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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, John Wiley & Sons, vol. 2015(1).
    • Handle: RePEc:wly:jnlaaa:v:2015:y:2015:i:1:n:797594
    DOI: 10.1155/2015/797594
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    References listed on IDEAS

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    1. 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.
    2. 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, John Wiley & Sons, vol. 2014(1).
    3. Vatan Karakaya & Kadri Doğan & Faik Gürsoy & Müzeyyen Ertürk, 2013. "Fixed Point of a New Three‐Step Iteration Algorithm under Contractive‐Like Operators over Normed Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    4. Vatan Karakaya & Kadri Doğan & Faik Gürsoy & Müzeyyen Ertürk, 2013. "Fixed Point of a New Three-Step Iteration Algorithm under Contractive-Like Operators over Normed Spaces," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-9, December.
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