Advances in Nonlinear Function Analysis: Multiple Zero Solutions
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Bibliographic record
Abstract
There is no doubt that there is plethora of optimal fourth-order iterative approaches available to estimate the simple zeros of nonlinear functions. We can extend these method/methods for multiple zeros but the main issue is to preserve the same convergence order. Therefore, numerous optimal and non-optimal modifications have been introduced in the literature to preserve the order of convergence. Such count of methods that can estimate the multiple zeros are limited in the scientific literature. With this point, a new optimal fourth-order scheme is presented for multiple zeros with known multiplicity. The proposed scheme is based on the weight function strategy involving functions in ratio. Moreover, the scheme is optimal as it satisfies the hypothesis of Kung–Traub conjecture. An exhaustive study of the convergence is shown to determine the fourth order of the methods under certain conditions. To demonstrate the validity and appropriateness for the proposed family, several numerical experiments have been performed. The numerical comparison highlights the effectiveness of scheme in terms of accuracy, stability, and CPU time
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.007 |
| Science and technology studies | 0.003 | 0.000 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.015 | 0.011 |
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 it