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New Improvement of the Domain of Parameters for Newton’s Method

Author

Listed:
  • Cristina Amorós

    (Escuela Superior de Ingeniería y Tecnología, UNIR, 26006 Logroño, Spain)

  • Ioannis K. Argyros

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

  • Daniel González

    (Escuela de Ciencias Físicas y Matemáticas, Universidad de las Americas, Quito 170517, Ecuador)

  • Ángel Alberto Magreñán

    (Departamento de Matemáticas y Computación, Universidad de la Rioja, 26004 Logroño, Spain)

  • Samundra Regmi

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

  • Íñigo Sarría

    (Escuela Superior de Ingeniería y Tecnología, UNIR, 26006 Logroño, Spain)

Abstract

There is a need to extend the convergence domain of iterative methods for computing a locally unique solution of Banach space valued operator equations. This is because the domain is small in general, limiting the applicability of the methods. The new idea involves the construction of a tighter set than the ones used before also containing the iterates leading to at least as tight Lipschitz parameters and consequently a finer local as well as a semi-local convergence analysis. We used Newton’s method to demonstrate our technique. However, our technique can be used to extend the applicability of other methods too in an analogous manner. In particular, the new information related to the location of the solution improves the one in previous studies. This work also includes numerical examples that validate the proven results.

Suggested Citation

  • Cristina Amorós & Ioannis K. Argyros & Daniel González & Ángel Alberto Magreñán & Samundra Regmi & Íñigo Sarría, 2020. "New Improvement of the Domain of Parameters for Newton’s Method," Mathematics, MDPI, vol. 8(1), pages 1-12, January.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:103-:d:306280
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    References listed on IDEAS

    as
    1. Cordero, Alicia & Gutiérrez, José M. & Magreñán, Á. Alberto & Torregrosa, Juan R., 2016. "Stability analysis of a parametric family of iterative methods for solving nonlinear models," Applied Mathematics and Computation, Elsevier, vol. 285(C), pages 26-40.
    2. Ioannis K. Argyros & Ángel Alberto Magreñán & Lara Orcos & Íñigo Sarría, 2019. "Unified Local Convergence for Newton’s Method and Uniqueness of the Solution of Equations under Generalized Conditions in a Banach Space," Mathematics, MDPI, vol. 7(5), pages 1-13, May.
    3. Lotfi, T. & Magreñán, Á.A. & Mahdiani, K. & Javier Rainer, J., 2015. "A variant of Steffensen–King’s type family with accelerated sixth-order convergence and high efficiency index: Dynamic study and approach," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 347-353.
    4. Ioannis K. Argyros & Á. Alberto Magreñán & Lara Orcos & Íñigo Sarría, 2019. "Advances in the Semilocal Convergence of Newton’s Method with Real-World Applications," Mathematics, MDPI, vol. 7(3), pages 1-12, March.
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    Cited by:

    1. Ioannis K. Argyros & Chirstopher Argyros & Michael Argyros & Johan Ceballos & Daniel González, 2022. "Extended Multi-Step Jarratt-like Schemes of High Order for Equations and Systems," Mathematics, MDPI, vol. 10(19), pages 1-9, October.

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