Rayleigh-Ritz Approximation For the Linear Response Eigenvalue Problem

Large scale eigenvalue computation is about approximating certain invariant subspaces associated with the interested part of the spectrum, and the interested eigenvalues are then extracted from projecting the problem by 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 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 solutions to linear response eigenvalue problems.