%In an experiment, the yield   z   is dependent on two reactants  x  and y
%(in mole percentage). The computed yields at 10 data points are given in a
%table with (z,x,y) values given below.
%It is proposed to use the following model that maximizes the yield (so as to
%minimize the least square error):
%z= a₀+ a₁x+ a₂x² + b₁y+ b₂y²
%Estimate the parameter vectors  a,b  from the data.
% matlab file with least squares problem follows:
z=[ 73 78 85 90 91 87 86 91 75 65]';
x=[
20
20
30
40
40
50
50
50
60
70
];
y=[10 10 15 22 22 27 27 27 32 40]';
n=size(y,1);
A=[ones(n,1) x x.*x y y.*y];
disp('data matrix A is ');
disp(num2str(A));
v=A\z;
disp(['least squares solution is v=A\z ']);
disp(num2str(v));
disp('verify the solution by checking the gradient: A''Av-A''z=0');
  A'*A*v-A'*z



%%%%%%%%%%%%%%%%%%%%%%output of file follows
data matrix A is 
1    20   400    10   100
1    20   400    10   100
1    30   900    15   225
1    40  1600    22   484
1    40  1600    22   484
1    50  2500    27   729
1    50  2500    27   729
1    50  2500    27   729
1    60  3600    32  1024
1    70  4900    40  1600
least squares solution is v=A\z 
   31.3806
   3.23385
-0.0490319
 -0.560854
  0.043766
verify the solution by checking the gradient: A'Av-A'z=0

ans =

   1.0e-09 *

   -0.0002
   -0.0146
         0
   -0.0073
   -0.2910
%