Fractional Newton-Raphson Method

TL;DR

This work extends the classical Newton-Raphson framework by incorporating fractional derivatives, enabling root finding for polynomials with complex roots from real initial conditions. By replacing the standard derivative with a fractional derivative (e.g., Riemann-Liouville), the Fractional Newton-Raphson method $x_{i+1}=x_i-(f^{(\alpha)}(x_i))^{-1}f(x_i)$ allows exploration of the complex plane and extraction of real and complex roots. The method yields at least linear convergence under a continuity assumption on the order $\alpha$, and its practical advantage lies in fixing $x_0$ while tuning $\alpha$ to reveal multiple roots, including complex conjugates. The results suggest broad potential for new fractional and hybrid iterative schemes that extend the reach of classical root-finding techniques.

Abstract

The Newton-Raphson (N-R) method is useful to find the roots of a polynomial of degree n. However, this method is limited since it diverges for the case in which polynomials only have complex roots if a real initial condition is taken. In the present work, we explain an iterative method that is created using the fractional calculus, which we will call the Fractional Newton-Raphson (F N-R) Method, which has the ability to enter the space of complex numbers given a real initial condition, which allows us to find both the real and complex roots of a polynomial unlike the classical Newton-Raphson method.

Figures (4)

• Figure 1: Illustration of the Newton-Raphson method.
• Figure 2: The R-L fractional derivatives of $f_0(x)$ and $f_1(x)$, with $\alpha\in[0,1]$.
• Figure 3: Illustration of some lines generated by the F N-R method.
• Figure 4: llustrations of some trajectories generated by the F N-R method for the same initial condition $x_0$ but with different values of $\alpha$.

Theorems & Definitions (11)

• Definition 1.1
• Definition 1.2
• proof
• Proposition 1.4
• proof
• Corollary 1.5
• Definition 2.1
• Proposition 2.2
• Example 3.1
• Example 3.2