Generalizations of the Rayleigh quotient iteration for the iterative refinement of the eigenvectors of real symmetric matrices

We present new algorithms for refining the estimates of the eigenvectors of a real symmetric matrix. Using techniques from calculus, we show that the algorithms converge locally cubically fast. By this we mean that locally all eigenvalue-eigenvector pairs simultaneously converge at a cubic rate. This is in contrast to well known shifted QR algorithms, which depending on the shifting strategy employed, have only one (or at most a small subset) of the eigenvalue-eigenvector pairs converging cubically at any one time. The algorithms are well suited to the situation where one needs to compute the eigenvectors of a perturbed matrix A + E based on a good estimate of the eigenvectors of a matrix A. Such a situation frequently appears in tracking applications.