clear

%%
% Here is a 4x4 Chebyshev differentiation matrix.
[t, Dx] = diffcheb(3, [-1 1]);
format rat, Dx

%%
% We again test the convergence rate.  
f = @(x) x + exp(sin(4*x));
dfdx = @(x) 1 + 4*exp(sin(4*x)).*cos(4*x);
d2fdx2 = @(x) 4*exp(sin(4*x)).*(4*cos(4*x).^2-4*sin(4*x));

n = (5:5:70)';
err1 = 0*n;
err2 = 0*n;

for k = 1:length(n)
    [t, Dx, Dxx] = diffcheb(n(k),[-1 1]);
    y = f(t);
    err1(k) = norm( dfdx(t) - Dx*y, inf );
    err2(k) = norm( d2fdx2(t) - Dxx*y, inf );
end

semilogy(n, [err1,err2],'.-')
hold on
axis tight
legend('f''','f''''')
xlabel('n')
ylabel('max error')
title('Convergence of Chebyshev derivatives')
grid on

%%
% Notice that the graph has a log-linear scale, not log-log. Hence the
% convergence is exponential, as we expect for a spectral method on a
% smooth function.