Bibliographic record
Abstract
Numerical homotopy continuation methods are known to be accurate and fast for obtaining roots of univariate polynomials with random coefficients. Due to a result of Kac (1943), which was extended by Edelman and Kostlan (1995), we know that the roots of such polynomials tend to be uniformly distributed on the unit circle, and due to the low condition numbers of such roots, offer a "best case" scenario for testing numerical root-finding algorithms. This paper considers the accuracy and computation cost of homotopy methods of average case polynomials such as the low degree Mandelbrot polynomials, and polynomials generated from random roots. For a worst case polynomial, we look at the Wilkinson polynomial with all positive roots. We take a novel approach in studying both numerical pseudozeros of the target polynomial, and the exact pseudozeros given by the homotopy. We confirm the practitioner's expectation that accuracy of high-speed homotopy methods are highly dependent on how well the start system is scaled to fit the target roots. Thus, the so-called Bézout start system used to find roots on the unit circle is nearly ideal. We show how to adapt these insights to work with other polynomials, including changing from the monomial basis to the Lagrange basis.
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How this classification was reachedexpand
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 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".