One Parameter Optimal Derivative-Free Family to Find the Multiple Roots of Algebraic Nonlinear Equations
Bibliographic record
Abstract
In this study, we construct the one parameter optimal derivative-free iterative family to find the multiple roots of an algebraic nonlinear function. Many researchers developed the higher order iterative techniques by the use of the new function evaluation or the first-order or second-order derivative of functions to evaluate the multiple roots of a nonlinear equation. However, the evaluation of the derivative at each iteration is a cumbersome task. With this motivation, we design the second-order family without the utilization of the derivative of a function and without the evaluation of the new function. The proposed family is optimal as it satisfies the convergence order of Kung and Traub’s conjecture. Here, we use one parameter a for the construction of the scheme, and for a=1, the modified Traub method is its a special case. The order of convergence is analyzed by Taylor’s series expansion. Further, the efficiency of the suggested family is explored with some numerical tests. The obtained results are found to be more efficient than earlier schemes. Moreover, the basin of attraction of the proposed and earlier schemes is also analyzed.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".