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SUMMARY In this paper, we propose a block Arnoldi method for solving the continuous low-rank Sylvester matrix equation AX + XB = EF T. We consider the case where both A and B are large and sparse real matrices, and E and F are real matrices with small rank. We first apply an alternating directional implicit preconditioner to our equation, turning it into a Stein matrix equation. We then apply a block Krylov method to the Stein equation to extract low-rank approximate solutions. We give some theoretical results and report numerical experiments to show the efficiency of this method. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a method for solving large-scale Sylvester equations is presented. The method is based on the sign function iteration and is particularly effective for Sylvester equations with factorized right-hand side. In this case, the solution will be computed in factored form as it is for instance required in model reduction. The hierarchical matrix format and the corresponding formatted arithmetic are integrated in the iteration scheme to make the method feasible for large-scale computations. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We consider the Sylvester equation AX-XB+C=0 where the matrix C∈ℂn×m is of low rank and the spectra of A∈ℂn×n and B∈ℂm×m are separated by a line. We prove that the singular values of the solution X decay exponentially, that means for any ℇ∈(0,1) there exists a matrix &Xtilde; of rank k=O(log(1/ℇ)) such that ∥X-&Xtilde;∥2⩽ℇ∥X∥2. As a generalization we prove that if A,B,C are hierarchical matrices then the solution X can be approximated by the hierarchical matrix format described in Hackbusch (Computing 2000; 62: 89–108). The blockwise rank of the approximation is again proportional to log(1/ℇ). Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]