A Class of Three‐Step Derivative‐Free Root Solvers with Optimal Convergence Order

Abstract

A class of three-step eighth-order root solvers is constructed in this study. Our aim is fulfilled by using an interpolatory rational function in the third step of a three-step cycle. Each method of the class reaches the optimal efficiency index according to the Kung-Traub conjecture concerning multipoint iterative methods without memory. Moreover, the class is free from derivative calculation per full iteration, which is important in engineering problems. One method of the class is established analytically. To test the derived methods from the class, we apply them to a lot of nonlinear scalar equations. Numerical examples suggest that the novel class of derivative-free methods is better than the existing methods of the same type in the literature.

On the construction of three step derivative free four-parametric methods with accelerated order of convergence

“…The proof would be similar to those already considered in [9,13,15,16,20], using the Taylor expansions of the function f in the mth iterative step. Hence, it is omitted.…”

“…A lot of derivative free without memory optimal root finding methods have been developed in recent years, for example (see [3,9,13,15,16,20,21]). Sometimes it is not possible to improve the convergence order and the efficiency index of without memory methods without additional functional evaluations based on free parameters [3].…”

On improved three-step schemes with high efficiency index and their dynamics

“…In this section, we check the effectiveness of the iterative class (2) by choosing their members (11), (13), and (15). Due to this, we have compared them with the following scheme, described in…”

“…The fractals are almost similar and the number of diverging points are high for all methods. Method (15) has the lowest number of diverging points whereas method (19) has the lowest mean iteration number of the converging points. On the whole, we see that our methods are better than the other methods in the literature.…”

“…To improve the local order of convergence different techniques have been used getting iterative schemes with and without memory (see, for instance [5][6][7]) and to upgrade the efficiency index, many optimal methods without memory have been proposed; see, for example, [3,[8][9][10][11][12][13][14][15][16], and the references therein.…”