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Local Convergence and Attraction Basins of Higher Order, Jarratt-Like Iterations

Author

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  • Janak Raj Sharma

    (Department of Mathematics, Sant Longowal Institute of Engineering & Technology, Longowal, Punjab 148106, India)

  • Deepak Kumar

    (Department of Mathematics, Sant Longowal Institute of Engineering & Technology, Longowal, Punjab 148106, India)

  • Ioannis K. Argyros

    (Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA)

Abstract

We studied the local convergence of a family of sixth order Jarratt-like methods in Banach space setting. The procedure so applied provides the radius of convergence and bounds on errors under the conditions based on the first Fréchet-derivative only. Such estimates are not proposed in the approaches using Taylor expansions of higher order derivatives which may be nonexistent or costly to compute. In this sense we can extend usage of the methods considered, since the methods can be applied to a wider class of functions. Numerical testing on examples show that the present results can be applied to the cases where earlier results are not applicable. Finally, the convergence domains are assessed by means of a geometrical approach; namely, the basins of attraction that allow us to find members of family with stable convergence behavior and with unstable behavior.

Suggested Citation

  • Janak Raj Sharma & Deepak Kumar & Ioannis K. Argyros, 2019. "Local Convergence and Attraction Basins of Higher Order, Jarratt-Like Iterations," Mathematics, MDPI, vol. 7(12), pages 1-16, December.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:12:p:1203-:d:295602
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    References listed on IDEAS

    as
    1. Moin-ud-Din Junjua & Saima Akram & Nusrat Yasmin & Fiza Zafar, 2015. "A New Jarratt-Type Fourth-Order Method for Solving System of Nonlinear Equations and Applications," Journal of Applied Mathematics, Hindawi, vol. 2015, pages 1-14, March.
    2. Alzahrani, Abdullah Khamis Hassan & Behl, Ramandeep & Alshomrani, Ali Saleh, 2018. "Some higher-order iteration functions for solving nonlinear models," Applied Mathematics and Computation, Elsevier, vol. 334(C), pages 80-93.
    3. Xiao, Xiaoyong & Yin, Hongwei, 2017. "Achieving higher order of convergence for solving systems of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 311(C), pages 251-261.
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