The authors propose a new alternating direction implicit (ADI)-based block Arnoldi preconditioned method for solving large Sylvester matrix equations of the form \[ AX +XB =EF^T, \] where the unknown matrix \(X \in \mathbb {R}^{n \times s}\), the coefficient matrices \(A \in \mathbb {R}^{n \times n}\), \(B \in \mathbb {R}^{s \times s}\) and \(E \in \mathbb {R}^{n \times r}\), \(F \in \mathbb {R}^{s \times r}\) are full rank with \(r\ll n,s\).
The new approach is used for accelerating the convergence of the block Krylov methods applied to the low-rank continuous Sylvester equations. More precisely, the proposed method is a combination of the block Arnoldi algorithm applied to Stein equations with the low-rank alternating direction implicit (LR-ADI) method. By incorporating some parameters provided by the ADI iterations, the initial Sylvester equation is transformed into a nonsymmetric Stein equation that is solved by the block Arnoldi method.
The method works generally well if the desired shifts are well chosen. The authors provide some theoretical results and reported numerical examples by comparing with some existing methods.