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In Fong, Saunders: LSMR: An iterative algorithm for sparse least-squares problems, section 3.3 one reads

Since only the last diagonal of $\tilde{R}_k$ and the bottom 2x2 part of $\hat{R}_k$ change each iteration, this estimate of $x_k$ can again be updated cheaply. The pseudocode, omitted here, can be derived as in section 3.1.1.

As a consequence of the omission, all LSMR codes I've found compute $\|x\|$ explicitly (and not cheaply) via a dot product.

Can anyone fill in the details here?

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