clear

%%
% We plot |Phi(x)| over the interval [-1,1] with equispaced nodes for
% different values of n. 
x = linspace(-1, 1, 1601)';
Phi = zeros(size(x));
for n = 10:10:50
    t = linspace(-1, 1, n+1)';
    for k = 1:length(x)
        Phi(k) = prod(x(k)-t);
    end
    semilogy(x, abs(Phi)) 
    hold on
end
axis tight
title('Effect of equispaced nodes')
xlabel('x')
ylabel('|\Phi(x)|')
grid on
legend('n=10', 'n=20', 'n=30', 'n=40', 'n=50', 'location', 'best')

%%
% (Each time Phi passes through zero at an interpolation node, the value
% on the log scale should go to -\infty, which explains the numerous
% cusps on the curves.) Two observations are important: First, the size of
% |Phi| decreases exponentially at each fixed location in the interval
% (because the spacing between curves is constant for constant increments
% of n). Second, |\Phi| is larger at the ends of the interval than in
% the middle, by an exponentially growing factor.