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

Abstract

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. To compare numerically the modified root-finding methods among them, we demonstrate their action for polynomial z3 − 1 on a complex plane.