Polynomiography for the polynomial infinity norm via Kalantari’s formula and nonstandard iterations

Abstract

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-Schröder 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.

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

“…Because the root finding process can be equivalently transformed into a fixed point problem [6] in recent years, researchers have studied the use of various iteration processes-known in fixed point theory-that are used to find fixed points in the root finding methods. Gdawiec et al in [12] proposed the use of ten different iteration methods, e.g., SP, Noor, Picard-S. Later, in [11], Gdawiec and Kotarski extended the list of iterations to seventeen different iterations. They also studied the dependencies between the iterations.…”

“…In [21], Kang et al used the S-iteration in Newton's method to obtain a variety of different polynomiographs. The iteration methods used in [11,12,21] were all explicit iteration schemes. Rafiq et al in [28] proposed the use of some implicit schemes in root finding, namely the Jungck, the Jungck-Mann, and Jungck-Ishikawa iterations.…”

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

“…In [4,5], the authors presented a survey of some modifications of Kalantari's results based on the classic Newton and the higher-order Newton-like root-finding methods and pseudo-methods for complex polynomials. The standard Picard iteration was replaced there by several different iteration processes.…”

Fixed Point Results for Fractal Generation in Extended Jungck–SP Orbit

“…After studies of Rani et al the use of various iteration procedures (used for an approximate finding of fixed points) in the generation of different types of fractals became very popular. Some types of fractals like Mandelbrot and Julia sets via different explicit iterations studied and generalized in [14]- [18] and [19], Iterated Function System fractals studied in [20] and [21], V-variable fractals and super-fractals demonstrated in [21] and [22], inversion fractals discussed in [23] and fractals arising from the root finding methods presented in [24]- [26] and [27]. The threshold escape radii for Jungck-Mann, Jungck-Ishikawa and Jungck-Noor iterations with the combination of s-convex function in second sense were proved in [28], [29].…”