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 α of the derivative is different from one. A simplified way of constructing t...

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Bibliographic Details
Published in:Axioms 2021, Vol.10 (2), p.47
Main Authors: Torres-Hernandez, A., Brambila-Paz, F., Iturrarán-Viveros, U., Caballero-Cruz, R.
Format: Article
Language:English
Subjects:
Aitken’s method
Algebra
Convergence
Derivatives
Fractional calculus
fractional derivative
Integrals
Iterative methods
Newton-Raphson method
Operators (mathematics)
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Summary: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 α 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.
ISSN:2075-1680
2075-1680
DOI:10.3390/axioms10020047