A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations

In this paper we present the cyclic low-rank Smith method, which is an iterative method for the computation of low-rank approximations to the solution of large, sparse, stable Lyapunov equations. It is based on a generalization of the classical Smith method and profits by the usual low-rank property of the right-hand side matrix. The requirements of the method are moderate with respect to both computational cost and memory. Furthermore, we propose a heuristic for determining a set of suboptimal alternating direction implicit (ADI) shift parameters. This heuristic, which is based on a pair of Arnoldi processes, does not require any a priori knowledge on the spectrum of the coefficient matrix of the Lyapunov equation. Numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the ADI parameters.