Perturbation Analysis of An Eigenvector-Dependent Nonlinear Eigenvalue Problem With Applications?
For researchers working on NEPv applications like electronic structure calculations, this work offers theoretical tools to assess solution sensitivity and error bounds, though it is an incremental theoretical contribution.
This paper provides a perturbation analysis for the eigenvector-dependent nonlinear eigenvalue problem (NEPv), deriving upper bounds for solution distances under perturbations and introducing a condition number. The results are validated on Kohn-Sham equation and trace ratio optimization examples.
The eigenvector-dependent nonlinear eigenvalue problem (NEPv) $A(P)V=VΛ$, where the columns of $V\in\mathbb{C}^{n\times k}$ are orthonormal, $P=VV^{\mathrm{H}}$, $A(P)$ is Hermitian, and $Λ=V^{\mathrm{H}}A(P)V$, 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.