On efficient approximation of quadratic irrationals

TL;DR

This paper introduces efficient algorithms for approximating quadratic irrationals, demonstrating that under certain conditions, their convergents form Chebyshev sequences and can be generated by Householder methods.

Contribution

The paper presents novel algorithms for computing convergents of quadratic irrationals and links their properties to Chebyshev sequences and Householder methods.

Findings

Decimations of convergents are signed Chebyshev sequences.

Convergents can be generated by Householder methods.

Results depend on Galois' refinement of Lagrange's theorem.

Abstract

We provide efficient algorithms to compute convergents of quadratic irrationals. We show that for square roots, provided Galois' refinement of Lagrange's theorem holds, certain decimations of the sequence of convergents are signed Chebyshev sequences, which can be also be generated by a Householder method.