Iterative minimization of the Rayleigh quotient by block steepest descent iterations

Abstract

SUMMARYThe topic of this paper is the convergence analysis of subspace gradient iterations for the simultaneous computation of a few of the smallest eigenvalues plus eigenvectors of a symmetric and positive definite matrix pair .A, M /. The methods are based on subspace iterations for A 1 M and use the Rayleigh-Ritz procedure for convergence acceleration. New sharp convergence estimates are proved by generalizing estimates, which have been presented for vectorial steepest descent iterations (see SIAM J. Matrix Anal. Appl., 32(2):443-456, 2011).

Convergence estimates of nonrestarted and restarted block‐Lanczos methods

“…For the restarted version, the estimate in eq. 2.22 in the work of Knyazev is generalized analogously to theorem 3.1 in the work of Neymeyr et al by applying the analysis for single‐vector restarted Krylov subspace iterations in the abovementioned theorem 3.1 to certain auxiliary Krylov subspaces inside a block‐Krylov subspace. Additionally, a sharp variant of the generalized estimate is derived based on the analysis in theorem 3.6 in the work of Zhou et al The corresponding bounds can be reached in certain limit cases.…”

“…Because this estimate only concerns one Ritz value (the sth one in an s-dimensional subspace iterate), a generalization to all Ritz values is of our interest. This is realized in this paper based on our previous results on block steepest descent iterations (see theorem 3.1 in the work of Neymeyr et al 19 ) and on single-vector restarted Krylov subspace iterations (see theorem 3.1 in the work of Neymeyr et al 11 ).…”

“…In the special case k =2, the analysis from Neymeyr et al for block steepest descent iterations can be applied and shows that with κ i =( μ i +1 − μ n )/( μ i − μ n ) for θ i > μ i +1 (cf. theorem 3.1 in the work of Neymeyr et al). This estimate is sharp in the sense that the equality can be attained in the limit case θ i → μ i within an invariant subspace associated with the eigenvalues μ i , μ i +1 , μ n .…”

“…Remark In order to generalize Knyazev's estimate (see eq. 2.22 in the work of Knyazev) in a similar form as , we use the basic idea for analyzing block steepest descent iterations from theorem 3.1 in the work of Knyazev . Certain auxiliary Krylov subspaces inside a block‐Krylov subspace are selected, to which we apply the estimate corresponding to theorem 3.1 in the work of Neymeyr et al for single‐vector restarted Krylov subspace iterations.…”

“…2.22 in the work by Knyazev cannot be generalized in a certain cluster‐robust form without further assumptions. Instead, a generalization based on the results from the work of Neymeyr et al on block steepest descent iterations and single‐vector restarted Krylov subspace iterations is derived and shown to be sharp. In Section 4, the new estimates derived in the two previous sections are reformulated for invert‐block‐Lanczos methods.…”