Numerical solution of the Lyapunov equation by approximate power iteration Original Research Article

We present the approximate power iteration (API) algorithm for the computation of the dominant invariant subspace of the solution X of large-order Lyapunov equations AX + XAT + Q = 0 without first computing the matrix X itself. The API algorithm is an iterative procedure that uses Krylov subspace bases in computing estimates of matrix-vector products Xv in a power iteration sequence. Application of the API algorithm requires that A + AT < 0; numberical experiments indicate that, if the matrix X admits a good low-rank solution, then API provides an orthogonal basis of a subspace that closely approximates the dominant X-invariant subspace of corresponding dimension. Analytical convergence results are also presented.