Numerical solution and perturbation theory for generalized Lyapunov equations

We discuss the numerical solution and perturbation theory for the generalized continuoustime Lyapunov equation E ⁄ XA+A ⁄ XE =iG with a singular matrix E. If this equation has a solution, it is not unique. We generalize a Bartels-Stewart method and a Hammarling method to compute a partial solution of the generalized Lyapunov equation with a special right-hand side. A spectral condition number is introduced and perturbation bounds for such an equation are presented. Numerical examples are given.

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