Given a constant unitary matrix $\mathbf{V}^*$ and a parameter diagonal matrix $\mathbf{D}$, can the QR factorization of many different $\mathbf{D}\mathbf{V}^*$ be performed efficiently? Here
$$\mathbf{D} = \operatorname{diag}\left(\sqrt{s_0^2 + \alpha^2} , \sqrt{s_1^2 + \alpha^2}, \sqrt{s_2^2 + \alpha^2}, \dots\right)$$
with the $s_i$ constants so that the only parameter is $\alpha$. Any ideas are welcome.
The original motivation is from my answer to this question, which I recognize as having some basis in statistics, though I do not remember the specific reference. (I do remember that two people came up with the form of the problem separately within the last few decades or so.)
