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We generalize the Rayleigh Quotient Iteration (RQI) to the problem of solvinga nonlinear equation where the variables are divided into two subsets, onesatisfying additional equality constraints and the other could be considered as(generalized nonlinear Lagrange) multipliers. This framework covers severalproblems, including the (linear\slash nonlinear) eigenvalue problems, theconstrained optimization problem, and the tensor eigenpair problem. Often, theRQI increment could be computed in two equivalent forms. The classical Rayleighquotient algorithm uses the {\it Schur form}, while the projected Hessianmethod in constrained optimization uses the {\it Newton form}. We link thecubic convergence of these iterations with a {\it constrained Chebyshev term},showing it is related to the geometric concept of {\it second covariantderivative}. Both the generalized Rayleigh quotient and the {\it Hessian of theretraction} used in the RQI appear in the Chebyshev term. We derive severalcubic convergence results in application and construct new RQIs for matrix andtensor problems.

Rayleigh Quotient Iteration, cubic convergence, and second covariant derivative