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 XA+BX=CXA+BX=C. The technique consists of orthogonal reduction of the matrix A to a block upper Hessenberg form PTAP=HPTAP=H and then solving the reduced equation, YH+BY=CYH+BY=C for YY through recurrence relation, where Y=XPY=XP, and C′=CPC′=CP. We then recover the solution of the original problem via the relation X=YPTX=YPT. 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.