%% MATLAB example for regression via least-squares 
%% Suppose given a 'complicated' function

clear, clf
f = @(x) 1+sin(10*x).*exp(x);
fplot(f,[-1,1]),grid on, shg

%%
% We'd like to approximate with a polynomial, say degree 10. Let's try do
% this via regression. We can do via interpolation:

n = 20+1;  % degree+1
Z = linspace(-1,1,n).'; % 11 sample points
hold on 
plot(Z,f(Z),'k.','markersize',15),shg

%%
V = fliplr(vander(Z)); % The matrix of Z^i
c = V\f(Z); % solve linear system to get polynomial 

p = @(x)c(1);
for ii = 1:length(c)-1
    p = @(x)p(x)+c(ii+1)*x.^ii;
end
fplot(p,[-1,1],'color','k'),grid on, shg

%%
% We see that the approximant (red) is looking mostly good, but not near
% the endpoint $x=1$. We can improve the situation by sampling at more
% points, then solving a least-squares problem. Least-squares makes the
% situation much more robust! 

Z = linspace(-1,1,100).'; % 100 sample points
hold on 
plot(Z,f(Z),'bo','markersize',5),shg

V = fliplr(vander(Z)); % The matrix of Z^i
% c = V(:,1:n)\f(Z); % solve linear system to get polynomial 
[Q,R] = qr(V(:,1:n),0); c = R\(Q'*f(Z));

p = @(x)c(1);
for ii = 1:length(c)-1
    p = @(x)p(x)+c(ii+1)*x.^ii;
end
fplot(p,[-1,1],'color','r'),grid on, shg