Chebyshev polynomials involved in the Householder's method for square roots

TL;DR

This paper explores how Chebyshev polynomials can be used to enhance the Householder's method for computing square roots, with extensions to nth roots, offering potentially more efficient root-finding algorithms.

Contribution

The paper introduces a novel approach linking Chebyshev polynomials with Householder's method for square root approximation and extends it to nth roots.

Findings

Chebyshev polynomials can be expressed in the context of Householder's method.

New algorithms for square root and nth root approximation are proposed.

Potential improvements in convergence or efficiency are suggested.

Abstract

The Householder's method is a root-find algorithm which is a natural extension of the methods of Newton and Halley. The current paper mostly focuses on approximating the square root of a positive real number based on these methods. The resulting algorithms can be expressed using Chebyshev polynomials. An extension to the nth root is also proposed.