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

Yann Dijoux

TL;DR

This work reveals that Chebyshev polynomials of multiple kinds underlie the iterative schemes used to compute square roots and nth roots via classical root-finding methods. By expressing the Babylonian/Newton, Halley, and Householder iterations through $T_n$, $U_n$, $V_n$, and $W_n$, it derives explicit closed forms, product representations, and rational forms for the iterates, along with parity-based and monomial decompositions. The paper also extends the framework to $p$-th roots using generalized binomial coefficients and roots-of-unity filters, and provides asymptotic expansions of Chebyshev-related functions linked to combinatorial objects like Dyck paths and symmetric Dyck paths, offering combinatorial interpretations of the coefficients. Together, these results deepen the connection between algebraic Chebyshev structures and efficient high-order root-finding methods, with implications for both numerical algorithms and analytic combinatorics. The findings highlight how Chebyshev polynomials organize iterative denominators and numerators, yielding transparent, high-order convergence schemes and new avenues for asymptotic analysis.

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.

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

TL;DR

This work reveals that Chebyshev polynomials of multiple kinds underlie the iterative schemes used to compute square roots and nth roots via classical root-finding methods. By expressing the Babylonian/Newton, Halley, and Householder iterations through , , , and , it derives explicit closed forms, product representations, and rational forms for the iterates, along with parity-based and monomial decompositions. The paper also extends the framework to -th roots using generalized binomial coefficients and roots-of-unity filters, and provides asymptotic expansions of Chebyshev-related functions linked to combinatorial objects like Dyck paths and symmetric Dyck paths, offering combinatorial interpretations of the coefficients. Together, these results deepen the connection between algebraic Chebyshev structures and efficient high-order root-finding methods, with implications for both numerical algorithms and analytic combinatorics. The findings highlight how Chebyshev polynomials organize iterative denominators and numerators, yielding transparent, high-order convergence schemes and new avenues for asymptotic analysis.

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.
Paper Structure (6 sections, 16 theorems, 93 equations, 1 figure, 3 algorithms)

This paper contains 6 sections, 16 theorems, 93 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Motivation
  2. Polynomial expressions in the Babylonian method
  3. Polynomial expressions in the Halley's method for square roots
  4. Polynomial expressions in the Householder's method for square roots
  5. A note on the Householder's method for nth roots
  6. A note on the asymptotic expansion of Chebyshev functions

Key Result

Theorem 1

Let $x$ be a real positive number and $\{u_n\}_{n\geq 0}$ be the sequence associated to the Babylonian method reminded in (babth), where $r$ is a real positive number. An explicit expression of $u_n$ is presented as follows for $n$ greater or equal than 1, based on the Chebyshev polynomials of the first kind $\{T_k\}_{k \geq 0}$: Furthermore, if we denote $b=\frac{x-r^2}{2}$ and $M=\frac{x+r^2}{

Figures (1)

  • Figure 1: List of Dyck paths of semilength 4

Theorems & Definitions (27)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • proof
  • Corollary 4
  • proof
  • Theorem 5
  • Lemma 6
  • proof
  • Theorem 7
  • ...and 17 more