naginterfaces.library.blas.dlhousg

naginterfaces.library.blas.dlhousg(alpha, x, tol)

dlhousg generates a real elementary reflection in the LINPACK (as opposed to NAG) style.

For full information please refer to the NAG Library document for f06fs

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f06/f06fsf.html

Parameters

alphafloat

The scalar $\alpha$.

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

The $n$-element vector $x$.

tolfloat

The value $\text{tol}$.

If $tol$ is not in the range $(0,1)$, the value $0$ is used for $\text{tol}$.

Returns

alphafloat

The scalar $\beta$.

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

The referenced elements are overwritten by details of the real elementary reflection.

z1float

The scalar $\zeta$.

Raises

NagValueError

(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.

dlhousg generates details of a real elementary reflection (Householder matrix), $P$, such that

$$\begin{array}{c}P\left(\begin{array}{c}\alpha \\ x\end{array}\right)=\left(\begin{array}{c}\beta \\ 0\end{array}\right)\end{array}$$

where $P$ is orthogonal, $\alpha$ and $\beta$ are real scalars, and $x$ is an $n$-element real vector.

$P$ is given in the form

$$\begin{array}{c}P=I-\frac{1}{\zeta }\left(\begin{array}{c}\zeta \\ z\end{array}\right)\left(\begin{array}{cc}\zeta & z^T\end{array}\right)\text{,}\end{array}$$

where $z$ is an $n$-element real vector and $\zeta$ is a real scalar. (This form is compatible with that used by LINPACK.)

If the elements of $x$ are all zero, or if the elements of $x$ are all less than $\text{tol}\times \left|\alpha \right|$ in absolute value, then $\zeta$ is set to $0$ and $P$ can be taken to be the unit matrix. Otherwise $\zeta$ always lies in the range $(1,2)$.