Use of near-breakdowns in the block Arnoldi method for solving large Sylvester equations

In this paper we construct an approximate solution to large Sylvester equations of the form AX+XB=CDT. The construction uses a new variant of the block Arnoldi algorithm which exploits the near-breakdowns, that is, the near singularities in the generated basis. As a consequence, the algorithm eliminates the directions which do not contribute to the approximate solution by keeping in the generated basis only the “active” vectors detected by a criterion based on the residual associated with the approximate solution. The effectiveness of the proposed algorithm is demonstrated on several examples, including the case where the matrix B has a small or a large size.