An Optimal Thirty-Second-Order Iterative Method for Solving Nonlinear Equations and a Conjecture

Juan Luis Varona

doi:10.1007/s12346-022-00572-3

Abstract

Many multipoint iterative methods without memory for solving non-linear equations in one variable are found in the literature. In particular, there are methods that provide fourth-order, eighth-order or sixteenth-order convergence using only, respectively, three, four or five function evaluations per iteration step, thus supporting the Kung-Traub conjecture on the optimal order of convergence. This paper shows how to find optimal high order root-finding iterative methods by means of a general scheme based in weight functions. In particular, we explicitly give an optimal thirty-second-order iterative method; as long as we know, an iterative method with that order of convergence has not been described before. Finally, we give a conjecture about optimal order multipoint iterative methods with weights.

Highlights

Introduction and Main ResultsSolving nonlinear equations is a basic and extremely valuable tool in all fields of science and engineering
Given a function f : D ⊂ C → C defined on a region D in the complex plane C, one of the most common methods for finding simple roots x∗ of a nonlinear equation f (x) = 0 is Newton’s method which, starting at an initial guess x0, iterates by means of xn+1 = N f := xn −
When x∗ is unknown, which is usual in practice, we can use the approximated computational order of convergence (ACOC), defined in [9], as ln |(xn+1 − xn)/(xn − xn−1)| ln |(xn − xn−1)/(xn−1 − xn−2)|