naginterfaces.library.blas.dutupd

naginterfaces.library.blas.dutupd(alpha, x, a)

dutupd performs a $QR$ factorization (as a sequence of plane rotations) of a real upper triangular matrix that has been augmented by a full row.

For full information please refer to the NAG Library document for f06qq

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/f06/f06qqf.html

Parameters

alphafloat

The scalar $\alpha$.

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

The vector $x$.

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

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

Returns

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

The referenced elements are overwritten by the tangents of the rotations $Q_{\text{k}}$, for $\text{k}=1,2,\dots ,n$.

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

The upper triangular matrix $R$.

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

The values $c_{\text{k}}$, the cosines of the rotations $Q_{\text{k}}$, for $\text{k}=1,2,\dots ,n$.

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

The values $s_{\text{k}}$, the sines of the rotations $Q_{\text{k}}$, for $\text{k}=1,2,\dots ,n$.

Raises

NagValueError

(errno $1$)

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

Constraint: $n\ge 0$.

(errno $4$)

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.

dutupd performs the factorization

$$\begin{array}{c}\left(\begin{array}{c}U\\ \alpha x^T\end{array}\right)=Q\left(\begin{array}{c}R\\ 0\end{array}\right)\end{array}$$

where $U$ and $R$ are $n\times n$ real upper triangular matrices, $x$ is an $n$-element real vector, $\alpha$ is a real scalar, and $Q$ is a real orthogonal matrix.

$Q$ is formed as a sequence of plane rotations

$$Q^T=Q_n\cdots Q_2{Q}_1$$

where $Q_k$ is a rotation in the $(k,n+1)$ plane, chosen to annihilate $x_k$.

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

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