瑞利商迭代、三次收敛性与第二协变导数
Rayleigh Quotient Iteration, cubic convergence, and second covariant derivative
AI总结：
本研究将瑞利商迭代（RQI）推广至带约束的非线性方程求解框架，关联其三次收敛性与约束切比雪夫项、第二协变导数的几何概念，推导多场景三次收敛结果并构造矩阵与张量问题的新RQI算法。
摘要：
我们将瑞利商迭代（Rayleigh Quotient Iteration，RQI）推广至求解变量分为两个子集的非线性方程问题：其中一个子集满足额外的等式约束，另一个子集可视为（广义非线性拉格朗日）乘子。该框架涵盖若干问题，包括（线性/非线性）特征值问题、约束优化问题以及张量特征对问题。通常，RQI的增量可通过两种等价形式计算：经典瑞利商算法采用“舒尔形式（Schur form）”，而约束优化中的投影海森方法采用“牛顿形式（Newton form）”。我们将这类迭代的三次收敛性与“约束切比雪夫项（constrained Chebyshev term）”相关联，证明其与“第二协变导数（second covariant derivative）”这一几何概念有关。广义瑞利商与RQI中使用的“收缩映射的海森矩阵（Hessian of the retraction）”均出现在切比雪夫项中。我们推导了若干应用中的三次收敛结果，并为矩阵与张量问题构造了新的RQI算法。
英文摘要：
We generalize the Rayleigh Quotient Iteration (RQI) to the problem of solving a nonlinear equation where the variables are divided into two subsets, one satisfying additional equality constraints and the other could be considered as (generalized nonlinear Lagrange) multipliers. This framework covers several problems, including the (linear\slash nonlinear) eigenvalue problems, the constrained optimization problem, and the tensor eigenpair problem. Often, the RQI increment could be computed in two equivalent forms. The classical Rayleigh quotient algorithm uses the {\it Schur form}, while the projected Hessian method in constrained optimization uses the {\it Newton form}. We link the cubic convergence of these iterations with a {\it constrained Chebyshev term}, showing it is related to the geometric concept of {\it second covariant derivative}. Both the generalized Rayleigh quotient and the {\it Hessian of the retraction} used in the RQI appear in the Chebyshev term. We derive several cubic convergence results in application and construct new RQIs for matrix and tensor problems.
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