A block GMRES method with deflated restarting for solving linear systems with multiple shifts and multiple right‐hand sides. Issue 5 (18th January 2018) - Sun, Dong‐Lin; Huang, Ting‐Zhu; Jing, Yan‐Fei; Carpentieri, Bruno
Summary: 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.