Searching coefficients of surface polinomial (equivalent to polyfit for surfaces) fixed degree

I'm looking for a the coefficients a1...a6 of the polinomial function

N1 = a1 + a2*x + a3*y + a4*x^2 + a5*x*y + a6*y^2

that goes through the following points. There is no a7, the degree is fixed.

x=[0   1   0   0.5 0.5 0  ]';
y=[0   0   1   0   0.5 0.5]';
z=[1   0   0   0   0   0  ]';

I know there generally is an infinite number of functions, yet the same applies in this case

a=[0 1 0 0.5 0.5 0];
b=[1 2 3 4 5 6];
N2=polyfit(a,b,2);
N2 =

   -7.3333    6.0000    3.3333

and MATLAB is able to work with that. I'm looking for something like polyfit for surfaces.

So far I tried:

N1 = scatteredInterpolant(x,y,z,'natural');

The function seems to work fine, but I cannot show the coefficients of the function, also I cannot influence the degree which is required in my case.

xq=0:0.5:1;
yq=0:0.5:1;
N1 = griddata(x,y,z,xq,yq);

It only returns the interpolated surface, not the function coefficients.

Answer 1

[Walter Roberson]

x=[0   1   0   0.5 0.5 0  ]';
y=[0   0   1   0   0.5 0.5]';
z=[1   0   0   0   0   0  ]';

coeffs_matrix = [ones(length(x),1), x, y, x.^2, x.*y, y.^2];
A_vals = coeffs_matrix \ z;

% coeffs_matrix * A_vals - z    %cross-check should be nearly 0

N1 = @(x,y) A_vals(1) + A_vals(2) .* x + A_vals(3) .* y + A_vals(4) .* x.^2 + A_vals(5) .* x .* y + A_vals(6) .* y.^2;

xq = 0:0.5:1;
yq = 0:0.5:1;
[XQ, YQ] = ndgrid(xq, yq);
N2 = N1(XQ, YQ);
surf(XQ, YQ, N2)

Answer 2

[Image Analyst]

If you have the Statistics and Machine Learning Toolbox, I believe fitlm() is the function you want. I think you need more points than 6 though. And I'm not sure what your z variable is used for -- it's not in your model.

[Walter Roberson]

If you treat z as being the N1 value for particular x, y, then you have 6 points and 6 coefficients, and a least-squares does fine.