Abstract
In this paper two hybrids of two algorithms, the Newton method and the Gradient Descent method, are presented in order to create polynomiographs. The first idea combines two methods as a convex combination, the second one as the two-step process. A polynomiograph is an image obtained as the result of roots finding method of a polynomial on a complex plain. In this paper the polynomiographs are also modified by using Mann and Ishikawa iterations instead of the standard Picard iteration. Coloring images by iterations reveals dynamic behavior of the used root-finding process and its speed of convergence. The paper joins, combines and modifies earlier results obtained by Kalantari, Zhang et al. and the authors. We believe that the results of the paper can inspire those who are interested in aesthetic patterns created automatically. They can also be used to increase functionality of the existing polynomiography software.
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References
Bahgat Mohamed, S.M., Hafiz, M.A.: Three-step iterative method with eighteenth order convergence for solving nonlinear equations. Int. J. Pure Appl. Math. 93(1), 85–94 (2014)
Berinde, V.: Iterative Approximation of Fixed Points, 2nd edn. Springer, Heidelberg (2007)
Brandwood, D.H.: A complex gradient operator and its application in adaptive array theory. Proc. IEE Commun. Radar Sig. Process. 130(1), 11–16 (1983)
Cheer, J., Daley, S.: A method of adaptation between steepest–descent and Newton’s algorithm for multi–channel active control of tonal noise and vibration. In: Proceedings of the 21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13–17 July 2014
Gdawiec K., Kotarski, W., Lisowska, A.: Polynomiography based on the nonstandard Newton-like root finding methods. Abstr. Appl. Anal. Article ID 797594 (2015)
Gdawiec, K.: Fractal patterns from the dynamics of combined polynomial root finding methods. Nonlinear Dyn. 90, 1–23 (2017)
Johnson, D.: Optimization Theory, Optimization Theory Page from the Connexions Project. http://cnx.org/content/m11240/latest/
Kalantari, B.: Polynomiography: from the fundamental theorem of algebra to art. Leonardo 38(3), 233–238 (2005)
Kalantari, B.: Polynomial Root-Finding and Polynomiography. World Scientific, Singapore (2009)
Li, H., Adali, T.: Optimization in the complex domain for nonlinear adaptive filtering. In: Proceeding of the 40th Asilomar Conference on Signals, Systems, and Computers, pp. 263-267 (2006)
Wu, H., Li, F., Li, Z., Zhang, Y.: Zhang fractals yielded via solving nonlinear equations by discrete-time complex-valued ZD. In: Proceedings of the IEEE International Conference on Automation and Logistics Zhengzhou, China (2012)
Zhang, Y., Li, Z., Li, W., Chen, P.: From Newton fractals to Zhang fractals yielded via solving nonlinear equations in complex domain. Control Intell. Syst. 41(4), 131–137 (2013)
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Kotarski, W., Lisowska, A. (2018). Polynomiography via the Hybrids of Gradient Descent and Newton Methods with Mann and Ishikawa Iterations. In: Rocha, Á., Adeli, H., Reis, L., Costanzo, S. (eds) Trends and Advances in Information Systems and Technologies. WorldCIST'18 2018. Advances in Intelligent Systems and Computing, vol 746. Springer, Cham. https://doi.org/10.1007/978-3-319-77712-2_43
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DOI: https://doi.org/10.1007/978-3-319-77712-2_43
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