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}
}