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Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

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  • Krzysztof Gdawiec
  • Wiesław Kotarski
  • Agnieszka Lisowska

Abstract

A survey of some modifications based on the classic Newton’s and the higher order Newton-like root finding methods for complex polynomials is presented. Instead of the standard Picard’s iteration several different iteration processes, described in the literature, which we call nonstandard ones, are used. Kalantari’s visualizations of root finding process are interesting from at least three points of view: scientific, educational, and artistic. By combining different kinds of iterations, different convergence tests, and different colouring we obtain a great variety of polynomiographs. We also check experimentally that using complex parameters instead of real ones in multiparameter iterations do not destabilize the iteration process. Moreover, we obtain nice looking polynomiographs that are interesting from the artistic point of view. Real parts of the parameters alter symmetry, whereas imaginary ones cause asymmetric twisting of polynomiographs.

Suggested Citation

  • Krzysztof Gdawiec & Wiesław Kotarski & Agnieszka Lisowska, 2015. "Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods," Abstract and Applied Analysis, Hindawi, vol. 2015, pages 1-19, February.
  • Handle: RePEc:hin:jnlaaa:797594
    DOI: 10.1155/2015/797594
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    Cited by:

    1. Bisheh-Niasar, Morteza & Gdawiec, Krzysztof, 2019. "Bisheh-Niasar–Saadatmandi root finding method via the S-iteration with periodic parameters and its polynomiography," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 160(C), pages 1-12.
    2. Deepak Kumar & Sunil Kumar & Janak Raj Sharma & Lorentz Jantschi, 2021. "Convergence Analysis and Dynamical Nature of an Efficient Iterative Method in Banach Spaces," Mathematics, MDPI, vol. 9(19), pages 1-16, October.
    3. Lateef Olakunle Jolaoso & Safeer Hussain Khan & Kazeem Olalekan Aremu, 2022. "Dynamics of RK Iteration and Basic Family of Iterations for Polynomiography," Mathematics, MDPI, vol. 10(18), pages 1-16, September.
    4. Lateef Olakunle Jolaoso & Safeer Hussain Khan, 2020. "Some Escape Time Results for General Complex Polynomials and Biomorphs Generation by a New Iteration Process," Mathematics, MDPI, vol. 8(12), pages 1-18, December.

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