Multiplicative perturbation analysis for QR factorizations
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Bibliographic record
Abstract
This paper is concerned with how the QR factors change when a real matrix $A$ suffers from a left or right multiplicative perturbation, where $A$ is assumed to have full column rank. It is proved that for a left multiplicative perturbation the relative changes in the QR factors in norm are no bigger than a small constant multiple of the norm of the difference between the perturbation and the identity matrix. One of common cases for a left multiplicative perturbation case naturally arises from the computation of the QR factorization. The newly established bounds can be used to explain the accuracy in the computed QR factors. For a right multiplicative perturbation, the bounds on the relative changes in the QR factors are still dependent upon the condition number of the scaled $R$-factor, however. Some ``optimized'' bounds are also obtained by taking into account certain invariant properties in the factors.
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Full frame distilled prediction
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it