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 - <p style="text-align: justify;">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.</p>