Some efficient fourth order multipoint methods for solving equations

A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics

“…In general, multipoint iterative methods [22,23] finding a zero α of a nonlinear equation f (x) = 0 can be written as Investigation of such dynamics clearly motivates our current analysis, which enables us to write the proposed method (2.1) in the following form:…”

Construction of fourth-order optimal families of iterative methods and their dynamics

“…From equations (2.4) and (2.5), we get 6) and in combination with Taylor series expansion of f x n − f (xn)…”

“…As fixed points satisfy O p (z, c) = z, it can be checked that z = 0 and z = ∞, associated to the roots of p(z), are fixed points and also the strange fixed points z = 1 and the roots of polynomial 6+(42+18α)z + (126 + 90α + 18α 2 )z 2 + (180 + 156α + 48α 2 + 6α 3 − α 3 β)z 3 + (126 + 90α + 18α 2 )z 4 + (42 + 18α)z 5 + 6z 6 , that will be called s i (α, β), i = 1, 2, . .…”

“…On the other hand, Jarratt [5,6], in 1966 and King [7], in 1975, proposed various optimal fourth-order multi-point methods. King further showed that Ostrowski's method was a particular case of his family.…”

A Variant of Sharma-Arora's Optimal Eighth-Order Family of Methods for Finding A Simple Root of Nonlinear Equation

“…In [9] Kung and Traub conjectured that the iterative method which requires n + 1 function evaluations per iteration can reach at most 2 n convergence order in general. The methods that satisfy KungTraub conjecture are known as optimal methods (see [10], [11], [12], [13], [14], [15], [16], [14], [18], [19]). This research presents a new family of such optimal eighth order methods.…”