Fractals via Ishikawa Iteration

Prasad, Bhagwati; Katiyar, Kuldip

doi:10.1007/978-3-642-19263-0_24

Cited by 25 publications

(14 citation statements)

References 11 publications

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“…This scheme is widely studied by P r a s a d and K a t i y a r [23,25] and interesting fractal patterns are generated in [24] using it. The following definition is motivated by Rani and Agarwal [26].…”

Section: Preliminariesmentioning

confidence: 99%

Stability and Fractal Patterns of Complex Logistic Map

Prasad

Katiyar

2014

Cybernetics and Information Technologies

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Section: Preliminariesmentioning

confidence: 99%

Stability and Fractal Patterns of Complex Logistic Map

Prasad

Katiyar

2014

Cybernetics and Information Technologies

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“…Usually, polynomiographs are colored based on the number of iterations needed to obtain the approximation of some polynomial roots with a given accuracy and a chosen iteration method. The description of polynomiography, its theoretical background and artistic applications are described in [5,6,7,9,10,12,13,15,16].…”

Section: Introductionmentioning

confidence: 99%

“…Kang et al [9], using the ideas from [10], have used the S-iteration in polynomiography. Earlier, the other types of iterations have been used in [15] for superfractals and in [13] for fractals generated by iterative function systems. Julia sets and Mandelbrot sets [1] and the antifractals [14] have been also investigated using Noor iteration instead of the standard Picard iteration.…”

Section: Introductionmentioning

confidence: 99%

Polynomiography via an iterative method corresponding to Simpsons 13 rule

Kang¹,

Ramay²,

Tanveer³

et al. 2016

J. Nonlinear Sci. Appl.

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The aim of this paper is to present some artwork produced via polynomiography of a few complex polynomials and a few special polynomials arising in science as well as a few considered to arrive at beautiful but anticipated designs. In this paper an iterative method corresponding to Simpson's 1 3 rule is used instead of Newton's method. The word "polynomiography" coined by Kalantari for that visualization process. The images obtained are called polynomiographs. Polynomiographs have importance for both the art and science aspects. By using an iterative method corresponding to Simpson's 1 3 rule, we obtain quite new nicely looking polynomiographs that are different from Newton's method. Presented examples show that we obtain very interesting patterns for complex polynomial equations, permutation matrices, doubly stochastic matrices, Chebyshev polynomial, polynomial arising in physics and Alexander polynomial in knot theory. We believe that the results of this paper enrich the functionality of the existing polynomiography software. c 2016 All rights reserved.

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“…Latif et al in [6], using the ideas from [5], have used the -iteration in polynomiography. Earlier, the other types of iterations have been used in [7] for superfractals and in [8] for fractals generated by IFS. Julia sets and Mandelbrot sets [9] and the antifractals [10] have been also investigated using Noor iteration instead of the standard Picard iteration.…”

Section: Introductionmentioning

confidence: 99%

Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

Gdawiec

Kotarski

Lisowska

2015

Abstract and Applied Analysis

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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.

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Fractals via Ishikawa Iteration

Cited by 25 publications

References 11 publications

Stability and Fractal Patterns of Complex Logistic Map

Stability and Fractal Patterns of Complex Logistic Map

Polynomiography via an iterative method corresponding to Simpsons 13 rule

Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

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