  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.
