On an Eigenvector-Dependent Nonlinear Eigenvalue Problem

Abstract

We first provide existence and uniqueness conditions for the solvability of an algebraic eigenvalue problem with eigenvector nonlinearity. We then present a local and global convergence analysis for a self-consistent field (SCF) iteration for solving the problem. The well-known $\sin\Theta$ theorem in the perturbation theory of Hermitian matrices plays a central role. The near-optimality of the local convergence rate of the SCF iteration revealed in this paper is demonstrated by examples from the discrete Kohn--Sham eigenvalue problem in electronic structure calculations and the maximization of the trace ratio in the linear discriminant analysis for dimension reduction.