Fifth-order iterative method for finding multiple roots of nonlinear equations

Abstract

This paper presents a fifth-order iterative method as a new modification of Newton's method for finding multiple roots of nonlinear equations with unknown multiplicity m. Its convergence order is analyzed and proved. Moreover, several numerical examples demonstrate that the proposed iterative method is superior to the existing methods. Keywords Nonlinear equation · Multiple roots · Newton-like method · High-order convergence · Iterative methodsWe consider the iterative method to solve multiple roots x * of a nonlinear equation f 0 (x) = 0, where f 0 : [a, b] ⊂ R → R is a nonlinear differential function on [a, b]. Schröder [1] has modified Newton's method to obtain a multiple root when the multiplicity m is known

The rate of multiplicity of the roots of nonlinear equations and its application to iterative methods

“…Even higher order methods can be found [10]. There is an extensive literature about methods for multiple roots (for instance, [1,2,[8][9][10]15,19,20]).…”

“…There is an extensive literature about methods for multiple roots (for instance, [1,2,[8][9][10]15,19,20]). In general, when m is known, one step methods for multiple roots are formulated in this form: (5) σ (x; m) depends on f(x) and, eventually, its derivatives.…”

An Optimal Order Method for Multiple Roots in Case of Unknown Multiplicity

“…In this portion, we employ the proposed three-step method Equation (15) (MM8) for solving five nonlinear equations and scrutinize them by the methods given by Sharma and Bahl Equation (9) (MM6), in [24], and the Li et al Equation (8) (MM5), in [23]. Displayed in Tables 1-3 are the absolute error in the root, the absolute value of the function and the absolute error in the approximation of unknown multiplicity, for three iterations.…”

“…All of the computations were done by using Mathematica 8. We mention below five test equations along with their exact roots ξ (the functions, as well as initial guesses are taken from [23]): , m = 7, ξ = 2.1478....…”

“…To get a higher order method, Li et al [23] proposed an iterative scheme for approximating multiple roots of f (x) = 0 with fifth-order convergence, by using the transformation f (x) = ψ(x n )/ψ (x n ), which is given as:…”