Perturbation Analysis of An Eigenvector-Dependent Nonlinear Eigenvalue Problem With Applications?

TL;DR

This paper conducts a perturbation analysis of eigenvector-dependent nonlinear eigenvalue problems, providing bounds and condition numbers that assess solution sensitivity, with applications in electronic structure calculations and linear discriminant analysis.

Contribution

It introduces a perturbation analysis framework, including condition numbers and error bounds, for NEPv, enhancing understanding of solution stability and accuracy in practical applications.

Findings

Upper bounds for solution perturbations are established.

A condition number for NEPv is defined, indicating sensitivity factors.

Numerical experiments validate theoretical bounds in applications.

Abstract

The eigenvector-dependent nonlinear eigenvalue problem (NEPv) $A(P)V=V\Lambda$, where the columns of $V\in\mathbb{C}^{n\times k}$ are orthonormal, $P=VV^{\mathrm{H}}$, $A(P)$ is Hermitian, and $\Lambda=V^{\mathrm{H}}A(P)V$, 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 NEPv, which gives upper bounds for the distance between the solution to the original NEPv and the solution to the perturbed NEPv. A condition number for the NEPv is introduced, which reveals the factors that affect the sensitivity of the solution. Furthermore, two computable error bounds are given for the NEPv, which can be used to measure the quality of an approximate solution. The theoretical results are validated by numerical experiments for the Kohn-Sham equation and the trace ratio optimization.