A modified block Newton iteration for approximating an invariant subspace of a symmetric matrix Original Research Article

In this paper we propose a Modified Block Newton Method (MBNM) for approximating an invariant subspace image and the corresponding eigenvalues of a symmetric matrix A. The method generates a sequence of matrices Z(k) which span subspaces imagek approximating image. The matrices Z(k) are calculated via a Newton step applied to a special formulation of the block eigenvalue problem for the matrix A, followed by a Rayleigh-Ritz step which also yields the corresponding eigenvalue approximations. We show that for sufficiently good initial approximations the subspaces imagek converge to image in the sense that sinphik with phik := angle-spherical(imagek,image)Q-quadratically converges to zero under appropriate conditions