Sharp error bounds for Ritz vectors and approximate singular vectors

Journal article

We derive sharp bounds for the accuracy of approximate eigenvectors (Ritz vectors) obtained by the Rayleigh-Ritz process for symmetric eigenvalue problems. Using information that is available or easy to estimate, our bounds improve the classical Davis-Kahan theorem by a factor that can be arbitrarily large, and can give nontrivial information even when the sinθ theorem suggests that a Ritz vector might have no accuracy at all. We also present extensions in three directions, deriving error bounds for invariant subspaces, singular vectors and subspaces computed by a (Petrov-Galerkin) projection SVD method, and eigenvectors of self-adjoint operators on a Hilbert space.

Authors: Nakatsukasa, Y

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: American Mathematical Society
• Journal: Mathematics of Computation
• Volume: 89
• Issue: 324
• Pages: 1843-1866
• Publication date: 2020-01-29
• Acceptance date: 2019-12-26
• DOI: 10.1090/mcom/3519
• ISSN: 0025-5718 and 1088-6842
• Language: English
• Keywords: eigenvector; self-adjoint operator; bounds; Davis-Kahan; FFR; error; Rayleigh-Ritz; singular vector