On the convergence of a new Rayleigh quotient method with applications to large eigenproblems

In this paper we propose a variant of the Rayleigh quotient method to compute an eigenvalue and cor-responding eigenvectors of a matrix. It is based on the observation that eigenvectors of a matrix with eigenvalue zero
are also singular vectors corresponding to zero singular values. Instead of computing eigenvector approximations by
the inverse power method, we take them to be the singular vectors corresponding to the smallest singular value of the
shifted matrix. If these singular vectors are computed exactly the method is quadratically convergent. However, ex-
act singular vectors are not required for convergence, and the resulting method combined with Golub–Kahan–Krylov
bidiagonalization looks promising for enhancement/refinement methods for large eigenvalue problems.