Givens Rotations

There is another type of rotations, which are called Givens rotations. These are like Jacobi rotations, but their purpose is not to annihilate one of the corner elements, a_pp, a_qq, a_qp, or a_pq, but instead to annihilate elements in the top row, i.e.,

$$\begin{eqnarray*}\boldsymbol{P}_{23} &\>\hbox{annihilates}\>& a_{31} \>\hbox{and... ...ol{P}_{25} &\>\hbox{annihilates}\>& a_{51} \>\hbox{and}\> a_{15} \end{eqnarray*}$$

and so on.

Because $a'_{rp} = a_{rp} - s\left(a_{rq} + \tau a_{rp}\right)$ and $a'_{rq} = a_{rq} + s\left(a_{rp} - \tau a_{rq}\right)$ for $r \neq p$and $r \neq q$, if a_rp and a_rq have been set to zero they will remain zero. So these Givens rotations are not unlike Householder rotations, but for normal filled matrices they are somewhat less efficient. However, they're actually a little more lightweight for tridiagonal matrices than Householder rotations, so we'll use them to effect the final diagonalisation.