发表机构
Moscow State University; Institute for Numerical Mathematics, Russian Academy of Sciences（莫斯科国立大学; 俄罗斯科学院数值数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究复射影空间上平方Rayleigh-残差泛函的负梯度流，刻画正规矩阵的临界集与振幅动力学，并证明低阶随机多项式滤波器可改善有限时间谱探索。
AI 中文摘要
我们研究了复射影空间上的平方Rayleigh-残差泛函及其负梯度流。对于任意复矩阵，每个正残差临界点都有一个严格下降的切方向；因此，局部极小值恰好是特征线。对于正规矩阵，我们通过谱点处的圆描述了临界集，并将振幅动力学识别为一个支付矩阵秩至多为四的复制方程。这给出了显式的对数首次积分和内部轨迹的简化表示。我们计算了临界分量附近的横向逃逸率，并分析了多项式重加权如何改变不稳定坐标。最后，我们区分了局部鞍点捕获与Haar分布初始态的集中，并推导了滤波系综的密度。具有随机谱和Jordan块的数值例子说明了收敛性、非单调Rayleigh运动以及低阶随机多项式滤波器对有限时间谱探索的改进。
英文摘要
We study the squared Rayleigh-residual functional on complex projective space and its negative gradient flow. For an arbitrary complex matrix, every positive-residual critical point has a strictly descending tangent direction; consequently, the local minima are precisely the eigenlines. For normal matrices, we describe the critical set in terms of circles through spectral points and identify the amplitude dynamics with a replicator equation whose payoff matrix has rank at most four. This gives explicit logarithmic first integrals and a reduced representation of interior trajectories. We compute transverse escape rates near critical components and analyze how polynomial reweighting changes unstable coordinates. Finally, we distinguish local saddle trapping from concentration of Haar-distributed initial states and derive the density of the filtered ensemble. Numerical examples with random spectra and a Jordan block illustrate convergence, nonmonotone Rayleigh motion, and the improvement of finite-time spectral exploration by low-degree random polynomial filters.
Comments20 pages, 4 figures
