Bibliographic record
Abstract
In order to approximate the integral $I(f)=\int_a^b f(x) dx$, where $f$ is a sufficiently smooth function, models for quadrature rules are developed using a given {\it panel} of $n\,\,(n\geq 2)$ equally spaced points. These models arise from the undetermined coefficients method, using a Newton's basis for polynomials. Although part of the final product is algebraically equivalent to the well known closed Newton-Cotes rules, the algorithms obtained are not the classical ones.In the basic model the most simple quadrature rule $Q_n$ is adopted (the so-called left rectangle rule) and a correction $\tilde E_n$ is constructed, so that the final rule $S_n=Q_n+\tilde E_n$ is interpolatory. The correction $\tilde E_n$, depending on the divided differences of the data, might be considered a {\em realistic correction} for $Q_n$, in the sense that $\tilde E_n$ should be close to the magnitude of the true error of $Q_n$, having also the correct sign. The analysis of the theoretical error of the rule $S_n$ as well as some classical properties for divided differences suggest the inclusion of one or two new points in the given panel. When $n$ is even it is included one point and two points otherwise. In both cases this approach enables the computation of a {\em realistic error} $\bar E_{S_n}$ for the {\it extended or corrected} rule $S_n$. The respective output $(Q_n,\tilde E_n, S_n, \bar E_{S_n})$ contains reliable information on the quality of the approximations $Q_n$ and $S_n$, provided certain conditions involving ratios for the derivatives of the function $f$ are fulfilled. These simple rules are easily converted into {\it composite} ones. Numerical examples are presented showing that these quadrature rules are useful as a computational alternative to the classical Newton-Cotes formulas.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.009 | 0.007 |
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 source (direct Gemma or distilled Codex), 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".