Derivatives of a Chebychev polynomial

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

• Why can't you just use a simple finite difference approximation to determine the derivatives? – cbcoutinho Nov 23 '17 at 18:45
• I apologise, I am new to this area of research, and do not know a great deal. I was just hoping to find the derivatives for T'_n(x), and I am not sure how. By finite difference, do you mean something like the gradient function in MATLAB? – Anup Mulay Nov 23 '17 at 19:10
• You can find explicit formulas for the derivatives of Chebyshev polynomials. In MATLAB, you might also want to check Chebfun. – nicoguaro Nov 23 '17 at 22:08

1 Answer

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.