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\begin{center}
  {\large \textbf{Mathematical Tripos Part IB: Easter 20001}}

  {\Large \textbf{Numerical Analysis -- Detailed Example
      5\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: Givens rotations}}
\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*}
  &&\Omega^{[1,2]}=\left[
    \begin{array}{rrrr}
      {\bf 0.4472} & {\bf -0.8944}  &   0.0000  &  0.0000\\
      {\bf 0.8944} &  {\bf 0.4472}  & 0.0000 & 0.0000\\
      0.0000 & 0.0000 & 1.0000 & 0.0000\\
      0.0000 & 0.0000 & 0.0000 & 1.0000
    \end{array}\right],\qquad R\leftarrow\Omega^{[1,2]}R=\left[
    \begin{array}{rrr}
      2.2361 &  0.4472 &  0.4472\\
      0.0000   & 0.8944 &   3.1305\\
      4.0000 &   2.0000 &   -3.0000\\
      -2.0000 &  1.0000 & 1.0000
    \end{array}\right],\\
  &&\Omega^{[1,3]}=\left[
    \begin{array}{rrrr}
      {\bf 0.4880} &  0.0000 & {\bf 0.8729} & 0.0000\\
      0.0000 &  1.0000 & 0.0000 & 0.0000\\
      {\bf -0.8729} & 0.0000 & {\bf 0.4880} & 0.0000\\
      0.0000 & 0.0000 & 0.0000 & 1.0000
    \end{array}\right], \qquad R\leftarrow\Omega^{[1,3]}R=\left[
    \begin{array}{rrr}
       4.5826 &  1.9640 &  -2.4004\\
       0.0000 &  0.8944 &   3.1305\\
       0.0000 &  0.5855 &  -1.8542\\
       -2.0000 & 1.0000 &   1.0000
    \end{array}\right],\\
  &&\Omega^{[1,4]}=\left[
    \begin{array}{rrrr}
      {\bf 0.9165} &  0.0000 &   0.0000 &  {\bf -0.4000}\\
      0.0000 &  1.0000 &   0.0000 &   0.0000\\
      0.0000 &  0.0000 &   1.0000 &   0.0000\\
      {\bf 0.4000} &  0.0000 &   0.0000 &   {\bf 0.9165}
    \end{array}\right], \qquad R\leftarrow\Omega^{[1,4]}R=\left[
    \begin{array}{rrr}
      5.0000  &  1.4000 &  -2.6000\\
      0.0000  &  0.8944 &   3.1305\\
      0.0000  &  0.5855 &  -1.8542\\
      0.0000  &  1.7021 &  -0.0436
    \end{array}\right],\\
  &&\Omega^{[2,3]}=\left[
    \begin{array}{rrrr}
      1.0000 &  0.0000  &  0.0000 &  0.0000\\
      0.0000 &  {\bf 0.8367}  &  {\bf 0.5477} &  0.0000\\
      0.0000 & {\bf -0.5477}  &  {\bf 0.8367} &  0.0000\\
      0.0000 &  0.0000  &  0.0000 &  1.0000
    \end{array}\right], \qquad R\leftarrow\Omega^{[2,3]}R=\left[
    \begin{array}{rrr}
      5.0000 &   1.4000 &  -2.6000\\
      0.0000 &   1.0690 &   1.6036\\
      0.0000 &   0.0000 &  -3.2660\\
      0.0000 &   1.7021 &  -0.0436
    \end{array}\right],\\
  &&\Omega^{[2,4]}=\left[
    \begin{array}{rrrr}
      1.0000 & 0.0000 & 0.0000 & 0.0000\\
      0.0000 & {\bf 0.5319} & 0.0000 & {\bf 0.8468}\\
      0.0000 & 0.0000 & 1.0000 & 0.0000\\
      0.0000 & {\bf -0.8468} & 0.0000 & {\bf 0.5319}
    \end{array}\right], \qquad R\leftarrow\Omega^{[2,4]}R=\left[
    \begin{array}{rrr}
      5.0000 &  1.4000 &  -2.6000\\
      0.0000 &  2.0100 &   0.8159\\
      0.0000 &  0.0000 &  -3.2660\\
      0.0000 &  0.0000 &  -1.3812
    \end{array}\right],\\
  &&\Omega^{[3,4]}=\left[
    \begin{array}{rrrr}
      1.0000 & 0.0000 & 0.0000 & 0.0000\\
      0.0000 & 1.0000 & 0.0000 & 0.0000\\
      0.0000 & 0.0000 & {\bf -0.9210} &  {\bf -0.3895}\\
      0.0000 & 0.0000 & {\bf  0.3895} & {\bf  -0.9210}
    \end{array}\right],\qquad R\leftarrow\Omega^{[3,4]}R=\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^{[3,4]}\Omega^{[2,4]}\Omega^{[2,3]}\Omega^{[1,4]}
  \Omega^{[1,3]} \Omega^{[1,2]})^{\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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