On efficient approximation of quadratic irrationals

TL;DR

The paper develops efficient algorithms for the convergents of quadratic irrationals by linking continued fractions to Chebyshev polynomials and their variants. It constructs fast, doubling-style recurrences using dilated and signed Chebyshev polynomials to compute convergents when Lagrange holds, with explicit formulas for the key traces and two concrete algorithms (ALG1 and ALG2) and illustrative examples. Under Galois conditions, certain decimations of convergents become signed Chebyshev sequences, enabling compact representations $p_{kl-1}=T^l_k(p_{l-1})$ and $q_{kl-1}=q_{l-1}U^l_{k-1}(p_{l-1})$ and related recurrences; this then extends to complex Hurwitz-like continued fractions. The final part connects these ideas to Householder's method, showing that high-order iterations yield the convergents via $H(x)=\dfrac{T^l_{d+1}(p)}{qU^l_d(p)}$, offering an $O(\log k)$ pathway to powerful, scalable computation of quadratic irrational approximants.

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.

Theorems & Definitions (14)

• Proposition 3
• proof
• Theorem 4
• proof
• Example 6
• Example 9
• Example 10
• Theorem 11
• proof
• Example 12