naginterfaces.library.blas.zutsrs

naginterfaces.library.blas.zutsrs(side, k1, k2, c, s, a)

zutsrs transforms a complex upper triangular matrix to an upper spiked matrix by applying a given sequence of plane rotations.

For full information please refer to the NAG Library document for f06tw

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f06/f06twf.html

Parameters

sidestr, length 1

Specifies whether $U$ is operated on from the left or the right.

$side=\text{'L'}$

$U$ is pre-multiplied from the left.

$side=\text{'R'}$

$U$ is post-multiplied from the right.

k1int

The value $k_1$.

k2int

The value $k_2$.

cfloat, array-like, shape $(k2-1)$

$c\left[\text{k}\right]$ must hold $c_{\text{k}}$, the cosine of the rotation $P_{\text{k}}$, for $\text{k}={\text{k}}_1,\dots ,{\text{k}}_2-1$.

scomplex, array-like, shape $(k2-1)$

$s\left[\text{k}\right]$ must hold $s_{\text{k}}$, the sine of the rotation $P_{\text{k}}$, for $\text{k}={\text{k}}_1,\dots ,{\text{k}}_2-1$.

acomplex, array-like, shape $(n,n)$

The $n\times n$ upper triangular matrix $U$. The imaginary parts of the diagonal elements must be zero.

Returns

scomplex, ndarray, shape $(k2-1)$

$s\left[\text{k}\right]$ holds a nonzero element of the spike of $H$: $h_{{\text{k}}_2,\text{k}}$ if $side=\text{'L'}$, or $h_{\text{k}+1,{\text{k}}_1}$ if $side=\text{'R'}$, for $\text{k}={\text{k}}_1,\dots ,{\text{k}}_2-1$.

acomplex, ndarray, shape $(n,n)$

The upper triangular part of the upper spiked matrix $H$. The imaginary parts of the diagonal elements are set to zero except for the $(k_2,k_2)$ element if $side=\text{'L'}$, or the $(k_1,k_1)$ element if $side=\text{'R'}$.

Raises

NagValueError

(errno $1$)

On entry, error in parameter $side$.

Constraint: $side=\text{'L'}$ or $\text{'R'}$.

(errno $2$)

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

Constraint: $n\ge 0$.

Notes

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

zutsrs transforms an $n\times n$ complex upper triangular matrix $U$ with real diagonal elements, to an upper spiked matrix $H$, by applying a given sequence of plane rotations from either the left or the right, in planes $k_1$ to $k_2$. $H$ has real diagonal elements except where the spike joins the diagonal.

If $side=\text{'L'}$, $H$ has a row spike, with nonzero elements $h_{k_2,k}$, for $k=k_1,k_1+1,\dots ,k_2-1$. The rotations are applied from the left:

$$H=PU\text{,}$$

where $P=P_{k_1}{P}_{k_1+1}\cdots P_{k_2-1}$ and $P_k$ is a rotation in the $(k,k_2)$ plane.

If $side=\text{'R'}$, $H$ has a column spike, with nonzero elements $h_{k+1,k_1}$, for $k=k_1,k_1+1,\dots ,k_2-1$. The rotations are applied from the right:

$$H{P}^H=R\text{,}$$

where $P=P_{k_2-1}\cdots P_{k_1+1}{P}_{k_1}$ and $P_k$ is a rotation in the $(k_1,k+1)$ plane.

The $2\times 2$ plane rotation part of $P_k$ has the form

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

with $c_k$ real.