Twice is enough for dangerous eigenvalues
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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 stable even when an eigenvalue is located near a pole of the filter. These dangerous eigenvalues contribute to large round-off errors in the first iteration, but are self-correcting in later iterations. 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.
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