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Fractional Newton-Raphson Method and Some Variants for the Solution of Non-linear Systems

A. Torres-Hernandez, F. Brambila-Paz

TL;DR

This work extends Newton-Raphson for nonlinear systems by introducing fractional-derivative-based updates with variable order $\alpha$ to enable complex roots from real initial data. It presents three methods: Fractional Newton-Raphson, Fractional Quasi-Newton, and Fractional Pseudo-Newton, each with distinct computational characteristics and convergence behavior. The Fractional Newton variant can achieve (at least) quadratic convergence but requires handling a fractional Jacobian; the Quasi-Newton and Pseudo-Newton variants ease computation at the cost of at most linear convergence, trading speed for simplicity. Collectively, these methods broaden the toolkit for locating multiple zeros (real and complex) of nonlinear systems, particularly when infinite or numerous roots are present, albeit with tradeoffs in order of convergence and implementation effort.

Abstract

The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow to find solutions for some non-linear systems in the complex space using real initial conditions. The origin of these methods is the fractional Newton-Raphson method but unlike the latter, the orders of fractional derivatives proposed here are functions. In the first method, a function is used to guarantee an order of convergence (at least) quadratic, and in the others, a function is used to avoid the discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible that the methods have at most an order of convergence (at least) linear.

Fractional Newton-Raphson Method and Some Variants for the Solution of Non-linear Systems

TL;DR

This work extends Newton-Raphson for nonlinear systems by introducing fractional-derivative-based updates with variable order to enable complex roots from real initial data. It presents three methods: Fractional Newton-Raphson, Fractional Quasi-Newton, and Fractional Pseudo-Newton, each with distinct computational characteristics and convergence behavior. The Fractional Newton variant can achieve (at least) quadratic convergence but requires handling a fractional Jacobian; the Quasi-Newton and Pseudo-Newton variants ease computation at the cost of at most linear convergence, trading speed for simplicity. Collectively, these methods broaden the toolkit for locating multiple zeros (real and complex) of nonlinear systems, particularly when infinite or numerous roots are present, albeit with tradeoffs in order of convergence and implementation effort.

Abstract

The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow to find solutions for some non-linear systems in the complex space using real initial conditions. The origin of these methods is the fractional Newton-Raphson method but unlike the latter, the orders of fractional derivatives proposed here are functions. In the first method, a function is used to guarantee an order of convergence (at least) quadratic, and in the others, a function is used to avoid the discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible that the methods have at most an order of convergence (at least) linear.

Paper Structure

This paper contains 11 sections, 2 theorems, 85 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Fixed Point Method
  3. Convergence and Order of Convergence
  4. Newton-Raphson Method
  5. Basic Definitions of the Fractional Derivative
  6. Introduction to the Definition of Riemann-Liouville
  7. Introduction to the Definition of Caputo
  8. Fractional Newton Method
  9. Fractional Quasi-Newton Method
  10. Fractional Pseudo-Newton Method
  11. Conclusions

Key Result

Proposition 1.6

Let $f: \Omega \subset \mathbb{R} \to \mathbb{R}$ be a function that is at least twice differentiable in $\xi \in \Omega$. So if $\xi$ is a zero of $f$ with algebraic multiplicity $m$ ($m \geq 2$), i.e., the Newton-Raphson method (one-dimensional) has an order of convergence (at least) linear.

Figures (2)

  • Figure 1: Illustration of some lines generated by the fractional Newton method, the red line corresponds to the case $\alpha = 1$.
  • Figure 2: llustrations of some trajectories generated by the fractional Newton method for the same initial condition $x_0$ but with different values of $\alpha$.

Theorems & Definitions (15)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.4
  • proof
  • Proposition 1.6
  • proof
  • Definition 2.1
  • Proposition 2.2
  • Example 3.1
  • Example 3.2
  • ...and 5 more