Computation of approximate null vectors of Sylvester and Lyapunov operators

This paper describes an effective algorithm for computing approximate null vectors of certain matrix operators associated with Sylvester or Lyapunov equations. The singular value decomposition and rank-revealing QR methods are two widely used stable algorithms for numerical determination of the rank and nullity of a matrix A. These methods, however, are not readily applicable to Sylvester and Lyapunov operators since they require on the order of n⁶ arithmetic operations on order n² data. For these problems, a variant of inverse power iteration is employed to compute orthonormal bases for singular subspaces associated with the small singular values. The method is practical since it relies only on the ability to solve a Sylvester or Lyapunov equation. Certain practical aspects are considered, and a direct refinement technique is proposed to enhance the convergence of the algorithm