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Sharp error bounds for Ritz vectors and approximate singular vectors


Author: Yuji Nakatsukasa
Journal: Math. Comp. 89 (2020), 1843-1866
MSC (2010): Primary 15A18, 15A42, 65F15
DOI: https://doi.org/10.1090/mcom/3519
Published electronically: January 29, 2020
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Abstract: 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 $ \sin \theta $ theorem by a factor that can be arbitrarily large, and can give nontrivial information even when the $ \sin \theta $ 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.


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Additional Information

Yuji Nakatsukasa
Affiliation: Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom
Email: nakatsukasa@maths.ox.ac.uk

DOI: https://doi.org/10.1090/mcom/3519
Keywords: Rayleigh-Ritz, eigenvector, Davis-Kahan, error bounds, singular vector, self-adjoint operator
Received by editor(s): October 5, 2018
Received by editor(s) in revised form: September 23, 2019
Published electronically: January 29, 2020
Additional Notes: This work was supported by JSPS grants No. 17H01699 and 18H05837, and JST grant JPMJCR1914.
Article copyright: © Copyright 2020 American Mathematical Society