The Projected Newton Method Has Order $1 + \sqrt 2 $ for the Symmetric Eigenvalue Problem

In their study of the classical inverse iteration algorithm, Peters and Wilkinson considered the closely related algorithm that consists of applying Newton’s method, followed by a 2-norm normalization, to the nonlinear system of equations consisting of the eigenvalue-eigenvector equation and an equation requiring the eigenvector to have the square of its 2-norm equal to one. They argue that in practice the $\infty $-norm is easier to work with, and they therefore replace the 2-norm normalization equation with a linear equation requiring that a particular component of the eigenvector be equal to one (effectively an $\infty $-norm normalization). Next, they observe that, because of the linearity of the normalization equation, the normalization step is no longer necessary; the algorithm thus reduces to Newton’s method and quadratic convergence follows from standard theory. Peters and Wilkinson choose to dismiss the 2-norm formulation in favor of the $\infty $-norm formulation; one factor in their choice seem...