I am using Chebychev collocation nodes for approximation, and my problem requires me to calculate derivatives of the polynomial. I have been reading from a few sources, but I am not sure I understand how to do it. Could anyone please help me with how I go about this?
Thank you very much.
Anup
A specific polynomial and any number $k$ of its derivatives can be evaluated using Horner's methods. The case of $k = 2$ is discussed here.
In the event that you need all Chebyshev polynomials of degree less than $n$ and their first derivatives, I suggest that you proceed directly from the defining recurrence relation.
In the case of Chebyshev polynomials of the first kind, we have $$T_0(x) = 1, \quad T_1(x) = x, \quad T_{k+1}(x) = 2xT_{k}(x) - T_{k-1}(x).$$ Let $Q_k$ denote the derivative of $T_k$ with respect to $x$. Then by the product rule of differentiation $$Q_0(x) = 0, \quad Q_1(x) = 1, \quad Q_{k+1}(x) = 2 T_k(x) + 2x Q_k(x) - Q_{k-1}(x)$$ It follows, that any code which computes $T_k$ for $k=0,1,2,\dotsc,n$ can be easily augmented to also produce the derivatives $Q_k$ for $k=0,1,2,\dotsc,n$.
If running error bounds are desired, then the principles applied here can be extended to cover you situation as well.
