Flexible and deflated variants of the block shifted GMRES method

The solution of linear systems with multiple shifts and multiple right-hand sides given simultaneously is required in many large-scale scientific and engineering applications. In this paper we introduce new flexible and deflated variants of the shifted block GMRES method for this problem class. The proposed methods solve the whole sequence of linear systems simultaneously, detecting effectively the linear systems convergence and allowing the use of variable preconditioning which may be particularly useful in some applications. Numerical experiments are illustrated to show the overall significant robustness of the iterative method for solving general sparse multi-shifted and multiple right-hand-side systems, and in realistic PageRank calculations. To the best of our knowledge, this is the first Krylov subspace method that combines deflation techniques and variable preconditioning for solving sequences of multi-shifted linear systems with multiple right-hand sides simultaneously.