Development of a robust iterative algorithm for contractive-like mapping in hyperbolic space: application to computational fractal imaging of polynomiographs
Submitted: 2025-09-18
|Accepted: 2026-05-26
|Published: 2026-06-22
Copyright (c) 2026 J. Grace Margrate Mary, R. Uthayakumar
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
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Keywords:
hyperbolic space, fixed-point convergence, stability analysis, Picard-new iteration process, polynomiographs, fractal images
Supporting agencies:
Abstract:
The primary objective is to establish the fixed-point convergence analysis of our newly proposed iteration process, within the framework of hyperbolic spaces, particularly in the presence of contractive-like mapping and further investigate its stability characteristics. To demonstrate its effectiveness, we present a numerical example that highlights the superiority of our iteration process in terms of dominance and faster convergence compared to existing iterative methods. Furthermore, we explore a practical application of our iteration process by employing polynomiographs, which are generated using traditional iteration schemes from the literature and visualized through fractal images. The scheme is implemented computationally using MATLAB, where a variety of visual outputs are produced to illustrate the structure and behavior of the convergence zones. We then compare these results with polynomiographs produced using our novel Picard-New iteration process, demonstrating its advantages in generating fractal images more effectively.
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