A Flexible Global GCRO-DR Method for Shifted Linear Systems and General Coupled Matrix Equations

In this paper, we mainly focus on the development and study of a new global GCRO-DR method that allows both the flexible preconditioning and the subspace recycling for sequences of shifted linear systems. The novel method presented here has two main advantages: firstly, it does not require the right-hand sides to be related, and, secondly, it can also be compatible with the general preconditioning. Meanwhile, we apply the new algorithm to solve the general coupled matrix equations. Moreover, by performing an error analysis, we deduce that a much looser tolerance can be applied to save computation by limiting the flexible preconditioned work without sacrificing the closeness of the computed and the true residuals. Finally, numerical experiments demonstrate that the proposed method illustrated can be more competitive than some other global GMRES-type methods.