Continuous methods for symmetric generalized eigenvalue problems

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.