TY - JOUR

T1 - The Rayleigh-Ritz method, refinement and Arnoldi process for periodic matrix pairs

AU - Chu, Eric King Wah

AU - Fan, Hung Yuan

AU - Jia, Zhongxiao

AU - Li, Tiexiang

AU - Lin, Wen Wei

N1 - Funding Information:
Third author’s work was supported by the National Basic Research Program of China 2011CB302400 and the National Science Foundation of China (Nos. 11071140 and 10771116 ). First, second and fourth author’s work were partially supported by the National Centre of Theoretical Sciences , of ROC in Taiwan.

PY - 2011/2/15

Y1 - 2011/2/15

N2 - We extend the Rayleigh-Ritz 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 Rayleigh-Ritz 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 Rayleigh-Ritz method with refinement. The numerical results illustrate that the refinement procedure produces excellent approximations to the original periodic eigenvectors.

AB - We extend the Rayleigh-Ritz 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 Rayleigh-Ritz 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 Rayleigh-Ritz method with refinement. The numerical results illustrate that the refinement procedure produces excellent approximations to the original periodic eigenvectors.

KW - Arnoldi process

KW - Periodic eigenvalues

KW - Periodic matrix pairs

KW - Rayleigh-Ritz method

KW - Refinement

KW - Ritz values

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U2 - 10.1016/j.cam.2010.11.014

DO - 10.1016/j.cam.2010.11.014

M3 - Article

AN - SCOPUS:79251596552

VL - 235

SP - 2626

EP - 2639

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 8

ER -