Polynomiography via Ishikawa and Mann Iterations

The aim of this paper is to present some modifications of complex polynomial roots finding visualization process. In this paper Ishikawa or Mann iterations are used instead of the standard Picard iteration. Kalantari introduced the name polynomiography for that visualization process and the obtained images he called polynomiographs. Polynomiographs are interesting both from educational and artistic point of view. By the use of different iterations we obtain quite new polynomiographs that look aestheatically pleasing comparing to the ones from standard Picard iteration. As examples we present some polynomiographs for complex polynomial equation z^3 - 1 = 0, permutation and doubly stochastic matrices. We believe that the results of this paper can inspire those who may be interested in aesthetic patterns created automatically. They also can be used to increase functionality of the existing polynomiography software

Polynomiography Based on the Non-standard Newton-like Root Finding Methods

In this paper a survey of some modifications based on the classic Newton's and the higher order Newton-like root finding methods for complex polynomials are presented. Instead of the standard Picard's iteration several different iteration processes, described in the literature, that we call as non-standard 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 multi-parameter iterations do not destabilize the iteration process. Moreover, we obtain nicely 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

Newton’s method with fractional derivatives and various iteration processes via visual analysis

The aim of this paper is to visually investigate the dynamics and stability of the process in which the classic derivative is replaced by the fractional Riemann–Liouville or Caputo derivatives in the standard Newton root-finding method. Additionally, instead of the standard Picard iteration, the Mann, Khan, Ishikawa and S iterations are used. This process when applied to polynomials on complex plane produces images showing basins of attractions for polynomial zeros or images representing the number of iterations required to achieve any polynomial root. The images are called polynomiographs. In this paper, we use the colouring according to the number of iterations which reveals the speed of convergence and dynamic properties of processes visualised by polynomiographs.Moreover, to investigate the stability of the methods, we use basins of attraction

Polynomiography for the Polynomial Infinity Norm via Kalantari's Formula and Nonstandard Iterations

In this paper, an iteration process, referred to in short as MMP, will be considered. This iteration is related to finding the maximum modulus of a complex polynomial over a unit disc on the complex plane creating intriguing images. Kalantari calls these images polynomiographs independently from whether they are generated by the root finding or maximum modulus finding process applied to any polynomial. We show that the images can be easily modified using different MMP methods (pseudo-Newton, MMP-Householder, methods from the MMP-Basic, MMP-Parametric Basic or MMP-Euler-Schroder Families of Iterations) with various kinds of non-standard iterations. Such images are interesting from three points of views: scientific, educational and artistic. We present the results of experiments showing automatically generated non-trivial images obtained for different modifications of root finding MMP-methods. The colouring by iteration reveals the dynamic behaviour of the used root finding process and its speed of convergence. The results of the present paper extend Kalantari's recent results in finding the maximum modulus of a complex polynomial based on Newton's process with the Picard iteration to other MMP-processes with various non-standard iterations

One more look on visualization of operation of a root-finding algorithm

Many algorithms that iteratively find solution of an equation require tuning. Due to the complex dependence of many algorithm’s elements, it is difficult to know their impact on the work of the algorithm. The article presents a simple root-finding algorithm with self-adaptation that requires tuning, similarly to evolutionary algorithms. Moreover, the use of various iteration processes instead of the standard Picard iteration is presented. In the algorithm’s analysis, visualizations of the dynamics were used. The conducted experiments and the discussion regarding their results allow to understand the influence of tuning on the proposed algorithm. The understanding of the tuning mechanisms can be helpful in using other evolutionary algorithms. Moreover, the presented visualizations show intriguing patterns of potential artistic applications

Mandelbrot- and Julia-like Rendering of Polynomiographs

Polynomiography is a method of visualization of complex polynomial root finding process. One of the applications of polynomiography is generation of aesthetic patterns. In this paper, we present two new algorithms for polynomiograph rendering that allow to obtain new diverse patterns. The algorithms are based on the ideas used to render the well known Mandelbrot and Julia sets. The results obtained with the proposed algorithms can enrich the functionality of the existing polynomiography software

Visual Analysis of Dynamics Behaviour of an Iterative Method Depending on Selected Parameters and Modifications

There is a huge group of algorithms described in the literature that iteratively find solutions of a given equation. Most of them require tuning. The article presents root-finding algorithms that are based on the Newton-Raphson method which iteratively finds the solutions, and require tuning. The modification of the algorithm implements the best position of particle similarly to the particle swarm optimisation algorithms. The proposed approach allows visualising the impact of the algorithm's elements on the complex behaviour of the algorithm. Moreover, instead of the standard Picard iteration, various feedback iteration processes are used in this research. Presented examples and the conducted discussion on the algorithm's operation allow to understand the influence of the proposed modifications on the algorithm's behaviour. Understanding the impact of the proposed modification on the algorithm's operation can be helpful in using it in other algorithms. The obtained images also have potential artistic applications

Biomorphs via Modified Iterations

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

Higher Order Methods of the Basic Family of Iterations via S-Iteration Scheme with s-Convexity

There are many methods for solving a polynomial equation and many different modifications of those methods have been proposed in the literature. One of such modifications is the use of various iteration processes taken from the fixed point theory. In this paper, we propose a modification of the iteration processes used in the Basic Family of iterations by replacing the convex combination with an s-convex one. In our study, we concentrate only on the S-iteration with s-convexity. We present some graphical examples, the so-called polynomiographs, and numerical experiments showing the dependency of polynomiograph’s generation time on the value of the s parameter in the s-convex combination