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Dynamic study of Schröder’s families of first and second kind

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Abstract

The Schröder iterative families of the first and second kind are of great importance in the theory and practice of iterative processes for solving nonlinear equations f(x) = 0. In both cases, the methods E r (first kind) and S r (second kind) converge locally to a zero α of f as O(|x k − α|r). Although characteristics of these families have been studied in many papers, their dynamic and chaotic behavior has not been completely investigated. In this paper, we compare convergence properties of both iterative schemes using the two methodologies: (i) comparison by numerical examples and (ii) comparison using dynamic study of methods by basins of attraction that enable their graphic visualization. Apart from the visualization of iterative processes, basins of attraction reveal very useful features on iterations such as consumed CPU time and average number of iterations, both as functions of starting points. We demonstrate by several examples that the Schröder family of the second kind S r possesses better convergence characteristics than the Schröder family of the first kind E r .

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Acknowledgements

This work was supported by the Serbian Ministry of Education and Science. The authors would like to thank Professor Beny Neta and anonymous referees for their valuable and constructive comments which helped to improve the manuscript.

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Authors and Affiliations

  1. Faculty of Electronic Engineering, University of Niš, 18000, Niš, Serbia

    Miodrag S. Petković

  2. Faculty of Mechanical Engineering, University of Niš, 18000, Niš, Serbia

    Ljiljana D. Petković

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  1. Miodrag S. Petković
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  2. Ljiljana D. Petković
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Correspondence to Miodrag S. Petković.

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Petković, M.S., Petković, L.D. Dynamic study of Schröder’s families of first and second kind. Numer Algor 78, 847–865 (2018). https://doi.org/10.1007/s11075-017-0403-0

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  • DOI: https://doi.org/10.1007/s11075-017-0403-0

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