Solving a system of nonlinear equations

I have about 1000 measurement which relates three variables to each other: x, y, and z. In other words, I have 1000 measurement values for x, y, and z. The relationship among them is as follows:

Could someone kindly tell me how I can find the coefficients c1, c2, and c3 using MATLAB? If there were no interaction between x and y (c3=0), then c1 and c2 could be easily found by the backslash operator in MATLAB in a least square way (A\b).

Answer 1

[Matt J]

If there were no interaction between x and y (c3=0), then c1 and c2 could be easily found by the backslash operator in MATLAB in a least square way (A\b)

Backslash should still work for least squares estimation,

   x=x(:); y=y(:); z=z(:);

   c = [x,y,x.*y].\z;  %c=[c1;c2;c3]

although, if you have measurement noise in x and y (regardless of c3=0), some would say you should be using total least squares instead of ordinary least squares,

      x=x(:); y=y(:); z=z(:);
    A=[x,y,x.*y,-z];
    A=bsxfun(@minus,A,mean(A,1));

    [~,~,V]=svd(A,0);

    c=V(1:end-1,end)./V(end);

[AP]

Thanks. The second method (for noise), using bsxfun, yields strange results for the coefficients.

[Matt J]

See if omitting the line

   A=bsxfun(@minus,A,mean(A,1));

makes a difference.