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.

New Family of Multi-Step Iterative Methods Based on Homotopy Perturbation Technique for Solving Nonlinear Equations

“…Remark 2. The definition of efficiency index in [16] is analyzed, which is given as EI = m √ s, where s is the order of the iterative method and m is the number of function evaluations in each iteration. Using this definition, the efficiency indices of our suggested Algorithms 2 and 3 are 1.587 and 1.681, respectively.…”

“…, 5, and the approximated root α of each equation F i (x) = 0. The table also provides information about the initial guess x 0 , the number of iterations (NI), the value of |x i+1 − x i |, the absolute value of the function |F(x i+1 )|, and the computational order of convergence (COC) , which is described in [16] and given as the following:…”

“…An essential aspect of iterative methods is their convergence rate, which determines how quickly the method converges to the true solution. Numerous studies have focused on the convergence analysis of iterative methods for nonlinear equations, resulting in various convergence criteria that can be used to improve the efficiency of iterative methods and optimize their convergence rate [11,[15][16][17].…”