Abstract
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.
Highlights
The linear systems (8) can be seen as generalized version of (2). erefore, based on the special structure of (1), we propose a new generalized global version of the generalized conjugate residual method with inner orthogonalization (GCRO)-DR method that combines with any type of the preconditioning and allows one to retain important spectral information generated during the solution of the i-th linear system and to exploit such information to accelerate convergence of the iterations when solving the subsequent i + 1-th system
We propose a global version of the FGMRES-DR method and analyze the relationship between two flexible global methods
A new flexible global GCRO with deflated restarting (GCRO-DR) method is proposed for addressing sequences of shifted linear systems and general coupled matrix equations. e proposed method does not require the right-hand sides to be related and can be compatible with general preconditioning
Summary
We focus on the solution of sequences of shifted linear systems of the form. Erefore, based on the special structure of (1), we propose a new generalized global version of the GCRO-DR method that combines with any type of the preconditioning and allows one to retain important spectral information generated during the solution of the i-th linear system and to exploit such information to accelerate convergence of the iterations when solving the subsequent i + 1-th system. We utilize the operator A to present our flexible global GCRO-DR algorithms for solving (1).
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