TY - JOUR
T1 - Continuous methods for symmetric generalized eigenvalue problems
AU - Gao, Xing-Bao
AU - Golub, Gene H.
AU - Liao, Li-Zhi
N1 - Funding Information:
The second author was supported in part by DOE-FC02-01ER4177. The third author was supported in part by grants from Hong Kong Baptist University and the Research Grant Council of Hong Kong. ∗ Corresponding author. E-mail addresses: [email protected] (X.-B. Gao), [email protected] (G.H. Golub), [email protected] (L.Z. Liao).
PY - 2008/1/15
Y1 - 2008/1/15
N2 - Generalized eigenvalue problems play a significant role in many applications. In this paper, continuous methods are presented to compute generalized eigenvalues and their corresponding eigenvectors for two real symmetric matrices. Our study only requires that the right-hand-side matrix is positive semi-definite. The main idea of our continuous methods is to convert the generalized eigenvalue problem into an optimization problem. Then a continuous method which includes both a merit function and an ordinary differential equation (ODE) is introduced for the resulting optimization problem. The strong convergence of the ODE solution is proved for any starting point. Both the generalized eigenvalues and their corresponding eigenvectors can be easily obtained under some mild conditions. Some numerical results are also presented.
AB - Generalized eigenvalue problems play a significant role in many applications. In this paper, continuous methods are presented to compute generalized eigenvalues and their corresponding eigenvectors for two real symmetric matrices. Our study only requires that the right-hand-side matrix is positive semi-definite. The main idea of our continuous methods is to convert the generalized eigenvalue problem into an optimization problem. Then a continuous method which includes both a merit function and an ordinary differential equation (ODE) is introduced for the resulting optimization problem. The strong convergence of the ODE solution is proved for any starting point. Both the generalized eigenvalues and their corresponding eigenvectors can be easily obtained under some mild conditions. Some numerical results are also presented.
KW - Continuous method
KW - Generalized eigenvalue
KW - Generalized eigenvector
UR - http://www.scopus.com/inward/record.url?scp=36048959857&partnerID=8YFLogxK
U2 - 10.1016/j.laa.2007.08.034
DO - 10.1016/j.laa.2007.08.034
M3 - Journal article
AN - SCOPUS:36048959857
SN - 0024-3795
VL - 428
SP - 676
EP - 696
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
IS - 2-3
ER -