Analyzing Vector Orthogonalization Algorithms
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
.Computer implementations of vector orthogonalization algorithms produce a sequence of supposedly orthogonal vectors, but rounding-errors can cause loss of orthogonality and rank. Nevertheless these computational algorithms can be very effective as parts of various methods. We develop a general theory based on the augmented orthogonal matrix developed in [SIAM J. Matrix Anal. Appl., 31 (2009), pp. 565–583] that can be applied to any such algorithm. This can be combined with a rounding-error analysis of the algorithm to analyze its finite-precision behavior. We apply this combination to prove that a particular Lanczos tridiagonalization of a Hermitian matrix always computes components for which backward-stable solutions to \(Ax\!=\!b\), \(A\!=\!A^H\), exist. If an appropriate rounding-error analysis is available, the approach can apparently be applied to any computation producing a sequence of supposedly orthogonal \(n\)-vectors, where a linear combination of these vectors is intended to approximate some quantity.Keywordsvector orthogonalizationfinite-precision computationsloss of orthogonalityLanczos processthe method of conjugate gradients (CG)Krylov subspace methodsconvergence of iterative solution of equationslarge sparse matricesMSC codes65F1065F2565F3065F5065G5015A2315A57
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 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