A method based on Rayleigh quotient gradient flow for extreme and interior eigenvalue problems

Abstract

Recently, a continuous method has been proposed by Golub and Liao as an alternative way to solve the minimum and interior eigenvalue problems. According to their numerical results, their method seems promising. This article is an extension along this line. In this article, firstly, we convert an eigenvalue problem to an equivalent constrained optimization problem. Secondly, using the Karush–Kuhn–Tucker conditions of this equivalent optimization problem, we obtain a variant of the Rayleigh quotient gradient flow, which is formulated by a system of differential-algebraic equations. Thirdly, based on the Rayleigh quotient gradient flow, we give a practical numerical method for the minimum and interior eigenvalue problems. Finally, we also give some numerical experiments of our method, the Golub and Liao method, and EIGS (a Matlab implementation for computing eigenvalues using restarted Arnoldi’s method) for some typical eigenvalue problems. Our numerical experiments indicate that our method seems promising for most test problems.