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How to compute the QR decomposition of matrix \begin{bmatrix} v^T \\ QR \end{bmatrix} in a stable way, where $QR$ represent the product of a real orthogonal matrix $Q \in \Bbb R^{m \times n}$ and upper triangular matrix $R \in \Bbb R^{n \times n}$ and $v \in \Bbb R^n$ is a vector?

I am allowed to use Matlab or Octave to solve this problem but I am not sure what would be a stable way to do this? I understand that $QR$ is already a product of required matrices, but I cannot come up with any idea to solve it.

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Golub and Van Loan's Matrix Computations is the standard reference. Chapter 7 presents a number of algorithms for the QR decomp and discusses stability of those algorithms.

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  • $\begingroup$ I did not find this one in particular though. I understand how to do QR decomposition otherwise $\endgroup$ – mandella May 5 '18 at 12:18

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