Fractional Newton-Raphson Method Accelerated with Aitken's Method

In the following paper, we present a way to accelerate the speed 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 Aitken's method is made and it is explained why it has the ability to accelerate the convergence of the iterative methods, in order to finally present the results that were obtained when implementing Aitken's method in the F N-R method, where it is shown that F N-R with Aitken's method converges faster than the simple F N-R.

Automated summary

This paper introduces a method to accelerate the convergence speed of the fractional Newton-Raphson method, as well as how Aitken's method can speed up the convergence of iterative methods. Experimental results show that implementing Aitken's method in the F N-R method can lead to faster convergence compared to the simple F N-R method.