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.

翻译：本文将瑞利商迭代（RQI）推广至求解一类非线性方程问题，其中变量分为两个子集：一个子集满足附加等式约束，另一个可视为（广义非线性拉格朗日）乘子。该框架涵盖多个问题，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增量可通过两种等价形式计算：经典瑞利商算法采用{\it Schur形式}，而约束优化中的投影Hessian方法则使用{\it Newton形式}。我们将这些迭代的三次收敛性与{\it 约束切比雪夫项}联系起来，表明其与几何概念{\it 二阶协变导数}相关。广义瑞利商及RQI中使用的{\it 收缩的Hessian矩阵}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性结果，并为矩阵与张量问题构造了新型RQI。