naginterfaces.library.blas.zsrotg

naginterfaces.library.blas.zsrotg(pivot, direct, alpha, x)

zsrotg generates a sequence of complex plane rotations.

For full information please refer to the NAG Library document for f06hq

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f06/f06hqf.html

Parameters

pivotstr, length 1

Specifies the plane rotated by $P_k$.

$pivot=\text{'V'}$ (variable pivot)

$P_k$ rotates the $(k,k+1)$ plane.

$pivot=\text{'F'}$ (fixed pivot)

$P_k$ rotates the $(1,k+1)$ plane if $direct=\text{'F'}$, or the $(k,n+1)$ plane if $direct=\text{'B'}$.

directstr, length 1

Specifies the sequence direction.

$direct=\text{'F'}$ (forward sequence)

$P=P_n\cdots P_2{P}_1$.

$direct=\text{'B'}$ (backward sequence)

$P=P_1{P}_2\cdots P_n$.

alphacomplex

The scalar $\alpha$.

xcomplex, array-like, shape $\left(n\right)$

The $n$-element vector $x$.

Returns

alphacomplex

The scalar $\beta$.

xcomplex, ndarray, shape $\left(n\right)$

The referenced elements are overwritten by details of the plane rotations.

cfloat, ndarray, shape $\left(n\right)$

The values $c_k$, the cosines of the rotations.

scomplex, ndarray, shape $\left(n\right)$

The values $s_k$, the sines of the rotations.

Raises

NagValueError

(errno $1$)

On entry, error in parameter $pivot$.

Constraint: $pivot=\text{'V'}$ or $\text{'F'}$.

(errno $2$)

On entry, error in parameter $direct$.

Constraint: $direct=\text{'F'}$ or $\text{'B'}$.

(errno $6$)

On entry, error in parameter $\text{incx}$.

Constraint: $\text{incx}>0$.

Notes

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

zsrotg generates the parameters of a complex unitary matrix $P$, of order $n+1$, chosen so as to set to zero the elements of a supplied $n$-element complex vector $x$.

If $pivot=\text{'F'}$ and $direct=\text{'F'}$, or if $pivot=\text{'V'}$ and $direct=\text{'B'}$,

$$\begin{array}{c}P\left(\begin{array}{c}\alpha \\ x\end{array}\right)=\left(\begin{array}{c}\beta \\ 0\end{array}\right)\text{;}\end{array}$$

If $pivot=\text{'F'}$ and $direct=\text{'B'}$, or if $pivot=\text{'V'}$ and $direct=\text{'F'}$,

$$\begin{array}{c}P\left(\begin{array}{c}x\\ \alpha \end{array}\right)=\left(\begin{array}{c}0\\ \beta \end{array}\right)\text{.}\end{array}$$

Here $\alpha$ and $\beta$ are complex scalars.

$P$ is represented as a sequence of $n$ plane rotations $P_k$, as specified by $pivot$ and $direct$; $P_k$ is chosen to annihilate $x_k$, and its $2\times 2$ plane rotation part has the form

$$\begin{array}{c}\left(\begin{array}{cc}{c}_k& {\overline{s}}_k\\ -s_k& c_k\end{array}\right)\text{,}\end{array}$$

with $c_k$ real. The tangent of the rotation, $t_k$, is overwritten on $x_k$.