On the accuracy and stability of algorithms most commonly used in the evaluation of Chebyshev polynomials of the first kind

This paper provides error analyses of the algorithms most commonly used for the evaluation of the Chebyshev polynomial of the first kind $T_N(x)$. Some of these algorithms are shown to be backward stable. This means that the computed value of $T_N(x)$ in floating point arithmetic by these algorithms can be interpreted as a slightly perturbed value of polynomial $T_N$, for slightly perturbed value of $x$.