naginterfaces.library.blas.dutsqr

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

dutsqr performs a $QR$ or $RQ$ factorization of the product of a real upper triangular matrix and a real matrix of plane rotations.

For full information please refer to the NAG Library document for f06qt

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f06/f06qtf.html

Parameters

sidestr, length 1

Specifies whether $P$ is applied from the left or the right in the transformation.

$side=\text{'L'}$

$P$ is applied from the left.

$side=\text{'R'}$

$P$ is applied 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 the cosine of the rotation $P_{\text{k}}$, for $\text{k}={\text{k}}_1,\dots ,{\text{k}}_2-1$.

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

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

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

The $n\times n$ upper triangular matrix $U$.

Returns

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

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

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

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

afloat, ndarray, shape $(n,n)$

The upper triangular matrix $R$.

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.

dutsqr performs one of the transformations

$$R\leftarrow PU{Q}^T\text{or}R\leftarrow QU{P}^T\text{,}$$

where $U$ is a given $n\times n$ real upper triangular matrix, $P$ is a given real orthogonal matrix, and $Q$ is a real orthogonal matrix chosen to make $R$ upper triangular. Both $P$ and $Q$ are represented as sequences of plane rotations in planes $k_1$ to $k_2$.

If $side=\text{'L'}$,

$$R\leftarrow PU{Q}^T\text{,}$$

where $P=P_{k_2-1}\dots P_{k_1+1}{P}_{k_1}$ and $Q=Q_{k_2-1}\dots Q_{k_1+1}{Q}_{k_1}$.

If $side=\text{'R'}$,

$$R\leftarrow QU{P}^T\text{,}$$

where $P=P_{k_1}{P}_{k_1+1}\dots P_{k_2-1}$ and $Q=Q_{k_1}{Q}_{k_1+1}\dots Q_{k_2-1}$.

In either case $P_k$ and $Q_k$ are rotations in the $(k,k+1)$ plane.

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

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