A fractional Traub method with $(2\alpha+1)$th-order of convergence and its stability

TL;DR

This paper introduces a new fractional Traub method with order $(2eta+1)$ for solving nonlinear equations, compares it with existing fractional Newton methods, and analyzes their stability and dependence on initial guesses.

Contribution

The paper presents a novel fractional Traub method with higher order of convergence and compares its performance and stability with fractional Newton methods.

Findings

The fractional Traub method achieves higher convergence order.

The methods' stability depends on initial estimations.

Comparative analysis shows the proposed method's effectiveness.

Abstract

Some fractional Newton methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper we introduce a fractional Newton method with order $\alpha+1$ and compare with another fractional Newton method with order $2\alpha$. We also introduce a fractional Traub method with order $2\alpha+1$ and compare with its first step (fractional Newton method with order $\alpha+1$). Some tests and analysis of the dependence on the initial estimations are made for each case.