naginterfaces.library.blas.dnhousg

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

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

For full information please refer to the NAG Library document for f06fr

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f06/f06frf.html

Parameters

alphafloat

The scalar $\alpha$.

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

The $n$-element vector $x$.

tolfloat

The value $\text{tol}$.

Returns

alphafloat

The scalar $\beta$.

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

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

zetafloat

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.

dnhousg 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

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

where $z$ is an $n$-element real vector and $\zeta$ is a real scalar.

If $x$ is such that

$$max\left(\left|x_i\right|\right)\le max(\text{tol},ϵ\left|\alpha \right|)$$

where $ϵ$ is the machine precision and $\text{tol}$ is a user-supplied tolerance, then $\zeta$ is set to $0$, and $P$ can be taken to be the unit matrix. Otherwise $1\le \zeta \le \sqrt{2}$.