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A Little Summary

Let us summarise what we have learnt about Jacobi and Householder rotations.

Jacobi rotations are designed to annihilate a selected off-diagonal element, say, apq. Given matrix elements aqq, app, and apq the rotation itself can be calculated as follows:

\begin{eqnarray*}\theta &=& \cot{2\varphi} = \frac{a_{qq} - a_{pp}}{2a_{pq}} \\ ...
...ght) \\
v'_{rq} &=& v_{rq} + s\left(v_{rp} - \tau v_{rq}\right)
\end{eqnarray*}


Householder rotations are designed to rotate the whole under diagonal column or the whole right-to-diagonal row onto its first direction. Given a subdiagonal column:

\begin{displaymath}\boldsymbol{x} =
\left(
\begin{array}{c}
a_{p+1,p} \\
a_{p+2,p} \\
\vdots \\
a_{n,p}
\end{array} \right)
\end{displaymath}

the rotation itself can be calculated as follows:

\begin{eqnarray*}\sigma &=& \boldsymbol{x}\cdot\boldsymbol{x} \\
\vert\boldsym...
...t
\boldsymbol{Q}_{j+1} \\
\boldsymbol{Q} &=& \boldsymbol{Q}_1
\end{eqnarray*}




Zdzislaw Meglicki
2001-02-26