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Local convergence of fourth and fifth order parametric family of iterative methods in Banach spaces

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Abstract

This paper deal with the study of local convergence of fourth and fifth order iterative method for solving nonlinear equations in Banach spaces. Only the premise that the first order Fréchet derivative fulfills the Lipschitz continuity condition is needed. Under these conditions, a convergence theorem is established to study the existence and uniqueness regions for the solution for each method. The efficacy of our convergence study is shown solving various numerical examples as a nonlinear integral equation and calculating the radius of the convergence balls. We compare the radii of convergence balls and observe that by our approach, we get much larger balls as existing ones. In addition, we also include the real and complex dynamic study of one of the methods applied to a generic polynomial of order two.

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Acknowledgements

Research supported in part by Séneca 20928/PI/18 and by MINECO PGC2018-095896-B-C21.

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Authors and Affiliations

  1. Department of Mathematics, Amrita School of Engineering, Amrita Vishwa vidyapeetham, Coimbatore, Tamilnadu, India

    P. Maroju

  2. Department of Mathematics, Universidad de La Rioja, Madre de Dios 53, 26006, Logroño, La Rioja, Spain

    Á. A. Magreñán

  3. Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja (UNIR), Avenida de La Paz 137, 26004, Logroño, La Rioja, Spain

    Í. Sarría

  4. Department of Mathematics, Lalit Narayan Mithila University, Darbhanga, India

    Abhimanyu Kumar

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  1. P. Maroju
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  2. Á. A. Magreñán
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  4. Abhimanyu Kumar
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Correspondence to Í. Sarría.

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Maroju, P., Magreñán, Á.A., Sarría, Í. et al. Local convergence of fourth and fifth order parametric family of iterative methods in Banach spaces. J Math Chem 58, 686–705 (2020). https://doi.org/10.1007/s10910-019-01097-y

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  • DOI: https://doi.org/10.1007/s10910-019-01097-y

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