Evaluation of Chebyshev polynomials by a three-term recurrence in floating-point arithmetic

摘要

This paper studies an approximation to the Chebyshev polynomial  $$T_n$$ T n computed via a three-term recurrence in floating-point arithmetic. It is shown that close to either endpoint of the interval $$[-1, 1]$$ [ - 1 , 1 ] , the numerical approximation coincides with the line tangent to $$T_n$$ T n at that endpoint. From this representation new upper and lower error bounds are derived.