An Efficient Three-step Iterative Methods Based on Bernstein Quadrature Formula for Solving Nonlinear Equations

Abstract

In this study, we suggest and analyze two new one-parameter families of an efficient iterative methods free from higher derivatives for solving nonlinear equations based on Newton theorem of calculus and Bernstein quadrature formula, Bernoulli polynomial basis, Taylor’s expansion and some numerical techniques. We prove that the new iterative methods reach orders of convergence ten with six and eight with four functional evaluations per iteration, which implies that the efficiency index of the new iterative methods is (10)1/6 1.4678 and (8)1/4 1.6818 respectively. Numerical examples are provided to show the efficiency and performance of our iterative methods, compare to Newton’s method and other relevant methods.