Rayleigh Quotient Iteration, cubic convergence, and second covariant derivative

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 (linearslash 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.