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**Some iterative methods free from derivatives and their basins of attraction for nonlinear equations.**
*(English)*
Zbl 1264.65076

Summary: First, we make the P. Jain’s derivative-free method [Appl. Math. Comput. 194, No. 2, 527–533 (2007; Zbl 1193.65063)]. optimal and subsequently increase its efficiency index from 1.442 to 1.587. Then, a novel three-step computational family of iterative schemes for solving single variable nonlinear equations is given. The schemes are free from derivative calculation per full iteration. The optimal family is constructed by applying the weight function approach alongside an approximation for the first derivative of the function in the last step in which the first two steps are the optimized derivative-free form of Jain’s method. The convergence rate of the proposed optimal method and the optimal family is studied. The efficiency index for each method of the family is 1.682. The superiority of the proposed contributions is illustrated by solving numerical examples and comparing them with some of the existing methods in the literature. In the end, we provide the basins of attraction for some methods to observe the beauty of iterative nonlinear solvers in providing fractals and also choose the best method in case of larger attraction basins.

### MSC:

65H05 | Numerical computation of solutions to single equations |

### Citations:

Zbl 1193.65063### Software:

Mathematica
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\textit{F. Soleimani} et al., Discrete Dyn. Nat. Soc. 2013, Article ID 301718, 10 p. (2013; Zbl 1264.65076)

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### References:

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