A randomized preconditioned Cholesky-QR algorithm
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911945293037568 |
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| author | Garrison, James E. Ipsen, Ilse C. F. |
| author_facet | Garrison, James E. Ipsen, Ilse C. F. |
| contents | We a present and analyze rpCholesky-QR, a randomized preconditioned Cholesky-QR algorithm for computing the thin QR factorization of real mxn matrices with rank n. rpCholesky-QR has a low orthogonalization error, a residual on the order of machine precision, and does not break down for highly singular matrices. We derive rigorous and interpretable two-norm perturbation bounds for rpCholesky-QR that require a minimum of assumptions. Numerical experiments corroborate the accuracy of rpCholesky-QR for preconditioners sampled from as few as 3n rows, and illustrate that the two-norm deviation from orthonormality increases with only the condition number of the preconditioned matrix, rather than its square -- even if the original matrix is numerically singular. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_11751 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A randomized preconditioned Cholesky-QR algorithm Garrison, James E. Ipsen, Ilse C. F. Numerical Analysis 65F35, 68W20, 60B20, 15A12, 15A18, 15A42, 15B10 We a present and analyze rpCholesky-QR, a randomized preconditioned Cholesky-QR algorithm for computing the thin QR factorization of real mxn matrices with rank n. rpCholesky-QR has a low orthogonalization error, a residual on the order of machine precision, and does not break down for highly singular matrices. We derive rigorous and interpretable two-norm perturbation bounds for rpCholesky-QR that require a minimum of assumptions. Numerical experiments corroborate the accuracy of rpCholesky-QR for preconditioners sampled from as few as 3n rows, and illustrate that the two-norm deviation from orthonormality increases with only the condition number of the preconditioned matrix, rather than its square -- even if the original matrix is numerically singular. |
| title | A randomized preconditioned Cholesky-QR algorithm |
| topic | Numerical Analysis 65F35, 68W20, 60B20, 15A12, 15A18, 15A42, 15B10 |
| url | https://arxiv.org/abs/2406.11751 |