naginterfaces.library.blas.zusqr

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

zusqr performs a $QR$ or $RQ$ factorization (as a sequence of plane rotations) of a complex upper spiked matrix.

For full information please refer to the NAG Library document for f06ts

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f06/f06tsf.html

Parameters

sidestr, length 1

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

$side=\text{'L'}$

$H$ is pre-multiplied from the left.

$side=\text{'R'}$

$H$ is post-multiplied from the right.

k1int

The value $k_1$.

k2int

The value $k_2$.

scomplex, array-like, shape $\left(k2\right)$

The nonzero elements of the spike of $H$: $s\left[\text{k}\right]$ must hold $h_{{\text{k}}_2,\text{k}}$ if $side=\text{'L'}$, and $h_{\text{k}+1,{\text{k}}_1}$ if $side=\text{'R'}$, for $\text{k}={\text{k}}_1,\dots ,{\text{k}}_2-1$.

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

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

Returns

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

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

scomplex, ndarray, shape $\left(k2\right)$

$s\left[\text{k}\right]$ holds $s_{\text{k}}$, the sine of the rotation $P_{\text{k}}$, for $\text{k}={\text{k}}_1,\dots ,{\text{k}}_2-1$; $s\left[{\text{k}}_2\right]$ holds $d_{k_2}$, the $k_2$th diagonal element of $D$, if $side=\text{'L'}$, or $d_{k_1}$, the $k_1$th diagonal element of $D$, if $side=\text{'R'}$.

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

The upper triangular matrix $R$. The imaginary parts of the diagonal elements are set to zero.

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.

zusqr transforms an $n\times n$ complex upper spiked matrix $H$ to upper triangular form $R$ by applying a complex unitary matrix $P$ from the left or the right. $H$ is assumed to have real diagonal elements except where the spike joins the diagonal; $R$ has real diagonal elements. $P$ is formed as a sequence of plane rotations in planes $k_1$ to $k_2$.

If $side=\text{'L'}$, $H$ is assumed to have a row spike, with nonzero elements $h_{{\text{k}}_2,\text{k}}$, for $\text{k}={\text{k}}_1,\dots ,{\text{k}}_2-1$. The rotations are applied from the left:

$$PH=R\text{,}$$

where $P=D{P}_{k_2-1}\cdots P_{k_1+1}{P}_{k_1}$, $P_k$ is a rotation in the $(k,k_2)$ plane and $D=diag(1,\dots ,1,d_{k_2},1,\dots ,1)$ with $\left|d_{k_2}\right|=1$.

If $side=\text{'R'}$, $H$ is assumed to have a column spike, with nonzero elements $h_{\text{k}+1,{\text{k}}_1}$, for $\text{k}={\text{k}}_1,\dots ,{\text{k}}_2-1$. The rotations are applied from the right:

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

where $P=D{P}_{k_1}{P}_{k_1+1}\cdots P_{k_2-1}$, $P_k$ is a rotation in the $(k_1,k+1)$ plane and $D=diag(1,\dots ,1,d_{k_1},1,\dots ,1)$ with $\left|d_{k_1}\right|=1$.

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.