A new method for estimating the real roots of real differentiable functions

TL;DR

This paper introduces a novel iterative method for finding real roots of differentiable functions that guarantees convergence to the nearest root without requiring derivative evaluations, improving efficiency and reliability.

Contribution

The paper presents a new root-finding method based on relaxed Krasnoselskii's theorem, ensuring convergence to the nearest root without derivative calculations, which is a significant advancement.

Findings

Method guarantees convergence to the nearest root within an interval.

No derivative evaluation needed, enhancing computational efficiency.

Numerical examples demonstrate the method's effectiveness.

Abstract

We introduce a new type of Krasnoselskii's result. Using a simple differentiability condition, we relax the nonexpansive condition in Krasnoselskii's theorem. More clearly, we analyze the convergence of the sequence $x_{n+1}=\frac{x_n+g(x_n)}{2}$ based on some differentiability condition of $g$ and present some fixed point results. We introduce some iterative sequences that for any real differentiable function $g$ and any starting point $x_0\in \mathbb [a,b]$ converge monotonically to the nearest root of $g$ in $[a,b]$ that lay to the right or left side of $x_0$. Based on this approach, we present an efficient and novel method for finding the real roots of real functions. We prove that no root will be missed in our method. It is worth mentioning that our iterative method is free from the derivative evaluation which can be regarded as an advantage of this method in comparison with many other methods. Finally, we illustrate our results with some numerical examples.