Fast Chebyshev second order differentiation

Fast computation of the second derivative of data located along Chebyshev points

fchd2(V) computes the first derivative of the data in V located along the N+1 Chebyshev–Gauss–Lobatto points cos(pi*(0:N)/N).


Example 1:
Use fchd2 to find the second derivative of the function f(x) = tan(x)
over [-1,1], and compare with f''(x) = 2 tan(x) sec(x)^2.

x = cos(pi*(0:10)/10); % create sparse Chebyshev-spaced grid of 11 points
xx = linspace(-1,1); % create dense, linearly spaced grid
plot(xx,2*tan(xx).*sec(xx).^2,x,fchd2(tan(x)));


Example 2:
To show the spectral convergence property of the Chebyshev derivative,
compute the error between the Chebyshev second derivative and the exact
second derivative of f(x) = tan(x) for several N.

N = 1:30;
err = zeros(1,length(N));

for n = N
x = cos(pi*(0:n)/n)'; % establish grid
err(n) = max(2*tan(x).*sec(x).^2 - fchd2(tan(x))); % compute error
end

loglog(N,err); %display