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

Abstract

The Householder's method is a root-find algorithm which is a natural extension of both the Newton's method and the Halley's method. The current paper 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 $n$th root is also proposed.