clear

%%
% We choose a function over the interval [0,1]. 
figure(1)
f = @(x) sin(exp(2*x));
fplot(f, [0 1])
xlabel('x')
ylabel('f(x)')
title('Test function')
grid on

%%
% We interpolate it at equally spaced nodes for increasing values of n. We
% will sample the interpolant at a large number of points in order to
% estimate the interpolation error. 
n = (5:5:60)';   
err = 0*n;
x = linspace(0, 1, 1001)';         % for measuring error
for k = 1:length(n) 
    t = linspace(0, 1, n(k)+1)';   % equally spaced nodes
    y = f(t);                      % interpolation data
    p = polyinterp(t,y);
    err(k) = norm(f(x)-p(x), inf );
end

figure(2)
semilogy(n,err,'.-')
xlabel('n')
ylabel('max error')
title('Polynomial interpolation error')
grid on

%%
% Initially the error decreases exponentially, i.e. as $O(K^{-n})$ for some
% $K>1$. However, around $n=30$ the error starts to _grow_ exponentially.