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Convergence Analysis and Dynamical Nature of an Efficient Iterative Method in Banach Spaces

Author

Listed:
  • Deepak Kumar

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

  • Sunil Kumar

    (Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Channai 601103, India)

  • Janak Raj Sharma

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

  • Lorentz Jantschi

    (Department of Physics and Chemistry, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
    Chemical Doctoral School, Babes-Bolyai University, 400028 Cluj-Napoca, Romania)

Abstract

We study the local convergence analysis of a fifth order method and its multi-step version in Banach spaces. The hypotheses used are based on the first Fréchet-derivative only. The new approach provides a computable radius of convergence, error bounds on the distances involved, and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives, which may not exist or may be very expensive or impossible to compute. Numerical examples are provided to validate the theoretical results. Convergence domains of the methods are also checked through complex geometry shown by drawing basins of attraction. The boundaries of the basins show fractal-like shapes through which the basins are symmetric.

Suggested Citation

  • Deepak Kumar & Sunil Kumar & Janak Raj Sharma & Lorentz Jantschi, 2021. "Convergence Analysis and Dynamical Nature of an Efficient Iterative Method in Banach Spaces," Mathematics, MDPI, vol. 9(19), pages 1-16, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:19:p:2510-:d:651046
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    References listed on IDEAS

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    1. Krzysztof Gdawiec & Wiesław Kotarski & Agnieszka Lisowska, 2015. "Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods," Abstract and Applied Analysis, Hindawi, vol. 2015, pages 1-19, February.
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