The nested block Krylov method based on GCR for solving Sylvester matrix equation

Document Type: Research Paper


1 Department of Applied Mathematics, Faculty of Mathematics & Computer Sciences, Shahid Bahonar University of Kerman

2 Department of Applied Mathematics, Shahid Bahonar University of Kerman, Iran.

3 Graduate University of Advanced Technology, Kerman, Iran.



In this paper, the block generalized conjugate residual method for solving the Sylvester equation is presented. This method involves two outer and inner iterations. A block generalized minimal residual method is used in the inner iteration to obtain a new search vector by solving a linear system of equation with multiple right-hand sides. In addition, a generalized conjugate residual approach is applied in the outer iteration to compute the optimal approximation on a given set of search vectors. In order to improve the convergence of the inner iteration, the block generalized minimal residual method is
combined with preconditioner. Finally, the efficiency of the proposed method is showed with some examples.