Numerical Algorithms

, Volume 20, Issue 1, pp 75–100

Solving stable generalized Lyapunov equations with the matrix sign function

  • Peter Benner
  • Enrique S. Quintana-Ortí

DOI: 10.1023/A:1019191431273

Cite this article as:
Benner, P. & Quintana-Ortí, E.S. Numerical Algorithms (1999) 20: 75. doi:10.1023/A:1019191431273


We investigate the numerical solution of the stable generalized Lyapunov equation via the sign function method. This approach has already been proposed to solve standard Lyapunov equations in several publications. The extension to the generalized case is straightforward. We consider some modifications and discuss how to solve generalized Lyapunov equations with semidefinite constant term for the Cholesky factor. The basic computational tools of the method are basic linear algebra operations that can be implemented efficiently on modern computer architectures and in particular on parallel computers. Hence, a considerable speed-up as compared to the Bartels–Stewart and Hammarling methods is to be expected. We compare the algorithms by performing a variety of numerical tests.

generalized Lyapunov equationsmathematical softwarematrix sign functionNewton iterationalgebraic Riccati equations65F1093B4093B51

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Peter Benner
    • 1
  • Enrique S. Quintana-Ortí
    • 2
  1. 1.Zentrum für Technomathematik, Fachbereich 3/Mathematik und InformatikUniversität BremenBremenGermany
  2. 2.Departamento de InformáticaUniversidad Jaime ICastellónSpain