6.1.1 Givens rotation

The transformation matrix from (6.2) will be slightly changed (again, we consider only the first step of the Gauß elimination)

$$\begin{pmatrix}c & s & 0 & \cdots \ - s & c & 0 & \cdots \ 0 ... ... \ \vdots & \vdots & & \ddots \end{pmatrix} \cdot \underline{b} \enspace ,$$ (6.3)

with the goal

$$-s \cdot a_{11} + c \cdot a_{21} \;=\; 0 \enspace .$$

The choice

$$s \;=\; \frac{a_{21}}{\sqrt{a_{11}^2+a_{21}^2}} \qquad c \;=\; \frac{a_{11}}{\sqrt{a_{11}^2+a_{21}^2}}$$ (6.4)

corresponds exactly to the entries $s=\sin\varphi$ und $c=\cos\varphi$ of a rotation matrix by normalization.
Instead of this, the original Gauß elimination uses a value $\tfrac{a_{21}}{a_{11}}=\tan\varphi$.

Advantage: $a_{11}=0$ $\Longrightarrow$ $s=1$, $c=0$ $\Longrightarrow$ numerically stable $\Longrightarrow$ no Pivot search necessary.

Since the main expenditure is with the computations in the transformation of matrix $A$, the application of the rotation to this is only regarded. In the following, vectors  $\underline{u}$ and  $\underline{v}$ denote those matrix rows which will be changed by the rotation and $\widehat{\underline{u}} \makebox[0pt]{}$ und  $\widehat{\underline{v}} \makebox[0pt]{}$ are the appropriate resulting matrix rows.

One single Givens rotation

$$\begin{split}\widehat{\underline{u}} \makebox[0pt]{} &\;:=\; ... ...;:=\; c \cdot \underline{v} - s \cdot \underline{u} \end{split}$$ (6.5)

is included in the BLAS1 library as one call (SROT/DROT).