A convergence condition for Newton-Raphson method

TL;DR

This paper establishes practical convergence conditions for the Newton-Raphson method, allowing reliable root-finding even from distant initial guesses by leveraging properties of the function and its derivatives.

Contribution

It introduces simple, easy-to-test convergence conditions for Newton-Raphson that work from far initial points, improving practical root-finding reliability.

Findings

Convergence guaranteed for initial points in [a,c] under certain conditions.

Similar convergence results for initial points in [c,b].

Enhanced ability to predict Newton-Raphson convergence regions.

Abstract

In this paper we study the convergence of Newton-Raphson method. For this method there exists some convergence results which are practically not very useful and just guarantee the convergence of this method when the first term of this sequence is very close to the guessed root \cite{sulimayer}. Khandani et al. introduced a new iterative method to estimate the roots of real-valued functions \cite{khandani}. Using this method we introduce some simple and easy-to-test conditions under which Newton-Raphson sequence converges to its guessed root even when the initial point is chosen very far from this root. More clearly, for a real-valued second differentiable function $f:[a,c]\to \mathbb R$ with $f^{''}f\ge 0$ on $(a,c)$ where $c$ is the unique root of $f$ in $[a,c]$, the Newton-Raphson sequence $f$ converges to $c$ for each $x_0\in[a,c]$ provided $f$ satisfies some other simple conditions on this interval. A similar result holds if $[a,c]$ be replaced with $[c,b]$. Our study will enable us to predict accurately where Newton-Raphson sequence converges.