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.

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一个与特征向量相关的非线性特征值问题的微扰分析与应用

依赖于特征向量的非线性特征值问题出现在许多重要应用中，例如电子结构计算中的离散化 Kohn-Sham 方程和线性判别分析中的痕量比问题。在本文中，我们对依赖于特征向量的非线性特征值问题进行了微扰分析，它给出了原始非线性特征值问题的解与受扰动的非线性特征值问题的解之间的距离的上限。引入了非线性特征值问题的条件数，它揭示了影响解灵敏度的因素。此外，为非线性特征值问题给出了两个可计算的误差界限，可用于衡量近似解的质量。实际问题的数值结果，