Abstract
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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Notes
Note that \(2k\le n\). By the CS decomposition [15, Chapter 1, Theorem 5.1], we know that there exist unitary matrices \({{\,\mathrm{diag}\,}}(U_1,U_2)\) and \({{\,\mathrm{diag}\,}}(U_3,U_4)\) with \(U_1,U_3\in \mathbb {C}^{k\times k}\) and \(U_2,U_4\in \mathbb {C}^{(n-k)\times (n-k)}\), such that \([\widetilde{V}_*, \widetilde{V}_c]=[V_*, V_c] {{\,\mathrm{diag}\,}}(U_1, U_2)\begin{bmatrix}\Gamma&-\Sigma&0\\ \Sigma&\Gamma&0 \\ 0&0&I\end{bmatrix}{{\,\mathrm{diag}\,}}(U_3,U_4)^{\mathrm {H}}\), where \(\Sigma \) and \(\Gamma \) are diagonal matrices and \(\Sigma ^2+\Gamma ^2=I_{k}\). Rewrite \([\widetilde{V}_*, \widetilde{V}_c]=[\widetilde{V}_*, \widetilde{V}_c] {{\,\mathrm{diag}\,}}(U_3U_1^{\mathrm {H}}Q_*\widetilde{Q}_*^{\mathrm {H}}, U_4U_2^{\mathrm {H}}Q_c\widetilde{Q}_c^{\mathrm {H}})\), (2.22) still holds. Then (2.23) follows immediately by setting \(Z=Q_c^{\mathrm {H}}U_2\left[ {\begin{matrix}\Sigma \Gamma ^{-1}\\ 0\end{matrix}}\right] U_1^{\mathrm {H}}Q_*\).
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Acknowledgements
We are grateful to the referees for their valuable comments and suggestions. We also would like to thank Prof. Ren-Cang Li from the University of Texas at Arlington for helpful comments on a preliminary draft of this paper.
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Communicated by Daniel Kressner.
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This work was supported by NSFC Nos. 11671023, 11671337, 11771188. The research of Zheng-Jian Bai was partially supported by the Fundamental Research Funds for the Central Universities (No. 20720180008). The research of Zhigang Jia is partially supported by the Priority Academic Program Development Project (PAPD) and the Top-notch Academic Programs Project (No. PPZY2015A013) of Jiangsu Higher Education Institutions, and the Natural Science Foundation of the Jiangsu Higher Educations of China (No.18KJA110001).
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Cai, Y., Jia, Z. & Bai, ZJ. Perturbation analysis of an eigenvector-dependent nonlinear eigenvalue problem with applications. Bit Numer Math 60, 1–29 (2020). https://doi.org/10.1007/s10543-019-00765-4
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DOI: https://doi.org/10.1007/s10543-019-00765-4