%A cookbook approach to linear, least-squares fitting in Matlab

% sample locations
x = (0:5)'
% simulated data
g = x.*x

% system matrix - value of terms in equaiton at sample locations
H = [x.*x x (x*0 + 1)]
% alternative way to fill constant column
H = [x.*x x ones(size(x))]

% try g = Hf --> 
%f = g/H
% oops, should be
f = H\g

% I usually write it
f = pinv(H)*g


% order doesn't matter as long as measurements 
% are in matching position to rows of H
%

% supppose
x = [x(3:end); x(1:2)]
g = [g(3:end); g(1:2)]

% have to make sure H is in proper order either by recreating as
% done here, or reordering
H = [x.*x x (x*0 + 1)]
f = pinv(H)*g

[x,y] = meshgrid( -1:.1:1 )
g = x.*x.*y + y.*y.*x + 1;
mesh(x,y,g)

% x, y and g are all 2D arrays, but we need vectors, that’s easy

x = x(:)
y = y(:)
g = g(:)

% Let’s fit to the equation 
% g = ax^2y + by^2x + cxy + dx^2 + ey^2 + fx + gy + h

H = [x.*x.*y y.*y.*x x.*y x.*x y.*y x y (y*0+1)]
f = pinv(H)*g

% add some noise uniformly distributed +/- 0.025
n = (rand(size(g))-.5)*.05
gn=g+n;
f = pinv(H)*g
fn = pinv(H)*gn