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Homework Help
Precalculus Mathematics Homework Help
Weighted Least Squares Solution
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[QUOTE="samgrace, post: 5006391, member: 501944"] [h2]Homework Statement [/h2] \begin{bmatrix} 3x_{1}& 7x_{2}& 4x_{3} \\ 3x_{1}& 4x_{2}& 5x_{3} \\ x_{1}& 10x_{2}& 8x_{3} \\ 8x_{1}& 8x_{2}& 6x_{3} \\ \end{bmatrix} = \begin{bmatrix} 26 \\ 16 \\ 33 \\ 46 \\ \end{bmatrix} the measurements represented by equations 1 and 3 above can be trusted more than those represented by equations 2 and 4 and are given twice the weight. Write down an explicit matrix form for the system of equations. Solve it using Matlab. However all I really need is to find the weighting factor, I can do the rest from there, struggling to see how I can weight the first and third rows by a factor of two, whilst simultaneously leaving the first and fourth alone. [h2]Homework Equations[/h2] I am going to use Ax = b e = W(Ax-b) so e^{T}e = (Ax-b)^T*W^T*W*(Ax-b) so A^T*W^T*W*A*x = A^T*W^T*W*b basically multiply that out and solve via guassian elimination for x [h2]The Attempt at a Solution[/h2] e = W(Ax-b) \begin{bmatrix} e \\ e \\ e \\ e \\ \end{bmatrix} = \begin{bmatrix} ?& ?& ?& ?& \\ ?& ?& ?& ?& \\ ?& ?& ?& ?& \\ ?& ?& ?& ?& \\ \end{bmatrix} * \begin{bmatrix} 3x_{1}& 7x_{2}& 4x_{3}& -26& \\ 3x_{1}& 4x_{2}& 5x_{3}& -16& \\ x_{1}& 10x_{2}& 8x_{3}& -33& \\ 8x_{1}& 8x_{2}& 6x_{3}& -46& \\ \end{bmatrix} I have tried various combinations of 4x4 matrices for the ? matrix (weighting matrix) that will result in the weighting factor needed, e.g a diagonal 4x4 matrix of 2's, works however the other rows get multiplied by this as well. Please inform me of how to find this. [/QUOTE]
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Weighted Least Squares Solution
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