naginterfaces.library.blas.drotg

naginterfaces.library.blas.drotg(a, b)

drotg generates a real Givens plane rotation.

For full information please refer to the NAG Library document for f06aa

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f06/f06aaf.html

Parameters

afloat

The value $a$, the first element of the vector which determines the rotation.

bfloat

The value $b$, the second element of the vector which determines the rotation.

Returns

afloat

The value $d$.

bfloat

The value $z$, from which $c$ and $s$ can be reconstructed.

cfloat

The value $c$, the cosine of the rotation.

sfloat

The value $s$, the sine of the rotation.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

drotg generates a real Givens plane rotation with parameters $c$ and $s$, such that, given real $a$ and $b$:

$$\begin{array}{c}\left(\begin{array}{cc}c& s\\ -s& c\end{array}\right)\left(\begin{array}{c}a\\ b\end{array}\right)=\left(\begin{array}{c}d\\ 0\end{array}\right)\text{.}\end{array}$$

The function computes $c$, $s$ and $d$ as follows:

$$d=\sigma \sqrt{a^2+b^2}\text{;}$$

$$\begin{array}{c}c=\{\begin{array}{cc}a/d\text{,}& \text{if}d\ne 0\text{,}\\ 1\text{,}& \text{if}d=0\text{,}\end{array}s=\{\begin{array}{cc}b/d\text{,}& \text{if}d\ne 0\text{,}\\ 0\text{,}& \text{if}d=0\text{,}\end{array}\end{array}$$

$$\begin{array}{c}\text{where}\sigma =\{\begin{array}{cc}sign\left(a\right)\text{,}& \text{if}\left|a\right|>\left|b\right|\text{,}\\ sign\left(b\right)\text{,}& \text{if}\left|a\right|\le \left|b\right|\text{.}\end{array}\end{array}$$

The function also computes the value $z$ defined as

$$\begin{array}{c}z=\{\begin{array}{cc}s\text{,}& \text{if}\left|s\right|<c\text{or}c=0\text{,}\\ 1/c\text{,}& \text{if}0<\left|c\right|\le s\text{.}\end{array}\end{array}$$

This enables $c$ and $s$ to be reconstructed from the single value $z$ as

$$\begin{array}{c}c=\{\begin{array}{cc}\sqrt{1-z^2}\text{,}& \text{if}\left|z\right|\le 1\text{,}\\ 1/z\text{,}& \text{if}\left|z\right|>1\text{,}\end{array}s=\{\begin{array}{cc}z\text{,}& \text{if}\left|z\right|\le 1\text{,}\\ \sqrt{1-c^2}\text{,}& \text{if}\left|z\right|>1\text{.}\end{array}\end{array}$$

To apply the plane rotation to a pair of real vectors, call drot(); to apply it to a pair of complex vectors, call zdrot().