Perturbation Analysis for Preconditioned Normal Equations in Mixed Precision
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914402326806528 |
|---|---|
| author | Garrison, James E. Ipsen, Ilse C. F. |
| author_facet | Garrison, James E. Ipsen, Ilse C. F. |
| contents | For real matrices of full column-rank, we analyze the conditioning of several types of normal equations that are preconditioned by a randomized preconditioner computed in lower precision. These include symmetrically preconditioned normal equations, half-preconditioned normal equations, seminormal equations and not-normal equations. Our perturbation bounds are realistic and informative, and suggest that the conditioning depends only mildly on the quality of the preconditioner; however, it does depend on the size of the least squares residual -- even if the normal equations do not originate from a least squares problem. We illustrate that a randomized preconditioner can deliver a solution accuracy comparable to that of Matlab's mldivide command, is efficient in practice, and well-suited to GPU implementations. For the computation of the preconditioner, we propose an automatic selection of the precision, based on a fast condition number estimation in lower precision. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16644 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Perturbation Analysis for Preconditioned Normal Equations in Mixed Precision Garrison, James E. Ipsen, Ilse C. F. Numerical Analysis For real matrices of full column-rank, we analyze the conditioning of several types of normal equations that are preconditioned by a randomized preconditioner computed in lower precision. These include symmetrically preconditioned normal equations, half-preconditioned normal equations, seminormal equations and not-normal equations. Our perturbation bounds are realistic and informative, and suggest that the conditioning depends only mildly on the quality of the preconditioner; however, it does depend on the size of the least squares residual -- even if the normal equations do not originate from a least squares problem. We illustrate that a randomized preconditioner can deliver a solution accuracy comparable to that of Matlab's mldivide command, is efficient in practice, and well-suited to GPU implementations. For the computation of the preconditioner, we propose an automatic selection of the precision, based on a fast condition number estimation in lower precision. |
| title | Perturbation Analysis for Preconditioned Normal Equations in Mixed Precision |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2603.16644 |