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Some Iterative Methods Free from Derivatives and Their Basins of Attraction for Nonlinear Equations

Author

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  • Farahnaz Soleimani
  • Fazlollah Soleymani
  • Stanford Shateyi

Abstract

First, we make the Jain's derivative-free method 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.

Suggested Citation

  • Farahnaz Soleimani & Fazlollah Soleymani & Stanford Shateyi, 2013. "Some Iterative Methods Free from Derivatives and Their Basins of Attraction for Nonlinear Equations," Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, pages 1-10, April.
    • Handle: RePEc:hin:jnddns:301718
    DOI: 10.1155/2013/301718
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    Cited by:

    1. Jian Li & Xiaomeng Wang & Kalyanasundaram Madhu, 2019. "Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction," Mathematics, MDPI, vol. 7(11), pages 1-15, November.
    2. Dilan Ahmed & Mudhafar Hama & Karwan Hama Faraj Jwamer & Stanford Shateyi, 2019. "A Seventh-Order Scheme for Computing the Generalized Drazin Inverse," Mathematics, MDPI, vol. 7(7), pages 1-10, July.

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