Remarks on numerical solution of the Lyapunov matrix equation

Direct solution of the equation A¿L + LA = ¿C may break down when A is ill conditioned or of large order. The reduction in order using a skew-symmetric matrix can sometimes be helpful, but methods relying on transformation of A into the Schwarz form encounter difficulty owing to ill-conditioning of the companion form. Expressing L in terms of a convergent infinite matrix series would seem to be the most useful form for practical purposes.