Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

Gdawiec, Krzysztof; Kotarski, W.; Lisowska, Agnieszka

doi:10.1155/2015/797594

Cited by 24 publications

(45 citation statements)

References 23 publications

(38 reference statements)

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“…The authors, basing on the papers on polynomiography [10,13] and inversion fractals [8], also think that further progress in biomorphs is possible by using different non-standard types of iterations known in the fixed point theory. Changing the real and/or complex values of parameters in different types of iterations should essentially enlarge the set of biomorphs.…”

Section: Discussionmentioning

confidence: 99%

Biomorphs via modified iterations

Gdawiec¹,

Kotarski²,

Lisowska³

2016

J. Nonlinear Sci. Appl.

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The aim of this paper is to present some modifications of the biomorphs generation algorithm introduced by Pickover in 1986. A biomorph stands for biological morphologies. It is obtained by a modified Julia set generation algorithm. The biomorph algorithm can be used in the creation of diverse and complicated forms resembling invertebrate organisms. In this paper the modifications of the biomorph algorithm in two directions are proposed. The first one uses different types of iterations (Picard, Mann, Ishikawa). The second one uses a sequence of parameters instead of one fixed parameter used in the original biomorph algorithm. Biomorphs generated by the modified algorithm are essentially different in comparison to those obtained by the standard biomorph algorithm, i.e., the algorithm with Picard iteration and one fixed constant.

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Section: Discussionmentioning

confidence: 99%

Biomorphs via modified iterations

Gdawiec¹,

Kotarski²,

Lisowska³

2016

J. Nonlinear Sci. Appl.

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“…This form of feedback iteration is also called the Picard iteration and it is used in many different algorithms, e.g., numerical polynomial root finding (Gdawiec et al, 2015). One of the most important applications of Picard's iteration is finding the fixed points of a contractive mapping using the Banach fixed point theorem.…”

Section: Iteration Processes and Dynamicsmentioning

confidence: 99%

“…In recent years, another interesting approach has been proposed, namely, the use of iteration processes from fixed point theory. The processes were successfully applied in the generation of generalised Mandelbrot and Julia sets (Ashish et al, 2014;Kang et al, 2015b), in inversion fractals (Gdawiec, 2017) or in polynomiography (patterns obtained from the polynomial root-finding process) (Gdawiec et al, 2015;Kang et al, 2015a).…”

Section: Introductionmentioning

confidence: 99%

Procedural generation of aesthetic patterns from dynamics and iteration processes

Gdawiec

2017

International Journal of Applied Mathematics and Computer Science

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Aesthetic patterns are widely used nowadays, e.g., in jewellery design, carpet design, as textures and patterns on wallpapers, etc. Most of the work during the design stage is carried out by a designer manually. Therefore, it is highly useful to develop methods for aesthetic pattern generation. In this paper, we present methods for generating aesthetic patterns using the dynamics of a discrete dynamical system. The presented methods are based on the use of various iteration processes from fixed point theory (Mann, S, Noor, etc.) and the application of an affine combination of these iterations. Moreover, we propose new convergence tests that enrich the obtained patterns. The proposed methods generate patterns in a procedural way and can be easily implemented on the GPU. The presented examples show that using the proposed methods we are able to obtain a variety of interesting patterns. Moreover, the numerical examples show that the use of the GPU implementation with shaders allows the generation of patterns in real time and the speed-up (compared with a CPU implementation) ranges from about 1000 to 2500 times.

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“…Recently, new methods of obtaining fractal patterns from root finding methods were presented. In the first method, the authors used different iteration methods known from fixed point theory [13,23,32]; then, in [11], a perturbation mapping was added to the feedback process. Finally, in [12] we can find the use of different switching processes.…”

Section: Introductionmentioning

confidence: 99%

Fractal patterns from the dynamics of combined polynomial root finding methods

Gdawiec

2017

Nonlinear Dyn

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Fractal patterns generated in the complex plane by root finding methods are well known in the literature. In the generation methods of these fractals, only one root finding method is used. In this paper, we propose the use of a combination of root finding methods in the generation of fractal patterns. We use three approaches to combine the methods: (1) the use of different combinations, e.g. affine and s-convex combination, (2) the use of iteration processes from fixed point theory, (3) multistep polynomiography. All the proposed approaches allow us to obtain new and diverse fractal patterns that can be used, for instance, as textile or ceramics patterns. Moreover, we study the proposed methods using five different measures: average number of iterations, convergence area index, generation time, fractal dimension and Wada measure. The computational experiments show that the dependence of the measures on the parameters used in the methods is in most cases a non-trivial, complex and non-monotonic function.

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Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

Cited by 24 publications

References 23 publications

Biomorphs via modified iterations

Biomorphs via modified iterations

Procedural generation of aesthetic patterns from dynamics and iteration processes

Fractal patterns from the dynamics of combined polynomial root finding methods

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