A hybrid block GMRES method for nonsymmetric systems with multiple right-hand sides

The block GMRES (BGMRES) method is a natural generalization of the GMRES algorithm for solving large non-symmetric systems with multiple right-hand sides. Unfortunately, its cost increases significantly per iteration, frequently rendering the method impractical. In this paper we propose a hybrid block GMRES method which offers significant performance improvements over BGMRES. This method uses the matrix polynomial obtained in the course of a BGMRES step and combines the advantages of the block approach with those of successful hybrid methods. We discuss the properties and several implementation variants of the method and report results from numerical experiments. We also describe how to use these techniques in order to solve multiply shifted systems with multiple right-hand sides.