On contractivity-preserving 2- and 3-step predictor-corrector series for ODEs
Bibliographic record
Abstract
New optimal, contractivity-preserving (CP), \(d\)-derivative, 2- and 3-step, predictor-corrector, Hermite-Birkhoff-Obrechkoff series methods, denoted by \(HBO(d,k,p)\), \(k=2,3\), with nonnegative coefficients are constructed for solving nonstiff first-order initial value problems \(y'=f(t,y)\), \(y(t_0)=y_0\). The upper bounds \(p_u\) of order \(p\) of \(HBO(d,k,p)\), \(k=2,3\) methods are approximately 1.4 and 1.6 times the number of derivatives \(d\), respectively. Their stability regions have generally a good shape and grow with decreasing \(p-d\). Two selected CP HBO methods: 9-derivative 2-step HBO of order 13, denoted by HBO(9,2,13), which has maximum order 13 based on the CP conditions, and 8-derivative 3-step HBO of order 14, denoted by HBO(8,3,14), compare well with Adams-Cowell of order 13 in PECE mode, denoted by AC(13), in solving standard N-body problems over an interval of 1000 periods on the basis of the relative error of energy as a function of the CPU time. They also compare well with AC(13) in solving standard N-body problems on the basis of the growth of relative error of energy and 10000 periods of integration. The coefficients of selected HBO methods are listed in the appendix.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.055 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".