Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fractional Newton-Raphson Method Accelerated with Aitken's Method

A. Torres-Hernandez, F. Brambila-Paz, U. Iturrarán-Viveros, R. Caballero-Cruz

TL;DR

The paper analyzes the convergence behavior of the Fractional Newton-Raphson (FNR) method, which uses a fractional Jacobian $f^{(\alpha)}$ defined via Riemann-Liouville derivatives to find zeros of $f$. It proves that the FNR method exhibits at least linear convergence for general $\alpha$ (with $\alpha\neq 1$) and can approach quadratic convergence as $\alpha \to 1$ under suitable derivative conditions, while also discussing a function-parameter approach to force improved convergence. It introduces the Aitken $\Delta^2$ acceleration and derives a Fractional Steffensen-like scheme $\Psi(\alpha,x)$ to accelerate convergence for arbitrary fractional iterations. The paper also surveys the RL and Caputo fractional derivatives, provides construction and examples, and demonstrates numerical results showing that combining FNR with Aitken acceleration yields substantially faster convergence and better robustness in locating multiple zeros. Overall, the work offers a practical framework for accelerating fractional-root-finding methods with clear guidance on when and how to apply Aitken acceleration to enhance performance in real and complex settings.

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 $α$ 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 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.

Fractional Newton-Raphson Method Accelerated with Aitken's Method

TL;DR

The paper analyzes the convergence behavior of the Fractional Newton-Raphson (FNR) method, which uses a fractional Jacobian defined via Riemann-Liouville derivatives to find zeros of . It proves that the FNR method exhibits at least linear convergence for general (with ) and can approach quadratic convergence as under suitable derivative conditions, while also discussing a function-parameter approach to force improved convergence. It introduces the Aitken acceleration and derives a Fractional Steffensen-like scheme to accelerate convergence for arbitrary fractional iterations. The paper also surveys the RL and Caputo fractional derivatives, provides construction and examples, and demonstrates numerical results showing that combining FNR with Aitken acceleration yields substantially faster convergence and better robustness in locating multiple zeros. Overall, the work offers a practical framework for accelerating fractional-root-finding methods with clear guidance on when and how to apply Aitken acceleration to enhance performance in real and complex settings.

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 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.

Paper Structure

This paper contains 13 sections, 10 theorems, 118 equations.

Table of Contents

  1. Fixed Point Method
  2. Order of Convergence
  3. Newton-Raphson Method
  4. Fractional Calculus
  5. Introduction to the Riemann-Liouville Fractional Derivative
  6. Construction of the Riemann-Liouville Fractional Derivative
  7. Examples of the Riemann-Liouville Fractional Derivative
  8. Introduction to the Caputo Fractional Derivative
  9. Fractional Newton-Raphson Method
  10. Convergence of the Fractional Newton-Raphson Method
  11. The Aitken's Method
  12. Results of the Fractional Newton-Raphson Method with the Aitken's Method
  13. Conclusions

Key Result

Corollary 1.4

Let $\Phi: \mathbb{R} ^n \to \mathbb{R} ^n$ be an iteration function. If $\Phi$ defines a sequence $\left\{x_i\right\} _{i=0}^\infty$ such that $x_i\to \xi$, and if the following condition is true then $\Phi$ has an order of convergence (at least) linear in $B(\xi;\delta)$.

Theorems & Definitions (24)

  • Definition 1.1
  • Definition 1.2
  • proof
  • Corollary 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Definition 3.1
  • ...and 14 more