A Hessenberg method for the numerical solutions to types of block Sylvester matrix equations

In this paper, we propose a new algorithm to solve the Sylvester matrix equation X A + B X = C . The technique consists of orthogonal reduction of the matrix A to a block upper Hessenberg form P T A P = H and then solving the reduced equation, Y H + B Y = C for Y through recurrence relation, where Y = X P , and C ′ = C P . We then recover the solution of the original problem via the relation X = Y P T . The numerical results show the accuracy and the efficiency of the proposed algorithm. In addition, how the technique described can be applied to other matrix equations was shown.