Some fourth order multipoint iterative methods for solving equations

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

“…If M 4 (x n , w n ) is some optimal fourth-order iterative scheme, then the error of the second step can be written in the form z n −α = A 4 e 4 n +A 5 e 5 n +A 6 e 6 n +A 7 e 7 n +A 8 e 8 n +O(e 9 n ), (10) for some constants A i , i ∈ {4, 5, 6, 7, 8}. By substituting (10) into (6), it is easy to note that…”

“…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.…”

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

“…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.…”

Efficient high-order methods based on golden ratio for nonlinear systems

“…Some methods existing in the literature are based on the use of interpolation quadrature formulas (see [1][2][3][4]), or include the second partial derivative of the function F or different estimations of it (see [5][6][7][8]), or are Steffensen's type methods (see [9]), etc. We also pay attention to the known Jarratt's method (J) (see [10]) whose efficiency is widely recognized. As the order of convergence of this method is four we will compare it with the new methods in the numerical section.…”