Fractional Newton-Raphson Method

Abstract

The Newton-Raphson (N-R) method is useful to find the roots of a polynomial of degree n, with n ∈ 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.

Sets of Fractional Operators and Numerical Estimation of the Order of Convergence of a Family of Fractional Fixed-Point Methods

“…which corresponds to the more general case of the fractional Newton-Raphson method [7,[16][17][18]. As a consequence, considering an iteration function Φ : (R \ Z) × R n → R n , the iteration function of a fractional iterative method may be written in general form as follows…”

“…Figure 1. Illustration of some lines generated by the fractional Newton-Raphson method for the same initial condition x 0 but with different orders α of the fractional operator implemented [16].…”

“…The initial condition does not necessarily have to be close to the sought values due to the non-local nature of fractional operators [5]. (ii) When working in a space of N dimensions, in the case that it is necessary to change the initial condition, unlike the classical iterative methods, where in the worst case, it is necessary to vary the N components of the initial condition until a suitable value is obtained; in the fractional fixed-point methods, it is enough to vary the parameter α of the fractional operators until an adequate value is found that allows generating a sequence that converges to a sought value [16].…”

“…To finish this section, it is necessary to mention that the applications of fractional operators have spread to different fields of science, such as finance [1,2], economics [3], number theory through the Riemann zeta function [4,5] and in engineering with the study of the manufacture of hybrid solar receivers [6,7]. It should be mentioned that there is also a growing interest in fractional operators and their properties for the solution of nonlinear algebraic systems [7][8][9][10][11][12][13][14][15][16][17][18], which is a classical problem in mathematics, physics and engineering that consists of finding the set of zeros of a function f :…”

Sets of fractional operators and numerical estimation of the order of convergence of a family of fractional fixed point methods

“…To finish this section, it is necessary to mention that the applications of fractional operators have spread to different fields of science such as finance [1,2], economics [3,4], number theory through the Riemann zeta function [5,6] and in engineering with the study for the manufacture of hybrid solar receivers [7][8][9][10]. It should be mentioned that there is also a growing interest in fractional operators and their properties for the solution of nonlinear systems [11][12][13][14][15][16], which is a classic problem in mathematics, physics and engineering, which consists of finding the set of zeros of a function…”