An Optimal Fourth Order Derivative-Free Numerical Algorithm for Multiple Roots

Abstract

A plethora of higher order iterative methods, involving derivatives in algorithms, are available in the literature for finding multiple roots. Contrary to this fact, the higher order methods without derivatives in the iteration are difficult to construct, and hence, such methods are almost non-existent. This motivated us to explore a derivative-free iterative scheme with optimal fourth order convergence. The applicability of the new scheme is shown by testing on different functions, which illustrates the excellent convergence. Moreover, the comparison of the performance shows that the new technique is a good competitor to existing optimal fourth order Newton-like techniques.

Memorizing Schröder’s Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity

“…There exist in the literature (see, for example, Reference [1][2][3][4][5][6][7][8]) numerous iterative methods without memory, involving or not derivatives, designed to estimate the multiple roots of a nonlinear equation f (x) = 0, but most of them need the knowledge of the multiplicity m of these roots.…”

Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

“…The major drawback of these schemes is the computation of the first-order derivative at each step, which consumes much time. To reduce this complexity, researchers [17][18][19][20][21][22] have worked on derivative-free schemes of multiple roots of scalar equations with the concept of the divided difference introduced by Traub-Steffensen [4]:…”

“…In 2020, Kumar et al [18] and Sharma et al [19][20][21] constructed derivative-free methods of second-order, fourth-order, and eighth-order convergence, respectively. Recently, Behl et al [6] proposed an optimal derivative-free Chebyshev-Halley family for multiple roots of a nonlinear equation.…”

On semilocal convergence of three-step Kurchatov method under weak condition

“…Ahmad et al [2] found the numerical solution of three-term time-fractional-order multi-dimensional diffusion equations by using an efficient local meshless method. Kumar et al [30] worked on fourth-order derivativefree iterative methods for finding a multiple root. Authors also shows the performance of the new technique is a good competitor to existing optimal fourth-order Newton-like techniques.…”