naginterfaces.library.blas.zuhqr

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

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

For full information please refer to the NAG Library document for f06tr

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f06/f06trf.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 $k_2$.

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

The nonzero subdiagonal elements of $H$: $s\left[\text{k}\right]$ must hold $h_{\text{k}+1,\text{k}}$, 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 Hessenberg matrix $H$.

Returns

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

$c\left[\text{k}\right]$ holds $c_{\text{k}}$, the cosine of the rotation $P_{\text{k}}$, for $\text{k}={\text{k}}_1,\dots ,{\text{k}}_2-1$; $c\left[k_2\right]$ 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 $side=\text{'R'}$.

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

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

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.

zuhqr transforms an $n\times n$ complex upper Hessenberg matrix $H$ to upper triangular form $R$ by applying a unitary matrix $P$ from the left or the right. $H$ is assumed to have real nonzero subdiagonal elements $h_{\text{k}+1,\text{k}}$, for $\text{k}={\text{k}}_1,\dots ,{\text{k}}_2-1$, only; $R$ has real diagonal elements. $P$ is formed as a sequence of plane rotations in planes $k_1$ to $k_2$.

If $side=\text{'L'}$, the rotations are applied from the left:

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

where $P=D{P}_{k_2-1}\cdots P_{k_1+1}{P}_{k_1}$ and $D=diag(1,\dots ,1,d_{k_2},1,\dots ,1)$ with $\left|d_{k_2}\right|=1$.

If $side=\text{'R'}$, the rotations are applied from the right:

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

where $P=D{P}_{k_1}{P}_{k_1+1}\cdots P_{k_2-1}$ and $D=diag(1,\dots ,1,d_{k_1},1,\dots ,1)$ with $\left|d_{k_1}\right|=1$.

In either case, $P_k$ is a rotation in the $(k,k+1)$ plane, chosen to annihilate $h_{k+1,k}$.

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

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

with $s_k$ real.