A Symbolic Algorithm to Obtain Low or High Degree Splines from Discrete Fourier Transforms
Bibliographic record
Abstract
Obtaining high-degree splines with the use of traditional spline interpolation methods is not an easy task; therefore, traditional spline interpolation is typically limited to cubic splines. In this paper, we present a symbolic and numeric algorithm to obtain splines of any degree, while providing detailed procedures and examples of how to use this algorithm so that it is immediately useful for an interested user. This method, which was initially developed by Beaudoin and Beauchemin [2, 3], works for splines of any degree and yields very accurate results when the boundary conditions are chosen wisely. It also provides approximations of higher order derivatives, something that is not available with the use of cubic splines. This paper presents formulas that can be used in a straightforward manner to obtain interpolation splines of first degree (linear splines), second degree (parabolic splines) and third degree (cubic splines). For splines of higher degree, a short but complete symbolic algorithm to compute the formulas is presented. The resulting formulas can be used in the same manner as those presented for splines of lower degree. A complete numerical example is included to show how the results are obtained and a link to the complete Maple source code is given.
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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 | 0.001 | 0.004 |
| 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 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.017 | 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".