# Rayleigh Quotient Iteration, cubic convergence, and second covariant derivative

## TL;DR

This paper extends Rayleigh Quotient Iteration to nonlinear equations with constraints, linking its cubic convergence to second covariant derivatives and applying it to matrix and tensor eigenproblems.

## Contribution

It generalizes RQI for constrained nonlinear problems, establishes a geometric interpretation of cubic convergence, and develops new RQIs for matrices and tensors.

## Key findings

- Established cubic convergence linked to second covariant derivatives.
- Derived new RQIs for matrix and tensor eigenproblems.
- Connected classical RQI forms with geometric concepts.

## Abstract

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.