  We analyze the stability of a class of eigensolvers that target interior
eigenvalues with rational filters. We show that subspace iteration with a
rational filter is robust even when an eigenvalue is near a filter's pole.
These dangerous eigenvalues contribute to large round-off errors in the first
iteration, but are self-correcting in later iterations. For matrices with
orthogonal eigenvectors (e.g., real-symmetric or complex Hermitian), two
iterations is enough to reduce round-off errors to the order of the unit-round
off. In contrast, Krylov methods accelerated by rational filters with fixed
poles typically fail to converge to unit round-off accuracy when an eigenvalue
is close to a pole. In the context of Arnoldi with shift-and-invert
enhancement, we demonstrate a simple restart strategy that recovers full
precision in the target eigenpairs.
