Page 141 - IJOCTA-15-3

A novel ninth-order root-finding algorithm for nonlinear equations with implementations in various ...
and computation time (in seconds). The results trade-off between computational time and accu-
clearly demonstrate that the proposed method racy, making it particularly well-suited for real-
efficiently handles increasing polynomial com- world applications that demand both precision
plexity while maintaining an exceptional balance and efficiency.
between accuracy and computational efficiency, In summary, the proposed iterative method
making it a highly reliable approach for solving stands out as a highly effective and reliable ap-
polynomial equations. proach for solving polynomial equations. By
seamlessly integrating numerical data with graph-
ical insights from polynomiographs, we confirm
Table 3. Dynamical analysis of the proposed that the method maintains exceptional accuracy,
method
exhibits a stable and controlled iteration growth
pattern, and efficiently handles increasing com-
Polynomial ANI CAI Time
putational demands. With further optimization,
p 1 2.6811 0.9902 31.021
such as adaptive iteration strategies or parallel
p 2 3.0728 0.9809 129.023
processing, the method could become even more
p 3 3.3573 0.9741 145.432
powerful, solidifying its position as an optimal
p 4 3.5553 0.9699 210.098
choice for complex mathematical computations.
Abbreviations: ANI, average number of
iterations; CAI, convergence area index.
6. Conclusion
To further highlight the advantages of the pro-
posed method, we analyzed both numerical re- Nonlinear phenomena appear in various fields, in-
sults and graphical representations using poly- cluding economics, engineering, and scientific re-
nomiographs in Figures 1-6. The ANI values search. With the rapid expansion of computa-
showed a controlled and gradual increase from tional science, the development of new numer-
2.6811 for p 1 to 3.5553 for p 4 , illustrating that ical schemes and the improvement of existing
the iterative method remains computationally ones remain crucial. These numerical methods
feasible even as polynomial complexity increases. should ideally offer higher convergence rates while
Unlike other methods that may require exces- being computationally efficient. In this study,
sive iterations for higher-degree polynomials, the we proposed a new three-step iterative algorithm
proposed method ensures stability without expo- for solving nonlinear scalar equations, addressing
nential growth in iterations. This efficiency was both convergence efficiency and computational
further confirmed by the polynomiographs, where cost. The proposed method was shown to have
the speed of convergence, depicted by the color ninth-order convergence while requiring only six
of each point, highlights the method’s ability to function evaluations per iteration. Several numer-
quickly approximate polynomial roots. ical examples were presented to demonstrate its
effectiveness and superior performance over clas-
Moreover, the CAI values remained impres- sical methods. The algorithm was implemented in
sively high, with only a slight decrease from Maple and Python, showcasing its practicality in
0.9902 for p 1 to 0.9699 for p 4 , proving that widely used computational tools. Additionally, it
the method consistently delivers accurate results can be extended to other software environments,
across all tested polynomials. These accuracy making it adaptable for various applications.
trends align with the polynomiographs, where Future work will focus on reducing computa-
regions with minimal color variation indicate a tional cost, extending the algorithm to systems of
steady and predictable convergence pattern. The nonlinear equations, and exploring adaptive step-
ability of the method to maintain high accuracy size techniques to improve robustness. The pro-
while efficiently converging further emphasizes its posed method provides a reliable and efficient ap-
superiority over traditional iterative techniques. proach to solving nonlinear equations, contribut-
ing to the advancement of numerical analysis and
Although computation time increases from computational mathematics.
31.021 s for p 1 to 210.098 s for p 4 , this is a nat-
ural consequence of the increasing complexity of
Acknowledgments
the polynomial equations. Importantly, the poly-
nomiographs further validate that the proposed None.
method successfully converges in all cases, demon-
strating its robustness and adaptability to vary-
Funding
ing polynomial structures. Compared to alterna-
tive approaches, this method ensures a favorable None.
513