Convergence Analysis of Gradient Iterations for the Symmetric Eigenvalue Problem

Abstract

Gradient iterations for the Rayleigh quotient are simple and robust solvers to determine a few of the smallest eigenvalues together with the associated eigenvectors of (generalized) matrix eigenvalue problems for symmetric matrices. Sharp convergence estimates for the Ritz values and Ritz vectors are derived for various steepest descent/ascent gradient iterations. The analysis shows that poorest convergence of the eigenvalue approximations is attained in a three-dimensional invariant subspace; explicit convergence estimates are then derived by means of a minidimensional analysis.