Numerical solution and perturbation theory for generalized Lyapunov equations (Q1611862)

The numerical solution and a perturbation theory for the generalized continuous-time Lyapunov equation \(E^*XA+ A^*XE= -G\) with a singular matrix \(E\) are presented. This equation may not have solutions and even if a solution exists, it is, in general, not unique. Generalizations of the Bartels-Stewart and Hammarling methods are proposed to compute a partial solution of the generalized Lyapunov equation with a special right-hand side. The numerical properties of these methods are discussed. A spectral condition number is introduced and perturbation bounds for the generalized Lyapunov equation are derived. Numerical examples are given.