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Unified Local Convergence for Newton’s Method and Uniqueness of the Solution of Equations under Generalized Conditions in a Banach Space

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  • Ioannis K. Argyros

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

  • Ángel Alberto Magreñán

    (Departamento de Matemáticas y Computación, Universidad de La Rioja, 26006 Logroño, Spain)

  • Lara Orcos

    (Facultad de Educación, Universidad Internacional de La Rioja, 26006 Logroño, Spain)

  • Íñigo Sarría

    (Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, 26006 Logroño, Spain)

Abstract

Under the hypotheses that a function and its Fréchet derivative satisfy some generalized Newton–Mysovskii conditions, precise estimates on the radii of the convergence balls of Newton’s method, and of the uniqueness ball for the solution of the equations, are given for Banach space-valued operators. Some of the existing results are improved with the advantages of larger convergence region, tighter error estimates on the distances involved, and at-least-as-precise information on the location of the solution. These advantages are obtained using the same functions and Lipschitz constants as in earlier studies. Numerical examples are used to test the theoretical results.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:5:p:463-:d:233774
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    References listed on IDEAS

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    1. Argyros, Ioannis K. & Magreñán, Á. Alberto, 2015. "On the convergence of an optimal fourth-order family of methods and its dynamics," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 336-346.
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    Cited by:

    1. 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.

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