Bibliographic record
Abstract
Abstract Sixty years ago Butcher [1] characterized a natural tabulation of the or- der conditions for Runge{Kutta methods as an isomorphism from the set of rooted trees having up to p nodes, and provided examples of explicit and implicit methods of several orders. Within a few years. Fehlberg [3] derived pairs of explicit methods of successive orders that could be implemented eciently by using the dierence of each pair of estimates to control the local error. Unfortunately, Fehlberg's pairs were decient for quadrature problems. Subsequently, this author [5],[6] derived para- metric families of explicit Runge{Kutta pairs of increasing orders 6 to 9 that avoided this problem altogether. These, and most known explicit methods, have been derived by exploiting certain 'simplifying conditions' suggested by Butcher [1] that imposed constraints on subsets of the co- ecients, and thereby simplied the solution of the order conditions for moderate to high order methods. 'Test 21', a MAPLE program developed recently by Butcher [2], was applied to derive known 13-stage pairs of orders 7 and 8. Unexpectedly, results of this application revealed the existence of some previously un- known methods - ie. some that satised most, but not all, of the previously known simplifying conditions. This present study develops formulas for directly computing exact coecients of these new pairs together with oth- ers lying within this new parametric family of (13,7-8) pairs. While the best of these new pairs falls short of the best of pairs already known, the properties discovered might be utilized to precisely characterize recently reported higher order methods found using other approaches by Khashin [4] and Zhang[7], and possibly lead to nding other Runge{Kutta and related yet unknown methods.
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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".