Fractional Newton-Raphson Method Accelerated with Aitken's 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.

Newton-Raphson Method, Fractional Calculus, Fractional Derivative of Riemann-Liouville, Method of Aitken. arXiv admin note: substantial text overlap with arXiv:1710.07634

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Cited by in corpus (6)

• An approximation to zeros of the Riemann zeta function using fractional calculus
• Sets of fractional operators and numerical estimation of the order of convergence of a family of fractional fixed point methods
• Introducción al Cálculo Fraccional
• Acceleration of the order of convergence of a family of fractional fixed point methods and its implementation in the solution of a nonlinear algebraic system related to hybrid solar receivers
• Proposal for Use the Fractional Derivative of Radial Functions in Interpolation Problems
• Numerical solution using radial basis functions for multidimensional fractional partial differential equations of type Black-Scholes