naginterfaces.library.blas.zrotgs

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

zrotgs generates a complex Givens plane rotation having real sine.

For full information please refer to the NAG Library document for f06cb

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f06/f06cbf.html

Parameters

acomplex

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

bcomplex

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

Returns

acomplex

The value $d$.

bcomplex

The value $t$, the tangent of the rotation.

ccomplex

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.

zrotgs generates a complex Givens plane rotation with parameters $c$ (complex) and $s$ (real), such that, given complex $a$ and $b$:

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

If $b$ is real, then $d$ is also real. On exit, $b$ is overwritten by $t$, the tangent of the rotation; $c$ and $s$ can be reconstructed from the single stored value $t$, by a subsequent call to zcsgs().

If $\left|b\right|<ϵ\left|a\right|$ and $Im\left(a\right)=0$, where $ϵ$ is the machine precision, the function sets $c=1$ and $s=\left|t\right|$.

If $\left|b\right|<ϵ\left|a\right|$ and $Im\left(a\right)\ne 0$, the function sets

$$c=\frac{sign\left(Re\left(a\right)\right)a}{\left|a\right|}\text{and}s=\left|t\right|$$

Note that $t$ is always set to $b/a$, unless overflow would occur, in which case the function returns the value of the complex expression

$$(\text{flmax}\times sign\left(Re\left(b\right)/a\right),\text{flmax}\times sign\left(Im\left(b\right)/a\right))\text{;}$$

$\text{flmax}$ is the float value given by $1/\left(\text{machine.real\_safe}\right)$.