Fractional Newton-Raphson Method

TL;DR

This paper introduces the Fractional Newton-Raphson method, an iterative approach based on fractional calculus, enabling the finding of both real and complex roots of polynomials from real initial conditions.

Contribution

The paper proposes a novel fractional calculus-based Newton-Raphson method that overcomes the divergence issue for complex roots with real initial guesses.

Findings

The method successfully finds complex roots from real initial conditions.

It extends the classical Newton-Raphson method to complex root spaces.

Demonstrates improved convergence properties for polynomials with complex roots.

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.