Bibliographic record
Abstract
The Hermite interpolation problem—defined in the article text—is more complicated than the Lagrange interpolation problem—also defined there—and occurs less frequently in practice. But it does occur, and solving it is occasionally useful. Solutions have been reinvented many times since the problem was first posed and solved in 1878 by Charles Hermite. This article shows how the barycentric forms of the solution, invented about a hundred years after Hermite, work. All one needs to do is to compute a partial fraction expansion by a numerically stable method, and this gives us numerically stable and efficient forms to evaluate the Hermite interpolational polynomial. I describe the Maple program BHIP and some of its ancillary routines (available for download from the Maple Cloud, by clicking on the link below the link to the article PDF, to the right of this abstract), and mention the equivalent Matlab versions genbarywts and hermiteeval. I also compare to some less numerically stable and less efficient approaches. I also show how to find the roots of a Hermite interpolational polynomial by constructing a companion matrix pencil with the routine CMP, which does not change the polynomial basis, and then using standard software to compute the generalized eigenvalues of the pencil, which then give us the roots of the polynomial.
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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 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.016 | 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".