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

AbstractIn this paper we propose a Modified Block Newton Method (MBNM) for approximating an invariant subspace J and the corresponding eigenvalues of a symmetric matrix A. The method generates a sequence of matrices Z(k) which span subspaces Jk approximating J. 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 Jk converge to J in the sense that sinϕk with ϕk := ∢(Jk,J)Q-quadratically converges to zero under appropriate condition