Sharp error bounds for Ritz vectors and approximate singular vectors
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 arbitraril...
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| Materyal Türü: | Journal article |
| Dil: | English |
| Baskı/Yayın Bilgisi: |
American Mathematical Society
2020
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| _version_ | 1826298983868268544 |
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| author | Nakatsukasa, Y |
| author_facet | Nakatsukasa, Y |
| author_sort | Nakatsukasa, Y |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T04:55:01Z |
| format | Journal article |
| id | oxford-uuid:d64b4586-45ea-4f76-991d-e3ef7a1a51a7 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T04:55:01Z |
| publishDate | 2020 |
| publisher | American Mathematical Society |
| record_format | dspace |
| spelling | oxford-uuid:d64b4586-45ea-4f76-991d-e3ef7a1a51a72022-03-27T08:32:27ZSharp error bounds for Ritz vectors and approximate singular vectorsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:d64b4586-45ea-4f76-991d-e3ef7a1a51a7EnglishSymplectic Elements at OxfordAmerican Mathematical Society2020Nakatsukasa, YWe 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. |
| spellingShingle | Nakatsukasa, Y Sharp error bounds for Ritz vectors and approximate singular vectors |
| title | Sharp error bounds for Ritz vectors and approximate singular vectors |
| title_full | Sharp error bounds for Ritz vectors and approximate singular vectors |
| title_fullStr | Sharp error bounds for Ritz vectors and approximate singular vectors |
| title_full_unstemmed | Sharp error bounds for Ritz vectors and approximate singular vectors |
| title_short | Sharp error bounds for Ritz vectors and approximate singular vectors |
| title_sort | sharp error bounds for ritz vectors and approximate singular vectors |
| work_keys_str_mv | AT nakatsukasay sharperrorboundsforritzvectorsandapproximatesingularvectors |