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On the local convergence of a third-order iterative scheme in Banach spaces

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Abstract

In a Banach space setting, we provide the local convergence analysis of a cubically convergent nonlinear system solver assuming that the first-order Fréchet derivative belongs to the Lipschitz class. The significance of this study is that it avoids the standard practice of Taylor expansion in the analysis of convergence and extends the applicability of the algorithm by using the theory based on the first-order derivative only. Also, our analysis provides the radii of convergence balls and computable error bounds along with the uniqueness of the solution. Furthermore, the generalization of this analysis using Hölder condition is studied. Different numerical tests confirm that the new technique produces better results, and it is useful in solving such problems where previous studies fail to solve.

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Acknowledgements

The first author would like to thank University Grants Commission of India for the financial support (ID: NOV2017-402662).

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  1. Department of Mathematics, International Institute of Information Technology Bhubaneswar, Bhubaneswar, Odisha, 751003, India

    Debasis Sharma & Sanjaya Kumar Parhi

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  1. Debasis Sharma
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  2. Sanjaya Kumar Parhi
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Correspondence to Debasis Sharma.

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Sharma, D., Parhi, S.K. On the local convergence of a third-order iterative scheme in Banach spaces. Rend. Circ. Mat. Palermo, II. Ser 70, 311–325 (2021). https://doi.org/10.1007/s12215-020-00500-x

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  • DOI: https://doi.org/10.1007/s12215-020-00500-x

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