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## Fast Chebyshev second order differentiation

version 1.1 (1.65 KB) by

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

Updated

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