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Visual Analysis of the Newton’s Method with Fractional Order Derivatives

Institute of Computer Science, University of Silesia, Bȩdzińska 39, 41-200 Sosnowiec, Poland
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Symmetry 2019, 11(9), 1143; https://doi.org/10.3390/sym11091143
Received: 28 July 2019 / Revised: 31 August 2019 / Accepted: 5 September 2019 / Published: 9 September 2019
The aim of this paper is to investigate experimentally and to present visually the dynamics of the processes in which in the standard Newton’s root-finding method the classic derivative is replaced by the fractional Riemann–Liouville or Caputo derivatives. These processes applied to polynomials on the complex plane produce images showing basins of attractions for polynomial zeros or images representing the number of iterations required to obtain polynomial roots. These latter images were called by Kalantari as polynomiographs. We use both: the colouring by roots to present basins of attractions, and the colouring by iterations that reveal the speed of convergence and dynamic properties of processes visualised by polynomiographs. View Full-Text
Keywords: fractional derivative; Newton method; root-finding; polynomiography fractional derivative; Newton method; root-finding; polynomiography
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Gdawiec, K.; Kotarski, W.; Lisowska, A. Visual Analysis of the Newton’s Method with Fractional Order Derivatives. Symmetry 2019, 11, 1143.

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