Perturbation analysis of an eigenvector-dependent nonlinear eigenvalue problem with applications

The eigenvector-dependent nonlinear eigenvalue problem arises in many important applications, such as the discretized Kohn-Sham equation in electronic structure calculations and the trace ratio problem in linear discriminant analysis. In this paper, we perform a perturbation analysis for the eigenvector-dependent nonlinear eigenvalue problem, which gives upper bounds for the distance between the solution to the original nonlinear eigenvalue problem and the solution to the perturbed nonlinear eigenvalue problem. A condition number for the nonlinear eigenvalue problem is introduced, which reveals the factors that affect the sensitivity of the solution. Furthermore, two computable error bounds are given for the nonlinear eigenvalue problem, which can be used to measure the quality of an approximate solution. Numerical results on practical problems, such as the Kohn-Sham equation and the trace ratio optimization, indicate that the proposed upper bounds are sharper than the state-of-the-art bounds.