# NAG FL Interfacef06hrf (zhousg)

## ▸▿ Contents

Settings help

FL Name Style:

FL Specification Language:

## 1Purpose

f06hrf generates a complex elementary reflection.

## 2Specification

Fortran Interface
 Subroutine f06hrf ( n, x, incx, tol,
 Integer, Intent (In) :: n, incx Real (Kind=nag_wp), Intent (In) :: tol Complex (Kind=nag_wp), Intent (Inout) :: alpha, x(*) Complex (Kind=nag_wp), Intent (Out) :: theta
#include <nag.h>
 void f06hrf_ (const Integer *n, Complex *alpha, Complex x[], const Integer *incx, const double *tol, Complex *theta)
The routine may be called by the names f06hrf or nagf_blas_zhousg.

## 3Description

f06hrf generates details of a complex elementary reflection (Householder matrix), $P$, such that
 $P ( α x )=( β 0 )$
where $P$ is unitary, $\alpha$ is a complex scalar, $\beta$ is a real scalar, and $x$ is an $n$-element complex vector.
$P$ is given in the form
 $P=I-γ ( ζ z ) ( ζ zH ) ,$
where $z$ is an $n$-element complex vector, $\gamma$ is a complex scalar such that $\mathrm{Re}\left(\gamma \right)=1$, and $\zeta$ is a real scalar. $\gamma$ and $\zeta$ are returned in a single complex value $\theta =\left(\zeta ,\mathrm{Im}\left(\gamma \right)\right)$. Thus $\zeta =\mathrm{Re}\left(\theta \right)$ and $\gamma =\left(1,\mathrm{Im}\left(\theta \right)\right)$.
If $x$ is such that
 $max(|Re(xi)|,|Im(xi)|)≤max(tol,εmax(|Re(α)|,|Im(α)|)),$
where $\epsilon$ is the machine precision and $\mathit{tol}$ is a user-supplied tolerance, then:
• either $\theta$ is set to $0$, in which case $P$ can be taken to be the unit matrix;
• or $\theta$ is set so that $\mathrm{Re}\left(\theta \right)\le 0$ and $\theta \ne 0$, in which case
 $P=( θ 0 0 I ) .$
Otherwise $1\le \mathrm{Re}\left(\theta \right)\le \sqrt{2}$.
None.

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of elements in $x$ and $z$.
2: $\mathbf{alpha}$Complex (Kind=nag_wp) Input/Output
On entry: the scalar $\alpha$.
On exit: the scalar $\beta$.
3: $\mathbf{x}\left(*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)×{\mathbf{incx}}\right)$.
On entry: the $n$-element vector $x$. ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left(1+\left(\mathit{i}-1\right)×{\mathbf{incx}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Intermediate elements of x are not referenced.
On exit: the referenced elements are overwritten by details of the complex elementary reflection.
4: $\mathbf{incx}$Integer Input
On entry: the increment in the subscripts of x between successive elements of $x$.
Constraint: ${\mathbf{incx}}>0$.
5: $\mathbf{tol}$Real (Kind=nag_wp) Input
On entry: the value $\mathit{tol}$.
6: $\mathbf{theta}$Complex (Kind=nag_wp) Output
On exit: the scalar $\theta$.

None.

Not applicable.