What is the difference between Gram-Smith QR decomposition procedure and qr.m function in Matlab?

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Osama Al-Shalali 2020-9-28

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评论： Christine Tobler 2020-10-22

采纳的回答： Christine Tobler

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Hi everyone!

I want to know what is the differnce between the Gram-Smith procedure and qr.m function in matlab. Why there is a fourth coulmn resulting from using qr.m function? The photo is attached for the Gram-Smith procedure .Any help will be so appreciated.

The code is
A=[1 -2 -1; 2 0 1; 2 -4 2;4 0 0];
[Q,R] = qr(A)
The result is
Q =
   -0.2000    0.4000    0.8000   -0.4000
   -0.4000   -0.2000   -0.4000   -0.8000
   -0.4000    0.8000   -0.4000    0.2000
   -0.8000   -0.4000    0.2000    0.4000
R =
   -5.0000    2.0000   -1.0000
         0   -4.0000    1.0000
         0         0   -2.0000
         0         0         0
         

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采纳的回答

Christine Tobler 2020-9-28

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MATLAB's QR decomposition is computed using Householder transformations, which is generally more numerically advantageous.

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f 2020-10-22

编辑：f 2020-10-22

Just a tiny detail: Matlab's qr does not ensure that det(Q)=1. The determinant of Q may be 1 or -1; it is data-dependent, since it is (-1)^(number_of_nontrivial_reflectors_used). So it cannot be used in a straightforward way to determine the sign of det(A).

Christine Tobler 2020-10-22

Good point, det(R) will only tell you the absolute value of the original matrix, otherwise you would have to construct the complete Q matrix to compute the sign of its determinant.

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