Rayleigh--Ritz Approximation For the Linear Response Eigenvalue Problem

Abstract

Large scale eigenvalue computation is about approximating certain invariant subspaces associated with the interesting part of the spectrum, and the interesting eigenvalues are then extracted through projecting the problem through approximate invariant subspaces into a much smaller eigenvalue problem. In the case of the linear response eigenvalue problem (aka the random phase eigenvalue problem), it is the pair of deflating subspaces associated with the first few smallest positive eigenvalues that needs to be computed. This paper is concerned with approximation accuracy relationships between a pair of approximate deflating subspaces and approximate eigenvalues extracted by the pair. Lower and upper bounds on eigenvalue approximation errors are obtained in terms of canonical angles between an exact and computed pair of deflating subspaces. These bounds can also be interpreted as lower/upper bounds on the canonical angles in terms of eigenvalue approximation errors. They are useful in analyzing numerical sol...