Rayleigh–Ritz Majorization Error Bounds with Applications to FEM

Abstract

The Rayleigh–Ritz (RR) 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 $\mathcal{X}$ and $\mathcal{Y}$ of the same finite dimension, such that $\mathcal{X}$ is A-invariant, the absolute changes in the Ritz values of A with respect to $\mathcal{X}$ compared to the Ritz values with respect to $\mathcal{Y}$ represent the RR absolute eigenvalue approximation error. Our first main result is a sharp majorization-type RR error bound in terms of the principal angles between $\mathcal{X}$ and $\mathcal{Y}$ for an arbitrary A-invariant $\mathcal{X}$, which was a conjecture in [SIAM J. Matrix Anal. Appl., 30 (2008), pp. 548–559]. Second, we prove a novel type of RR error bound that deals with the products of the errors, rather than the sums. Third, we establish majorization bounds for the relative errors. We extend our bounds to the case $\dim\mathcal{X}\leq\dim\mathcal{Y}<\infty$ in Hilbert spaces and apply them in the context of the finite element method.