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\noindent
The Bartels-Golub-Reid update uses a product of elementary matrices
of the form
\[
   M      = \pmat{1 &   \\ \mu &   1} \quad \mathrm{or} \quad
  \Mtilde = \pmat{  & 1 \\   1 & \mu},
\]
where each elementary matrix operates on two rows of $U$.
The choice between $M$ and $\Mtilde$ is made to keep
$|\mu| \le \updatetol = 10$, 5, or 2 (say) while allowing some
freedom to have the ``1'' operate on the sparsest row of $U$.
(That is, to use the sparsest row of $U$ as pivot row.)
The update procedure should be stable as long as the
elementary matrices are well-conditioned, or at least
behave like well-conditioned matrices.

\begin{enumerate}

\item Suppose $\mu > 0$.
With the help of \Matlab, complete this table in terms of $\mu$:
      \[
         \cond\pmat{1 & \\ \mu & 1} \approx
           \left\{
             \begin{array}{ll}
                \dots,  & \mu \ll  1,
             \\ 2.6180, & \mu  = \,1,
             \\ \dots,  & \mu \gg  1.
             \end{array}
           \right.
      \]
      (Fill in the dots \dots{} with expressions involving $\mu$.)

\item If $M_k = \pmat{1 & \\ \mu_k & 1}$, what is $M_1 M_2$?

\item If $M_1 = \pmat{1 & \\ 90 & 1}$, $M_2 = \pmat{1 & \\ 90 & 1}$,
      what are $M_1 M_2$ and $\cond(M_1 M_2)$?

\item How does the previous $\cond(M_1 M_2)$ compare with the bound
      \[
        \cond(M_1 M_2) \le \cond(M_1)\cond(M_2)?
      \]

\item If
      \[
        L_1 = \pmat{1 \\ &1 \\ & &1 \\ 10&  &  &1}, \quad
        L_2 = \pmat{1 \\ &1 \\ & &1 \\   &10&  &1}, \quad
        L_3 = \pmat{1 \\ &1 \\ & &1 \\   &  &10&1},
      \]
      what are $L_1 L_2 L_3$ and  $\cond(L_1 L_2 L_3)$?

\item How does the previous $\cond(L_1 L_2 L_3)$ compare with the bound
      \[
        \cond(L_1 L_2 L_3) \le \cond(L_1)\cond(L_2)\cond(L_3)?
      \]

\end{enumerate}

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