clear
clc

%% 
% We plot a cardinal Lagrange polynomial for n=5 and k=2. 
t = [ 1, 1.5, 2, 2.25, 2.75, 3 ];
n = 5;  
k = 2;

not_k = [0:k-1 k+1:n];   % all except the kth node

%%
% Whenever we index into the vector t, we have to add 1 since our
% mathematical index starts at zero. 
phi = @(x) prod(x - t(not_k + 1));
ell_k = @(x) phi(x) ./ phi(t(k + 1));

fplot(ell_k, [1 3])
hold on 
grid on
plot(t(not_k+1), 0*t(not_k+1), 'o')
plot(t(k+1), 1, 'rx')
xlabel('x') 
ylabel('ell_2(x)')
title('Lagrange cardinal function')

%%
% Observe that ell_k is _not_ between zero and one everywhere between the
% nodes.