SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 84-111, March 2024.
Abstract. This paper concerns a class of monotone eigenvalue problems with eigenvector nonlinearities (mNEPv). The mNEPv is encountered in applications such as the computation of joint numerical radius of matrices, best rank-one approximation of third-order partial-symmetric tensors, and distance to singularity for dissipative Hamiltonian differential-algebraic equations. We first present a variational characterization of the mNEPv. Based on the variational characterization, we provide a geometric interpretation of the self-consistent field (SCF) iterations for solving the mNEPv, prove the global convergence of the SCF, and devise an accelerated SCF. Numerical examples demonstrate theoretical properties and computational efficiency of the SCF and its acceleration.

中文翻译：

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单调非线性特征向量问题的变分表征和自洽场迭代的几何

《SIAM 矩阵分析与应用杂志》，第 45 卷，第 1 期，第 84-111 页，2024 年 3 月。
摘要。本文涉及一类具有特征向量非线性 (mNEPv) 的单调特征值问题。mNEPv 在计算矩阵的联合数值半径、三阶偏对称张量的最佳一阶近似以及耗散哈密顿微分代数方程的奇点距离等应用中会遇到。我们首先提出 mNEPv 的变异特征。基于变分表征，我们提供了用于求解 mNEPv 的自洽场 (SCF) 迭代的几何解释，证明了 SCF 的全局收敛性，并设计了加速 SCF。数值例子证明了 SCF 及其加速的理论特性和计算效率。