Two New Variants of the Simpler Block GMRES Method with Vector Deflation and Eigenvalue Deflation for Multiple Linear Systems

In this paper, two effective methods based on the simpler block GMRES method are established in order to solve the linear systems of equations with multiple right-hand sides. The first method is derived from the simpler block GMRES method with vector deflation restarting (SBGMRES-DR). The second method is constructed from a combination of SBGMRES-DR with the eigenvalue deflation technique, which is called the deflated simpler block GMRES method with vector deflation restarting (D-SBGMRES-DR). To be more specific, SBGMRES-DR is capable of removing linearly or almost linearly dependent vectors created by the block Arnoldi process. On the other hand, D-SBGMRES-DR not only deletes linearly or almost linearly dependent vectors but also retains harmonic Ritz vectors associated with the smallest harmonic Ritz values in magnitude, and adds them to the new search subspace at the time of restart. Finally, a wide range of practical experiments are carried out to assess the efficiency of the proposed methods. The numerical results indicate that the D-SBGMRES-DR method outperforms the compared methods with respect to the number of matrix–vector products and the computational time.