A slight generalization of Steffensen Method for Solving Non Linear Equations

TL;DR

This paper introduces a modified Steffensen-based iterative method for solving nonlinear equations that converges quadratically without requiring derivative evaluations, offering advantages over Newton's and original Steffensen's methods.

Contribution

A slight modification of Steffensen's method is proposed, providing a derivative-free quadratic convergence approach for solving nonlinear equations.

Findings

The method converges quadratically.

It does not require derivative evaluations.

Numerical tests show advantages over Newton's and Steffensen's methods.

Abstract

In this article, we present an iterative method to find simple roots of nonlinear equations, that is, to solving an equation of the form $f(x) = 0$. Different from Newton's method, the method we purpose do not require evaluation of derivatives. The method is based on the classical Steffensen's method and it is a slight modification of it. The proofs of theoretical results are stated using Landau's Little o notation and simples concepts of Real Analysis. We prove that the method converges and its rate of convergence is quadratic. The method present some advantages when compared with Newton's and Steffesen's methods as ilustrated by numerical tests given.