TWICE IS ENOUGH FOR DANGEROUS EIGENVALUES

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 are 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.

Automated summary

We analyze the stability of eigensolvers that use rational filters to target interior eigenvalues. The results show that subspace iteration with a rational filter remains robust even when an eigenvalue is close to a filter's pole, and these dangerous eigenvalues self-correct in subsequent iterations.