The primary focus of this study is to introduce a new three-step iterative method without memory for root finding by merging two different existing techniques. Based on the computational cost, the proposed method acquires optimal eight-order convergence with four functional evaluations (three evaluations for the function and one computation of the first derivative). Furthermore, the suggested scheme supports Kung-Traub’s Conjecture with an efficiency index of $8^\frac{1}{4}=1.682$. We also established the convergence criteria developed for the root-finding technique and demonstrated the fact that the suggested approach is eighth-order convergent. In order to demonstrate the efficacy as well as application of the constructed root-finding technique, we addressed a few practical engineering models and some non-linear functions. In contrast to several existing approaches, this particular method converges more quickly. Finally, several forms of complex functions are taken into consideration under basins of attraction in order to observe the overall fractal behavior of the proposed technique.
Adullah, S. , Choubey, N. and Dara, S. (2025). A new three-step optimal without memory iterative scheme for solving non-linear equations with basins of attraction. Computational Methods for Differential Equations, 13(4), 1289-1305. doi: 10.22034/cmde.2024.59161.2514
MLA
Adullah, S. , , Choubey, N. , and Dara, S. . "A new three-step optimal without memory iterative scheme for solving non-linear equations with basins of attraction", Computational Methods for Differential Equations, 13, 4, 2025, 1289-1305. doi: 10.22034/cmde.2024.59161.2514
HARVARD
Adullah, S., Choubey, N., Dara, S. (2025). 'A new three-step optimal without memory iterative scheme for solving non-linear equations with basins of attraction', Computational Methods for Differential Equations, 13(4), pp. 1289-1305. doi: 10.22034/cmde.2024.59161.2514
CHICAGO
S. Adullah , N. Choubey and S. Dara, "A new three-step optimal without memory iterative scheme for solving non-linear equations with basins of attraction," Computational Methods for Differential Equations, 13 4 (2025): 1289-1305, doi: 10.22034/cmde.2024.59161.2514
VANCOUVER
Adullah, S., Choubey, N., Dara, S. A new three-step optimal without memory iterative scheme for solving non-linear equations with basins of attraction. Computational Methods for Differential Equations, 2025; 13(4): 1289-1305. doi: 10.22034/cmde.2024.59161.2514
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