Fractional Newton-Raphson Method Accelerated with Aitken's Method

TL;DR

This paper analyzes the convergence of the Fractional Newton-Raphson method, introduces a simplified fractional operator construction, and demonstrates how Aitken's method accelerates its convergence.

Contribution

It provides a new analysis of the fractional Newton-Raphson method's convergence order and combines it with Aitken's acceleration technique for improved performance.

Findings

Fractional Newton-Raphson has at least linear convergence for non-one fractional orders.

A simplified construction of Riemann-Liouville fractional operators is proposed.

Aitken's method effectively accelerates the fractional Newton-Raphson method.

Abstract

In the following document, we present a way to obtain the order of convergence of the Fractional Newton-Raphson (F N-R) method, which seems to have an order of convergence at least linearly for the case in which the order $\alpha$ of the derivative is different from one. A simplified way of constructing the Riemann-Liouville (R-L) fractional operators, fractional integral and fractional derivative, is presented along with examples of its application on different functions. Furthermore, an introduction to the Aitken's method is made and it is explained why it has the ability to accelerate the convergence of the iterative methods, to finally present the results that were obtained when implementing the Aitken's method in the F N-R method.