A block GMRES method with deflated restarting for solving linear systems with multiple shifts and multiple right-hand sides

The restarted block generalized minimum residual method (BGMRES) with deflated restarting (BGMRES-DR) was proposed by Morgan to dump the negative effect of small eigenvalues from the convergence of the BGMRES method. More recently, Wu et al. introduced the shifted BGMRES method (BGMRES-Sh) for solving the sequence of linear systems with multiple shifts and multiple right-hand sides. In this paper, a new shifted block Krylov subspace algorithm that combines the characteristics of both the BGMRES-DR and the BGMRES-Sh methods is proposed. Moreover, our method is enhanced with a seed selection strategy to handle the case of almost linear dependence of the right-hand sides. Numerical experiments illustrate the potential of the proposed method to solve efficiently the sequence of linear systems with multiple shifts and multiple right-hand sides, with and without preconditioner, also against other state-of-the-art solvers.