Sub-ODEs Simplify Taylor Series Algorithms for Ordinary Differential Equations
Bibliographic record
Abstract
Abstract. A Taylor method for solving an ordinary differential equation initial-value problem [Formula: see text], [Formula: see text], computes the Taylor series (TS) of the solution at the current point, truncated to some order, and then advances to the next point by summing the TS with a suitable stepsize. A standard ODE method (e.g., Runge–Kutta) treats function [Formula: see text] as a black box, but a Taylor solver requires [Formula: see text] to be preprocessed into a code-list of elementary operations that it interprets as operations on (truncated) TS. The trade-off for this extra work includes arbitrary order, typically enabling much larger stepsizes. For a standard function, such as [Formula: see text], this means evaluating [Formula: see text], where [Formula: see text] are TS. The sub-ODE method applies the ODE [Formula: see text], obeyed by [Formula: see text], to in-line this operation as [Formula: see text]. This gives economy of implementation: each function that satisfies a simple ODE goes into the “Taylor library” with a few lines of code—not needing a separate recurrence relation, which is the typical approach. Mathematically, however, the use of sub-ODEs generally transforms the original ODE into a differential-algebraic system, making it nontrivial to ensure a sound system of recurrences for Taylor coefficients. We prove that, regardless of how many sub-ODEs are incorporated into [Formula: see text], this approach guarantees a sound system. We introduce our sub-ODE-based MATLAB ODE solver and show that its performance compares favorably with solvers from the MATLAB ODE suite.
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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 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.009 | 0.005 |
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".