A software package has been developed to solve efficiently the Sylvester-type matrix equation AXB T + CXD T = E . A transformation method is used which employs the QZ algorithm to structure the equation in such a way that it can be solved columnwise by a back substitution technique. The algorithm is an extension of the Bartels-Stewart method and the Hessenberg-Schur method. The numerical performance of the algorithms and software is demonstrated by application to near-singular systems.

This paper documents a software package for solving the Sylvester matrix equation (1) AXB T + CXD T = e All quantities are real matrices; A and C are m x n ; B and D are m x n ; and X and E are m x n . The unknown is X . Two symmetric forms of Eq. (1) are treated separately for efficiency. They are the continuous-time symmetric Sylvester equation (2) AXE T + EXA T + C = 0 and the discrete time equation (3) AXA T + C = 0, for which A , E , and C is symmetric. The software also provides a means for estimating the condition number of these three equations. The algorithms employed are more fully described in an accompanying paper [3].

Iterative refinement reduces the roundoff errors in the computed solution to a system of linear equations. Only one step requires higher precision arithmetic. If sufficiently high precision is used, the final result is shown to be very accurate.