Rayleigh-ritz Majorization Error Bounds with Applications to Fem and Subspace Iterations

The Rayleigh-Ritz method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator A. If the trial subspace is A-invariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspaces X and Y of the same finite dimension, such that X is A-invariant, the absolute changes in the Ritz values of A with respect to X compared to the Ritz values with respect to Y represent the absolute eigenvalue approximation error. A recent paper [SIAM J. Matrix Anal. Appl., 30 (2008), pp. 548-559] by M. Argentati et al. bounds the error in terms of the principal angles between X and Y using weak majorization, e.g., a sharp bound is proved if X corresponds to a contiguous set of extreme eigenvalues of A. In this paper, we extend this sharp bound to dimX ≤ dimY and to the general case of an arbitrary A-invariant subspace X, which was a conjecture in this previous paper. We present our Rayleigh-Ritz majorization error bound in the context of the finite element method (FEM), and show how it can improve known FEM eigenvalue error bounds. We derive a new majorization-type convergence rate bound of subspace iterations and combine it with the previous result to obtain a similar bound for the block Lanczos method.