Numerical solution of the Lyapunov equation by approximate power iteration

AbstractWe 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