TY - JOUR

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

AU - Chu, King-Wah

AU - Fan, Hung-Yuan

AU - Jia, Zhongxiao

AU - Li, Tiexiang

AU - Lin, Wen-wei

PY - 2011

Y1 - 2011

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

UR - http://www.sciencedirect.com/science/article/pii/S037704271000631X

U2 - 10.1016/j.cam.2010.11.014

DO - 10.1016/j.cam.2010.11.014

M3 - Article

SN - 0377-0427

VL - 235

SP - 2626

EP - 2639

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

IS - 8

ER -