Algorithm 705; a FORTRAN-77 software package for solving the Sylvester matrix equation AXB ^T + CXD ^T = E

Abstract

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].

On the numerical solution of large-scale sparse discrete-time Riccati equations

“…This can be done using the Bartels-Stewart algorithm [5,6] or an extension to the case E = I [27,28,48]. The Bartels-Stewart algorithm is the standard direct method for the solution of Stein equations of small to moderate size.…”

Solving a Quadratic Matrix Equation by Newton's Method with Exact Line Searches

“…To solve (3.2) we can adapt methods for solving the generalized Sylvester equation described by Golub, Nash, and Van Loan [21] and Epton [13] (see also Chu [6] and Gardiner et al [17], [18] …”

Least-Squares Approximate Solution of Overdetermined Sylvester Equations

“…The numerical solution of generalized Lyapunov equations AXB + BXA + C = 0 may be achieved through the use of a QZ decomposition [27] of the matrix pencil (A, B) [7], [8]; low-rank approximate solution techniques may be applied to these problems in a fashion analogous to the standard case (1.1).…”