetric solutions of the general Sylvester matrix equations in a finite number of iterations in the absence of round-off errors. Finally, the comparative numerical results are carried out to support that the current algorithm may be more efficient than other iterative algorithms.