A Block Rational Arnoldi Method for Solving Large-Scale Sylvester Equations

This paper is concerned with the low-rank approximate solution of the large-scale Sylvester matrix equations. A simple block rational Arnoldi decomposition is derived, and the large-scale Sylvester matrix equations are projected into the block rational Krylov subspace. The projected problems are solved by using some direct methods based on matrix factorizations. Some estimates for the residual and error norms associated with the approximate solution are provided, and a reasonable and effective rule is proposed for selecting shift parameters. Experimental results are included to demonstrate the efficiency of the proposed method.