An Hermite–Obreshkov Method for 2nd order linear initial-value problems for ODE with special attention paid to the Mathieu equation.
Bibliographic record
Abstract
Abstract The numerical solution of initial-value problems (IVP) for ordinary differentialequations (ODE) is at this time a mature subject, with many high-quality codes freely available.Second-order linear equations without singularities are an especially simple class ofproblems to solve, even more so if only a single scalar equation such as the Mathieu equation y′′+(a−2qcos2x)y=0 is being considered. Nonetheless, the topic is not yet exhausted, andthis paper considers the case of writing an efficient arbitrary-precision code for the solutionof such equations. For this purpose, an implicit Hermite–Obreshkov method attains nearlyspectral accuracy at at cost only polynomial in the number of bits of accuracy requested.This is interesting for the Mathieu equation in particular because the solutions can be highlyoscillatory of variable frequency and be highly ill-conditioned. This paper reports on the details of the prototype Maple implementation of the method,and summarizes the approximation theoretic results justifying the choice of a balancedHermite–Obreshkov method including its backward stability and decent Lebesgue constants.This method may be of especial interest for the solution of so-called D-finite equations,for which Taylor series coefficients up to degree m are available at cost only O(m),instead of the more usual O(m2). This paper celebrates the happy occasion of the 90th birthday of John C. Butcher. Mathematics Subject Classification (2000) 65L04 · 33F05 · 65D15
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".