Accurate evaluation of Chebyshev polynomials in floating-point arithmetic

Abstract

It is well-known that the Chebyshev polynomials $T_n$ and $U_{n-1}$ can be recursively expressed as linear combinations of $T_{n-1}$ and $U_{n-2}$. We study floating-point implementations of these recurrences for arguments in the interval $[-\,1,1]$. We demonstrate that the maximum error of the resulting approximation to $T_n$ is $\mathcal{O}(un)$, where u is the unit roundoff. In contrast, a commonly used three-term recurrence for $T_n$ has the maximum error within a constant factor from $un^2$.