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I was looking for some help with derivatives of Chebychev polynomials at zero points. The recursive expression,

$$ T_{(j+1)}(x) = 2xT_j(x) - T_{(j-1)}(x) $$

has the derivative $$ T'_{j+1}(x) = 2T_j(x) + 2xT'_j(x) - T'_{j-1}(x) $$

with $T'_0=0$ and $T'_1=1$.

But I was looking at Trefethen's book (Chapter 6, Page 54) where he uses the following code to construct a derivative matrix for Chebychev polynomials at extreme points:

% CHEB compute D = differentiation matrix, x = Chebyshev grid 
function [D,x] = cheb(N) 
if N==0, D=0; x=1; return, end 
x = cos(pi*(0:N)/N)'; 
c = [2; ones(N-1,1); 2].*(-1).^(0:N)'; 
X = repmat(x,1,N+1); 
dX = X-X'; 
D = (c*(1./c)')./(dX+(eye(N+ 1))) ; % off-diagonal entries 
D = D - diag(sum(D')); 

How are these two derivative calculations related? Or is it that the derivatives at extreme points and zero points are calculated differently, as above? Any references I can read to better understand this?

Thank you.

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  • $\begingroup$ One is the derivative in physical space, and the other in spectral space. It is quite similar to performing a derivative in Fourier space (with the difference that in your case the derivative operator is not diagonal). $\endgroup$ – davidhigh Apr 6 '18 at 9:14
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On p. 53 of the book of Trefethen, you can read "D_{N} contains the derivative of the degree N polynomial interpolant p_j(x) to the delta function supported at x_j [= cos(j*pi/N)], sampled at the grid {x_i}" So the entries of D are the derivatives of the Lagrange interpolatory polynomials and not the Chebyshev polynomials.

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