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Dynamics of RK Iteration and Basic Family of Iterations for Polynomiography

Author

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  • Lateef Olakunle Jolaoso

    (Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 94, Medunsa 0204, South Africa
    School of Mathematical Sciences, University of Southampton, Highfield SO17 3AS, UK)

  • Safeer Hussain Khan

    (Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54000, Pakistan)

  • Kazeem Olalekan Aremu

    (Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 94, Medunsa 0204, South Africa
    Department of Mathematical Sciences, Usmanu Danfodiyo University, Sokoto 840004, Nigeria)

Abstract

In this paper, we propose some modifications of the basic family of iterations with a new four-step iteration called RK iteration and its s -convexity. We present some graphical examples showing the dynamics of the new iteration in the colouring and shapes of the obtained polynomiographs compared to the ones from the basic family only. Moreover, the computational results reveal that the value of s in the s -convex combination of the RK iteration has a significant impact on the time taken by the iteration process for approximating the roots of the polynomials. The obtained results are interesting from an artistic and computational point of view.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:18:p:3324-:d:914017
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. T. Lotfi & F. Soleymani & S. Sharifi & S. Shateyi & F. Khaksar Haghani, 2014. "Multipoint Iterative Methods for Finding All the Simple Zeros in an Interval," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-13, July.
    4. Gdawiec, Krzysztof & Kotarski, Wiesław, 2017. "Polynomiography for the polynomial infinity norm via Kalantari’s formula and nonstandard iterations," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 17-30.
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    Cited by:

    1. Tanveer, Muhammad & Nazeer, Waqas & Gdawiec, Krzysztof, 2023. "On the Mandelbrot set of zp+logct via the Mann and Picard–Mann iterations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 209(C), pages 184-204.

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