Convergence analysis of the extended Krylov subspace method for the Lyapunov equation (Q634615)

The paper is devoted to the numerical solution of the large scale Lyapunov matrix equation \[ A X+X A^{*}+B B^{*}=0 \] for a positive definite matrix \(A\) and a tall and slim matrix \(B\). The authors analyse a recently developed Galerkin-type method, which projects the original problem onto an enriched space that extends the Krylov subspace recurrence to inverse powers of \(A\). This extended Krylov subspace is given by \[ \mathbf{EK}_m (A,b)=K_m(A,b)+K_m(A^{-1},A^{-1}b) \] (\(b\) is a column vector). The spaces \(\mathbf{EK}_m (A, b)\) in the sequence are nested and can be iteratively generated by means of a recurrence similar to that used for computing the standard Krylov subspaces \(K_m(A,b)\). The paper provides general estimates for the convergence rate of the extended Krylov subspace method for real nonsymmetric \(A\) with field of values in the right half-plane. Efficiency of the procedure is shown by numerical results.