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Abstract and Applied Analysis
Volume 2015 (2015), Article ID 797594, 19 pages
http://dx.doi.org/10.1155/2015/797594
Research Article

Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

Institute of Computer Science, University of Silesia, Będzińska 39, 41-200 Sosnowiec, Poland

Received 17 December 2014; Accepted 5 February 2015

Academic Editor: Naseer Shahzad

Copyright © 2015 Krzysztof Gdawiec et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.