Binomial expansion of Newton's method

TL;DR

This paper extends Newton's method to include a binomial expansion, analyzing its convergence properties and comparing it with the original method through theoretical and numerical examples.

Contribution

It introduces a binomial expansion of Newton's method and compares their convergence behaviors, including curvature and convexity considerations.

Findings

Quadratic and linear convergence regimes identified.

Binomial expansion converges faster in certain cases.

Numerical examples illustrate theoretical results.

Abstract

We extend the Newton's method and show the extended Newton's method leads to the binomial expansion of Newton's method that the convergences become the quadratic and linearly. In case of the quadratic convergence, we give the convergence comparison of the binomial expansion of Newton's method and Newton's method. And we give convergence comparisons of the binomial expansion of Newton's method and Newton's method using the curvature and convex-concave of curve. Next we give examples of numerical calculations of the binomial expansion of Newton's method.