Accurate evaluation of Chebyshev polynomials in floating-point arithmetic

摘要

It is well-known that the Chebyshev polynomials Tn and Un-1 can be recursively expressed as linear combinations of Tn-1 and Un-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 Tn is O(un), where u is the unit roundoff. In contrast, a commonly used three-term recurrence for Tn has the maximum error within a constant factor from un(2).