Abstract
Two efficient methods for solving generalized Lyapunov equations and their implementations in FORTRAN 77 are presented. The first one is a generalization of the Bartels–Stewart method and the second is an extension of Hammarling's method to generalized Lyapunov equations. Our LAPACK based subroutines are implemented in a quite flexible way. They can handle the transposed equations and provide scaling to avoid overflow in the solution. Moreover, the Bartels–Stewart subroutine offers the optional estimation of the separation and the reciprocal condition number. A brief description of both algorithms is given. The performance of the software is demonstrated by numerical experiments.
Similar content being viewed by others
References
E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov and D. Sorenson, LAPACK Users' Guide (SIAM, Philadelphia, PA, 1992).
R.H. Bartels and G.W. Stewart, Solution of the equation AX +XB = C, Comm. ACM 15 (1972) 820–826.
R. Byers, A LINPACK-style condition estimator for the equation AX ? XB T = C, IEEE Trans. Automat. Control 29 (1984) 926–928.
K.-W.E. Chu, The solution of the matrix equation AXB?CXD = Y and (YA?DZ, YC?BZ) = (E,F), Linear Algebra Appl. 93 (1987) 93–105.
J.J. Dongarra, J.R. Bunch, C.B. Moler and G.W. Stewart, LINPACK Users' Guide (SIAM, Philadelphia, PA, 1979).
B.S. Garbow, J.M. Boyle, J.J. Dongarra and C.B. Moler, Matrix Eigensystem Routines – EISPACK Guide Extension, Lecture Notes in Computer Science 51 (Springer, New York, 1977).
J.D. Gardiner, A.J. Laub, J.J. Amato and C.B. Moler, Solution of the Sylvester matrix equation AXB T + CXD T = E, ACM Trans. Math. Software 18 (1992) 223–231.
J.D. Gardiner, M.R. Wette, A.J. Laub, J.J. Amato and C.B. Moler, A FORTRAN-77 software package for solving the Sylvester matrix equation AXB T + CXD T = E, ACM Trans. Math. Software 18 (1992) 232–238.
G.H. Golub and C.F. Van Loan, Matrix Computations (Johns Hopkins University Press, Baltimore, MD, 3rd ed., 1996).
W.W. Hager, Condition estimates, SIAM J. Sci. Statist. Comput 5 (1984) 311–316.
S.J. Hammarling, Numerical solution of the stable, non-negative definite Lyapunov equation, IMA J. Numer. Anal. 2 (1982) 303–323.
S.J. Hammarling, Numerical solution of the discrete-time, convergent, non-negative definite Lyapunov equation, Systems Control Lett. 17 (1991) 137–139.
N.J. Higham, Perturbation theory and backward error for AX ?XB = C, BIT 33 (1993) 124–136.
N.J. Higham, FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation, ACM Trans. Math. Software 14 (1988) 381–396.
B. Kågström and L. Westin, Generalized Schur methods with condition estimators for solving the generalized Sylvester equation, IEEE Trans. Automat. Control 34 (1989) 745–751.
B. Kågström and P. Poromaa, LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs, ACM Trans. Math. Software 22 (1996) 78–103.
P. Lancaster and M. Tismenetsky, The Theory of Matrices (Academic Press, Orlando, 2nd ed., 1985).
J. Snyders and M. Zakai, On non-negative solutions of the equation AD + DA T = ?C, SIAM J. Appl. Math. 18 (1970) 704–714.
Working Group on Software (WGS), Implementation and documentation standards, WGS-Report 90-1, Eindhoven/Oxford (1990).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Penzl, T. Numerical solution of generalized Lyapunov equations. Advances in Computational Mathematics 8, 33–48 (1998). https://doi.org/10.1023/A:1018979826766
Issue Date:
DOI: https://doi.org/10.1023/A:1018979826766