Continuous methods for symmetric generalized eigenvalue problems

Xing-Bao Gao*, Gene H. Golub, Li-Zhi Liao

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

10 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)676-696
Number of pages21
JournalLinear Algebra and Its Applications
Issue number2-3
Publication statusPublished - 15 Jan 2008

Scopus Subject Areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

User-Defined Keywords

  • Continuous method
  • Generalized eigenvalue
  • Generalized eigenvector


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