A composite fourth-order iterative method for solving non-linear equations

Certain improvements of Newton’s method with fourth-order convergence

“…The informational efficiency of the above methods is unity. The efficiency index of those methods is 1.4142 for (1) and (8) but I ¼ 1:442 for (34) and (9).…”

“…The fourthorder methods in the literatute usually require three evaluations of the given function and its first derivative per iteration, and it was shown that they can compete with Newton's method, see [1,9,13,16,20] and the references therein. Other than the above-mentioned methods, various types of improvements of Newton's method are available in the literature [6][7][8][9][10][11][12][13][14][15] and the references therein. Among these methods it is noteworthy to mention that the method of undetermined coefficients was successfully applied in [11] to show that many methods in the literature can be derived from each other, and so proving their equivalence.…”

“…Traub [1]) that this method is quadratically convergent. Some modifications of Newton's method to achieve higher order and better efficiency have been suggested and analyzed using several different techniques such as quadrature rules [2][3][4][5][6][7][8][9][10][11][12][13], decomposition [14,15] and homotopy techniques [16,17].…”

Solving Nonsmooth Equations Using Derivative-Free Methods

“…For a given x 0 , compute approximates solution by the iterative schemes Algorithm 2.1 has fourth-order convergence, which was obtained by [13]. Homeier [3] derived the following cubically convergent iteration scheme.…”

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.…”