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Convergence Analysis of Weighted-Newton Methods of Optimal Eighth Order in Banach Spaces

Author

Listed:
  • Janak Raj Sharma

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

  • Ioannis K. Argyros

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

  • Sunil Kumar

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

Abstract

We generalize a family of optimal eighth order weighted-Newton methods to Banach spaces and study their local convergence. In a previous study, the Taylor expansion of higher order derivatives is employed which may not exist or may be very expensive to compute. However, the hypotheses of the present study are based on the first Fréchet-derivative only, thereby the application of methods is expanded. New analysis also provides the radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions of derivatives of higher order. Moreover, the order of convergence for the methods is verified by using computational order of convergence or approximate computational order of convergence without using higher order derivatives. Numerical examples are provided to verify the theoretical results and to show the good convergence behavior.

Suggested Citation

  • Janak Raj Sharma & Ioannis K. Argyros & Sunil Kumar, 2019. "Convergence Analysis of Weighted-Newton Methods of Optimal Eighth Order in Banach Spaces," Mathematics, MDPI, vol. 7(2), pages 1-14, February.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:2:p:198-:d:207369
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    References listed on IDEAS

    as
    1. Sharma, Janak Raj & Arora, Himani, 2016. "A new family of optimal eighth order methods with dynamics for nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 924-933.
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