%% 
clear
format compact
seed = 10;
rng(seed);

%%
n = 10;

sc = 0.1;
t = linspace(0, 3, n)';
y = sin(t) + sc*randn(n,1);

clf
plot(t, y, 'o')
hold on
fplot(@sin, [0 3])
xlabel('t') 
ylabel('y')
legend('noisy data', 'noiseless data', 'location', 'best')
grid on
hold off

%%
% A polynomial interpolant can be used to fit the data. Here we build one using
% a Vandermonde matrix. 
V = t.^0;      % vector of ones
k = n - 1;
%k = 3;
for j = 1:k
    V(:,j+1) = t.*V(:,j);
end

c = V\y;
p = @(x) polyval(flipud(c), x);

plot(t, y, 'o')
hold on 
fplot(p, [0 3]) 
fplot(@sin, [0 3])
grid on
legend('noisy data', 'interpolation', 'noiseless data', 'location', 'best')
hold off