A Grassmann-Rayleigh quotient iteration for computing invariant subspaces

The classical Rayleigh quotient iteration (RQI) computes a 1-dimensional invariant subspace of a symmetric matrix A with cubic convergence. We propose a generalization of the RQI which computes a p-dimensional invariant subspace of A. The geometry of the algorithm on the Grassmann manifold Gr(p,n) is developed to show cubic convergence and to draw connections with Newton algorithms on Riemannian manifolds.