Twice is enough for dangerous eigenvalues

Journal article

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.

Authors: Horning, A; Nakatsukasa, Y

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Matrix Analysis and Applications
• Volume: 43
• Issue: 1
• Pages: 68-93
• Publication date: 2022-01-11
• Acceptance date: 2021-09-01
• DOI: 10.1137/20M1385330
• EISSN: 1095-7162
• ISSN: 0895-4798
• Language: English
• Keywords: FFR