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\begin{center}
  {\large \textbf{Mathematical Tripos Part IB: Easter 2001}}

  {\Large \textbf{Numerical Analysis -- Detailed Example
      6\footnote{Corrections and suggestions to these notes should
        be emailed to \texttt{a.shadrin@damtp.cam.ac.uk}. All handouts
        are available on the WWW at the URL 
        \texttt{http://www.damtp.cam.ac.uk/user/na/PartIB/Handouts.html}.}}}

  \vspace{8pt}
  \textbf{\textit{\LARGE QR factorization: Householder reflections}}
\end{center}

Let
\begin{displaymath}
  A=\left[
    \begin{array}{rrr}
      1 & 1 & 3\\
      -2 & 0 & 1\\
      4 & 2 & -3\\
      -2 & 1 & 1
    \end{array}\right].
\end{displaymath}
Then $R\rightarrow A$,
\begin{Eqnarray*}
 && \MM{u}=\left[
    \begin{array}{r}
      6\\-2\\4\\-2
    \end{array}\right],\qquad
  \Omega_1=I-2\frac{\MM{u}\MM{u}^{\CC{T}}}{\|\MM{u}\|^2} =\left[
    \begin{array}{rrrr}
      -0.2000  &  0.4000 &  -0.8000 &   0.4000\\
      0.4000  &  0.8667  &  0.2667 &  -0.1333\\
      -0.8000 &   0.2667 &   0.4667 &   0.2667\\
      0.4000 &  -0.1333  &  0.2667  &  0.8667
    \end{array}\right],\\
  &&\hspace*{100pt} R\rightarrow \Omega_1R=\left[
    \begin{array}{rrr}
      -5.0000 &  -1.4000 &   2.6000\\
      0.0000 &   0.8000  &  1.1333\\
      0.0000 &   0.4000 &  -3.2667\\
      0.0000  &  1.8000 &   1.1333
    \end{array}\right],\\
  && \MM{u}=\left[
    \begin{array}{r}
      0.0000\\ 2.8100\\ 0.4000\\ 1.8000
    \end{array}\right],\qquad
  \Omega_2=I-2\frac{\MM{u}\MM{u}^{\CC{T}}}{\|\MM{u}\|^2} =\left[ 
    \begin{array}{rrrr}
       1.0000   & 0.0000 & 0.0000 & 0.0000\\
       0.0000 &   -0.3980 &  -0.1990 &  -0.8955\\
       0.0000 &   -0.1990 &   0.9717 &  -0.1275\\
       0.0000 &  -0.8955 &  -0.1275 &   0.4263
     \end{array}\right],\\
   &&\hspace*{100pt} R\rightarrow \Omega_2R=\left[
     \begin{array}{rrr}
       -5.0000 &  -1.4000 &   2.6000\\
       0.0000 &  -2.0100 &  -0.8159\\
       0.0000 & 0.0000 &  -3.5441\\
       0.0000 &  0.0000 &  -0.1153
     \end{array}\right],\\
    && \MM{u}=\left[
      \begin{array}{r}
        0\\0\\ -7.0902\\ -0.1153
      \end{array}\right],\qquad
    \Omega_3=I-2\frac{\MM{u}\MM{u}^{\CC{T}}}{\|\MM{u}\|^2} =\left[ 
      \begin{array}{rrrr}
        1.0000 & 0.0000 & 0.0000 & 0.0000\\
        0.0000 & 0.0000 & 0.0000 & 0.0000\\
        0.0000 & 0.0000 & -0.9995 & -0.0325\\
        0.0000 & 0.0000 & -0.0325 & 0.9995
      \end{array}\right],\\
    &&\hspace*{100pt} R\rightarrow \Omega_3R=\left[
      \begin{array}{rrr}
        -5.0000  & -1.4000  &  2.6000\\
        0.0000  & -2.0100 &  -0.8159\\
        0.0000  &  0.0000 &   3.5460\\
        0.0000  &  0.0000 &   0.0000
      \end{array}\right].
\end{Eqnarray*}
Finally,
\begin{displaymath}
  Q=(\Omega_3\Omega_2\Omega_1)^{\CC{T}}=\left[
    \begin{array}{rrrr}
      -0.2000 &  -0.3582 &   0.9102 &  -0.0561\\
      0.4000 &  -0.2786 &  -0.0754 &  -0.8699\\
      -0.8000 &  -0.4378 &  -0.3602 &  -0.1964\\
      0.4000 &  -0.7761 &  -0.1899 &   0.4490
    \end{array}\right].
\end{displaymath}

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