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TY - JOUR AU - Romadani, Moh. Nafis Husen PY - 2025 DA - 2025/12/14 TI - From Theory to Code: Transforming Classical Root-Finding Methods into Efficient Python Implementations JO - ICCK Journal of Applied Mathematics T2 - ICCK Journal of Applied Mathematics JF - ICCK Journal of Applied Mathematics VL - 1 IS - 3 SP - 154 EP - 189 DO - 10.62762/JAM.2025.840767 UR - https://www.icck.org/article/abs/JAM.2025.840767 KW - numerical methods KW - nonlinear equations KW - root-finding algorithms KW - python implementation KW - computational mathematics AB - This study conducts a comparative evaluation of seven numerical methods for finding roots of nonlinear equations: Bisection, Regula-Falsi, Fixed-Point Iteration, Newton-Raphson, Secant, Aitken's \(\Delta^2\), and Steffensen. The aim is to analyze the effectiveness of each method based on convergence speed, numerical accuracy, stability, and computational time efficiency. Algorithm implementation was carried out in the Python programming language using NumPy, SymPy, Pandas, and Matplotlib libraries. Test functions included polynomial, trigonometric, exponential, and mixed functions to represent diverse functional characteristics. The results indicate that Steffensen and Newton-Raphson achieved the fastest convergence in terms of iteration count, while Secant excelled in execution time efficiency. Bracketing methods such as Bisection and Regula-Falsi guaranteed convergence stability despite being slower. Fixed-Point Iteration was highly sensitive to the choice of iteration function, whereas Aitken's \(\Delta^2\) served effectively as an accelerator. The study concludes that method selection should be tailored to function characteristics and practical needs, such as derivative availability, computational complexity, and error tolerance. The implication is that this research provides an empirical guide for researchers and practitioners in selecting optimal numerical algorithms for scientific and engineering applications. SN - 3068-5656 PB - Institute of Central Computation and Knowledge LA - English ER -
@article{Romadani2025From,
author = {Moh. Nafis Husen Romadani},
title = {From Theory to Code: Transforming Classical Root-Finding Methods into Efficient Python Implementations},
journal = {ICCK Journal of Applied Mathematics},
year = {2025},
volume = {1},
number = {3},
pages = {154-189},
doi = {10.62762/JAM.2025.840767},
url = {https://www.icck.org/article/abs/JAM.2025.840767},
abstract = {This study conducts a comparative evaluation of seven numerical methods for finding roots of nonlinear equations: Bisection, Regula-Falsi, Fixed-Point Iteration, Newton-Raphson, Secant, Aitken's \(\Delta^2\), and Steffensen. The aim is to analyze the effectiveness of each method based on convergence speed, numerical accuracy, stability, and computational time efficiency. Algorithm implementation was carried out in the Python programming language using NumPy, SymPy, Pandas, and Matplotlib libraries. Test functions included polynomial, trigonometric, exponential, and mixed functions to represent diverse functional characteristics. The results indicate that Steffensen and Newton-Raphson achieved the fastest convergence in terms of iteration count, while Secant excelled in execution time efficiency. Bracketing methods such as Bisection and Regula-Falsi guaranteed convergence stability despite being slower. Fixed-Point Iteration was highly sensitive to the choice of iteration function, whereas Aitken's \(\Delta^2\) served effectively as an accelerator. The study concludes that method selection should be tailored to function characteristics and practical needs, such as derivative availability, computational complexity, and error tolerance. The implication is that this research provides an empirical guide for researchers and practitioners in selecting optimal numerical algorithms for scientific and engineering applications.},
keywords = {numerical methods, nonlinear equations, root-finding algorithms, python implementation, computational mathematics},
issn = {3068-5656},
publisher = {Institute of Central Computation and Knowledge}
}
Copyright © 2025 by the Author(s). Published by Institute of Central Computation and Knowledge. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 
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