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

Abstract

In this paper, we propose a general class of fourth-order optimal multi-point methods without memory for obtaining simple roots. This class requires only three functional evaluations (viz. two evaluations of function f (x n ), f (y n ) and one of its first-order derivative f (x n )) per iteration. Further, we show that the well-known Ostrowski's method and King's family of fourth-order procedures are special cases of our proposed schemes. One of the new particular subclasses is a biparametric family of iterative methods. By using complex dynamics tools, its stability is analyzed, showing stable members of the family. Further on, one of the parameters is fixed and the stability of the resulting class is studied. On the other hand, the accuracy and validity of new schemes is tested by a number of numerical examples by comparing them with recent and classical optimal fourth-order methods available in the literature. It is found that they are very useful in high precision computations.

Sixteenth-Order Optimal Iterative Scheme Based on Inverse Interpolatory Rational Function for Nonlinear Equations

“…Additionally, we want to check what the outcomes will be if we assume different numerical examples and staring guesses that are not suggested in their articles. Therefore, we assume another numerical example from Behl et al [27]. For the detailed information of the considered examples or test functions, please see Table 4.…”

An Optimal Family of Eighth-Order Iterative Methods With an Inverse Interpolatory Rational Function Error Corrector for Nonlinear Equations

“…, (ii) Next, we shall choose another optimal family of fourth-order methods proposed by Behl et al in [2]. Then, we further yield another new optimal family of eighth-order methods, which is given by…”

Generating Function Approach to the Derivation of Higher-Order Iterative Methods for Solving Nonlinear Equations

“…We note that this general class of optimal fourth-order iterations is also included in our methods with or ω = 1 − β 6 α 3 + α 2 − 2α, then the iterations (1) with τ n = H(θ n ) given by (8) reduce to (3.8) and (3.10) in [12], respectively. This shows that our class of optimal fourth-order methods is wider than that of [12]. So, we have obtained an optimal fourth-order convergence family of iterative methods with three degrees of freedom based on the generating function method.…”

“…Recently, Behl et al [12] proposed a general class of fourthorder optimal methods that includes the well-known Ostrowski's and King's family as special cases. We note that this general class of optimal fourth-order iterations is also included in our methods with or ω = 1 − β 6 α 3 + α 2 − 2α, then the iterations (1) with τ n = H(θ n ) given by (8) reduce to (3.8) and (3.10) in [12], respectively. This shows that our class of optimal fourth-order methods is wider than that of [12].…”