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Volume 28, Issue 2
Nonlinear Rank-One Modification of the Symmetric Eigenvalue Problem

Xin Huang, Zhaojun Bai & Yangfeng Su

J. Comp. Math., 28 (2010), pp. 218-234.

Published online: 2010-04

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  • Abstract

Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In this paper, we first study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method (SLAM). The global convergence of the SLAM is proven under some mild assumptions. Numerical examples illustrate that the SLAM is the most robust method.

  • Keywords

Nonlinear eigenvalue problem, Rank-one modification, Rank-one damping, Low-rank damping, Picard, Successive linear approximation method, Nonlinear Rayleigh quotient iteration, Safeguard, Global convergence.

  • AMS Subject Headings

65F15, 65H17, 15A18, 35P30, 65Y20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-28-218, author = {}, title = {Nonlinear Rank-One Modification of the Symmetric Eigenvalue Problem}, journal = {Journal of Computational Mathematics}, year = {2010}, volume = {28}, number = {2}, pages = {218--234}, abstract = {

Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In this paper, we first study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method (SLAM). The global convergence of the SLAM is proven under some mild assumptions. Numerical examples illustrate that the SLAM is the most robust method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2009.10-m1002}, url = {http://global-sci.org/intro/article_detail/jcm/8516.html} }
TY - JOUR T1 - Nonlinear Rank-One Modification of the Symmetric Eigenvalue Problem JO - Journal of Computational Mathematics VL - 2 SP - 218 EP - 234 PY - 2010 DA - 2010/04 SN - 28 DO - http://doi.org/10.4208/jcm.2009.10-m1002 UR - https://global-sci.org/intro/article_detail/jcm/8516.html KW - Nonlinear eigenvalue problem, Rank-one modification, Rank-one damping, Low-rank damping, Picard, Successive linear approximation method, Nonlinear Rayleigh quotient iteration, Safeguard, Global convergence. AB -

Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In this paper, we first study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method (SLAM). The global convergence of the SLAM is proven under some mild assumptions. Numerical examples illustrate that the SLAM is the most robust method.

Xin Huang, Zhaojun Bai & Yangfeng Su. (1970). Nonlinear Rank-One Modification of the Symmetric Eigenvalue Problem. Journal of Computational Mathematics. 28 (2). 218-234. doi:10.4208/jcm.2009.10-m1002
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