# Updating the qr factorization and the least squares problem

This should make it easier for people who read this article to go to the actual literature.

However it does make for some awkwardness in a few places.

Other techniques that are found in the literature that could also be used include Given's reflections, Householder reflections, fast Givens, Gentleman's methods, etc.

Given's rotations are used in this paper because they have low operation counts on problems with the structure that survey networks show, and are relatively easy to describe and implement.

There are, in fact, several equivalent methods of forming the QR factorization.

This article will discuss using Given's rotations for the task.

Numerical analysts have many methods appropriate to least squares problems.

The redundancy of the problem r is the number of rows minus the number of columns (m-n).

Not all shots are of equal accuracy, and they are not normally weighted the same.

Any good book about adjusting survey networks will cover the details of the problem setup, so only a quick summary of the usual background is given below.

A survey network consists of shots that connect points, along with weights for those shots.

### Search for updating the qr factorization and the least squares problem:

I'm trying to understand how to solve a least squares problem of the form: $$\begin A& B \end\beginx\\y\end = [b]$$ where I only explicitly solve for $y$ and not $x$.

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