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Volume 28, Issue 6
On Smooth LU Decompositions with Applications to Solutions of Nonlinear Eigenvalue Problems

Hua Dai & Zhong-Zhi Bai

DOI: 10.4208/jcm.1004-m0009

J. Comp. Math., 28 (2010), pp. 745-766.

Published online: 2010-12

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  • Abstract

We study the smooth LU decomposition of a given analytic functional $\lambda$-matrix $A(\lambda)$ and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of $A(\lambda)$, and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.

  • Keywords

Matrix-valued function, Smooth LU decomposition, Pivoting, Nonlinear eigenvalue problem, Multiple eigenvalue, Newton method.

  • AMS Subject Headings

15A18, 15A23, 65F15.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-28-745, author = {Hua Dai and Zhong-Zhi Bai}, title = {On Smooth LU Decompositions with Applications to Solutions of Nonlinear Eigenvalue Problems}, journal = {Journal of Computational Mathematics}, year = {2010}, volume = {28}, number = {6}, pages = {745--766}, abstract = {

We study the smooth LU decomposition of a given analytic functional $\lambda$-matrix $A(\lambda)$ and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of $A(\lambda)$, and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1004-m0009}, url = {http://global-sci.org/intro/article_detail/jcm/8548.html} }
TY - JOUR T1 - On Smooth LU Decompositions with Applications to Solutions of Nonlinear Eigenvalue Problems AU - Hua Dai & Zhong-Zhi Bai JO - Journal of Computational Mathematics VL - 6 SP - 745 EP - 766 PY - 2010 DA - 2010/12 SN - 28 DO - http://doi.org/10.4208/jcm.1004-m0009 UR - https://global-sci.org/intro/article_detail/jcm/8548.html KW - Matrix-valued function, Smooth LU decomposition, Pivoting, Nonlinear eigenvalue problem, Multiple eigenvalue, Newton method. AB -

We study the smooth LU decomposition of a given analytic functional $\lambda$-matrix $A(\lambda)$ and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of $A(\lambda)$, and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.

Hua Dai and Zhong-Zhi Bai. (2010). On Smooth LU Decompositions with Applications to Solutions of Nonlinear Eigenvalue Problems. Journal of Computational Mathematics. 28 (6). 745-766. doi:10.4208/jcm.1004-m0009
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