Barycentric Hermite Interpolants for Event Location in Initial-Value Problems
Bibliographic record
Abstract
Continuous extensions are now routinely provided by many IVP solvers, for graphical output, error control, or event location. Recent developments suggest that a uniform, stable and convenient interpolant may be provided directly by value and derivative data (Hermite data), because a new companion matrix for such data allows stable, robust and convenient root-finding by means of (usually built-in) generalized eigenvalue solvers such as eig inMatlab or Eigenvalues inMaple. Even though these solvers are not as efficient as a special-purpose Hermite interpolant root-finder might be, being O(d3) in cost instead of O(d2), for low or moderate degrees d they are efficient enough. More, because all roots are found, the first root (and hence the event) is guaranteed to be found. Further, the excellent conditioning properties (compared to the monomial basis or to divided differences) suggest that the results will be as accurate as possible. The techniques of this paper apply to polynomial or rational interpolants such as the shape-preserving interpolants of Brankin and Gladwell. We give a sketch of barycentric Hermite interpolation and a sketch of the theory of con-ditioning of such interpolants. Moreover, we present the construction of the Hermite interpolation polynomial companion matrix pencil and a discussion of scaling and precom-putation. We remark that the Bézout matrix can be used used to solve more complicated event location problems involving more than one polynomial, via polynomial eigenvalue problems.
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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 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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".