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Ball Comparison between Three Sixth Order Methods for Banach Space Valued Operators

Author

Listed:
  • Ramandeep Behl

    (Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia)

  • Ioannis K. Argyros

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

  • Jose Antonio Tenreiro Machado

    (Department of Electrical Engineering, ISEP-Institute of Engineering, Polytechnic of Porto, 431 4294-015 Porto, Portugal)

Abstract

Three methods of sixth order convergence are tackled for approximating the solution of an equation defined on the finitely dimensional Euclidean space. This convergence requires the existence of derivatives of, at least, order seven. However, only derivatives of order one are involved in such methods. Moreover, we have no estimates on the error distances, conclusions about the uniqueness of the solution in any domain, and the convergence domain is not sufficiently large. Hence, these methods have limited usage. This paper introduces a new technique on a general Banach space setting based only the first derivative and Lipschitz type conditions that allow the study of the convergence. In addition, we find usable error distances as well as uniqueness of the solution. A comparison between the convergence balls of three methods, not possible to drive with the previous approaches, is also given. The technique is possible to use with methods available in literature improving, consequently, their applicability. Several numerical examples compare these methods and illustrate the convergence criteria.

Suggested Citation

  • Ramandeep Behl & Ioannis K. Argyros & Jose Antonio Tenreiro Machado, 2020. "Ball Comparison between Three Sixth Order Methods for Banach Space Valued Operators," Mathematics, MDPI, vol. 8(5), pages 1-12, April.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:5:p:667-:d:351293
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    References listed on IDEAS

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    1. Artidiello, Santiago & Cordero, Alicia & Torregrosa, Juan R. & Vassileva, Maria P., 2015. "Multidimensional generalization of iterative methods for solving nonlinear problems by means of weight-function procedure," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 1064-1071.
    2. Esmaeili, H. & Ahmadi, M., 2015. "An efficient three-step method to solve system of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 1093-1101.
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    Cited by:

    1. Ramandeep Behl & Ioannis K. Argyros & Fouad Othman Mallawi & Samaher Khalaf Alharbi, 2022. "Extending the Applicability of Highly Efficient Iterative Methods for Nonlinear Equations and Their Applications," Mathematics, MDPI, vol. 11(1), pages 1-18, December.

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