A geometric theory for preconditioned inverse iteration applied to a subspace

Authors:
Klaus Neymeyr
Affiliations:
Mathematisches Institut der Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Venue:
Mathematics of Computation
Year:
2002

Citing 3
Cited 2

Computing an Eigenvector with Inverse Iteration

SIAM Review
The symmetric eigenvalue problem

The symmetric eigenvalue problem
A geometric theory for preconditioned inverse iteration applied to a subspace

Mathematics of Computation

A geometric theory for preconditioned inverse iteration applied to a subspace

Mathematics of Computation
Accelerated Block Preconditioned Gradient method for large scale wave functions calculations in Density Functional Theory

Journal of Computational Physics

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Abstract

The aim of this paper is to provide a convergence analysis for a preconditioned subspace iteration, which is designated to determine a modest number of the smallest eigenvalues and its corresponding invariant subspace of eigenvectors of a large, symmetric positive definite matrix. The algorithm is built upon a subspace implementation of preconditioned inverse iteration. i.e., the well-known inverse iteration procedure, where the associated system of linear equations is solved approximately by using a preconditioner. This step is followed by a Rayleigh-Ritz projection so that preconditioned inverse iteration is always applied to the Ritz vectors of the actual subspace of approximate eigenvectors. The given theory provides sharp convergence estimates for the Ritz values and is mainly built on arguments exploiting the geometry underlying preconditioned inverse iteration.