MATLAB: Lscov (weighted least squares) not deweighing when weights are altered

least squareslscovminimization

I am currently trying to use weighted least squares via lscov, to minimize using the design matix,
   A = [-0.7867    0.0464   -0.6155    1.0000;
        -0.3751    0.4299   -0.8213    1.0000;
         0.0447    0.4895   -0.8708    1.0000;
        -0.5946    0.8029    0.0424    1.0000]
and the error vector
    b = [26.3019  0   4.4677    4.6455]'
However, using lscov, the results are the same.
    x1 = lscov(A,(b),[.001 1 1 1])
    x2 = lscov(A,(b),[1 1 1 1])
    x1 = x2 = [26.2071  -73.2683   43.6825   77.2045]'
I have realized that this is because A is a square matrix. Is there a way to minimize with a weighting on a residual vector in the case of square matrices.
I have considered multiplying the error vector by a weight but I am not sure that is a reasonable approach.

Best Answer

Is there a way to minimize with a weighting on a residual vector in the case of square matrices.
You already are doing so. It's just that, if A is non-singular, the weighting will not matter because the solution A\b always achieves ideal zero residuals regardless of the weights. If A is singular, then weighting will have an impact, but the solution will be under-determined.
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