Numerical solution of very large, sparse Lyapunov equations through approximate power iteration

The authors present an algorithm for the solution of large order (1000⩽n) Lyapunov equations AX+XA´+Q =0. The algorithm, approximate power iteration, attempts to compute directly an orthogonal basis of the dominant eigenspace of the solution X. It is shown that if the dominant eigenvalues λ₁ and λ₂ of X are sufficiently well separated (λ₁≫λ₂), then a special case of the approximate power iteration algorithm has at least one fixed point υ that is near to the dominant eigenvector u₁ of X, and that there is a small attractive region in IR ⁿ containing both u₁ and υ