Abstract
This chapter develops the basic theories on the existence and uniqueness of solutions of the Sylvester and Lyapunov equations, discusses perturbation theories, and then describes computational methods. The continuous-time Lyapunov equation and the continuous-time Sylvester equation are referred to as the Lyapunov and Sylvester equations, respectively. The chapter also discusses the Schur methods for the Lyapunov equations, the Hessenberg-Schur Method for the Sylvester equations, the modified Schur methods for the Cholesky factors of the Lyapunov equations, and a Hessenberg method, based on Hessenberg decomposition only, for the Sylvester equation. These methods have excellent numerical properties and are recommended for use in practice. Hessenberg method is more efficient than the Hessenberg-Schur method, but numerical stability of the Hessenberg method method has not been investigated yet. At present, the method is mostly of theoretical interest only. The chapter further discusses why solving the Lyapunov and Sylvester equations via the Jordan canonical form or a companion form of the matrix “A” cannot be recommended for use in practice.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
