Journal
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Volume 21, Issue 4, Pages 1401-1418Publisher
SIAM PUBLICATIONS
DOI: 10.1137/S1064827598347666
Keywords
ADI iteration; Smith method; iterative methods; Lyapunov equation; matrix equations
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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.
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