Abstract
In this work we provide analytic and graphic arguments to explain the behaviour of Chebyshev’s method applied to cubic polynomials in the complex plane. In particular, we study the parameter plane related to this method and we compare it with other previously known, such as Newton’s or Halley’s methods. Our specific interest is to characterize “bad” polynomials for which Chebyshev’s method presents convergence to points distinct from the roots (i.e. the root-finding algorithm fails). In particular, we prove the existence of polynomials for which Chebyshev’s method has superattracting n-cycles and the existence of polynomials for which Chebyshev’s method has superattracting extraneous fixed points. The first fact is shared with other root-finding methods, such as Newton’s or Halley’s, but the second one is an established dynamic feature of Chebyshev’s method. Here we go depth on the study of the dynamics related to the superattracting n-cycles of Chebyshev’s method and its relationships with the superattracting extraneous fixed points, providing some analytical and geometric arguments to explain the related parameter plane. In particular, we prove the existence of a sequence of parameters for which the corresponding Chebyshev’s iterative method has a superattracting n-cycle.
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Acknowledgements
This research was partially supported by Ministerio de Ciencia, Innovación y Universidades under Grants PGC2018-095896-B-C21 and PGC2018-096504-B-C32. The authors want to acknowledge the helpful, constructive and detailed revision of the manuscript by the referees, that has allowed us to improve the final version of the paper.
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Gutiérrez, J.M., Varona, J.L. Superattracting Extraneous Fixed Points and n-cycles for Chebyshev’s Method on Cubic Polynomials. Qual. Theory Dyn. Syst. 19, 54 (2020). https://doi.org/10.1007/s12346-020-00390-5
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DOI: https://doi.org/10.1007/s12346-020-00390-5