An Iterative Method with Asymptotic Orthogonality for Simultaneous Eigenpair Computation

The simultaneous computation of a cluster of eigenpairs with mutually orthogonal eigenvectors is a basic task in scientific computing. We develop a predictor--corrector discretization of the quasi-Grassmannian gradient flow for simultaneous eigenpair computation. The proposed iteration requires neither orthogonal initial data nor any orthogonalization operation: the predictor preserves the current Gram matrix, whereas the corrector reduces the orthogonality error, so that the iterates approach orthogonality asymptotically. We establish well-posedness of the discrete scheme and prove invariant-subspace nonexpansion, asymptotic orthogonality, energy decrease, and convergence to the target eigenspace. Numerical experiments with the discrete equations solved approximately show that the numerical iterates converge to eigenpair approximations whose eigenvectors are numerically orthogonal to high accuracy, while the energy and gradient norm decrease over the iterations.