>\[\begin{equation}
    \hat{\textbf{y}} + (\textbf{q}_k^T\textbf{y})\textbf{q}_k = \hat{\textbf{y}} + (\textbf{q}_k^T\textbf{r})\textbf{q}_k,\nonumber
\end{equation}\]</div>
<p>and thus the residual sum-of-squares is reduced by</p>
<div class="arithmatex">\[\begin{equation}
    (\textbf{q}_k^T\textbf{r})^2.\nonumber
\end{equation}\]</div>
<p>Therefore, at each step <span class="arithmatex">\(q\)</span> we are going to choose <span class="arithmatex">\(k\)</span> such that</p>
<div class="arithmatex">\[\beg