Chapter 8 - Numerical Solutions and Conditioning of Lyapunov and Sylvester Equations

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.