clear

%%
% This function has infinitely many continuous derivatives on the entire
% real line and looks very easy to approximate over [-1,1].  
f = @(x) 1./(x.^2 + 16);
x = linspace(-1, 1, 1601)';
n = [4 10 16 40];
for k = 1:length(n) 
    theta = linspace(0, pi, n(k)+1)';
    t = -cos(theta);
    p = polyinterp(t, f(t));
    semilogy(x, abs(f(x)-p(x)));
    hold on
    axis tight
end
axis tight
title('Error for degrees 4,10,16,40')
xlabel('x')
ylabel('|f(x)-p(x)|')
grid on
legend('n=4', 'n=10', 'n=16', 'n = 40', 'location', 'best')

%%
% By degree 16 the error is uniformly within machine epsilon. Note that
% even as the degree continues to increase, the error near the ends does
% not grow as with the Runge phenomenon for equally spaced nodes.