Some Iterative Methods Free from Derivatives and Their Basins of Attraction for Nonlinear Equations

Abstract

First, we make the Jain's derivative-free method optimal and subsequently increase its efficiency index from 1.442 to 1.587. Then, a novel three-step computational family of iterative schemes for solving single variable nonlinear equations is given. The schemes are free from derivative calculation per full iteration. The optimal family is constructed by applying the weight function approach alongside an approximation for the first derivative of the function in the last step in which the first two steps are the optimized derivative-free form of Jain's method. The convergence rate of the proposed optimal method and the optimal family is studied. The efficiency index for each method of the family is 1.682. The superiority of the proposed contributions is illustrated by solving numerical examples and comparing them with some of the existing methods in the literature. In the end, we provide the basins of attraction for some methods to observe the beauty of iterative nonlinear solvers in providing fractals and also choose the best method in case of larger attraction basins.

Some Matrix Iterations for Computing Matrix Sign Function

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“…Soleymani et al [11] presented derivative-free iterative methods without memory with convergence orders of eight and sixteen for solving nonlinear equations. Soleimani et al [12] proposed a optimal family of three-step iterative methods with a convergence order of eight by using a weight function alongside an approximation for the first derivative. Soleymani et al [13] gave a class of four-step iterative schemes for finding solutions of one-variable equations.…”

A Class of Optimal Eighth-Order Steffensen-Type Iterative Methods for Solving Nonlinear Equations and their Basins of Attraction

“…In order to compare different iterative methods of same order the classical efficiency index of an iterative process in [3] given by p 1 n , where p is the rate of convergence and n is the total number of functional evaluations per iteration. More recently, many researchers have focused to make existing iterative methods free from derivatives, interested researcher can follow [11]- [17] . In many of the science and engineering problem, the evaluation of derivative is difficult and time consuming.…”