When I evaluate the following Chebyshev series of the first kind in two different ways, I get two very different results:
N[ChebyshevT[100, Cos[Pi/7]], 8]
N[ChebyshevT[100, Cos[Pi/7]]]
Out[1] = 0.62348980
Out[2] = 3.71097*10^18
Clearly the first result is the correct one. Could someone please explain why the difference occurs e.g. is it perhaps because of a peculiar way that the long Chebyshev polynomial is numerically evaluated? The problem does not arise for shorter length Chebyshev polynomials.
N
without any precision specified works in machine precision with precision tracking switched off. For high-order polynomials, this is a bad idea. When a precision is specified, the working precision is adjusted so as to ensure that all of the requested digits are correct. $\endgroup$ChebyshevT[100, N@Cos[Pi/7]]
is accurate and fast, no doubt because stable algorithms are used to compute built-in functions with machine precision inputs. $\endgroup$