The RayleighRitz method, refinement and Arnoldi process for periodic matrix pairs

Eric King-Wah Chu Hung-Yuan Fan Zhongxiao Jia Tiexiang Li Wen-Wei Lin

Numerical Linear Algebra mathscidoc:1912.43749

Journal of computational and applied mathematics, 235, (8), 2626-2639, 2011.2
We extend the RayleighRitz method to the eigen-problem of periodic matrix pairs. Assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero, we show that there exist periodic Ritz values that converge to the desired periodic eigenvalues unconditionally, yet the periodic Ritz vectors may fail to converge. To overcome this potential problem, we minimize residuals formed with periodic Ritz values to produce the refined periodic Ritz vectors, which converge under the same assumption. These results generalize the corresponding well-known ones for RayleighRitz approximations and their refinement for non-periodic eigen-problems. In addition, we consider a periodic Arnoldi process which is particularly efficient when coupled with the RayleighRitz method with refinement. The numerical results illustrate that the refinement procedure produces excellent
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@inproceedings{eric2011the,
  title={The RayleighRitz method, refinement and Arnoldi process for periodic matrix pairs},
  author={Eric King-Wah Chu, Hung-Yuan Fan, Zhongxiao Jia, Tiexiang Li, and Wen-Wei Lin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205512397807313},
  booktitle={Journal of computational and applied mathematics},
  volume={235},
  number={8},
  pages={2626-2639},
  year={2011},
}
Eric King-Wah Chu, Hung-Yuan Fan, Zhongxiao Jia, Tiexiang Li, and Wen-Wei Lin. The RayleighRitz method, refinement and Arnoldi process for periodic matrix pairs. 2011. Vol. 235. In Journal of computational and applied mathematics. pp.2626-2639. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205512397807313.
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